# THE PHYSICS ASSOCIATED WITH NEUTRINO MASSES

EDITED BY : Diego Aristizabal Sierra, Frank Franz Deppisch and Alexander Merle PUBLISHED IN : Frontiers in Physics

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ISSN 1664-8714 ISBN 978-2-88963-351-7 DOI 10.3389/978-2-88963-351-7

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# THE PHYSICS ASSOCIATED WITH NEUTRINO MASSES

Topic Editors:

Diego Aristizabal Sierra, Federico Santa María Technical University, Chile Frank Franz Deppisch, University College London, United Kingdom Alexander Merle, Max Planck Institute for Physics, Germany

Citation: Sierra, D. A., Deppisch, F. F., Merle, A., eds. (2020). The Physics Associated with Neutrino Masses. Lausanne: Frontiers Media SA. doi: 10.3389/978-2-88963-351-7

# Table of Contents


Martina Gerbino and Massimiliano Lattanzi

	- Yi Cai, Tao Han, Tong Li and Richard Ruiz

# GUT and Flavor Models for Neutrino Masses and Mixing

#### Davide Meloni\*

Dipartimento di Matematica e Fisica, Università di Roma Tre, Rome, Italy

In the recent years neutrino experiments have studied in detail the phenomenon of neutrino oscillations and most of the oscillation parameters have been measured with a good accuracy. However, in spite of many interesting ideas, the problem of flavor in the lepton sector remains an open issue. In this review, we discuss the state of the art of models for neutrino masses and mixing formulated in the context of flavor symmetries, with particular emphasis on the role played by grand unified gauge groups.

Keywords: grand unified theory, neutrino mass, flavor symmetries, discrete symmetries, group theory

## 1. INTRODUCTION

#### Edited by:

Alexander Merle, Max Planck Institute for Physics (MPG), Germany

#### Reviewed by:

Hugo Serodio, Lund University, Sweden Michael Andreas Schmidt, The University of Sydney, Australia

> \*Correspondence: Davide Meloni davide.meloni@uniroma3.it

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 21 July 2017 Accepted: 06 September 2017 Published: 10 October 2017

#### Citation:

Meloni D (2017) GUT and Flavor Models for Neutrino Masses and Mixing. Front. Phys. 5:43. doi: 10.3389/fphy.2017.00043 In the course of the last two decades, valuable experimental evidences for three families of massive neutrinos and flavor neutrino oscillations were obtained in various experimental channels, and the parameters which characterize the mixing are now known with a relatively high precision. As a consequence, the existence of non-vanishing neutrino masses and mixing have been firmly established. In spite of the huge amount of available data, many properties of the neutrino physics are yet poorly known or even completely unknown as, just to mention some of them, whether the massive neutrinos are Dirac or Majorana particles [1], what kind of spectrum the neutrino masses obeys, what is the absolute scale of neutrino masses, what is the octant for the atmospheric mixing angle θ<sup>23</sup> and what are the values of the CP violating phases in the leptonic sector. In a unified description of fermion masses and mixing, the above-mentioned features must be somehow linked to quark properties which, however, appear so dissimilar to make such a connection very hard to find; this is the well-known flavor problem. Let us take the mixing angles as an example. Quark and neutral leptonic mixings are described by the Cabibbo-Kobayashi-Maskawa matrix VCKM [2, 3] and the Pontecorvo-Maki-Nakagawa-Sakata matrix UPMNS [4–7], respectively. Although one can assume an identical parametrization, **Figure 1** shows that the absolute values of the matrix elements are quite different: the VCKM is an almost diagonal matrix, with the largest deviation from 1 coming from the Cabibbo angle in the (12) position while the UPMNS exhibits a pattern where all but the (13) entry are of the same order of magnitude of O(1). Since at the end of the day the VCKM and UPMNS matrices all come from the Yukawa matrices of the theory, one would naively expect no sort of relations among their entries, which is obviously the case. Unless one decides to take seriously the numerical quark-lepton complementarity relation [8–11] that connects the solar θ<sup>12</sup> and atmospheric θ<sup>23</sup> leptonic angles to the Cabibbo angle θC, θ<sup>12</sup> + θ<sup>C</sup> ∼ π/4. In this case (and also for other similar relations), Grand Unified Theories (GUT) supplemented with the help of family symmetries could provide a simple explanation so that their role in deciphering the flavor problem cannot be neglected. In fact, while GUT groups relate the properties of particles belonging to different species, thus establishing a connections among mass matrices of leptons and quarks, flavor symmetries act on the members of particles of the same species but different families, enabling a strong connection between the matrix elements of a given mass matrix. Thus, one can arrange the theory in such a way that flavor symmetries are mainly responsible for a definite mixing

**4**

pattern in the neutrino sector and that GUT symmetries introduce the Cabibbo angle in the leptonic sector as a correction to the UPMNS given by the diagonalization of the charged lepton mass matrix (somehow related to the down quark masses).

Notice that the additional degree of symmetry involved in these theories allows a substantial decrease of the number of independent parameters compared to the Standard Model case (which amounts to 19) and, quite often, the model produces observable predictions that can be verified by experiments. The typical example in GUT theories is related to the mean life of the proton τp; since the new colored gauge bosons and scalars implied by the larger symmetry can mediate proton decay at a rate faster than the age of the Universe, many variations have been ruled out based on the predicted upper limit on τp. On the other hand, the less freedom in the elements of the mass matrices subsequent to the imposition of flavor symmetries allowed in the past to derive patterns of leptonic mixing in very good agreement with the old neutrino data which unfortunately do not resist to the comparison with the more precise measurements as we currently have. The typical example is provided by the socalled Tribimaximal mixing (TBM [12–16], more on this and other patterns later in section 4) which predicts θ<sup>13</sup> = 0 and requires ad-hoc large corrections to fall over acceptable ranges. Given the vastness of the scientific production in terms of models employing flavor symmetries, we restrict ourselves here to nonabelian discrete symmetries and abelian U(1)'s. While the latter have been inspired by the Froggatt and Nielsen mechanism [17], the former answers to the necessity of explaining the existence of three generations of fermions or at least to unify two of them (that is why non-abelian group), avoiding at the same time the presence of Goldstone and gauge bosons coming from their spontaneous symmetry breaking (that is why discrete). Discrete symmetries can be inspired by different extensions of the Standard Model (SM); for example, one can start with an SU(3) invariant theory and then break it into its discrete groups using large Higgs representations [18]; or one can consider extra dimensional theories [19] (also string inspired), where the new dimensions are properly compactified and the discrete group appears as a remnant of the n-dimensional space-time symmetry [19].

Although the combination GUT ⊕ flavor seems to be even more restrictive in terms of free parameters, the aim of this short review is to show that several attempts in this direction have been done that produced good results. But, before arriving at this conclusion, we will devote section 3 to the understanding of the main prediction for neutrino masses in GUT theories and section 4 on the role played by flavor. Only in section 5 we will investigate the physics opportunity given by the union of these two different types of symmetries.

### 2. REMARKS ON NEUTRINO MASSES

#### 2.1. Dirac Mass Term

Dirac neutrino masses can be generated by the same Higgs mechanism that gives masses to quarks and charged leptons in the SM. To this aim, we need to introduce SM singlet fermions νRi and the related Yukawa couplings with the Higgs field; after spontaneous symmetry breaking, the Lagrangian containing the lepton mass terms is given by:

$$\mathcal{L}\_{\text{mass}} = -\frac{\nu}{\sqrt{2}} \sum\_{\alpha,\beta=e,\mu,\tau} \left< \overline{v}\_{\alpha L} Y\_{\alpha\beta}^{\nu} v\_{\beta R} + \text{h.c.} \right> $$

$$ -\frac{\nu}{\sqrt{2}} \sum\_{\alpha,\beta=e,\mu,\tau} \left< \overline{\ell}\_{\alpha L} Y\_{\alpha\beta}^{\ell} \ell\_{\beta R} + \text{h.c.} \right>, \tag{1} $$

where ℓ<sup>α</sup> represents the charged lepton fields, v is the vacuum expectation value (vev) of the Higgs field and Y ν and Y ℓ are the Yukawa couplings of neutrinos and charged leptons, respectively, accommodated in 3×3 matrices. The diagonalization of Y ν,ℓ can be performed with a biunitary transformation:

$$U\_L^{\nu^\dagger} Y^\nu U\_R^\nu = Y'^\nu \quad \text{with} \quad Y\_{i\bar{j}}^{\prime\nu} = y\_i^{\prime\nu} \delta\_{i\bar{j}} \,, \tag{2}$$

$$U\_L^{\ell^\dagger} Y^\ell U\_R^\ell = Y'^\ell \quad \text{with} \quad Y\_{\alpha\beta}^{\prime\ell} = \jmath\_\alpha^{\prime\ell} \delta\_{\alpha\beta} \,, \tag{3}$$

and, consequently, the left and right-handed components of the fields with definite mass are as follows:

$$\upsilon\_{kL} = \sum\_{\beta=e,\mu,\tau} (U\_L^{\nu^\dagger})\_{k\beta} \,\upsilon\_{\beta L} \,, \quad \upsilon\_{kR} = \sum\_{\beta=e,\mu,\tau} (U\_R^{\nu^\dagger})\_{k\beta} \,\upsilon\_{\beta R} \,, \tag{4}$$

$$\ell\_{\alpha L}^{\prime} = \sum\_{\beta = e, \mu, \tau} (U\_L^{\ell^\dagger})\_{\alpha \beta} \, \ell\_{\beta L} \,, \quad \ell\_{\alpha R}^{\prime} = \sum\_{\beta = e, \mu, \tau} (U\_R^{\ell^\dagger})\_{\alpha \beta} \, \ell\_{\beta R} \, \cdot \tag{5}$$

In terms of the mass states defined in Equations (4) and (5), the Lagrangian in (1) can be rewritten as:

$$\mathcal{L}\_{\text{mass}} = -\sum\_{k=1,2,3} \frac{\nu \nu\_k^{\prime \prime}}{\sqrt{2}} (\overline{v}\_{kL} \nu\_{kR} + \text{h.c.})$$

$$-\sum\_{\alpha=c,\mu,\tau} \frac{\nu \nu\_\alpha^{\prime \ell}}{\sqrt{2}} (\overline{\ell}\_{\alpha L}^{\prime} \ell\_{\alpha R}^{\prime} + \text{h.c.}) = \tag{6}$$

$$=-\sum\_{k=1,2,3} \frac{\nu \mathcal{V}\_k^{\prime\prime}}{\sqrt{2}} \overline{\nu}\_k \nu\_k - \sum\_{\alpha=e,\mu,\tau} \frac{\nu \mathcal{V}\_\alpha^{\prime\ell}}{\sqrt{2}} \overline{\ell}\_\alpha^{\prime} \ell\_\alpha^{\prime},\tag{7}$$

with

$$\nu\_k = \nu\_{kL} + \nu\_{kR} \,, \qquad \ell'\_{\alpha} = \ell'\_{\alpha L} + \ell'\_{\alpha R} \,.$$

More importantly, the mixings driven by U ν,ℓ L enter in the leptonic charged current expressed in terms of mass eigenstates as

$$J\_{\rm CC}^{\mu} = \sum\_{k=1,2,3} \sum\_{\alpha=e,\mu,\tau} \nabla\_{kL} \boldsymbol{\nu}^{\mu} (\boldsymbol{U}\_{L}^{\nu\dagger} \boldsymbol{U}\_{L}^{\ell})\_{k\alpha} \boldsymbol{\ell}\_{\alpha L}^{\prime} \,, \tag{8}$$

and give rise to the well known PMNS matrix:

$$U\_{\rm PMNS} = U\_L^{\ell^\dagger} U\_L^v \,. \tag{9}$$

This unitary matrix is generally parametrized in terms of three mixing angles and one CP-violating phase, in a way similar to that used for VCKM:

$$U\_{\rm PMINS} = \begin{pmatrix} \mathfrak{c}12\mathfrak{c}13 & \mathfrak{s}12\mathfrak{c}13 & \mathfrak{s}13\mathfrak{c}^{-i\delta} \\ -\mathfrak{s}12\mathfrak{c}23 - \mathfrak{c}12\mathfrak{s}23\mathfrak{s}13\mathfrak{c}^{i\delta} & \mathfrak{c}12\mathfrak{c}23 - \mathfrak{s}12\mathfrak{s}23\mathfrak{s}13\mathfrak{c}^{i\delta} & \mathfrak{s}23\mathfrak{c}13 \\ \mathfrak{s}12\mathfrak{s}23 - \mathfrak{c}12\mathfrak{c}23\mathfrak{s}13\mathfrak{c}^{i\delta} & -\mathfrak{c}12\mathfrak{c}23 - \mathfrak{s}12\mathfrak{c}23\mathfrak{s}13\mathfrak{c}^{i\delta} & \mathfrak{c}23\mathfrak{c}13 \end{pmatrix}, \tag{10}$$

where cij = cos(θij), sij = sin(θij) and θij are the mixing angles (0 ≤ θij ≤ π/2). δ is the Dirac CP-violating phase ranging in the interval 0 ≤ δ < 2π.

The current best-fit values and the allowed 1σ and 3σ ranges for the oscillation parameters as well as for the two independent mass differences 1m<sup>2</sup> kj <sup>=</sup> <sup>m</sup><sup>2</sup> <sup>k</sup> <sup>−</sup> <sup>m</sup><sup>2</sup> j , as obtained from the flavor transition experiments, are summarized in **Table 1**. Normal Ordering refers to the situation in which m<sup>1</sup> < m<sup>2</sup> < m3, whereas for the Inverted Ordering we mean m<sup>3</sup> < m<sup>1</sup> < m2.

The reported values are obtained from the global analysis of Esteban et al. [20].

#### 2.2. Majorana Mass Terms

With the minimal particle content of the SM, namely leptons L<sup>i</sup> and the Higgs doublet H:

$$L\_i = \begin{pmatrix} \upsilon \\ e \end{pmatrix}\_{iL}, \qquad H = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \tag{11}$$

one can generate dimension five operators of the form:

$$\mathcal{L}\_5 \sim \frac{\mathcal{Y}\_{ij}}{\Lambda} \,\overline{L}\_i L\_j^c \tilde{H} \tilde{H}^T,\tag{12}$$

TABLE 1 | Value of the oscillation parameters obtained from a global analysis from Esteban et al. [20].


For the squared mass difference in the last line, ℓ = 1 in the Normal Ordering and ℓ = 2 in the Inverted Ordering.

where 3 can be understood as the scale where new physics probably sets in and H˜ = −i τ2H<sup>∗</sup> . In fact, two SM singlets are built from the product of four SU(2)<sup>L</sup> doublets as [21]:

$$2 \otimes 2 \otimes 2 \otimes 2 = (3 \oplus 1) \otimes (3 \oplus 1),\tag{13}$$

either via the product of two triplets or by the product of two singlets. Since L and H are different fields, we have four possible combinations that can give an overall SU(2)<sup>L</sup> singlet:

$$\begin{aligned} \mathcal{O}\_1 &= (L\_i H)\_1 \left( L\_j H \right)\_1 & \mathcal{O}\_2 &= \left( L\_i L\_j \right)\_1 \left( H H \right)\_1 \\ \mathcal{O}\_3 &= \left( L\_i L\_j \right)\_3 \left( H H \right)\_3 & \mathcal{O}\_4 &= \left( L\_i H \right)\_3 \left( L\_j H \right)\_3 \end{aligned}$$

where the subscript 1, 3 refer to the SU(2)<sup>L</sup> representation. Since (HH)<sup>1</sup> = 0 due to the antisymmetry under the exchange of the two doublets, only O1,3,4 contribute to neutrino masses. In particular, the explicit form of the bilinear are as follows:

$$\left(L\_i L\_j\right)\_1 \sim \upsilon\_i e\_j - e\_i \upsilon\_j \qquad \qquad \left(L\_i L\_j\right)\_3 \sim \begin{pmatrix} \upsilon\_i \upsilon\_j \\ \upsilon\_i e\_j + e\_i \upsilon\_j \\ e\_i e\_j \end{pmatrix} \tag{14}$$

$$(L\_i H)\_1 \sim \upsilon\_i \phi^0 - e\_i \phi^+ \qquad (L\_i H)\_3 \sim \begin{pmatrix} \upsilon\_i \phi^+ \\ \upsilon\_i \phi^0 + e\_i \phi^+ \\ e\_i \phi^0 \end{pmatrix} \tag{15}$$

$$(H H)\_3 \sim \begin{pmatrix} \phi^+ \phi^+ \\ \phi^+ \phi^0 + \phi^0 \phi^+ \\ \phi^0 \phi^0 \end{pmatrix} (\text{l.6})\_3$$

from which we realize that O1, O<sup>3</sup> and O<sup>4</sup> all contain the combination of fields νiνj(φ 0 ) 2 that generate neutrino masses after electroweak spontaneous symmetry breaking. However, giving their different contractions of the SU(2)<sup>L</sup> indices, O<sup>1</sup> has a tree-level realization in terms of the interchange of a heavy SM singlet νR, the type-I see-saw mechanism [22–26], whereas heavy triplets are needed to realize O<sup>3</sup> and O4, either with the interchange of a scalar particle (the type-II see-saw mechanism [27]) or of a fermion field (the type-III mechanism [28]), see **Figure 2**.

In the first case, the introduction of three right-handed neutrinos N<sup>i</sup> ≡ νR<sup>i</sup> allows for an invariant mass Lagrangian of the form [29]:

$$\mathcal{L}\_m = -Y\_{\vec{ij}}\vec{L}\_i(\tilde{H}N\_{\vec{j}}) + \frac{1}{2}\bar{N}\_i^c M\_{\vec{ij}} N\_{\vec{j}} + \text{ h.c.} \tag{17}$$

The first term in this equation is known as the Dirac mass term and it is essentially a copy of the mass term "employed" by the charged fermions and quarks to get their masses. The second term, instead, is a pure Majorana contribution to the neutrino mass. After spontaneous symmetry breaking, L<sup>m</sup> gives rise to the Dirac mass matrix (mD)ij ≡ YijhHi, which is non-hermitian and non-symmetric, and to the Majorana mass matrix M which is symmetric. Assuming all N<sup>i</sup> to be very heavy, one can integrate them away so that the resulting light neutrino mass matrix reads:

$$
\delta m\_{\upsilon} = -m\_{\mathrm{D}} \mathcal{M}^{-1} m\_{\mathrm{D}}^T. \tag{18}
$$

The type-I see-saw mechanism shows that the light neutrino masses depend quadratically on the Dirac masses but are inversely proportional to the large Majorana mass, so that the scale of new physics is clearly 3 = M.

In the case of type-II mechanism, at least one scalar SU(2)<sup>L</sup> triplet must be added to the field content of the SM; for values of the weak hypercharge equal to +1, the triplet has the following components:

$$
\Delta\_L = \begin{pmatrix} \Delta^{++} \\ \Delta^+ \\ \Delta^0 \end{pmatrix}, \tag{19}
$$

and the Lagrangian terms that accommodate the new states and are relevant for neutrino masses are:

$$\begin{split} \mathcal{L}\_{\Delta} & \sim \left( k\_{\hat{\boldsymbol{y}}} \bar{\boldsymbol{L}}\_{i} \left( \boldsymbol{\sigma} \cdot \boldsymbol{\Delta}\_{L}^{\dagger} \right) \boldsymbol{L}\_{\hat{\boldsymbol{y}}}^{\boldsymbol{c}} - \mu\_{\Delta} \tilde{\boldsymbol{H}}^{T} \left( \boldsymbol{\sigma} \cdot \boldsymbol{\Delta}\_{L} \right) \tilde{\boldsymbol{H}} + \text{h.c.} \right) \\ & \quad + m\_{\Delta}^{2} \left| \boldsymbol{\Delta}\_{L} \right|^{2}, \end{split} \tag{20}$$

where σ<sup>i</sup> are the Pauli matrices and kij the new Yukawa couplings induced by the presence of 1L. Assuming that the scalar potential has a minimum in the direction h1Li = (0, 0, v1) (as well as in the standard vacuum hHi = (0, v)) and that the hierarchy m2 <sup>1</sup> ≫ µ1v is valid, then the light neutrino mass matrix is:

$$(m\_{\upsilon})\_{\vec{ij}} \sim \frac{\mu\_{\Delta} \nu^2}{m\_{\Delta}^2} k\_{\vec{ij}};\tag{21}$$

in this case, the scale of new physics is approximately given by <sup>3</sup> <sup>∼</sup> <sup>m</sup><sup>2</sup> 1 /µ1.

In the last case of type-III see-saw mechanism, the triplet hyperchargeless fermions 6 can be arranged in the following form:

$$
\Sigma = \begin{pmatrix} \Sigma^0 / \sqrt{2} & \Sigma^+ \\ \Sigma^- & -\Sigma^0 / \sqrt{2} \end{pmatrix}, \tag{22}
$$

and the related Lagrangian reads:

$$\mathcal{L}\_{\Sigma} - \sim k\_{ij}^{\Sigma} \, \vec{L}\_{i} \tilde{H} \Sigma\_{j} + (m\_{\Sigma})\_{ij} Tr\left(\bar{\tilde{\Sigma}}\_{i}^{\epsilon} \Sigma\_{j}\right),\tag{23}$$

where again k 6 ij is a Yukawa coupling matrix. Under the hypothesis that m<sup>6</sup> ≫ k <sup>6</sup>v, the light mass matrix assumes the form

$$m\_{\boldsymbol{\nu}} \sim -k^{\Sigma} \frac{1}{m\_{\Sigma}} \left(k^{\Sigma}\right)^{T} \boldsymbol{\nu}^{2},\tag{24}$$

which is very similar to Equation (18) since, for the purposes of neutrino masses, the state 6<sup>0</sup> acts like a right-handed neutrino.

It has to be noted that the Majorana nature of neutrinos modifies the PMNS matrix of Equation (10) to take into account two more independent CP violating phases α and β that cannot be eliminated by a rotation of the neutrino fields; a possible convention for the new UPMNS is as follows:

$$U'\_{\rm PMNS} = U\_{\rm PMNS} \times \text{diag}\{1, e^{i\alpha/2}, e^{i\beta/2}\}.\tag{25}$$

Neutrino oscillation data cannot determine whether the massive neutrinos are Dirac or Majorana particles because the new phases cancel out of the oscillation amplitudes.

### 3. NEUTRINO MASSES AND MIXING IN GUT THEORIES

The possibility to generate non-zero neutrino masses through the see-saw mechanism, which requires quite a large B − L scale, fit rather naturally in grand unified models based on the gauge group SO(10) [30]. Putting aside Supersymmetry (SUSY) for the moment, the experimental constraints from the lifetime of the proton and from the weak mixing angle sin<sup>2</sup> θ<sup>W</sup> impose that SO(10) breaks to the SM at least in two or more steps [31, 32]. In a minimal setup which allows for a two-step breaking, the intermediate gauge groups (typically a Pati-Salam group SU(4)× SU(2)<sup>L</sup> × SU(2)<sup>R</sup> ≡ 4<sup>C</sup> 2<sup>L</sup> 2<sup>R</sup> [33]) is broken down to the SM at a scale around 10<sup>12</sup> GeV, which is usually also the scale of the Majorana masses. To accomplish this program, the Higgs sector must be carefully chosen in such a way to avoid bad mass relations of the SU(5) type [34]. Let us discuss an example. Consider the following chain:

$$\begin{aligned} \text{SO(10)} \stackrel{M\_U - \mathbf{210\_{H}}}{\longrightarrow} 4\_C \ 2\_L \ 2\_R \stackrel{M\_I - \mathbf{126\_{H}}}{\longrightarrow} \text{SM} \stackrel{M\_Z - \mathbf{10\_{H}}}{\longrightarrow} \\ \text{SU(3)}\_C \ U(1)\_{EM} \end{aligned} \tag{26}$$

where the three mass scales refer to the scale where SO(10) is broken down to the PS (MU), where PS is broken to the SM (MI) and finally where the SM group is broken down to the electromagnetism (MZ). The SO(10) representations used to perform the various stages of symmetry breaking are also indicated. With fermions in the **16** representation, the Yukawa Lagrangian contains two terms:

$$\mathcal{L} = \mathbf{16} \left( h \, \mathbf{10}\_H + f \, \overline{\mathbf{126}}\_H \right) \mathbf{16},\tag{27}$$

where the couplings h and f are 3 × 3 symmetric matrices in flavor space. In terms of their PS quantum numbers, the Higgses in Equation (26) decompose as:

$$\begin{aligned} \mathbf{10}\_H &= \begin{pmatrix} 1,2,2 \end{pmatrix} \oplus \begin{pmatrix} 6,1,1 \end{pmatrix}, \\ \mathbf{126}\_H &= \begin{pmatrix} 6,1,1 \end{pmatrix} \oplus \begin{pmatrix} \overline{10},3,1 \end{pmatrix} \oplus \begin{pmatrix} 10,1,3 \end{pmatrix} \oplus \begin{pmatrix} 15,2,2 \end{pmatrix}. \end{aligned}$$

Of all the previous sub-multiplets, the ones useful for generating neutrino (and fermion) masses are the (1, 2, 2) ≡ 8 ∈ **10**<sup>H</sup> entering the last breaking in Equation (26) and that contains an SU(2)<sup>L</sup> doublet, the (10, 1, 3) ≡ 1<sup>R</sup> ∈ **126**<sup>H</sup> to allow for righthanded Majorana masses and the (15, 2, 2) ≡ 6 ∈ **126**<sup>H</sup> which also contains an SU(2)<sup>L</sup> doublet. Using the extended survival hypothesis [31], we assume that both 1<sup>R</sup> and 6 have masses around M<sup>I</sup> , and all other multiplets are close to the GUT scale<sup>1</sup> .

A comment here is in order. The (1, 2, 2) of the **10**<sup>H</sup> representation can be decomposed into

$$(1,2,2) = (1,2,+\frac{1}{2}) \oplus (1,2,-\frac{1}{2}) \equiv H\_{\underline{u}} \oplus H\_{\underline{d}}\tag{28}$$

under the SM group; if 10<sup>H</sup> = 10<sup>∗</sup> H then H∗ <sup>u</sup> = H<sup>d</sup> as in the SM but, as it has been shown in Bajc et al. [37], in the limit Vcb = 0 the ratio mt/m<sup>b</sup> should be close to 1, in contrast with the experimental fact that at the GUT scale mt/m<sup>b</sup> ≫ 1. On the other hand, even though the **10**<sup>H</sup> is a real representation from the SO(10) point of view, one can choose its components to be either real or complex. In the latter case, 10<sup>H</sup> 6= 10<sup>∗</sup> H and then H∗ u 6= Hd. In order to keep the parameter space at an acceptable level, it is a common practice to introduce an extra symmetry (for instance, the Peccei-Quinn U(1)PQ [38]) to avoid the Yukawa couplings related to 10∗ H .

For the vev values of the **10**<sup>H</sup> components we will use the following short-hand notation:

$$k\_u \equiv \langle (1, 2, 2)\_{10}^u \rangle \not\equiv k\_d \equiv \langle (1, 2, 2)\_{10}^d \rangle. \tag{29}$$

For the vev of the **126**H, instead, one can take full advantage of the fact that a vev for the doublet 6 (that we call vu,d) can be induced by a term in the scalar potential of the form [39]:

$$V = \lambda \, 12\mathbf{6}\_H \, \overline{12\mathbf{6}\_H} \, 12\mathbf{6}\_H \, 1\mathbf{0}\_H \to \lambda \, \Delta\_R \, \overline{\Delta\_R} \, \Sigma \, \Phi,$$

which gives:

$$\nu\_{\mathfrak{u},d} \sim \lambda \frac{\nu\_{\mathbb{R}}^2}{M\_{(15,2,2)}^2} \, k\_{\mathfrak{u},d},\tag{30}$$

where v<sup>R</sup> = h(10, 1, 3)i. According to this, the fermion mass matrices of the model assume the form:

$$\begin{aligned} M\_u &= h \, k\_u + f \, \nu\_u, &\quad M\_d &= h \, k\_d + f \, \nu\_d \\ M\_\upsilon^D &= h \, k\_u - \mathfrak{H} \, \nu\_u, &\quad M\_l &= h \, k\_d - \mathfrak{H} \, \nu\_d, &\quad M\_\upsilon^M &= f \, \nu\_R. \end{aligned} \tag{31}$$

These relations clearly show why the Yukawa sector requires more than the **10**H; in fact, in the absence of the **126**<sup>H</sup> (or **120**H) one would get M<sup>d</sup> ≡ M<sup>l</sup> , which is phenomenologically wrong. The role of the **126**<sup>H</sup> in SO(10) theories is exactly to break the wrong mass relations and the factor of 3 appearing in Equation (31), derived from the vev of 6 of the **126**H, is the equivalent of the Georgi-Jarlskog factor of the non-minimal SU(5) [40].

Under the hypothesis that the type-I see-saw mechanism is responsible for the light neutrino masses, a fit can be performed which fixes the entries of the h and f couplings to reproduce the low energy observables in the flavor sector (also in the supersymmetric case) in the full three-flavor approach [35, 41, 42]. This partially contradicts the conclusions derived in the two-flavor limit, where the type-I see-saw mechanism has been shown to be incompatible with a large atmospheric mixing. To show this, let us approximate M<sup>D</sup> <sup>ν</sup> ≈ M<sup>u</sup> and work in the basis where the charged leptons are diagonal; assuming a small up and down quark mixings λ<sup>C</sup> (of the order of the Cabibbo angle), Equation (18) tells us that

$$m\_{\upsilon} \sim 4 \, r\_R \begin{pmatrix} m\_c^2 / (m\_s - m\_\mu) & \lambda\_C \\ \lambda\_C & m\_t^2 / (m\_b - m\_\pi) \end{pmatrix},\tag{32}$$

so that two non-degenerate eigenvalues can be generated whose squared difference can be made of the correct order of magnitude <sup>∼</sup>10−<sup>3</sup> eV<sup>2</sup> , but the atmospheric mixing angle is suppressed by λC, thus making this construction incompatible with the data.

Relations of the form (31) are also obtained in the minimal SU(5) scenario with a **5**<sup>H</sup> and fermions in the reducible **5**¯ ⊕ **10** representation. With this minimal Higgs content, the prediction at the GUT scale is again M<sup>d</sup> ≡ M<sup>l</sup> . To solve this problem, the scheme proposed in Georgi and Jarlskog [40] involved a slightly more complicated Higgs structure due to the presence of the **45**<sup>H</sup> representation. It replaces the above wrong relations with the more appropriate m<sup>d</sup> = 3m<sup>e</sup> and 3m<sup>s</sup> = mµ, which can be derived from the following textures [43]:

$$Y\_u = \begin{pmatrix} 0 & p & 0 \\ p & 0 & q \\ 0 & q & v \end{pmatrix}, \ Y\_d = \begin{pmatrix} 0 & r & 0 \\ r & s & 0 \\ 0 & 0 & t \end{pmatrix}, \ Y\_e = \begin{pmatrix} 0 & r & 0 \\ r & -3s & 0 \\ 0 & 0 & t \end{pmatrix}, \tag{33}$$

and whose flavor structure can be obtained, for example, by means of additional symmetries (discussed later). In the context of SO(10), the textures in Equation (33) have been obtained in Harvey et al. [44, 45], in a model with three families of lefthanded fermions, 161,2,3, two real **10**H's, three **126**<sup>H</sup> and one **45**H. Equally successful phenomenological attempts where instead all quark and lepton mass matrices have the same zero texture with vanishing (1,1), (1,3) and (3,3) entries have been proposed in Matsuda et al. [46].

Going beyond the type-I see-saw mechanism for neutrino masses, it has been shown that there exists a very elegant connection between the large atmospheric angle θ<sup>23</sup> and the relation m<sup>b</sup> = m<sup>τ</sup> , if the type-II see-saw is the dominant one [47, 48]. To show this, let us allow the (10, 3, 1) component of the **126**<sup>H</sup> to take a large vev vL. This generates a "left" mass matrix for

<sup>1</sup>One can safely estimate that the colored states 1<sup>R</sup> and 6 do not give a catastrophic contribution to proton decay [35, 36].

the Majorana neutrinos M<sup>L</sup> <sup>ν</sup> = f v<sup>L</sup> so that the total light neutrino mass matrix is given by <sup>m</sup><sup>ν</sup> <sup>=</sup> <sup>M</sup><sup>L</sup> <sup>ν</sup> <sup>−</sup> <sup>m</sup><sup>T</sup> D (M<sup>M</sup> ν ) <sup>−</sup>1mD. Under the hypothesis of the dominance of type-II, in the basis where the charged leptons are diagonal we easily get:

$$m\_{\boldsymbol{\nu}} = M\_{\boldsymbol{\nu}}^L \approx M\_d - M\_l \approx \begin{pmatrix} m\_s - m\_{\mu} & \theta\_D \\ \theta\_D & m\_b - m\_{\boldsymbol{\pi}} \end{pmatrix},\tag{34}$$

(θ<sup>D</sup> being a small down quark mixing) and a maximal atmospheric mixing necessarily requires a cancellation between m<sup>b</sup> and m<sup>τ</sup> . However, SM extrapolation of the fermion masses from the electroweak scale up to the GUT scale (but see [49, 50] for the effects of the intermediate mass scales in the running) shows that m<sup>b</sup> ∼ 1.7m<sup>τ</sup> [51], so this mechanism does not seem to fit well with a non-SUSY SO(10) GUT with the **10**<sup>H</sup> ⊕ **126**<sup>H</sup> Higgs sector [52]. This conclusion is not altered when the fit takes into account the three families of fermions. On the other hand, in the SUSY case the relation m<sup>b</sup> = m<sup>τ</sup> is roughly fulfiled for low tan β ∼ <sup>O</sup>(1) with no threshold corrections but also for larger tan β ∼ <sup>O</sup>(40) with significant threshold corrections. The quality of the full three-family fits in these cases is comparable.

If we insist on minimality in the Higgs sector, the next combinations are the **120**<sup>H</sup> ⊕ **126**<sup>H</sup> and **10**<sup>H</sup> ⊕ **120**H. Both of them make use of the **120**<sup>H</sup> representation which, according to the following decomposition under the PS gauge group, contains several bi-doublets useful for fermion masses:

$$\mathbf{120}\_H = (10 + \overline{10}, 1, 1) \oplus (6, 3, 1) \oplus (6, 1, 3) \oplus (15, 2, 2) \oplus (1, 2, 2).$$

Models of the first kind (**120**<sup>H</sup> ⊕ **126**H) have been considered predictive when restricted to the second and third generations [37]. However, the predicted ratio mb/m<sup>τ</sup> ∼ 3 strongly disfavors a SM (for which mb/m<sup>τ</sup> ∼ 2) and SUSY (for which mb/m<sup>τ</sup> ∼ 1) fits with neither type-I nor type-II see-saw dominance. The second combination, **10**H⊕**120**<sup>H</sup> [53], in spite of being compatible with the b − τ unification [54], produces either down-quark mass or top-quark mass unrealistically small.

In the case of a non-minimal Higgs content with **10**<sup>H</sup> ⊕ **120**<sup>H</sup> ⊕ **126**H, the Yukawa sector contains a large number of independent parameters but, except the supersymmetric case, the use of the **120**<sup>H</sup> does not improve the fits in the type-II seesaw dominated case. On the other hand, the fits obtained for the type-I scenario, including neutrino observables, are considerably better than the corresponding SUSY as well as better of the **10**<sup>H</sup> ⊕ **126**<sup>H</sup> non-SUSY case.

### 4. NEUTRINO MASSES AND MIXING FROM FLAVOR SYMMETRIES

#### 4.1. Lepton Mixing from Discrete Symmetry

The general strategy to get the leptonic mixing matrix UPMNS from symmetry consideration is to assume that at some large energy scale the theory is invariant under the action of a flavor symmetry group G<sup>f</sup> ; the scalar sector is then built in a suitable way as to be broken to different subgroups in the neutrino sector G<sup>ν</sup> , and in the charged lepton sector, Gℓ. The lepton mixing originates then from the mismatch of the embedding of G<sup>ℓ</sup> and G<sup>ν</sup> into G<sup>f</sup> . Let us assume that

$$\mathcal{G}\_{\ell} \subset \mathcal{G}\_{f} \qquad \mathcal{G}\_{\nu} \subset \mathcal{G}\_{f} \qquad \mathcal{G}\_{\ell} \cap \mathcal{G}\_{\nu} = \emptyset. \tag{35}$$

For Majorana particles, we can write the action of the elements of the subgroups of G<sup>f</sup> on the mass matrix as<sup>2</sup>

$$Q^\dagger M\_\ell^\dagger M\_\ell Q = M\_\ell^\dagger M\_\ell \qquad Q \in \mathcal{G}\_\ell \tag{36a}$$

$$Z^{\overline{T}}M\_{\boldsymbol{\upsilon}}Z = M\_{\boldsymbol{\upsilon}} \qquad Z \in \mathcal{G}\_{\boldsymbol{\upsilon}}.\tag{36b}$$

For Dirac neutrinos the last relation must be modified as:

$$Z^{\uparrow}M\_{\nu}^{\uparrow}M\_{\nu}Z = M\_{\nu}^{\uparrow}M\_{\nu} \qquad Z \in \mathcal{G}\_{\mathcal{V}}.\tag{37}$$

If we restrict ourselves to matrices Z with det Z = 1 and to Majorana neutrinos, then the maximal invariance group of the neutrino mass matrix which leave the neutrino masses unconstrained is the Klein group V = Z<sup>2</sup> ⊗ Z<sup>2</sup> [55–58]. The charged leptonic subgroup G<sup>ℓ</sup> could be either a cyclic group Zn, with the index n ≥ 3, or a product of cyclic symmetries like, for example, Z<sup>2</sup> ⊗ Z2. We discard in the discussion possible residual non-abelian symmetries because their character would result in a partial or complete degeneracy of the mass spectrum, and thus incompatible with the current data on charged lepton masses. For the same reason we assume that Z ∈ <sup>G</sup><sup>ν</sup> decomposes into three inequivalent representations under Gℓ.

The diagonalization of the mass matrices is equivalent, using (36), to a rotation of the group elements Q and Z through unitary matrices as:

$$Q^{\text{diag}} = U\_{\ell}^{\dagger} Q U\_{\ell} \tag{38a}$$

$$Z^{\text{diag}} = U^{\dagger}\_{\nu} Z U\_{\nu},\tag{38b}$$

because both G<sup>ℓ</sup> and G<sup>ν</sup> are abelian. The matrices U<sup>ℓ</sup> and U<sup>ν</sup> are determined up to unitary diagonal Kℓ,<sup>ν</sup> and permutation Pℓ,<sup>ν</sup> matrices:

$$U\_{\ell} \longrightarrow U\_{\ell} P\_{\ell} K\_{\ell} \tag{39a}$$

$$U\_{\boldsymbol{\nu}} \longrightarrow U\_{\boldsymbol{\nu}} P\_{\boldsymbol{\nu}} K\_{\boldsymbol{\nu}}.\tag{39b}$$

Thus, up to Majorana phases and permutations of rows and columns, the lepton mixing matrix UPMNS is given by:

$$U\_{\rm PMNS} = U\_{\ell}^{\dagger} U\_{\nu}.\tag{40}$$

Notice that, as a consequence of the fact that UPMNS is not completely determined, the mixing angles are fixed up to a small number of degeneracies. For the same reason, the Dirac CP phase δ is determined up to a factor π and the Majorana phases cannot be predicted because the matrix Mν remains unconstrained in this setup. In **Figure 3** we have pictorially summarized the above procedure.

It is remarkable that, under particular assumptions on the residual symmetry groups in the neutrino and charged lepton

<sup>2</sup>The charged lepton mass matrix M<sup>ℓ</sup> is written in the right-left basis.

sectors<sup>3</sup> , the construction we have just discussed allow for model (and mass)-independent predictions on the mixing angles (or columns of UPMNS). As it has been shown in Grimus [59], Hernandez and Smirnov [60, 61], if only a cyclic group from each sector is a subgroup of the full flavor group G<sup>f</sup> , then it is possible to derive non-trivial relations between the mixing matrix in terms of the symmetry transformations which, in turn, provoke the appearance of well-defined connections among different mixing angles, also called sum rules. In particular, non-zero θ13, deviations from maximal mixing for θ<sup>23</sup> and predictions for the CP Dirac phase [62, 63] are relevant predictions in (quasi perfect) agreement with the current data. An intersting and useful classification of all possible mixing matrices completely determined by residual symmetries (originated from a finite flavor symmetry group) can be found in Fonseca and Grimus [58].

Since the family symmetry G<sup>f</sup> has to be broken to generate the observed pattern of masses and mixing, the models generally consider an enlarged Higgs sector where Higgs-type fields, called flavons φ, are neutral under the SM gauge group and break spontaneously the family symmetry by acquiring a vev

$$
\epsilon = \frac{\langle \phi \rangle}{\Lambda},
\tag{41}
$$

where 3 denotes a high energy mass scale. If the scale of the vev is smaller (or at least of the same order of magnitude) than 3, one can consider ǫ as a small expansion parameter which can be used to derive Yukawa matrices with built-in hierarchies and/or precise relations among their entries. In order to do that, it is often necessary that all three lepton families are grouped into triplet irreducible representations, so that the possible choices for G<sup>f</sup> are U(3) and subgroups. To give an example, in the case of SU(3) and for the Weinberg operator of Equation (12), one can consider lepton doublets into a triplet of SU(3) and the Higgs doublet H in a singlet of G<sup>f</sup> [64, 65]; the lowest dimensional SU(3) invariant operator is built using a pair of flavon fields transforming in the **3** of SU(3). For a generic flavon alignment hφi ∝ (a, b,c) T , the neutrino mass matrix is then proportional to

$$
\begin{pmatrix} a^2 & ab \ ac \\ ba \ b^2 & bc \\ ca & cb \ c^2 \end{pmatrix} \begin{array}{c} \\ \\ \\ \end{array} \tag{42}
$$

Special mixing patterns, as the ones discussed below, are obtained assuming particular flavon alignments in the flavor space which, quite frequently, imply well defined relations among the mixing angles and the Dirac CP-violating phase [66–72].

For a model to be consistent, the alignment must descend from the minimization of the scalar potential, without adhoc assumptions on the potential parameters. Widely used ingredients for this type of constructions are:


In SUSY frameworks, both flavons and driving fields are neede to derive the superpotential w of the model. In the limit of unbroken SUSY, the minimum of the related scalar potential V is given by the derivatives of w with respect to the components of the driving fields, which determine a set of equations for the components of the flavon fields. A detailed account of such a procedure has been given in Altarelli and Feruglio [73], to which we refer the interested reader. Here we limit ourselves to a simple representative example, extracted from de Medeiros Varzielas et al. [74]. Suppose that the SM singlet pair (ϕ0, ϕ) is made up of a driving (ϕ0) and a flavon (ϕ) triplet fields in such a way that terms like ϕ0ϕ and ϕ0ϕ 2 are flavor invariant; thus, the most general renormalizable superpotential is given by:

$$
\omega = M(\varphi\_0 \varphi) + \mathfrak{g}(\varphi\_0 \varphi \varphi). \tag{43}
$$

The vacuum minimization conditions for the ϕ field are then:

$$\begin{aligned} \frac{\partial \boldsymbol{w}}{\partial \varphi\_{01}} &= M\varphi\_1 + \boldsymbol{g}\varphi\_2 \varphi\_3 = \mathbf{0}, \\ \frac{\partial \boldsymbol{w}}{\partial \varphi\_{02}} &= M\varphi\_2 + \boldsymbol{g}\varphi\_3 \varphi\_1 = \mathbf{0}, \\ \frac{\partial \boldsymbol{w}}{\partial \varphi\_{03}} &= M\varphi\_3 + \boldsymbol{g}\varphi\_1 \varphi\_2 = \mathbf{0}, \end{aligned} \tag{44}$$

which are solved by:

$$
\varphi = \nu(1, 1, 1), \qquad \nu = -\frac{M}{g}.\tag{45}
$$

Thissimple case does not obviously exhaust all possible situations arising after the minimization procedure; in more complicated

<sup>3</sup>For instance, one can impose relations between the generators of these residual groups and/or force the determinants to assume specific values.

cases, it could happen that some of the vevs depends on unknown parameters which are not related to the parameters appearing in w. This indicates that there are flat directions in the flavon potential, as one could check by analyzing the flavons and driving fields mass spectrum in the SUSY limit. SUSY breaking effects and radiative corrections are eventually important to give mass to the modes associated to these flat directions.

The presence of driving fields is not a necessary condition for obtaining the correct vacuum alignment. While this implies to deal with longer and more complicated potentials [75–77], one can avoid intricated calculations formulating flavor models in extra dimensions where the scalar fields live in the bulk of the higher-dimensional space [78]. The vacuum alignment is then achieved by the boundary conditions of the scalar fields and the physics at low energy is described by massless zero modes which break the flavor symmetries [79].

#### 4.2. Typical Discrete Patterns

The use of discrete symmetries was first suggested to explain a simplified form of the neutrino mass matrix called Tri-Bi-Maximal mixing (TBM) [12–16]:

$$U\_{\rm TB} = \begin{pmatrix} \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} & 0\\ -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} \end{pmatrix},\tag{46}$$

which implies s 2 <sup>12</sup> = 1/3, s 2 <sup>23</sup> = 1/2 and s<sup>13</sup> = 0. In this case the matrix mν takes the form:

$$m\_{\nu} = \begin{pmatrix} \varkappa & \varkappa & \varkappa \\ \varkappa \ x + \nu & \jmath - \nu \\ \nu \ \jmath - \nu & \varkappa + \nu \end{pmatrix},\tag{47}$$

(x, y and v are complex numbers) which can also parametrized as:

$$m\_{\upsilon} = m\_1 \Phi\_1 \Phi\_1^T + m\_2 \Phi\_2 \Phi\_2^T + m\_3 \Phi\_3 \Phi\_3^T,\tag{48}$$

where

$$\Phi\_1^T = \frac{1}{\sqrt{6}} \langle 2, -1, -1 \rangle, \quad \Phi\_2^T = \frac{1}{\sqrt{3}} \langle 1, 1, 1 \rangle, \quad \Phi\_3^T = \frac{1}{\sqrt{2}} \langle 0, -1, 1 \rangle \tag{49}$$

are the respective columns of UTB and m<sup>i</sup> are the neutrino mass eigenvalues given by the simple expressions m<sup>1</sup> = x − y, m<sup>2</sup> = x + 2y and m<sup>3</sup> = x − y + 2v [80].

Notice that, in the basis where charged leptons are diagonal, the mass matrix for TBM mixing is the most general matrix which is invariant under the so-called 2-3 (or µ − τ ) symmetry [81, 82] under which

$$m\_{\vee} = A\_{23} m\_{\vee} A\_{23},\tag{50}$$

where A<sup>23</sup> is given by:

$$A\_{23} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},\tag{51}$$

and, in addition, under the action of a unitary symmetric matrix STB which commutes with A23:

$$
\Delta m\_{\upsilon} = \mathcal{S}\_{TB} m\_{\upsilon} \mathcal{S}\_{TB},
\tag{52}
$$

where STB is given by:

$$\mathcal{S}\_{\text{TB}} = \frac{1}{3} \begin{pmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{pmatrix} . \tag{53}$$

In practice, the matrices A<sup>23</sup> and STB realize the action of Z ∈ <sup>G</sup><sup>ν</sup> .

For bimaximal (BM) mixing [83], instead, we have s 2 <sup>12</sup> = s 2 23 = 1/2 and accordingly:

$$U\_{\rm BM} = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}} \end{pmatrix} . \tag{54}$$

The respective mass matrix is of the form:

$$m\_{\upsilon} = \begin{pmatrix} \varkappa & \varkappa & \varkappa \\ \varkappa & z & \varkappa - z \\ \varkappa & \varkappa - z & z \end{pmatrix} \,, \tag{55}$$

that is

$$m\_{\boldsymbol{\nu}} = m\_1 \boldsymbol{\Phi}\_1 \boldsymbol{\Phi}\_1^T + m\_2 \boldsymbol{\Phi}\_2 \boldsymbol{\Phi}\_2^T + m\_3 \boldsymbol{\Phi}\_3 \boldsymbol{\Phi}\_3^T,\tag{56}$$

where

$$\Phi\_1^T = \frac{1}{2}(\sqrt{2}, 1, 1), \quad \Phi\_2^T = \frac{1}{2}(-\sqrt{2}, 1, 1), \quad \Phi\_3^T = \frac{1}{\sqrt{2}}(0, -1, 1). \tag{57}$$

The resulting matrix is characterized by the invariance under the action of A<sup>23</sup> and also under the application of the real, unitary and symmetric matrix SBM of the form

$$m\_{\upsilon} = S\_{BM} m\_{\upsilon} S\_{BM},\tag{58}$$

with SBM given by:

$$\mathbf{S}\_{BM} = \begin{pmatrix} 0 & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{\sqrt{2}} & -\frac{1}{2} & \frac{1}{2} \end{pmatrix}. \tag{59}$$

In this case, are the matrices A<sup>23</sup> and SBM that realize the action of Z ∈ <sup>G</sup><sup>ν</sup> on the neutrino mass matrix.

Other examples of special patterns can be found in the literature; among them, a vast production has been devoted to the Golden Ratio mixing (GR), of which two slightly different versions have attracted much attention: in one of them [84–87] the solar angle is given by tan θ<sup>12</sup> = 1/φ, where φ = (1 + √ 5)/2 is the golden ratio, which implies θ<sup>12</sup> = 31.7◦ ; in the other one, suggested in Rodejohann [88], cos θ<sup>12</sup> = φ/2 and θ<sup>12</sup> = 36◦ .

Since these special patterns mainly differ for the value of the solar angle, we report in **Figure 4** the predictions for sin<sup>2</sup> θ<sup>12</sup> of GR and TBM and compare them with three different fit results coming from Capozzi et al. [89] (labeled as CLMMP), Forero et al. [90] (labeled as FTV) and Gonzalez-Garcia et al. [91] (labeled as GMS). See the caption for more details.

The neutrino mass matrices analyzed so far have been derived in the basis where charged leptons are diagonal; then one can ask which features the matrix Q of Equation (36) must have in order to maintain the hermitian product M † <sup>ℓ</sup> M<sup>ℓ</sup> diagonal; observing that the most general diagonal M † <sup>ℓ</sup> M<sup>ℓ</sup> is left invariant under the action of a diagonal phase matrix with 3 different phase factors, one can easily see that if Q <sup>n</sup> <sup>=</sup> 1 then the matrix <sup>Q</sup> generates a cyclic group Zn. Examples for n = 3 and n = 4 are the following:

$$\begin{aligned} Q\_{TB} &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{pmatrix}, \quad \omega^3 = 1 \\\\ Q\_{BM} &= \begin{pmatrix} -1 & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & i \end{pmatrix}. \end{aligned} \tag{60}$$

We stress again that a realistic flavor model that reproduces all experimental features of neutrino masses and mixing can be realized from a theory invariant under the spontaneously broken symmetry described by G<sup>f</sup> which, in turn, must contain at least the S and Q transformations. These generate the subgroups G<sup>ν</sup> and Gℓ, respectively. The breaking of G<sup>f</sup> must be arranged in such a way that it is broken down to G<sup>ν</sup> in the neutrino mass sector and to G<sup>ℓ</sup> in the charged lepton mass sector. In some cases also the

since the allowed region is the same for both orderings) performed in Capozzi et al. [89] (labeled as CLMMP), Forero et al. [90] (labeled as FTV) and Gonzalez-Garcia et al. [91] (labeled as GMS). The white vertical lines inside the boxes are the best fit values, the gray boxes the 1σ confidence regions and the gray lines the 3σ allowed regions.

symmetry under A<sup>23</sup> is part of G<sup>ℓ</sup> and then must be preserved in the neutrino sector or it can arise as a consequence of the breaking of Gℓ.

Notice that it is not strictly necessary to deal with diagonal charged leptons because the special patterns analyzed so far can be considered as a good first approximation of the data and suitable corrections, for example coming explicitly from the charged leptons, must be taken into account [92–94].

Many discrete groups with the previous properties have been studied and their potentialities to describe neutrino masses and mixings scrutinized in detail. Just to give some examples, the groups A4, S<sup>4</sup> and T ′ are commonly utilized to generate TBM mixing (see, for example, [73, 95–103]); the group S<sup>4</sup> can also be used to generate BM mixing [83, 104, 105]; A<sup>5</sup> can be utilized to generate GR mixing [84–87] and the groups D<sup>10</sup> and D<sup>12</sup> can lead to another type of GR [88, 106] and to hexagonal mixing [107, 108]. Excellent reviews in this sector can be found, for instance, in King [65], Altarelli and Feruglio [80], Ishimori et al. [109] and Grimus and Ludl [110].

### 4.3. TBM and BM from Discrete Symmetries

To make a direct connection with the procedure outlined in section 4.1, we study here two examples on how to get the TBM and BM patterns from <sup>G</sup><sup>f</sup> = S4. This is the permutation group of order four, it has 4! = 24 elements and it is isomorphic to the symmetry group of the cube. The algebra contains two generators, S and T, that satisfy the condition S <sup>2</sup> <sup>=</sup> <sup>T</sup> <sup>4</sup> <sup>=</sup> (ST) <sup>3</sup> <sup>=</sup> 1. The group contains five irreducible representations: two singlets **1** and **1** ′ , one doublet **2** and two triplets **3** and **3** ′ . The (non trivial) tensor products are

$$\begin{aligned} \mathbf{1'} \otimes \mathbf{1'} &= \mathbf{1} \\ \mathbf{1'} \otimes \mathbf{2} &= \mathbf{2} \\ \mathbf{1'} \otimes \mathbf{3} &= \mathbf{3'} \end{aligned}$$

$$\mathbf{1'} \otimes \mathbf{3'} = \mathbf{3}$$

$$\mathbf{2} \otimes \mathbf{2} = \mathbf{1}\_{\mathbf{s}} \oplus \mathbf{2}\_{\mathbf{s}} \oplus \mathbf{1'}\_{\mathbf{a}}$$

$$\mathbf{2} \otimes \mathbf{3} = \mathbf{2} \otimes \mathbf{3'} = \mathbf{3} \oplus \mathbf{3'}$$

$$\begin{aligned} \mathbf{3} \otimes \mathbf{3} &= \mathbf{3'} \otimes \mathbf{3'} = \mathbf{1}\_{\mathbf{s}} \oplus \mathbf{2}\_{\mathbf{s}} \oplus \mathbf{3'}\_{\mathbf{s}} \oplus \mathbf{3}\_{\mathbf{a}} \\ \mathbf{3} \otimes \mathbf{3'} &= \mathbf{1'} \oplus \mathbf{2} \oplus \mathbf{3} \oplus \mathbf{3'}, \end{aligned}$$

where the subscript s (a) denotes symmetric (antisymmetric) combinations. The S<sup>4</sup> elements can be classified by the order h of each element, where ω <sup>h</sup> <sup>=</sup> <sup>e</sup> (see **Table 2** where the five conjugacy



classes and their characters are summarized. As expected, we have 1 + 3 + 6 + 6 + 8 = 24 elements in each class and the superscript indicates the order of each element in the conjugacy classes). A possible choice for the three dimensional generators is

$$S = \frac{1}{2} \begin{pmatrix} 0 & \sqrt{2} & \sqrt{2} \\ \sqrt{2} & -1 & 1 \\ \sqrt{2} & 1 & -1 \end{pmatrix} \qquad T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{i\pi/2} & 0 \\ 0 & 0 & e^{i3\pi/2} \end{pmatrix} . \tag{62}$$

The group S<sup>4</sup> contains another three dimensional representation, whose generators are related to those in Equation (62) through {S, T} → {−S, −T}. The abelian subgroups of S<sup>4</sup> are four Klein groups V, four Z<sup>3</sup> groups and three different Z4. These are summarized in **Table 3**.

The patterns of interest can be obtained using the following choices of subgroups:

$$\bullet \text{ } \mathcal{G}\_{\ell} = Z\_3 \text{ and } \mathcal{G}\_{\upsilon} = V$$

These subgroups are useful to reproduce the TBM only. We assume C<sup>3</sup> ∈ Z<sup>3</sup> and K<sup>1</sup> ∈ V as representative algebra. The absolute value of the PMNS matrix is therefore given by:

$$\|U\_{\rm PMNS}\| = U\_{\rm TBM} = \frac{1}{\sqrt{6}} \begin{pmatrix} 2 \ \sqrt{2} & 0 \\ 1 \ \sqrt{2} & \sqrt{3} \\ 1 \ \sqrt{2} & \sqrt{3} \end{pmatrix} . \tag{63}$$

Notice that the Jarlskog invariant JCP [112], defined as:

$$J\_{\rm CP} \equiv \Re \left[ \{ U\_{\rm PMNS} \}\_{11} \{ U\_{\rm PMNS} \}\_{13}^\* \{ U\_{\rm PMNS} \}\_{31}^\* \{ U\_{\rm PMNS} \}\_{33} \right] = $$
 
$$ \frac{1}{8} \sin 2\theta\_{12} \sin 2\theta\_{23} \sin 2\theta\_{13} \cos \theta\_{13} \sin \delta, \tag{64} $$

is zero. To obtain a realistic mixing pattern with θ<sup>13</sup> ∼ 9 ◦ we need to include large corrections.

• <sup>G</sup><sup>ℓ</sup> = Z<sup>4</sup> and <sup>G</sup><sup>ν</sup> = V

In this case only the BM pattern is possible; therefore both θ<sup>12</sup> and θ<sup>23</sup> are maximal. Next to leading order corrections of roughly the same order of magnitude of the Cabibbo angle are needed to reproduce the data as discussed, for instance, in Altarelli et al. [105].

• <sup>G</sup><sup>ℓ</sup> = V and <sup>G</sup><sup>ν</sup> = V

TABLE 3 | Possible independent algebras of S4 subgroups (same classification as the one adopted in de Adelhart Toorop et al. [111]).


This case, discussed in Lam [113], produces a BM mixing pattern. A representative choice for the subalgebras for G<sup>ℓ</sup> is K<sup>1</sup> and for G<sup>ν</sup> is K2.

### 4.4. Other LO Patterns

The fact that the value of the reactor angle is non-zero with high accuracy opens the possibility to use discrete symmetries to enforce the LO leptonic mixing patterns to structures where θ<sup>13</sup> is different from zero from the beginning. The various realizations all differ by the amount of the NLO needed to reconcile the theoretical predictions with the experimental data. Some of the new patterns, that have been obtained and studied in specific model realizations, are the following:


$$
\sin \theta\_{12} = \sin \theta\_{23} = -\frac{1}{\sqrt{2}}, \quad \sin \theta\_{13} = \frac{1}{3}, \tag{65}
$$

which corresponds to the following mixing matrix:

$$U\_{\rm TP} \sim \frac{1}{3} \begin{pmatrix} 2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2 \end{pmatrix} . \tag{66}$$

• the Bi-trimaximal (BT) mixing, introduced in King et al. [118] and corresponding to the mixing matrix:

$$U\_{\rm BT} = \begin{pmatrix} a\_+ & \frac{1}{\sqrt{3}} & a\_- \\ -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\\ a\_- & -\frac{1}{\sqrt{3}} & a\_+ \end{pmatrix},\tag{67}$$

where a<sup>±</sup> = (1± <sup>√</sup> 1 3 )/2, and leads to the following predictions:

$$
\sin \theta\_{12} = \sin \theta\_{23} = \sqrt{\frac{8 - 2\sqrt{3}}{13}} \approx 0.591 \quad \text{( $\theta\_{12} = \theta\_{23} \approx 36.2^\circ$ )},
$$

$$
\sin \theta\_{13} = a\_- \approx 0.211 \quad \text{( $\theta\_{13} \approx 12.2^\circ$ )}.\tag{68}
$$

### 4.5. Discrete Symmetries and Invariance Under CP

Let us now enlarge the symmetry content of the theory assuming, in addition to the invariance under the discrete group, also invariance under CP [119–121].

As in section 4.1, we consider that the residual symmetry in the charged sector <sup>G</sup><sup>ℓ</sup> is a cyclic group Zn, n ≥ 3, or the product Z<sup>2</sup> ⊗ Z2. Under the action of CP, a generic field 8 transforms as [122–124]:

$$\Phi(\mathfrak{x}) \longrightarrow \Phi'(\mathfrak{x}) = X\Phi^\star(\mathfrak{x}\_{\mathbb{CP}}),\tag{69}$$

where X is the representations of the CP operator in field space and xCP is the space-time coordinate transformed under the usual CP transformation x → xCP = (x 0 , −**x**). The invariance of the field under G<sup>f</sup> is expressed as:

$$\Phi(\mathfrak{x}) \longrightarrow \Phi'(\mathfrak{x}) = A\Phi(\mathfrak{x}),\tag{70}$$

where A is an element of a non-abelian discrete symmetry group. X can be chosen as a constant unitary symmetric matrix<sup>4</sup> :

$$XX^\dagger = XX^\star = 1,\tag{71}$$

in such a way that the square of the CP transformation is the identity, X <sup>2</sup> <sup>=</sup> 1. The action of <sup>X</sup> on the mass matrices, before the symmetry breaking, is given by

$$X^\star M\_\ell^\dagger M\_\ell X = (M\_\ell^\dagger M\_\ell)^\star \tag{72a}$$

$$X M\_{\boldsymbol{\upsilon}} X = M\_{\boldsymbol{\upsilon}}^{\star},\tag{72b}$$

if neutrinos are Majorana particles. If instead neutrinos are Dirac particles, (72b) has to be modified to

$$X^\star M\_\upsilon^\dagger M\_\upsilon X = (M\_\upsilon^\dagger M\_\upsilon)^\star. \tag{73}$$

The fact that the theory is invariant under the flavor symmetry group G<sup>f</sup> requires that for the generators of the group A the representations X in the field space must satisfy the following relation:

$$(X^{-1}AX)^\star = A' \qquad A, A' \in \{\mathcal{G}\_f\},\tag{74}$$

where in general A 6= A ′ . Notice that if X is a solution of (71) and (74) also e <sup>i</sup>ρX, with ρ being an arbitrary phase, is a solution.

Let us now specify this framework to the case where the residual symmetry <sup>G</sup><sup>ν</sup> is Z<sup>2</sup> ⊗ CP, with Z<sup>2</sup> contained in the flavor group; the matrix Z representing the generator of the former symmetry and the CP transformation X have to fulfil the constraint

$$XZ^\star - ZX = 0,\tag{75}$$

which is invariant under (74). In the neutrino sector, the light neutrino mass matrix satisfies both relations:

$$Z^T M\_\upsilon Z = M\_\upsilon \tag{76a}$$

$$X M\_{\boldsymbol{\upsilon}} X = M\_{\boldsymbol{\upsilon}}^{\star}.\tag{76b}$$

Notice that it is always possible to choose a basis where

$$X = \Omega \Omega^T \quad Z\_{\mathfrak{c}} = \Omega^\dagger Z \Omega \quad Z\_{\mathfrak{c}} = \text{diag}\left\{ (-1)^{z\_1}, (-1)^{z\_2}, (-1)^{z\_3} \right\}, \tag{77}$$

with z<sup>i</sup> = 0, 1. Since Z generates a Z<sup>2</sup> symmetry, two of the three parameters z<sup>i</sup> have to coincide and the combination TMν is constrained to be block-diagonal and real. Thus, this matrix can be diagonalized using a rotation R(θ) in the ij-plane of degenerate eigenvalues of Z, where θ is an unconstrained parameter that can be fixed to describe the neutrino mixing parameters. The positiveness of the light neutrino masses is ensured by the diagonal matrix K<sup>ν</sup> with elements equal to ±1 or ±i. In this way the matrix Mν can be diagonalized with unitary matrix defined as

$$U\_{\upsilon} \equiv \mathfrak{Q} \mathcal{R}\_{\vec{\eta}}(\theta) K\_{\upsilon}. \tag{78}$$

The mass spectrum is not fixed and thus permutations of columns are admitted. The inclusion of the charged leptons into the game proceeds as discussed in section 4.1. So, called Uℓ the matrix diagonalizing M † <sup>ℓ</sup>Mℓ, the full UPMNS is given by:

$$U\_{\rm PMNS} \equiv U\_{\ell}^{\dagger} U\_{\upsilon} = U\_{\ell} \Omega \mathcal{R}\_{ij}(\theta) \mathcal{K}\_{\upsilon},\tag{79}$$

up to permutations of rows and columns. To give an explicit example [125], let us assume that U<sup>ℓ</sup> = 1 and take to be

$$
\Omega = \frac{1}{\sqrt{2}} \begin{pmatrix}
\sqrt{2}\cos\varphi & -\sqrt{2} \ i\sin\varphi & 0 \\
\sin\varphi & i\cos\varphi & -1 \\
\sin\varphi & i\cos\varphi & 1
\end{pmatrix};
\tag{80}
$$

this matrix fulfils (77) for Z and X chosen as (Z, X) = (T 2 ST<sup>3</sup> ST<sup>2</sup> , SX0), with X<sup>0</sup> ≡ A23. Since z<sup>1</sup> and z<sup>3</sup> of the diagonal combination † Z are equal, the indices ij of the rotation matrix Rij(θ) in (79) are {i, j} = {1, 3}. Thus, the PMNS mixing matrix simply reads

$$U\_{\rm PMNS} = \Omega \, R\_{13}(\theta) \, K\_{\nu} \, . \tag{81}$$

Extracting the mixing angles from (81) we find:

$$\begin{aligned} \sin^2 \theta\_{12} &= \frac{2}{2 + (3 + \sqrt{5})\cos^2 \theta}, \\ \sin^2 \theta\_{13} &= \frac{1}{10} \left( 5 + \sqrt{5} \right) \sin^2 \theta \end{aligned}$$

$$\begin{aligned} \sin^2 \theta\_{23} &= \frac{1}{2} - \frac{\sqrt{2 \left( 5 + \sqrt{5} \right)} \sin 2\theta}{7 + \sqrt{5} + \left( 3 + \sqrt{5} \right) \cos 2\theta} \end{aligned} \tag{82}$$

which also call for an exact sum rule among the solar and the reactor mixing angles:

$$
\sin^2 \theta\_{12} = \frac{\sin^2 \varphi}{1 - \sin^2 \theta\_{13}} \approx \frac{0.276}{1 - \sin^2 \theta\_{13}} \,\,. \tag{83}
$$

Using for sin<sup>2</sup> θ<sup>13</sup> its best fit value (sin<sup>2</sup> θ13) bf <sup>=</sup> 0.0217, we find for the solar mixing angle sin<sup>2</sup> θ<sup>12</sup> ≈ 0.282 which is within its 3 σ range, see **Table 1**.

Models that explore the predictability of the CP symmetry in conjunction with non-abelian discrete symmetries have been massively explored in the very recent years; for example, the interplay between S<sup>4</sup> and CP has been studied, among others, in Mohapatra and Nishi [126], Feruglio et al. [127], Luhn [128], and

<sup>4</sup>The requirement that X is a symmetric matrix has been shown in Bajc et al. [119] to be a necessary condition, otherwise the neutrino mass spectrum would be partially degenerate.

Penedo et al. [129], while the role of A<sup>5</sup> has been elucidated in Li and Ding [130], Ballett et al. [131] and Turner [132] and that of several 1 groups in de Medeiros Varzielas and Emmanuel-Costa [133], Bhattacharyya et al. [134], Ma [135], Hagedorn et al. [136] and Ding and King [137].

### 4.6. The Use of Abelian Symmetries

Let us now investigate the possibility to construct SUSY models where the only flavor symmetry is a continuous U(1) [17]; thus the following procedure can be used:


$$\mathcal{L}\_{Y} = (Y\_{\varepsilon})\_{\dot{\imath}\dot{\jmath}} L\_{i} H\_{d} \, e\_{\dot{\jmath}}^{\varepsilon} \left(\frac{\theta}{\Lambda}\right)^{\mathbb{P}\varepsilon} + (Y\_{\nu})\_{\dot{\imath}\dot{\jmath}} \, \frac{L\_{i} L\_{\dot{\jmath}} H\_{\scriptscriptstyle\tt\tt\tt\tt\tt\tt\tt\tt}}{\Lambda\_{\mathcal{L}}} \left(\frac{\theta}{\Lambda}\right)^{\mathbb{P}\varepsilon} + \text{H.c.}\tag{84}$$

where 3 is the cut-off of the effective flavor theory and 3<sup>L</sup> the scale of the lepton number violation, in principle distinct from 3. Here (Ye)ij and (Y<sup>ν</sup> )ij are free complex parameters with modulus of O(1) while p<sup>e</sup> and p<sup>ν</sup> are appropriate powers of the ratio θ/3 needed to compensate the U(1) charges for each Yukawa term. Without loss of generality, we can fix n<sup>θ</sup> = −1; consequently, n L,R 1 , n L,R <sup>2</sup> > 0 for the Lagrangian expansion to make sense. For the neutrino masses we consider that they are described by the effective Weinberg operator, while the extension to see-saw mechanisms is straightforward.


$$
\epsilon = \frac{\langle \theta \rangle}{\Lambda} < 1.
$$

The lepton charges assignments reported in **Table 4**, some of them already studied in Altarelli et al. [138], give rise to the following mass matrices [139]:

$$A:\quad Y\_{\varepsilon} = \begin{pmatrix} \epsilon^3 & \epsilon^2 & 1\\ \epsilon^3 & \epsilon^2 & 1\\ \epsilon^3 & \epsilon^2 & 1 \end{pmatrix},\quad Y\_{\upsilon} = \begin{pmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{pmatrix},$$

$$A\_{\mu\tau}:\quad Y\_{\varepsilon} = \begin{pmatrix} \epsilon^4 & \epsilon^3 & \epsilon\\ \epsilon^3 & \epsilon^2 & 1\\ \epsilon^3 & \epsilon^2 & 1 \end{pmatrix},\quad Y\_{\upsilon} = \begin{pmatrix} \epsilon^2 & \epsilon & \epsilon\\ \epsilon & 1 & 1\\ \epsilon & 1 & 1 \end{pmatrix},\tag{85}$$

$$H: \quad Y\_{\mathfrak{e}} = \begin{pmatrix} \epsilon^7 \ \epsilon^5 \ \epsilon^2 \\ \epsilon^6 \ \epsilon^4 \ \epsilon \\ \epsilon^5 \ \epsilon^3 \ 1 \end{pmatrix}, \ Y\_{\mathbb{V}} = \begin{pmatrix} \epsilon^4 \ \epsilon^3 \ \epsilon^2 \\ \epsilon^3 \ \epsilon^2 \\ \epsilon^2 \ \epsilon \ 1 \end{pmatrix},$$

TABLE 4 | Examples of charge assignment under U(1).


With anarchy we refer to models where no symmetry at all is acting on the neutrino sector [140–142] and so the charge of the lepton doublets is vanishing.

$$\begin{aligned} A': \quad Y\_{\varepsilon} &= \begin{pmatrix} \epsilon^3 & \epsilon & 1\\ \epsilon^3 & \epsilon & 1\\ \epsilon^3 & \epsilon & 1 \end{pmatrix}, \ Y\_{\upsilon} = \begin{pmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{pmatrix},\\ H': \quad Y\_{\varepsilon} &= \begin{pmatrix} \epsilon^{10} & \epsilon^6 & \epsilon^2\\ \epsilon^9 & \epsilon^5 & \epsilon\\ \epsilon^8 & \epsilon^4 & 1 \end{pmatrix}, \ Y\_{\upsilon} = \begin{pmatrix} \epsilon^4 & \epsilon^3 & \epsilon^2\\ \epsilon^3 & \epsilon^2 & \epsilon\\ \epsilon^2 & \epsilon & 1 \end{pmatrix}. \end{aligned} \tag{86}$$

As already remarked, the coefficients in front of ǫ n are complex numbers with absolute values of O(1) and arbitrary phases. Considering that Yν is a symmetric matrix, the total number of undetermined parameters that arise in this type of constructions is 30 plus the unknown value of ǫ. In order to establish which models adapt better to the data of **Table 1**, one cannot use a χ 2 based analysis because the minimum is always very close to zero for every (Ye, Y<sup>ν</sup> ) pairs; thus, a meaningful comparison of two models is better achieved with the help of a Bayesian analysis. This has been done in Bergstrom [139] and the results of the Bayes factor between all models and A ′ are reported in **Figure 5**.

The relevant features of such an analysis can be summarized as follows: when using only the neutrino data, the hierarchical models are all weakly preferred over the anarchical ones. When also the charged lepton data are taken into account in the analysis, the A model turns out to be strongly disfavored. Adding in the comparison also the H′ and A ′ models, the former is the best one: it is moderately better than Aµτ and A ′ , and weakly preferred over H.

Other possibilities in the direction of using U(1) rely on the fact that the U(1) charges are not completely arbitrary but are determined by an underlying symmetry of the type L<sup>e</sup> − L<sup>µ</sup> − L<sup>τ</sup> for lepton doublets and arbitrary right-handed charges [143– 145]. In the limit of exact symmetry, the neutrino mass matrix has the following structure:

$$m\_{\mathbb{V}} = m\_0 \begin{pmatrix} 0 & 1 & \boldsymbol{x} \\ 1 & 0 & 0 \\ \boldsymbol{x} & 0 & 0 \end{pmatrix},\tag{87}$$

which leads to a spectrum of inverted type and mixing angles as θ<sup>12</sup> = π/4, tan θ<sup>23</sup> = x (i.e., large atmospheric mixing for x ∼ <sup>O</sup>(1)) and θ<sup>13</sup> = 0. An important limitation of such a texture is that two eigenvalues have the same absolute values and the solar mass difference cannot be reproduced. Successful tentatives to describe also 1m<sup>2</sup> <sup>21</sup> have been presented, for instance, in Lavoura and Grimus [146] and Grimus and Lavoura [147] where, however, either the reactor angle was almost vanishing or the

solar angle was too large with respect to its current value. Corrections of O(λC) from the charged lepton sector [92–94] could be invoked to properly shift θ<sup>12</sup> from maximal mixing and θ<sup>13</sup> from zero, thus allowing a sizable reactor angle, but at the prize of a too large solar-to-atmospheric mass ratio r. A possible solution to the previous issues was discussed in Meloni [145], where the U(1) flavor symmetry was broken by the vevs of two complex fields φ and θ (instead of one) of charges Q<sup>φ</sup> = 1 and Q<sup>θ</sup> = −1/2. An appropriate breaking of L<sup>e</sup> − L<sup>µ</sup> − L<sup>τ</sup> in the neutrino sector assures the correct value of r ∼ λ 2 C and preserves the leading order (LO) prediction of large θ23, whereas the necessary deviations for the solar and reactor angles are instead obtained from the charged lepton mass matrix with complex entries.

### 5. WHERE GUT MEETS FLAVOR

The importance of the discovery of neutrino masses and mixing angles is that they provide interesting information on the problem of understanding the origin of three families of quarks and leptons and their mixing parameters. In this respect, as we have already outlined before, the relevance of GUT groups resides on the fact that some of the mass matrices of different fermions are related in a non-trivial way, see for example Equation (31), whereas family symmetries impose stringent constraints on the matrix elements of the same mass matrix. **Figure 6** summarizes in concise way how GUT and family symmetries act on the observable fermions (see caption for more details).

The next obvious step is to merge these two different type of symmetries in order to construct a flavor sector with very few free parameters. As it was the case for the special patterns of lepton mixing, also in the case with GUT one needs to identify which features of the data are really relevant for the formulation of a model. In this sense, the fact that the reactor angle θ<sup>13</sup> is approximately related to the Cabibbo angle θ<sup>C</sup> by the relation θ<sup>13</sup> ∼ θC/ √ 2 may be a hint of a connection between leptonic and quark mixing [9]. And this is not restricted to the reactor angle only. In fact, as shown in **Figure 7**, the experimental value of sin<sup>2</sup> θ<sup>12</sup> is related to the predictions of exact TBM or GR by a jump of order λ 2 C , or of order λ<sup>C</sup> in the case of BM.

This idea seems to agree with the empirical observation that θ<sup>12</sup> + θ<sup>C</sup> ∼ π/4, a relation known as quark-lepton complementarity [8]–[11], sometimes replaced by θ<sup>12</sup> + <sup>O</sup>(θC) ∼ π/4 (weak complementarity). If we want to realize in a complete model the previous relations, one possibility is to start from BM and generate universal corrections to the mixing angles of order λC, arriving at the following relations:

$$
\sin^2 \theta\_{12} \sim \frac{1}{2} + \mathcal{O}(\lambda\_C) \quad \sin^2 \theta\_{23} \sim \frac{1}{2} + \mathcal{O}(\lambda\_C) \quad \sin \theta\_{13} \sim \mathcal{O}(\lambda\_C),
\tag{88}
$$

which are all in agreement with the experimental data. These corrections can be appropriately fabricated by charged lepton rotations which differ from the identity by off-diagonal elements whose magnitude is obviously of order of the Cabibbo angle. The game becomes highly non-trivial in GUT theories which demand that also masses for the quarks and the CKM matrix are reproduced at the same time. An example based on SU(5) that permits to realize the program of having the BM structure in the neutrino sector and then to correct it by terms arising from the diagonalization of the charged lepton mass matrix is built as follows [149] (but see [150] for a variant using the A<sup>4</sup> family group). The construction is a SUSY SU(5) model in 4+1 dimensions [151, 152] with a flavor symmetry S<sup>4</sup> ⊗ Z<sup>3</sup> ⊗ U(1)<sup>R</sup> ⊗ U(1) [105, 149], where U(1) is the Froggatt-Nielsen (FN) symmetry that leads to the hierarchies of fermion masses and U(1)<sup>R</sup> is the usual R-symmetry. The particle assignments are displayed in **Table 5** where, for the sake of simplicity, we have not reported the driving fields needed to realize the wanted symmetry breaking pattern. From the table we see that the three 5 are grouped into the S<sup>4</sup> triplet F, while the tenplets T1,2,3 are assigned to the singlet of S4. The breaking of the S<sup>4</sup> symmetry is ensured by a set of SU(5)-invariant flavon supermultiplets, which are three triplets ϕℓ, ϕ<sup>ν</sup> (31), χ<sup>ℓ</sup> (32) and one singlet ξ<sup>ν</sup> . The alignment in flavor space of their vevs along appropriate directions will be the source of the BM lepton mixing. The GUT Higgs fields H<sup>5</sup> and H<sup>5</sup> are singlets under S<sup>4</sup> but equally charged under Z3, so that they are distinguished only by their SU(5) transformation properties. The tenplets T<sup>1</sup> and T<sup>2</sup> are charged under the U(1) flavor group which is spontaneously broken by the vevs of the θ and θ ′ fields, both carrying U(1) charges −1 and transforming as a singlet of S4.

As a result of symmetries and field assignments to the irreducible representations of SU(5) × S4, the charged lepton masses are diagonal at LO and exact BM is achieved for neutrinos. Higher dimension vertices in the Lagrangian, suppressed by powers of a large scale 3, generate corrections to the diagonal

masses have been multiplied or divided by the factors on top of each columns.

charged leptons and to exact BM. We adopt the definitions:

$$\frac{\nu\_{\varphi\_{\ell}}}{\Lambda} \sim \frac{\nu\_{\chi}}{\Lambda} \sim \frac{\nu\_{\varphi\_{\nu}}}{\Lambda} \sim \frac{\nu\_{\xi}}{\Lambda} \sim \frac{\langle \theta \rangle}{\Lambda} \sim \frac{\langle \theta' \rangle}{\Lambda} \sim s \sim \lambda\_{C} \tag{89}$$

where s = 1 √ πR3 is the volume suppression factor and vφ are the vevs of the flavon fields listed in **Table 5**. This simple (and democratic) choice leads to a good description of masses and mixing. In fact, the charged lepton mass matrix turns out to be:

$$m\_{\epsilon} \sim \begin{pmatrix} a\_{11}\lambda\_C^5 & a\_{21}\lambda\_C^4 & a\_{31}\lambda\_C^2\\ a\_{12}\lambda\_C^4 & -c\lambda\_C^3 & \dots & \dots\\ a\_{13}\lambda\_C^4 & c\lambda\_C^3 & a\_{33}\lambda\_C \end{pmatrix} \lambda\_C,\tag{90}$$

where the aij are generic complex coefficients of modulus of O(1) not predicted by the theory. The corresponding lepton rotation is thus:

$$U\_{\ell} \sim \begin{pmatrix} 1 & u\_{12}\lambda\_C & u\_{13}\lambda\_C \\ -u\_{12}^\*\lambda\_C & 1 & 0 \\ -u\_{13}^\*\lambda\_C & -u\_{12}^\*u\_{13}^\*\lambda\_C^2 & 1 \end{pmatrix},\tag{91}$$

(uij again of O(1)) so that θ ℓ <sup>23</sup> = 0.

The neutrino masses are obtained by Weinberg operators of the form:

$$\{\text{(FF)}\_1\{H\_5H\_5\}, \{FF\}\_{3\_1}H\_5H\_5\varphi\_\nu, \{FF\}\_{3\_1}H\_5H\_5\xi\_\nu,\tag{92}$$

which are diagonalized by exact BM, so the mixing angles are easily derived:

$$\begin{aligned} \sin^2 \theta\_{12} &= \frac{1}{2} - \frac{1}{\sqrt{2}} \operatorname{Re} (\mu\_{12} + \mu\_{13}) \lambda\_C \quad \sin^2 \theta\_{23} = \frac{1}{2} + \mathcal{O}(\lambda\_C^2), \\\sin \theta\_{13} &= \frac{1}{\sqrt{2}} |\mu\_{12} - \mu\_{13}| \lambda\_C. \end{aligned}$$

We observe that the model produces at the same time the "weak" complementarity relation and the empirical fact that sin θ<sup>13</sup> is of the same order than the shift of sin<sup>2</sup> θ<sup>12</sup> from the BM value of 1/2, both of order λC.

It is important to stress that the predictions of GUT models are valid at the GUT scale and, in order to compare with the experimental results, the evolution of the Yukawa matrices down to the electroweak scale must be performed [153, 154]. Although the final values depend somehow on the details of the model, it is known that in the case of a quasi-degenerate neutrino mass


TABLE 5 | Matter and Higgs assignment of the model.

The symbol br(bu) indicates that the corresponding fields live on the brane (bulk).

spectrum, the renormalization group corrections to the neutrino parameters can be dramatically large [155, 156]. However, as it has been elucidated in Antusch et al. [157, 158], in SUSY models small tan β and small neutrino Yukawa couplings are sufficient conditions for having the corrections to the mixing angles (and CP phases) are under control.

The requirement of having a BM mixing as a starting point is not a necessary ingredient to get a good description of fermion observables; as pointed out in Hagedorn et al. [159], even from the TBM at LO one can conceive a model where the corrections to the reactor angle are large enough to meet the experimental value, maintaining at the same time the solar and atmospheric mixing at acceptable values. Also the choice of the discrete group is not restricted to S4; examples where a large θ<sup>13</sup> is obtained after substantial corrections from higher order operators can be found, for example, in King et al. [118], Antusch et al. [154], Cooper et al. [160], Marzocca et al. [161], Antusch et al. [162, 163], Björkeroth et al. [164], Antusch and Hohl [165], Gehrlein et al. [166], and Meroni et al. [167], which employ the A4, A5, T ′ and 1(96) groups, respectively, within an SU(5) framework.

If the gauge group is enlarged to SO(10), we loose the advantages of using the SU(5)-singlet right-handed neutrinos since one generation of fermion belongs to the **16**-dimensional representation. One possible strategy to separate neutrinos from the charged fermions is to assume the dominance of type-II see-saw with respect to the more usual type-I see-saw.

As we have already seen, in models of this type neutrino masses are described by <sup>M</sup><sup>ν</sup> ∼ fvL, where v<sup>L</sup> is the vev of the B − L = 2 triplet in the **126**<sup>H</sup> Higgs field and f is its Yukawa coupling matrix with the **16**. Since one can decide to work in a basis where the matrix f is diagonalized by the BM or by TBM matrices, the results of a fit of the model parameters on the fermion observables performed in one basis lead to the same χ 2 than the fit in the other basis, thus a χ 2 analysis cannot decide whether TBM or BM is a better starting point [148]. This is confirmed by the plot in **Figure 8**, where it is shown that, within uncertainties, the χ 2 as a function of the reactor angle is equal in the two cases, and this is true also for values of sin θ<sup>13</sup> different than the measured value. In particular, the minimum χ 2 value, χ <sup>2</sup> <sup>=</sup> 0.003, is obtained for sin<sup>2</sup> θ<sup>13</sup> ∼ 0.015, just a bit below the experimental value sin<sup>2</sup> θ<sup>13</sup> ∼ 0.022. Nevertheless, as the minimum χ 2 is quite shallow for sin<sup>2</sup> θ<sup>13</sup> < 0.1, the fit does not exhibit any strongly preferred value of θ13.

FIGURE 8 | χ <sup>2</sup> as a function of sin<sup>2</sup> θ13 in the type-II see-saw SO(10) models obtained when starting in the TBM or BM basis.

Having established that the χ 2 is not the best variable to decide whether TBM or BM is better, one can consider to measure the amount of fine-tuning needed to fit a set of data by means of the parameter dFT introduced in Altarelli and Blankenburg [168]:

$$d\_{FT} = \sum \left| \frac{p\_i}{e\_i} \right|,\tag{93}$$

where e<sup>i</sup> is the "error" of a given parameter p<sup>i</sup> defined as the shift from the best fit value that changes the χ <sup>2</sup> by one unit, with all other parameters fixed at their best fit values. In **Figure 9** we report a study of the fine tuning parameter when the fit is repeated with the same data except for sin<sup>2</sup> θ13. It clearly shows that:


A closer inspection of the dFT parameter reveals a series of interesting features: first of all, that the large values are

predominantly driven by the smallness of the electron mass; then, due to the presence of mixing, the dFT coming from the 33 component of h (mainly responsible for the top mass) is actually one of the largest contributions to the global dFT because of its relevance to the electron mass in both TBM and BM scenarios. Although this might be surprising, one has to take into account that the dependence of the observables on the parameters is quite complicated due to the off-diagonal elements of the mass matrices.

Other classes of renormalizable and non-renormalizable SO(10) models supplemented by discrete and continuous symmetries have been discussed in the literature. In Altarelli and Blankenburg [168] a model comparison based on a χ 2 analysis and on the values of dFT has been carried out with sufficient details to allow for a discrimination in terms of performance in the description of the data. **Table 6** has been extracted from Altarelli and Blankenburg [168] and reports the results of such a comparison. The model called BSV [47] (no flavor symmetries involved here) has a minimal Yukawa sector with **10**<sup>H</sup> and **126**<sup>H</sup> and has been compared with the data in Bertolini et al. [52], where the type-I and mixed type-I and type-II cases were considered. As it is well known, the restricted Higgs content calls for complex h and f matrices. Even increasing the number of free parameters, with type-II dominance no good fit of the data can be obtained. The situation changes if one introduces the **120**<sup>H</sup> of Higgs, as in the model with type-II see-saw dominance introduced by Joshipura and Kodrani (JK) [169]. The relevant feature of this model is the existence of a broken µ−τ symmetry in addition to the parity symmetry which causes hermitian mass matrices. Similarly, Grimus and Kuhbock [170] (GK) also have an extended Higgs sector with **10**H, **126** and **120**<sup>H</sup> but their model is based on type-I see-saw dominance.

In the class of non-renormalizable SO(10) theories, we can cite the model of Dermisek and Raby (DR) [171, 172]; it contains Higgses in the **10**H, **45**<sup>H</sup> and **16**H, and it is based on the flavor symmetry S<sup>3</sup> × U(1) × Z<sup>2</sup> × Z2. In the symmetric S<sup>3</sup> limit only


Above the double lines mark we report the non-renormalizable models whereas below we list the renormalizable models considered in this paper. Adapted from Altarelli and Blankenburg [168].

the masses of the third generation are non-vanishing while the second and first generation masses are generated by a symmetry breaking stage. The neutrino masses are obtained through a type-I see-saw mechanism with a hierarchical Majorana mass matrix. Enough freedom to reproduce the observed neutrino properties is guarantee by new SO(10)-singlet neutrino and new scalar fields.

A similar Higgs sector with **10**H, (**16** + **16**)<sup>H</sup> and **45**<sup>H</sup> representations and a few SO(10) singlets constitute the scalar sector of the model by Albright, Babu and Barr (ABB) [173, 174]. However, this model is based on a flavor symmetry U(1)×Z2×Z<sup>2</sup> which is mainly used to select the desired terms which in the Lagrangian and reject those that would not help in reproducing the data. A modification of this model has been proposed by Ji, Li, Mohapatra (JLM) [175]; the charged lepton and the down quark mass matrices are the same as in the ABB model but the up and Dirac neutrino mass matrices are modified thanks to new dimension five and six vertices introduced in the theory. The model is based on type-I see-saw and the new operators provide a sufficient number of free parameters to fit the leptonic mixing angles.

The relevant feature of the results presented in **Table 6** is that the realistic SO(10) models which are non-renormalizable with type-I see-saw (DR, ABB, JLM), have a χ 2 /d.o.f. smaller than 1 and a moderate level of fine tuning dFT, if compared with the relatively more constrained BSV, JK and GK. They all have a large amount of fine tuning and, with the exception of the GK model, a worst χ 2 . The larger fine tuning arises from the more pronounced difficulty of fitting the light first generation of charged fermion masses, together with the neutrino mass differences and mixing angles.

More recently, successful attempts to completely describe neutrino data within S<sup>4</sup> and 1(27) have been presented in Björkeroth et al. [176–178], where also the ability to provide a framework for the leptogenesis mechanism has been addressed [178].

Beside the models with complete unification at the GUT scale, one can also consider the possibility of supplementing with flavor symmetries models with partial unification, that is theories where the gauge group at the GUT scale is not an unique group. Good examples in this direction are those based on the Pati-Salam group SU(4)<sup>c</sup> ⊗ SU(2)<sup>L</sup> ⊗ SU(2)<sup>R</sup> (PS), as discussed in de Adelhart Toorop et al. [179], where S<sup>4</sup>

was employed to recover the quark-lepton complementarity at LO and in King [180, 181], which explores the capabilities of A<sup>4</sup> to describe quark and lepton masses, mixing and CP violation<sup>5</sup> . As usual, these models also need the presence of additional discrete (or continuous U(1)) symmetries to forbid or suppress unwanted operators. In **Figure 10**, modified from King [181], we sketch a possible particle assignment for models with - PS ⊗ permutation ⊗ discrete groups, where it is understood that the permutation group contains triplet representations. In both panels, the red, blue and green colors represent the SU(3) triplets, which are accompanied with the light gray particles to complete the fundamental **4** representation of SU(4)<sup>c</sup> . The left-handed families are assigned to triplet presentations of the permutation groups and are doublets under SU(2)L, left panel. On the right panel we consider that the right-handed families are distinguished by different charges of the discrete group and are doublets of SU(2)R.

# 6. CONCLUSIONS

The question of the theoretical understanding of the experimental numbers of fermion masses and mixing is a very old story. Although neutrinos were considered as a promising tool to access the fundamental properties of particle interactions, the new data helped to discard some theoretical model on lepton mixing (mainly those based on θ<sup>13</sup> = 0 at the LO) but many other still offer a viable solution, spanning a wide range of possibilities going from a situation with no structure and no symmetry in the neutrino sector (anarchy) to a maximum of symmetry for the models based on discrete non-abelian flavor groups.

In this respect, neutrinos have not offered so far any crucial insight on the problem of flavor. The extension to include GUT (or Partial Unification) symmetry exacerbates the difficulties in the model building, as also the quark properties must be taken into account and the larger symmetry reduces the useful number of free parameters.

If one is driven by the fact that the quark-lepton complementarity is a real feature of Nature, then models based on SU(5) with a broken S<sup>4</sup> symmetry emerge as one among the most viable and predictive theory, in which fermion masses and mixing are all well reproduced inside their experimental ranges at the prize of small fine-tunings in very few model parameters.

As we have seen in **Table 1**, the octant of the atmospheric angle, the value of the CP violating phase δ and the neutrino mass orderings are features of the neutrinos that have not been clearly addressed so far. Thus, from the model building point of view, the results coming from the running (for instance, NOνA [183] and T2K [184]) and planned experiments (like DUNE [185]) can certainly help in selecting the class of models that, more than others, will be able to incorporate the new information. In this respect, the emerging indication of δ ∼ 3/2π seems to exclude the whole class of models predicting CP-conserving Dirac phase, as many do of those listed in section 4.5.

On the other hand, the uncertainties affecting the already measured mixing angle and mass differences are expected to be reduced to a sub-percent level in the next 5–10 years (as it is the case for the solar parameters measured by the JUNO detector [186]) and, in a framework where the mixing parameters are obtained from a LO neutrino mass texture corrected by charge lepton rotations, this can influence in a critical manner which LO mass matrix is the most useful starting point; with more precise measurements, the jumps described in **Figure 7**, needed to reconcile the LO predictions with the data, must be chosen more carefully.

## AUTHOR CONTRIBUTIONS

The author confirms being the sole contributor of this work and approved it for publication.

## ACKNOWLEDGMENTS

The author is strongly indebted with Erica Vagnoni and Andrea Di Iura for useful discussions.

<sup>5</sup> See King [182] for an example of a PS model where, instead of a discrete group, the continuos SO(3) gauged family symmetry has been employed.

### REFERENCES


**Conflict of Interest Statement:** The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Meloni. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Multicomponent Dark Matter in Radiative Seesaw Models

#### Mayumi Aoki\*, Daiki Kaneko and Jisuke Kubo

*Institute for Theoretical Physics, Kanazawa University, Kanazawa, Japan*

We discuss radiative seesaw models, in which an exact *Z*<sup>2</sup> × *Z* ′ 2 symmetry is imposed. Due to the exact *Z*<sup>2</sup> × *Z* ′ 2 symmetry, neutrino masses are generated at a two-loop level and at least two extra stable electrically neutral particles are predicted. We consider two models: one has a multi-component dark matter system and the other one has a dark radiation in addition to a dark matter. In the multi-component dark matter system, non-standard dark matter annihilation processes exist. We find that they play important roles in determining the relic abundance and also responsible for the monochromatic neutrino lines resulting from the dark matter annihilation process. In the model with the dark radiation, the structure of the Yukawa coupling is considerably constrained and gives an interesting relationship among cosmology, lepton flavor violating decay of the charged leptons and the decay of the inert Higgs bosons.

#### Edited by:

*Frank Franz Deppisch, University College London, United Kingdom*

#### Reviewed by:

*Arindam Das, Korea Institute for Advanced Study, South Korea Michael Duerr, Deutsches Elektronen-Synchrotron (HZ), Germany*

#### \*Correspondence:

*Mayumi Aoki mayumi@hep.s.kanazawa-u.ac.jp*

#### Specialty section:

*This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics*

Received: *04 August 2017* Accepted: *10 October 2017* Published: *01 November 2017*

#### Citation:

*Aoki M, Kaneko D and Kubo J (2017) Multicomponent Dark Matter in Radiative Seesaw Models. Front. Phys. 5:53. doi: 10.3389/fphy.2017.00053* Keywords: neutrino mass, dark matter, radiative seesaw mechanism, flavor physics, dark radiation

### 1. INTRODUCTION

Neutrino oscillation experiments show that the neutrinos have tiny masses and mix with each other. It is a clear evidence for physics beyond the standard model (SM), since the SM has no mechanism for giving masses to the neutrinos. The global fit [1] shows that the mass-squared differences of the neutrinos are 1m<sup>2</sup> <sup>21</sup> <sup>=</sup> 7.50+0.19 −0.17×10−<sup>5</sup> eV<sup>2</sup> and 1m<sup>2</sup> <sup>31</sup> <sup>=</sup> 2.524+0.039 <sup>−</sup>0.040 (−2.514+0.038 −0.041)×10−<sup>3</sup> eV<sup>2</sup> for normal (inverted) mass hierarchy. The cosmological data, on the other hand, gives the upper bound of the sum of the neutrino masses as 6jmν<sup>j</sup> < 0.23 eV [2], a scale twelve orders of magnitudes smaller than the electroweak scale. It is one of the most important problems of particle physics to reveal the origin of the tiny masses for the neutrinos.

Type-I seesaw mechanism [3–6] is one of the attractive way to realize the tiny masses of the neutrinos, where the right-handed neutrinos are introduced to the SM. If the neutrino Yukawa coupling for the Dirac neutrino mass is O(1), the mass of the right-handed neutrino has to be around O(1015) GeV to obtain eV-scale neutrinos. The mass scale of O(1015) GeV is obviously beyond the reach of collider experiments. Even for the mass of the right-handed neutrinos around O(1) TeV, the direct search of the right-handed neutrinos would be difficult because of the tiny neutrino Yukawa couplings of O(10−<sup>6</sup> ).

Another attractive way to give the neutrino masses is a radiative generation (the so-called radiative seesaw model). The original idea of radiatively generating neutrino masses due to TeVscale physics has been proposed by Zee [7], in which the neutrino masses are induced at the one-loop level because of the addition of an isospin doublet scalar field and a charged singlet field to the SM. Another possibility for generating neutrino masses via the new scalar particles is e.g., the Zee-Babu model [8, 9], in which the neutrino masses arise at the two-loop level.

A further extension with a TeV-scale right-handed neutrino has been proposed in Krauss et al. [10]. In this model the neutrino masses are induced at the three-loop level, where the Dirac neutrino mass at the tree level is forbidden due to an exact Z<sup>2</sup> symmetry. The right-handed neutrino is odd under the Z<sup>2</sup> and becomes a candidate of dark matter (DM). The idea of simultaneous explanation for the neutrino masses via the radiative seesaw mechanism and the stability of DM by introducing an exact discrete symmetry has been discussed in many models (see e.g., [11–46] and the recent review [47] and references therein).

The model proposed by Ma [11] is one of the simplest radiative seesaw model with DM candidates. The model has the Z2-odd right-handed neutrinos N<sup>k</sup> and the inert doublet scalar field η = (η +, η 0 <sup>R</sup> + iη 0 I ) T . The neutrino masses are generated at the one-loop level, in which the Yukawa interactions Y ν ikLiηN<sup>k</sup> and the scalar interaction (λ5/2)(H†η) 2 contribute to the neutrino mass generation. The mass matrix is expressed as

$$\begin{split} (\mathcal{M}\_{\boldsymbol{\nu}})\_{\boldsymbol{ij}} &= \sum\_{k} \frac{Y\_{ik}^{\boldsymbol{\nu}} Y\_{jk}^{\boldsymbol{\nu}} M\_{k}}{32\pi^{2}} \left[ \frac{m\_{\eta\_{R}}^{2}}{m\_{\eta\_{R}}^{2} - M\_{k}^{2}} \ln \left( \frac{m\_{\eta\_{R}^{0}}}{M\_{k}} \right)^{2} \right] \\ &- \frac{m\_{\eta\_{I}^{0}}^{2}}{m\_{\eta\_{I}^{0}}^{2} - M\_{k}^{2}} \ln \left( \frac{m\_{\eta\_{I}^{0}}}{M\_{k}} \right)^{2} \Bigg], \end{split}$$

where M<sup>k</sup> is the Majorana mass of the k-th generation of righthanded neutrino, m<sup>η</sup> 0 R and m<sup>η</sup> 0 I are the mass of the η 0 R and η 0 I , respectively. In this model, we have two scenarios with respect to the DM candidate; the lightest right-handed neutrino N<sup>1</sup> or the lighter Z2-odd neutral scalar field (η 0 R or η 0 I ). The phenomenology of the model is studied in Kubo et al. [48], Bouchand and Merle [49], Merle and Platscher [50], Ma and Raidal [51], Suematsu et al. [52], Aoki and Kanemura [53], Suematsu et al. [54], Schmidt et al. [55], Hehn and Ibarra [56], Toma and Vicente [57], Ibarra et al. [58], Merle et al. [59], Lindner et al. [60], Hessler et al. [61], Aristizabal Sierra et al. [62], Gelmini et al. [63], and Ma [64].

A DM candidate can be made stable by an unbroken symmetry. The simplest possibility of such a symmetry is Z<sup>2</sup> symmetry as in the above models. However, if the DM stabilizing symmetry is larger than Z2: Z<sup>N</sup> (N ≥ 4) or a product of two or more Z2s, the DM is consisting of stable multi-DM particles (multicomponent DM system). A supersymmetric extension of the radiative seesaw model of Ma [11] is an example [41– 46]. Other possibilities with multicomponent DM are widely discussed in Ma and Sarkar [34], Kajiyama et al. [35, 37], Wang and Han [36], Baek et al. [38], Aoki et al. [39, 40], Bhattacharya et al. [65], Berezhiani and Khlopov [66, 67], Hur et al. [68], Zurek [69], Batell [70], Dienes and Thomas [71, 72], Ivanov and Keus [73], Dienes et al. [74], D'Eramo et al. [75], Gu [76], Bhattacharya et al. [77, 78], Geng et al. [79], Boddy et al. [80], Geng et al. [81], Esch et al. [82], Geng et al. [83], Arcadi et al. [84], DiFranzo and Mohlabeng [85], Aoki and Toma [86], and Aoki et al. [87].

In this paper we study two models of the two-loop extension of the model by Ma [11], we call them as "model A" and "model B," in which due to the Z<sup>2</sup> × Z ′ 2 symmetry a set of stable particles can exist. Introducing two new scalar fields, the λ<sup>5</sup> term is generated radiatively in the model A [39, 40]. In this model we discuss the three component DM system in which the two new scalar fields and a right-handed neutrino are the DM candidate. Such case has been discussed in Aoki et al. [40], however, we reanalyze the model since the benchmark points studied in Aoki et al. [40], where the masses of both new scalars are several hundred GeV, has been excluded by the recent results of the direct detection DM experiments. In this paper we focus on the case where the mass of one of the scalar DMs is close to the half of the Higgs boson mass to satisfy the constraints from the direct detection. In the model B the right-handed neutrinos have the mass radiatively generated through the one loop of internal new fermion and scalar fields. We identify the lightest right-handed neutrino as dark radiation.

We start in section 2 by writing down a set of the Boltzmann equations of the multicomponent DM system. The model A is discussed in section 3 by following Aoki et al. [40]. In section 4 we discuss the model B and relate dark radiation with the lepton flavor violating decay of the charged leptons and the decay of the inert Higgs bosons. Summery and discussion are given in section 5.

### 2. MULTICOMPONENT DARK MATTER SYSTEMS

In the case of one-component DM the relic density of DM χ is determined by the Boltzmann equation

$$
\dot{n}\_{\chi} + \mathfrak{Z}H n\_{\chi} = - \left< \sigma\_{\chi\chi \to XX'} \nu \right> (n\_{\chi}^2 - \bar{n}\_{\chi}^2), \tag{1}
$$

where n<sup>χ</sup> is the DM number density, n¯<sup>χ</sup> is the equilibrium number density and hσχχ→XX′vi is the thermally averaged cross section for χχ → XX′ . Here X and X ′ stand for the SM particles. The Hubble parameter H is given by H = 1.66 × g 1/2 <sup>∗</sup> T 2 /MPL, where g<sup>∗</sup> is the total number of effective degrees of freedom, T and MPL are the temperature and the Planck mass, respectively. It is convenient to rewrite the equation in terms of dimensionless quantities; the number per comoving volume Y<sup>χ</sup> = n<sup>χ</sup> /s and the inverse temperature x = m/T. Here s is the entropy density s = (2π 2 /45)g∗T 3 and m is the mass of the DM particle. Using the replacement of dx/dt = H|T=m/x, we obtain

$$\frac{dY\_\chi}{d\chi} = -0.264 \, g\_\*^{1/2} \frac{mM\_{\rm PL}}{\chi^2} \, \langle \sigma\_{\chi\chi \to X\bar{X}'} \nu \rangle \left( Y\_\chi Y\_\chi - \bar{Y}\_\chi \bar{Y}\_\chi \right) . \tag{2}$$

The thermally averaged cross section <sup>h</sup>σχχ→XX′v<sup>i</sup> of <sup>O</sup>(10−<sup>9</sup> ) GeV with a DM mass of 100 GeV gives <sup>Y</sup><sup>χ</sup> <sup>≃</sup> <sup>10</sup>−12, which is consistent with the observed value of the relic abundance h 2 ≃ 0.12 [88].

In the multicomponent DM system three types of processes enter in the Boltzmann equations<sup>1</sup> :

$$\chi\_i \chi\_i \rightsquigarrow XX' \quad \text{(standard annihilation)}, \tag{3}$$


<sup>1</sup> Semi-annihilation processes also exist in one-component DM systems when DM is a Z<sup>3</sup> charged particle [89–95] or a vector boson [96–99].

See **Figure 1** for a depiction of three types of processes. Here we assume that none of the DM particles have the same quantum number with respect to the DM stabilizing symmetry. The Boltzmann equations for the DM particle χ<sup>i</sup> with mass m<sup>i</sup> are

$$\begin{split} \frac{dY\_i}{dx} &= -0.264 \, g\_\*^{1/2} \, \frac{\mu M\_{\rm PL}}{\chi^2} \Big\{ \langle \sigma\_{\dot{X} \circ \bar{X} \to \mathcal{X} \mathcal{X}'} \nu \rangle \left( Y\_i Y\_i - \bar{Y}\_i \bar{Y}\_i \right) \\ &+ \sum\_{i>j} \langle \sigma\_{\dot{X} \circ \bar{X} \to \mathcal{X} \mathcal{X} \mathcal{Y}} \nu \rangle \Big( Y\_i Y\_i - \frac{Y\_j Y\_j}{\bar{Y}\_j \bar{Y}\_j} \bar{Y}\_i \bar{Y}\_i \Big) \\ &- \sum\_{j>i} \langle \sigma\_{\dot{X} \circ \bar{X} \to \mathcal{X} \mathcal{X} \mathcal{X}} \nu \rangle \left( Y\_j Y\_j - \frac{Y\_i Y\_i}{\bar{Y}\_i \bar{Y}\_j} \bar{Y}\_j \bar{Y}\_j \right) \\ &+ \sum\_{j,k} \langle \sigma\_{\dot{X} \circ \bar{X} \to \mathcal{X} \mathcal{X} \mathcal{X} \mathcal{Y}} \nu \rangle \left( Y\_i Y\_j - \frac{Y\_k}{\bar{Y}\_k} \bar{Y}\_i \bar{Y}\_j \right) \\ &- \sum\_{j,k} \langle \sigma\_{\dot{X} \circ \bar{X} \to \mathcal{X} \mathcal{X} \mathcal{Y}} \nu \rangle \left( Y\_j Y\_k - \frac{Y\_i}{\bar{Y}\_i} \bar{Y}\_j \bar{Y}\_k \right) \Bigg\}. \tag{6} \end{split}$$

Here x = µ/T and 1/µ = ( P i m −1 i ) is the reduced mass of the system. The contributions of non-standard annihilations have been discussed in e.g., Aoki et al. [87] for two and three component DM system with a Z<sup>2</sup> × Z ′ 2 symmetry.

#### 3. MODEL A

In the following, by extending the one-loop model in Ma [11] we study two of the two-loop radiative seesaw models with Z<sup>2</sup> × Z ′ 2 symmetry. We refer to them as "model A" and "model B." Owing to the Z<sup>2</sup> × Z ′ 2 symmetry, there exist at least two extra stable electrically neutral particles. The multicomponent DM system is realized in the model A, while one of the stable particles plays as the dark radiation in the model B.

The matter content of the model A is shown in **Table 1**. In addition to the matter content of the SM model, we introduce the right-handed neutrino N<sup>k</sup> , an SU(2)<sup>L</sup> doublet scalar η, and two SM singlet scalars χ and φ. Note that the lepton number L ′ of N is zero. The Z<sup>2</sup> × Z ′ <sup>2</sup> × L ′ -invariant Yukawa sector and Majorana mass term for N can be described by

$$\mathcal{L}\_Y = Y\_{ij}^e H^\dagger L\_i l\_{\text{Rj}}^\epsilon + Y\_{ik}^\upsilon L\_i \epsilon \eta N\_k - \frac{1}{2} M\_k N\_k N\_k + h.c. \tag{7}$$

where i, j, k (= 1, 2, 3) stand for the flavor indices. The scalar potential V is written as V = V<sup>λ</sup> + Vm, where

$$V\_{\lambda} = \lambda\_1 (H^\dagger H)^2 + \lambda\_2 (\eta^\dagger \eta)^2 + \lambda\_3 (H^\dagger H)(\eta^\dagger \eta) + \lambda\_4 (H^\dagger \eta)(\eta^\dagger H)$$

$$+ \gamma\_1 \chi^4 + \gamma\_2 (H^\dagger H) \chi^2 + \gamma\_3 (\eta^\dagger \eta) \chi^2 + \gamma\_4 |\phi|^4 + \gamma\_5 (H^\dagger H) |\phi|^2$$

$$+ \gamma\_6 (\eta^\dagger \eta) |\phi|^2 + \gamma\_7 \chi^2 |\phi|^2 + \frac{\kappa}{2} [(H^\dagger \eta) \chi \phi + h.c.] \,, \tag{8}$$

$$\begin{aligned} \left[\cdots \cdots \cdots \cdots \right] &= \left[\cdots \right] + \sum\_{1}^{2} \left[\cdots \right] + \cdots \\ \left[\left.V\_{m} = m\_{1}^{2}H^{\dagger}H + m\_{2}^{2}\eta^{\dagger}\eta + \frac{1}{2}m\_{3}^{2}\chi^{2} + m\_{4}^{2}|\phi|^{2} + \frac{1}{2}m\_{5}^{2}|\phi|^{2} \right. \\ &\left. + (\phi^{\*})^{2}\right]. \end{aligned} \tag{9}$$

The Z<sup>2</sup> × Z ′ 2 is the unbroken discrete symmetry while the lepton number L ′ is softly broken by the last term in the potential Vm, the φ mass tem. In the absence of this term, there will be no neutrino mass. Note that the "λ<sup>5</sup> term," (λ5/2)(H†η) 2 , is also forbidden by L ′ . A small λ<sup>5</sup> of the original model of Ma [11] is "natural" according to 't Hooft [100], because the absence of λ<sup>5</sup> implies an enhancement of symmetry. In fact, if λ<sup>5</sup> is small at some scale, it remains small for other scales as one can explicitly verify [49, 50]. Here we attempt to derive the smallness of λ<sup>5</sup> dynamically, such that the λ<sup>5</sup> term becomes calculable.

The Higgs doublet field H, the inert doublet field η and the singlet scalar φ are respectively parameterized as

$$\begin{aligned} H &= \begin{pmatrix} H^+ \\ (\upsilon\_h + h + i\mathcal{G})/\sqrt{2} \end{pmatrix}, & \eta &= \begin{pmatrix} \eta^+ \\ (\eta\_R^0 + i\eta\_I^0)/\sqrt{2} \end{pmatrix}, \\ \phi &= (\phi\_R + i\phi\_I)/\sqrt{2} \end{aligned} \tag{10}$$

TABLE 1 | The matter content and the corresponding quantum numbers of the model A.


where v<sup>h</sup> is the vacuum expectation value. The tree-level masses of the scalars are given by

$$\left\|m\_h^2 = 2\lambda\_1 \nu\_h^2\right\|,\tag{11}$$

$$m\_{\eta^{\pm}}^2 = m\_2^2 + \lambda\_3 \nu\_h^2 / 2 \,, \tag{12}$$

$$m\_{\eta\_R^0}^2 = m\_{\eta\_I^0}^2 = m\_2^2 + (\lambda\_3 + \lambda\_4)\nu\_h^2/2 \,, \tag{13}$$

$$m\_{\phi\_R}^2 = m\_4^2 + m\_5^2 + \wp\_5 \nu\_h^2,\tag{14}$$

$$m\_{\phi\_I}^2 = m\_4^2 - m\_5^2 + \chi\_5 \nu\_h^2,\tag{15}$$

$$m\_{\chi}^2 = m\_3^2 + \wp\_2 \nu\_h^2. \tag{16}$$

Although the tree-level mass of η 0 R is the same as that of η 0 I as shown in Equation (13), the degeneracy is lifted at the one-loop level via the effective λ<sup>5</sup> term:

$$
\lambda\_5^{\text{eff}} \sim -\frac{\kappa^2}{128\pi^2} \frac{m\_5^2}{m\_{\phi\_R}^2 - m\_\chi^2} \left[ 1 - \frac{m\_\chi^2}{m\_{\phi\_R}^2 - m\_\chi^2} \ln \frac{m\_{\phi\_R}^2}{m\_\chi^2} \right] \tag{17}
$$

$$
\text{for } m\_5 \ll m\_{\phi\_R} \text{ .} \tag{17}
$$

This correction is embedded into the two-loop diagram to generate the neutrino mass (see **Figure 2**). The 3 × 3 neutrino mass matrix M<sup>ν</sup> can be given by

$$\begin{aligned} (\mathcal{M}\_{\boldsymbol{\nu}})\_{\vec{\boldsymbol{\nu}}} &= \left(\frac{1}{16\pi^{2}}\right)^{2} \frac{\kappa^{2} v\_{h}^{2}}{16} \\ &\sum\_{k} Y\_{ik}^{\boldsymbol{\nu}} Y\_{jk}^{\boldsymbol{\nu}} M\_{k} \int\_{0}^{\infty} d\boldsymbol{x} \, \frac{\boldsymbol{x}}{(\boldsymbol{x} + m\_{\boldsymbol{\eta}}^{2})^{2} (\boldsymbol{x} + M\_{k}^{2})} \\ &\int\_{0}^{1} d\boldsymbol{z} \ln \left[\frac{z m\_{\boldsymbol{\chi}}^{2} + (1 - z) m\_{\boldsymbol{\phi} \boldsymbol{\eta}}^{2} + z(1 - z) \boldsymbol{x}}{z m\_{\boldsymbol{\chi}}^{2} + (1 - z) m\_{\boldsymbol{\phi} \boldsymbol{\eta}}^{2} + z(1 - z) \boldsymbol{x}}\right], \tag{18} \end{aligned}$$

where we have assumed that m<sup>η</sup> <sup>0</sup> = m<sup>η</sup> 0 R ≃ m<sup>η</sup> 0 I .

Using λ eff 5 given in Equation (17), the neutrino mass matrix can be approximated as

$$(\mathcal{M}\_{\upsilon})\_{ij} \sim \frac{\lambda\_5^{\text{eff}} \nu\_h^2}{32\pi^2} \sum\_k \frac{Y\_{ik}^{\upsilon} Y\_{jk}^{\upsilon} M\_k}{m\_{\eta^0}^2 - M\_k^2} \left[ 1 - \frac{M\_k^2}{m\_{\eta^0}^2 - M\_k^2} \ln \frac{m\_{\eta^0}^2}{M\_k^2} \right]. \tag{19}$$

We see from Equations (17) and (19) that the neutrino mass matrix <sup>M</sup><sup>ν</sup> is proportional to |Y νκ| <sup>2</sup>m<sup>2</sup> 5 . When m<sup>χ</sup> , mφ<sup>R</sup> , m<sup>η</sup> 0 , <sup>M</sup><sup>k</sup> <sup>∼</sup> <sup>O</sup>(10<sup>2</sup> ) GeV, for instance, implies that |Y <sup>ν</sup>κ|m<sup>5</sup> <sup>∼</sup> O(10−<sup>1</sup> ) GeV to obtain the neutrino mass scale of O(0.1) eV. With the same set of the parameter values we find that λ eff 5 ∼ 10−<sup>6</sup> , where the smallness λ eff 5 is a consequence of the radiative generation of this coupling. As we will see, the product |Y νκ| enters into the semi-annihilation of DM particles which produces monochromatic neutrinos, while the upper bound of |Y ν | follows from the µ → eγ constraint.

#### 3.1. Multicomponent Dark Matter System

In the model A there are three type of dark matter candidates ; N<sup>1</sup> (the lightest among N<sup>k</sup> 's) or η 0 R (or η 0 I ) with (Z2, Z ′ 2 ) = (−, +), χ with (Z2, Z ′ 2 ) = (+, −) and φ<sup>R</sup> (or φI) with (Z2, Z ′ 2 ) = (−, −). For (Z2, Z ′ 2 ) = (−, +) there are two candidates. In the following discussions we assume that N<sup>1</sup> is a DM candidate [40]. The other possibility, η 0 R -DM, is discussed in Aoki et al. [39].

We discuss the three DM system of N1, φR, χ. There are three types of DM annihilation process:

$$\begin{aligned} \text{Standard annihilation} &: N\_1 N\_1 \to XX', & \phi\_R \phi\_R \to XX',\\ & \chi \chi \to XX', \end{aligned} \tag{20}$$

$$\begin{aligned} \chi \chi &\rightarrow X\mathcal{X}',\\ \text{DM conversion}: \phi\_{\mathbb{R}}\phi\_{\mathbb{R}} &\rightarrow \chi \chi,\end{aligned} \tag{20}$$

$$\text{Semi-annihilation}: N\_1 \phi\_\mathbb{R} \to \stackrel{\frown}{\chi} \nu, \ \chi N\_1 \to \phi\_\mathbb{R} \nu,$$

$$
\phi\_R \stackrel{\cdots}{\chi} \to N\_1 \stackrel{\cdots}{\nu} \stackrel{\cdots}{\cdot} \tag{22}
$$

Here we have assumed mφ<sup>R</sup> > m<sup>χ</sup> and mφ<sup>R</sup> + m<sup>χ</sup> < M2,3. Moreover, since the mass difference between φ<sup>R</sup> and φ<sup>I</sup> is controlled by the lepton-number breaking mass m5, which is assumed to be much smaller than mφ<sup>R</sup> . Then mφ<sup>R</sup> and mφ<sup>I</sup> are practically degenerate and the contribution of φ<sup>I</sup> to the annihilation processes during the decoupling of DMs is nonnegligible. The diagrams for annihilation processes which enter the Boltzmann equation are shown in **Figures 3**–**5**. Since the reaction rate of the conversion between φ<sup>R</sup> and φ<sup>I</sup> can reach chemical equilibrium during the decoupling of DMs, we can sum up the number densities of φ<sup>R</sup> and φ<sup>I</sup> and compute the relic abundance of φ<sup>R</sup> h 2 [40].

In the multicomponent DM scenario, the effective cross section off the nucleon is given by

$$
\sigma\_i^{\text{eff}} = \sigma\_i \left( \frac{\Omega\_i h^2}{\Omega\_{\text{total}} h^2} \right) \,. \tag{23}
$$

In our model, only χ and φ<sup>R</sup> scatter with the nucleus, and the right-handed neutrino N<sup>1</sup> does not interact with nucleus at tree level. So we can neglect the N<sup>1</sup> contribution at the lowest order in perturbation theory. The effective cross sections of φ<sup>R</sup> and χ are expressed as

$$
\sigma\_\chi^{\rm eff} = \sigma\_\chi \left( \frac{\Omega\_\chi h^2}{\Omega\_{\rm total} h^2} \right), \quad \sigma\_{\phi\_\mathbb{R}}^{\rm eff} = \sigma\_{\phi\_\mathbb{R}} \left( \frac{\Omega\_{\phi\_\mathbb{R}} h^2}{\Omega\_{\rm total} h^2} \right), \tag{24}
$$

where σ<sup>χ</sup> and σφ<sup>R</sup> are the spin independent cross sections and given by

$$
\sigma\_{\chi} = \frac{1}{\pi} \left( \frac{\nu\_2 \hat{f} m\_N}{m\_{\chi} m\_h^2} \right)^2 \left( \frac{m\_N m\_{\chi}}{m\_N + m\_{\chi}} \right)^2,\tag{25}
$$

$$
\sigma\_{\phi\_{\mathbb{R}}} = \frac{1}{\pi} \left( \frac{(\wp\_5/2)\hat{f}m\_N}{m\_{\phi\_{\mathbb{R}}}m\_h^2} \right)^2 \left( \frac{m\_N m\_{\phi\_{\mathbb{R}}}}{m\_N + m\_{\phi\_{\mathbb{R}}}} \right)^2. \tag{26}
$$

Here ˆ f ∼ 0.3 is the usual nucleonic matrix element [101] and m<sup>N</sup> is nucleon mass.

The upper bounds on the cross section off the nucleon is obtained by LUX [102] and XENON1T [103]. In the cases of onecomponent DM system of a real or complex scalar boson, those experimental results give the strong constraint on the masses of those DM particles; the allowed DM mass region is ≃mh/2 and <sup>&</sup>gt;∼1 TeV [104–106]. In the model A with the multicomponent DM system, the constrains on the cross sections off the nucleon for χ and φ<sup>R</sup> are also relatively severe. As a benchmark we take the mass of χ as m<sup>χ</sup> = mh/2 while vary the mass of φ<sup>R</sup> in the following analysis<sup>2</sup> .

In the original one-loop neutrino mass model in Ma [11], the relic density of N<sup>1</sup> tends to be larger than the observational value [48]. The additional contributions coming from the semiannihilation can enhance the annihilation rate for N<sup>1</sup> so that the N<sup>1</sup> DM contribution to the total relic abundance can be suppressed. This situation is realized for M<sup>1</sup> > mφ<sup>R</sup> , mχ as can be seen later.

As the benchmark set we take the following values for the parameters.

$$m\_{\chi} = 62 \text{ GeV}, \; M\_1 = 300 \text{ GeV}, \tag{27}$$

$$m\_{\eta\_R^0} = m\_{\eta\_I^0} = m\_{\eta^+} = m\_\chi + m\_{\phi\_R} - 10\,\text{GeV},\tag{28}$$

$$m\_{\phi\_l} = m\_{\phi\_R} + 5\text{ GeV},\tag{29}$$

$$
\gamma\_2 = 0.004, \ \gamma\_5 = 0.05, \ \gamma\_7 = 0.17,\tag{30}
$$

$$
\kappa = 0.4, \ Y\_{ij}^{\upsilon} = 0.01. \tag{31}
$$

<sup>2</sup>Two singlet scalar DM scenario in <sup>Z</sup><sup>2</sup> <sup>×</sup> <sup>Z</sup> ′ <sup>2</sup> model has been explored in detail in Bhattacharya et al. [65].

The masses of heavier right-handed neutrinos are M<sup>2</sup> = M<sup>3</sup> = 1 TeV. The mass differences between m<sup>η</sup> 0 R and the sum of m<sup>χ</sup> and mφ<sup>R</sup> are so chosen that no resonance appears in the s-channel of the semi-annihilation in **Figure 5** (right). The benchmark set satisfies the constraints from the perturbativeness, the stability conditions of the scalar potential [39, 40], the lepton flavor violation (LFV) such as µ → eγ [107] and the electroweak precision measurements [108, 109]. It is noted that κ is bounded as |κ| . 0.4 by the perturbativeness and the stability conditions [39, 40].

**Figure 6** shows the relic abundances of <sup>χ</sup> h 2 , φ<sup>R</sup> h 2 , and N<sup>1</sup> h 2 and the total relic abundance totalh 2 (= <sup>χ</sup> h <sup>2</sup> <sup>+</sup>φ<sup>R</sup> h 2 + N<sup>1</sup> h 2 ) as a function of mφ<sup>R</sup> for the benchmark set. The horizontal dashed line stands for the observed value obsh <sup>2</sup> <sup>∼</sup> 0.12. It is shown that the relic abundance of the χ is <sup>χ</sup> ≃ obs/2. When φ<sup>R</sup> is lighter than N1, the semi-annihilation tends to decrease the relic abundance of N1. For the benchmark set, the total relic abundance is consistent with the observed value around mφ<sup>R</sup> ≃ 280 GeV.

The left panel in **Figure 7** shows the contour plot for the mφ<sup>R</sup> − γ<sup>5</sup> plane where the total relic density of DM can be made consistent with the observed value obsh <sup>2</sup> <sup>∼</sup> 0.12. We take two values, 10 GeV (black line) and 1 GeV (red line), for the mass difference between m<sup>η</sup> 0 R and m<sup>χ</sup> + mφ<sup>R</sup> in Equation (28). The other parameters are taken as the same in Equations (27–31). We can see the scalar coupling γ<sup>5</sup> increases drastically as mφ<sup>R</sup> increases for mφ<sup>R</sup> <sup>&</sup>gt;<sup>∼</sup> 290 GeV. It is because the relic density of the N<sup>1</sup> DM, N<sup>1</sup> h 2 , becomes significant for mφ<sup>R</sup> >∼ 290 GeV, so that φ<sup>R</sup> h 2 should be drastically suppressed. The scalar couplings of DM particles with the SM Higgs boson, γ<sup>2</sup> and γ5, and the DM masses are constrained by the DM direct detection experiments. For the χ DM, the effective cross section off nucleon σ eff χ in Equation (24) is σ eff <sup>χ</sup> <sup>∼</sup> <sup>10</sup>−<sup>47</sup> cm<sup>2</sup> for the benchmark set. It is an order of magnitude smaller than the current experimental bound. For the φ<sup>R</sup> DM, the right panel in **Figure 7** shows the relation between mφ<sup>R</sup> and the effective cross

abundance total*h* <sup>2</sup> as a function of *<sup>m</sup>*φ*<sup>R</sup>* . The relevant masses and couplings are taken as in Equations (27–31). The horizontal dashed line stands for the observed value obs*h* <sup>2</sup> <sup>∼</sup> 0.12.

section σ eff φR for (m<sup>χ</sup> + mφ<sup>R</sup> ) − m<sup>η</sup> 0 R =10 GeV (black line) and 1 GeV (red line), where the DM relic abundance is consistent with the observation. The plot corresponds to the parameter space in the left panel in **Figure 7**. The dot and dashed lines indicate the upper bounds of LUX and XENON1T, respectively. The hatched region is excluded by perturbativity. Although the scalar coupling γ<sup>5</sup> becomes large for mφ<sup>R</sup> <sup>&</sup>gt;<sup>∼</sup> 290 GeV and then the cross sections off the nucleon σφ<sup>R</sup> becomes large, the effective cross section σ eff φR decreases for mφ<sup>R</sup> <sup>&</sup>gt;<sup>∼</sup> <sup>M</sup>1(<sup>=</sup> 300 GeV), since the abundance of φ<sup>R</sup> decreases. For the case of (m<sup>χ</sup> + mφ<sup>R</sup> ) − m<sup>η</sup> 0 R = 10 GeV, it can be seen that the mass region 288 GeV <sup>&</sup>lt;<sup>∼</sup> <sup>m</sup>φ<sup>R</sup> is excluded by LUX and XENON1T data. On the other hand, there are no constraints from the direct DM search experiments on the mass of φ<sup>R</sup> for the case of (m<sup>χ</sup> + mφ<sup>R</sup> ) − m<sup>η</sup> 0 R = 1 GeV. This is because the relic abundance of φ<sup>R</sup> becomes much smaller by the large contribution from the s-channel process of the semi-annihilation. **Figure 8** shows the same as in **Figure 7** but for M<sup>1</sup> = 500 GeV and γ<sup>7</sup> = 0.28. >From the right panel in **Figure 8**, we see that 485 (490) GeV <sup>&</sup>lt;<sup>∼</sup> <sup>m</sup>φ<sup>R</sup> <sup>&</sup>lt;<sup>∼</sup> 510 (502) GeV is excluded by the direct detection experiments for (m<sup>χ</sup> + mφ<sup>R</sup> ) − m<sup>η</sup> 0 R = 10 (1) GeV.

Before we go to discuss indirect detection, we summarize the parameter space, in which a correct value of the total relic DM abundance totalh 2 can be obtained without contradicting the constraint from the direct detection experiments. As in the case of the single SM singlet DM, the constraint is in fact very severe: The mass of χ has to be very close to mh/2, and γ<sup>2</sup> (the Higgs portal coupling) also has to be close to 0.004 for an adequate amount of <sup>χ</sup> . However, as for mφ<sup>R</sup> and γ5, there exist a certain allowed region. The allowed region in the mφ<sup>R</sup> − γ<sup>5</sup> plane is controlled by the semi-annihilation [especially, the last diagram in **Figure 5**, which is sensitive to the mass relation (28)] and the DM conversion (especially the right diagram in **Figure 4**, which is sensitive to γ7). If we increase the mass of the right-handed neutrino DM, the mass of φ<sup>R</sup> increases, but how the allowed range in the mφ<sup>R</sup> -γ<sup>5</sup> plane emerges remains the same. If we take the larger γ7, e.g., γ<sup>7</sup> = 0.28, in **Figure 7**, the allowed region for mφ<sup>R</sup> becomes narrower as 295 GeV <sup>&</sup>lt;<sup>∼</sup> <sup>m</sup>φ<sup>R</sup> <sup>&</sup>lt;<sup>∼</sup> 300 GeV. The smaller mφ<sup>R</sup> (<<sup>∼</sup> 295 GeV) is excluded by total < obs due to the larger DM conversion i.e., the larger annihilation process of φRφ<sup>R</sup> → χχ → XX.

#### 3.2. Indirect Detection

For indirect detections of DM the SM particles produced by the annihilation of DM are searched. Because the semi-annihilation produces a SM particle, this process can serve for an indirect detection. In our model, especially, the SM particle from the semi-annihilation process as shown in **Figure 5** is neutrino which has a monochromatic energy spectrum. Therefore, we consider below the neutrino flux from the Sun [110–120] as a possibility to detect the semi-annihilation process of DMs.

The DM particles are captured in the Sun losing their kinematic energy through scattering with the nucleus. Then captured DM particles annihilate each other. The time dependence of the number of DM n<sup>i</sup> in the Sun is given by Griest and Seckel [114], Ritz and Seckel [115], Bertone et al. [116], Silk et al. [117], Kamionkowski [118], Kamionkowski et al.

FIGURE 7 | (Left) Contour plot for the total relic density total*h* <sup>2</sup> <sup>∼</sup> 0.12. (Right) The relation between the *<sup>m</sup>*<sup>φ</sup> and the effective cross sections given in Equation (24). The black dot and dashed lines show the upper limit of the spin independent cross section off the nucleon given by LUX [102] and XENON1T [103], respectively. The hatched region is excluded by perturbativity. In both panels, we take two values, 10 GeV (black line) and 1 GeV (red line), for the mass difference between *<sup>m</sup>*<sup>η</sup> 0 *R* and *m*<sup>χ</sup> + *m*φ*<sup>R</sup>* .

[119], and Jungman et al. [120]

$$\begin{aligned} \dot{n}\_i &= \mathbf{C}\_i - \mathbf{C}\_A (ii \to \text{ SM}) n\_i^2 \\ &- \sum\_{m\_i > m\_j} \mathbf{C}\_A (ii \to jj) n\_i^2 - \mathbf{C}\_A (ij \to k\nu) n\_i n\_j \end{aligned} \tag{32}$$

where i, j, k = χ, φR, N<sup>1</sup> and C<sup>i</sup> is the capture rates in the Sun:

$$C\_{\chi} \simeq 2.5 \times 10^{18} \text{s}^{-1} f(m\_{\chi}) \left(\frac{\hat{f}}{0.3}\right)^{2} \left(\frac{\chi\_{2}}{0.004}\right)^{2} \left(\frac{60 \text{ GeV}}{m\_{\chi}}\right)^{2}$$

$$\left(\frac{125 \text{ GeV}}{m\_{\text{h}}}\right)^{4} \left(\frac{\Omega\_{\chi} h^{2}}{\Omega\_{\text{total}} h^{2}}\right),\tag{33}$$

$$\mathbf{C}\_{\phi\mathbf{k}} \simeq 6.2 \times 10^{17} \text{s}^{-1} f(m\_{\phi\mathbf{k}}) \left(\frac{\hat{f}}{0.3}\right)^{2} \left(\frac{\chi\_{5}}{0.02}\right)^{2}$$

$$\begin{split} & \left(\frac{300 \text{ GeV}}{m\_{\phi\mathbf{k}}}\right)^{2} \left(\frac{125 \text{ GeV}}{m\_{\text{h}}}\right)^{4} \left(\frac{\Omega\_{\phi\mathbf{k}} h^{2}}{\Omega\_{\text{total}} h^{2}}\right), \\ & \mathbf{C}\_{\mathbf{N}\_{1}} = \mathbf{0} \end{split} \tag{34}$$

and CA's are the annihilation rate:

$$\mathcal{C}\_{A}(\vec{y}\rightarrow\bullet)=\frac{\langle\sigma\,(\vec{y}\rightarrow\bullet)\nu\rangle}{V\_{\vec{y}}}\,,$$

$$V\_{\vec{y}}=5.7\times10^{27}\left(\frac{100\,\text{GeV}}{\mu\_{\vec{y}}}\right)^{3/2}\text{cm}^{3}\,.\tag{36}$$

Here f(mi) depends on the form factor of the nucleus, elemental abundance, kinematic suppression of the capture rate, etc., varying <sup>O</sup>(0.01 − 1) depending on the DM mass [118–120]. Vij is an effective volume of the Sun with µij = 2mimj/(m<sup>i</sup> +mj) in the non-relativistic limit. In the Equation (32) we have neglected the DM production processes such as jj → ii and jk → iX because the kinetic energy of the produced particle i is much larger than that corresponding to the escape velocity from the Sun, i.e., <sup>∼</sup> <sup>10</sup><sup>3</sup> km/s [114, 121, 122]. Consequently, the number of the righthand neutrino DM cannot increase in the Sun, and hence the semi-annihilation process, φRχ → N1ν, is the only neutrino production process <sup>3</sup> , where its reaction rate as a function of t is given by Ŵ(φRχ → Nν;t) = CA(φRχ → N1ν)nφ<sup>R</sup> (t)nχ (t).

**Figure 9** shows the mφ<sup>R</sup> dependence of the neutrino production rate today Ŵ(φRχ → Nν;t0), where t<sup>0</sup> = 1.45 × 1017s is the age of the Sun, for the same parameter space as in **Figure 7** (**Figure 9**, left) and in **Figure 8** (**Figure 9**, right). The hatched region is excluded by perturbativity. Arrows indicate the excluded regions by the direct detection experiments. For mφ<sup>R</sup> <sup>&</sup>gt;<sup>∼</sup> <sup>M</sup><sup>1</sup> where the relic abundance of <sup>N</sup><sup>1</sup> dominates that of <sup>φ</sup>R, the neutrino production rate decreases since the capture rate of the φ<sup>R</sup> becomes small. As we can see from **Figure 5** a resonance effect for the s-channel annihilation process can be achieved if mη 0 R ≃ mφ<sup>R</sup> + m<sup>χ</sup> . Then the smaller neutrino mass difference mη 0 R − (mφ<sup>R</sup> + m<sup>χ</sup> ) gives the larger neutrino production rate. For the case of m<sup>η</sup> 0 R −(mφ<sup>R</sup> +m<sup>χ</sup> )= 1 GeV, the rate Ŵ(φRχ → Nν;t0) reaches about 10<sup>18</sup> s −1 at mφ<sup>R</sup> ≃ 290 GeV for M<sup>1</sup> = 300 GeV and <sup>4</sup> <sup>×</sup> <sup>10</sup><sup>17</sup> <sup>s</sup> −1 at mφ<sup>R</sup> ≃ 490 GeV for M<sup>1</sup> = 500 GeV, respectively.

The upper limits on the DM DM → XX′ from the Sun are given by IceCube experiment [123]. For instance, the upper limit on the annihilation rate of the DM of 250 (500) GeV into <sup>W</sup>+W<sup>−</sup> is 1.13 <sup>×</sup> <sup>10</sup><sup>21</sup> (2.04 <sup>×</sup> <sup>10</sup>20) s−<sup>1</sup> and that into τ +τ − is 5.99×10<sup>20</sup> (7.96×1019) s−<sup>1</sup> , which is at least 10<sup>2</sup> times larger than the rate Ŵ(ν) shown in **Figure 9**. Note however that the energy spectrum of the neutrino flux produced by the W or τ decay is different from the monochromatic neutrino. With an increasing resolution of energy and angle the chance for the observation of the semi-annihilation and hence of a multicomponent nature of DM can increase.

### 4. MODEL B

Neutrinos have always played consequential roles in cosmology (see [124], and also [125] and references therein). While they play a role as hot dark matter, the mechanism of their mass generation is directly connected to cosmological problems such as baryon asymmetry of the Universe [126] and dark matter [10– 36, 39–48]. Resent cosmological observations with increasing accuracy [88, 127–129] provide useful hints on how to extend the neutrino sector. Here we propose an extension of the neutrino sector such that the tensions among resent different cosmological observations can be alleviated. The tensions have emerged since the first Planck result [88] in the Hubble constant H<sup>0</sup> and in the density variance σ<sup>8</sup> in spheres of radius 8h <sup>−</sup><sup>1</sup> Mpc: The Planck values of 1/H<sup>0</sup> and σ<sup>8</sup> are slightly larger than those obtained from the observations of the local Universe such as Cepheid variables [128] and the Canada-France- Hawaii Telescope Lensing Survey [130], respectively. The Planck galaxy cluster counts [131] and also the Sloan Digital Sky Survey data [127] yield a smaller σ8.

It has been recently suggested [131–139] that these tensions can be alleviated if the number Neff of the relativistic species in the young Universe is slightly larger than the standard value 3.046 and the mass of the extra relativistic specie is of O(0.1) eV [139]. Here we suggest a radiative generation mechanism of the neutrino mass, which is directly connected to the existence of a stable DM particle and also a non-zero 1Neff = Neff − 3.046.

The matter content of the model is shown in **Table 2**. It is a slight modification of the model A: χ in this model is a Majorana fermion. The Z<sup>2</sup> × Z ′ <sup>2</sup> × L ′ -invariant Yukawa sector (the quark sector is suppressed) is described by the Lagrangian

$$\mathcal{L}\_Y = Y\_{ij}^\epsilon H^\dagger L\_i l\_{Rj}^c + Y\_{ij}^\nu L\_i \epsilon \eta N\_j + Y\_{ij}^\chi N\_i \chi\_j \phi - \frac{1}{2} M\_{\chi\_k} \chi\_k \chi\_k + h.c. \tag{37}$$

where i, j, k = 1, 2, 3, and we have assumed without loss of generality that the χ mass term is diagonal. We also assume that Y e ij have only diagonal elements. The most general renormalizable form of the Z<sup>2</sup> × Z ′ <sup>2</sup> × L ′ -invariant scalar potential is given by

$$\begin{split} V\_{\lambda} &= \lambda\_1 (H^\dagger H)^2 + \lambda\_2 (\eta^\dagger \eta)^2 + \lambda\_3 (H^\dagger H)(\eta^\dagger \eta) + \lambda\_4 (H^\dagger \eta)(\eta^\dagger H) \\ &+ \frac{\lambda\_5}{2} [(H^\dagger \eta)^2 + h.c.] \\ &+ \chi\_2 (H^\dagger H) |\phi|^2 + \chi\_3 (\eta^\dagger \eta) |\phi|^2 + \chi\_4 |\phi|^4, \end{split} \tag{38}$$

<sup>3</sup>There are also neutrinos having continuous energy spectrum from the decay of SM particles produced by the standard annihilation. The upper bounds for the production rates of the SM particles are given in Agrawal et al. [121], Andreas et al. [122], and Aartsen et al. [123].


TABLE 2 | The matter content of the model B and the corresponding quantum numbers.

and the mass terms are

$$W\_m = m\_1^2 H^\dagger H + m\_2^2 \eta^\dagger \eta + m\_3^2 |\phi|^2 - \frac{m\_4^2}{2} [|\phi^2 + (\phi^\*)^2|],\tag{39}$$

where the m<sup>4</sup> term in Equation (39) breaks L ′ softly. The scalar fields H, η and φ are defined in Equation (10). Since we assume that the discrete symmetry Z<sup>2</sup> × Z ′ 2 is unbroken, the scalar fields above do not mix with other, so that their tree-level masses can be simply expressed:

$$m\_{\eta^{\pm}}^2 = m\_2^2 + \lambda\_3 \nu\_h^2 / 2\,, \tag{40}$$

$$m\_{\eta\_R^0}^2 = m\_2^2 + \left(\lambda\_3 + \lambda\_4 + \lambda\_5\right) \nu\_h^2 / 2 \,, \tag{41}$$

$$m\_{\eta\_I^0}^2 = m\_2^2 + \left(\lambda\_3 + \lambda\_4 - \lambda\_5\right) \nu\_h^2 / 2 \,, \tag{42}$$

$$m\_{\phi\_R}^2 = m\_3^2 - m\_4^2 + \chi\_2 \upsilon\_h^2 / 2 \,, \tag{43}$$

$$m\_{\phi\_I}^2 = m\_3^2 + m\_4^2 + \wp\_2 \nu\_h^2 / 2 \,. \tag{44}$$

The two-loop diagram for the neutrino mass is shown in **Figure 10**. Because of the soft breaking of the dimension two operator φ 2 , the propagator between φ and φ can exist, generating the mass of N:

$$(M\_N)\_{\vec{\eta}} = \frac{1}{32\pi^2} \sum\_k (Y\_{ik}^{\chi})^\* (Y\_{jk}^{\chi})^\* M\_{\chi\_k} \left[ \frac{m\_{\phi\_R}^2}{m\_{\phi\phi\_R}^2 - M\_{\chi\_k}^2} \ln \left( \frac{m\_{\phi\phi\_R}}{M\_{\chi\_k}} \right)^2 \right]$$

$$-\frac{m\_{\phi\_I}^2}{m\_{\phi\_I}^2 - M\_{\chi\_k}^2} \ln \left( \frac{m\_{\phi\_I}}{M\_{\chi\_k}} \right)^2 \Big]. \tag{45}$$

The 3 × 3 two-loop neutrino mass matrix <sup>M</sup><sup>ν</sup> is given by

$$(\mathcal{M}\_{\boldsymbol{v}})\_{ij} = \frac{1}{(32\pi^2)^2} \sum\_{l,k} Y\_{il}^{\boldsymbol{v}} Y\_{jm}^{\boldsymbol{v}} (Y\_{lk}^{\boldsymbol{X}})^\* (Y\_{mk}^{\boldsymbol{X}})^\* M\_{\boldsymbol{\chi}k} (m\_{\boldsymbol{\eta}\_l}^2 - m\_{\boldsymbol{\eta}\_k}^2)$$

$$\int\_0^\infty dz \, \frac{1}{(\boldsymbol{x} + m\_{\boldsymbol{\eta}\_R}^2)^2 (\boldsymbol{x} + m\_{\boldsymbol{\eta}\_l}^2)}$$

$$\int\_0^1 dz \ln \left[ \frac{zM\_{\boldsymbol{\chi}\_k}^2 + (1 - z)m\_{\boldsymbol{\phi}\_l}^2 + z(1 - z)x}{zM\_{\boldsymbol{\chi}\_k}^2 + (1 - z)m\_{\boldsymbol{\phi}\_R}^2 + z(1 - z)x} \right]. \tag{46}$$

We can also use (45) to obtain an approximate formula for the neutrino mass

$$(\mathcal{M}\_{\upsilon})\_{ij} \sim \frac{1}{32\pi^2} \sum\_{k} Y\_{ik}^{\prime\upsilon} Y\_{jk}^{\prime\upsilon} M\_k \ln \frac{m\_{\eta\_R^0}^2}{m\_{\eta\_I^0}^2}, \ Y\_{jk}^{\prime\upsilon} = Y\_{jl}^{\upsilon} U\_{lk}^N,\tag{47}$$

cross means the soft breaking mass term *m*<sup>2</sup> 4 , which should indicate that there are φ*R* and φ*I* loops in the inner one-loop diagram. The lower cross indicates the chirality flip of χ. The result (Equation 46) is obtained by using the exact propagators of φs and χs.

where U <sup>N</sup> is the unitary matrix diagonalizing the mass matrix (MN)ij with the eigenvalues M<sup>k</sup> and the mass eigenstates N ′ k , and we have used the fact that M<sup>k</sup> ≪ m<sup>η</sup> 0 R ≃ m<sup>η</sup> 0 I . In the following discussions we choose the theory parameters so as to be consistent with the global fit [1]:

$$\begin{aligned} \Delta m\_{21}^2 &= 7.50\_{-0.17}^{+0.19} \times 10^{-5} \text{ eV}^2, \\ \Delta m\_{31}^2 &= 2.524\_{-0.040}^{+0.039} \text{ (-2.514}\_{-0.041}^{+0.038}) \times 10^{-3} \text{ eV}^2, \\ \sin^2 \theta\_{12} &= 0.306 \pm 0.012, \sin^2 \theta\_{23} = 0.441\_{-0.021}^{+0.027} \text{ (0.587}\_{-0.024}^{+0.020}), \\ \sin^2 \theta\_{13} &= 0.02166 \pm 0.00075 \text{ (0.02179 \pm 0.00076)}, \end{aligned}$$

where the values in the parenthesis are those for the inverted mass hierarchy.

#### 4.1. Dark Radiation

According to the discussion at the beginning of this section, we identify the lightest right-handed neutrino with dark radiation contributing to 1Neff 4 . Without los of generality we may assume it is N ′ <sup>1</sup> with mass <sup>&</sup>lt;<sup>∼</sup> 0.24 eV. The upper bound on the mass is obtained together with 3.10 < Neff < 3.42 in Feng et al. [139]. To simplify the situation, we require that the heavier right-handed neutrinos N ′ 2 and N ′ 3 decay above the decoupling temperature T dec N of N ′ 1 . Their decay widths are given by

$$\begin{array}{c} \langle \Gamma \{ N'\_{2\langle 3 \rangle} \to N'\_1 \nu \bar{\upsilon} \} \rangle \\ + \Gamma \{ N'\_{2\langle 3 \rangle} \to \bar{N'}\_1 \nu \bar{\upsilon} \} \rangle \end{array} = \frac{1}{3072\pi^3} \frac{M\_{2\langle 3 \rangle}^5}{m\_{\eta^0}^4} \sum\_{i,j} |Y\_{i2\langle 3 \rangle}^{\prime \upsilon}|^2 |Y\_{j1}^{\prime \upsilon}|^2, \tag{49}$$

<sup>4</sup>Within a similar framework of radiative seesaw mechanism, the lightest righthanded neutrino has been regarded as stable warm dark matter in Aristizabal Sierra et al. [62]. In the models proposed in Kajiyama et al. [37] and Baek et al. [38], the topology of the two loop to generate the neutrino mass is basically the same as that of **Figure 10**. But the matter content of our model is much simpler; we have only two additional extra fields compared with the one-loop model of Ma [11], while in these papers five and four additional ones have to be introduced. Apart from this difference, they have not considered the lightest right-handed neutrino as dark radiation. In Baek et al. [38], however, the Nambu-Goldstone boson associated with the spontaneous breaking of U(1)B−<sup>L</sup> is regarded as dark radiation.

where we have used m<sup>η</sup> <sup>0</sup> = m<sup>η</sup> 0 R ≃ m<sup>η</sup> 0 I and neglected the mass of N ′ 1 and νLs. Therefore, N ′ 2 and N ′ 3 can decay above T dec N if

$$\langle \Gamma \{ N\_{2(3)}' \to N\_1' \nu \bar{\nu} \} + \Gamma \{ N\_{2(3)}' \to \bar{N}'\_1 \nu \bar{\nu} \} \rangle \gtrsim H \langle T\_N^{\text{dec}} \rangle \tag{50}$$

is satisfied, where H(T) is the Hubble constant at temperature T, and g∗s(T) is the relativistic degrees of freedom at T. To obtain the effective number of the light relativistic species Neff [125, 140], we have to compute the energy density of N ′ 1 at the time of the photon decoupling, where we denote the decoupling temperature of γ , ν<sup>L</sup> and N ′ 1 by T<sup>γ</sup> <sup>0</sup>, T dec ν and T dec N , respectively. Further, Tν<sup>0</sup> (TN0) stands for the temperature of ν<sup>L</sup> (N ′ 1 ) at the decoupling of γ . The most important fact is that the entropy per comoving volume is conserved, so that sa<sup>3</sup> is constant, where s is the entropy density, and a is the scale factor. The effective number Neff follows from ρr(T<sup>γ</sup> <sup>0</sup>) = (π 2 /15)(1 <sup>+</sup> (7/8)(4/11)4/3Neff) <sup>T</sup> 4 γ 0 and is given by Kolb and Turner [125], Steigman [140], Anchordoqui and Goldberg [141], Steigman et al. [142], Anderhalden et al. [143, 145], Anchordoqui et al. [144], and Weinberg [146]

$$N\_{\rm eff} = 3.046 + \left(\frac{g\_{\ast s}(T\_{\upsilon}^{\rm dec})}{g\_{\ast s}(T\_N^{\rm dec})}\right)^{4/3} \tag{51}$$

for N<sup>ν</sup> = 3, where ρ<sup>r</sup> is the energy density of relativistic species. Since g∗s(T dec ν ) = (11/2) + (7/4)N<sup>ν</sup> = 10.75, we need to compute the decoupling temperature T dec N to obtain g∗s(T dec N ) and hence <sup>N</sup>eff. For 0.05 <sup>&</sup>lt;<sup>∼</sup> <sup>1</sup>Neff <sup>&</sup>lt;<sup>∼</sup> 0.38 [139] we find 101 <sup>&</sup>gt;<sup>∼</sup> g∗s(T dec N ) <sup>&</sup>gt;<sup>∼</sup> 22 and also <sup>T</sup> dec <sup>N</sup> ≃ 165 MeV to obtain g∗s(T dec N ) ≃ 30 (which gives 1Neff = 0.25). To estimate T dec N , we compute the annihilation rate ŴN(T) of N ′ 1 at T, which is given by

$$\begin{split} \Gamma\_{N}(T) &= n\_{N}(T) \left[ \langle \sigma\_{N\_{1}^{\prime} N\_{1}^{\prime} \rightarrow \upsilon\_{L} \upsilon\_{L}} \nu \rangle(T) + \langle \sigma\_{N\_{1}^{\prime} \bar{N}\_{1}^{\prime} \rightarrow \upsilon\_{L} \bar{\upsilon}\_{L}} \nu \rangle(T) \right] \\ &= \frac{\pi^{5}}{9 \zeta(3)} \left( \frac{7}{120} \right)^{2} \sum\_{i,j} |Y\_{i1}^{\prime\upsilon}|^{2} |Y\_{j1}^{\prime\upsilon}|^{2} \frac{T^{5}}{(m\_{\eta^{0}})^{4}} \,, \end{split} \tag{52}$$

where ζ (3) ≃ 1.202 . . . and nN(T) is the number density of N ′ 1 . Then we calculate T dec N from ŴN(T dec N ) = H(T dec N ), which can be rewritten as <sup>5</sup>

$$\left(\frac{T\_N^{\text{dec}}}{164.2 \text{ MeV}}\right)^3 \left(\frac{29.9}{g\_{\ast\imath}(T\_N^{\text{dec}})}\right)^{\frac{1}{2}} = \left(\frac{m\_{\eta^0}}{200 \text{ GeV}} \frac{0.0409}{Y^{\upsilon}}\right)^4 \text{(53)}$$

with (Y ν ) 2 = P i |Y ′ν i1 | 2 .

It turns out that M2,3 ∼ <sup>O</sup>(10) GeV to obtain 1Neff ∼ 0.25 while satisfying <sup>M</sup><sup>1</sup> <sup>&</sup>lt;<sup>∼</sup> 0.24 eV. To see this, we first find that

$$\left(\frac{m\_{\eta^0}^2}{\sum\_{i}|Y\_{i1}^{\prime\prime}|^2}\right) \sim 2.4 \times 10^7 \text{ GeV}^2,\tag{54}$$

which follows from Equation (53) for 1Neff ∼ 0.25. Further we can estimate a part of Equation (49) from the neutrino mass Equation (47) with M<sup>ν</sup> ∼ 0.05 eV:

$$\frac{M\_{2(3)}}{m\_{\eta^0}^2} \sum\_{i} |Y\_{i2(3)}'|^2 \sim 2.7 \times 10^{-15} |\lambda\_5|^{-1} \,\text{GeV}^{-1},\tag{55}$$

where we have used m<sup>2</sup> η 0 R <sup>−</sup> <sup>m</sup><sup>2</sup> η 0 I ≃ λ5v 2 h . Then using Equation (50) with T ≃ 165 MeV (which corresponds to 1Neff ≃ 0.25), we obtain

$$M\_{2,3} \lesssim 17 \left| \lambda\_5 \right|^{1/4} \text{ GeV} \,. \tag{56}$$

Note that this is an order of magnitude estimate, and indeed M2(3) can not be smaller than 10 GeV to satisfy <sup>1</sup>Neff <sup>&</sup>lt;<sup>∼</sup> 0.38.

Since we require that <sup>M</sup><sup>1</sup> <sup>&</sup>lt;<sup>∼</sup> 0.24 eV, there exists a huge hierarchy in the right-handed neutrino mass. This has a consequence on the Yukawa coupling matrix Y ′ν : To obtain realistic neutrino masses with the mixing parameters given in Equation (48),

$$|Y\_{i1}^{\prime\upsilon}| \gg |Y\_{i2(3)}^{\prime\upsilon}|\tag{57}$$

has to be satisfied. Note that only |Y ′ν i1 | enters into the thermally averaged annihilation cross section of N ′ 1 , as we can see from Equation (52). Because of <sup>1</sup>Neff <sup>&</sup>lt;<sup>∼</sup> 0.38, on the other hand, <sup>|</sup><sup>Y</sup> ′ν i1 | can not be made arbitrarily large. The hierarchy (Equation 57) has effects on the LFV radiative decays of the type l<sup>i</sup> → ljγ , so that the LFV decays and 1Neff are related, as we will see below. In the limit m<sup>j</sup> ≪ m<sup>i</sup> , where m<sup>i</sup> and m<sup>j</sup> stand for the mass of l<sup>i</sup> and l<sup>j</sup> , respectively, the ratio of the partial decay width Bˆ(l<sup>i</sup> → ljγ ) = Ŵ(l<sup>i</sup> → ljγ )/ Ŵ(l<sup>i</sup> → νieν¯e) can be written as Ma and Raidal [51]

$$\hat{B}(l\_i \to l\_j \gamma) = \left(\frac{\alpha}{768\pi G\_F^2}\right) \frac{\left|\sum\_k (Y\_{ik}^{\prime\prime})^\* Y\_{jk}^{\prime\prime}\right|^2}{m\_{\eta^\pm}^4}.\tag{58}$$

Here mη<sup>±</sup> and Y ′ν ik are defined in Equations (40) and (47), respectively, and the current upper bounds on the branching fraction of these processes [107, 148] require

$$\mu \to \varepsilon \nu : \left| \sum\_{k} (Y\_{2k}^{\prime \upsilon})^{\*} Y\_{1k}^{\prime \upsilon} \right| \lessapprox 2.5 \times 10^{-4} \left( \frac{m\_{\eta^{\pm}}}{220 \,\text{GeV}} \right)^{2}, \tag{59}$$

$$\tau \to \mu \gamma : \left| \sum\_{k} (Y\_{3k}^{\prime \prime})^{\*} Y\_{2k}^{\prime \prime} \right| \lessapprox 8.1 \times 10^{-2} \left( \frac{m\_{\eta^{\pm}}}{220 \,\mathrm{GeV}} \right)^{2}, \quad \text{(60)}$$

$$\forall \tau \to e\nu : \left| \sum\_{k} (Y\_{3k}^{\prime \prime})^{\*} Y\_{1k}^{\prime \prime} \right| \lesssim 7.0 \times 10^{-2} \left( \frac{m\_{\eta^{\pm}}}{220 \,\mathrm{GeV}} \right)^{2} . \tag{61}$$

From Equation (59) we find that Y ′ν <sup>31</sup> is not constrained by the stringent constraint from µ → eγ , which will be crucial in obtaining a realistic Neff without having any contradiction with Equations (59–61). Furthermore, if Y ′ν <sup>31</sup> is large compared with others and the hierarchy (Equation 57) is satisfied, the ratio R =

<sup>5</sup>We use the relation between <sup>T</sup> and <sup>g</sup>∗<sup>s</sup> given in Husdal [147] to solve Equation (53) for T dec N .

Bˆ(τ → µγ )Bˆ(τ → eγ )/Bˆ(µ → eγ ) is ∼ |Y ′ν 31| 2 , and from the same reason 1Neff depends mostly on Y ′ν <sup>31</sup>. A benchmark set of the input parameters is given by

$$Y\_{ij}^{\prime\upsilon} = \begin{pmatrix} -0.0382 & 2.510 \times 10^{-5} & 3.349 \times 10^{-5} \\ 0.00129 & -1.183 \times 10^{-6} & 1.081 \times 10^{-4} \\ 0.0154 & -7.723 \times 10^{-5} & 9.334 \times 10^{-5} \end{pmatrix},\tag{62}$$

$$M\_1 = 0.147\,\text{eV},\quad M\_2 = M\_3 = 9.55\,\text{GeV},\tag{63}$$

$$m\_{\eta^{\pm}} = 220 \text{ GeV}, \quad m\_{\eta\_{R}^{0}} = 200 \text{ GeV}, \quad m\_{\eta\_{I}^{0}} = 207 \text{ GeV}, \quad \text{(64)}$$

which yields

$$\begin{aligned} \sin^2 \theta\_{12} &= 0.305, \quad \sin^2 \theta\_{23} = 0.441, \quad \sin^2 \theta\_{13} = 0.0213, \\\\ \Delta m\_{21}^2 &= 7.50 \times 10^{-5} \,\text{GeV}^2, \; \Delta m\_{31}^2 = 0.00248 \,\text{GeV}^2, \end{aligned} \tag{65}$$

$$
\Delta m\_{21}^2 = 7.50 \times 10^{-5} \,\text{GeV}^2, \,\,\Delta m\_{31}^2 = 0.00248 \,\text{GeV}^2,\tag{66}
$$

$$
\dots \quad \dots \quad \dots
$$

$$
\hat{B}(\mu \to e\gamma) = 2.30 \times 10^{-14}, \,\hat{B}(\tau \to \mu\gamma) = 3.75 \times 10^{-15},
$$

$$
\hat{B}(\tau \to e\gamma) = 3.31 \times 10^{-12},\tag{67}
$$

where we have assumed that Y ′ν ij are all real so that there is no CP phase. These values are consistent with Equations (48), (59– 61). With the same input parameters we find: The lhs of (50) = 5.46×10−<sup>21</sup> (1.78×10−20) GeV for <sup>N</sup><sup>2</sup> (N3), where the rhs is <sup>H</sup> <sup>=</sup> 2.10×10−<sup>20</sup> GeV with <sup>T</sup> dec <sup>N</sup> = 166.8 MeV and g∗s(T dec N ) = 30.83, and 1Neff = 0.245.

In **Figure 11** we plot R 1/2 against 1Neff with mη<sup>±</sup> = 240 GeV and m<sup>η</sup> 0 R = 220 GeV, where we have varied m<sup>η</sup> 0 I between 221 and 227 GeV. In the black region of **Figure 11** the differences of the neutrino mass squared and the neutrino mixing angles are consistent with Equation (48) for the normal hierarchy, and the constraints M<sup>1</sup> < 0.24 eV, (Equations 50 and 59–61) are satisfied. If 1Neff and R <sup>1</sup>/<sup>2</sup> would depend on Y ′ν <sup>31</sup> only, we would obtain a line in the 1Neff − R <sup>1</sup>/<sup>2</sup> plane. The Y ′ν <sup>11</sup> and Y ′ν <sup>21</sup> dependence in R 1/2 cancels, but this is not the case for 1Neff. This is the reason

why we have an area instead of a line in **Figure 11**. We see from **Figure 11** that the predicted region for <sup>1</sup>Neff <sup>&</sup>lt;<sup>∼</sup> 0.1 is absent. The main reason is that we have assumed that <sup>M</sup>2, <sup>M</sup><sup>3</sup> <sup>&</sup>lt;<sup>∼</sup> <sup>16</sup> GeV. This has also a consequence on the difference between m2 η 0 R and m<sup>2</sup> η 0 I , because the mass difference changes the overall scale of the neutrino mass (47). To obtain a larger M2,3, we can decrease the mass difference, thereby implying an increase of the degree of fine-tuning. Further, the difference between m<sup>2</sup> η 0 R and m<sup>2</sup> η 0 I can not be made arbitrarily large, because it requires a smaller M2,3, which due to H(T) ∝ T 2 in turn implies that the decoupling temperature T dec N has to decrease to satisfy the constraint [Equation (50)]. A smaller T dec N , on the other hand, means a larger 1Neff which is constrained to be below 0.38. This is why m<sup>η</sup> 0 is varied only in a small interval in **Figure 11**.

I Since the current upper bound on B(µ → eγ ) ≃ Bˆ(µ → eγ ) is 4.2 <sup>×</sup> <sup>10</sup>−<sup>13</sup> [107], the model B predicts

$$\begin{aligned} \left[ \mathcal{B}(\mathfrak{r} \to \mu\mathcal{\gamma}) \mathcal{B}(\mathfrak{r} \to e\mathcal{\gamma}) \right]^{1/2} &\simeq \left[ \frac{\hat{\mathcal{B}}(\mathfrak{r} \to \mu\mathcal{\gamma})}{0.17} \frac{\hat{\mathcal{B}}(\mathfrak{r} \to e\mathcal{\gamma})}{0.18} \right]^{1/2} \\ &\lessapprox 1.2 \times 10^{-10}, \end{aligned} \tag{68}$$

which is about two orders of magnitude smaller than the current experimental bounds [148].

Another consequence of the hierarchy (Equation 57) is that the total decay width of η<sup>R</sup> depends on P i,j |Y ′ ij| 2 , which is approximately P i |Y ′ i1 | 2 (we assume that η<sup>R</sup> is the lightest among ηs). Therefore, 1Neff is basically a function of the decay width. In **Figure 12** we show 1Neff against Ŵη<sup>R</sup> /m<sup>η</sup> 0 R , the decay width of η 0 R over m<sup>η</sup> 0 R , where we have used the same parameters as for **Figure 11**. η 0 R decays almost 100 percent into neutrinos and dark radiation N ′ 1 , which is invisible. In contrast to this, η + can decay into a charged lepton and N ′ 1 , and the decay width over mη<sup>±</sup> is the same as Ŵη<sup>R</sup> /m<sup>η</sup> 0 R . Ŵη<sup>R</sup> should be compared with the decay width for η <sup>+</sup> → W+∗ η 0 <sup>R</sup>,<sup>I</sup> → f ¯ f ′ N ′ 1 <sup>ν</sup>, which is <sup>∼</sup>10−8mη<sup>±</sup> for the same parameter space as for **Figure 12**, where f and f ′ stand for the SM fermions (except the top quark). Therefore, η + decays almost 100 percent into a charged lepton and missing energy. In Aristizabal Sierra [62], a similar hierarchical spectrum of the right-handed neutrinos in the model of Ma [11] has been assumed (the lightest one has been regarded as a warm dark matter) and collider physics has been discussed. How the inert Higgs bosons can be produced via s-channel exchange of a virtual photon and Z boson [149, 150] is the same, but the decay of the inert Higgs bosons is different because of the hierarchy Equation (57) of the Yukawa coupling constants. As it is mentioned above, the η ± decays in the present model almost only into the lightest one N ′ 1 and a charged lepton. Therefore, the cascade decay of the heavier right-handed neutrinos into charged leptons will not be seen at collider experiments, because they can be produced only as a decay product of η ±. The decay width of η ± into an individual charged lepton depends of course on the value of Y ′ν i1 . In the parameter space we have scanned we cannot make any definite conclusion on the difference.

### 4.2. Cold Dark Matter and Its Direct and Indirect Detection

Since the lightest N is dark radiation and the masses of the heavier ones are O(10) GeV (as we have seen in the previous subsection), η 0 R,I can not be DM candidates, because they decay into N and ν. So, DM candidates are χ and the lightest component of φ 6 . In the case that ηs are lighter than φ<sup>R</sup> and the lightest component of φ (which is assumed to be φR) is DM, a correct relic abundance φ<sup>R</sup> h <sup>2</sup> <sup>=</sup> 0.1204 <sup>±</sup> 0.0027 [88] can be easily obtained, because <sup>γ</sup><sup>3</sup> for the scalar coupling (η †η)|φ<sup>|</sup> 2 is an unconstrained parameter so far. So, in the following discussion we assume that φ<sup>R</sup> is DM.

Because of the Higgs portal coupling γ2, the direct detection of φ<sup>R</sup> is possible. The current experimental bound of XENON1T [103] of the spin-independent cross section σSI off the nucleon requires <sup>|</sup>γ2<sup>|</sup> <sup>&</sup>lt;<sup>∼</sup> 0.05 <sup>∼</sup> 0.14 for <sup>m</sup>φ<sup>R</sup> <sup>=</sup> <sup>250</sup> <sup>∼</sup> 500 GeV. Since γ<sup>2</sup> is allowed only below an upper bound (which depends on the DM mass mφ<sup>R</sup> ), γ<sup>3</sup> can vary in a certain interval for a given DM mass.

With this remark, we note that the capture rate of DM in the Sun is proportional to σSI, while its annihilation rate in the Sun is proportional to the thermally averaged annihilation cross section, hvσ(φRφ<sup>R</sup> → η +η −, η 0 R η 0 R , η 0 I η 0 I )i [110–120]. If a pair of φRs annihilates into η 0 R η 0 R and also η 0 I η 0 I , a pair of ν<sup>L</sup> and ν¯<sup>L</sup> will be produced, which may be observed on the Earth [121, 122]. The signals will look very similar to those coming from W±, which result from DM annihilation. The annihilation rate as a function of time t is given by Jungman et al. [120]

$$
\Gamma(\phi\_R \phi\_R \to \eta\_R^0 \eta\_R^0, \eta\_I^0 \eta\_I^0; t) = \Gamma(\phi\_R \phi\_R \to \eta^0 \eta^0; t)
$$

$$
= \frac{1}{2} \frac{\mathcal{C}\_{\phi\mathbb{R}} \mathcal{C}\_A(\eta^0 \eta^0)}{\mathcal{C}\_A(\eta^+ \eta^-) + \mathcal{C}\_A(\eta^0 \eta^0) + \mathcal{C}\_A(\mathcal{X}\mathcal{X}')} \tanh^2
$$

$$
\left[ t\sqrt{(\mathcal{C}\_A(\eta^+ \eta^-) + \mathcal{C}\_A(\eta^0 \eta^0) + \mathcal{C}\_A(\mathcal{X}\mathcal{X}')) \mathcal{C}\_{\phi\mathbb{R}}} \right], \tag{69}
$$

<sup>6</sup>Both together can not be DM, because the heavier one decays into N ′ 1 + lighter one.

[103] upper bound on σSI for *m*φ*<sup>R</sup>* = 250 (black) and 500 (red) GeV.

where Cφ<sup>R</sup> is the capture rate in the Sun,

$$C\_{\Phi\_R} \cong 1.4 \times 10^{20} f(m\_{\Phi\_R}) \left(\frac{\hat{f}}{0.3}\right)^2 \left(\frac{\mathcal{V}\_2}{0.1}\right)^2 \left(\frac{200 \text{ GeV}}{m\_{\Phi\_R}}\right)^2$$
 
$$\left(\frac{125 \text{ GeV}}{m\_h}\right)^4,\tag{70}$$

and C<sup>A</sup> is given by

$$\mathbf{C}\_{A}(\bullet) = \left(\frac{\langle \sigma\_{\phi\mathbf{n}\#\mathbb{R}\rightarrow\bullet}\nu \rangle}{5.7 \times 10^{27} \text{cm}^3} \right) \left(\frac{m\_{\phi\mathbf{n}}}{100 \text{ GeV}}\right)^{3/2} \text{s}^{-1}$$
 
$$\text{with } \bullet = \eta^{+} \eta^{-}, \ \eta^{0} \eta^{0}, \text{ and } \mathbf{X} \mathbf{X}^{\prime}. \tag{71}$$

We have used f(250 GeV) ≃ 0.5 and f(500 GeV) ≃ 0.2 [120], and we have assumed that all the ηs have the same mass and therefore CA(η 0η 0 ) = CA(η +η −). In **Figure 13** we plot the annihilation rate Ŵ(φRφ<sup>R</sup> → η 0η 0 ;t0) today (t<sup>0</sup> <sup>=</sup> 1.45×10<sup>17</sup> s) against <sup>σ</sup>SI for mφ<sup>R</sup> = 250 and 500 GeV with m<sup>η</sup> fixed at 230 GeV and 0.117 < φ<sup>R</sup> h <sup>2</sup> < 0.123. The vertical dashed lines are the XENON1T upper bound on σSI [103]. The peak of Ŵ(φRφ<sup>R</sup> → η 0η 0 ;t0) for <sup>m</sup>φ<sup>R</sup> <sup>=</sup> 250 (500) GeV appears at <sup>σ</sup>SI <sup>=</sup> 4.2 (4.7) <sup>×</sup> <sup>10</sup>−<sup>46</sup> cm<sup>2</sup> and is <sup>≃</sup> 1.7 (0.7) <sup>×</sup> <sup>10</sup><sup>18</sup> <sup>s</sup> −1 , which is two to three orders of magnitude smaller than the upper bound on the DM annihilation rate into W± in the Sun [123] .

#### 5. CONCLUSION

We have discussed the extensions of the Ma model by imposing a larger unbroken symmetry Z<sup>2</sup> × Z ′ 2 . Thanks to the symmetry, at least two stable particles exit. We have studied two models, model A and model B, where the stable particles form a multicomponent DM system in the model A, while they are a DM and dark radiation in the model B.

The model A is an extension of the model of Ma such that the lepton-number violating "λ<sup>5</sup> coupling," which is O(10−<sup>6</sup> ) to obtain small neutrino masses for Y <sup>ν</sup> <sup>∼</sup> 0.01, is radiatively generated. Consequently, the neutrino masses are generated at the two-loop level, where the unbroken Z<sup>2</sup> × Z ′ 2 symmetry acts to forbid the generation of the one-loop mass. Such larger unbroken symmetry implies that the model involves a multicomponent DM system. We have considered the case of the three-component DM system: two of them are SM singlet real scalars and the other one is a right-handed neutrino. The DM conversion and semi-annihilation in addition to the standard annihilation are relevant to the DM annihilation processes. We have found that the non-standard processes have a considerable influence on the DM relic abundance. We also have discussed the monochromatic neutrinos from the Sun as the indirect signal of the semi-annihilation of the DM particles. In the cases of one-component DM system of a real scalar boson or of a Majorana fermion the monochromatic neutrino production by the DM annihilation is strongly suppressed due to the chirality of the left-handed neutrino. However, such suppression is absent when DM is a complex scalar boson or a Dirac fermion. Also in a multicomponent DM system, the neutrino production is unsuppressed if it is an allowed process. We have found that the rate for the monochromatic neutrino production in the model A is very small compared with the current IceCUBE [123] sensitivity. However, the resonant effect in the s-channel process of the semi-annihilation can be expected to enhance the rate.

In the model B, the mass of the right-handed neutrinos are produced at the one-loop level. Then the radiative seesaw mechanism works at the two-loop level. Thanks to Z<sup>2</sup> × Z ′ 2 there exist at least two stable DM particles; a dark radiation N ′ <sup>1</sup> with a mass of O(1) eV and the other one, DM, is the real part of φ. The dark radiation contributes to 1Neff < 1 such that the

#### REFERENCES


tensions in cosmology that exist among the observations in the local Universe (CMB temperature fluctuations and primordial gravitational fluctuations) can be alleviated. Because of the hierarchy M2,3 ≫ T dec <sup>N</sup> ≃ <sup>O</sup>(100) MeV ≫ MN<sup>1</sup> <sup>O</sup>(1) eV, we are able to relate to the ratio of the lepton flavor violating decays to 1Neff. The indirect signal of the neutrino from the Sun has also been discussed. It is found that the predicted annihilation rate of the neutrinos is two to three orders of magnitude smaller than the current bound [123]. We have also expressed 1Neff as a function of the decay width of η 0 R (which is assumed to be lightest among ηs). It decays 100 percent into left- and righthanded neutrinos, where the heavier right-handed neutrinos decay further into dark radiation (the lightest among them). Dark radiation appears as a missing energy in collider experiments. We also have found that η + decays 100 percent into a charged lepton and the missing energy. This is a good example in which, through the generation mechanism of the neutrino masses, cosmology and collider physics are closely related.

### AUTHOR CONTRIBUTIONS

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

The work of MA is supported in part by the Japan Society for the Promotion of Sciences Grant-in-Aid for Scientific Research (Grants No. 16H00864 and No. 17K05412). JK is partially supported by the Grant-in-Aid for Scientific Research (C) from the Japan Society for Promotion of Science (Grant No.16K05315).


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Aoki, Kaneko and Kubo. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Value of the Axial-Vector Coupling Strength in β and ββ Decays: A Review

Jouni T. Suhonen\*

*Department of Physics, University of Jyvaskyla, Jyvaskyla, Finland*

In this review the quenching of the weak axial-vector coupling strength, *g*A, is discussed in nuclear β and double-β decays. On one hand, the nuclear-medium and nuclear many-body effects are separated, and on the other hand the quenching is discussed from the points of view of different many-body methods and different β-decay and double-β-decay processes. Both the historical background and the present status are reviewed and contrasted against each other. The theoretical considerations are tied to performed and planned measurements, and possible new measurements are urged, whenever relevant and doable. Relation of the quenching problem to the measurements of charge-exchange reactions and muon-capture rates is pointed out.

#### Edited by:

*Alexander Merle, Max Planck Institute for Physics (MPG), Germany*

#### Reviewed by:

*Eligio Lisi, National Institute for Nuclear Physics, Italy Fedor Simkovic, Comenius University, Slovakia*

\*Correspondence:

*Jouni T. Suhonen jouni.suhonen@phys.jyu.fi*

#### Specialty section:

*This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics*

Received: *19 June 2017* Accepted: *17 October 2017* Published: *16 November 2017*

#### Citation:

*Suhonen JT (2017) Value of the Axial-Vector Coupling Strength in* β *and* ββ *Decays: A Review. Front. Phys. 5:55. doi: 10.3389/fphy.2017.00055* Keywords: double beta decays, Gamow-Teller beta decays, forbidden beta decays, axial-vector coupling strength, beta spectra, charge-exchange reactions, strength functions, muon capture

## 1. INTRODUCTION

The neutrinoless double beta (0νββ) decays of atomic nuclei are of great experimental and theoretical interest due to their implications of physics beyond the standard model of electroweak interactions. Since these processes occur in nuclei, nuclear-structure effects play an important role and they may affect considerably the decay rates. The nuclear effects are summarized as the nuclear matrix elements (NMEs) containing information about the initial and final states of the nucleus and the action of the 0νββ transition operator on them. The NMEs, in turn, are computed numerically using some nuclear-theory framework suitable for the nuclei under consideration. The possible future detection of the 0νββ decay in the next generation of ββ experiments constantly drives nuclear-structure calculations toward better performance. Accurate knowledge of the NMEs is required in order that the data will be optimally used to obtain information about the fundamental nature and mass of the neutrino [1–7]. In addition, the 0νββ decay relates also to the breaking of lepton-number symmetry and the baryon asymmetry of the Universe [8, 9]. A number of nuclear models, including configuration-interaction based models like the interacting shell model (ISM), and various mean field models, have been adopted for the calculations. The resulting computed NMEs have been analyzed in the review articles [4, 10–12]. Most of the calculations have been done by the use of the proton-neutron quasiparticle random-phase approximation (pnQRPA) [13].

The performed 0νββ-decay calculations, as also those of the two-neutrino double beta (2νββ) decay, indicate that the following nuclear-structure ingredients affect the values of NMEs:


major shells that are separated by large energy gaps. The gaps occur at "magic numbers" of nucleons and have sometimes drastic effects on nuclear properties.


At the nuclear level, β decay can be considered as a mutual interaction of the hadronic and leptonic currents mediated by massive vector bosons W± [25]. The leptonic and hadronic currents can be expressed as mixtures of vector and axialvector contributions [26–28]. The weak vector and axial-vector coupling strengths g<sup>V</sup> and g<sup>A</sup> enter the theory when the hadronic current is renormalized at the nucleon level [29]. The conserved vector-current hypothesis (CVC) [26] and partially conserved axial-vector-current hypothesis (PCAC) [30, 31] yield the free-nucleon values g<sup>V</sup> = 1.00 and g<sup>A</sup> = 1.27 [25] but inside nuclear matter the value of g<sup>A</sup> is affected by manynucleon correlations and a quenched or enhanced value might be needed to reproduce experimental observations [32–35]. Precise information on the effective value of g<sup>A</sup> is crucial when predicting half-lives of neutrinoless double beta decays since the half-lives are proportional to the fourth power of g<sup>A</sup> [1, 36].

Since the vector bosons W± have large mass and thus propagate only a short distance, the hadronic current and the leptonic current can be considered to interact at a point-like weak-interaction vertex with an effective coupling strength GF, the Fermi constant. The parity non-conserving nature of the weak interaction forces the hadronic current to be written at the quark level (up quark u and down quark d) as a mixture of vector and axial-vector parts:

$$J\_H^{\mu} = \bar{u}(\mathbf{x})\gamma^{\mu}(1-\chi\_5)d(\mathbf{x}),\tag{1}$$

where γ <sup>µ</sup> are the usual Dirac matrices and <sup>γ</sup><sup>5</sup> <sup>=</sup> <sup>i</sup><sup>γ</sup> 0γ 1γ 2γ 3 . Renormalization effects of strong interactions and energy scale of the processes must be taken into account when moving from the quark level to the hadron level. Then the hadronic current between nucleons (neutron n and proton p) takes the rather complex form

$$J\_H^{\mu} = \bar{p}(\mathbf{x})[V^{\mu} - A^{\mu}]n(\mathbf{x}),\tag{2}$$

where the vector-current part can be written as

$$V^{\mu} = \text{gv}(q^2)\boldsymbol{\gamma}^{\mu} + \text{ig}\mathbf{M}(q^2)\frac{\sigma^{\mu\upsilon}}{2m\_{\text{N}}}q\_{\upsilon} \tag{3}$$

and the axial-vector-current part as

$$A^{\mu} = \text{g}\_{\text{A}}(q^2)\boldsymbol{\gamma}^{\mu}\boldsymbol{\gamma}\_5 + \text{g}\_{\text{P}}(q^2)q^{\mu}\boldsymbol{\gamma}\_5. \tag{4}$$

Here q <sup>µ</sup> is the momentum transfer, q 2 its magnitude, m<sup>N</sup> the nucleon mass (roughly 1 GeV) and the weak couplings depend on the magnitude of the exchanged momentum. For the vector and axial-vector couplings one usually adopts the dipole approximation

$$g\_{\mathcal{V}}(q^2) = \frac{\mathcal{S}^{\mathcal{V}}}{\left(1 + q^2/M\_{\mathcal{V}}^2\right)^2}; g\_{\mathcal{A}}(q^2) = \frac{\mathcal{g}\_{\mathcal{A}}}{\left(1 + q^2/M\_{\mathcal{A}}^2\right)^2},\tag{5}$$

where g<sup>V</sup> and g<sup>A</sup> are the weak vector and axial-vector coupling strengths at zero momentum transfer (q <sup>2</sup> <sup>=</sup> 0), respectively. For the vector and axial masses one usually takes M<sup>V</sup> = 84 MeV [37] and M<sup>A</sup> ∼ 1 GeV [37–39] coming from the acceleratorneutrino phenomenology. For the weak magnetism term one can take gM(q 2 ) = (µ<sup>p</sup> −µn)gV(q 2 ) and for the induced pseudoscalar term it is customary to adopt the Goldberger-Treiman relation [40] gP(q 2 ) = 2mNgA(q 2 )/(q <sup>2</sup> <sup>+</sup> <sup>m</sup><sup>2</sup> π ), where mπ is the pion mass and µ<sup>p</sup> − µ<sup>n</sup> = 3.70 is the anomalous magnetic moment of the nucleon. It should be noted that the β decays and 2νββ decays are low-energy processes (few MeV) involving only the vector [first term in Equation (3)] and axial-vector [first term in Equation (4)] parts at the limit q <sup>2</sup> <sup>=</sup> 0 so that the <sup>q</sup> dependence of Equation (5) does not play any role in the treatment of these processes in this review. Contrary to this, the 0νββ decays and nuclear muoncapture transitions involve momentum transfers of the order of 100 MeV and the full expression (2) is active with slow decreasing trend of the coupling strengths according to Equation (5).

### 2. EFFECTIVE VALUES OF GA: PREAMBLE

The effective value of g<sup>A</sup> can simply be characterized by a renormalization factor q (in case of quenching of the value of g<sup>A</sup> it is customarily called quenching factor):

$$q = \frac{\text{gA}}{\text{g}\_{\text{A}}^{\text{free}}},\tag{6}$$

where

$$g\_{\rm A}^{\rm free} = 1.2723(23) \tag{7}$$

is the free-nucleon value of the axial-vector coupling measured in neutron beta decay [41] and g<sup>A</sup> is the value of the axialvector coupling derived from a given theoretical or experimental analysis. This derived g<sup>A</sup> can be called the effective g<sup>A</sup> so that from (6) one obtains for its value

$$\mathbf{g}\_{\rm A}^{\rm eff} = q \mathbf{g}\_{\rm A}^{\rm free}.\tag{8}$$

Equations (6)−(8) constitute the basic definitions used in this review.

The effective value of g<sup>A</sup> can be derived from several different experimental and theoretical analyses. In these analyses it is mostly impossible to separate the different sources of renormalization affecting the value of gA: (i) the meson-exchange currents (many-body currents) that are beyond the onenucleon impulse approximation (only one nucleon experiences the weak decay without interference from the surrounding nuclear medium), usually assumed in the theoretical calculations, (ii) other nuclear medium effects like interference from nonnucleonic degrees of freedom, e.g., the 1 isobars and (iii) the deficiencies in the nuclear many-body approach that deteriorate the quality of the wave functions involved in the decay processes.

The effects (i) and (ii) can be studied by performing calculations using meson-exchange models and allowing nonnucleonic degrees of freedom in the calculations. These calculations that go beyond the nucleonic impulse approximation are described in section 3 in the context of Gamow-Teller β decays for which the related effects are measurable. The calculations yield a fundamental quenching factor q<sup>F</sup> and the related fundamentally renormalized effective g<sup>A</sup> for the space components (µ = 1, 2, 3) of the axial current (4) via the effects of the virtual pion cloud around a nucleon. The time component of µ = 0, the axial charge ρ5, is, however, fundamentally enhanced by, e.g., heavy meson exchange and the corresponding effective coupling g eff A (γ5) is discussed in section 8.2, in the context of first-forbidden 0<sup>+</sup> ↔ 0 − transitions for which the effect is measurable.

The ISM has the longest history behind it in studies of the axial quenching in Gamow-Teller β decays. The reason for this is the success of the ISM to describe nuclear spectroscopy of light nuclei and the rather large amount of data on these type of allowed β decays. The results of these studies are presented in section 5. In the same section the ISM results are compared with those obtained by the use of the pnQRPA. In section 6.2 the effective value of g<sup>A</sup> is analyzed for the first-forbidden unique β decays for which there are some experimental data available. In section 7 this study is extended to higher-forbidden unique β decays where no experimental data are available and one has to resort to mere theoretical speculations. In section 8 the forbidden non-unique β decays are discussed. Experimentally, there are available data for the above-mentioned first-forbidden non-unique 0<sup>+</sup> ↔ 0 − and other β transitions. For the higher-forbidden non-unique transitions, discussed in section 9, there are scattered half-life and β-spectrum data but more measurements are urgently needed, in particular for the shapes of the β spectra. Unfortunately, in all these studies it is not possible to completely disentangle the nuclear-medium effects (i) and (ii) from the nuclear-model effects (iii).

In the last two sections, 11, 12 more exotic methods to extract the in-medium value of g<sup>A</sup> are presented: The spin-multipole strength functions and nuclear muon capture. Measurements of the spin-multipole strength functions, in particular the location of the corresponding giant resonances, help theoretical calculations fine-tune the parameters of the model Hamiltonians such that the low-lying strength of, say 2− states, is closer to reality. Hence, more such measurements are called for. The nuclear muon capture probes the axial current (4) at 100 MeV of momentum transfer and thus suits perfectly for studies of the renormalization of the NMEs related to 0νββ decays. This means that muon-capture experiments for medium-heavy nuclei are urgently needed.

The renormalization of g<sup>A</sup> which stems from the nuclearmodel effects (iii) depends on the nuclear-theory framework chosen to describe the nuclear many-body wave functions involved in the weak processes, like β and ββ decays. This is why the effective values of g<sup>A</sup> can vary from one nuclear model to the other. On the other hand, the different model frameworks can give surprisingly similar results as witnessed in section 9 in the context of the comparison of the measured β spectra with the computed ones. The renormalization of g<sup>A</sup> can also depend on the process in question. For the zero-momentum-exchange (q <sup>2</sup> <sup>=</sup> 0) processes, like <sup>β</sup> and 2νββ decays, the renormalization can be different from the high-momentum-exchange (q <sup>2</sup> <sup>∼</sup> <sup>100</sup> MeV) processes, like 0νββ decays (in section 9 the related g<sup>A</sup> is denoted as g eff A,0ν ) or nuclear muon captures.

This introduction to the many-faceted renormalization of the axial-vector coupling is supposed to enable a "soft landing" into the review that follows. As can be noticed, the renormalization issue is far from being solved and lacks a unified picture thus far. There is not yet a coherent effort to solve the issue, but rather some sporadic attempts here and there. The most critical issue may be the nuclear many-body deficiencies (iii) that hinder a quantitative assessment of the nuclear-medium effects (i) and (ii) in light, medium-heavy and heavy nuclei. Only gradually this state of affairs will improve with the progress in the ab-initio nuclear methods extendable to nuclei beyond the very lightest ones. Hence, the lack of perfect nuclear many-body theory is reflected in this review as a wide collection of different effective g<sup>A</sup> variants, different for different theory frameworks and processes and not necessarily connected to each other (yet). The hope is that in the future the different studies would point to one common low-energy renormalization of g<sup>A</sup> for the β and twoneutrino ββ decays and that we would have some idea about the renormalization mechanisms at work in the case of the neutrinoless ββ decays.

On the other hand, there are some attempts to disentangle the nuclear medium effects from the nuclear many-body effects. Examples are the fundamental quenching elaborated in section 3 and the nuclear-medium-independent quenching factor k introduced in section 5.2 for the Gamow-Teller β decays, and in sections 6.2, 7 for the unique-forbidden β transitions. This factor is designed to give hints about the impact of the changes in the complexity of the nuclear model on the value of the effective axial coupling. Also the previously mentioned method based on the examination of β spectra in section 9 is largely nuclearmodel independent and seems to be a reasonable measure of the nuclear-medium effects (i) and (ii). More measurements of the β spectra are thus urgently called for.

### 3. NUCLEAR-MEDIUM EFFECTS

Based on the early shell-model studies of Gamow-Teller β decays, effects of 1 resonances and meson-exchange currents on the weak axial-vector coupling strength of the space part, **A**, of the axial current A <sup>µ</sup> (4) is expected to be quenched in nuclear medium and finite nuclei. Contrary to this, the coupling strength of the time part, A 0 , of (4) is expected to be enhanced by, e.g., the contributions coming from exchanges of heavy mesons. Many of these modifications in the strengths of the axial couplings stem from processes beyond the impulse approximation where only one nucleon at a time is experiencing a weak process, e.g., β decay, without interference from the surrounding nuclear medium. In fact, based on general arguments concerning softpion amplitudes [42] the space part of A <sup>µ</sup> is quenched and the time part of A <sup>µ</sup> is enhanced relative to the single-particle processes of the impulse approximation.

The origin of the quenching of the space part of A <sup>µ</sup> is not completely known and various mechanisms have been proposed for its origin: studied have been the 1-isobar admixture in the nuclear wave function [43], shifting of Gamow-Teller strength to the 1-resonance region, and renormalization effects of mesonexchange currents. The β − and β + Gamow-Teller strengths were related to the 1-isobar region e.g., in Delorme et al. [44] and sizable 1-resonance effects on β decays of low-lying nuclear states by tensor forces were reported in Oset and Rho [45] and Bohr and Mottelson [46]. In Towner and Khanna[47, 48] very simple nuclear systems were used to study the tensor force and related effects in order to minimize the impact of nuclear many-body complexities. Studied were the tensor effects and their interference with the 1-isobar current and meson-exchange currents in building up corrections to the Gamow-Teller matrix elements. Also relativistic corrections to the Gamow-Teller operator were included. Large cancellations among the various contributions were recorded and corrections below some 20% were obtained for the light (simple) nuclei. However, recent experimental studies of (p,n) and (n,p) reactions [49] report that the 1-nucleon-hole admixtures into low-lying nuclear states play only a minor role in the quenching of gA, in line with the results of Suhonen [43]. Also extended sum rules have been derived for relating g<sup>A</sup> to pion-proton total cross sections [50–52], or the method of QCD sum rules has been utilized [53].

In Wilkinson [54] the renormalization of the β-decay operator by the two- or many-nucleon correlations, in terms of internucleonic and intra-nucleonic mesonic currents, leads to the notion effective "fundamentally" renormalized axial coupling gAeF. The quenching of g<sup>A</sup> is then described by the fundamental quenching factor q<sup>F</sup> such that q<sup>F</sup> > q since q contains, in addition, the quenching stemming from the inadequate treatment of the nuclear many-body problem. From here on the above notation is adopted for the renormalization of g<sup>A</sup> stemming from the (fundamental) mesonic-current effects.

In the early study of Ericson [55] of the sum rule for Gamow-Teller matrix elements a (fundamental) quenching of roughly

$$q\_{\rm F} = 0.9 \tag{9}$$

was obtained for very light nuclei (A ≤ 17) by the examination of the effects of meson-exchange currents on the pion-nucleon interaction vertex and extending the result to a sum rule for Gamow-Teller matrix elements. This (practically) modelindependent study produces the following (fundamentally) renormalized value of the axial coupling strength

$$\text{g}\_{\text{AeF}} = 0.9 \times 1.27 = 1.1.\tag{10}$$

The above result does not necessarily apply to individual Gamow-Teller transitions between low-energy nuclear states.

The work of Ericson [55] was followed by the works [56, 57] where it was found that the renormalization should be universal for all transitions, in particular applicable to the mentioned Gamow-Teller transitions at low nuclear excitations. The procedure bases on the fact that the partially conserved axial current (PCAC) hypothesis [30, 31] enables one to calculate the full axial-current matrix element in terms of a pion-nucleus vertex [58]. At the low-momentum-exchange limit, relevant for the nuclear β decays, the PCAC leads to the Goldberger-Treiman relation [40, 59] which relates the effective value of g<sup>A</sup> to the effective value of the pionic coupling constant gπ by Ericson [55]

$$\frac{g\_{\rm A}^{\rm eff}}{g\_{\pi}^{\rm eff}} = \frac{g\_{\rm A}^{\rm free}}{g\_{\pi}^{\rm free}} = \frac{f\_{\pi}}{\sqrt{2}m\_{\rm N}},\tag{11}$$

where m<sup>N</sup> is the nucleon mass and f<sup>π</sup> = 0.932m<sup>π</sup> is the pion decay constant, mπ being the pion mass. The pionic coupling constant is, in turn, renormalized by the effects on the virtual pion field by the presence of other nucleons. For large nuclei (surface effects can be omitted) the renormalization arises from nucleonic short-range correlations leading to voids between nucleons and the renormalization can be understood via an electromagnetic analog: an electric dipole in a correlated dielectric medium is renormalized in a similar way as the pionic coupling constant. There is also a connection to the low-energy scattering of pions on nuclei: the short-range correlations quench the p-wave pion-nucleon amplitude by the same amount as the dielectric effect. For finite nuclei a model-dependent surface factor has to be taken into account [55]. The size renormalization emerges from the nuclear surface layer of a thickness of the order of the pion Compton wavelength and thus the quenching of g<sup>A</sup> increases with increasing nuclear radius and, as a consequence, with increasing nuclear mass.

In Rho [57] the pion-nucleus vertex was calculated and the related quenched g<sup>A</sup> agreed with the one of Ericson [55] to leading order. In infinite nuclear matter This quenching turns out to be [57]

$$q\_{\rm F}^{\infty} = 0.76 \quad \text{(infinite nuclear matter)} \tag{12}$$

leading to the quenched effective axial coupling strength

$$g\_{\rm AeF}^{\infty} = 0.76 \times 1.27 = 0.96. \quad \text{(infinite nuclear matter)} \quad \text{(13)}$$

in infinite nuclear matter.

The works of Ericson [55, 56] and Rho [57] were used by Wilkinson [54] to bridge the gap between the infinite nuclear matter and finite nuclei. In Wilkinson [54] it was argued that the fundamental quenching can be described by the formula

$$q\_{\rm F} = \sqrt{(q\_{\rm F}^{\infty})^2 + \left[1 - (q\_{\rm F}^{\infty})^2\right]/A^{0.17}} \tag{14}$$

for finite nuclei of mass number A. This formula includes the short-range correlation effect and the finite-size factor [56, 57] and gives for the fundamental quenching, using (12), between A = 50 − 150 the value q<sup>F</sup> = 0.88. This means that the fundamental quenching is practically constant over the range of nuclei of interest to the double beta decay. The corresponding fundamentally quenched value of the axial-vector strength is plotted in **Figure 2**, and its value is practically 1.1 through the whole range of interest.

In Siiskonen et al. [60] the renormalization of the axial current (and vector and induced pseudoscalar terms of the nucleonic current) was studied for several nuclear systems as a function of transition energy by including effective transition operators up to second order in perturbation theory. Thus, the renormalization of g<sup>A</sup> contains both the fundamental and nuclear many-body aspects. It was found that the renormalization was practically constant up to 60 MeV in transition energy, in agreement with the q dependence of g<sup>A</sup> in relation (5). The obtained quenchings are as follows

$$\mathrm{g}\_{\mathrm{A}}^{\mathrm{eff}} = 1.0 \, (1s0d \, \mathrm{shell}); \, 0.98 \, (1p0f \, \mathrm{shell}); \, 0.71 \, (^{56}\mathrm{Ni}); \, 0.52 \, (^{100}\mathrm{Sn}). \tag{15}$$

The results (15), obtained by using the nuclear-mediumcorrected transition operators have been repeated in **Table 1** of section 5.1 and **Figure 3** of section 5.2 in order to compare them with the more phenomenological shell-model results. Effective operators have also been used in the connection with the calculations for the double beta decays in a solvable model [69] and for the nucleus <sup>92</sup>Mo [70] and the nuclei <sup>76</sup>Ge and <sup>82</sup>Se [71, 72] in the framework of the interacting shell model.

As speculated in Wilkinson [54], the mesonic effects (mesonexchange currents) show up as effective two-body contributions to the β-decay operators. These two-body currents quench g<sup>A</sup> and this quenching was first estimated in Menéndez [73], in the framework of the chiral effective field theory (cEFT) where both the weak currents and nuclear forces can be described on the same footing and to a given order of approximation (leading order, next-to-leading order, etc.) In Menéndez [73] the twobody currents were replaced by an effective one-body current derived from the cEFT, leading to a momentum-dependent effective coupling g eff A (q 2 ), renormalized with respect to the bare axial coupling of (5). It turned out that the additional quenching is caused by the short-range nucleon-nucleon coupling present in the original two-body current. The additional quenching decreases with increasing q, being the strongest at the zeromomentum-transfer limit, affecting mostly the nuclear β and 2νββ decays. In fact, the strength of the short-range nucleonnucleon coupling in the two-body current can be adjusted such as to reproduce the empirical quenching of the Gamow-Teller β decays discussed in section 5. As the 0νββ decay is a highmomentum-transfer process (q ∼ 100 MeV) it is expected that the two-body currents have not such a drastic effect on the onebody current (4) for the 0νββ decay. Here it should be noted that the one-body current (2) has been fully taken into account in all 0νββ-decay calculations and the two-body currents introduce a renormalization, g eff A (q 2 ), that deviates from the one-body dipole gA(q 2 ) of (5) the less the higher the momentum exchange q is. The quenching caused by the two-body currents could probably be measured by using charge-exchange reactions [49] in advanced nuclear-physics infrastructures.

In Menéndez [73] it was estimated, by using the ISM manybody framework in the mass range A = 48 − 136, that the effect of the two-body currents on the value of the 0νββ NME is between −35 and 10% depending on the (uncertain) values of the cEFT parameters, the smallest corrections occurring for A = 48. In Engel [74] the effect of the two-body currents was studied in the framework of the pnQRPA in the mass range A = 48 − 136, and a quenching effect of 10–22% was obtained for the 0νββ NMEs, the 10% effect pertaining to the case of <sup>48</sup>Ca. A more complete calculation, including three-nucleon forces and consistent treatment of the two-body currents and the nuclear Hamiltonian, was performed in Ekström [75]. Application to the Gamow-Teller β decays in <sup>14</sup>C and 22,24O nuclei yielded the quenching q = 0.92 − 0.96 by comparison of the computed strengths to that of the Ikeda 3(N − Z) sum rule [35, 76]. This <10% quenching is in line with the trend observed in the studies [73, 74] where the quenching approaced the 10% limit for light nuclei. It should be noted that the twobody meson-exchange currents appear also in neutrino-nucleus scattering [77] but at energies where two nucleons are ejected as a result of the scattering (the so-called two-particle-twohole exchange currents). The higher energy evokes considerable difficulties in handling the two-body meson-echange currents, as demonstrated in Simo et al. [78].

The meson-exchange currents can cause also enhancement phenomena, like in the case of the renormalization of the onebody weak axial charge density ρ<sup>5</sup> [time part of A <sup>µ</sup> in (4)] in the case of the 0<sup>−</sup> ↔ 0 + nuclear β transitions [42, 79]. In this case the γ <sup>5</sup> operator mediates the first-forbidden nonunique β transition and the corresponding axial-vector coupling strength is enhanced quite strongly. In the work of Kirchbach and Reinhardt [79] the effects of a pionic two-body part of ρ<sup>5</sup> was studied for 4 nuclear masses and the corresponding leading single-particle transitions. This work was extended by Kirchbach et al. [80] and Towner [81] by taking into account also the heavymeson exchanges. In Towner [81] 6 nuclear masses and a number of single-particle transitions were computed by using nuclear wave functions from the ISM. An interesting investigation of the role of the two-particle-two-hole excitations in the A = 16 nuclei was performed in Towner and Khanna [82]. The renormalization of the weak axial charge by the meson-exchange currents had to be taken into account in order to explain the measured rates of both the 0<sup>−</sup> → 0 <sup>+</sup> β decay and the 0<sup>+</sup> → 0 − muon capture. The axial-charge enhancement is elaborated further, quantitatively, in section 8.2.

Very recently break-through results in the calculations of the axial charge and axial-vector form factors have been achieved in the lattice QCD (quantum chromodynamics) calculations [83–85]. In the work [84] the result

$$\mathbf{g}\_{\rm A}^{\rm free} = 1.278(21)(26) \qquad \text{(lattice calculation)} \tag{16}$$

was obtained, where the first uncertainty is statistical and the second comes from the extrapolation systematics. This computed value is quite compatible with the measured free value of g<sup>A</sup> in (7). Also the lattice QCD calculations of the double beta decay are advancing in the two-nucleon (toy) systems (see [86]).

### 4. NUCLEAR-MODEL EFFECTS

The studies on the effective value of the axial-vector coupling strength, gA, have mainly been performed for β decays in established nuclear many-body frameworks. Also the magnetic moments of nuclei have been studied [87, 88] for simple oneparticle and one-hole nuclei in order to pin down the effects of the tensor force in shifting low-energy strength of Gamow-Teller type to higher energies, and thus effectively quenching the spin-isospin operator for Gamow-Teller decays. The used many-body frameworks encompass the interacting shell model (ISM) [89] and the pnQRPA [13, 90]. Also the frameworks of the microscopic interacting boson model (IBM-2) [91] and the interacting boson-fermion-fermion model, IBFFM-2 [92], have been used. Let us discuss next the various many-body aspects of these models that may affect the (apparent) renormalization of the magnitude of gA. It is appropriate to note here that in all these studies the nuclear many-body framework can be considered more or less deficient and thus the many-body effects cannot be disentangled from the nuclear-medium effects, discussed in section 3.

### 4.1. Many-Body Aspects of the ISM

The ISM is a many-body framework that uses a limited set of single-particle states, typically one harmonic-oscillator major shell or one nuclear major shell, to describe nuclear wave functions involved in various nuclear processes. The point of the ISM is to form all the possible many-nucleon configurations in the given single-particle space, each configuration described by one Slater determinant, and diagonalize the nuclear (residual) Hamiltonian in the basis formed by these Slater determinants. In this way the many-body features are taken into account exactly but only in a limited set of single-particle states. The problem is to extend the single-particle space beyond the one-shell description due to the factorially increasing size of the sparse Hamiltonian matrix to be diagonalized. In this way only the low-energy features of a nucleus can be described, leaving typically the giantresonance region out of reach. The other problem with the ISM is to find a suitable (renormalized) nucleon-nucleon interaction to match the limited single-particle space. Since this space is small, the renormalization effects of the two-body interaction become substantial. Typically, mostly in the early works, all the matrix elements of the two-body interaction were fitted such that the computed observables, energies, electromagnetic decays, etc., are as close as possible to the corresponding measured ones (see section 5.1). In some works also perturbative approaches through particle-hole excitations from the valence to the excluded space have been considered (see, e.g., [93–95] and the references therein).

From early on there have been difficulties for the ISM to reproduce the measured β-decay rates [96]. This has lead to a host of investigations of the effective (quenched) value of g<sup>A</sup> in the ISM framework (see section 5.1 below). The main limitation of the ISM is its confinement to small single-particle spaces, typically comprising one oscillator major shell or a magic shell, leaving one or two spin-orbit partners out of the model space. From, e.g., pnQRPA calculations [15, 16] and perturbative ISM calculations [72, 97] one knows that inclusion of all spin-orbit partners in the single-particle model space is quite essential. This has been noticed also in the extended ISM calculations where the missing spin-orbit partners have been included at least in an effective way [20, 98]. Even extension of the ISM to include two harmonic-oscillator shells (1s0d and 1p0f shells) has been done for the calculation of the 0νββ decay of <sup>48</sup>Ca [99].

Several advanced shell-model methods have been devised in order to include larger single-particle spaces into the calculations. One can try to find clever ways to select the most important configurations affecting the observables one is interested in. Such an established algorithm is the Monte Carlo shell model (MCSM) where statistical sampling of the Slater determinants is used [100, 101]. One can also use importance-truncation schemes [102] or very advanced ab initio methods, like the coupled-cluster theory, where the two- and three-body interactions can be derived from the chiral effective field theory (cEFT) [103]. One can also use the in-medium similarity renormalization group (IM-SRG) method, like in Bogner [104], where an ab initio construction of a nonperturbative 1s0d-shell Hamiltonian, based on cEFT two- and three-body forces, has been done. Another new method is the density matrix renormalization group (DMRG) algorithm [105], which exploits optimal ordering of the proton and neutron single-particle orbitals and concepts of quantum-information theory.

All the new methods extend the traditionally used ISM model spaces and the future β-decay calculations using these methods will either confirm or reduce the amount of quenching of g<sup>A</sup> observed in the older ISM calculations, described in section 5.1 below. The ab initio methods are already available for the light nuclei, occupying the 0p and 1s0d shells, and later for the medium-heavy and heavy nuclei dwelling in the higher oscillator shells. The quenching problem can only be solved by using manybody methods with error estimates, including a systematic way to improve their accuracy. At the same time the two- and threebody forces used in the calculations should be produced on the same footing as the many-body framework itself, preferably from ab initio principles. One should not forget that also the operators used in the computations should be made effective operators that match the adopted single-particle valence spaces. Using these prescriptions one can eliminate the deficiencies of the nuclear many-body framework and obtain information about the quenching of g<sup>A</sup> in the nuclear medium (see section 3), beyond the effects caused by the deficiencies of a nuclear model.

## 4.2. Many-Body Aspects of the pnQRPA

The random-phase approximation (RPA) is an extension of the Tamm-Dancoff model (TDM) in the description of magic nuclei (at closed major shells) by particle-hole excitations across the magic gaps between closed nuclear major shells [35, 106]. In the RPA the simple particle-hole vacuum, with the singleparticle orbitals fully occupied up to the Fermi surface at the magic gap, is replace by the correlated vacuum, containing twoparticle–two-hole, four-particle–four-hole, etc. excitations across the magic gap. The use of the correlated vacuum in the RPA enhances the strength of collective transitions [35, 106]. Its quasiparticle version, quasiparticle RPA (QRPA) describes openshell nuclei, outside the closures of magic shells, by replacing the particle-hole excitations by two-quasiparticle excitations. Usually these quasiparticles are generated by the use of the Bardeen-Cooper-Schrieffer (BCS) theory [107] from the shortrange interaction part of the nuclear Hamiltonian in an eveneven reference nucleus. The quasiparticles can be viewed as partly particles and partly holes, inducing fractional occupancies of the nuclear single-particle orbitals and leading to a smeared Fermi surface for protons and/or neutrons for open-shell nuclei. The proton-neutron version of the QRPA (pnQRPA) uses twoquasiparticle excitations that are built from a proton and a neutron quasiparticle. This enables description of odd-odd nuclei starting from the even-even BCS reference nucleus.

The strong point of the pnQRPA theory is that it can include large single-particle valence spaces in the calculations. There are no problems associated with leaving spin-orbit-partner orbitals out of the computations. On the other hand, the pnQRPA has a limited configuration space, essentially including twoquasiparticle excitations on top of a correlated ground state [35]. Deficiencies of the pnQRPA formalism have been analyzed against the ISM formalism, e.g., in Menéndez [21] by using a seniority-based scheme (seniority was defined earlier, at point (c) in section (1). In that work the pnQRPA was considered to be a low-seniority approximation of the ISM. But on the other hand, the ground-state correlations of the pnQRPA introduce higher-seniority components to the pnQRPA wave functions and the deficiencies stemming from the incomplete seniority content of the pnQRPA should not be so bad [108]. Also the renormalization problems of the two-body interaction are not so severe as in the ISM due to the possibility to use large single-particle model spaces. On the other hand, it is harder to find a perturbative scheme for the effective Hamiltonian due to the incompleteness of the available many-body configuration space. Due to this, schematic or G-matrix-based boson-exchange Hamiltonians have widely been used (see section 5.2).

In any case, the configuration content of the pnQRPA is limited and extensions and improvements of the theory framework are wanted in order to see how the quenching problem of g<sup>A</sup> evolves with these extensions and improvements. Such extensions have been devised, including, e.g., the renormalized QRPA (RQRPA) [109, 110] and similar "fully" renormalized schemes [111–113]. Another possible improvement of the pnQRPA is the relativistic quasiparticle time-blocking approximation (RQTBA), in particular its protonneutron version, the pn-RQTBA, advocated in Robin and Litvinova [114]. It shows good promise for improvements over the β-decay calculations of the ordinary pnQRPA the use of which clearly points out to need for a quenched value of g<sup>A</sup> in β-decay calculations, as discussed in section 5.2.

The (charge-conserving) QRPA framework, with linear combinations of proton-proton and neutron-neutron quasiparticle pairs, phonons [35], can be used to describe (collective) excitations of even-even nuclei (collectivity is where the name phonon stems from). These, in turn, can be used as reference nuclei in building the excitations of the neighboring odd-mass (odd-proton or odd-neutron) nuclei by coupling the QRPA phonons with proton or neutron quasiparticles. This phonon-quasiparticle coupling can be carried out in a microscopic way, based on a realistic effective residual Hamiltonian. This has been achieved, e.g., in the microscopic quasiparticle-phonon model (MQPM) [115, 116] where a microscopic effective Hamiltonian based on the Bonn G matrix has been used to produce the one- and three-quasiparticle states in odd-mass nuclei. This extension of the QRPA has been used to describe β decays, and in particular in connection with the renormalization problem of gA, as discussed in section 9.

It should be noted that odd-mass nuclei can also be described by starting from an odd-odd reference nucleus, described by the pnQRPA phonons [35]. By coupling either proton or neutron quasiparticles with pnQRPA phonons one can, again, create the states of either a neutron-odd or a proton-odd nucleus. This approach was coined the proton-neutron MQPM (pnMQPM) and was used to describe forbidden beta decays in Mustonen and Suhonen [117]. Although the pnQRPA-based phonons better take into account the Ikeda sum rule [35, 76] and the Gamow-Teller giant-resonance region of the β −-type strength function, the pnMQPM lacks the important three-proton-quasiparticle and three-neutron-quasiparticle contributions, essential for good reproduction of the low-energy spectra of odd-mass nuclei. This is why its use in β-decay calculations has been very limited.

## 4.3. Many-Body Aspects of the IBM

In its simplest version, the interacting boson model (IBM), the theory framework consists of s and d bosons which have as their microscopic paradigms the 0+ and 2+ coupled collective Fermion pairs present in nuclei. Even a mapping of the collective Fermion pairs to these bosons can be devised [91]. An extension of the IBM is the microscopic IBM (IBM-2) where the proton and neutron degrees of freedom are explicitly separated. The IBM and IBM-2 are sort of phenomenological versions of the ISM, containing the seniority aspect and the restriction to one magic shell in terms of the single-particle valence space. The Hamiltonian and the transition operators are constructed from the s and d bosons as lowest-order boson expansions with coupling coefficients to be determined by fits to experimental data or by relating them to the underlying fermion valence space through a mapping procedure [118, 119]. Thus, the IBM and its extensions use more or less phenomenological operators mimicking the renormalized operators used in the ISM (see section 4.1).

The two versions of the IBM can be extended to include higher-multipole bosons, like g bosons, as well. Further extension concerns the description of odd-mass nuclei by the use of the interacting boson-fermion model (IBFM) and its extension, the microscopic IBFM (IBFM-2) [92]. The IBM concept can also be used to describe odd-odd nuclei by using the interacting bosonfermion-fermion model (IBFFM) and its proton-neutron variant, the proton-neutron IBFFM (IBFFM-2) [120]. Here the problems arise from the interactions between the bosons and the one or two extra fermions in the Hamiltonian, and from the transition operators containing a host of phenomenological parameters to be determined in some way. The IBM-2 and the IBFFM-2 have been used to access the renormalization of gA, as described in section 10.2.

### 5. EFFECTIVE VALUE OF GA IN ALLOWED GAMOW-TELLER β DECAYS

Gamow-Teller decays are mediated by the Pauli spin operator σ and they are thus able to change the initial nuclear spin J<sup>i</sup> by one unit. In the renormalization studies the simplest Gamow-Teller transitions are selected, namely the ground-state-to-ground-state ones. In **Figure 1** are depicted Gamow-Teller ground-state-toground-state β − and β +/EC transitions between even-even 0+ and odd-odd 1<sup>+</sup> ground states in the A = 100 Zr-Nb-Mo-Tc-Ru region. Shown are three different situations with a cascade pattern (left panel), lateral feeding to a middle nucleus (middle panel), and lateral feeding from a middle nucleus (right panel). All these transitions are mediated by a Gamow-Teller NME, MGT, of the Pauli spin operator, defined, e.g., in Suhonen [35]. The corresponding β-decay data can be obtained from ENSDF <sup>1</sup> . In the figure this NME is denoted by M<sup>L</sup> (MR) in the case it is to the left (right) of the central nucleus. The corresponding reduced transition probability BGT can be written as

$$B\_{\rm GT} = \frac{g\_{\rm A}^2}{2f\_i + 1} \left| M\_{\rm GT} \right|^2,\tag{17}$$

where J<sup>i</sup> is the spin of the ground state of the initial nucleus, g<sup>A</sup> is the weak axial-vector coupling strength, substituted by the effective coupling strength g eff A of Equation (8) in practical calculations of the β-decay rates involving nuclear levels of low excitation energy [Hence, the coupling strength g<sup>A</sup> is probed at the q <sup>2</sup> <sup>→</sup> 0 limit in (5)]. It is worth noting that the Gamow-Teller decays probe only gA, not g<sup>V</sup> which is carried by the vector part (Fermi spin-zero operator) of the β transitions, not active for the here discussed 1<sup>+</sup> ↔ 0 + transitions due to the conservation of angular momentum.

The comparative half-lives (log ft values) of the 1<sup>+</sup> ↔ 0 + Gamow–Teller transitions are given in terms of the reduced transition probabilities as given in Suhonen [35]

$$\log ft = \log\_{10}(f\_0 t\_{1/2} \text{[s]}) = \log\_{10}\left(\frac{6147}{B\_{\text{GT}}}\right) \tag{18}$$

for the β +/EC or β − type of transitions. The half-life of the initial nucleus, t1/2, has been given in seconds.

Next we inspect the evolution of the quenching concept, based on (17) and (18), in nuclear-structure calculations performed during the last four decades.

#### 5.1. Interacting Shell Model

Traditionally the renormalization of the axial-vector coupling strength has been addressed in the context of the ISM in a wealth of calculations pertaining to Gamow-Teller β decays of very light (p-shell), light (sd-shell), and medium-heavy (pf-shell and sdgshell) nuclei. In these calculations it appears that the value of g<sup>A</sup> is quenched. As indicated by the ISM results below, the quenching factor (6) is roughly a decreasing function of the nuclear mass number A, implying stronger quenching with increasing nuclear mass. The studies can be grouped according to the mass regions as follows.

<sup>1</sup>ENSDF at NNDC site, http://www.nndc.bnl.gov/

#### 5.1.1. Results for the 0p-Shell Nuclei

A thorough study of the Gamow-Teller β decays of the 0p-shell nuclei was performed in Chou et al. [61]. A 0p − 1s0d crossshell Hamiltonian derived by Warburton and Brown [121] was used in the calculations. The thus derived phenomenological (the fundamental, section 3, and nuclear-model induced renormalization cannot be disentangled) quenching factor [see Equation (6)] (from a least-squares fit, with one standard deviation error) assumed the value

$$q = 0.82 \pm 0.02,\tag{19}$$

when using the then adopted value g free <sup>A</sup> = 1.26 in contrast to the presently adopted value of Equation (7). Since the presently adopted free value of g<sup>A</sup> is a bit larger, the quenching increases slightly and for the effective value (8) of g<sup>A</sup> we have to use

$$\mathbf{g\_A^{eff}} = (0.82 \pm 0.02) \times \frac{1.26}{1.27} \times 1.27 = 1.03^{+0.03}\_{-0.02}, \qquad \text{(20)}$$

leading to an effective quenched value of g<sup>A</sup> close to unity.

#### 5.1.2. Results for the 1s0d-Shell Nuclei

A pioneering early work of Wilkinson [122] investigated Gamow-Teller β decays in the 0p shell and lower 1s0d shell for the quenching of gA. In this work Wilkinson obtained a quenching factor which was slightly corrected in Wilkinson [54] based on new experimental data. The corrected value reads (from a least-squares fit, with one standard deviation error)

$$q = 0.899 \pm 0.035,\tag{21}$$

when using the then adopted value g free <sup>A</sup> = 1.25. Using again the correction for g free <sup>A</sup> we have

$$\mathbf{g}\_{\rm A}^{\rm eff} = (0.899 \pm 0.035) \times 1.25 = 1.12^{+0.05}\_{-0.04}.\tag{22}$$

The same quenching was obtained in Brown et al. [123] by using a different ISM effective Hamiltonian indicating that the quenching is not very sensitive to the detailed aspects of the shellmodel analysis. In Wilkinson [54] the empirical result (21) was combined with relativistic corrections to yield

$$q = 0.927 \pm 0.038. \quad \text{(with relativistic corrections)} \tag{23}$$

This yields

$$\lg\_{\rm A}^{\rm eff} = (0.927 \pm 0.038) \times 1.25 = 1.18 \pm 0.05. \tag{24}$$

when including the relativistic corrections.

In Wilkinson [122] and Wilkinson [54] it was speculated that the renormalization effects of the Gamow-Teller transitions at low nuclear excitation are of the order expected from fundamental mesonic effects [55–57] (nuclear medium effect, see section 3) or from the lifting of Gamow-Teller strength to higher energies by the nuclear tensor force [87, 88] (nuclear model effect, see section 4). Indeed, by using sum-rule arguments of Ericson [55] the expected quenching by the meson-exchange effects would be around q = 0.93 for nuclei in the vicinity of A = 16. This is in very good agreement with the relativistically corrected empirical result (23).

A full sd-shell analysis of the quenching was performed in Wildenthal [62] with a new set of wave functions derived from a Hamiltonian reproducing the global spectroscopic features of the 1s0d-shell nuclei. The least-squares study (with one standar deviation error) yielded the (empirical) quenching factor q = 0.77 ± 0.02 and thus leads to the global g free A -corrected 1s0d-shell effective axial-vector coupling of

$$g\_{\rm A}^{\rm eff} = (0.77 \pm 0.02) \times 1.25 = 0.96\_{-0.02}^{+0.03},\tag{25}$$

which is notably smaller than (22) obtained for the lower 1s0d shell. In the least-squares-fit studues, like this and the one of Chou et al. [61] (see section 5.1.1), the separation of the fundamental quenching (see section 3) from the total quenching is impossible.

### 5.1.3. Results for the 1p0f(0g9/2)-Shell Nuclei

In the work [63] 64 Gamow-Teller β decays for the nuclear mass range A = 41 − 50 were studied. This mass range covers the lower part of the 1p0f shell. The shell-model work was based on Caurier et al. [124] and KB3 two-body interaction was adopted. In Martínez-Pinedo et al. [63] the experimental values of Gamow-Teller matrix elements (extracted by using the free value of gA) were compared with their computed values by plotting them against each other in an xy plane. The plot was well described by a line with the slope giving a phenomenological quenching factor. From the slope and its error the quenching factor

$$q = 0.744 \pm 0.015\tag{26}$$

was derived, when using the their adopted value g free <sup>A</sup> = 1.26. Then the g free A -corrected lower pf-shell quenching amounts to

$$\mathbf{g}\_{\rm A}^{\rm eff} = (0.744 \pm 0.015) \times 1.26 = 0.937\_{-0.018}^{+0.019}.\tag{27}$$

It is interesting to note that with this value of g eff A the half-life of the 2νββ decay of <sup>48</sup>Ca could be predicted [125] in perfect agreement with the later measured value [126]. In the work [127] it was confirmed that the value q = 0.77 reasonably describes the quenching in the A = 48 region. The quenching in the 1s0d and 1p0f shells was also studied in Auerbach et al. [128] for the nucleus <sup>26</sup>Mg (1s0d model space) and for the nuclei <sup>54</sup>Fe and <sup>56</sup>Ni (1p0f model space) by using both the random-phase approximation and the ISM. The computed β + Gamow-Teller strengths were compared with those derived from the (n,p) charge-exchange reactions. This comparison implied a phenomenological quenched value of g eff <sup>A</sup> ∼ 0.98, not far from the value (25), extracted in the 1s0d shell by Wildenthal [62] and the value (27), extracted in the 1p0f shell.

The upper 1p0f(0g9/2)-shell Gamow-Teller transitions were analyzed in Honma [65] in the 0f5/21p0g9/<sup>2</sup> valence space using a renormalized G-matrix-based two-body interaction, fitted in the

mass region A = 63 − 96. A rough phenomenological quenching factor

$$q = 0.6\tag{28}$$

was adopted in the subsequent calculations of the 2νββ-decay rates of <sup>76</sup>Ge and <sup>82</sup>Se. This, in turn, leads to an upper 1p0f(0g9/2)-shell effective coupling strength of

$$\text{g}\_{\text{A}}^{\text{eff}} = 0.6 \times 1.26 = 0.8,\tag{29}$$

which is considerably smaller than (27) obtained for the lower 1p0f shell.

### 5.1.4. Results for the 0g7/21d2s0h11/2-Shell Nuclei

In Caurier et al. [66] an analysis of the Gamow-Teller β decays in the (incomplete) sdg shell (for A = 128 − 130) was performed using the 0g7/21d2s0h11/<sup>2</sup> single-particle space. A model Hamiltonian based on a renormalized Bonn-C G-matrix with a subsequent fitting of about 300 energy levels of some 90 nuclei in the 0g7/21d2s0h11/<sup>2</sup> shell was used in the calculations. The resulting phenomenological quenching factor was

$$q = 0.57,\tag{30}$$

implying a 0g7/21d2s0h11/2-shell effective coupling strength of

$$g\_{\rm A}^{\rm eff} = 0.57 \times 1.26 = 0.72,\tag{31}$$

which is a bit smaller than those obtained in the 1p0f(0g9/2) shell.

In Caurier et al. [66] also the case of A = 136 was discussed for the 2νββ decay of <sup>136</sup>Xe using the above-mentioned singleparticle space. Comparing the experimentally available [129] (p,n) type of strength function on <sup>136</sup>Xe (up to excitation energies of 3.5 MeV in <sup>136</sup>Cs) with the computed one, the authors concluded a phenomenological quenching factor

$$q = 0.45\tag{32}$$

for A = 136. This leads to a heavily quenched effective axialvector coupling strength of

$$g\_{\rm A}^{\rm eff}(A=136) = 0.45 \times 1.26 = 0.57,\tag{33}$$

for the A = 136 region of the 0g7/21d2s0h11/<sup>2</sup> shell. On the other hand, more recent calculations by Horoi et al. [68, 130] for the 2νββ NMEs of <sup>130</sup>Te and <sup>136</sup>Xe suggest a milder quenching and a larger value g eff A (A = 130 − 136) = 0.94 [68] for the effective coupling strength. This is in a rather sharp tension with the results (31) and (33) of Caurier et al. [66].

In Juodagalvis et al. [67] a cross-shell study for the mass region A = 90 − 97 was performed in the single-particle space 1p1/20g9/<sup>2</sup> for protons and 0g7/21d0s0h11/<sup>2</sup> for neutrons by using a Bonn-CD-based potential with perturbative renormalization. Again, lack of the full space of spin-orbit partners lead to a strong phenomenological Gamow-Teller quenching

$$q = 0.48,\tag{34}$$

leading to a cross pf − sdg-shell effective coupling strength of

$$g\_{\rm A}^{\rm eff} = 0.48 \times 1.26 = 0.60.\tag{35}$$

The above-derived quenching is not far from the quenching <sup>q</sup> <sup>=</sup> 0.5 derived in Brown [131] for nuclei in the <sup>100</sup>Sn region using a 0f5/21p0g9/<sup>2</sup> proton-hole space and 0g7/21d0s0h11/<sup>2</sup> neutron-particle space.

A quite recent ISM analysis of the nuclei within the mass range 52 ≤ A ≤ 80 was performed by Kumar et al. [64]. There the 1p0f-shell nuclei, 52 ≤ A ≤ 67, were treated by using the KB3G interaction, and the comparison with the experimental β −-decay half-lives produced a phenomenological quenching factor leading to the effective coupling strength

$$g\_{\rm A}^{\rm eff} = 0.838\_{-0.020}^{+0.021} \quad \text{(52} \le A \le 67\text{)}.\tag{36}$$

The 0f5/21pg9/2-shell nuclei, 67 ≤ A ≤ 80, were computed by using the JUN45 interaction, producing the effective coupling strength

$$g\_{\rm A}^{\rm eff} = 0.869 \pm 0.019 \quad \text{(67} \le A \le 80\text{)}.\tag{37}$$

In this work the error estimation is given by the slopes-of-thelines method [63], discussed in the context of Equation (26) above.

TABLE 1 | Mass ranges and effective values of *g*A extracted from the works of the last column.


*RC in lines 2 and 3 denotes relativistic corrections.*

All the results of the ISM analyses have been collected in **Table 1**. There the mass range (magic shell), value of g eff A , and the author information are given. Also the results of Siiskonen et al. [60], from section 3, obtained by the use of effective operators in the nuclear medium, have been given for comparison. In addition, the ISM results (adding the <sup>100</sup>Sn results of Siiskonen et al. [60]) for masses 60 ≤ A ≤ 136 have been visualized in **Figure 3** of section 5.2. In the figure the results of Honma et al. [65], Caurier et al. [66], Horoi and Neacsu [68], Juodagalvis et al. [67], Kumar et al. [64], and Siiskonen et al. [60] (see the discussions above) have been plotted against the background (the hatched region of **Figure 3**) of the results of the pnQRPA analyses performed in section 5.2. Looking at the figure makes it obvious that the ISM results of the aforementioned references are commensurate with the results of the (global) analyses of Gamow-Teller transitions performed in the framework of the pnQRPA.

Finally, it is of interest to point out to the recent work [132] where no-core-configuration-interaction formalism, rooted in multireference density functional theory, was used to compute the Gamow-Teller NMEs for T = 1/2 mirror nuclei (pairs of nuclei where either a neutron or a proton is added to an eveneven N = Z core nucleus) in the 1s0d and 1p0f shells. The computations were performed in a basis of 10 or 12 spherical harmonic-oscillator shells by using two different Skyrme forces. The computed quenching factors coincide surprisingly closely with those of the ISM quoted in (25) (Wildenthal et al. [62]) for the 1s0d shell and in (26) (Martinez-Pinedo et al. [63]) for the 1p0f shell, despite the big differences in the two nuclear models. This would point to the possibility that the quenching in the 1s0d and 1p0f shells is not so much related to the deficiencies of the nuclear models but rather to omission of effects coming from the nuclear medium, like from the two-body currents and other mesonic effects discussed in section 3.

### 5.2. Quasiparticle Random-Phase Approximation

Only recently the important aspect of the effective value of g<sup>A</sup> has been addressed within the framework of the pnQRPA. The situation with pnQRPA is more involved than in the case of the ISM since the adopted schematic or realistic interactions are usually renormalized separately in the particle-hole (gph parameter) and particle-particle (gpp parameter) [133–136] channels. Typically the particle-hole parameter, gph, is fitted to reproduce the centroid of the Gamow-Teller giant resonance (GTGR) obtained from the semi-empirical formula [135, 136]

$$
\Delta E\_{\rm GT} = E(1\_{\rm GTGR}^{+}) - E(0\_{\rm gs}^{+}) = \left[1.444\left(Z + \frac{1}{2}\right)A^{-1/3}\right]$$

$$- 30.0 (N - Z - 2)A^{-1} + 5.57 \right] \text{MeV}.\tag{38}$$

The above formula indicates that the difference 1EGT between the GTGR and the ground state of the neighboring even-even reference nucleus depends on the proton and neutron numbers (Z, N) of the reference nucleus, as well as on its mass number. For the particle-particle parameter, gpp, there is no unique way to fix its value, as criticized in Suhonen [137]. Furthermore, the exact value of gpp depends on the size of the active single-particle model space. In this review several ways how this can be done are discussed. As a result of the gpp problems and problems with systematic renormalization of the two-body interactions, the fundamental quenching (see section 3) cannot be disentangled from the nuclear-model effects, discussed in section 4.

The first pnQRPA attempts were inspired by a simultaneous description of β and 2νββ decays, as elaborated more in section 10. In Delion and Suhonen [138] 9 isobaric systems, with A = 70, 78, 100, 104, 106, 110, 116, 128, 130, of the type displayed in the right panel of **Figure 1** were analyzed by using a spherical pnQRPA with schematic particle-hole and particleparticle forces. The pnQRPA calculations were performed in the even-even reference nuclei. For each GTGR-fixed gph the value of gpp was varied in order to reproduce the experimentally known ratio MR/M<sup>L</sup> which is independent of the value of gA. The value of g<sup>A</sup> was then determined by requiring MR(th)/MR(exp) = 1. This produced the mean value

$$\lg\_{\text{A}}^{\text{eff}} = 0.27\tag{39}$$

and the approximate mass dependence gpp ≈ 0.5/ √ A. By using this dependence of gpp and the above value (39) for g<sup>A</sup> the experimental β +/EC and β − NMEs of 218 Gamow-Teller transitions were quite well reproduced in Delion and Suhonen [138]. The quite low value obtained for g eff A implies that a larger quenching is required than in the ISM due to the simple schematic form of the adopted Hamiltonian in the pnQRPA calculations. in other words, the quenching coming from the many-body effects is stronger for the pnQRPA calculation than for the ISM calculation which is more realistic in terms of twobody interactions and configuration space. In this analysis the effects coming from the nuclear medium (section 3) cannot be disentangled from the many-body effects, unfortunately.

In Pirinen and Suhonen [139] an analysis of 26 β − and 22 β +/EC Gamow-Teller transitions of the type depicted in **Figure 1** in the mass range A = 100 − 136 was performed. In this study the geometric mean

$$
\bar{M}\_{\rm GT} = \sqrt{M\_{\rm L} M\_{\rm R}} \tag{40}
$$

of the extracted experimental NMEs was compared with that computed by the use of the pnQRPA with realistic effective forces based on the gph- and gpp-renormalized Bonn-A G matrix. The use of the geometric mean of the left and right NMEs stabilizes the values of the mean NMEs and smoother trends can be obtained. This is based on the fact that the NME for the β − branch is a decreasing function of gpp and the NME for the β <sup>+</sup>/EC branch is an increasing function of gpp. Thus, the product of the NMEs of these branches remains essentially constant over a wide range of gpp values (see the figures in Ejiri and Suhonen [140]).

Like in Delion and Suhonen [138], the pnQRPA calculations of Pirinen and Suhonen [139] were performed in the eveneven reference nuclei. The value of gph was fixed by the phenomenological centroid (38) of the GTGR separately for each nucleus. In the calculations it turned out that the value gpp = 0.7 represents a reasonable global value for the particleparticle interaction strength in the model spaces used in the calculations: at least one oscillator major shell above and below those oscillator shells where the proton and neutron Fermi surfaces lie. Furthermore, an average piece-wise linear behavior

$$\mathcal{g}\_{\rm A}^{\rm eff} = \begin{cases} 0.02A - 1.6 \text{ for } A \le 120 \\ \frac{1}{60}A - \frac{43}{30} & \text{for } A \ge 122 \end{cases} \tag{41}$$

of g<sup>A</sup> was found in the calculations. These derived values of g eff A , plotted in **Figure 2**, were used, in turn, to describe the Gamow-Teller and 2νββ decay rates to the ground state and lowest excited states in the even-even reference nucleus in the A = 100 − 136 mass region. These results were compared with those obtained by the use of the average value

$$g\_{\rm A}^{\rm eff}(\text{ave}) = 0.6\tag{42}$$

for g eff A . The average value reproduced surprisingly well the experimentally known 2νββ half-lives in this mass region.

The work of Pirinen and Suhonen [139] was extended in Deppisch and Suhonen [141] to a wider range of nuclei (A = 62 − 142) and to a more refined statistical analysis of the results. The same renormalized Bonn-A G matrix as in Pirinen and Suhonen [139] was adopted for the pnQRPA calculations, along with the scaling with the gph and gpp parameters. A Markov chain Monte Carlo statistical analysis of 80 Gamow-Teller transitions

(41) and the "fundamentally" quenched *g*A, Equation (14), are plotted for

in 47 isobaric decay triplets of the kind depicted in **Figure 1** was performed. The analysis was also extended to 28 longer isobaric chains and the results were compared with those obtained for the isobaric triplets. Also the measured half-lives of 2νββ decays occurring in the isobaric chains were analyzed. A roughly linearly increasing trend of g eff A as a function of the mass number A could be extracted from the analysis of the isobaric triplets for A ≥ 100, in accordance with the result of Pirinen and Suhonen [139]. Similar features were seen also in the fits to longer multiplets. For the range 100 ≤ A ≤ 136 the average (42) was roughly obtained in both analyses.

In contrast to Pirinen and Suhonen [139] also the value of gpp was kept as a free parameter, the same for the left and right NMEs of transitions in the triplets like in **Figure 1**, and different for each even-even reference nucleus in the longer chains. Both types of analysis yield a rough average of gpp ≈ 0.7 for the particle-particle strength parameter in the mass range 100 ≤ A ≤ 136 (see the last column of **Table 2**), in accordance with the value used in the analysis of Pirinen and Suhonen [139]. At this point it should be noted that the adopted single-particle model spaces used in the calculations correspond to those of Pirinen and Suhonen [139] for 100 ≤ A ≤ 136: at least one oscillator major shell above and below those oscillator shells where the proton and neutron Fermi surfaces lie.

A slightly different analysis of the Gamow-Teller transitions in the mass range 62 ≤ A ≤ 142 was carried out in Ejiri and Suhonen [140]. This is the same mass range as analyzed in Deppisch and Suhonen [141]. Again the gph- and gpp-renormalized Bonn-A G matrix was used in a pnQRPA framework, and the geometric mean (40) was used in the analysis to smooth the systematics. The mass range was divided

comparison.


TABLE 2 | Mass ranges, the corresponding leading *pn* configurations and average effective values (45) of *g*A extracted from three different works.

*The numbers of column 4 (5) are obtained from fits to isobaric triplets (multiplets). The last column shows the averaged values of gpp deduced from Deppisch and Suhonen [141]. In all studies the same single-particle model spaces have been used (see the text). The second last line shows the averages in the mass interval* 98 ≤ *A* ≤ 136*. In all analyses the arithmetic mean with a standard deviation from it has been given.*

in 5 sub-ranges according to the leading proton-neutron (pn) configuration influencing the Gamow-Teller decay rate. The reduction of the NME in the chain M¯ qp → M¯ pnQRPA → M¯ exp was followed, where M¯ qp is the mean two-quasiparticle NME (40) for the leading pn configuration, M¯ pnQRPA is the pnQRPAcomputed mean NME, and M¯ exp is the mean experimental NME, extracted from the experimental decay-half-life data by using g free <sup>A</sup> = 1.27. The ratio

$$k = \frac{\bar{M}\_{\text{pnQRPA}}}{\bar{M}\_{\text{qp}}} \tag{43}$$

is a measure of the quenching of the NME when going from a rudimentary many-body approach toward a more sophisticated one. This ratio is independent of the nuclear-matter effects and is usually nuclear-mass dependent (the results of the analysis [140] are quoted in the second column of **Table 4** in section 7). The quenching of g<sup>A</sup> by the nuclear-medium and many-body (inseparable!) effects was incorporated in the ratio

$$k\_{\rm NM} = \langle \bar{M}\_{\rm exp} / \bar{M}\_{\rm pnQRPA} \rangle,\tag{44}$$

representing an average of the ratio M¯ exp/M¯ pnQRPA of the experimental NME and the pnQRPA-computed NME over each sub-range of masses. The resulting effective g<sup>A</sup> can be extracted from kNM by using the simple relation

$$\mathbf{g}\_{\rm A}^{\rm eff} = \mathbf{g}\_{\rm A}^{\rm free} k\_{\rm NM} = 1.27 k\_{\rm NM}.\tag{45}$$

The resulting values of g eff A , along with the mass ranges and leading pn configurations are listed in **Table 2**. The pnQRPA results were obtained by fitting the gph parameter to the phenomenological centroid (38) of the GTGR separately for each nucleus, and by adopting gpp = 0.67, in line with the analyses of Pirinen and Suhonen [139] and Deppisch and Suhonen [141]. Again, the adopted single-particle model spaces correspond to those of Deppisch and Suhonen [141]: at least one oscillator major shell above and below those oscillator shells containing the proton and neutron Fermi surfaces.

In **Table 2** also the averaged results of Deppisch and Suhonen [141] and Pirinen and Suhonen [139] are shown for comparison. For Deppisch and Suhonen [141] are shown the results of both the isobaric triplet (tripl.) and multiplet (mult.) fits, as also the averaged gpp values, extracted from the analysis of the triplet fits of Deppisch and Suhonen [141]. In all the analyses the same single-particle model spaces were used: at least one oscillator major shell above and below those oscillator shells containing the proton and neutron Fermi surfaces. The triplet and multiplet fits of Deppisch and Suhonen [141] are quite consistent, excluding the multiplet fit of mass range 78–82 (the result in parenthesis) which has two fitted multiplets, the other rendering an ambiguous result. The results of Deppisch and Suhonen [141] are very close to those of Pirinen and Suhonen [139] and Ejiri and Suhonen [140]. Most of the (quite small) differences between the various calculations stem from the different ways of treating the value of the particle-particle strength gpp, which for the studies of Ejiri and Suhonen [140] and Pirinen and Suhonen [139] was kept constant (gpp = 0.67 and gpp = 0.7, respectively) but was allowed to vary in the work of Deppisch and Suhonen [141] (see the last column of **Table 2**).

The numbers of **Table 2** have been visualized in **Figure 2**. Also the linear fit (41) and the "fundamentally" quenched gA, Equation (14), are plotted for comparison. The plot reveals quite a simple structure of the ranges of g eff <sup>A</sup> within different mass regions. The numbers of Ejiri et al. [140] are given as dark-hatched regions while the light-hatched regions contain the results of Ejiri et al. plus the results of Pirinen et al. [139] and Deppisch et al. [141]. A general decreasing trend of the ranges of g eff A (the hatched boxes) can be seen, except for the heaviest masses A ≥ 138. It is noteworthy that there is a small shift in the values of g eff A at A = 120 indicated by all the pnQRPA analyses (both light and hatched boxes). Also the linear fit (41) indicates a discontinuity close to this mass number. The most probable cause for this displacement is the change in the nuclear wave functions from the 0g-orbital dominated to the 1d-orbital dominated proton-neutron configuration, as seen in **Table 2**. A similar, even more drastic, displacement is seen between A = 70–78 where the dominating proton-neutron configuration of the nuclear wave functions shifts from the 1p orbitals to the 0g orbitals.

The obtained pnQRPA ranges can be compared with results obtained by performing combined g eff A analyses of β and 2νββ

decay rates in the pnQRPA and other models: The lighthatched regions of **Figure 2** have been plotted in **Figure 18** for comparison with the results of section 10. The result of the linear fit (41) is not included in that plot since the hatched regions are a better way to describe the (large) spread of the pnQRPA results for different masses A. This large spread is not perceivable in the linear fit.

In **Figure 3** the light-hatched regions of **Figure 2** (combined results of the pnQRPA analyses) have been plotted as a background against the results of the ISM of section 5.1. As can be seen in the figure, the ISM results and the pnQRPA results are in excellent agreement with each other. This is a non-trivial result considering the quite different premises of these two different calculation frameworks. For the masses A ≥ 138 there is no comparison between the two approaches since mid-shell heavy nuclei, with increasing deformation, are hard to access by the ISM due to an overwhelming computational burden.

### 6. QUENCHING OF G<sup>A</sup> IN FORBIDDEN UNIQUE β DECAYS

The forbidden unique β transitions are the simplest ones that mediate β decays between nuclear states of large angularmomentum difference 1J. In particular, if one of the states is a 0 <sup>+</sup> state, then for a Kth forbidden (K = 1, 2, 3, ...) unique beta decay the angular momentum of the other involved state is J = K + 1. At the same time the parity changes in the odd-forbidden and remains the same in the even-forbidden decays [35]. The change in angular momentum and parity for different degrees of forbiddenness is presented in **Table 3**, and they obey the simple rule

$$(-1)^{\Delta f} \Delta \pi = -1. \quad \text{(Forbidden unique decays)} \qquad \text{(46)}$$

Here it is interesting to note that also the Gamow-Teller decays obey the rule (46) if one of the involved nuclear states has the multipolarity 0+.

#### 6.1. Theoretical Considerations

The theoretical half-lives t1/<sup>2</sup> of Kth forbidden unique β decays can be expressed in terms of reduced transition probabilities BK<sup>u</sup> and phase-space factors fKu. The BK<sup>u</sup> is given by the NME, which in turn is given by the single-particle NMEs and one-body transition densities. Then (for further details see [35])

$$\text{At}\_{1/2} = \frac{\kappa}{f\_{\text{Ku}}B\_{\text{Ku}}}; \quad B\_{\text{Ku}} = \frac{g\_{\text{A}}^2}{2I\_{\text{i}} + 1} \left| M\_{\text{Ku}} \right|^2,\tag{47}$$

TABLE 3 | Change in angular momentum and parity in *K*th forbidden unique β decays with a 0+ state as an initial or final nuclear state.


where J<sup>i</sup> is the angular momentum of the mother nucleus and κ is a constant with value [142]

$$\kappa = \frac{2\pi^3 \hbar^7 \ln 2}{m\_e^5 c^4 (\text{G}\_\text{F} \cos \theta\_\text{C})^2} = 6147 \text{ s},\tag{48}$$

with G<sup>F</sup> being the Fermi constant and θ<sup>C</sup> being the Cabibbo angle. The phase-space factor fK<sup>u</sup> for the Kth forbidden unique β ± decay can be written as

$$f\_{\rm Ku} = \left(\frac{3}{4}\right)^K \frac{(2K)!!}{(2K+1)!!} \int\_1^{\rm w0} C\_{\rm Ku}(\omega\_\varepsilon) p\_\varepsilon \nu\_\varepsilon (\nu\_0 - \nu\_\varepsilon)^2 F\_0(Z\_f, \nu\_\varepsilon) d\nu\_\varepsilon,\tag{49}$$

where CK<sup>u</sup> is the shape function for Kth forbidden unique β decays which can be written as (see, e.g., [35, 143])

$$C\_{\rm Ku}(\boldsymbol{\omega}\_{\varepsilon}) = \sum\_{k\_{\varepsilon} + k\_{\upsilon} = K + 2} \frac{\lambda\_{k\_{\varepsilon}} p\_{\varepsilon}^{2(k\_{\varepsilon} - 1)} (\boldsymbol{\omega}\_0 - \boldsymbol{\omega}\_{\varepsilon})^{2(k\_{\upsilon} - 1)}}{(2k\_{\varepsilon} - 1)! (2k\_{\upsilon} - 1)!},\tag{50}$$

where the indices k<sup>e</sup> and k<sup>ν</sup> (both k = 1, 2, 3...) come from the partial-wave expansion of the electron (e) and neutrino (ν) wave functions. Here w<sup>e</sup> is the total energy of the emitted electron/positron, p<sup>e</sup> is the electron/positron momentum, Z<sup>f</sup> is the charge number of the daughter nucleus and F0(Z<sup>f</sup> ,we) is the Fermi function taking into account the coulombic attraction/repulsion of the electron/positron and the daughter nucleus<sup>2</sup> . The factor λk<sup>e</sup> contains the generalized Fermi function Fke−<sup>1</sup> [144] as the ratio

$$\lambda\_{k\_{\varepsilon}} = \frac{F\_{k\_{\varepsilon}-1}(Z\_f, \boldsymbol{w}\_{\varepsilon})}{F\_0(Z\_f, \boldsymbol{w}\_{\varepsilon})}. \tag{51}$$

The integration is performed over the total (by electron rest-mass) scaled energy of the emitted electron/positron, w<sup>0</sup> being the endpoint energy, corresponding to the maximum electron/positron energy in a given transition.

The NME in (47) can be expressed as

$$M\_{\rm Ku} = \sum\_{ab} M^{(\rm Ku)}(ab) (\psi\_f || [c\_a^\dagger \tilde{c}\_b]\_{K+1} || \psi\_i), \tag{52}$$

where the factors M(Ku)(ab) are the single-particle matrix elements and the quantities (ψ<sup>f</sup> ||[c † ac˜b ]K+1||ψi) are the one-body transition densities with ψ<sup>i</sup> being the initial-state wave function and ψ<sup>f</sup> the final-state wave function. The operator c † <sup>a</sup> is a creation operator for a nucleon in the orbital a and the operator c˜<sup>a</sup> is the corresponding annihilation operator. The single-particle matrix elements are given (in the Biedenharn-Rose phase convention) by

$$M\_{\rm Ku}(ab) = \sqrt{4\pi} \left( a ||r^K[Y\_K \sigma]\_{K+1} i^K || b \right), \tag{53}$$

where Y<sup>K</sup> is a spherical harmonic of rank K, r the radial coordinate, and a and b stand for the single-particle orbital quantum numbers. The NME (53) is given explicitly in Suhonen [35].

<sup>2</sup>For positron emission the change <sup>Z</sup><sup>f</sup> → −Z<sup>f</sup> has to be performed in <sup>F</sup>0(Z<sup>f</sup> ,we) and Fke−<sup>1</sup> (Z<sup>f</sup> ,we), Equation (51) below.

### 6.2. First-Forbidden Unique β Decays

The first-forbidden unique β transitions are mediated by a rank-2 (i.e., having angular-momentum content 2) parity-changing spherical tensor operator [a special case of the operator (53)], schematically written as O(2−). For these decays it is customary to modify the general structure of Equations (47)–(49) by replacing the phase-space factor fK=1,u of (49) by a 12 times larger phase-space factor f1u, i.e.,

$$f\_{\rm lu} = 12f\_{\rm K=1,u},\tag{54}$$

yielding a factor log 12 = 1.079 times larger comparative halflives (18) than in the standard definition (47).

In the quenching studies it is advantageous to use the simplest first-forbidden transitions, namely the ground-state-toground-state ones. In **Figure 4** are depicted the first-forbidden unique ground-state-to-ground-state β − and β +/EC transitions between even-even 0+ and odd-odd 2− ground states in the A = 84 Kr-Rb-Sr isobaric chain. Shown is the lateral feeding from a middle odd-odd nucleus to adjacent even-even ground states. In the figure, as also in **Figure 1** for the Gamow-Teller transitions, the NME is denoted by M<sup>L</sup> (MR) in case it is to the left (right) of the central nucleus.

In the early work [145] a systematic schematic analysis of the first-forbidden unique β decays was performed from the point of view of suppression factors stemming from the effect of E1 (electric dipole) giant resonance in the final odd-odd nucleus. In Towner et al. [146] the suppression mechanism of the firstforbidden and third-forbidden β decays of light nuclei (A ≤ 50) was studied by using simple shell-model estimates and first-order perturbation theory. The hindrance was traced to the repulsive T = 1 (isospin 1) particle-hole force.

In the work [147] 19 first-forbidden unique ground-state-toground-state β-decay transitions were studied. The interesting transitions are the ones where both M<sup>L</sup> and M<sup>R</sup> NMEs are known experimentally, like in the case of **Figure 4**. The experimental

values of the NMEs can be deduced by using Equations (47) and (48) and by adopting the free value of the axial-vector coupling strength<sup>3</sup> . In this case one can use the geometric mean (40) of the left and right NMEs in the analysis, making the analysis more stable. In Ejiri et al. [147] a gph- and gpp-renormalized Bonn-A G matrix was used as the two-nucleon interaction in a pnQRPA framework. The two-quasiparticle and pnQRPA NMEs were compared with the ones extracted from the measured comparative half-lives. Again the relations (44) and (45) can be used to obtain the value

$$g\_{\rm A}^{\rm eff} \approx 0.45 \times 1.27 = 0.57\tag{55}$$

for the effective axial-vector coupling strength using the pnQRPA wave functions. The average of the values of the leading twoquasiparticle NMEs gives in turn

$$g\_{\rm A}^{\rm eff}(2\,\mathrm{qp}) \approx 0.18 \times 1.27 = 0.23,\tag{56}$$

implying the ratio

$$k = \frac{\bar{M}\_{\text{pnQRPA}}}{\bar{M}\_{\text{qp}}} = 0.4 \tag{57}$$

and thus a drastic nuclear many-body effect when going from the two-quasiparticle level of approximation to the pnQRPA level. The 2qp-NME to pnQRPA-NME comparison is the only one where a clean separation between the nuclear-medium effects and the nuclear-model effects can be achieved, the nuclear-model effect being responsible for the (in this case large) shift in the values of the NMEs.

### 7. HIGHER-FORBIDDEN UNIQUE β DECAYS

Early studies of the quenching in the second- and third-forbidden unique β decays were performed in Towner et al. [146] and Warburton et al. [149]. The work of Towner et al. [146] was discussed in section 6.2. In Warburton et al. [149] these β decays were studied using a simple ISM and the unified model (deformed shell model) for six β transitions in the A = 10, 22, 26, 40 nuclei. The interest for these studies derived from nuclear-structure considerations: how to explain in a nuclear model the hindrance phenomena occurring in certain measured β transitions. Beyond this, the incentive to study the Gamow-Teller (section 5), first-forbidden unique (section 6.2), and higher-forbidden unique (this section) β decays stems from their relation to the Gamow-Teller type of NME involved in 0νββ decays. The 0νββ decays proceed via virtual intermediate states of all multipolarities J <sup>π</sup> due to the multipole expansion of the Majorana-neutrino propagator (see, e.g., [1–3, 150–155]). Studies

<sup>3</sup> In Ejiri et al. [147] the Bohr-Mottelson (BM) formulation [148] of first-forbidden decays is used. The difference between the present and the BM formulation can be crystallized into the following relations: M(BM) = M1u/ √ 4π, B(BM) = B1u/(4πg 2 A ), f1(BM) = 3f1u/4. In addition, since gV(BM) = GFg<sup>V</sup> and gA(BM) = GFgA, one has to make replacements gA(BM) → g<sup>A</sup> and gV(BM) → 1 in order to go from the BM formulation to the present one.

of the quenching of these two-leg ("left-leg" and "right-leg" transitions illustrated in the schematic **Figure 5** for the 0νββ decay of <sup>116</sup>Cd to <sup>116</sup>Sn via the virtual intermediate states in <sup>116</sup>In) virtual transitions is of paramount importance to, e.g., estimate the sensitivities of the present and future neutrino experiments to the Majorana-neutrino mass. The possible quenching of these intermediate multipole transitions in the GT type of 0νββ NME can be, in a simplistic approach, condensed into an effective axial coupling, g eff A,0ν , multiplying the NME:

$$M\_{\rm GTGT}^{(0\upsilon)} = (\mathcal{g}\_{\rm A,0\upsilon}^{\rm eff})^2 \sum\_{f^\pi} (\mathbf{0}\_f^+ || \mathcal{O}\_{\rm GTGT}^{(0\upsilon)} (f^\pi) || \mathbf{0}\_i^+),\tag{58}$$

where O (0ν) GTGT denotes the transition operator mediating the 0νββ transition through the various multipole states J π , 0+ i denotes the initial ground state, and the final ground state is denoted by 0 + f (here, for simplicity, we assume a ground-state-to-groundstate transition). The effective axial coupling relevant for 0νββ decay is denoted as g eff A,0ν to emphasize that its value may deviate from the one determined in single beta and 2νββ decays. The remarkable feature of Equation (58) is that the effective axial coupling strength is raised to 2nd power making the value of g eff A,0ν play an extremely important role in determining the 0νββ-decay rate which is (neglecting the smaller double Fermi and tensor contributions) proportional to the squared NME and thus to the 4th power of the coupling:

$$\alpha \text{\textit{0}} \boldsymbol{\upnu} \boldsymbol{\upbeta} \boldsymbol{\upbeta} - \text{rate} \sim \left| \boldsymbol{M}\_{\text{GTGT}}^{\text{(0\upsilon)}} \right|^2 = \boldsymbol{g}\_{\text{A,0\upsilon}}^4 \left| \sum\_{f^{\pi}} \text{(\hspace{0}}^+\_f ||\mathcal{O}\_{\text{GTGT}}^{\text{(0\upsilon)}} (f^{\pi})|| \mathbf{0}\_i^+ \right|^2 . \tag{59}$$

The quenching related to the left-leg and right-leg β transitions of **Figure 5** can be studied by using the theoretical machinery

of section 6.1. In Kostensalo and Suhonen [156] this machinery was applied to 148 potentially measurable second-, third-, fourth- , fifth-, sixth- and seventh-forbidden unique beta transitions. The calculations were done using realistic single-particle model spaces and G-matrix-based microscopic two-body interactions. The results of Kostensalo and Suhonen [156] could shed light on the magnitudes of the NMEs corresponding to the highforbidden unique 0<sup>+</sup> ↔ J <sup>π</sup> <sup>=</sup> <sup>3</sup> +, 4−, 5+, 6−, 7+, 8− virtual transitions taking part in neutrinoless double beta decay, as shown in **Figure 5**.

In Kostensalo and Suhonen [156] the ratio k, Equation (62) below, of the NMEs, calculated by the pnQRPA, MpnQRPA, and a two-quasiparticle model, Mqp, was studied and compared with earlier calculations for the allowed Gamow-Teller 1+ [140] and first-forbidden spin-dipole (SD) 2− [147] transitions. Based on this comparison the expected half-lives of the studied β-decay transitions were predicted. An example case of the expected half-lives of second-, fourth-, and seventh-forbidden β decays is shown in **Figure 6**. The computed NMEs are corrected by the use of the ratio of the geometric means (40) of the experimental and pnQRPA NMEs,

$$k\_{\rm NM} = \frac{\bar{M}\_{\rm exp}}{\bar{M}\_{\rm pnQRPA}},\tag{60}$$

extracted from the GT work of Ejiri and Suhonen [140], to predict the transition half-lives of the figure. In the figure one sees that the expected half-lives range from 4 years to the astronomical 9 <sup>×</sup> <sup>10</sup><sup>29</sup> years. It is expected that the decays to and from isomeric states are not measurable and the decays between the nuclear ground states are masked by transitions to excited states with lesser degree of forbiddenness. Only in some cases the high-forbidden β decay exhausts 100% of the decay rate between two nuclear ground states; one example being the second-forbidden β <sup>−</sup> transition <sup>54</sup>Mn(3<sup>+</sup> gs) <sup>→</sup> <sup>54</sup>Fe(0<sup>+</sup> gs), with a half-life 4.2(9) <sup>×</sup> <sup>10</sup><sup>5</sup> years, shown in **Figure 7**. Even in this case the measurement will be challenging due to the Gamow-Teller type of electron-capture feeding of the first excited 2+ state of <sup>54</sup>Cr, taking practically 100% of the feeding intensity.

The geometric mean of the EC/β + and β − NMEs, defined in (40), can be generalized to a geometric mean of n NMEs, M<sup>i</sup> , i = 1, 2, ... n, of successive β transitions with a common mother or daughter nucleus:

$$
\bar{M} = \left(\prod\_{i=1}^{n} M\_i\right)^{1/n}.\tag{61}
$$

Here the aim, as in the case of (40), is to reduce the fluctuations in the computed NMEs by exploiting the compensating trends of the β − and β +/EC branches of decay when changing the value of the particle-particle interaction parameter gpp of the pnQRPA. One can now define the ratio

$$k = \frac{\bar{M}\_{\text{pnQRPA}}}{\bar{M}\_{\text{qp}}} \tag{62}$$

FIGURE 6 | Predicted half-lives and their error estimates (in parenthesis) for β <sup>−</sup> and EC (electron-capture) transitions in the isobaric chain *A* = 116. The spin-parity assignments, decay energies (*Q* values) and life-times of the nuclear ground (gs) and isomeric (isom) states are experimental data and taken from ENSDF (http://www. nndc.bnl.gov/). The 2νββ half-life is taken from Barabash [157]. In addition to the half-lives the degree of forbiddenness and the leading single-particle transition are shown.

of the pnQRPA-calculated mean NME, M¯ pnQRPA, and the mean two-quasiparticle NME, M¯ qp, computed by using (61). The ratio k is a measure of the evolution of the nuclear-model dependent many-body effects on the computed NME. The ratio (62) is independent of the nuclear-medium effects (the fundamental quenching of section 3) and gives an idea of how the quenching of g<sup>A</sup> depends on the degree of complexity of the adopted nuclear model.

In Kostensalo and Suhonen [156] the β transitions were divided in two groups: GROUP 1 contained only non-magic even-even reference nuclei (i.e., nuclei where the pnQRPA and the associated BCS (Bardeen-Cooper-Schrieffer) calculation were performed), whereas GROUP 2 contained (semi)magic reference nuclei. The transitions in GROUP 2 were left out from the analysis of the ratio k of (62) since the BCS results tend to be unstable at magic shell closures. In **Figure 8** the ratio k is shown for transitions belonging to GROUP 1. The same k distribution is shown in terms of division to β − and EC/β + decays in **Figure 9**.

From **Figure 8** it is visible that the second- and fourthforbidden β transitions are distributed to masses below A = 62 and above masses A = 92, whereas the third-forbidden decays occupy the mass range 74 ≤ A ≤ 90. The sixth-forbidden decays occur within the range 92 ≤ A ≤ 110 and the seventh-forbidden decays occur above A = 116. The fifth-forbidden decays occur in a scattered way above A = 84. From **Figure 9** one observes that most of the β <sup>−</sup> decays are concentrated above mass A = 118 where also quite low values of k can be obtained. The EC/β + decays, on the other hand, are more concentrated in the middle-mass region 82 ≤ A ≤ 116.

**Figure 8** suggests that the values of the ratio (62) can be classified in terms of three mass regions, namely A = 50 − 88 (k∼0.4), A = 90 − 120 (values of k have a scattered, decreasing trend), and A = 122 − 146 (a low-k region with k∼0.2). The ratios k for the three mass regions and for various degrees of forbiddenness K are shown in **Table 4** for β transitions belonging to GROUP 1. A comparison is made to the GT results of Ejiri and Suhonen [140] and SD results of Ejiri et al. [147]. The ratios are also plotted in **Figure 10** for illustrative purposes. In the figure

color and shape of the symbol.


TABLE 4 | Ratio (62) for three mass regions and for various degrees of forbiddenness *K* for β transitions belonging to GROUP 1.

*The results of the Gamow-Teller (GT, see section 5.2) and first-forbidden (K* = 1*, see section 6.2) decays are quoted for comparison.*

one can see that the trend, in terms of the mass number A, is a bit different for the Gamow-Teller and the forbidden (K ≥ 2) transitions. For most of the forbidden transitions, namely K = 3, 4, 5, k has a decreasing tendency as a function of A, in particular for the k = 4 transitions. For K = 2 and K = 7 a slightly increasing tendency is observed. It seems, on average, that the quenching of the forbidden transitions is somewhat stronger than that of Gamow-Teller transitions in the mass region A = 90 − 146.

The numbers in **Table 4** suggest that, in the gross, k is independent of the degree of forbiddenness and thus the (low-energy) forbidden unique contributions [obeying the simple rule (46)] to the 0νββ NME (59) should be roughly uniformly quenched. If these conclusions can be generalized to include also the non-unique β transitions, obeying the rule (−1)1J1π = +1, one can then speak about an effective axial coupling, g eff A,0ν , in front of the 0νββ NME in (58), at least for low intermediate excitation energies. The quenching for these low intermediate excitation energies could then be deduced from the hatched regions of **Figure 2**, implying the effective axial couplings listed in **Table 5** for the three mass regions of interest for 0νβ−β −-decay calculations in the pnQRPA framework. At this point it has to be noted that a "low" excitation energy is still an undefined notion that has to be investigated in future works.

Finally, it should be stressed that the use of the low-energy effective axial coupling, g eff A,0ν , is particular to the pnQRPA many-body framework and reflects the deficiencies of pnQRPA in calculating the magnitudes of the NMEs of the allowed and forbidden unique β-decay transitions. It is not directly related to the more fundamental quenching of the axial-vector

TABLE 5 | Values of the low-energy effective axial coupling, *g* eff A,0ν of (58), for the three mass regions of interest for 0νβ−β −-decay calculations in the pnQRPA framework.


coupling strength gA, related to the meson-exchange currents, delta isobars, two-body weak currents, etc., discussed in section 3, but it is rather a nuclear-model effect, discussed in section 4.

### 8. QUENCHING OF G<sup>A</sup> IN FORBIDDEN NON-UNIQUE β DECAYS

The general theory of forbidden beta decays is outlined in Behrens and Buhring [144] and Schopper [158]. Streamlined version of those is given in Mustonen et al. [159].

#### 8.1. Theoretical Considerations

In the forbidden non-unique β decay the half-life can be given, analogously to (47), in the form

$$t\_{1/2} = \kappa / \bar{C},\tag{63}$$

where C˜ is the dimensionless integrated shape function, given by

$$\tilde{\mathbf{C}} = \int\_{1}^{\omega\_{0}} \mathbf{C}(\boldsymbol{\omega}\_{\varepsilon}) p \boldsymbol{\omega}\_{\varepsilon} (\boldsymbol{\omega}\_{0} - \boldsymbol{\omega}\_{\varepsilon})^{2} \mathbf{F}\_{0}(\mathbf{Z}\_{f}, \boldsymbol{\omega}\_{\varepsilon}) \mathrm{d}\boldsymbol{\omega}\_{\varepsilon}, \tag{64}$$

with the notation explained in section 6.1. The general form of the shape factor of Equation (64) is a sum

$$\begin{aligned} C(\boldsymbol{w}\_{\varepsilon}) &= \sum\_{k\_{\boldsymbol{\varepsilon}},k\_{\boldsymbol{\nu}},K} \lambda\_{k\_{\boldsymbol{\varepsilon}}} \Bigg| M\_{K}(k\_{\boldsymbol{\varepsilon}},k\_{\boldsymbol{\nu}})^2 \\ &+ m\_{K}(k\_{\boldsymbol{\varepsilon}},k\_{\boldsymbol{\nu}})^2 - \frac{2\gamma\_{k\_{\boldsymbol{\varepsilon}}}}{k\_{\boldsymbol{\varepsilon}}\omega\_{\varepsilon}} M\_{K}(k\_{\boldsymbol{\varepsilon}},k\_{\boldsymbol{\nu}}) m\_{K}(k\_{\boldsymbol{\varepsilon}},k\_{\boldsymbol{\nu}}) \Bigg], \end{aligned} (65)$$

where the factor λk<sup>e</sup> was given in (51) and Z<sup>f</sup> is the charge number of the final nucleus. The indices k<sup>e</sup> and k<sup>ν</sup> (k = 1, 2, 3...) are related to the partial-wave expansion of the electron (e) and neutrino (ν) wave functions, K is the order of forbiddenness of the transition, and γk<sup>e</sup> = q k 2 <sup>e</sup> − (αZ<sup>f</sup> ) 2 , α ≈ 1/137 being the fine-structure constant. The nuclear-physics information is hidden in the factors MK(ke, k<sup>ν</sup> ) and mK(ke, k<sup>ν</sup> ), which are complicated combinations of the different NMEs and leptonic phase-space factors. For more information on the integrated shape function, see [144, 159].

The quite complicated shape factor (65) can be simplified in the so-called ξ approximation when the coulomb energy of the emitted β particle at the nuclear surface is much larger than the endpoint energy, i.e., ξ = αZ<sup>f</sup> /2R ≫ w0, where R is the nuclear radius. Then the forbidden non-unique transition can be treated as a unique one of the same 1J. Applicability of this approximation has recently been criticized in Mougeot [160].

### 8.2. First-Forbidden Non-unique β Decays

For the first-forbidden non-unique β decays the shape factor (65) has to be supplemented with a 1J = |J<sup>i</sup> − J<sup>f</sup> | = 0 term C (1)(we) [144, 158, 161, 162], where J<sup>i</sup> (J<sup>f</sup> ) is the initial-state (final-state) spin of the mother (daughter) nucleus. Then the shape factor can be cast in the simple form [144, 158, 163]

$$C(\boldsymbol{w}\_{\boldsymbol{e}}) = K\_0 + K\_1 \boldsymbol{w}\_{\boldsymbol{e}} + K\_{-1}/\boldsymbol{w}\_{\boldsymbol{e}} + K\_2 \boldsymbol{w}\_{\boldsymbol{e}}^2,\tag{66}$$

where the factors K<sup>n</sup> contain the NMEs (6 different, altogether) of transition operators O of angular-momentum content (rank of a spherical tensor) O(0−), O(1−), and O(2−), where the parity indicates that the initial and final nuclear states should have opposite parities according to **Table 3**. In the leading order these operators contain the pieces [148]

$$\mathcal{O}(0^{-}): \mathbf{g}\_{\rm A}(\boldsymbol{\nu}^{\,5}) \frac{\boldsymbol{\sigma} \cdot \mathbf{p}\_{\rm e}}{M\_{\rm N}}; \ \mathrm{ig}\_{\rm A} \frac{\alpha Z\_{\rm f}}{2R}(\boldsymbol{\sigma} \cdot \mathbf{r}),\tag{67}$$

$$\mathcal{O}(1^{-}): \mathbb{g}\_{\text{V}} \frac{\mathbf{p}\_{\text{t}}}{M\_{\text{N}}}; \text{g}\_{\text{A}} \frac{\alpha Z\_{\text{f}}}{2R} (\boldsymbol{\sigma} \times \mathbf{r}); \text{ig}\_{\text{V}} \frac{\alpha Z\_{\text{f}}}{2R} \mathbf{r},\tag{68}$$

$$\mathcal{O}(2^{-}): \frac{\mathrm{i}}{\sqrt{3}} \mathrm{g}\_{\mathrm{A}} \, [\sigma \, \mathbf{r}]\_{2} \sqrt{\mathbf{p}\_{\varepsilon}^{2} + \mathbf{q}\_{\nu}^{2}},\tag{69}$$

where **p**<sup>e</sup> (**q**<sup>ν</sup> ) is the electron (neutrino) momentum, **r** the radial coordinate, and the square brackets in (69) denote angularmomentum coupling. The matrix elements of the operators (67) and (68) are suppressed relative to the Gamow-Teller matrix elements by the small momentum **p**<sup>e</sup> of the electron and the large nucleon mass M<sup>N</sup> or the small value of the fine-structure constant α. The matrix element of (69) is suppressed by the small electron and neutrino momenta. The axial operator σ · **p**<sup>e</sup> and vector operator **r** trace back to the time component of the axial current A <sup>µ</sup> in (4) and vector current V <sup>µ</sup> in (3), and the rest of the operators stem from the space components of V <sup>µ</sup> and A <sup>µ</sup>. The renormalization of these pieces is discussed next.

The ξ approximation to the first-forbidden non-unique transitions has been discussed, e.g., [144, 148, 158]. One of the first analyses of first-forbidden non-unique transitions in this approximation was done in Bohr and Mottelson [148] for nuclei around <sup>208</sup>Pb, based on the work of Damgaard and Winther [164]. Assuming certain dominant single-particle configurations around the double-closed shell at A = 208, Bohr and Mottelson obtained two sets of values for the effective vector and axialvector coupling when analyzing the decay rates mediated by the rank-1 operators O(1−) in (68). Combining the two obtained values we obtain

$$\mathcal{g}\_{\rm A}^{\rm eff}(\rm sp) = (0.5 - 0.6) \times 1.18 = 0.46 - 0.56,\tag{70}$$

where the symbol sp refers to single-particle estimate for the states involved in the β decays in odd-A nuclei. It is interesting that also an effective value for the vector coupling was derived:

$$g\_V^{\text{eff}}(\text{sp}) = 0.3 - 0.7.\tag{71}$$

This deviates quite much from the canonical value g<sup>V</sup> = 1 dictated by the CVC hypothesis [26]. Hence, strong nuclearmodel dependent effects are recorded in this case. In the case of the axial-vector strength the numbers of (70) can be compared with the ones extracted from the first-forbidden unique decays in the two-quasiparticle approximation for odd-odd nuclei. There, in Equation (56), a value g eff A (2qp) ∼ 0.2 was obtained, implying that for the odd-odd systems the quenching is more drastic than for the odd-mass systems. All in all, a proper many-body treatment should reduce the quenching markedly, as shown by the factor k in (57), describing the transition from the twoquasiparticle approximation to the pnQRPA level in the case of the unique-forbidden β transitions.

In Ejiri et al. [145] a schematic study of the six NMEs corresponding to the operators (67)–(69) was performed. The hindrance factors associated with the NMEs were related to the E1 (electric dipole) giant resonance in a semi-quantitative way. The nuclear medium effect, in the form of the meson-exchange currents, on the <sup>σ</sup> · **<sup>p</sup>**<sup>e</sup> part of <sup>O</sup>(0−) in (67) was discussed in Kubodera et al. [42], Kirchbach and Reinhardt [79] and Towner [81]. This is the well-known (fundamental) enhancement of the γ <sup>5</sup> NME (axial charge ρ5, the time component of the axial current, see section 3), stemming from the renormalization of the pion-decay constant and the nucleon mass M<sup>N</sup> in nuclear medium [165] and exchange of heavy mesons [80, 81]. In this review the corresponding coupling strength is coined g eff A (γ 5 ) for short. In Kirchbach and Reinhardt [79] a simple nuclear approach to the meson-exchange renormalization g eff A (γ 5 ) = (1 + δ)g free A gave the following values of g eff A (γ 5 ) (below are given the studied nuclear masses and the corresponding active single-particle transitions):

$$\begin{aligned} \text{g}\_{\text{A}}^{\text{eff}}(\text{\textdegree }) &= 1.90 \quad A = 16 \quad \text{(1}s\_{1/2} \rightarrow 0 \text{p}\_{1/2}\text{)}\\ \text{g}\_{\text{A}}^{\text{eff}}(\text{\textdegree }) &= 1.96 \quad A = 18 \quad \text{(1}s\_{1/2} \rightarrow 0 \text{p}\_{1/2}\text{)}\\ \text{g}\_{\text{A}}^{\text{eff}}(\text{\textdegree }) &= 1.84 \quad A = 96 \quad \text{(2}s\_{1/2} \rightarrow 1 \text{p}\_{1/2}\text{)}\\ \text{g}\_{\text{A}}^{\text{eff}}(\text{\textdegree }) &= 1.78 \quad A = 206 \text{ (2}\text{p}\_{1/2} \rightarrow 2 \text{p}\_{1/2}\text{)} \end{aligned} \tag{72}$$

The work of Kirchbach and Reinhardt [79] was extended by Towner [81] to include 6 nuclear masses and several singleparticle transitions for each mass. The resulting renormalization by the meson-exchange currents amounted to

$$\lg\_{\text{A}}^{\text{eff}}(\nu^{\text{5}}) = 2.0 - 2.3 \quad \text{(A} = 16 - 208\text{)}\tag{73}$$

for the masses A = 16 − 208.

The above fundamental renormalization of the axial charge was contrasted with the nuclear-model dependent many-body effects by using the framework of the interacting shell model in several studies in the past. For very low masses, A = 11 [166] and A = 16 [167], some 40 − 50% enhancement of the axial charge was obtained leading to g eff A (γ 5 ) = 1.8−1.9. A further study [168] of the A = 11 − 16 nuclei indicated an enhanced axial charge of

$$g\_{\rm A}^{\rm eff}(\gamma^5) = 2.04 \pm 0.04, \quad \text{(\$A = 11 - 16)}\tag{74}$$

where the uncertainties come solely from the experimental errors, not from the uncertainties associated with the theoretical analyses. A general study of the first-forbidden non-unique decays was carried on in Warburton et al. [169] for 34 ≤ A ≤ 44, and a further comparison [170] with the measured rate of the β − decay of <sup>50</sup>K indicated an enhanced value of

$$g\_{\rm A}^{\rm eff}(\nu^5) = 1.93 \pm 0.09, \quad (A = 50) \tag{75}$$

where the uncertainty is purely experimental.

A thorough shell-model treatment of the mass A = 205 − 212 nuclei in the lead region was carried out in Warburton [171–173]. There a rather strongly enhanced value of

$$\lg\_{\text{A}}^{\text{eff}}(\text{y}^{5}) = 2.55 \pm 0.07 \quad \text{(\$A = 205 - 212)}\tag{76}$$

was obtained for the axial charge. The uncertainty comes from the least-squares fit to 18 measured β-decay transitions in the indicated mass region. For the σ · **r** operator (space component of A <sup>µ</sup>) essentially no renormalization (quenching, since space components tend to be quenched opposite to the enhancement of the time component, see beginning of section 3) was obtained: gA/g free A (0−) = 0.97 ± 0.06. The value (76) is notably larger than those obtained for the lower masses and also larger than the leadregion results of Towner (73). However, in Kubodera and Rho [165] the theoretical result

$$g\_{\rm A}^{\rm eff}(\gamma^5) = 2.5 \pm 0.3 \quad \text{(\$A = 205-212)}\tag{77}$$

was obtained by adopting an effective Lagrangian incorporating approximate chiral and scale invariance of QCD. This seems to confirm the phenomenological result of Warburton [172, 173]. For further information see the review [174].

For the <sup>O</sup>(1−) operator <sup>σ</sup> <sup>×</sup> **<sup>r</sup>** in (68) the analyses of Warburton [171, 173] yielded the effective values

$$\text{g}\_{\text{A}}^{\text{eff}} (1^{-}) \sim 0.6; \quad \text{g}\_{\text{V}} (1^{-}) \sim 0.6 \quad \text{(Warburton)} \tag{78}$$

due to core-polarization effects caused by the limited model space used. In the work Rydstrom [175] a shell-model study of the firstforbidden transition <sup>205</sup>Tl(1/2 + gs) <sup>→</sup> <sup>205</sup>Pb(1/<sup>2</sup> −) yielded the effective values

$$\mathrm{g}\_{\mathrm{A}}^{\mathrm{eff}}(1^{-}) \sim 0.43 - 0.65; \; \mathrm{g}\_{\mathrm{V}}(1^{-}) \sim 0.38 - 0.85. \; \text{(Rydström } et \, al.) \tag{79}$$

The shell-model analysis of Suzuki et al. [163] of the N = 126 isotones suggests a large quenching for g eff A (1−) but a large quenching of g eff V (1−) is not necessarily needed for most of the studied cases, contrary to (78) and (79), in accordance with the CVC hypothesis [26].

In the work [176] half-lives of a number of nuclei at the magic neutron numbers N = 50, 82, 126 were analyzed by comparing results of large-scale shell-model calculations with experimental data. Both Gamow-Teller and first-forbidden β decays were included in the analysis. By performing a least-squares fit to the experimental data the following quenched weak couplings were extracted: For the enhanced γ <sup>5</sup> matrix element the value g eff A (γ 5 ) = 1.61 was obtained and for the σ · **r** part the quenching gA/g free A (0−) = 0.66 was obtained. For the 1<sup>−</sup> part the quenched values read

$$\text{g}\_{\text{A}}^{\text{eff}}(1^{-}) \sim 0.48; \quad \text{g}\_{\text{V}}(1^{-}) \sim 0.65. \quad \text{(Zhi et al.)}\tag{80}$$

Interestingly, also for the first-forbidden unique operator O(2−) of (69) a quenching

$$\mathrm{g}\_{\mathrm{A}}^{\mathrm{eff}}(2^{-}) \sim 0.53 \quad \text{(Zhi et al.)}\tag{81}$$

was obtained. This is not far from the result g eff <sup>A</sup> ∼ 0.57 [see Equation (55)] obtained in the analysis of the first-forbidden unique β decays in Ejiri et al. [147].

The above considerations for the vector coupling coefficient g<sup>V</sup> are in conflict with the CVC hypothesis [26] and the findings of [177] where the shape of the computed β-electron spectrum was compared with that of the measured one for the fourthforbidden β <sup>−</sup> decay of <sup>113</sup>Cd. This comparison confirmed an unquenched value g<sup>V</sup> = 1.0 for the vector coupling coefficient, in accordance with the CVC hypothesis. For more discussion of the related method for highly-forbidden β decays, see section 9.

### 9. HIGHER-FORBIDDEN NON-UNIQUE β DECAYS

The shape functions of forbidden non-unique beta decays are rather complex combinations of different NMEs and phase-space factors. Furthermore, their dependence on the weak coupling strengths g<sup>V</sup> (vector part) and g<sup>A</sup> (axial-vector part) is very nontrivial. In fact, the shape factor C(we) (65) can be decomposed into vector, axial-vector and mixed vector-axial-vector parts in the form [177]

$$\mathcal{C}(\boldsymbol{\omega}\_{\varepsilon}) = \mathcal{g}\_{\mathrm{V}}^{2}\mathrm{C}\_{\mathrm{V}}(\boldsymbol{\omega}\_{\varepsilon}) + \mathcal{g}\_{\mathrm{A}}^{2}\mathrm{C}\_{\mathrm{A}}(\boldsymbol{\omega}\_{\varepsilon}) + \mathcal{g}\_{\mathrm{V}}\mathrm{g}\_{\mathrm{A}}\mathrm{C}\_{\mathrm{VA}}(\boldsymbol{\omega}\_{\varepsilon}).\tag{82}$$

Integrating equation (82) over the electron kinetic energy, we obtain an analogous expression for the integrated shape factor (64)

$$
\hat{\mathbf{C}} = \mathbf{g}\_{\text{V}}^{2} \hat{\mathbf{C}}\_{\text{V}} + \mathbf{g}\_{\text{A}}^{2} \hat{\mathbf{C}}\_{\text{A}} + \mathbf{g}\_{\text{V}} \mathbf{g}\_{\text{A}} \hat{\mathbf{C}}\_{\text{VA}},\tag{83}
$$

where the factors C˜ i in Equation (83) are just constants, independent of the electron energy.

In Haaranen et al. [177] it was proposed that the shapes of β-electron spectra could be used to determine the values of the weak coupling strengths by comparing the computed spectrum with the measured one for forbidden non-unique β decays. This method was coined the spectrum-shape method (SSM). In this study also the next-to-leading-order corrections to the β-decay shape factor were included. In Haaranen et al. [177] the β-electron spectra were studied for the 4th-forbidden non-unique ground-state-to-ground-state β − decay branches <sup>113</sup>Cd(1/2 <sup>+</sup>) <sup>→</sup> <sup>113</sup>In(9/<sup>2</sup> <sup>+</sup>) and <sup>115</sup>In(9/2 <sup>+</sup>) <sup>→</sup> <sup>115</sup>Sn(1/<sup>2</sup> +) using the microscopic quasiparticle-phonon model (MQPM) [115, 116] and the ISM. It was verified by both nuclear models that the β spectrum shapes of both transitions are highly sensitive to the values of g<sup>V</sup> and g<sup>A</sup> and hence comparison of the calculated spectrum shape with the measured one opens a way to determine the values of these coupling strengths. As a by-product it was found that for all values of g<sup>A</sup> the best fits to data were obtained by using the canonical value g<sup>V</sup> = 1.0 for the vector coupling strength. This result is in conflict with those obtained by analyzing first-forbidden non-unique β decays in section 8.2, where strongly quenched values of g<sup>V</sup> were obtained.

The work of Haaranen et al. [177] on the <sup>113</sup>Cd and <sup>115</sup>In decays was extended in Haaranen et al. [178] to include an analysis made by using a third nuclear model, the microscopic interacting boson-fermion model (IBFM-2) [92]. At the same time the next-to-leading-order corrections to the β-decay shape factor were explicitly given and their role was thoroughly investigated. A striking feature of the SSM analysis was that the three models yield a consistent result, g<sup>A</sup> ≈ 0.92, when the SSM is applied to the available experimental β spectrum [179] of <sup>113</sup>Cd. The result is illustrated in **Figure 11** where the three curves overlap best at the values g eff <sup>A</sup> = 0.92 (MQPM), g eff <sup>A</sup> = 0.90 (ISM), and g eff <sup>A</sup> = 0.93 (IBFM-2). The agreement of the β-spectrum shapes computed in the three different nucleartheory frameworks speaks for the robustness of the SSM in determining the effective value of gA. For completeness, in **Figure 12** are shown the three components (82) as functions of the electron energy for the three different nuclear models used to compute the spectrum shapes of <sup>113</sup>Cd in **Figure 11**. It is seen that for the whole range of electron energies the two components, CV(we) and CA(we) are roughly of the same size whereas the magnitude of the component CVA(we) is practically the sum of the previous two, but with opposite sign. Hence, for the whole

FIGURE 11 | Comparison of the computed β spectra of <sup>113</sup>Cd with the experiment. The next-to-leading-order corrections to the shape factor have been included, and only the best matches are shown in the figure. The canonical value *g*<sup>V</sup> = 1.0 is used for the vector coupling strength. The areas under the curves are normalized to unity.

range of electron energies there is a delicate balance between the three terms, and their sum is much smaller than the magnitudes of its constituent components.

The works [177, 178] were continued by the work [180] where the evolution of the β spectra with changing value of g<sup>A</sup> was followed for 26 first-, second-, third-, fourth- and fifth-forbidden β − decays of odd-A nuclei by calculating the associated NMEs by the MQPM. The next-to-leading-order contributions were taken into account in the β-decay shape factor. It was found that the spectrum shapes of the third- and fourth-forbidden non-unique decays depend strongly on the value of gA, whereas the first- and second-forbidden decays were practically insensitive to the variations in gA. Furthermore, the gA-driven evolution of the normalized β spectra seems to be quite universal, largely insensitive to small changes of the nuclear mean field and the adopted residual many-body Hamiltonian. These features were also verified in the follow-up work [181], where the ISM was used as the nuclear-model framework. This makes SSM a robust tool for extracting information on the effective values of the weak coupling strengths. This also means that if SSM really is largely nuclear-model independent there is a chance to access the fundamental renormalization factor q<sup>F</sup> of section 3 for (highly) forbidden β transitions. It is also worth noting that in the works [180, 181] several new experimentally interesting decays for the SSM treatment were discovered.

Results of the investigations of Kostensalo et al. [180] and Kostensalo and Suhonen [181] are summarized in **Tables 6**, **7**, and in **Figures 13**–**15**. **Figure 13** displays the β spectra of the second-forbidden non-unique transitions <sup>94</sup>Nb(6+) <sup>→</sup> <sup>94</sup>Mo(4+) (left panel) and <sup>98</sup>Tc(6+) <sup>→</sup> <sup>98</sup>Ru(4+) (right panel) calculated by using the ISM [181]. It is obvious that the shape of the spectra depends sensitively on the value of g<sup>A</sup> but not as strongly as the transitions associated with the mother nuclei <sup>113</sup>Cd and <sup>115</sup>In, as shown in the figures of Haaranen et al. [177]. It is to be noted that both of the transitions have been observed experimentally since the branching is 100%, but the electron spectra are not yet available.

In **Figure 14** a comparison of the MQPM (left panel) and ISM (right panel) calculations [181] for the β spectrum of the second-forbidden non-unique decay transition <sup>99</sup>Tc(9/2 <sup>+</sup>) → <sup>99</sup>Ru(5/2 +) is shown. Again there is clear sensitivity to the value of gA, at the level of the <sup>94</sup>Nb and <sup>98</sup>Tc transitions, but the remarkable thing is that the spectrum shapes computed by the two nuclear models agree almost perfectly, giving further evidence in favor of the robustness of the SSM. Again, experimentally, the branching to this decay channel is practically 100% so that the β spectrum is potentially well measurable.

Finally, In **Figure 15** the β spectrum of the second-forbidden non-unique decay-transition <sup>137</sup>Cs(7/2 <sup>+</sup>) <sup>→</sup> <sup>137</sup>Ba(3/<sup>2</sup> +) is shown. Here the spectrum shape is quite independent of the value of g<sup>A</sup> and has exactly the same computed shape for the two applied nuclear-model frameworks: the MQPM and the ISM [181]. The robustness of the β-spectrum shape against variations in g<sup>A</sup> and the calculational scheme makes the measurement of this spectrum interesting in terms of testing the basic framework of high-forbidden non-unique β decays. The cause of the inertia against variations of g<sup>A</sup> is seen in **Table 6** in the decomposition (83) of the dimensionless integrated shape function C˜ for the decays of both <sup>135</sup>Cs and <sup>137</sup>Cs. It is seen that for these two decays all the components of C˜ are of the same sign, thus adding coherently. Hence, changes in the value of g<sup>A</sup> do not affect the spectrum shape, contrary to those decays where there is a destructive interference between the axial-vector and mixed components of (83), like in the cases of **Figures 11**–**14**, further analyzed in **Table 8**.

**Table 7** summarizes the exploratory works of Haaranen et al. [177, 178], Kostensalo et al. [180] and Kostensalo and Suhonen [181] in terms of listing the studied decay-transition candidates and their potential for future measurements. Here only the studied non-unique β-decay transitions are listed since the unique forbidden transitions are practically gA-independent even when the next-to-leading-order terms are included in the β-decay shape factor [177]. The most favorable cases for measurements are the ones that have a strong dependence on g<sup>A</sup> and the branching to the final state of interest is close to 100%. By

TABLE 6 | Dimensionless integrated shape functions *C*˜ (83) and their vector *C*˜ V, axial-vector *C*˜ A, and mixed components *C*˜ VA for the forbidden non-unique β decays of <sup>135</sup>Cs and <sup>137</sup>Cs.


*The forbiddenness K and the nuclear model used to calculate C is give* ˜ *n. For the total integrated shape factor C the values of the coupling strengths were set to g* ˜ *<sup>V</sup>* = *g<sup>A</sup>* = 1.0*.*


*Here J<sup>i</sup> (Jf ) is the angular momentum of the initial (final) state,* π*<sup>i</sup> (*π*f ) the parity of the initial (final) state, and K the degree of forbiddenness. The initial state is always the ground state (gs, column 2) of the mother nucleus and the final state is either the ground state (gs) or the n<sup>f</sup>* : *th, n<sup>f</sup>* <sup>=</sup> *1,2,3,4,5, excited state (column 3) of the daughter nucleus. Column 4 gives the branching to this particular decay channel [with boldface if (almost) 100%], column 5 indicates the sensitivity to the value of g<sup>A</sup> (with boldface if strong), and the last column lists the nuclear models which have been used (thus far) to compute the* β*-spectrum shape.*

these criteria the best candidates for measurements are the nonunique transitions <sup>94</sup>Nb(6+) <sup>→</sup> <sup>94</sup>Mo(4+) (second-forbidden), <sup>98</sup>Tc(6+) <sup>→</sup> <sup>98</sup>Ru(4+) (second-forbidden), <sup>99</sup>Tc(9/<sup>2</sup> <sup>+</sup>) → <sup>99</sup>Ru(5/2 <sup>+</sup>) (second-forbidden), <sup>113</sup>Cd(1/2 <sup>+</sup>) <sup>→</sup> <sup>113</sup>In(9/<sup>2</sup> +) (fourth-forbidden), and <sup>115</sup>In(9/2 <sup>+</sup>) <sup>→</sup> <sup>115</sup>Sn(1/<sup>2</sup> +) (fourthforbidden). Plans for accurate measurements of (some) of these transitions are on-going in the DAMA (V. Tretyak, private communication) and COBRA collaborations (K. Zuber, private communication). It should be noted that also the transition <sup>87</sup>Rb(3/2 <sup>−</sup>) <sup>→</sup> <sup>87</sup>Sr(9/<sup>2</sup> +) could be of interest for measurements since it has a 100% branching and the corresponding β spectrum is moderately sensitive to gA.

In **Table 8** the dimensionless integrated shape functions C˜ (83) have been decomposed into their vector C˜ <sup>V</sup>, axial-vector C˜ <sup>A</sup> and mixed vector-axial-vector components C˜ VA for the experimentally most promising forbidden non-unique β decays

FIGURE 13 | Normalized ISM-computed electron spectra for the second-forbidden non-unique decays of <sup>94</sup>Nb and <sup>98</sup>Tc. The value *<sup>g</sup>*<sup>V</sup> <sup>=</sup> 1.0 was assumed and the color coding represents the value of *g*A.

of forbiddenness K of **Table 7**. In the table also the nuclear model used to calculate C˜ is given. A characteristic of the numbers of **Table 8** is that the magnitudes of the vector, axial-vector, and mixed components are of the same order of magnitude, and the vector and axial-vector components have the same sign whereas the mixed component has the opposite sign. This makes the three components largely cancel each other and the resulting magnitude of the total dimensionless integrated shape function is always a couple of orders of magnitude smaller than its components. Thus, the integrated shape function becomes extremely sensitive to the value of gA, as seen in **Figure 13** for the decays of <sup>94</sup>Nb and <sup>98</sup>Tc, and in **Figure 14** for the decay of <sup>99</sup>Tc.

For the beta spectrum of the decays of <sup>113</sup>Cd and <sup>115</sup>In there are calculations available in three different nuclear-theory frameworks as shown in **Figure 11** and **Tables 7**, **8**. As visible in **Table 8**, an interesting feature of the components of the integrated shape functions C˜ is that the MQPM and ISM results are close to each other whereas the numbers produced by IBM-2 are clearly smaller. Surprisingly enough, the total value of C˜ is roughly the same in all three theory frameworks. This is another indication of the robustness of the SSM.

There are indirect ways to access the quenching of highforbidden β-decay transitions. One of them is to study electromagnetic decays of analogous structure. In Jokiniemi et al. [182] magnetic hexadecapole (M4) γ transitions in odd-A medium-heavy nuclei were studied by comparing the singlequasiparticle NMEs against the MQPM-computed NMEs to learn about the quenching in the analogous third-forbidden unique β decays (parity change with angular-momentum content 4). The MQPM calculations suggest a strong quenching g<sup>A</sup> ∼ 0.33g free <sup>A</sup> ∼ 0.4 for these transitions. This strong quenching could be an artifact of the MQPM framework since


TABLE 8 | Dimensionless integrated shape functions *C*˜ (83) and their vector *C*˜ V, axial-vector *C*˜ A and mixed components *C*˜ VA for the experimentally most promising forbidden non-unique β decays of forbiddenness *K*.

*Also the nuclear model used to calculate C is give* ˜ *n. For the total integrated shape factor C the values of the coupling strengths were set to g* ˜ *<sup>V</sup>* = *g<sup>A</sup>* = *1.0.*

there the excitations of an odd-A nucleus are formed by coupling BCS quasiparticles to excitations of the neighboring even-even reference nucleus. Thus, the predicted M4 giant resonance in the odd-A nucleus might not be strong enough to draw lowlying M4 strength to higher excitation energies, around the giant-resonance region.

### 10. QUENCHING OF G<sup>A</sup> IN 2νββ DECAYS

The 2νββ decay rate can be compactly written as

$$\left[t\_{1/2}^{(2\upsilon)}(\mathbf{0}\_i^+ \rightarrow \mathbf{0}\_f^+)\right]^{-1} = \mathbf{g}\_{\rm A}^4 \mathbf{G}\_{2\upsilon} \left|\mathcal{M}^{(2\upsilon)}\right|^2,\tag{84}$$

where G2<sup>ν</sup> represents the leptonic phase-space factor (without including gA) as defined in Kotila and Iachello [183]. The initial ground state is denoted by 0+ i and the final ground state by 0+ f . The 2νββ NME M(2ν) can be written as

$$M^{(2\upsilon)} = \sum\_{m,n} \frac{M\_{\rm L}(1\_m^+)M\_{\rm R}(1\_n^+)}{D\_m},\tag{85}$$

where the quantity D<sup>m</sup> is the energy denominator and the NMEs M<sup>L</sup> and M<sup>R</sup> correspond to left-leg and right-leg virtual Gamow-Teller transitions depicted in **Figure 16**. The summation is in general over all intermediate 1+ states, not just the first one as implied by the very schematic **Figure 5**. On the other hand, the summation in (85) can be dominated by one transition, usually through the lowest 1+ state if it happens to be the ground state of the intermediate nucleus. In this case one speaks about singlestate dominance. This dominance has been addressed in several works (e.g., [184–186]).

The 2νββ decay rate (84) and 0νββ decay rate (59) share the same strong dependence on gA. It is thus essential to study the renormalization of g<sup>A</sup> in beta and 2νββ decays before entering studies of the 0νββ decay. These studies touch only the 1+ contribution to the 0νββ decay. However, it is known that contributions from higher multipoles are also very important for the 0νββ decay (see section 7). It is challenging to relate the results emanating from the β and 2νββ decay studies to the value of the 0νββ NME: the former two involve momentum transfers of a few MeV whereas the latter involves momentum exchanges of the order of 100 MeV through the virtual Majorana neutrino. The high exchanged momenta in the 0νββ decay allow for the possibility that the effective value of g<sup>A</sup> acquires momentum dependence, as discussed in section 3. In addition, the high exchanged momenta induce substantial contributions from the higher J π states to the 0νββ decay rate [155]. The renormalization of g<sup>A</sup> for these higher-lying states could be different from the renormalization for the low-lying states, the subject matter of this review.

After this preamble we now proceed to discuss the possible renormalization of the axial-vector coupling strength [at zeromomentum limit q → 0 in (5)] as obtained from the combined β-decay and 2νββ-decay analyses performed in different theoretical approaches. It is important to be aware that in all the studies of the present section it is impossible to disentangle between the fundamental, nuclear-matter affected, and the many-body, nuclear-model affected, contributions to the renormalization of gA.

### 10.1. Quasiparticle Random-phase Approximation

The simultaneous analysis of both β and 2νββ decays opens up new vistas in attempts to pin down the effective value of the weak axial-vector coupling strength. Indeed, analysis of these two decay modes is possible for few nuclear systems where both the β-decay data (http://www.nndc.bnl.gov/) and 2νββ-decay data [157, 187] are available. The involved transitions, with the available data, are depicted schematically in **Figure 17**. The aim in using the three pieces of data available for the three isobaric systems is to gain information on the effective value of g<sup>A</sup> and the value of the particle-particle interaction parameter gpp of pnQRPA in the mass regions A = 100,116,128.

The first work to address the quenching in both β and 2νββ decays was [188] where both the beta-decay and 2νββ decay data were analyzed for the A = 100, 116 systems in the framework of the pnQRPA using the method of least squares to fit the pair (gpp,gA) to the available three pieces of data, namely the log ft

values of the left- and right-branch β decays, and the 2νββ halflife (see **Figure 17**). Realistic model spaces (large and small basis) and a phenomenologically renormalized microscopic G-matrixbased Hamiltonian was used in the investigations. In Faessler et al. [188] the best fit values g eff <sup>A</sup> = 0.74 (A = 100) and g eff <sup>A</sup> = 0.84 (A = 116) were obtained in the large single-particle model space. Furthermore, it is interesting to note that in the first version [189] of the paper [188] also results for the A = 128 system were included. There the result g eff <sup>A</sup> = 0.39 (A = 128) was quoted. These values of g eff A have been quoted in **Table 9** and plotted in **Figure 18** in section 10.2.

In Suhonen and Civitarese [192, 193] realistic single-particle bases and a G-matrix-based microscopic interaction was used to analyze the A = 100, 116, 128 systems of β and ββ decays. A slightly different approach to the one of Faessler et al. [188, 189] was adopted: by taking the left and right branches of β-decay data of **Figure 17** one can fix the pair (gpp,gA(β)) by reproducing the available log ft values. By using the just determined value of gpp one can compute the 2νββ NME and half-life and compare with the experimental halflife. This comparison produces a new value of g eff A , which can be denoted as gA(ββ). In an ideal case the two effective values of gA, namely gA(β) and gA(ββ), are the same but the over-constrained nature of the problem tends to yield different values to these parameters. The thus obtained values of gA(β) and gA(ββ) are quoted in **Table 9** and plotted in **Figure 18** in section 10.2.

### 10.2. Interacting Shell Model and Interacting Boson Model

A monotonic behavior of gA(ββ) was parametrized in Barea et al. [191] by analyzing the magnitudes of 2νββ NMEs produced by the microscopic interacting boson model (IBM-2) [91] and the ISM. In this study the obtained gA-vs.-A slopes were very flat, having the analytic expressions

$$\mathrm{g}\_{\mathrm{A}}^{\mathrm{eff}} \text{(IBM-2)} = 1.269 A^{-0.18}; \quad \mathrm{g}\_{\mathrm{A}}^{\mathrm{eff}} \text{(ISM)} = 1.269 A^{-0.12}.\tag{86}$$

These curves have been plotted in **Figure 18** together with the results obtained in the pnQRPA analyses of ββ decays in section 10.1. The results of these analyses, together with the the original numbers for gA(ββ) produced in the IBM-2 calculations of Barea et al. [191] are quoted in **Table 9**. The IBM-2 numbers are given in the last column of the table and the first two lines refer to the use of the single-state dominance (SSD) hypothesis in the IBM-2 calculations. Based on the analysis in Suhonen and Civitarese [192] this assumption is approximately valid since the magnitudes of the first 1+ contribution and the final 2νββ NME are practically the same for the decays of <sup>100</sup>Mo and <sup>116</sup>Cd. The last number of the IBM-2 column in **Table 9** refers to the assumption of closure approximation (CA) in the IBM-2 calculation. It is well established [1, 23] that such an approximation does not work for the 2νββ decays and thus this number could be dubious. Indeed, in a later publication [190] a more consistent theoretical framework was used (the interacting

boson-fermion-fermion model, IBFFM-2 [120]) and in the case of A = 128 values of g<sup>A</sup> were obtained that differ notably from the ones obtained in Barea et al. [191]. The IBFFM-2 numbers, based on analysis of both the β and 2νββ decay, are presented in columns 5 and 6 of **Table 9**. One can see that the IBFFM-2 values of gA(β) and gA(ββ) are quite close to those of the pnQRPAbased calculations. The combined β and ββ results of Yoshida and Iachello [190] have also been depicted in **Figure 18**.

Recent ISM calculations [68, 130] for the 2νββ NMEs of <sup>130</sup>Te and <sup>136</sup>Xe, and a subsequent comparison with the experimental NMEs (updated comparison performed in Horoi and Neacsu [68]) suggest a mild quenching and a rather large value of for the effective coupling strength:

$$g\_{\rm A}^{\rm eff}(A = 130 - 136) = 0.94.\tag{87}$$

This result was already discussed in section 5.1 and it was included in **Table 1** of that section. The result (87) was also illustrated in **Figure 3** of section 5.2.

From **Figure 18** one sees that the results of the pnQRPA analyses of Faessler et al. [188, 189] and Suhonen et al. [192] are consistent with each other, and are in agreement with the 2νββ results of the ISM (the upper dotted curve in **Figure 18**) for the masses A = 100, 116. For the mass A = 128 both the pnQRPA and IBFFM-2 results deviate strongly from the ISM result, coming closer to the IBM-2 results. Both the ISM and IBM-2 curves follow, in average, the trend of the pnQRPA results of the β-decay analyses of section 5.2 (the light-hatched regions of **Figure 18**), except for the very heavy masses, A ≥ 138. The differences in the results of the β-decay and 2νββ-decay analyses are not drastic but they still exist. The differences may stem from

al. [191] [curves (86)] and the vertical segments display the results of the combined β and 2νββ analyses of Faessler et al. [188, 189] (solid line) and Suhonen and Civitarese [192] (dashed lines). Also the (combined) IBFFM-2 result of Yoshida and Iachello [190] is depicted.

TABLE 9 | Extracted values of *g*A for three isobaric chains hosting a 2νββ transition.


*The values are obtained in the pnQRPA, in the IBFFM-2, and in the IBM-2 theory frameworks. In the last column SSD denotes single-state dominance, CA denotes closure approximation, and the errors in parentheses stem from the error limits of the adopted data. The intervals in column 2 correspond to the 1*σ *errors quoted in Faessler et al. [188, 189] and the ranges in the third and fourth columns stem from the experimental errors of the adopted data. The range in the fifth column stems from the different obtained values for the* β − *and* β +*/EC branches, respectively.*

the fact that not only one 1+ state takes part in most of the 2νββ decays. Contributions from the 1+ states above the lowest 1+ state, sometimes the ground state, interfere with each other and the contribution coming from the lowest one. These interferences have been discussed (e.g., [184, 186]).

## 11. SPIN-MULTIPOLE STRENGTH FUNCTIONS, GIANT RESONANCES AND THE RENORMALIZATION OF G<sup>A</sup>

As discussed in sections 5 and 6 the low-lying Gamow-Teller and higher isovector spin-multipole strengths, in particular the 2− strength, are quenched against nuclear-model calculations. The low-lying spin-multipole strength represents the low-energy tail of the corresponding spin-multipole giant resonance (SMGR). Usually only the low-energy part, with excitation energies E ≤ 5 MeV, of the spin-multipole strength function is experimentally known, and only for low multipoles, like for Gamow-Teller strength [194] or spin-dipole 2− strength [195]. These strength functions have been measured using charge-exchange reactions at low momentum transfers, like the (p,n), (3He,t), (n,p), and (d, <sup>2</sup>He) reactions [196–198]. As an example, in **Figure 19** is shown the strength for isovector spin-dipole excitations from the 0 <sup>+</sup> ground state of <sup>76</sup>Ge to the 0−, 1−, and 2<sup>−</sup> states in <sup>76</sup>As. The centroid energies of the corresponding giant resonances are roughly 24 MeV (0−), 20 MeV (1−), and 18 MeV (2−) [199].

The measured strength functions for Gamow-Teller transitions can be pestered by the isovector spin monopole (IVSM) contributions at high energies [200, 201]. The location (38) of the Gamow-Teller giant resonance, GTGR, dictates partly the amount of strength remaining at low energies [at zero-momentum limit q → 0 in (5)], and thus the quenching of the axial-vector coupling strength g<sup>A</sup> in model calculations [202]. These calculations have mostly been performed in the framework of the pnQRPA which represents well the centroids of the strong Gamow-Teller peaks, and extensions of the pnQRPA to two- plus four-quasiparticle models, like the proton-neutron microscopic anharmonic vibrator approach (pnMAVA) [203, 204], does not alter the picture very much.

Like in the case of the Gamow-Teller strength, also the location of the SMGRs affect the low-lying strength of, e.g., isovector spin-dipole (J <sup>π</sup> <sup>=</sup> <sup>0</sup> −, 1−, 2−, see **Figure 19**) and spinquadrupole (J <sup>π</sup> <sup>=</sup> <sup>1</sup> +, 2+, 3+) excitations [199, 205]. This is why measurements of such giant resonances could help in solving the quenching problems associated to g<sup>A</sup> at low energies.

### 12. EFFECTIVE G<sup>A</sup> FROM NUCLEAR MUON CAPTURE

The (ordinary, non-radiative) nuclear muon capture is a transition between nuclear isobars such that

$$
\mu^- + \langle A, Z, N \rangle \to \upsilon\_\mu + \langle A, Z - 1, N + 1 \rangle,\tag{88}
$$

where a negative muon is captured from an atomic s orbital and as a result the nuclear charge decreases by one unit and a muon neutrino is emitted. The process is schematically depicted in **Figure 20** for the capture on <sup>76</sup>Se, with the final states in <sup>76</sup>As. Here also the nucleus <sup>76</sup>Ge is depicted since it ββ decays to <sup>76</sup>Se. Properties of the µ-mesonic atoms have been treated theoretically in Ford and Wells [206] and experimentally in e.g., [207–210]. Due to the heavy mass of the muon (m<sup>µ</sup> = 105 MeV) the process has a momentum exchange of the order

shown in the intermediate nucleus <sup>76</sup>As. These states are populated by the ordinary muon capture (OMC) transitions from <sup>76</sup>Se.

of q ∼ 100 MeV and is thus similar to the neutrinoless ββ decay where a Majorana neutrino of a similar momentum is exchanged. This means that contrary to β decays all the terms of the hadronic current (2) are activated and that the contributions from the forbidden transitions J > 1 are not suppressed relative to the allowed ones, just like in the case of 0νββ decays. Since the induced currents in (2) are activated the theoretical expressions for the individual capture transitions are rather complex [211–215] whereas the total capture rates are much easier to calculate [216, 217].

Most of the theoretical attempts to describe the muon capture to individual nuclear states have concentrated on very light nuclei, A ≤ 20 [207, 213, 218–222] or to the mass region A = 23–40 [209, 210, 214, 215, 223–229]. Also studies in the 1s − 0d and 1p − 0f shells have been performed [230, 231]. Heavier nuclei, involved in ββ-decays, have been treated in Kortelainen and Suhonen [232, 233]. Interestingly enough, the muon-capture transitions can be used to probe the right-leg virtual transitions of 0νββ decays [231–233], but they can also give information on the in-medium renormalization of the axial current (4) in the form of an effective g<sup>A</sup> [210, 223, 225] and an effective induced pseudoscalar coupling g<sup>P</sup> (in fact the ratio gP/gA) [209, 213–215, 219, 220, 223–225, 227, 228] at high (100 MeV) momentum transfers, relevant for studies of the virtual transitions of the 0νββ decays. A recent review on the renormalization of g<sup>P</sup> is given in Gorringe and Fearing [234].

More experimental data on partial muon-capture rates to nuclear states are needed for heavier nuclei in order to access the renormalization of g<sup>A</sup> and g<sup>P</sup> for momentum transfers of interest for the 0νββ decay. The present (see e.g., [235]) and future experimental muon-beam installations should help solve this problem.

### 13. CONCLUSIONS

The quenching of the weak axial-vector coupling strength, gA, is an important issue considering its impact on the detectability of the neutrinoless double beta decay. The quenching appeared in old shell-model calculations as a way to reconcile the measured and calculated β-decay rates and strength functions. Later such quenching was studied in other nuclear-model frameworks, like the quasiparticle random-phase approximation and the IBM. The quenching of g<sup>A</sup> can be observed in allowed Gamow-Teller decays as also in forbidden β decays. The origins of the quenching seem to be both

### REFERENCES


the nuclear-medium effects and deficiencies in the nuclear many-body approaches, but a clean separation of these two aspects is formidably difficult. Different quenchings have been obtained in different calculations, based on different manybody frameworks. There is not yet a coherent approach to the quenching problem and many different separate studies have been performed. However, when analyzed closer, the obtained quenching of g<sup>A</sup> is surprisingly similar in different many-body schemes for different physical processes (e.g., for Gamow-Teller β transitions, for electron spectra of forbidden non-unique β decays) in the mass range from light to medium-heavy nuclei.

Different ways to access the quenching have been proposed, like comparisons with Gamow-Teller β-decay and two-neutrino double-β-decay data. In a promising new method, the SSM, the comparison of the computed and measured electron spectra of high-forbidden non-unique β decays is proposed. The robustness of the method is based on the observations that the computed spectra seem to be relatively insensitive to the adopted meanfield and nuclear models. Measurements of such electron spectra for certain key transitions are encouraged. Also the relation of the quenching problem to the low-lying strength for Gamow-Teller and higher isovector spin-multipole excitations is worth stressing, as also the relation to the corresponding giant resonances, accessible in present and future charge-exchangereaction experiments. The development of high-intensity muon beams makes measurements of nuclear muon-capture rates easier and enables access to the renormalization of the axial current at momentum exchanges relevant for the neutrinoless ββ decay.

### AUTHOR CONTRIBUTIONS

The author confirms being the sole contributor of this work and approved it for publication.

### FUNDING

University of Jyväskylä and Suomen Akatemia.

### ACKNOWLEDGMENTS

This work was supported by the Academy of Finland under the Finnish Center of Excellence Program 2012–2017 (Nuclear and Accelerator Based Program at JYFL).


**Conflict of Interest Statement:** The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Suhonen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# The λ Mechanism of the 0νββ-Decay

Fedor Šimkovic1, 2, 3 \*, Dušan Štefánik <sup>1</sup> and Rastislav Dvornický 1, 4

<sup>1</sup> Department of Nuclear Physics and Biophysics, Comenius University, Bratislava, Slovakia, <sup>2</sup> Boboliubov Laboratory of Theoretical Physics, Dubna, Russia, <sup>3</sup> Institute of Experimental and Applied Physics, Czech Technical University in Prague, Prague, Czechia, <sup>4</sup> Dzhelepov Laboratory of Nuclear Problems, Dubna, Russia

The λ mechanism (W<sup>L</sup> − W<sup>R</sup> exchange) of the neutrinoless double beta decay (0νββ-decay), which has origin in left-right symmetric model with right-handed gauge boson at TeV scale, is investigated. The revisited formalism of the 0νββ-decay, which includes higher order terms of nucleon current, is exploited. The corresponding nuclear matrix elements are calculated within quasiparticle random phase approximation with partial restoration of the isospin symmetry for nuclei of experimental interest. A possibility to distinguish between the conventional light neutrino mass (W<sup>L</sup> − W<sup>L</sup> exchange) and λ mechanisms by observation of the 0νββ-decay in several nuclei is discussed. A qualitative comparison of effective lepton number violating couplings associated with these two mechanisms is performed. By making viable assumption about the seesaw type mixing of light and heavy neutrinos with the value of Dirac mass m<sup>D</sup> within the range 1 MeV < m<sup>D</sup> < 1 GeV, it is concluded that there is a dominance of the conventional light neutrino mass mechanism in the decay rate.

### Edited by:

Diego Aristizabal Sierra, Federico Santa María Technical University, Chile

#### Reviewed by:

Janusz Gluza, University of Silesia of Katowice, Poland Juan Carlos Helo, University of La Serena, Chile

#### \*Correspondence:

Fedor Šimkovic simkovic@fmph.uniba.sk

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 01 August 2017 Accepted: 27 October 2017 Published: 16 November 2017

#### Citation:

Šimkovic F, Štefánik D and Dvornický R (2017) The λ Mechanism of the 0νββ-Decay . Front. Phys. 5:57. doi: 10.3389/fphy.2017.00057 Keywords: majorana neutrinos, neutrinoless double beta decay, right-handed current, left-right symmetric models, nuclear matrix elements, quasiparticle random phase approximation

## 1. INTRODUCTION

The Majorana nature of neutrinos, as favored by many theoretical models, is a key for understanding of tiny neutrino masses observed in neutrino oscillation experiments. A golden process for answering this open question of particle physics is the neutrinoless double beta decay (0νββ-decay) [1–3],

$$(A, Z) \to (A, Z + 2) + 2e^-,\tag{1}$$

in which an atomic nucleus with Z protons decays to another one with two more protons and the same mass number A, by emitting two electrons and nothing else. The observation of this process, which violates total lepton number conservation and is forbidden in the Standard Model, guaranties that neutrinos are Majorana particles, i.e., their own antiparticles [4].

The searches for the 0νββ-decay have not yielded any evidence for Majorana neutrinos yet. This could be because neutrinos are Dirac particles, i.e., not their own antiparticles. In this case we will never observe the decay. However, it is assumed that the reason for it is not sufficient sensitivity of previous and current 0νββ-decay experiments to the occurrence of this rare process.

Due to the evidence for neutrino oscillations and therefore for 3 neutrino mixing and masses the 0νββ-decay mechanism of primary interest is the exchange of 3 light Majorana neutrinos interacting through the left-handed V-A weak currents (mββ mechanism). In this case, the inverse 0νββ-decay half-life is given by Vergados et al. [1], DellOro et al. [2] and Vergados et al. [3]

$$\left[T\_{1/2}^{0\upsilon}\right]^{-1} = \left(\frac{m\rho\beta}{m\_{\varepsilon}}\right)^{2} \mathcal{g}\_{A}^{4} \mathcal{M}\_{\upsilon}^{2} \,\mathrm{G}\_{01},\tag{2}$$

where G01, g<sup>A</sup> and M<sup>ν</sup> represent an exactly calculable phase space factor, the axial-vector coupling constant and the nuclear matrix element (whose calculation represents a severe challenge for nuclear theorists), respectively. m<sup>e</sup> is the mass of an electron. The effective neutrino mass,

$$m\_{\beta\beta} = \left| U\_{e1}^2 m\_1 + U\_{e2}^2 m\_2 + U\_{e3}^2 m\_3 \right|,\tag{3}$$

is a linear combination of the three neutrino masses m<sup>i</sup> , weighted with the square of the elements Uei of the first row of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix. The measured value of mββ would be a source of important information about the neutrino mass spectrum (normal or inverted spectrum), absolute neutrino mass scale and the CP violation in the neutrino sector. However, that is not the only possibility.

There are several different theoretical frameworks that provide various 0νββ-decay mechanisms, which generate masses of light Majorana neutrinos and violate the total lepton number conservation. One of those theories is the left-right symmetric model (LRSM) [5–9], in which corresponding to the left-handed neutrino, there is a parity symmetric right-handed neutrino. The parity between left and right is restored at high energies and neutrinos acquire mass through the see-saw mechanism, what requires presence of additional heavy neutrinos. In general one cannot predict the scale where the left-right symmetry is realized, which might be as low as a few TeV—accessible at Large Hadron Collider, or as large as GUT scale of 10<sup>15</sup> GeV.

The LRSM, one of the most elegant theories beyond the Standard Model, offers a number of new physics contributions to 0νββ-decay, either from right-handed neutrinos or Higgs triplets. The main question is whether these additional 0νββmechanisms can compete with the mββ mechanism and affect the 0νββ-decay rate significantly. This issue is a subject of intense theoretical investigation within the TeV-scale left-right symmetry theories [10–14]. In analysis of heavy neutrino mass mechanisms of the 0νββ-decay an important role plays a study of related lepton number and lepton flavor violation processes in experiments at Large Hadron Collider [2, 15–19].

The goal of this article is to discuss in details the W<sup>L</sup> − W<sup>R</sup> exchange mechanism of the 0νββ-decay mediated by light neutrinos (λ mechanism) and its coexistence with the standard mββ mechanism. For that purpose the corresponding nuclear matrix elements (NMEs) will be calculated within the quasiparticle random phase approximation with a partial restoration of the isospin symmetry [20] by taking the advantage of improved formalism for this mechanism of the 0νββ-decay of Štefánik [21]. A possibility to distinguish mββ and λ mechanisms in the case of observation of the 0νββ-decay on several isotopes will be analyzed. Further, the dominance of any of these two mechanisms in the 0νββ-decay rate will be studied within seesaw model with right-handed gauge boson at TeV scale. We note that a similar analysis was performed by exploiting a simplified 0νββdecay rate formula and different viable particle physics scenarios in Tello et al. [10], Barry and Reodejohann [11], Bhupal Dev et al. [12], Deppisch et al. [13] and Borah et al. [14].

### 2. DECAY RATE FOR THE NEUTRINOLESS DOUBLE-BETA DECAY

Recently, the 0νββ-decay with the inclusion of right-handed leptonic and hadronic currents has been revisited by considering exact Dirac wave function with finite nuclear size and electron screening of emitted electrons and the induced pseudoscalar term of hadron current, resulting in additional nuclear matrix elements [21]. In this section we present the main elements of the revisited formalism of the λ mechanism of the 0νββdecay briefly. Unlike in Štefánik et al. [21] the effect the weakmagnetism term of the hadron current on leading NMEs is taken into account.

If the mixing between left and right vector bosons is neglected, for the effective weak interaction hamiltonian density generated within the LRSM we obtain

$$H^{\beta} = \frac{G\_{\beta}}{\sqrt{2}} \left[ j\_L^{\rho} J\_{L\rho}^{\dagger} + \lambda j\_R^{\rho} J\_{R\rho}^{\dagger} + h.c. \right]. \tag{4}$$

Here, G<sup>β</sup> = G<sup>F</sup> cos θC, where G<sup>F</sup> and θ<sup>C</sup> are Fermi constant and Cabbibo angle, respectively. The coupling constant λ is defined as

$$
\lambda = (M\_{W\_L} / M\_{W\_R})^2. \tag{5}
$$

Here, MW<sup>L</sup> and MW<sup>R</sup> are masses of the Standard Model lefthanded W<sup>L</sup> and right-handed W<sup>R</sup> gauge bosons, respectively. The left- and right-handed leptonic currents are given by

$$j\_L^{\rho} = \bar{\varepsilon}\gamma\_\rho (1 - \gamma\_5)\nu\_{eL}, \qquad j\_R^{\rho} = \bar{\varepsilon}\gamma\_\rho (1 + \gamma\_5)\nu\_{eR}.\tag{6}$$

The weak eigenstate electron neutrinos νeL and νeR are superpositions of the light and heavy mass eigenstate Majorana neutrinos ν<sup>j</sup> and N<sup>j</sup> , respectively. We have

$$\nu\_{eL} = \sum\_{j=1}^{3} \left( U\_{\varepsilon j} \nu\_{jL} + S\_{\varepsilon j} (N\_{jR})^C \right), \ \nu\_{eR} = \sum\_{j=1}^{3} \left( T\_{\varepsilon j}^\* (\nu\_{jL})^C + V\_{\varepsilon j}^\* N\_{jR} \right) (7)$$

Here, U, S, T, and V are the 3 × 3 block matrices in flavor space, which constitute a generalization of the Pontecorvo-Maki-Nakagawa-Sakata matrix, namely the 6 × 6 unitary neutrino mixing matrix [22]

$$\mathcal{U} = \begin{pmatrix} U \ S \\ T \ V \end{pmatrix}. \tag{8}$$

The nuclear currents are, in the non-relativistic approximation, [23]

$$\begin{aligned} f\_{L,\mathbb{K}}^{\rho}(\mathbf{x}) &= \sum\_{n} \tau\_{n}^{+} \delta(\mathbf{x} - \mathbf{r}\_{n}) \\ &\left[ \left( \mathbf{g}\_{V} \mp \mathbf{g}\_{A} \mathbf{C}\_{\mathrm{il}} \right) \mathbf{g}^{\rho 0} + \mathbf{g}^{\rho k} \left( \pm \mathbf{g}\_{A} \sigma\_{n}^{k} - \mathbf{g}\_{V} \mathbf{D}\_{n}^{k} \mp \mathbf{g}^{p} q\_{n}^{k} \frac{\vec{\sigma}\_{n} \cdot \mathbf{q}\_{n}}{2m\_{\mathrm{N}}} \right) \right]. \end{aligned} \tag{9}$$

Here, m<sup>N</sup> is the nucleon mass. q<sup>V</sup> ≡ qV(q 2 ), q<sup>A</sup> ≡ qA(q 2 ), q<sup>M</sup> ≡ qM(q 2 ) and q<sup>P</sup> ≡ qP(q 2 ) are, respectively, the vector, axialvector, weak-magnetism and induced pseudoscalar form-factors. The nucleon recoil terms are given by

$$\mathbf{C}\_{n} = \frac{\vec{\sigma} \cdot \left(\mathbf{p}\_{n} + \mathbf{p}\_{n}^{\prime}\right)}{2m\_{N}} - \frac{\mathcal{g}^{p}}{\mathcal{g}\_{A}} \left(E\_{n} - E\_{n}^{\prime}\right) \frac{\vec{\sigma} \cdot \mathbf{q}\_{n}}{2m\_{N}},$$

$$\mathbf{D}\_{n} = \frac{\left(\mathbf{p}\_{n} + \mathbf{p}\_{n}^{\prime}\right)}{2m\_{N}} - i\left(1 + \frac{\mathcal{g}\_{M}}{\mathcal{g}^{V}}\right) \frac{\vec{\sigma} \times \mathbf{q}\_{n}}{2m\_{N}},\tag{10}$$

where **q**<sup>n</sup> = **p**<sup>n</sup> − **p** ′ n is the momentum transfer between the nucleons. The initial neutron (final proton) possesses energy E ′ n (En) and momentum **p** ′ n (**p**n). **r**n, τ + n and σEn, which act on the n-th nucleon, are the position operator, the isospin raising operator and the Pauli matrix, respectively.

By assuming standard approximations [21] for the 0νββdecay half-life we get

$$\left[T\_{1/2}^{0\upsilon}\right]^{-1} = \eta\_{\upsilon}^{2}\,\mathrm{C}\_{mm} + \eta\_{\lambda}^{2}\,\mathrm{C}\_{\lambda\lambda} + \eta\_{\upsilon}\,\eta\_{\lambda}\,\cos\psi\,\,\mathrm{C}\_{m\lambda}.\tag{11}$$

The effective lepton number violating parameters η<sup>ν</sup> (W<sup>L</sup> − W<sup>L</sup> exchange), η<sup>λ</sup> (W<sup>L</sup> − W<sup>R</sup> exchange) and their relative phase Ψ are given by

$$\begin{aligned} \eta\_{\boldsymbol{\nu}} &= \frac{m\beta\boldsymbol{\rho}}{m\_{\boldsymbol{\epsilon}}}, \quad \eta\_{\boldsymbol{\lambda}} = \boldsymbol{\lambda}|\sum\_{j=1}^{3} U\_{\boldsymbol{\epsilon}j} T\_{\boldsymbol{\epsilon}j}^{\*}|,\\ \boldsymbol{\psi} &= \arg\Big[\left(\sum\_{j=1}^{3} m\_{j} U\_{\boldsymbol{\epsilon}j}^{2}\right) \left(\sum\_{j=1}^{3} U\_{\boldsymbol{\epsilon}j} T\_{\boldsymbol{\epsilon}j}^{\*}\right)^{\*}\Big].\end{aligned} \tag{12}$$

The coefficients C<sup>I</sup> (I = mm, mλ and λλ) are linear combinations of products of nuclear matrix elements and phase-space factors:

$$\begin{aligned} \mathcal{C}\_{mm} &= \mathcal{g}\_A^4 M\_\nu^2 \mathcal{G}\_{01}, \\ \mathcal{C}\_{m\lambda} &= -\mathcal{g}\_A^4 M\_\nu \left( M\_{2-} \mathcal{G}\_{03} - M\_{1+} \mathcal{G}\_{04} \right), \\ \mathcal{C}\_{\lambda\lambda} &= \mathcal{g}\_A^4 \left( M\_{2-}^2 \mathcal{G}\_{02} + \frac{1}{9} M\_{1+}^2 \mathcal{G}\_{011} - \frac{2}{9} M\_{1+} M\_{2-} \mathcal{G}\_{010} \right), \text{(13)} \end{aligned}$$

The explicit form and calculated values of phase-space factors G0<sup>i</sup> (i = 1, 2, 3, 4, 10 and 11) of the 0νββ-decaying nuclei of experimental interest are given in Štefánik et al. [21]. The NMES, which constitute the coefficients C<sup>I</sup> in Equation (13), are defined as follows:

$$\begin{aligned} M\_{\upsilon} &= M\_{GT} - \frac{M\_F}{\text{g}\_A^2} + M\_T, & M\_{\upsilon\omega} &= M\_{GT\omega} - \frac{M\_{F\alpha}}{\text{g}\_A^2} + M\_{T\alpha} \\ M\_{1+} &= M\_{qGT} + 3\frac{M\_{qF}}{\text{g}\_A^2} - 6M\_{qT}, & M\_{2-} &= M\_{\upsilon\omega} - \frac{1}{9}M\_{1+}.\end{aligned} \tag{14}$$

The partial nuclear matrix elements M<sup>I</sup> , where I = GT, F, T, ωF, ωGT, ωT, qF, qGT, and qT are given by

$$M\_{F,GT,T} = \sum\_{rs} \langle A\_f | h\_{F,GT,T}(r\_-) O\_{F,GT,T} | A\_i \rangle$$

$$M\_{oF,oGT,oT} = \sum\_{rs} \langle A\_f | h\_{oF,oGT,oT}(r\_-) O\_{F,GT,T} | A\_i \rangle$$

$$M\_{qF,qGT,qT} = \sum\_{rs} \langle A\_f | h\_{qF,qGT,qT}(r\_-) O\_{F,GT,T} | A\_i \rangle. \tag{15}$$

Here, OF,GT,<sup>T</sup> are the Fermi, Gamow-Teller and tensor operators 1, σE<sup>1</sup> · Eσ<sup>2</sup> and 3(σE<sup>1</sup> · ˆr12)(σE<sup>2</sup> · ˆr12). The two-nucleon exchange potentials hI(r) with I = F, GT, T, ωF, ωGT, ωT, qF, qGT, and qT can be written as

$$h\_{I}(r) = \frac{2R}{\pi} \int f\_{I}(q, r) \frac{q \, dq}{q + \vec{E}\_{n} - (E\_{i} + E\_{f})/2},\tag{16}$$

where

fGT = j0(q,r) g 2 A g 2 A (q 2 ) − gA(q 2 )gP(q 2 ) m<sup>N</sup> q 2 3 + g 2 P (q 2 ) 4m<sup>2</sup> N q 4 3 + 2 g 2 <sup>M</sup>(q 2 ) 4m<sup>2</sup> N q 2 3 , f<sup>F</sup> = g 2 V (q 2 )j0(qr), f<sup>T</sup> = j2(q,r) g 2 A gA(q 2 )gP(q 2 ) m<sup>N</sup> q 2 3 − g 2 P (q 2 ) 4m<sup>2</sup> N q 4 3 + g 2 <sup>M</sup>(q 2 ) 4m<sup>2</sup> N q 2 3 , <sup>f</sup>qF <sup>=</sup> rg<sup>2</sup> V (q 2 )j1(qr)q, fqGT = g 2 A (q 2 ) g 2 A q + 3 g 2 P (q 2 ) g 2 A q 5 4m<sup>2</sup> N + gA(q 2 )gP(q 2 ) g 2 A q 3 m<sup>N</sup> rj1(qr), fqT = r 3 g 2 A (q 2 ) g 2 A q − gP(q 2 )gA(q 2 ) 2g 2 A q 3 m<sup>N</sup> j1(qr) − 9 g 2 P (q 2 ) 2g 2 A q 5 20m<sup>2</sup> N - 2j1(qr)/3 − j3(qr) , (17)

and

$$h\_{a\bar{o}GT,F,T} = \frac{q}{\left(q + \bar{E}\_n - (E\_i + E\_f)/2\right)} h\_{GT,F,T}.\tag{18}$$

Here, E<sup>i</sup> , E<sup>f</sup> and E¯ <sup>n</sup> are energies of the initial and final nucleus and averaged energy of intermediate nuclear states, respectively. **r** = (**r**<sup>r</sup> − **r**s), **r**r,<sup>s</sup> is the coordinate of decaying nucleon and ji(qr) (i = 1, 2, 3) denote the spherical Bessel functions. **p**<sup>r</sup> + **p** ′ <sup>r</sup> ≃ 0, E<sup>r</sup> − E ′ <sup>r</sup> ≃ 0 and **p**<sup>r</sup> − **p** ′ <sup>r</sup> ≃ **q**, where **q** is the momentum exchange. The form factors gV(q 2 ), gA(q 2 ), gM(q 2 ) and gP(q 2 ) are defined in Simkovic et al. [24]. We note that factor 4 in definition of the two-nucleon exchange potentials hI(r) with I = ωF, ωGT, and ωT in Equation (48) of Štefánik et al. [21] needs to be replaced by factor 2.

#### 3. RESULTS AND DISCUSSION

The nuclear matrix elements are calculated in proton-neutron quasiparticle random phase approximation with partial restoration of the isospin symmetry for <sup>48</sup>Ca, <sup>76</sup>Ge, <sup>82</sup>Se, <sup>96</sup>Zr, <sup>100</sup>Mo, <sup>110</sup>Pd, <sup>116</sup>Cd, <sup>124</sup>Sn, <sup>130</sup>Te and <sup>136</sup>Xe, which are of experimental interest. In the calculation the same set of nuclear structure parameters is used as in Simkovic et al. [20]. The pairing and residual interactions as well as the two-nucleon short-range correlations derived from the realistic nucleonnucleon Argonne V18 potential are considered [26]. The closure approximation for intermediate nuclear states is assumed with (E¯ <sup>n</sup> − (E<sup>i</sup> + E<sup>f</sup> )/2) = 8 MeV. The free nucleon value of axial-vector coupling constant (g<sup>A</sup> = 1.25 − 1.27) is considered.

TABLE 1 | The nuclear matrix elements of the 0νββ-decay associated with <sup>m</sup>ββ and <sup>λ</sup> mechanisms and the coefficients <sup>C</sup>mm, <sup>C</sup>m<sup>λ</sup> and <sup>C</sup>λλ (in 10−<sup>14</sup> years−<sup>1</sup> ) of the decay rate formula (see Equation 11).


The nuclear matrix elements are calculated within the quasiparticle random phase approximation with partial restoration of the isospin symmetry. The G-matrix elements of a realistic Argonne V18 nucleon-nucleon potential are considered [20]. The phase-space factors are taken from Štefánik et al. [21]. fλ<sup>m</sup> <sup>=</sup> <sup>C</sup>λλ/Cmm, f<sup>G</sup> <sup>λ</sup><sup>m</sup> = G02/G<sup>01</sup> and g<sup>A</sup> = 1.269 is assumed. Qββ is the Q-value of the double beta decay in MeV.

In **Table 1** the calculated NMEs are presented. The values of MF,GT,<sup>T</sup> and M<sup>ν</sup> differ slightly (within 10%) with those given in Simkovic et al. [20], which were obtained without consideration of the closure approximation. By glancing **Table 1** we see that MFω,GTω,T<sup>ω</sup> ≃ MF,GT,<sup>T</sup> and Mνω ≃ M<sup>ν</sup> as for the average neutrino momentum q = 100 MeV and used average energy of intermediate nuclear states we have q/ q + E¯ <sup>n</sup> − (E<sup>i</sup> + E<sup>f</sup> )/2 ≃ 1. The absolute value of MFq,GTq,Tq is smaller in comparison with MF,GT,<sup>T</sup> by about 50% for Fermi NMEs and by about factor two in the case of Gamow-Teller and tensor NMEs. From **Table 1** it follows that there is a significant difference between results of this work and the QRPA NMEs of Muto et al. [25], especially in the case of <sup>100</sup>Mo. This difference can be attributed to the progress achieved in the 0νββ-decay formalism due to inclusion of higher order terms of nucleon currents [21, 24], the way of adjusting the parameters of nuclear Hamiltonian [27], description of short-range correlations [26] and restoration of the isospin symmetry [20].

Nuclear matrix elements M2<sup>−</sup> , M1<sup>+</sup> (λ mechanism) and Mν (mββ mechanism) for 10 nuclei under consideration are given in **Table 1** and displayed in **Figure 1**. We note a rather good agreement between M2<sup>−</sup> and M<sup>ν</sup> for all calculated nuclear systems. It is because the contribution of M1<sup>+</sup> to M2<sup>−</sup> is suppressed by factor 9 and as a result M2<sup>−</sup> is governed by the Mνω contribution (see Equation 14). Values of M1<sup>+</sup> exhibit similar systematic behavior in respect to considered nuclei as values of M<sup>ν</sup> and M2<sup>−</sup> , but they are suppressed by about factor 2–3 (with exception of <sup>48</sup>Ca).

The importance of the mββ and λ mechanisms depends, respectively, not only on values of η<sup>ν</sup> and η<sup>λ</sup> parameters, which are unknown, but also on values of coefficients C<sup>I</sup> (I = mm, mλ, λλ), which are listed for all studied nuclei in **Table 1**. They have been obtained by using improved values of phase-space factors G0<sup>k</sup> (k = 1, 2, 10 and 11) from Štefánik et al. [21]. We note that the squared value of MGT and fourth power of axial-vector coupling constant g<sup>A</sup> are included in the definition of coefficient C<sup>I</sup> unlike

in Štefánik et al. [21]. We see that Cλλ is always larger when compared with Cmm. The absolute value of Cm<sup>λ</sup> is significantly smaller than Cmm and Cλλ. This fact points out on less important contribution to the 0νββ-decay rate from the interference of mββ and λ mechanisms.

For 10 nuclei of experimental interest the decomposition of coefficient Cλλ (see Equation 11) on partial contributions C 0k I associated with phase-space factors G0<sup>k</sup> (k = 2, 10, and 11) is shown in **Figure 2**. By glancing the plotted ratio C 0k I /C<sup>I</sup> we see that Cλλ is dominated by a single contribution associated with the phase-space factor G02. From this and above analysis it follows that 0νββ-decay half-life to a good accuracy can be written as

$$\begin{aligned} \left[T\_{1/2}^{0\upsilon}\right]^{-1} &= \left(\eta\_{\upsilon}^{2} + \eta\_{\lambda}^{2}f\_{\lambda m}\right) \ \mathrm{C}\_{mm} \\ &\simeq \left(\eta\_{\upsilon}^{2} + \eta\_{\lambda}^{2}f\_{\lambda m}^{G}\right) \ g\_{A}^{4} \ M\_{\upsilon}^{2} \ \mathrm{G}\_{01} \end{aligned} \tag{19}$$

with

$$f\_{\lambda m} = \frac{C\_{\lambda \lambda}}{C\_{mm}} \simeq f\_{\lambda m}^{G} = \frac{G\_{02}}{G\_{01}}.\tag{20}$$

For a given isotope the factor fλ<sup>m</sup> reflects relative sensitivity to the mββ and λ mechanisms and f G λm is its approximation, which does not depend on NMEs. The values fλ<sup>m</sup> and f G λm are tabulated in **Table 1** and plotted as function of Qββ in **Figure 3**. We see that fλ<sup>m</sup> depends only weakly on involved nuclear matrix elements (apart for the case of <sup>48</sup>Ca) what follows from a comparison of fλ<sup>m</sup> with f G λm . The value of fλ<sup>m</sup> is mainly determined by the Qvalue of double beta decay process. From 10 analyzed nuclei the largest value of fm<sup>λ</sup> is found for <sup>48</sup>Ca and the smallest value for <sup>76</sup>Ge. A larger value of fλ<sup>m</sup> means increased sensitivity to mββ mechanism in comparison to λ mechanism and vice versa.

Upper bounds on the effective neutrino mass mββ and right-handed current coupling strength η<sup>λ</sup> are deduced from experimental half-lives of the 0νββ-decay by using the coefficients Cmm, Cm<sup>λ</sup> and Cλλ of **Table 1**. The maximum and the value on axis (mββ = 0 or η<sup>λ</sup> = 0) are listed in **Table 2**. The decays of <sup>136</sup>Xe and <sup>76</sup>Ge set the sharpest limit <sup>m</sup>ββ <sup>≤</sup> 0.13 eV and 0.18 eV, and <sup>η</sup><sup>λ</sup> <sup>≤</sup> 1.7 10−<sup>7</sup> and 3.1 10−<sup>7</sup> , respectively. These are more stringent than those deduced from other experimental sources.

It is well known that by measuring different characteristics, namely energy and angular distributions of two emitted electrons, it is possible to identify which of mββ and λ mechanisms is responsible for 0νββ-decay [21, 23]. It might be achieved only by some of future 0νββ-decay experiments, e.g., the SuperNEMO [37] or NEXT [38]. A relevant question is whether the underlying mββ or λ mechanism can be revealed by observation of the 0νββ-decay in a series of different isotopes. In **Figure 4** this issue is addressed by an illustrative case of observation of the 0νββ-decay of <sup>136</sup>Xe with halflife T 0ν <sup>1</sup>/<sup>2</sup> <sup>=</sup> 6.86 10<sup>26</sup> years, which can be associated with <sup>m</sup>ββ <sup>=</sup> 50 meV or <sup>η</sup><sup>λ</sup> = 9.8 10−<sup>8</sup> . The 0νββ-decay halflife predictions associated with a dominance of mββ and λ mechanisms exhibit significant difference for some nuclear systems. We see that by observing, e.g., the 0νββ-decay of <sup>100</sup>Ge and <sup>100</sup>Mo with sufficient accuracy and having calculated relevant NMEs with uncertainty below 30%, it might be possible to conclude, whether the 0νββ-decay is due to mββ or λ mechanism.

Currently, the uncertainty in calculated 0νββ-decay NMEs can be estimated up to factor of 2 or 3 depending on the considered isotope as it follows from a comparison of results of different nuclear structure approaches [3]. The improvement of the calculation of double beta decay NMEs is a very important and challenging problem. There is a hope that due to a recent progress in nuclear structure theory (e.g., ab initio methods) and increasing computing power the calculation of the 0νββ-decay


TABLE 2 | Upper bounds on the effective Majorana neutrino mass mββ and parameter ηλ associated with right-handed currents mechanism imposed by current constraints on the 0νββ-decay half-life for nuclei of experimental interest.

The calculation is performed with NMEs obtained within the QRPA with partial restoration of the isospin symmetry (see Table 1). The upper limits on mββ and η<sup>λ</sup> are deduced for a coexistence of the mββ and λ mechanisms (Maximum) and for the case η<sup>λ</sup> = 0 or η<sup>ν</sup> = 0 (On axis). g<sup>A</sup> = 1.269 and CP conservation (ψ = 0) are assumed.

NMEs with uncertainty of about 30 % might be achieved in future.

### 4. THE LEPTON NUMBER VIOLATING PARAMETERS WITHIN THE SEESAW AND NORMAL HIERARCHY

The 6 × 6 unitary neutrino mixing matrix <sup>U</sup> (see Equation 8) can be parametrized with 15 rotational angles and 10 Dirac and 5 Majorana CP violating phases. For the purpose of study different LRSM contributions to the 0νββ-decay the mixing matrix U is usually decomposed as follows [22]

$$\mathcal{U} = \begin{pmatrix} \mathbf{1} \ \mathbf{0} \\ \mathbf{0} \ U\_0 \end{pmatrix} \begin{pmatrix} A \ R \\ S \ B \end{pmatrix} \begin{pmatrix} V\_0 \ \mathbf{0} \\ \mathbf{0} \ \mathbf{1} \end{pmatrix}. \tag{21}$$

Here, **0** and **1** are the 3×3 zero and identity matrices, respectively. The parametrization of matrices A, B, R and S and corresponding orthogonality relations are given in Xing [22].

If A = **1**, B = **1**, R = **0** and S = **0**, there would be a separate mixing of light and heavy neutrinos, which would participate only in left and right-handed currents, respectively. In this case we get η<sup>λ</sup> = 0, i.e., the λ mechanism is forbidden.

If masses of heavy neutrinos are above the TeV scale, the mixing angles responsible for mixing of light and heavy neutrinos are small. By neglecting the mixing between different generations of light and heavy neutrinos, the unitary mixing matrix U takes the form

$$\mathcal{U} = \begin{pmatrix} U\_0 & \frac{m\_D}{m\_{LNV}} \mathbf{1} \\ -\frac{m\_D}{m\_{LNV}} \mathbf{1} & V\_0 \end{pmatrix} . \tag{22}$$

Here, m<sup>D</sup> represents energy scale of charged leptons and mLNV is the total lepton number violating scale, which corresponds to masses of heavy neutrinos. We see that U = U<sup>0</sup> can be identified to a good approximation with the PMNS matrix and V<sup>0</sup> is its analogue for heavy neutrino sector. Due to unitarity condition we find V<sup>0</sup> = U † 0 . Within this scenario of neutrino mixing the effective lepton number violating parameters η<sup>ν</sup> (mββ mechanism) and η<sup>λ</sup> (λ mechanism) are given by

$$\eta\_{\upsilon} = \frac{m\_{D}}{m\_{\varepsilon}} \frac{m\_{D}}{m\_{LNV}} \zeta\_{m}, \quad \eta\_{\lambda} = \left(\frac{M\_{W\_{L}}}{M\_{W\_{R}}}\right)^{2} \frac{m\_{D}}{m\_{LNV}} \zeta\_{\lambda} \tag{23}$$

with

$$\zeta\_m = \left| \sum\_{j=1}^3 U\_{\epsilon j}^2 \frac{m\_j m\_{LNV}}{m\_D^2} \right|, \quad \zeta\_\lambda = \left| \sum\_{j=1}^3 U\_{\epsilon j} \right| = 0.14 - 1.5. \tag{24}$$

The importance of mββ or λ-mechanism can be judged from the ratio

$$\frac{\eta\_{\lambda}}{\eta\_{\upsilon}} = \left(\frac{M\_{W\_L}}{M\_{W\_R}}\right)^2 \frac{m\_e}{m\_D} \frac{\xi\_{\lambda}}{\xi\_m}.\tag{25}$$

It is naturally to assume that ζ<sup>m</sup> ≈ 1 and to consider the upper bound for the factor ζλ, i.e., there is no anomaly cancellation among terms, which constitute these factors. Within this approximation ηλ/η<sup>ν</sup> does not depend on scale of the lepton number violation mLNV and is plotted in **Figure 5**. The Dirac mass m<sup>D</sup> is assumed to be within the range 1 MeV < m<sup>D</sup> < 1 GeV. The flavor and CP-violating processes of kaons and Bmesons make it possible to deduce lower bound on the mass of the heavy vector boson MW<sup>2</sup> > 2.9 TeV [12]. From **Figure 5** it follows that within accepted assumptions the λ mechanism is practically excluded as the dominant mechanism of the 0νββdecay.

In this section the light-heavy neutrino mixing of the strength mD/mLNV is considered. However, we note that there are models with heavy neutrinos mixings where strength of the mixing decouples from neutrino masses [39–44]. This subject goes beyond the scope of this paper.

### 5. SUMMARY AND CONCLUSIONS

The left-right symmetric model of weak interaction is an attractive extension of the Standard Model, which may manifest itself in the TeV scale. In such case the Large Hadron Collider can determine the right-handed neutrino mixings and heavy neutrino masses of the seesaw model. The LRSM predicts new physics contributions to the 0νββ half-life due to exchange of light and heavy neutrinos, which can be sizable.

In this work the attention was paid to the λ mechanism of the 0νββ-decay, which involves left-right neutrino mixing through mediation of light neutrinos. The recently improved formalism of the 0νββ-decay concerning this mechanism was considered. For 10 nuclei of experimental interest NMEs were calculated within the QRPA with a partial restoration of the isospin symmetry. It was found that matrix elements governing the conventional mββ and λ mechanisms are comparable and that the λ contribution to the decay rate can be associated with a single phase-space factor. A simplified formula for the 0νββdecay half-life is presented (see Equation 19), which neglects the suppressed contribution from the interference of both mechanisms. In this expression the λ contribution to decay rate is weighted by the factor fλm, which reflects relative sensitivity to the mββ and λ mechanisms for a given isotope and depends only weakly on nuclear physics input. It is manifested that measurements of 0νββ-decay half-life on multiple isotopes with largest deviation in the factor fλ<sup>m</sup> might allow to distinguish both considered mechanisms, if involved NMEs are known with sufficient accuracy.

Further, upper bounds on effective lepton number violating parameters mββ (η<sup>ν</sup> ) and η<sup>λ</sup> were deduced from current lower limits on experimental half-lives of the 0νββ-decay. The ratio ηλ/η<sup>ν</sup> was studied as function of the mass of heavy vector boson MW<sup>R</sup> assuming that there is no mixing among different generations of light and heavy neutrinos. It was found that if the value of Dirac mass m<sup>D</sup> is within the range 1 MeV < m<sup>D</sup> < 1 GeV, the current constraint on MW<sup>R</sup> excludes the dominance

#### REFERENCES


of the λ mechanism in the 0νββ-decay rate for the assumed neutrino mixing scenario.

### AUTHOR CONTRIBUTIONS

FŠ: calculation of nuclear matrix elements and preparation of manuscript; RD: analysis of lepton number violating parameters and preparation of manuscript; DŠ: derivation of the formalism of neutrinoless double beta decay, and analysis of obtained results.

### FUNDING

This work is supported by the VEGA Grant Agency of the Slovak Republic un- der Contract No. 1/0922/16, by Slovak Research and Development Agency under Contract No. APVV-14-0524, RFBR Grant No. 16-02-01104, Underground laboratory LSM— Czech participation to European-level research infrastructue CZ.02.1.01/0.0/0.0/16 013/0001733.


double beta decay with SuperNEMO. Eur Phys J C (2010) **70**:927–43. doi: 10.1140/epjc/s10052-010-1481-5


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Šimkovic, Štefánik and Dvornický. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# From the Trees to the Forest: A Review of Radiative Neutrino Mass Models

#### Yi Cai 1, 2, Juan Herrero García<sup>3</sup> \*, Michael A. Schmidt <sup>4</sup> \*, Avelino Vicente<sup>5</sup> and Raymond R. Volkas <sup>2</sup>

<sup>1</sup> School of Physics, Sun Yat-sen University, Guangzhou, China, <sup>2</sup> ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Melbourne, VIC, Australia, <sup>3</sup> ARC Centre of Excellence for Particle Physics at the Terascale, Department of Physics, The University of Adelaide, Adelaide, SA, Australia, <sup>4</sup> ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, Sydney, NSW, Australia, 5 Instituto de Física Corpuscular (CSIC-Universitat de València), Valencia, Spain

#### Edited by:

Frank Franz Deppisch, University College London, United Kingdom

#### Reviewed by:

Mayumi Aoki, Kanazawa University, Japan Koji Tsumura, Kyoto University, Japan Hiroaki Sugiyama, Toyama Prefectural University, Japan

#### \*Correspondence:

Juan Herrero García juan.herrero-garcia@coepp.org.au Michael A. Schmidt michael.schmidt@sydney.edu.au

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 18 July 2017 Accepted: 14 November 2017 Published: 04 December 2017

#### Citation:

Cai Y, Herrero García J, Schmidt MA, Vicente A and Volkas RR (2017) From the Trees to the Forest: A Review of Radiative Neutrino Mass Models. Front. Phys. 5:63. doi: 10.3389/fphy.2017.00063 A plausible explanation for the lightness of neutrino masses is that neutrinos are massless at tree level, with their mass (typically Majorana) being generated radiatively at one or more loops. The new couplings, together with the suppression coming from the loop factors, imply that the new degrees of freedom cannot be too heavy (they are typically at the TeV scale). Therefore, in these models there are no large mass hierarchies and they can be tested using different searches, making their detailed phenomenological study very appealing. In particular, the new particles can be searched for at colliders and generically induce signals in lepton-flavor and lepton-number violating processes (in the case of Majorana neutrinos), which are not independent from reproducing correctly the neutrino masses and mixings. The main focus of the review is on Majorana neutrinos. We order the allowed theory space from three different perspectives: (i) using an effective operator approach to lepton number violation, (ii) by the number of loops at which the Weinberg operator is generated, (iii) within a given loop order, by the possible irreducible topologies. We also discuss in more detail some popular radiative models which involve qualitatively different features, revisiting their most important phenomenological implications. Finally, we list some promising avenues to pursue.

Keywords: neutrino masses, lepton flavor violation, lepton number violation, beyond the standard model, effective field theory, model building, LHC, dark matter

# 1. INTRODUCTION

The discovery of neutrino oscillations driven by mass mixing is one of the crowning achievements of experimental high-energy physics in recent decades. From its beginnings as the "solar neutrino problem"—a deficit of electron neutrinos from the Sun compared to the prediction of the standard solar model, an anomaly first discovered by the Homestake experiment—through the emergence of the "atmospheric neutrino problem" and its eventual confirmation by SuperKamiokande, to terrestrial verifications by long baseline and reactor neutrino experiments, the existence of nonzero and non-degenerate neutrino masses is now well established [1–17]. In addition, the existence of oscillations proves that the weak eigenstate neutrinos νe, νµ, and ν<sup>τ</sup> are not states of definite mass themselves, but rather non-trivial, coherent superpositions of mass eigenstate fields called

**86**

simply ν1, ν2, and ν3, with masses m1, m2, and m3, respectively<sup>1</sup> . The dynamical origin of neutrino mass is at present unknown, including whether neutrinos are Dirac or Majorana fermions. In the former case, neutrinos and antineutrinos are distinct and have a total of four degrees of freedom, exactly as do the charged leptons and quarks. Majorana fermions, on the other hand, are their own antiparticles, and they have just two degrees of freedom corresponding to left- and right-handed helicity. Dirac neutrinos preserve total lepton number conservation, while Majorana neutrino masses violate lepton number conservation by two units. The purpose of this review is to survey one class of possible models, where neutrino masses arise at loop order and are thus called "radiative." Almost all of the models we examine are for the Majorana mass case. Before turning to a discussion of possible models, we should summarize the experimental data the models are trying to understand or at least accommodate.

The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix (Uαi) [18, 19] defines the relationship between the weak and mass eigenstates, through

$$\upsilon\_{\alpha} = \sum\_{i} U\_{\alpha i} \upsilon\_{i\flat} \tag{1}$$

where α = e,µ, τ and i = 1, 2, 3. The PMNS matrix U is unitary, and may be parameterized by three (Euler) mixing angles θ12, θ23, and θ13, a CP-violating Dirac phase δ that is analogous to the phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, and two Majorana phases α2,3 if neutrinos are Majorana fermions. The standard parametrization is

where i = 1, 2 depending on the sign of the atmospheric squaredmass difference (see Forero et al. [21] and Capozzi et al. [22] for earlier fits). The sign of 1m<sup>2</sup> <sup>21</sup> has been measured because the Mikheyev-Smirnov-Wolfenstein or MSW effect [23, 24] in the Sun depends on it. The sign of the atmospheric equivalent is, however, not currently known, and is a major target for future neutrino oscillation experiments. Because of this ambiguity, there are two possible neutrino mass orderings: m<sup>1</sup> < m<sup>2</sup> < m<sup>3</sup> which is called either "normal ordering" or "normal hierarchy", and m<sup>3</sup> < m<sup>1</sup> < m<sup>2</sup> which is termed "inverted." The global fit results for the other parameters depend somewhat on which ordering is assumed. In Equations (4) and (5) we quote results that leave the ordering as undetermined. See Esteban et al. [20] for a discussion of these subtleties, but they will not be important for the rest of this review. Note that the convention is i = 1 in Equation (5) for normal ordering and i = 2 for inverted ordering.

At the 3σ level, the CP-violating phase δ can be anything. However, there is a local minimum in χ 2 at δ ∼ −π/2, which is tantalizing and very interesting. It hints at large CP-violation in the lepton sector, and the specific value of −π/2 is suggestive of a group theoretic origin (but beware that the definition of this phase is convention dependent). As with the mass ordering, the discovery of CP violation in neutrino oscillations is a prime goal for future experiments. One strong motivation for this is the cosmological scenario of baryogenesis via leptogenesis [25], and even if other sources of leptonic CP-violation are involved, it is important to experimentally establish the general phenomenon in the lepton sector. At present, we do not know if neutrinos are Dirac or Majorana fermions, so there is no information about the

$$U = \begin{pmatrix} c\_{12}c\_{13} & s\_{12}c\_{13} & s\_{13}e^{-i\delta} \\ -s\_{12}c\_{23} - c\_{12}s\_{23}s\_{13}e^{i\delta} & c\_{12}c\_{23} - s\_{12}s\_{23}s\_{13}e^{i\delta} & s\_{23}c\_{13} \\ s\_{12}s\_{23} - c\_{12}c\_{23}s\_{13}e^{i\delta} & -c\_{12}s\_{23} - s\_{12}c\_{23}s\_{13}e^{i\delta} & c\_{23}c\_{13} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{i\frac{\alpha\_2}{2}} & 0 \\ 0 & 0 & e^{i\frac{\alpha\_3}{2}} \end{pmatrix},\tag{2}$$

where cij ≡ cos θij and sij ≡ sin θij. The neutrino oscillation lengths are set by the ratio of squared-mass differences and energy, while the amplitudes are governed by the PMNS mixing angles and the Dirac phase. The Majorana phases do not contribute to oscillation probabilities. The angles θ12, θ23, and θ<sup>13</sup> are sometimes referred to as the solar, atmospheric and reactor angles, respectively, because of how they were originally or primarily measured. The "solar" and "atmospheric" oscillation length parameters are, respectively,

$$
\Delta m\_{21}^2 \equiv m\_2^2 - m\_1^2, \quad \Delta m\_{32}^2 \equiv m\_3^2 - m\_2^2 \sim \Delta m\_{31}^2 \equiv m\_3^2 - m\_1^2,\tag{3}
$$

where the distinction between the two atmospheric quantities will be discussed below.

A recent global fit [20] obtains the following 3σ ranges for the mixing angle and 1m<sup>2</sup> parameters:

$$
\sin^2 \theta\_{12} \in [0.271, 0.345], \quad \sin^2 \theta\_{23} \in [0.385, 0.638],
$$

$$
\sin^2 \theta\_{13} \in [0.01934, 0.02397], \tag{4}
$$

$$
\Delta m\_{21}^2 \in \text{ [7.03, 8.09]} \times 10^{-5} \text{ eV}^2, \quad \Delta m\_{3i}^2 \in \text{ [-2.629, -2.405]}
$$

$$
\cup \text{ [2.407, 2.643]} \times 10^{-3} \text{ eV}^2, \tag{5}
$$

possible Majorana phases α2,3. Neutrinoless double-beta decay is sensitive to these parameters, as is standard leptogenesis.

The final parameter to discuss is the absolute neutrino mass scale. The square root of the magnitude of the atmospheric 1m<sup>2</sup> provides a lower bound of 0.05 eV on at least one of the mass eigenvalues. Laboratory experiments performing precision measurements of the tritium beta-decay end-point spectrum currently place a direct kinematic upper bound of about 2 eV on the absolute mass scale [26–28] as quantified by an "effective electron-neutrino mass" mν<sup>e</sup> ≡ q |Uei| <sup>2</sup>m<sup>2</sup> i , independent of whether the mass is Dirac or Majorana, and the sensitivity of the currently running KATRIN experiment is expected to be about 0.2 eV [29]. With appropriate caution because of model dependence, cosmology now places a strong upper bound on the sum of neutrino masses of about 0.2 eV [30], with the precise number depending on exactly what data are combined. If the neutrino mass sum was much above this figure, then its effect on large-scale structure formation—washing out structure on small scales—would be strong enough to cause disagreement with observations. For Majorana masses, neutrinoless double beta-decay experiments have determined an upper bound on an effective mass defined by

<sup>1</sup>The possibility of additional neutrino-like states will be discussed below.

$$|m\_{\beta\beta}| \equiv |\sum\_{i} U\_{ei}^{2} m\_{i}|\tag{6}$$

of 0.15 − 0.33 at 90% C.L., depending on nuclear matrix element uncertainties [31] 2 . We can thus see that experimentally and observationally, we are closing in on a determination of the absolute mass scale.

The fact that the laboratory and cosmological bounds require the absolute neutrino mass scale to be so low strongly motivates the hypothesis that neutrinos obtain their masses in a different manner from the charged leptons and quarks. A number of approaches have been explored in the literature, with one of them being the main topic of this review: radiative neutrino mass generation. Other approaches will also be briefly commented on, to place radiative models into the overall context of possible explanations for why neutrino masses are so small.

This completes a summary of the neutrino mass and mixing data that any model, including radiative models, must explain or accommodate. As noted above, future experiments and observational programs have excellent prospects to determine the mass ordering, discover leptonic CP violation, observe neutrinoless double beta-decay (0νββ) and hence the violation of lepton number by two units, and measure the absolute neutrino mass scale. In addition, the determination of the θ<sup>23</sup> octant—whether or not θ<sup>23</sup> is less than or greater than π/4 is an important goal of future experiments. Before turning to a discussion of neutrino mass models, we should review some interesting experimental anomalies that may imply the existence of light sterile neutrinos<sup>3</sup> in addition to the active flavors νe,µ,<sup>τ</sup> (see Gariazzo et al. [32], Kopp et al. [33] for phenomenological fits).

There are three anomalies. The first is > 3σ evidence from the LSND [34, 35] and MiniBooNE [36, 37] experiments of ν¯<sup>e</sup> appearance in a ν¯<sup>µ</sup> beam, with MiniBooNE also reporting a ν<sup>e</sup> signal in a ν<sup>µ</sup> beam. Interpreted through a neutrino oscillation hypothesis, these results indicate an oscillation mode with a 1m<sup>2</sup> or order 1 eV<sup>2</sup> . This cannot be accommodated with just the three known active neutrinos simultaneously with the extremely well-established solar and atmospheric modes that require much smaller 1m<sup>2</sup> parameters. This hypothesis thus only works if there are four or more light neutrino flavors, and the additional state or states must be sterile to accord with the measured Z-boson invisible width<sup>4</sup> . The Icecube neutrino telescope has recently tested the sterile neutrino oscillation explanation of these anomalies through the zenith angle dependence of muon track signals and excludes this hypothesis at about the 99% C.L. [38].

The next two anomalies concern ν<sup>e</sup> and ν¯<sup>e</sup> disappearance. Nuclear reactors produce a ν¯<sup>e</sup> flux that has been measured by several experiments. When compared to the most recent computation of the expected flux [39, 40], a consistent deficit of a few percent is observed, a set of results known as the "reactor anomaly" [41]. The Gallium anomaly arose from neutrino calibration source measurements by the Gallex and SAGE radiochemical solar neutrino experiments, also indicating a deficit [42–45]. Both deficits are consistent with very short baseline transitions driven by eV-scale sterile neutrinos, and a significant number of experiments are underway to test the oscillation explanation. It should be noted that a recent analysis by the Daya Bay collaboration points to the problem being with the computation of the reactor ν¯<sup>e</sup> flux rather than being an indication of very short baseline oscillations [46]. The key point is that if a sterile neutrino was responsible, one should observe the same deficit for all neutrinos from the reactor fuel, independent of nuclear species origin, but this was observed to not be the case. There is also a tension between the appearance and disappearance anomalies when trying to fit both with a selfconsistent oscillation scheme [32, 33], and there is a cosmological challenge of devising a mechanism to prevent the active-sterile transitions from thermalizing the sterile neutrino in the early universe, as thermalization would violate the ∼ 0.2 eV bound on the sum of neutrino masses.

Because the situation with the above anomalies is unclear, and there are challenges to explaining them with oscillations, this review will focus on neutrino mass models that feature just the three known light active neutrinos. If any of the above anomalies is eventually shown to be due to oscillations, then all neutrino mass models will need to be extended to incorporate light sterile neutrinos, including the radiative models that are our subject in this review.

The rest of this review is structured as follows: section 2 provides a general discussion of schemes for neutrino mass generation and attempts a classification. The structure of radiative neutrino mass models is then described in section 3. Section 4 covers phenomenological constraints and search strategies, including for cosmological observables. Detailed descriptions of specific models are then given in section 5, with the examples chosen so as to exemplify some of the different possibilities that the radiative mechanisms permit. We conclude in section 6, where we discuss some research directions for the future. Appendix gives further details on the relative contributions of the different operators to neutrino masses.

### 2. SCHEMES FOR NEUTRINO MASSES AND MIXINGS

In this section, we survey the many different general ways that neutrinos can gain mass, and attempt a classification of at least most of the proposed schemes. As part of this, we place both the tree-level and radiative models in an overarching context a systematic approach, if you will, or at least as systematic as we can make it. The number of different kinds of models can seem bewildering, so there is some value in understanding the broad structure of the neutrino mass "theory space."

Under the standard model (SM) gauge group GSM ≡ SU(3)c× SU(2)<sup>L</sup> × U(1)Y, the left-handed neutrinos feature as the upper isospin component of

<sup>2</sup>The effective mass mββ depends on the Majorana phases and thus provides a unique probe for them.

<sup>3</sup> Sterile neutrinos are not charged under the SM gauge group.

<sup>4</sup>MiniBooNE also has a mysterious excess in their low-energy bins that cannot be explained by any oscillation hypothesis.

$$L = \begin{pmatrix} \nu\_{\rm L} \\ \varepsilon\_{\rm L} \end{pmatrix} \sim \langle 1, 2, -\frac{1}{2} \rangle,\tag{7}$$

where on the right-hand (RH) side the first entry denotes the representation with respect to the color group SU(3)c, the second SU(2)<sup>L</sup> (weak-isospin), and the third hypercharge Y, normalized so that electric charge is given by Q = I<sup>3</sup> + Y. In the minimal standard model, there is no way to generate non-zero neutrino masses and mixings at the renormalizable level. Dirac masses are impossible because of the absence of RH neutrinos,

$$
\upsilon\_{\mathbb{R}} \sim (1, 1, 0), \tag{8}
$$

as are Majorana masses because there is no scalar isospin triplet

$$
\Delta \sim \text{(1,3,1)}\tag{9}
$$

to which the lepton bilinear L <sup>c</sup>L could have a Yukawa coupling. Thus, the family-lepton numbers Le, L<sup>µ</sup> and L<sup>τ</sup> are (perturbatively) conserved because of three accidental global U(1) symmetries. The discovery of neutrino oscillations means that the family-lepton number symmetries must be broken. If they are broken down to the diagonal subgroup generated by total lepton number L ≡ L<sup>e</sup> + L<sup>µ</sup> + L<sup>τ</sup> , then the neutrinos must be Dirac fermions. If total lepton number is also broken, then the neutrinos are either fully Majorana fermions or pseudo-Dirac<sup>5</sup> .

The question of whether neutrinos are Dirac or Majorana (or possibly pseudo-Dirac) is one of the great unknowns. The answer is vital for model building, as well as for some aspects of phenomenology. If neutrinos are Majorana, then it is not necessary to add RH neutrinos to the SM particle content. In fact, many of the radiative models we shall review below do not feature them. If RH neutrinos do not exist, then a possible deep justification could be SU(5) grand unification, which is content with a <sup>5</sup>¯ <sup>⊕</sup> 10 structure per family<sup>6</sup> . But another logical possibility, motivated by quark-lepton symmetry and SO(10) grand unification, is that RH neutrinos exist but have large (SM gauge invariant) Majorana masses, leading to the extremely well-known type-I seesaw model [47–51]. On the other hand, if neutrinos are Dirac, then RH neutrinos that are singlets under the SM gauge group, as per Equation (8), are mandatory and they must not have Majorana masses even though such terms are SM gauge invariant and renormalizable. Thus, at the SM level, something like total lepton-number conservation must be imposed by hand. Most of the radiative models we shall discuss lead to Majorana neutrinos, though we shall also briefly review the few radiative Dirac models that have been proposed.

The choice of Dirac or Majorana is thus a really important step in model building. It is perhaps fair to say that theoretical prejudice, as judged by number of papers, favors the Majorana possibility. There are a couple of reasons for this. One is simply that Majorana fermions are permitted by the Poincaré group, so it might be puzzling if they were never realized in nature, and the fact is that they constitute the simplest spinorial representation. (Recall that a Dirac fermion is equivalent to two CP-conjugate, degenerate Majorana fermions). Another was already discussed above: even if RH neutrinos exist, at the SM level they can have gauge-invariant Majorana masses, leading to Majorana mass eigenstates overall. Yet another reason is a connection between Majorana masses and an approach to understanding electric charge quantization using classical constraints and gauge anomaly cancellation [52, 53]. Nevertheless, theoretical prejudice or popularity in the literature is not necessarily a reliable guide to how nature actually is, so the Dirac possibility should be given due consideration.

### 2.1. Dirac Neutrino Schemes

The simplest way to obtain Dirac neutrinos is by copying the way the charged-fermions gain mass. Right-handed neutrinos are added to the SM particle content, producing the gauge-invariant, renormalizable Yukawa term

$$
\mathcal{Y}\_{\boldsymbol{\V}} \overline{\boldsymbol{L}} \tilde{H} \boldsymbol{\nu}\_{\mathbb{R}} + \text{H.c.}, \tag{10}
$$

where the Higgs doublet H transforms as (1, 2, 1/2) with H˜ ≡ iτ2H<sup>∗</sup> . The Dirac neutrino mass matrix is then

$$\mathcal{M}\_{\boldsymbol{\nu}} = \mathcal{Y}\_{\boldsymbol{\nu}} \langle H^0 \rangle = \mathcal{Y}\_{\boldsymbol{\nu}} \frac{\boldsymbol{\nu}}{\sqrt{2}} \,, \tag{11}$$

To accommodate the O(0.1) eV neutrino mass scale, one simply takes <sup>y</sup><sup>ν</sup> <sup>∼</sup> <sup>10</sup>−13. The price to pay for this simple and obvious model is a set of tiny dimensionless parameters, some six or seven orders of magnitude smaller than the next smallest Yukawa coupling constant (that for the electron), and smaller even than the value a fine-tuned θQCD needs to be from the upper bound on the neutron electric-dipole moment. This is of course logically possible, and it is also technically natural in the 't Hooft sense [54] because taking yν to zero increases the symmetry of the theory. Nevertheless, it seems unsatisfactory to most people. The really tiny neutrino masses strongly suggest that the generation of neutrino mass proceeds in some different, less obvious manner, one that provides a rationale for why the masses are so small. As well as the Dirac vs. Majorana question, the explanation of the tiny masses has dominated model-building efforts in the literature.

So, how may one produce very light Dirac neutrinos? We highlight three possibilities, but there may be others: (i) a Dirac seesaw mechanism, (ii) radiative models, and (iii) extradimensional theories.

#### 2.1.1. Dirac Seesaw Mechanism

In addition to the ν<sup>L</sup> that resides inside the doublet L, and the standard RH neutrino of Equation (8), we introduce a vectorlike heavy neutral fermion NL,R ∼ (1, 1, 0) and impose total lepton-number conservation with νL,R and NL,R assigned lepton numbers of 1. In addition, we impose a Z<sup>2</sup> discrete symmetry under which ν<sup>R</sup> and a new gauge-singlet real scalar S are odd,

<sup>5</sup>Pseudo-Dirac neutrinos are a special case of Majorana neutrinos where the masses of two Majorana neutrinos are almost degenerate and the breaking of lepton number is small. However, they should not be confused with Dirac neutrinos. <sup>6</sup>RH neutrinos could obviously be added as a singlet of SU(5).

with all other fields even. With these imposed symmetries, the most general Yukawa and fermion bare mass terms are

$$\mathcal{Y}\mathcal{N}\overline{L}\tilde{H}\mathcal{N}\_{\mathbb{R}} + \mathcal{Y}\mathcal{N}\_{\mathbb{L}}\overline{\mathcal{N}\_{\mathbb{R}}}\mathcal{S} + \mathcal{M}\_{\mathcal{N}}\overline{\mathcal{N}\_{\mathbb{L}}}\mathcal{N}\_{\mathbb{R}} + \text{H.c.}\tag{12}$$

leading to the neutral-fermion mass matrix

$$\left(\begin{array}{c}\overline{\text{vL}} \end{array}\overline{\text{N}}\right)\left(\begin{array}{c}0\\m\_{\text{R}}\end{array}\begin{array}{c}m\_{\text{L}}\\ \end{array}\right)\left(\begin{array}{c}\nu\_{\text{R}}\\ \text{N}\_{\text{R}}\end{array}\right) + \text{H.c.},\tag{13}$$

where

$$m\_L = \wp\_N \frac{\nu}{\sqrt{2}} \quad \text{and} \quad m\_R = \wp\_R \langle \mathbf{S} \rangle \,. \tag{14}$$

We now postulate the hierarchy m<sup>L</sup> ≪ m<sup>R</sup> ≪ M<sup>N</sup> on the justification that the bare mass term has no natural scale so could be very high, and that the symmetry breaking scale of the new, imposed Z<sup>2</sup> should be higher than the electroweak scale. The light neutrino mass eigenvalue is thus

$$m\_{\psi} \sim m\_L \frac{m\_R}{M\_N},\tag{15}$$

and the eigenvector is dominated by the ν<sup>L</sup> admixture so does not violate weak universality bounds. The inverse relationship of the light neutrino mass with the large mass M<sup>N</sup> is the seesaw effect, with the postulated small parameter mR/M<sup>N</sup> causing m<sup>ν</sup> to be much smaller than the electroweak-scale mass mL. The above structure is the minimal one necessary to illustrate the Dirac seesaw mechanism (and has a cosmological domain wall problem because of the spontaneously broken Z2), but the most elegant implementation is in the left–right symmetric model [55]. Under the extended electroweak gauge group SU(2)<sup>L</sup> ×SU(2)<sup>R</sup> × U(1)B−L, the RH neutrino sits in an SU(2)<sup>R</sup> doublet with B−L = −1, while NL,R remains as gauge singlets. The scalars are a left– right symmetric pair of doublets HL,R with B−L = 1. The usual scalar bidoublet is not introduced. The Z<sup>2</sup> symmetry is then a subgroup of SU(2)R, and S is embedded in the RH scalar doublet. The mass and symmetry breaking hierarchy is then hH 0 L i ≪ hH 0 R i ≪ MN. The absence of the bidoublet ensures the zero in the top-left entry of the mass matrix<sup>7</sup> . Several treelevel Dirac neutrino mass models have been discussed in Ma and Popov [56]: The SM singlet Dirac fermion N<sup>L</sup> + N<sup>R</sup> can be obviously replaced by an electroweak triplet. Alternatively a neutrinophilic two Higgs doublet model [57, 58] is an attractive possibility to obtain small Dirac neutrino masses.

#### 2.1.2. Radiative Dirac Schemes

A generalization of the symmetry structure of the Z<sup>2</sup> Dirac seesaw model discussed above provides us with one perspective on the construction of radiative Dirac neutrino mass models. A basic structural issue with such models is the prevention of the treelevel term generated by the renormalizable Yukawa interaction of Equation (10). Some new symmetry must be imposed that forbids that term, but that symmetry must also be spontaneously or softly broken in such a way that an effective νLν<sup>R</sup> operator is produced. In the case of radiative models, this must be made to happen at loop order. One obvious possibility is to demand that "RH neutrino number" is conserved, meaning that invariance under

$$
\nu\_{\mathbb{R}} \to e^{i\theta} \nu\_{\mathbb{R}},\tag{16}
$$

with all other SM fields as singlets, is imposed. One may then introduce a complex scalar ρ that transforms, for example, as

$$
\rho \to e^{-i\theta/n}\rho,\tag{17}
$$

whose non-zero expectation value spontaneously breaks the symmetry. The effective operator

$$\frac{1}{\Lambda^n} \,\,\overline{L}\tilde{H}\nu\_{\mathbb{R}}\rho^n,\tag{18}$$

produced by integrating out new physics at mass scale 3, is both SM gauge invariant and invariant under the imposed symmetry <sup>8</sup> . It generates a neutrino Dirac mass of order

$$m\_{\upsilon} \sim \nu \left(\frac{\langle \rho \rangle}{\Lambda} \right)^{n} \tag{19}$$

which will be small compared to the weak scale when <sup>h</sup>ρ<sup>i</sup> <sup>3</sup> ≪ 1. If this operator is "opened up"—derived from an underlying renormalizable or ultraviolet (UV) complete theory—at looplevel, then a radiative neutrino Dirac-mass model is produced. Note that in a loop-level completion, the parameter 1/3<sup>n</sup> depends on powers of renormalizable coupling constants and a 1/16π <sup>2</sup> per loop as well as the actual masses of new, exotic massive particles. See Ma and Popov [56] for a recent systematic study of 1-loop models based on this kind of idea. Note that the Dirac seesaw model discussed earlier is obtained as a truncated special case: the U(1) symmetry with n = 1 is replaced with its Z<sup>2</sup> subgroup, the complex scalar field ρ is replaced with the real scalar field S, and the effective operator LH˜ νRS is opened up at tree-level.

Obviously, the phase part of ρ will be a massless Nambu-Goldstone boson (NGB), but its phenomenology might be acceptable because it only couples to neutrinos. If one wishes to avoid this long range force, one could find a way to make the new U(1) anomaly-free and then gauge it so that the NGB gets eaten, or one may use a discrete subgroup of the U(1) to forbid Equation (10). See Wang and Han [60] for a discussion of the Z<sup>2</sup> case for 1-loop models that also include a dark matter candidate.

The above is simply an example of the kind of thinking that has to go into the development of a radiative Dirac neutrino model—we are not claiming it is the preferred option. To our knowledge, a thorough analysis of symmetries that can prevent a tree-level Dirac mass and thus guide the construction of complete theories has not yet been undertaken in the literature. That is one of the reasons this review will discuss Majorana models at greater length than Dirac models.

<sup>7</sup> If one does not impose left–right discrete symmetry on the Lagrangian, then there will be no cosmological domain wall problem. The Dirac seesaw mechanism does not require this discrete symmetry.

<sup>8</sup>This construction resembles the well-known Froggatt-Nielsen mechanism [59].

#### 2.1.3. Extra-Dimensional Theories

One way or another, the effective coefficient in front of LH˜ ν<sup>R</sup> must be made small. Seesaw models achieve this by exploiting powers of a small parameter given by the ratio of symmetry breaking and/or mass scales. Radiative models augment the seesaw feature with 1/16π 2 loop factors and products of perturbative coupling constants. In warped or Randall-Sundrum extra-dimensional theories [61, 62], the geometry of fermion localization in the bulk [63, 64] can lead to the suppression of Dirac neutrino masses through having a tiny overlap integral between the profile functions for the neutrino chiral components and the Higgs boson [63, 65–68]. The phenomenological implications of Dirac neutrinos in extra-dimensional set-ups have been studied in De Gouvea et al. [69], where it is shown that these effects can be encoded in specific dimension-six effective operators.

One can also have a "clockwork" mechanism [70, 71] to generate exponentially suppressed Dirac masses. In the same way, it is also useful to have low-scale seesaw [72]. This mechanism can be implemented with a discrete number of new fields or via an extra spatial dimension [73].

### 2.2. Majorana Neutrino Schemes

We now come to our main subject: radiative Majorana neutrino mass generation. We also briefly review tree-level seesaw schemes, both for completeness and for the purposes of comparison and contrast to the loop-level scenarios. In the course of the discussion below, an attempt will be made to classify the different kinds of radiative models. This is a multidimensional problem: no single criterion can be singled out as definitely the most useful discriminator between models. Instead, we shall see that several overlapping considerations emerge, including 1L = 2 effective operators, number of loops, number of Higgs doublets, nature of the massive exotic particles, whether or not there are extended symmetries and gauge bosons, distinctive phenomenology, and whether or not the models address problems or issues beyond just neutrino mass (e.g., dark matter, grand unification, ...).

The main distinctive feature of Majorana neutrino mass is, of course, that it violates lepton-number conservation by two units. It is thus extremely useful to view the possibilities for the new physics responsible from a bottom-up perspective, meaning SM gauge-invariant, 1L = 2 low-energy effective operators that are to be derived from integrating out new physics that is assumed to operate at scales higher than the electroweak. This approach permits the tree-level seesaw [47–51, 74–80] and radiative models to be seen from a unified perspective.

Taking the particle content of the minimal SM, it is interesting that the simplest and lowest mass-dimension effective operator one can produce is directly related to Majorana neutrino mass generation. This is the famous Weinberg operator [81]

$$O\_1 = LLHH,\tag{20}$$

where the SU(2) indices and Lorentz structures are suppressed (one can check that there is only one independent invariant even though there are three different ways to contract the SU(2) indices of the four doublets.). We say the singular "operator" for convenience, but it is to be understood that there are also family indices so we really have a set of operators. This is a mass dimension five operator, so enters the Lagrangian with a 1/3 coefficient, where 3 is the scale of the new physics that violates lepton number by two units. Replacing the Higgs doublets with their vacuum expectation values (VEVs), one immediately obtains the familiar Majorana seesaw formula,

$$m\_{\upsilon} \sim \frac{\nu^2}{\Lambda},\tag{21}$$

displaying the required suppression of mν with respect to the weak scale v when ǫ ≡ v/3 ≪ 1, so that the 1L = 2 new physics operates at a really high scale.

The Weinberg operator can be immediately generalized to the set

$$O\_1^{\prime \cdots \prime} = L L H H (H^\dagger H)^n,\tag{22}$$

where the number of primes is equal to n. One obtains ever more powerful seesaw suppression,

$$m\_{\nu} \sim \nu \epsilon^{2n+1},\tag{23}$$

as n increases.

The task now is to derive, from an underlying renormalizable or UV complete theory, one of the Weinberg-type operators as the leading contribution to neutrino mass. This process has come to be termed "opening up the operator." The choices one makes about which operator (what value of n) is to dominate and how it is to be opened up determine the type of theory one obtains. Here are some possible choices:


Option 1 leads, in its simplest form, precisely to the familiar type-I [47–51], type-II [74–79] and type-III [80] seesaw mechanisms, as we review in the next subsection. Option 2 leads to a certain kind of radiative model, to be contrasted with that arising from option 3. The difference between the two can be expressed in terms of the matching conditions used to connect an effective theory below the scale 3 of the 1L = 2 new physics to the full theory above that scale, as outlined in **Figure 1**. For scenario 2, the effective Weinberg operator has a non-zero Wilson coefficient at 3, and for all scales below that. In scenario 3, on the other hand, the Weinberg operator has a coefficient at scale 3 that is loop-suppressed compared to the Wilson coefficients of other, non-Weinberg-type <sup>1</sup><sup>L</sup> <sup>=</sup> 2 operators<sup>9</sup> at that scale, where these

<sup>9</sup>The other <sup>1</sup><sup>L</sup> <sup>=</sup> 2 operators also play an important role in the classification of radiative neutrino mass models and will be discussed in detail in section 2.2.2.

other operators are obtained by integrating out the heavy fields only. If the matching is performed at tree-level approximation, then the coefficient of the Weinberg operator at 3 in fact vanishes. Under renormalization group mixing, the non-zero 1L = 2 operators will, however, generate an effective Weinberg operator as the parameters are run to scales below 3. If the matching is performed at loop-level, then the Weinberg operator will have a non-zero coefficient at scale 3, but it will be loopsuppressed compared to the coefficients of the relevant non-Weinberg operators. Below 3, the Weinberg operator coefficient will, once again, receive corrections from the renormalization group running and operator mixing. Option 3 will be a major topic in this review, and it motivates the enumeration of all SM gauge-invariant 1L = 2 operators, not just those in the Weinberg class, since the non-Weinberg operators describe the dominant 1L = 2 processes at scale 3. Opening up the non-Weinberg operators at tree-level then provides a systematic method of constructing a large class of theories that generate neutrino masses at loop order.

Options 4-6 obviously repeat the exercise, but with two more powers of ǫ which help suppress the neutrino mass. With these options, one needs to ensure that O ′···′ 1 generated from the new physics dominates over O<sup>1</sup> and all lower-dimensional operators O ′···′ 1 . Option 6 is similar to 3 in that the effective theory between the weak and new physics scales contains some non-Weinberg type of 1L = 2 operator(s) that dominate at scale 3.

#### 2.2.1. Tree-Level Seesaw Mechanisms

The three familiar seesaw models may be derived in a unified way by opening up the Weinberg operator O<sup>1</sup> at tree level in the simplest possible way, using as the heavy exotics only scalars or fermions. The available renormalizable interactions are then just of Yukawa and scalar-scalar type. The opening-up process is depicted in **Figure 2**. The type-I and type-III seesaw models are obtained by Yukawa coupling LH with the two possible choices of (1, 1, 0) and (1, 3, 0) fermions, both of which can have gaugeinvariant bare Majorana masses. The type-II model is the unique theory obtained from Yukawa coupling the fermion bilinear LL ≡ L <sup>c</sup>L to a (1, 3, 1) scalar multiplet, which in turn couples to H†H† , a cubic interaction term in the scalar potential10. The seesaw effect is obtained in this case by requiring a positive quadratic term for the triplet in the scalar potential, that on its own would cause the triplet's VEV to vanish, but which in combination with the cubic term induces a small VEV for it.

As is clear from **Figure 2**, there are two interaction vertices for all three cases, and there is only one type of exotic per case. An interesting non-minimal tree-level seesaw model realizing option 4 is obtained by allowing four vertices instead of two, and two exotic multiplets: a (1, 4, −1/2) scalar that couples to HHH† and a (1, 5, 0) massive fermion that Yukawa couples to the exotic scalar quadruplet and the SM lepton doublet [82– 84]. The resulting model produces the generalized Weinberg operator O ′′ <sup>1</sup> <sup>=</sup> LLHH(H†H) <sup>2</sup> which has mass-dimension nine. This model is a kind of hybrid of the type-II and type-III seesaw mechanisms, because it features both a small induced VEV for the quadruplet and a seesaw suppression from mixing with the fermion quintuplet.

<sup>10</sup>Note that the LL <sup>∼</sup> (1, 1, <sup>−</sup>1) option is irrelevant for tree level mechanisms because it does not produce the required ν <sup>c</sup>ν bilinear.

### 2.2.2. Radiative Schemes and Their Classification

As noted above, there are many different kinds of radiative neutrino mass models and there is probably no single classification scheme that is optimal for all purposes. We thus discuss a few different perspectives, some much more briefly than others. Two will be treated at length: (i) the 1L = 2 effective operator approach, and (ii) classification by loop-order openings of the Weinberg operator.

The massive exotic is a (1, 3, 0) fermion 6 whose middle component mixes with the left-handed neutrino.

A. Standard model 1L = 2 effective operators. This approach can be considered as stemming from the observations made about options 3 and 6 in section 2.2: when both light SM particles and heavy exotics appear in the neutrino mass loop graph, it is useful to first consider integrating out the heavy exotics at tree-level. This produces effective 1L = 2 operators that are of non-Weinberg type. They must be of different type, because if they were not, then the heavy exotics would produce the Weinberg operator without participation by light SM particles, leading either to a class 1 model (if O<sup>1</sup> is produced at tree-level) or a class 2 model (if O<sup>1</sup> is produced at loop level). An exhaustive list of gauge-invariant, non-Weinberg 1L = 2 operators is thus needed.

Such a list was provided by Babu and Leung (BL) [85], based on the following assumptions: (i) the gauge group is that of the SM only, (ii) no internal global symmetries are imposed apart from baryon number, (iii) the external lines are SM quarks, SM leptons and a single Higgs doublet, and (iv) no operators of mass dimension higher than 11 were considered. We first comment on these assumptions. Clearly, if the gauge symmetry was extended beyond that of the SM, then some combination of effective operators might be restricted to having a single coefficient, and others might be forced to vanish, compared to the SMgauge-group-only list. Similar observations follow for imposed global symmetries. It is sensible to impose baryon number conservation, because otherwise phenomenological constraints will force the new physics to such high scales that obtaining neutrino masses of the required magnitude (at least one at 0.05 eV) will be impossible. The case of a single Higgs doublet can readily be generalized to multiple Higgs doublets, given that the gauge quantum numbers are the same. This would obviously enrich the phenomenology of the resulting models, and if additional symmetries were also admitted, then it would change the model-building options. The point is simply that H†H is invariant under all possible internal symmetries, while H † <sup>1</sup>H<sup>2</sup> is not. (Admitting additional Higgs doublets is also interesting for generalized-Weinberg-operator models, because then a symmetry reason can exist for, say, LLH1,2H1,2(H † <sup>1</sup>H2) being generated without also generating what would otherwise be dominant LLH1,2H1,2 operators.) The addition of nondoublet scalar multiplets into the external lines is a more serious complication. Some discussion of the possible roles of additional scalars that gain non-zero VEVs that contribute to neutrino mass generation will be given in later sections. Another restriction worth noting in the BL list is the absence of the gauge-singlet RH neutrinos. In assumption (iv), the point to highlight is the absence of SM gauge fields. Babu and Leung did actually write down the mass-dimension-7 operators containing gauge fields, and Bhattacharya and Wudka [86] further examined them. As far as we know, however, no complete analysis has been undertaken for the dimension-9 and -11 cases. Finally, it is sensible to stop at dimension 11 because at any higher order the contribution to neutrino mass will be insufficiently large. The BL list, as enumerated from O<sup>1</sup> to O60, took operators that could be thought of as products of lower-dimension operators with the SM invariants HH† and the three dimension-4 charged-fermion Yukawa terms as implicit. de Gouvea and Jenkins [87] extended their list by explicitly including the latter cases, thereby augmenting the operator count to O75.

Operators meeting all of these requirements exist at all odd mass dimensions [85, 88, 89], starting with the Weinberg operator O<sup>1</sup> as the unique dimension-5 case (up to family indices). The dimension-7 list is as follows:

$$O\_2 = L^l L^l L^k \epsilon^c H^l \epsilon\_{ij} \epsilon\_{kl}, \quad O\_{3a} = L^l L^j Q^k d^c H^l \epsilon\_{ij} \epsilon\_{kl}, \quad O\_{3b} = L^l L^j Q^k d^c H^l \epsilon\_{ik} \epsilon\_{jl}, \tag{24}$$

$$O\_{4a} = L^l L^j \bar{Q}\_l \bar{u}^c H^k \epsilon\_{jk}, \quad O\_{4b} = L^l L^l \bar{Q}\_k \bar{u}^c H^k \epsilon\_{jl}, \qquad O\_8 = L^l \bar{\epsilon}^c \bar{u}^c d^c H^l \epsilon\_{ij}. \tag{24}$$

We follow the BL numbering scheme, which was based on tracking the number of fermion fields in the operator rather than the mass dimension. The operators are separated in three groups with 2, 4, and 6 fermions. Some comments now need to be made about the schematic notation and what features are suppressed. The field-string defining each operator above completely defines the flavor content of that operator. Thus L ∼ (1, 2, −1/2) is the lepton doublet, Q ∼ (3, 2, 1/6) is the quark doublet, e <sup>c</sup> <sup>∼</sup> (1, 1, 1) is the isosinglet charged antilepton, d <sup>c</sup> <sup>∼</sup> (3, 1, 1 ¯ /3) is the isosinglet anti-down, <sup>u</sup> <sup>c</sup> <sup>∼</sup> (3, 1, ¯ −2/3) is the isosinglet anti-up, and H ∼ (1, 2, 1/2) is

O<sup>9</sup> = L i L j L k e c L l e c ǫijǫkl, O<sup>10</sup> = L i L j L k e cQ l d c ǫijǫkl, O11<sup>a</sup> = L i L jQ k d cQ l d c ǫijǫkl, O11<sup>b</sup> = L i L jQ k d cQ l d c ǫikǫjl, O12<sup>a</sup> = L i L jQ¯ iu¯ cQ¯ ju¯ c , O12<sup>b</sup> = L i L jQ¯ ku¯ cQ¯ <sup>l</sup>ǫijǫ kl , O<sup>13</sup> = L i L jQ¯ iu¯ c L k e c ǫjk, O14<sup>a</sup> = L i L jQ¯ ku¯ cQ k d c ǫij, O14<sup>b</sup> = L i L jQ¯ iu¯ cQ k d c ǫjk, O<sup>15</sup> = L i L j L k d c L¯ iu¯ c ǫjk, O<sup>16</sup> = L i L j e¯ c d c e¯ c u c ǫij, O<sup>17</sup> = L i L j d c d c ¯d c u¯ c ǫij, O<sup>18</sup> = L i L j d c u c u¯ c u¯ c ǫij, O<sup>19</sup> = L iQ j d c d c e¯ c u¯ c ǫij, O<sup>20</sup> = L i d cQ¯ iu¯ c e¯ c u¯ c

the Higgs doublet. The color indices and the different possible Lorentz structures are suppressed. In general, there are a number of independent operators corresponding to each flavor-string. For the dimension-7 list, operators O<sup>3</sup> and O<sup>4</sup> each have two independent possibilities for the contraction of the isospin indices, as explicitly defined above, but obviously a unique color contraction. Babu and Leung specify the independent internalindex contractions, but only make general remarks on the Lorentz structures, and we shall follow suit. To assist the reader to understand the notation, we write out the above operators more completely in standard 4-component spinor notation, but for scalar and pseudoscalar Lorentz structures only and with isospin indices suppressed:

$$\begin{aligned} O\_{\mathbb{Z}} &= LLe^{\varepsilon}H = \left[\overline{(L\_{\mathbb{L}})^{\varepsilon}}L\_{\mathbb{L}}\right]\left[\overline{e\_{\mathbb{R}}}L\_{\mathbb{L}}\right]H, \\ O\_{\mathbb{S}} &= LLQd^{\varepsilon}H = \left[\overline{(L\_{\mathbb{L}})^{\varepsilon}}L\_{\mathbb{L}}\right]\left[\overline{d\_{\mathbb{R}}}Q\_{\mathbb{L}}\right]H \text{ or } \left[\overline{(L\_{\mathbb{L}})^{\varepsilon}}Q\_{\mathbb{L}}\right]\left[\overline{d\_{\mathbb{R}}}L\_{\mathbb{L}}\right]H, \\ O\_{\mathbb{L}} &= LL\bar{Q}\bar{u}^{\varepsilon}H = \left[\overline{(L\_{\mathbb{L}})^{\varepsilon}}L\_{\mathbb{L}}\right]\left[\overline{Q\_{\mathbb{L}}}\mu\_{\mathbb{R}}\right]H, \\ O\_{\mathbb{R}} &= L\bar{e}^{\varepsilon}\bar{u}^{\varepsilon}d^{\varepsilon}H = \left[\overline{d\_{\mathbb{R}}}L\_{\mathbb{L}}\right]\left[\overline{(e\_{\mathbb{R}})^{\varepsilon}}\mu\_{\mathbb{R}}\right]H. \end{aligned} \tag{25}$$

Of course, these operators feature quark and charged-lepton fields in addition to neutrinos and Higgs bosons, so they do not by themselves produce neutrino masses. The charged fermion fields have to be closed off in a loop or loops to produce a neutrino self-energy graph which then generates a Weinbergtype operator, as per options 3 and 6. In fact, using this procedure and naive dimensional analysis one can estimate their matching contribution to the Weinberg operator, as done in de Gouvea and Jenkins [87]. In addition, every dimension-7 operator in Equation (24) may be multiplied by H†H to produce a dimension-9 generalization of that operator, just as O ′ 1 is a generalization of O1. At dimension 9, there are many more operators. Six of the flavor strings feature four fermion fields and three Higgs doublets:

$$\begin{aligned} \mathcal{O}\_5 &= L^i L^j Q^k d^c H^l H^m H^\dagger\_i \epsilon\_{jl} \epsilon\_{km}, & \mathcal{O}\_6 &= L^i L^j \bar{Q}\_k \bar{u}^c H^l H^k H^k\_i \epsilon\_{jl}, \\ \mathcal{O}\_7 &= L^i Q^j \bar{\mathcal{e}}^c \bar{Q}\_k H^k H^l H^m \epsilon\_{il} \epsilon\_{jm}, & \mathcal{O}\_{61} &= L^i L^j H^k H^l L^r \epsilon^c H^\dagger\_r \epsilon\_{ik} \epsilon\_{jl}, \\ \mathcal{O}\_{66} &= L^i L^j H^k H^l Q^l d^c H^\dagger\_r \epsilon\_{ik} \epsilon\_{jl}, & \mathcal{O}\_{71} &= L^i L^j H^k H^l Q^r u^c H^s \epsilon\_{ik} \epsilon\_{jl} \epsilon\_{rs}, \end{aligned} \tag{26}$$

Note that the operators O61,66,71 are the products of O<sup>1</sup> and the three SM Yukawa operators. Another 12 are six-fermion operators:

$$\begin{aligned} &\tilde{\epsilon}^{\varepsilon}\tilde{\epsilon}^{\varepsilon}\epsilon^{\varepsilon}\epsilon\_{ij}, \\ &\epsilon^{\varepsilon}\tilde{u}^{\varepsilon}\tilde{u}^{\varepsilon}\epsilon\_{ij}, \\ &\tilde{\epsilon}^{\varepsilon}\tilde{\epsilon}^{\varepsilon}\tilde{u}^{\varepsilon} \\ &\text{or} \\ &\text{or} \\ &\text{by } \text{In }\epsilon^{\varepsilon}\tilde{d}^{\varepsilon}\tilde{d}^{\varepsilon}\epsilon^{\varepsilon}e^{\varepsilon}, \text{ which generates the correct neutrino mass scale }\epsilon \\ &\text{g. only for a very low lepton-number violation scale. In case it is} \\ &\text{an consists entirely of the first generation SM fermions it is strongly} \\ &\text{an constrained by } 0\upsilon\beta\beta \text{ (generated at tree level by this operator)} \\ &\text{or} \qquad \text{The large number of dimension-1 1 operators can be found listed} \end{aligned}$$

in Babu and Leung [85] and de Gouvea and Jenkins [87]. de Gouvea and Jenkins [87] and Angel et al. [90] performed general analyses of diagram topologies for opening up these operators at tree-level using massive exotic scalars and either vector-like or Majorana fermion exotics, and consequently producing neutrino mass at various loop levels. The operators

$$\begin{array}{ccccccccc} \text{O}\_2, & \text{O}\_{3b}, & \text{O}\_{4a}, & \text{O}\_5, & \text{O}\_6, & \text{O}\_{61}, & \text{O}\_{66}, & \text{O}\_{71} \end{array} \tag{28}$$

can give rise to 1-loop neutrino mass models, while

$$\begin{array}{ccccccccc} \text{O}\_2, & \text{O}\_{3a}, & \text{O}\_{3b}, & \text{O}\_{4a}, & \text{O}\_{4b}, & \text{O}\_{5-10}, & \text{O}\_{11b}, & \text{O}\_{12a}, & \text{O}\_{13}, & \text{O}\_{14b},\\ & & & \text{O}\_{61}, & \text{O}\_{66}, & \text{O}\_{71} \end{array} \tag{29}$$

can produce 2-loop models. The set

$$O\_{11a}, \ O\_{12b}, \ O\_{14a}, \ O\_{15-20} \tag{30}$$

can form the basis for neutrino mass to be generated at three or more loops.

In each of these cases, one may derive an indicative upper bound on the scale of new physics from the requirement that at least one neutrino mass be at least 0.05 eV in magnitude. For example, for operators involving first generation<sup>11</sup> quarks this bound can be estimated as follows: Operator O19, which can be opened up to give a 3-loop neutrino mass contribution,

<sup>11</sup>The bound on the scale of new physics is generally higher for operators involving heavier quarks.

has the lowest upper bound on the new physics scale of about 1 TeV (apart from u cu c ¯d c ¯d c e c e c ). The highest is about 4 <sup>×</sup> <sup>10</sup><sup>9</sup> TeV for the 1-loop case of O4a. These estimates come from an examination of the loop contribution to neutrino mass only, and do not take into account other phenomenological constraints that will exist for each complete model. As part of that, any unknown coupling constants, such as Yukawas that involve the exotic fermions and/or scalars were set to unity. In a realistic theory, many of these constants would be expected to be less than one, which would bring the scale of new physics to lower values. In any case, one can see that the required new physics, even for 1-loop models, is typically more testable than the type-I, II, and III seesaw models. Some high loop models, as the O<sup>19</sup> case demonstrates, have very low scales of new physics and some may even be ruled out already. At the dimension-11 operator level, so not explicitly discussed here, there are even examples which can at best produce a 5-loop neutrino mass contribution. Those models are definitely already excluded. Examples of full models that are associated with specific operators will be presented in later sections.

B. Number of loops. A complementary perspective on the spectrum of possible radiative neutrino mass models is provided by adopting the number of loops as the primary consideration rather than the type of 1L = 2 effective operator that dominates the new physics. Equations (28–30) already form the basis for such a classification for type 3 and type 6 scenarios, but a more general analysis will also capture the type 2 and type 5 possibilities.

At j-loop order, neutrino masses are typically given by

$$m\_{\nu} \sim C \left(\frac{1}{16\pi^2}\right)^j \frac{\nu^2}{\Lambda} \tag{31}$$

for the O<sup>1</sup> associated options 2 and 3, and

$$m\_{\nu} \sim C \left(\frac{1}{16\pi^2}\right)^j \frac{\nu^4}{\Lambda^3} \tag{32}$$

for the O ′ 1 cases of options 5 and 6, where v ≡ √ <sup>2</sup>hH<sup>0</sup> i ≃ 100 GeV, and 3 is the new-physics scale where lepton number is violated by two units. All coupling constants, and for some models also certain mass-scale ratios, are absorbed in the dimensionless coefficient C. In order to explain the atmospheric mass splitting lower bound of 0.05 eV, we obtain an upper limit on the new physics scale 3 of 10<sup>5</sup> C TeV for 3-loop models and 10 C TeV for 5-loop models corresponding to the O<sup>1</sup> cases, and 10 C <sup>1</sup>/<sup>3</sup> TeV for the O ′ 1 case at 3-loop order. Constraints from flavor physics severely constrain the scale of new physics and the couplings entering in C. In addition, in models which feature explicit 1L = 2 lepton-number violation through trilinear scalar interactions, the latter cannot be arbitrarily large because otherwise they have issues with naturalness (see Herrero-García et al. [91] for the case of the Zee model) and charge/color breaking minima (see Frere et al. [92], Alvarez-Gaume et al. [93] and Casas and Dimopoulos [94] for studies in the context of supersymmetry and Herrero-Garcia et al. [95] for the case of the Zee-Babu model). Thus, apart from a few 4-loop models [96–98] which compensate the loop suppression by a high multiplicity of particles in the loop, the vast majority of radiative neutrino mass models generate neutrino mass at 1-, 2-, or 3-loop level. We therefore focus on these cases.

1-loop topologies for O<sup>1</sup> = LLHH. The opening up of the Weinberg operator at 1-loop level has been systematically studied in Ma [99] and Bonnet et al. [100]. The authors of Bonnet et al. [100] identified 12 topologies which contribute to neutrino mass. Among all the topologies and possible Lorentz structures, topology T2 cannot be realized in a renormalizable theory. For the other topologies, the expression for neutrino mass and the possible particle content for electroweak singlet, doublet, and triplet representations is listed in the appendix of Bonnet et al. [100]. The divergent ones, T4-1-i, T4-2-ii, T4-3-ii, T5 and T6, need counter-terms to absorb the divergences, which are indeed tree-level realizations of the Weinberg operators. Furthermore, for T4-1-ii, there is no mechanism to forbid or suppress the tree-level contribution from Weinberg operator, such as extra discrete symmetry or U(1). Therefore, there are in total six topologies which generate neutrino mass via a genuine<sup>12</sup> 1 loop diagram: T1-i, T1-ii, T1-iii, T3, T4-2-i, T4-3-i, which are depicted in **Figure 3**. Depending on the particle content, the topologies do not rely on any additional symmetry. However, the topologies T4-x-i require a discrete Z<sup>2</sup> symmetry in addition to demanding Majorana fermions in the loop with lepton-number conserving couplings. This is difficult to achieve in a field theory, as lepton-number is necessarily broken by neutrino mass. For example, in topology T4-2-i the scalar connected to the two Higgs doublets H is necessarily an electroweak triplet and thus its direct coupling to two lepton doublets L is unavoidable. This coupling induces a type-II seesaw tree-level contribution to neutrino mass. Similar arguments hold for the other topologies T4-x-i.

1-loop topologies for O′ <sup>1</sup> <sup>=</sup> LLHH(H†H). A similar analysis has been performed for 1-loop topologies that give rise to the dimension-7 generalized Weinberg operator [101]. Of the 48 possible topologies, only the eight displayed in **Figure 4** are relevant for genuine 1-loop models. For specific cases, not all of these eight diagrams will be realized. The three-point vertices can be Yukawa, gauge or cubic scalar interactions, while the four-point vertices only contain scalar and gauge bosons.

2-loop topologies for O<sup>1</sup> = LLHH. A systematic analysis of 2-loop openings of O<sup>1</sup> was performed in Aristizabal Sierra et al. [102]. **Figure 5** displays the topologies identified in this study as able to contribute to genuine 2-loop models. There are additional 2-loop diagrams – that were termed "class II" – that have the form of one of the 1-loop topologies of **Figure 3** with one the vertices expanded into a 1-loop subgraph. They remark the class II topologies may be useful for justifying why a certain vertex has an unusually small magnitude.

C. Other considerations. We now briefly survey other perspectives on classifying or discriminating between neutrino mass models.

<sup>12</sup>In a genuine n-loop neutrino mass model, only diagrams starting from n-loop order contribute to neutrino mass. There are no tree level or lower order loop contributions.

FIGURE 3 | Feynman diagram topologies for 1-loop radiative neutrino mass generation with the Weinberg operator O<sup>1</sup> = LLHH. Dashed lines could be scalars or gauge bosons if allowed.

One suggested criterion is complexity [103]. While recognizing that sometimes nature appears to favor minimal possibilities (in an Occam's razor approach), and at other times not (e.g., the old problem of why there are three families), it does make sense to rank neutrino mass models on some sensible measure of how complex they are. Law and McDonald [103]

proposes a hierarchy based on (i) whether or not the model relies on the imposition of ad hoc symmetries, (ii) the number of exotic multiplets required, and (iii) the number of new parameters. Interestingly, they construct radiative models that are even simpler, on the basis of these criteria, than the 1-loop Zee-Wolfenstein model [104, 105]. However, like the Zee-Wolfenstein model, while these models generate non-zero neutrino masses, they fail phenomenologically. Thus, we must conclude that if nature utilizes the radiative mechanism, it will be non-minimal.

Another consideration for Majorana mass models is the important phenomenological connection to 0νββ decay [106–108]. Just as Majorana neutrino mass models may be systematically constructed through opening up 1L = 2 effective operators, models for 0νββ decay can be analysed by opening up the u¯udd ¯ e¯e¯ family of operators. The neutrino mass and 0νββ decay considerations are of course connected, but the nature of the relationship is model-dependent. An interesting situation would emerge in a hypothetical future where 0νββ decay is observed, but the standard Majorana neutrino exchange contribution through mββ is contradicted by, for example, cosmological upper bounds on the absolute neutrino mass scale. That would point to a non-minimal framework, which may be connected with radiative neutrino mass generation.

A further interesting aspect is the existence or otherwise of a deep theoretical reason for a given radiative model. At first sight, each such model looks random. However, some of them can be connected with, for example, grand unified theories (GUTs). One simple point to make is that exotics, such as scalar leptoquarks, that often feature in radiative models can be components of higher-dimension multiplets of SU(5) and SO(10). Also, by contributing to renormalization group running, some of them can assist with gauge coupling constant unification [109]. If they are to be light enough to play these roles, while other exotics within the multiplets have, for example, GUT-scale masses, then we face a similar issue to the famous doublet-triplet splitting problem. Nevertheless, this is a starting point for investigating the possible deeper origin of some of the required exotics. Another interesting GUT-related matter was analysed in depth in de Gouvêa et al. [88]. A necessary condition for a 1L = 2 operator of a certain mass dimension to be consistent with a GUT origin is that it occurs as a term in an effective operator of the same mass dimension derived with grand unified gauge invariance imposed. For example, the dimension-7 operator O3<sup>a</sup> from Equation (24) does not appear as a component in any SU(5) operator of the same dimension. On the other hand, other SM operators are embedded in the same GUT operator, with only one of them being able of giving the dominant contribution to neutrino masses. In addition to the question of the mere existence of SM-level operators in GUT decompositions, grand unification also imposes relations between SM-level operators, including some that violate baryon number and generate B−L violating nucleon decays and/or neutron-antineutron oscillations, leading to additional constraints. In the end, the authors of de Gouvêa et al. [88] conclude that only a small subset of SM 1L = 2 operators are consistent with grand unification.

Another strategy for uncovering a deeper origin for a radiative model is by asking if a given model has some close connection with the solution of important particle physics problems beyond just the origin of neutrino mass. One that has been explored at length in the literature is a possible connection to dark matter. Examples of such models will be given in more detail in later sections. Here, we simply mention some systematic analyses of what new symmetries can be imposed in radiative models to stabilize dark matter [110, 111]. Farzan et al. [110] classified the symmetries Gν that can be imposed in order to ensure that the first non-zero contribution to O<sup>1</sup> occurs at a given loop order, by forbidding all potential lower-order contributions. All standard model particles are singlets under Gν , implying that the lightest of the exotics that do transform under this symmetry must be stable if the symmetry remains exact, establishing a connection with dark matter. Restrepo et al. [111] performed a systematic analysis ofradiative models in a certain class in order to find those that have viable dark matter candidates. The considered models are those that generate mass at 1-loop level using exotics that are at most triplets under weak isospin, and where the stabilizing symmetry is Z2. They found 35 viable models. A similar analysis, but requiring 2-loop neutrino mass generation, can be found in Simoes and Wegman [112].

Besides dark matter, radiative neutrino mass models may also be connected to other physics beyond the SM such as the anomalous magnetic moment of the muon, the strong CP problem, the baryon asymmetry of the Universe or B-physics anomalies, among others. Phenomenology related to radiative neutrino mass models is briefly discussed in section 4 in general and an example of a possible connection to the recent B-physics anomalies is presented in section 5.4.

### 3. RADIATIVE GENERATION OF NEUTRINO MASSES

We adopt the classification of radiative neutrino mass models according to their Feynman diagram topology13, but refer to the other classification schemes where appropriate. In particular, we indicate the lowest-dimensional non-trivial 1L = 2 operator which is generated beyond the Weinberg operator LLHH. These 1L = 2 operators capture light particles which are in the loop to generate neutrino mass and are very useful to identify relevant low-energy phenomenology.

In the subsections 3.1–3.3 we classify Majorana neutrino mass models proposed in the literature according to their topology and specifically discuss models with SM gauge bosons in the loop in section 3.4. In section 3.5 we review Dirac neutrino mass models and briefly comment on models based on the gauge group SU(3)<sup>c</sup> × SU(3)<sup>L</sup> × U(1)<sup>X</sup> in section 3.6.

### 3.1. 1-Loop Majorana Neutrino Mass Models

This section is divided into several parts: (i) 1-loop UV completions of the Weinberg operator, (ii) 1-loop seesaws, (iii) UV completions with additional VEV insertions, (iv) 1-loop UV completions of the higher dimensional operators and (v) other 1-loop models. Notice that the first part includes models with multi-Higgs doublets, while the second part discusses external fields which transform under an extended symmetry. Besides the genuine topologies discussed in section 2, there are models based on the non-genuine 1-loop topologies in **Figure 6**.

#### 3.1.1. Weinberg Operator LLHH

We follow the general classification of UV completions of the Weinberg operator at 1-loop [100] discussed in section 2.2.2. The six genuine topologies are shown in **Figure 3**. Analytic expressions for all 1-loop topologies are listed in the appendix of Bonnet et al. [100].

Here we list the theories falling into respective categories. As the topologies stay the same while incorporating multiple Higgs doublets, theories with more than one Higgs doublet will also be listed here. Models in which the generation of neutrino mass relies on additional VEVs connected to the neutrino mass loop diagram are discussed in section 3.1.3. We first discuss the models based on topology T3, the only one with a quartic scalar interaction, before moving on to the other topologies.

### **T3**

Topology T3 is one of the most well-studied. It was first proposed in Ma [99] and its first realization, the scotogenic model with a second electroweak scalar doublet and sterile fermion singlets (at least two) both odd under a Z<sup>2</sup> symmetry, was later proposed in Ma [113]. See section 5.3 for a detailed discussion of the model. Its appeal lies in the simultaneous explanation of dark matter, which is stabilized by a Z<sup>2</sup> symmetry. A crucial ingredient is the quartic scalar interaction (H†η) 2 (see Equation 95) of the SM Higgs boson H with the electroweak scalar doublet η in the loop. This scalar interaction splits the masses of the neutral scalar and pseudoscalar components of η. Neutrino masses vanish in the limit of degenerate neutral η scalar masses. Several variants of the scotogenic model have been proposed in the literature: with triplet instead of singlet fermions [114–116], an extension with an additional singlet scalar [117], one fermionic singlet and two additional electroweak scalar doublets [118], scalar triplets [119], colored scalars and fermions [120, 121], a vector-like fermionic lepton doublet, a triplet scalar, and a neutral [122, 123] or charged [124] singlet scalar, vector-like doublet and singlet fermions and doublet scalar, which contains a doubly charged scalar [125], higher SU(2) representations [126– 129], an extended discrete symmetry with Z<sup>2</sup> × Z<sup>2</sup> [130, 131] or Z<sup>2</sup> × CP [132], a discrete flavor symmetry based on S<sup>3</sup> [133], A<sup>4</sup> [134–137], 1(27) [138, 139], which is either softly-broken or via electroweak doublets, and its embedding in (grand) unified theories [137, 140–143]. Finally, the authors of Megrelidze and Tavartkiladze [144] proposed the generation of neutrino mass via lepton-number-violating soft supersymmetry-breaking terms. In particular the generation of the dimension-4 term (LH˜ u) <sup>2</sup> with left-handed sleptons L˜ leads to models based on the topology T-3 with supersymmetric particles in the loop. Another variant involves a global continuous dark symmetry [145], Hagedorn, (in prep), termed the generalized scotogenic model.

### **T1-i**

Ma [146] discusses a supersymmetrized version of the scotogenic model, which is based on topology T3 and we discuss in detail in section 5.3. The topology necessarily differs from T3 because the term (H†η) 2 is not introduced by D-terms. An embedding of this model in SU(5) is given in Ma [147]. In a non-supersymmetric context, the same topology is discussed in Farzan [117], which introduces one real singlet scalar, in the context of a (dark) left– right symmetric model [148, 149], and in Budhi et al. [150], Kashiwase and Suematsu [151], and Budhi et al. [152], which introduce multiple singlet scalars to connect the two external Higgs fields. The term (H†η) 2 , which is essential to generate topology T3, is neglected in Budhi et al. [150], Kashiwase and Suematsu [151], and Budhi et al. [152] and thus neutrino mass is generated via topology T1-i. One of the singlet scalars in the neutrino mass model can be the inflaton via a nonminimal coupling with the Ricci-scalar. The term (H†η) 2 can be explicitly forbidden by imposing a U(1) symmetry, which is softly broken by the CP-violating mass term χ <sup>2</sup> of a complex scalar field χ [153]. Finally the authors of Lu and Gu [154] proposed a model with electroweak singlet and triplet scalars as well as fermions and study the dark matter phenomenology and leptogenesis.

#### **T1-ii**

Among the models based on the topology T1-ii, there are four possible operators which models are based on. Besides models

<sup>13</sup>Note that diagrams with scalar or vector bosons are equivalent from a topological point of view.

with only heavy new particles, there are models with SM charged leptons, down-type quarks, or up-type quarks in the loop, which are based on the operators O<sup>2</sup> and O3, respectively. We first discuss the models based on operator O2. The first radiative Majorana neutrino mass model, the Zee model [104], is based on this operator. See section 5.1.1 for a detailed discussion of its phenomenology. Several variants of the Zee model exist in the literature. The minimal Zee-Wolfenstein model [105] with a Z<sup>2</sup> symmetry to forbid tree-level FCNCs has been excluded by neutrino oscillation data [155, 156], while the general version with both Higgs doublets coupling to the leptons is allowed [91, 157]. Imposing a Z<sup>4</sup> symmetry [158] allows to explain neutrino data and forbid tree-level FCNCs in the quark sector. Previously in Aranda et al. [159] a flavor-dependent Z<sup>4</sup> symmetry was used to obtain specific flavor structures in the quark and lepton sector. A supersymmetric version of the Zee model has been proposed in Leontaris and Tamvakis [160], Haba et al. [161], Cheung and Kong [162], and Kanemura et al. [163]. Its embedding into a grand unified theory has been discussed in Zee [104], Tamvakis and Vergados [164] and Fileviez Perez and Murgui [165], and in models with extra dimensions in Chang and Ng [166] and Chang et al. [167].

Other flavor symmetries beyond Z<sup>4</sup> have been studied in Babu and Mohapatra [168, 169], Koide and Ghosal [170], Kitabayashi and Yasue, [171], Adhikary et al. [172], Fukuyama et al. [173], Aranda et al. [174, 175], and Okamoto and Yasue [176] studied the Zee model when the third generation transforms under a separate SU(2) × U(1) group. Babu and Mohapatra [168, 169] studied large transition magnetic moments of the electron neutrino, which was an early, now excluded, explanation for the solar neutrino anomaly. General group theoretic considerations about the possible particle content in the loop are discussed in Ma [99].

Models with multiple leptoquarks, which mix among each other, also generate neutrino mass via topology T1-ii. We discuss this possibility in more detail in section 5.4.1. They induce the operator O<sup>3</sup> if the leptoquark couples down-type quarks to neutrinos. Well-studied examples of leptoquarks are down-type squarks in R-parity violating SUSY models, which generate neutrino masses, as was first demonstrated in Hall and Suzuki [177]. Specific examples with multiple leptoquarks which mix with each other were discussed in Nieves [178], Chua et al. [179], Mahanta [180], Aristizabal Sierra et al. [181], Helo et al. [182], Päs and Schumacher [183], Cheung et al. [184], Doršner et al. [185]. There are several supersymmetric models [179, 186– 190] which generate neutrino mass via different down-type quarks or charged leptons in the loop and consequently induce the operators O<sup>3</sup> and O2, respectively. Finally, there are models with only heavy particles in the loop such as the inert Zee model [191] or supersymmetric models with R-parity conservation [192, 193].

### **T1-iii**

This topology was first proposed in Ma [99] and it naturally appears in the supersymmetrized version of the scotogenic model [146, 147, 194–203] together with topology T1-i. The topology can be used to implement the radiative inverse seesaw [204–206], which resembles the structure of the inverse seesaw [207, 208]. This model has been extended by a softlybroken non-Abelian flavor symmetry group [209–211] in order to explain the flavor structure in the lepton sector. The SUSY model in Ma and Sarkar [212] generates neutrino mass via sneutrinos and neutralinos in the loop. This mechanism was first pointed out in Hirsch et al. [213]. In the realization of Ma and Sarkar [212], the masses of the real and imaginary parts of the sneutrinos are split by the VEV of a scalar triplet, which only couples to the sneutrinos via a soft-breaking term and thus does not induce the ordinary type-II seesaw. Similarly it has been used in a model with vector-like downtype quarks [214, 215], which requires mixing of the SM quarks with the new vector-like quarks. This model leads to the operator O3.

#### 3.1.2. 1-Loop Seesaws and Soft-Breaking Terms

For completeness we also include the two possible 1-loop seesaw topologies T4-2-i and T4-3-i which have been identified in Bonnet et al. [100]. Topology T4-2-i always involves a electroweak scalar triplet like in the type-II seesaw mechanism and topology T4-3-i contains an electroweak singlet or triplet fermion like in the type-I or type-III seesaw mechanism, respectively. Based on our knowledge, there are currently no models based on topologies T4-2-i and T4-3-i in the literature.

Finally, although the topology T4-2-ii shown in **Figure 6C** has been discarded in Bonnet et al. [100], because it is generally accompanied by the tree-level type-II seesaw mechanism, there are three models based on this topology [216–218]. They break lepton number softly by a dimension-2 term and thus there is no tree-level contribution by forbidding the "hard-breaking" dimension-4 terms which are required for the type-II seesaw mechanism. Similar constructions may be possible for other topologies and lead to new interesting models.

### 3.1.3. Additional VEV Insertions

The above discussed classification technically does not cover models with additional scalar fields, which contribute to neutrino mass via their vacuum expectation value in contrast to being a propagating degree of freedom in the loop. Inspired by the above classification, we similarly classify these new models according to the topologies in **Figure 3** by disregarding the additional VEV insertions.

#### **T1-i**

There are several radiative neutrino mass models which are based on a U(1) symmetry, which is commonly broken to a remnant Z<sup>2</sup> symmetry: there are models based on a global Peccei-Quinn U(1)PQ symmetry [219, 220], which connects neutrino mass to the strong CP problem, a local U(1)B−<sup>L</sup> symmetry [221–223] and local dark U(1) symmetry [224–226]. The authors of Ho et al. [221] systematically study radiative neutrino mass generation at 1-loop (but also 2-, and 3-loop) level based on a gauged U(1)B−<sup>L</sup> symmetry, which is broken to a Z<sup>N</sup> symmetry. The models in Chang and Wong [224], Dasgupta et al. [219], Lindner et al. [225], Adhikari et al. [227], Kownacki and Ma [226] also have a contribution to neutrino mass at 2-loop order based on a Cheng-Li-Babu-Zee (CLBZ) topology.

#### **T1-ii**

All of the models with additional VEV insertions rely on the breaking of a symmetry: left–right symmetry [228– 230], a more general SU(2)<sup>1</sup> × SU(2)<sup>2</sup> symmetry [231], a flavor symmetry [232–234], U(1)B−<sup>L</sup> [235], and dilation symmetry [236]. All these models lead to the operator O2. Foot et al. [236] discusses in particular the following two 1-loop models: the scale-invariant Zee model and a scale-invariant model with leptoquarks which induces O3. Finally, there is the inert Zee model with a flavor symmetry [237, 238].

#### **T1-iii**

The model in Nomura et al. [239] relies on the VEVs of a scalar triplet and a septuplet which are subject to strong constraints from electroweak precision tests in particular from the T (or ρ) parameter. The minimization of the potential is not discussed, but the VEVs can in principle be introduced via the linear term in the scalar potential, which leads to the operator O ′′′′ 1 at 2 loop level, because the linear term for the septuplet is only induced at the 1-loop level. The topology can also be generated by new heavy lepton-like doublets and sterile fermions, which are charged under a new gauged dark U(1) in addition to a Z<sup>2</sup> symmetry [240].

#### **T3**

There are several variants of the scotogenic model with additional VEV insertions. Most of them are based on an extended symmetry sector, such as a discrete Z<sup>3</sup> instead of a Z<sup>2</sup> symmetry [241, 242], dilation symmetry [236, 243–245], a gauged U(1)B−<sup>L</sup> [246–250], global U(1)B−<sup>L</sup> [251], a general gauged U(1) [252–254], continuous U(1) flavor symmetry [255, 256], a discrete flavor symmetry based on D<sup>6</sup> [257], A<sup>4</sup> [258– 262] or S<sup>4</sup> [263], and different LR symmetric models without a bidoublet [264]. Apart from additional symmetries, the mixing of the fermionic singlet with a fermionic triplet in the loop requires the VEV of an electroweak triplet with vanishing hypercharge [265–267]. Finally, the two models discussed in Okada and Yagyu [268, 269] rely on a similar topology as the scotogenic model, but with triplet VEVs instead of electroweak doublet VEVs.

#### **T4-2-i**

Based on our knowledge, there are currently no models based on topology T4-2-i in the literature.

### **T4-3-i**

Wang and Han [270] proposed a model which reduces to topology T4-3-i after breaking of the U(1)B−<sup>L</sup> symmetry. As the Majorana mass term for the fermionic pure singlet is not introduced, there is no inverse seesaw contribution to neutrino mass after the breaking of the U(1)B−<sup>L</sup> symmetry and neutrino masses are generated at 1-loop level.

### **T4-1-i/ii**

These types of models contain a triplet scalar which couples to the lepton doublet as per the tree-level type-II seesaw. However, the neutral component of the triplet scalar gets an induced VEV at 1-loop and thus generates neutrino masses effectively at 1-loop. The model in Nomura and Okada [271] is based on topology T4-1-i shown in **Figure 6A**, which is finite due to additional VEV insertions on the fermion line. The model in Kanemura and Sugiyama [272] is based on topology T4-1-ii shown in **Figure 6B**. The tree-level contribution is forbidden by a discrete symmetry and renormalizability of the theory. However, at looplevel neutrino mass is generated by a dimension-7 operator LLHHs<sup>2</sup> <sup>1</sup> with two additional SM singlet fields s1. Note in both cases an extra symmetry such as U(1)B−<sup>L</sup> or a discrete symmetry and lepton number is needed to forbid the contribution from the tree-level type-II seesaw. Topology T4-1-ii is also induced in the SUSY model in Figueiredo [273] and Franceschini and Mohapatra [201] after the breaking of SUSY and the discrete Z<sup>4</sup> symmetry.

#### 3.1.4. Higher-Dimensional Weinberg-Like Operators

Apart from UV completions of the Weinberg operator, there are a few models which induce one of the higher dimensional operators with additional Higgs doublets at 1-loop level.

#### **Dimension-7 (**O ′ 1 **)**

The first model which induced the dimension-7 operator O ′ 1 at 1-loop level in a two Higgs doublet model was proposed in Kanemura and Ota [274]. It was realized using at most adjoint representations and an additional softly-broken Z<sup>5</sup> symmetry and an exact Z<sup>2</sup> symmetry and thus allows to use the topologies T12 (**Figure 4E**) and T31 (**Figure 7**), which would otherwise be accompanied by the dimension-5 operator O1. If the Zee model is extended by a triplet Majoron [275, 276] the operator O ′ 2 = LLLecH(H†H) is induced at tree-level. After closing the loop of charged leptons via topology T3 (**Figure 4B**), the dimension-7 operator O ′ 1 is obtained. Cepedello et al. [101] systematically studies the possible 1-loop topologies of O ′ 1 and explicitly shows

several models: the only genuine model without representations beyond the adjoint of SU(2) is based on topology T11, while the other models use quadruplets or even larger representations to realize the other genuine topologies.

#### **Dimension-9 (**O ′′ 1 **)**

In Law and McDonald [277] and Baldes et al. [278] neutrino masses are generated via a radiative inverse seesaw. The mass of the additional SM singlets is induced at tree-level and then first transmitted to the neutral components of new electroweak doublets via a 1-loop diagram, before it induces neutrino mass via the seesaw. It leads to the dimension-9 operator O ′′ 1 via the four VEV insertions on the scalar line of the 1-loop diagram. There is also a 2-loop contribution, which may dominate neutrino mass depending on the masses of the new particles.

#### **Dimension-11 (**O ′′′ 1 **)**

The model proposed in Aranda and Peinado [279] relies on the VEV of a 7-plet χ, which is induced via a non-renormalizable coupling, linear in χ, to six electroweak Higgs doublets.

As can be seen from the discussion above, in order to generate Weinberg-like effective operators at dimension larger than five, typically extra symmetries (in some cases large discrete symmetries), new large representations, a large number of fields or a combination of all the previous need to be invoked. This makes the model-building of such scenarios much more involved than for the case of the Weinberg operator.

#### 3.1.5. Other 1-Loop Models

Apart from the models in the general classification [100], it is possible to generate neutrino mass via a radiative inverse seesaw mechanism shown in **Figure 8** at 1-loop order, which has been proposed in Ahriche [280]. Tree-level contributions are forbidden by a softly-broken Z<sup>4</sup> symmetry. The soft-breaking is indicated by the cross on the scalar line. Note the cross on the fermion line in the loop denotes a Majorana mass term, while the other two denote Dirac mass terms.

Finally we would like to comment on one further possibility to generate neutrino mass at 1-loop order. If the neutrino masses vanish at tree-level in type-I seesaw model, then 1-loop electroweak corrections give the leading contribution [281]:<sup>14</sup>

non-zero neutrino masses are induced by finite 1-loop diagrams with either a Z-boson or a Higgs boson. The UV divergent part of the 1-loop corrections to the Weinberg operator cancel due to the absence of a tree-level contribution. This has been explicitly shown in Pilaftsis [281] with a calculation in the mass basis. In terms of the classification of 1-loop topologies, these diagrams correspond to the topologies T3 and T1-iii for the Higgs and Z-boson in the loop, respectively. The vanishing of the treelevel contribution can be achieved using a specific texture in the seesaw model with SM singlet fermions S [285] in addition to the right-handed neutrinos N

$$\begin{pmatrix} 0 & m\_D & 0\\ \cdot & \mu\_R & M\_N^T\\ \cdot & \cdot & \mu\_S \end{pmatrix} \tag{33}$$

in the basis (ν, N, S). In the limit µ<sup>S</sup> → 0 the treelevel contribution to the active neutrinos exactly vanishes and neutrino masses are generated at 1-loop order. This construction has been denoted minimal radiative inverse seesaw [285].

This texture can be obtained by imposing a U(1) symmetry under which S is charged. After it is spontaneously broken by the VEV of a SM singlet scalar η, the Yukawa interaction SNη generates the term M<sup>N</sup> without generating a Majorana mass term µ<sup>S</sup> for the fermionic singlets S or a coupling of S to the SM lepton doublets L at the renormalizable level.

### 3.2. 2-Loop Majorana Neutrino Mass Models

The possible 2-loop topologies of the Weinberg operator have been discussed in Aristizabal Sierra [102]. We will closely follow this classification. All possible genuine 2-loop topologies are shown in **Figure 5**. Analytic expressions for the 2-loop diagrams are summarized in the appendix of Aristizabal Sierra [102] and are based on the results in McDonald and McKellar [286] and Angel et al. [287]. Most topologies can be considered as variations of a few 2-loop models discussed in the literature: (i) variations of the Cheng-Li-Babu-Zee (CLBZ) topology [76, 288, 289], (ii) the Petcov-Toshev-Babu-Ma (PTBM) topology [290–292], and the so-called rainbow (RB) topology [102]. In the following we will further distinguish between fermion and scalar lines and show in **Figures 9**, **10B** the relevant diagrams of genuine topologies and the internal-scalar-correction (ISC)-type topology which are used in the following discussion. The first two subsections discuss

<sup>14</sup>The finite 1-loop corrections to the active neutrino mass matrix in the seesaw model were first discussed in Grimus and Neufeld [282] with an arbitrary number of right-handed neutrinos, left-handed lepton doublets, and Higgs doublets. The finite 1-loop corrections are particularly important in case of delicate cancellations

in the tree-level neutrino mass terms, which have been studied in Aristizabal Sierra and Yaguna, [283] using the result of Grimus and Lavoura [284].

models based on genuine topologies, the third one models based on non-genuine topologies, and the last one models based on multiple topologies.

### 3.2.1. Genuine 2-Loop Topologies

The relevant diagrams for the genuine topologies are shown in **Figure 9**.

### **CLBZ-1**

The topology CLBZ-1 is displayed in **Figure 9A**. The first model was independently proposed and studied by Zee [288] and Babu [289], and is commonly called Zee-Babu model (See a more detailed discussion in section 5.1.2). It also leads to the operator O9. A scale-invariant version of the model has been proposed in Foot et al. [236]. It has been extended to include a softlybroken continuous L<sup>e</sup> − L<sup>µ</sup> − L<sup>τ</sup> flavor symmetry [293, 294] or discrete flavor symmetry [295], and has been embedded in a SUSY model [296, 297]. The same topology has also been used for models with quarks instead of charged leptons inside the loop. They rely on the introduction of a leptoquark and a diquark [298– 300] and lead to operator O11. Similarly, there is a version without light fields in the loop [221, 301–303]. The models in Ho et al. [221] are part of a systematic study of models based on a gauged U(1)B−<sup>L</sup> which is broken to a Z<sup>N</sup> symmetry.

### **CLBZ-3**

Topology CLBZ-3 is depicted in **Figure 9B** and only differs from topology CLBZ-1 in the way how the Higgs VEVs are attached to the loop diagram: Topology CLBZ-3 has the Higgs VEVs attached to two of the scalar lines, while they are attached to the internal fermion lines for CLBZ-1. Cheng and Li [76] listed several possible neutrino mass models, including the first 2-loop model which was based on topology CLBZ-3 with an effective scalar coupling. A possible UV completion was presented with an electroweak quintuplet scalar. This UV completion leads to the operator O<sup>33</sup> = ¯e c e¯ cL iL j e c e <sup>c</sup>HkH<sup>l</sup> ǫikǫjl (with an additional VEV insertion from an electroweak quintuplet scalar). All models [221, 304–307] based on topology CLBZ-3 only contain heavy fields.

#### **CLBZ-8**

The topology is shown in **Figure 9C**. Variants of the Zee-Babu model have also been embedded in grand unified theories [308].

In case of SU(5), there is a 5-plet of matter particles in the loop which leads to the effective operators O<sup>9</sup> and O11.

#### **CLBZ-9**

Topology CLBZ-9 which is displayed in **Figure 9D** has been utilized in a model with two diquarks [215].

#### **CLBZ-10**

The same paper also introduces another model with two diquarks which is based on topology CLBZ-10, shown in **Figure 9E**.

#### **PTBM-1**

The first model to utilize the topology **Figure 9F**, although in presence of a tree-level contribution, was presented in Petcov and Toshev [290], Babu and Ma [291], Branco et al. [292] and Babu and Ma [309]. Neutrino mass receives a 2-loop correction via the exchange of two W-bosons as shown in **Figure 9F**. This idea has been recently revived and experimentally excluded in the context of extra chiral generations [310], but the mechanism can still work in the case of vector-like leptons. Lepton number is violated by the SM singlet Majorana fermion N in the center of the diagram and thus there is a tree-level contribution in addition to the 2-loop contribution to neutrino mass. Lepton number can equally well be broken by the type-III seesaw, when the fermionic singlet is replaced by a fermionic triplet [311]. The model in Babu and Julio [312] has one of the W-bosons replaced by scalar leptoquarks and it is consequently not accompanied by a tree-level contribution. The 1-loop contribution induced by the mixing of the leptoquarks vanishes, because the left-chiral coupling of one of the leptoquarks is switched off [313]. All models with W bosons will lead to operators with derivatives in the classification according to 1L = 2 operators. Finally, Angel et al. [287] proposed a model with a scalar leptoquark and colored octet fermion.

#### 3.2.2. Genuine Topologies with Additional VEV Insertions

Similar to the 1-loop models, we also categorize the models with additional VEV insertions following the classification of Aristizabal Sierra et al. [102].

#### **CLBZ-1**

There are several models based on topology CLBZ-1 (shown in **Figure 9A**), which all induce the operator O9. Bamba et al. [314] discusses a possible connection of neutrino mass with dark energy. Porto and Zee [315] proposed a model with one electroweak Higgs doublet field per lepton generation, an extension of the so-called private Higgs scenario. Finally, Lindner et al. [316] discusses an extension of the Zee-Babu model by a global U(1)B−<sup>L</sup> symmetry, which is spontaneously broken to a Z<sup>2</sup> subgroup. This implies the existence of a Majoron and a DM candidate.

#### **CLBZ-3**

Chang and Keung [317] proposed a variant of the Zee-Babu model with an additional triplet Majoron, which is based on topology CLBZ-3 which is displayed in **Figure 9B**.

#### **CLBZ-9**

The topology CLBZ-9 is depicted in **Figure 9D**. The model in Guo [318] is based on a dark U(1) symmetry with only heavy fields in the loop.

#### **RB-2**

The model proposed in Kajiyama et al. [319] is based on U(1)B−L, which is broken to Z2. Apart from the VEV breaking U(1)B−L, neutrino mass is generated by a diagram with topology RB-2 which is shown in **Figure 9G**.

#### 3.2.3. Non-genuine Topologies

The relevant non-genuine 2-loop topologies are shown in **Figure 10**.

#### **NG-RB-1**

The non-genuine topology NG-RB-1 (**Figure 10A**) is generated in Nomura and Okada [320]. There are no lower-order contributions due to the U(1) symmetry, which is broken to Z<sup>2</sup> as in the above-mentioned models.

#### **Other non-genuine topologies**

There are several models which generate vertices or masses of particles at loop level. The models in Aoki et al. [321, 322] realize an ISC-type topology which is shown in **Figure 10B** by softly breaking lepton number with a dimension-2 scalar mass insertion in the internal scalar loop. Similarly, Ma and Sarkar [323] discusses a supersymmetric model where the scalarquartic coupling is induced after supersymmetry is softly broken and thus an ISC-type topology is induced for neutrino mass. The models in Kajiyama et al. [324] and Baek et al. [325] have only heavy particles in the loop and can be considered as a 1 loop scotogenic model, where the Majorana mass term for the SM singlet fermions is generated at 1-loop order. Thus, neutrino mass is effectively generated at 2-loop order. It can be considered as an RB-type topology. In contrast to the topology RB-2, the SM Higgs fields are attached to the outer scalar line (the one on the left in **Figure 9G**). Both models break U(1)B−<sup>L</sup> to a discrete Z<sup>N</sup> subgroup. Ghosh et al. [326] proposes another model based on an RB-type topology, where the Higgs fields couple to the fermions in the outer loop. The model features a stable dark matter candidate due to an imposed Z<sup>2</sup> symmetry. Moreover, neutrino mass relies on the spontaneous breaking of an extended lepton number symmetry to a discrete Z<sup>2</sup> subgroup. The models in Ma [327], Nasri and Moussa [328], Chao [329], Ma and Wudka [330] and Nomura and Okada [331] realize the type-I seesaw by generating the Dirac mass terms at 1-loop order, and the model in Okada and Orikasa [332] generates a radiative type-II seesaw contribution by generating the triplet VEV at 1-loop level, and thus the Weinberg operator at 2-loop level. Finally, Witten [333] and Arbelàez Rodríguez et al. [334] firstly generate the right-handed neutrino mass at 2-loop level in the context of a GUT, which induces the active neutrino mass via the usual seesaw mechanism. Similarly Law and McDonald [335] and Baldes et al. [278] realize a radiative inverse seesaw. The mass of additional singlets is generated at 2-loop order. The model is based on an additional gauged U(1) symmetry (which is spontaneously broken to its Z<sup>2</sup> subgroup) to forbid the generation of neutrino mass at tree-level via the seesaw mechanism. The model can explain the matter-antimatter asymmetry of the universe, but not account for the dark matter abundance [278].

#### 3.2.4. Models Based on Several Topologies

Several models in the literature [144, 221, 336–351] are based on multiple 2-loop topologies. We highlight three examples. Megrelidze and Tavartkiladze [144] proposed to generate neutrino mass via lepton-number-violating soft supersymmetrybreaking terms using the so-called type-II-B soft seesaw with electroweak triplet superfields. Integrating out the scalar components of the electroweak triplets leads to the dimension-5 lepton-number-violating term (L˜H˜ u) 2 . Neutrino mass is generated at 2-loop via a diagram based on topology CLBZ-1 and diagrams which generate the couplings of the scalar component of the electroweak triplet superfield to two lepton doublets L on the one hand and the two electroweak Higgs doublets H<sup>u</sup> on the other hand at the 1-loop level. Another interesting class of models are based on internal electroweak gauge bosons, which are based on CLBZ-type topologies and discussed in Chen et al. [345, 346], del Aguila et al. [347], Chen et al. [348], del Aguila et al. [349], King et al. [350] and Geng and Tsai [351]. All of them introduce a doubly-charged scalar and a coupling of the doubly-charged scalar to two W-bosons, which can be achieved via a mixing of the doubly-charged scalar with the doubly-charged scalar in an electroweak triplet scalar. Neutrino mass is typically generated via topologies CLBZ-1 and CLBZ-3 and induces the operator

$$\mathcal{O}^{\text{RR}} = \bar{e}\_{\text{R}} e\_{\text{R}}^{\epsilon} (H^{\dagger} D^{\mu} \tilde{H}) (H^{\dagger} D\_{\mu} \tilde{H}) \ . \tag{34}$$

This possibility is further discussed in section 3.4. Gauge bosons similarly can play an important role in the generation of neutrino mass in extended technicolor (ETC) models as discussed in Appelquist and Shrock [352, 353] and Appelquist et al. [354]. These models contain many SM singlet fermions and only a few elements of the neutral fermion mass matrix are directly generated by condensates, while many elements are generated at 1-loop (or higher loop) level via loop diagrams with ETC gauge bosons. In particular the relevant Dirac mass terms relevant for the active neutrino masses are generated at 1-loop level and thus neutrino mass is effectively generated at 2-loop (or even higher loop) level.

### 3.3. 3-Loop Majorana Neutrino Mass Models

Unlike 1-loop and 2-loop topologies, there is no systematic classification of all 3-loop topologies. Thus, we restrict ourselves to the existing 3-loop models in the literature and do not consider other topologies or different fermion flow for the given topologies. Most of the existing 3-loop models can be categorized in four basic types of diagrams shown in **Figure 11** where we do not specify the Higgs insertions. The remaining models are either based on a combination of the listed topologies or the combination of a loop-induced vertex at 1- or 2-loop inside a loop diagram.

#### 3.3.1. The KNT Models

The first 3-loop radiative neutrino mass model was proposed in Krauss et al. [355] with the topology shown in **Figure 11A** by Krauss, Nasri and Trodden (KNT) and it leads to the operator O9. We refer to radiative neutrino mass models sharing the same topology as KNT models and discuss them in more detail in section 5.2. A systematic study with several different variants can be found in Chen et al. [356]. The models of Chen et al. [357], Ahriche and Nasri [358], Ahriche et al. [359], Chen et al. [356], Ahriche et al. [360–364] also generate the operator O9, the models of Chen et al. [356], Nomura et al. [365] and Cheung et al. [366] the operator O<sup>11</sup> with down-type quarks, while the models in Okada and Okada [367], Chen et al. [356], Okada and Yagyu [368] and Cheung et al. [369] only have new heavy states in the loop.

### 3.3.2. AKS-Type Models

Neutrino mass can also arise at 3-loop order from the diagram shown in **Figure 11C**. The first model of such topology was proposed by Aoki, Kanemura, and Seto (AKS) in Aoki et al. [370] and is based on the operator e¯ c e¯ cH i 1H j 2H k 1H l 2 ǫijǫkl with two Higgs doublets H<sup>i</sup> . We will refer to it as the AKS model and more generally to models based on this topology as AKS-type models. It contains a second Higgs doublet and several SU(2)<sup>L</sup> singlets. The exotic particles can also be all electroweak singlets [221, 371]. The model in Gu [371] leads to the operator O9. Other variants include colored exotic particles such as leptoquarks [356,

372, 373], which generate the operators O11,12, or electroweak multiplets [356, 374, 375] generating the operators O1,9. Note cross diagrams may be allowed in specific models.

#### 3.3.3. Cocktail Models

The third class of models are based on the two cocktail diagrams shown in **Figures 11B,D**. The name for the diagram has been coined by Gustafsson et al. [376], which proposed a 3-loop model with two W-bosons based on topology **Figure 11B** and consequently generated the operator ORR, which are discussed in more detail in section 3.4. The same model has also been studied in Geng et al. [377]. The models in Hatanaka et al. [378] and Alcaide et al.[379] are based on the same topology, but with W bosons replaced by scalars. While Alcaide et al. [379] induces operator ORR, the model of Hatanaka et al. [378] leads to operator O9. Finally, the fermionic cocktail topology **Figure 11D** is used in the models of Nishiwaki et al. [380],and Kanemura et al. [381], both of which generate operator O9.

Apart from the three classes of models, there are a few models which do not uniquely fit in any of the three classes. The model in Jin et al. [382] is based on topologies **Figures 11A,C** with two Wbosons and thus generates the operator ORR. Nomura et al. [383] generates the mass of new exotic fermions at 2-loop level via a CLBZ-type diagram, which in turn generate neutrino mass at 1 loop. Geng and Huang [384] studies a 2-loop model based on the operator O8, which itself is generated at 1-loop order.

Most of the 3-loop models need to impose extra discrete symmetries such as Z<sup>2</sup> or a continuous U(1) symmetry to forbid lower-loop or tree-level contributions, unless accidental symmetries exist and thus partly require other VEV insertions. One example is to employ higher dimensional representation of SU(2)<sup>L</sup> [362], e.g., septuplet, in the spirit of minimal dark matter [385, 386] such that undesirable couplings are forbidden by the SM gauge group alone. Due to the existence of the extra imposed or accidental symmetries, 3-loop models serve as a natural playground for DM physics.

### 3.4. Models with Gauge Bosons

The first model [290–292] with gauge bosons in the loop uses the topology PTBM-1 and leads to operators built from two lepton doublets including covariant derivatives. However, it also has a tree-level contribution, while models based on operators with right-handed charged leptons are genuine radiative neutrino mass models.

In del Aguila [107] two LNV effective operators with gauge bosons, i.e., present in covariant derivatives, were considered, which allowed to have neutrinoless double beta decay rates generated at tree level thanks to new couplings to the SM leptons15. Interestingly, depending on the chirality of the outgoing leptons in 0νββ, there are two new operators (beyond the standard contribution from the Weinberg operator which involves left-handed electrons). For left–right (LR) chiralities of the outgoing electrons, there is a dimension-7 operator:<sup>16</sup>

$$\mathcal{O}^{\rm LR} = (H^{\dagger}D^{\mu}\tilde{H})(H^{\dagger}\overline{e\_{\mathbb{R}}}\mathcal{Y}\_{\mu}\tilde{L})\,. \tag{35}$$

For right-right (RR) chiralities, there is a dimension-9 operator ORR as define in Equation (34). After electroweak symmetry breaking, these operators generate the relevant vertices for 0νββ at tree level: W− µ eRγ µν c L and W− <sup>µ</sup> W−µeRe c R , respectively. The contributions of OLR and ORR to 0νββ are depicted in **Figures 12B,C** respectively, where the red point denotes the effective operator insertion.

The lowest order contributions from these operators to neutrino masses occur at 1- and 2-loop orders, respectively, via the diagrams of **Figure 13**. The dominant contributions come from matching (see also Babu and Leung [85], de Gouvêa et al. [87], Angel et al. [90] and de Gouvêa et al.[88] for estimates of the matching contributions to neutrino masses of LNV

<sup>15</sup>In general, 0νββ is generated in these models at a lower order than neutrino masses.

<sup>16</sup>There are other operators which, however, are simultaneously generated with the Weinberg operator, which dominates as it is dimension 5 [107].

FIGURE 13 | Lowest order contributions of <sup>O</sup>LR (left, at 1-loop order) and <sup>O</sup>RR (right, at 2-loop order) to neutrino masses. The red dot indicates the <sup>1</sup><sup>L</sup> <sup>=</sup> <sup>2</sup> effective vertex. Figure reproduced following del Aguila [107].

operators), which using dimensional analysis can be estimated to be given by del Aguila et al. [107]:

$$(m\_{\upsilon})\_{ab}^{\mathrm{LR}} \cong \frac{\nu}{16\pi^2 \Lambda} \left( m\_a \mathcal{C}\_{ab}^{\mathrm{LR}} + m\_b \mathcal{C}\_{ba}^{\mathrm{LR}} \right) \tag{36}$$

for OLR and by

$$\left(m\_{\upsilon}\right)\_{ab}^{\text{RR}} \simeq \frac{1}{(16\pi^2)^2 \Lambda} m\_a \mathcal{C}\_{ab}^{\text{RR}} m\_b \tag{37}$$

for ORR . Notice that the appearance of the chirality-flipping charged lepton masses is expected in order to violate lepton number in the LH neutrinos, which naturally generates textures in the neutrino mass matrix.

Possible tree-level UV completions which have new contributions to 0νββ at tree level were outlined in del Aguila et al. [107]. See also del Aguila [349], which provides a summary of two examples of models generating OLR and ORR , respectively. The UV model of ORR [347] generates 0νββ at tree level, while neutrino masses are generated as expected at 2-loop order. It includes a doubly-charged singlet, a Y = 1 triplet scalar and a real singlet. In order to prevent tree-level neutrino masses as in type-II seesaw via the latter field, a discrete Z<sup>2</sup> symmetry, which was spontaneously broken by the VEV of the singlet, was added. Recently a variation has been studied, in which the Z<sup>2</sup> symmetry is exact, such that there is a good dark matter candidate, which is a mixture of singlet and triplet [379]. In this case, the contributions to 0νββ and to neutrino masses are further shifted by one extra loop, i.e., they are generated at 1- and 3-loop orders, respectively. Gustafsson et al. [376, 387] studied also a specific model with a dark matter candidate, named the cocktail model, which generated ORR at 1-loop order, i.e., 0νββ at 1-loop order and therefore neutrino masses at 3-loop order. It includes a singly-charged singlet, a doubly-charged singlet and a Y = 1 scalar doublet, together with a discrete symmetry Z<sup>2</sup> under which all the new fields except the doubly-charged are odd. Other models generating ORR were presented in Chen et al. [345, 346, 348], King et al. [350], Geng and Tsai [351], and Liu and Gu [388].

### 3.5. Radiative Dirac Neutrino Mass Models

Although Majorana neutrinos are the main focus of research, Dirac neutrinos are a viable possibility to explain neutrino mass. It is noteworthy that the first radiative neutrino mass model [389] was based on Dirac neutrinos. In recent years, there has been an increased interest in Dirac neutrinos and, in particular, there are a few systematic studies on the generation of Dirac neutrino mass beyond the simple Yukawa interaction, which include both treelevel and loop-level realizations, besides several newly-proposed radiative Dirac neutrino mass models, which we will outline below.

Ma and Popov [56] and Wang and Han [60] performed a study of Dirac neutrino mass according to topology at treelevel and 1-loop level. There are only two possible one-particleirreducible topologies for the Dirac Yukawa coupling at 1 loop, which are shown in **Figure 14**. The simplest radiative Dirac neutrino mass models are based on a softly-broken Z<sup>2</sup> symmetry, which is required to forbid the tree-level contribution, and generate the topologies in **Figure 14**. Wang et al. [390] studied scotogenic-type models with a U(1)B−<sup>L</sup> symmetry at

1- and 2-loop order. Finally, Kanemura et al. [391] takes a model-independent approach and discusses the possible flavor structures of the induced Dirac mass term under a number of constraints: The fermion line only contains leptons and each lepton type can appear at most once.

#### 1-Loop Models

Many of the proposed 1-loop models are realized in a left–right symmetric context without SU(2) triplet scalars [56, 389, 392– 397]. Rajpoot [398] attempted the generation of Dirac neutrino masses in the context of a model where hypercharge emerges as diagonal subgroup of U(1)<sup>L</sup> × U(1)R. To our knowledge the generation of Dirac neutrino mass at 1-loop level with a softly broken Z<sup>2</sup> was first suggested in Kanemura et al. [399] based on topology **Figure 14B**. Farzan and Ma [400] implements the first scotogenic model of Dirac neutrino mass by using a dark Z<sup>2</sup> and softly-broken Z2. Both of these possibilities have been studied in more detail in the systematic studies outlined above. Another way to explain the smallness of Dirac neutrino mass is via a small loop-induced VEV [401]. Finally, Borah and Dasgupta [402] discusses a left–right symmetric model with pseudo-Dirac neutrinos. The tree-level Majorana mass terms are not allowed, because the bidoublet is absent and the coupling of the left-handed triplet to leptons is forbidden by a discrete symmetry.

#### 2-Loop Models

Two explicit models of Dirac neutrino mass have been discussed in Bonilla et al. [403] and Kanemura et al. [404] apart from the general classification [390]. They are both based on a U(1) symmetry, a dark U(1) and lepton number, respectively. The U(1) symmetry is broken to a discrete subgroup and thus both models feature a stable dark matter candidate.

#### 3-Loop Model

Finally, a Dirac neutrino mass term can also be induced via a global chiral anomaly term [405]. The five-dimensional anomaly term aFµνF˜µν with the pseudo-scalar a and the (dual) field strength tensor Fµν (F˜µν ) is induced at 1-loop level and leads to a Dirac mass term at 2-loop order, being effectively a 3-loop contribution.

### 3.6. 331 Models

Another interesting class of models is based on the extended gauge group SU(3)<sup>c</sup> × SU(3)<sup>L</sup> × U(1)X. The SM gauge group can be embedded in several different ways and is determined by how the hypercharge generator is related to the generator T<sup>8</sup> of SU(3)<sup>L</sup> and the generator X of U(1)X,

$$Y = \beta \ T\_8 + X \ , \tag{38}$$

where β is a continuous parameter. In addition to one radiative Dirac neutrino mass model [406], several radiative Majorana neutrino mass models have been proposed at 1-loop level [407– 423], 2-loop order [424–429], 3-loop order [430], and even at 4 loop order[98]. As lepton number violation (LNV) in 331 models and in particular neutrino mass generation has been discussed in a recent review [431], we refer the interested reader to it for a detailed discussion. However, we highlight one model based on gauged lepton number violation [419–421], which generates neutrino mass via lepton number violation in the 1-loop diagram shown in **Figure 15** with the SU(3)L×U(1)<sup>X</sup> gauge bosons, where H<sup>i</sup> denotes the SM Higgs doublets, hχi the VEV in the third component of SU(3)<sup>L</sup> and N c the third partner of ν<sup>L</sup> in the triplet of SU(3)L. Note that lepton number is broken by the mixing of the gauge bosons in the vertex at the top of the diagram.

### 4. PHENOMENOLOGY

In this section we revisit the most relevant phenomenological implications of radiative neutrino mass models. The possible signals are very model dependent, as each radiative model has its own particularities that should be studied on a case-by-case basis. However, in the following we will try to discuss generic predictions of these models, making use of simplified scenarios and/or of effective operators, and referring to particular examples when necessary.

### 4.1. Universality Violations and Non-standard Interactions

In the SM, leptonic decays mediated by gauge interactions are universal. Several scenarios of physics beyond the SM have universality violations, that is, decays into different families (up

to phase space-factors) are no longer identical17. These may or may not be related to neutrino masses, as lepton number is not violated in these interactions. Indeed, for instance a two Higgs doublet model with general Yukawa interactions breaks universality, irrespective of neutrino masses. In tree-level neutrino mass models, there are also violations of universality, mediated by the (singly) charged scalar boson in the type-II seesaw model, or due to the non-unitarity of the leptonic mixing matrix in type-I and type-III seesaw models when the extra neutral fermions are heavy [432, 433].

In some of the radiative models there can be violations of universality. One illustrative example of this case is due to the presence of a singly-charged singlet h with mass mh<sup>+</sup> (as in the Zee and Zee-Babu models, see section 5.1). The relevant interaction is Lf L ˜ h +, where f is an antisymmetric matrix in flavor space and <sup>L</sup>˜ <sup>≡</sup> <sup>i</sup>τ2CL<sup>c</sup> <sup>=</sup> <sup>i</sup>τ2CL¯<sup>T</sup> . Integrating out the singlet, one obtains the following dimension 6 effective operator [434]

$$\mathcal{L}\_{\rm eff} \subset \frac{1}{m\_{h^{+}}^2} (\overline{e\_{\rm L}} f^{\dagger} \,\nu\_{\rm L}^{\epsilon})(\overline{\nu\_{\rm L}^{\epsilon}} \, f \, e\_{\rm L}) \,. \tag{39}$$

One can see that this operator involves left-handed leptons, like charged currents in the SM18. This implies that it interferes constructively with the W boson, modifying among others the muon decay rate [435]. Therefore, the Fermi constant which is extracted from muon decay in the SM, G SM µ , and that in a model with a singly-charged singlet, G h µ , are different, i.e., G SM µ 6= G h µ . Their ratio obeys to leading order in f :

$$\left(\frac{G\_{\mu}^{\text{h}}}{G\_{\mu}^{\text{SM}}}\right)^{2} \simeq 1 + \frac{\sqrt{2}}{G\_{F}m\_{\text{h}^{+}}^{2}} |f^{\epsilon\mu}|^{2} \,. \tag{40}$$

The new Fermi constant G h µ is subject to different constraints. For example, from measurements of the unitarity of the CKM matrix, as the Fermi constant extracted from hadronic decays should be equivalent to that from leptonic decays, we can bound feµ:

$$|V\_{\rm ud}^{\rm exp}|^2 + |V\_{\rm us}^{\rm exp}|^2 + |V\_{\rm ub}^{\rm exp}|^2 = \left(\frac{G\_{\mu}^{\rm SM}}{G\_{\mu}^{\rm h}}\right)^2 = 1 - \frac{\sqrt{2}}{G\_{F}m\_{h^+}^2}|f^{e\mu}|^2. \tag{41}$$

Also leptonic decays which in the SM are mediated by chargedcurrent interactions are not universal anymore. The ratio of leptonic decays among the different generations can be tested via the effective couplings given by

$$\left(\frac{G\_{\mathfrak{r}\to\mathfrak{e}}^{\mathrm{h}}}{G\_{\mu\to\mathfrak{e}}^{\mathrm{h}}}\right)^{2} \approx 1 + \frac{\sqrt{2}}{G\_{F}m\_{\mathrm{h}^{+}}^{2}} \left(|f^{\mathrm{er}}|^{2} - |f^{\mathrm{er}}|^{2}\right),\tag{42}$$

$$\left(\frac{G\_{\mathfrak{r}\to\mu}^{\mathrm{h}}}{G\_{\mu\to\epsilon}^{\mathrm{h}}}\right)^{2} \approx 1 + \frac{\sqrt{2}}{G\_{F}m\_{\mathrm{h}^{+}}^{2}} \left(|f^{\mu\pi}|^{2} - |f^{\epsilon\mu}|^{2}\right),\tag{43}$$

$$\left(\frac{G\_{\mathbf{r}\rightarrow\mu}^{\mathbf{h}}}{G\_{\mathbf{r}\rightarrow e}^{\mathbf{h}}}\right)^{2} \approx 1 + \frac{\sqrt{2}}{G\_{F}m\_{h^{+}}^{2}}\left(|f^{\mu\tau}|^{2} - |f^{e\tau}|^{2}\right). \tag{44}$$

All these lead to strong limits on the f couplings depending on the mass on the singlet [95].

Furthermore, the new singly-charged scalar via the effective operator in Equation (39) induces neutrino interactions that cannot be described by W-boson exchange and are termed nonstandard neutrino interactions (NSIs). Equation (39) is usually rewritten after a Fierz identity as

$$\mathcal{L}\_{d=6}^{\rm NSI} = 2\sqrt{2}G\_F \varepsilon\_{\alpha\beta}^{\rho\sigma} \left( \overline{v\_\alpha} \gamma^\mu \mathbf{P}\_\mathbf{L} \upsilon\_\beta \right) \left( \overline{e\_\rho} \gamma\_\mu \mathbf{P}\_\mathbf{L} e\_\sigma \right), \tag{45}$$

where ε ρσ αβ are the NSI parameters given by

$$\varepsilon^{\rho\sigma}\_{\alpha\beta} = \frac{f\_{\sigma\beta}(f\_{\rho\alpha})^\*}{\sqrt{2}G\_F m\_{h^+}^2} \,. \tag{46}$$

These could be in principle probed at neutrino oscillation experiments. However, typically whenever NSIs are induced, lepton flavor violating (LFV) processes are also generated, which are subject to stronger constraints. This is particularly the case for the four-lepton dimension 6 operators, due to gauge invariance. Models with large NSI are difficult to construct, and typically involve light mediators [436, 437]. We refer the reader to Davidson et al. [438], Ibarra et al. [439], Gavela et al. [440], Biggio et al. [441, 442], Antusch et al. [443] and Ohlsson [444] for studies of NSIs and their theoretical constraints.

### 4.2. Lepton Flavor Violation

One of the common predictions shared by most neutrino mass models (radiative or not) is the existence of LFV processes involving charged leptons with observable rates in some cases. Indeed, neutrino oscillations imply that lepton flavors are violated in neutrino interactions, and as in the SM neutrinos come in SU(2) doublets together with the charged leptons, also violations of lepton flavors involving the latter are expected. Which is the most constraining LFV observable is, however, a model-dependent question. It is thus convenient to use a

<sup>17</sup>Higher order effects break universality in a tiny amount due to Higgs interactions, i.e, by the charged lepton Yukawa couplings.

<sup>18</sup>In models with an extra Higgs doublet coupled to the leptons, other operators can be formed by integrating out the second Higgs doublet. In those cases, the electrons involved are right-handed and therefore there is no interference with the W boson. An example is the Zee model, see Herrero-García et al. [91].

parametrization that allows for a model-independent description of these processes. For each of the models one can then compute the relevant coefficients and apply the following formalism. We follow the notation and conventions of Porod et al. [445] 19 .

The general LFV Lagrangian can be written as

$$\mathcal{L}\_{\text{LFV}} = \mathcal{L}\iota\_{\ell\ell} + \mathcal{L}\iota\_{\ell}\mathbf{z} + \mathcal{L}\_{\ell\ell h} + \mathcal{L}\_{4\ell} + \mathcal{L}\_{2\ell 2q} \,. \tag{47}$$

The first term contains the ℓ−ℓ−γ interaction Lagrangian, given by

$$\begin{split} \mathcal{L}\_{\ell\ell\gamma} &= e \, \bar{\ell}\_{\beta} \left[ \gamma^{\mu} \left( \mathbf{K}\_{1}^{L} \mathbf{P}\_{\mathcal{L}} + \mathbf{K}\_{1}^{R} \mathbf{P}\_{\mathcal{R}} \right) \\ &+ i m\_{\ell\_{\mathcal{Q}}} \sigma^{\mu\nu} q\_{\upsilon} \left( \mathbf{K}\_{2}^{L} \mathbf{P}\_{\mathcal{L}} + \mathbf{K}\_{2}^{R} \mathbf{P}\_{\mathcal{R}} \right) \right] \ell\_{\alpha} A\_{\mu} + \text{H.c.}, \end{split} \tag{48}$$

where e is the electric charge, q is the photon momentum, PL,R = 1 2 (1 ∓ γ5) are the standard chirality projectors and the indices {α, β} denote the lepton flavors. The first term in Equation (48) corresponds to the monopole interaction between a photon and a pair of leptons whereas the second is a dipole interaction term. In this parametrization the form factors K L,R 1 vanish when the photon is on-shell, i.e., in the limit of q <sup>2</sup> <sup>→</sup> 0. Similarly, the interaction Lagrangians with the Z and Higgs bosons are given by<sup>20</sup>

$$\mathcal{L}\_{\ell\ell Z} = \bar{\ell}\_{\beta} \left[ \boldsymbol{\gamma}^{\mu} \left( \mathbf{R}\_{1}^{L} \mathbf{P}\_{\mathcal{L}} + \mathbf{R}\_{1}^{R} \mathbf{P}\_{\mathcal{R}} \right) + \boldsymbol{p}^{\mu} \left( \mathbf{R}\_{2}^{L} \mathbf{P}\_{\mathcal{L}} + \mathbf{R}\_{2}^{R} \mathbf{P}\_{\mathcal{R}} \right) \right] \ell\_{\alpha} \boldsymbol{Z}\_{\mu} \,, \tag{49}$$

where p is the ℓ<sup>β</sup> 4-momentum, and

$$\mathcal{L}\_{\ell\ell h} = \bar{\ell}\_{\beta} \left( \mathcal{S}\_{L} \mathcal{P}\_{\mathcal{L}} + \mathcal{S}\_{\mathcal{R}} \mathcal{P}\_{\mathcal{R}} \right) \ell\_{\alpha} \, h \tag{50}$$

with the SM Higgs h. The general 4-lepton interaction Lagrangian can be written as

$$\mathcal{L}\_{4\ell} = \sum\_{I=\text{S.V.T} \atop X, Y=\text{L.R}} A\_{XY}^I \bar{\ell}\_{\beta} \Gamma\_I \mathbf{P}\_X \ell\_\alpha \bar{\ell}\_\delta \Gamma\_I \mathbf{P}\_Y \ell\_\gamma + \text{H.c.}, \qquad (51)$$

where in this case the indices {α, β, γ , δ} denote the lepton flavors and we have defined Ŵ<sup>S</sup> = 1, Ŵ<sup>V</sup> = γ<sup>µ</sup> and Ŵ<sup>T</sup> = σµν . It is clear that the Lagrangian in Equation (51) contains all possible terms allowed by Lorentz invariance. Finally, the general 2ℓ2q 4 fermion interaction Lagrangian (at the quark level) can be split in two pieces

$$
\mathcal{L}\_{2\ell 2\underline{q}} = \mathcal{L}\_{2\ell 2\underline{d}} + \mathcal{L}\_{2\ell 2\underline{u}},\tag{52}
$$

where

$$\mathcal{L}\_{2\ell 2d} = \sum\_{I \subseteq \mathcal{S}, \mathcal{V}, \mathcal{T} \atop \mathcal{V} \subseteq \mathcal{V} - \mathcal{U} \atop \mathcal{D}} B\_{XY}^I \bar{\mathcal{E}}\_\beta \Gamma\_I \mathbf{P}\_X \ell\_\alpha \bar{d}\_\mathcal{V} \Gamma\_I \mathbf{P}\_Y d\_\mathcal{V} + \text{H.c.}, \quad \text{(53)}$$

$$\mathcal{L}\_{2\ell 2\mu} = \begin{array}{c} \text{\(X, Y=1,R\\ \mathcal{L}\_{2\ell 2d}|\_{d \to \mu, B \to C} \text{\(}.\tag{54} \end{array} \tag{54}$$

Here γ denotes the d-quark flavor and we are neglecting the possibility of quark flavor violation, which is beyond the scope of this review <sup>21</sup> .

The parametrization used implies that the operators appearing in Equations (51), (53), and (54) have canonical dimension six. Therefore, the Wilson coefficients A I XY, B I XY and C I XY scale as 1/3<sup>2</sup> , where 3 is the new physics energy scale at which they are generated. Note this scale is unrelated to the scale at which lepton number is violated. The same comment applies to the dipole coefficients K L,R 2 in Equation (48). In contrast, the rest of the coefficients discussed in this section, K L,R 1 , R L,R 1,2 and SL,R, are dimensionless (although their leading new physics contribution appears at order v 2 /3<sup>2</sup> ). If we restrict the discussion to flavor violating coefficients, they all vanish in the SM. Therefore, they encode the effects induced by the new degrees of freedom present in specific models.

It should be noted that all operators in the general LFV Lagrangian in Equations (48–54) break gauge invariance. For instance, they contain new charged lepton interactions, but not the analogous new interactions for the neutrinos, their SU(2)<sup>L</sup> doublet partners which are partly discussed in the previous subsection. This type of parametrization of LFV effects is correct at energies below the electroweak symmetry breaking scale, but it may miss relevant correlations between operators that are connected by gauge invariance in the underlying new physics theory. See for instance Pruna and Signer [450] for a discussion of LFV in terms of gauge-invariant operators.

We now proceed to discuss the LFV processes with the most promising experimental perspectives in the near future. We will provide simple analytical expressions in terms of the coefficients of the general LFV Lagrangian and highlight some radiative neutrino mass models with specific features leading to nonstandard expectations for these processes. By no means this will cover all the models constrained by these processes, but will serve as a review of the novel LFV scenarios in radiative neutrino mass models.

Note, however, that there are other processes, which may yield stringent constraints in particular models: for instance in models with leptoquarks, the latter can mediate semi-leptonic τ -decays and leptonic meson decays at tree level. The LFV decays Z → ℓαℓ¯ β have also been investigated in several radiative models, although they typically have very low rates, see for instance Ghosal et al. [451] and Li et al. [452].

### 4.2.1. ℓ<sup>α</sup> → ℓβγ

The most popular LFV process is ℓ<sup>α</sup> → ℓβγ . There are basically two reasons for this: (1) for many years, the experiments looking for the radiative process µ → eγ have been leading the experimental developments, with the publication of increasingly tighter bounds, and (2) in many models of interest these are the processes where one expects the highest rates. In fact, many phenomenological studies have completely focused on these decays, neglecting other LFV processes that may also be relevant.

<sup>19</sup>See Lee et al. [446], Lee and Shrock [447] and Marciano and Sanda [448] for pioneering work on LFV processes.

<sup>20</sup>Note the different choice of Lorentz structures in Equations (48), (49). The two forms can be related via the Gordon-identity.

<sup>21</sup>Carpentier and Davidson [449] provides a comprehensive collection of constraints on quark flavor violating operators.

TABLE 1 | Current experimental bounds and future sensitivities for ℓ<sup>α</sup> → ℓ<sup>β</sup> γ branching ratios.


The experimental situation in radiative LFV decays is summarized in **Table 1**. As one can easily see in this table, muon observables have the best experimental limits. This is due to the existing high-intensity muon beams. The current limit for the µ → eγ branching ratio has been obtained by the MEG experiment, BR(<sup>µ</sup> <sup>→</sup> <sup>e</sup><sup>γ</sup> ) <sup>&</sup>lt; 4.2 · <sup>10</sup>−<sup>13</sup> [453], slightly improving the previous bound also obtained by the same collaboration. This bound is expected to be improved by about one order of magnitude in the MEG-II upgrade [454]. The bounds in τ decays are weaker, with the branching ratios bounded to be below ∼ 10−<sup>8</sup> , and some improvements are expected as well in future B-factories [456].

radiative models the tight connection between neutrino masses and LFV implies suppressed ℓ<sup>α</sup> → ℓβγ rates. This is the case of bilinear R-parity violating models [461–463], see section 5.5 for a detailed discussion of this type of supersymmetric neutrino mass models.

### 4.2.2. ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup>

We now consider the ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> 3-body decays. One can distinguish three categories: ℓ<sup>α</sup> → ℓβℓβℓβ, ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> (with β 6= δ) and ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> (also with β 6= δ). These processes have received less attention even though the experimental limits on their branching ratios are of the same order as for the analogous ℓ<sup>α</sup> → ℓβγ decays. We summarize the current experimental bounds and future sensitivities for the ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> 3-body decays in **Table 2**. We note that an impressive improvement of four orders of magnitude is expected in the µ → eee branching ratio sensitivity thanks to the Mu3e experiment at PSI [464].

The ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> decay width receives contributions from several operators of the general LFV Lagrangian. In the case of the first category, ℓ<sup>α</sup> → ℓβℓβℓβ, the decay width is given by Porod et al.[445]

Ŵ ℓ<sup>α</sup> → ℓβℓβℓ<sup>β</sup> = m5 ℓα 512π<sup>3</sup> " e 4 K L 2 2 + K R 2 2 16 3 ln <sup>m</sup>ℓ<sup>α</sup> mℓ<sup>β</sup> − 22 3 ! + 1 24 A S LL 2 + A S RR 2 + 1 12 A S LR 2 + A S RL 2 + 2 3 Aˆ V LL 2 + Aˆ V RR 2 + 1 3 Aˆ V LR 2 + Aˆ V RL 2 + 6 A T LL 2 + A T RT 2 + e 2 3 K L 2A S∗ RL + K R <sup>2</sup> A S∗ LR + H.c. − 2e 2 3 K L <sup>2</sup>A<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> RL + K R <sup>2</sup> <sup>A</sup><sup>ˆ</sup> <sup>V</sup><sup>∗</sup> LR <sup>+</sup> H.c. − 4e 2 3 K L <sup>2</sup>A<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> RR + K R <sup>2</sup> <sup>A</sup><sup>ˆ</sup> <sup>V</sup><sup>∗</sup> LL <sup>+</sup> H.c. − 1 2 A S LLA T∗ LL + A S RRA T∗ RR <sup>+</sup> H.c. − 1 6 A S LRA<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> LR + A S RLA<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> RL <sup>+</sup> H.c. (56)

The decay width for ℓ<sup>α</sup> → ℓβγ is given by Hisano et al. [457]

$$
\Gamma \left( \ell\_{\alpha} \to \ell\_{\beta} \chi \right) = \frac{\alpha m\_{\ell\_{\alpha}}^5}{4} \left( |K\_2^L|^2 + |K\_2^R|^2 \right), \tag{55}
$$

where α is the fine structure constant. Only the dipole coefficients K L,R 2 , defined in Equation (48), contribute to this process. General expressions for these coefficients can be found in Lavoura [458].

The µ → eγ limit is typically the most constraining one in most radiative neutrino mass models. One can usually evade it by adopting specific Yukawa textures that reduce the µ − e flavor-violating entries (see for example Schmidt et al. [459]) or simply by globally reducing the Yukawa couplings by increasing the new physics scale. However, in some cases this is not possible. A simple example of such situation is the scotogenic model [113] with a fermionic dark matter candidate. The singlet fermions in the scotogenic model only couple to the SM particles via the Yukawa couplings. Therefore, these Yukawa couplings must be sizable in order to thermally produce singlet fermions in the early universe in sufficient amounts so as to reproduce the observed DM relic density. This leads to some tension between the DM relic density requirement and the current bounds on LFV processes, although viable regions of the parameter space still exist [459, 460]. In contrast, in other in case of the second category, ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> (with β 6= δ), the expression is given by Abada et al. [467]

Ŵ ℓ<sup>α</sup> → ℓ<sup>β</sup> ℓδℓ<sup>δ</sup> = m5 ℓα 512π<sup>3</sup> " e 4 K L 2 2 + K R 2 2 16 3 ln <sup>m</sup>ℓ<sup>α</sup> mℓ<sup>γ</sup> − 8 ! + 1 12 A S LL 2 + A S RR 2 + 1 12 A S LR 2 + A S RL 2 + 1 3 Aˆ V LL 2 + Aˆ V RR 2 + 1 3 Aˆ V LR 2 + Aˆ V RL 2 + 4 A T LL 2 + A T RR 2 − 2e 2 3 K L <sup>2</sup>A<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> RL +K R <sup>2</sup> <sup>A</sup><sup>ˆ</sup> <sup>V</sup><sup>∗</sup> LR +K L <sup>2</sup>A<sup>ˆ</sup> <sup>V</sup><sup>∗</sup> RR +K R <sup>2</sup> <sup>A</sup><sup>ˆ</sup> <sup>V</sup><sup>∗</sup> LL <sup>+</sup>H.c. (57)

whereas for the third category, ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> (with β 6= δ), the decay width is given by Abada et al. [467]

$$\begin{split} \Gamma\left(\ell\_{\alpha} \rightarrow \overline{\ell\_{\beta}}\ell\_{\delta}\ell\_{\delta}\right) &= \frac{m\_{\ell\_{\alpha}}^{5}}{512\pi^{3}} \left[ \frac{1}{24} \left( \left|A\_{LL}^{\mathbb{S}}\right|^{2} + \left|A\_{RR}^{\mathbb{S}}\right|^{2} \right) \right] \\ &+ \frac{1}{12} \left( \left|A\_{LR}^{\mathbb{S}}\right|^{2} + \left|A\_{RL}^{\mathbb{S}}\right|^{2} \right) + \frac{2}{3} \left( \left|\hat{A}\_{LL}^{\mathbb{V}}\right|^{2} + \left|\hat{A}\_{RR}^{\mathbb{V}}\right|^{2} \right) \right] \\ &+ \frac{1}{3} \left( \left|\hat{A}\_{LR}^{\mathbb{V}}\right|^{2} + \left|\hat{A}\_{RL}^{\mathbb{V}}\right|^{2} \right) + 6 \left( \left|A\_{LL}^{\mathbb{T}}\right|^{2} + \left|A\_{RR}^{\mathbb{T}}\right|^{2} \right) \end{split}$$

TABLE 2 | Current experimental bounds and future sensitivities for ℓ<sup>α</sup> → ℓ<sup>β</sup> ℓδℓ<sup>δ</sup> branching ratios.


$$-\left[\frac{1}{2}\left(A\_{LL}^{\mathcal{S}}A\_{LL}^{T\*} + A\_{RR}^{\mathcal{S}}A\_{RR}^{T\*} + \text{H.c.}\right)\right.$$

$$-\frac{1}{6}\left(A\_{LR}^{\mathcal{S}}\hat{A}\_{LR}^{V\*} + A\_{RL}^{\mathcal{S}}\hat{A}\_{RL}^{V\*} + \text{H.c.}\right)\right].\tag{58}$$

Here we have defined

$$
\hat{A}\_{XY}^V = A\_{XY}^V + e^2 K\_1^X \qquad (X, Y = L, R) \; . \tag{59}
$$

The masses of the leptons in the final state have been neglected in Equations (56–58), with the exception of the contributions given by the dipole coefficients K L,R 2 , where infrared divergences would otherwise occur.

The dipole coefficients K L,R 2 , which contribute to ℓ<sup>α</sup> → ℓβγ , also contribute ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> . It is easy to see how: the Feynman diagram contributing to ℓ<sup>α</sup> → ℓβγ can always be supplemented with a flavor-conserving ℓ<sup>δ</sup> −ℓ<sup>δ</sup> −γ additional vertex resulting in a diagram contributing to ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> <sup>22</sup>. In fact, such diagrams have been shown to be dominant in many models, the most popular example being the Minimal Supersymmetric Standard Model (MSSM). In this case, known as dipole dominance scenario, a simple proportionality between the decays widths of both LFV decays can be established. For example, in the β = δ case this proportionality leads to

$$\text{BR}(\ell\_{\alpha} \to \ell\_{\beta} \overline{\ell\_{\beta}} \ell\_{\beta}) \simeq \frac{\alpha}{3\pi} \left( \ln \left( \frac{m\_{\alpha}^{2}}{m\_{\beta}^{2}} \right) - \frac{11}{4} \right) \text{BR}(\ell\_{\alpha} \to \ell\_{\beta} \chi) \,, \tag{60}$$

which implies BR ℓ<sup>α</sup> → ℓβℓβℓ<sup>β</sup> ≪ BR ℓ<sup>α</sup> → ℓβγ , making the radiative decay the most constraining process.

The dipole dominance assumption is present in many works discussing LFV phenomenology. However, it can be easily broken in many radiative neutrino mass models. This can happen in two ways:<sup>23</sup>

• **Due to tree-level LFV:** In many radiative neutrino mass models the 4-lepton operators receive contributions at treelevel. The most prominent example of such models is the Zee-Babu model, in which the doubly-charged scalar k ++ mediates unsuppressed ℓ<sup>α</sup> → ℓβℓβℓ<sup>β</sup> decays. In such case one can easily find regions of parameter space where BR ℓ<sup>α</sup> → ℓβℓβℓ<sup>β</sup> ≫ BR ℓ<sup>α</sup> → ℓβγ , see Herrero-Garcia et al. [95] for a recent study.

• **Due to loop-level LFV:** Kubo et al. [468], Aristizabal Sierra et al. [469], Suematsu et al. [470], and Adulpravitchai et al. [471] explored the LFV phenomenology of the scotogenic model but only considered µ → eγ . However, this assumption has been shown to be valid only in some regions of the parameter space. In fact, box diagrams contributing to 4-lepton coefficients can actually dominate, dramatically affecting the phenomenology of the scotogenic model [460, 472]. Qualitatively similar results have been found in other variants of the scotogenic model [129, 266] <sup>24</sup>. In fact, this feature is not specific of the scotogenic model and its variants: one can find other radiative neutrino mass models with loop contributions dominating over the dipole. For instance, Z-penguin contributions have been found to be dominant in the angelic model [90] and RνMDM models [473].

This clearly shows that radiative neutrino mass models typically have a very rich LFV phenomenology with new (sometimes unexpected) patterns and correlations.

### 4.2.3. µ − e Conversion

The most spectacular improvements in the search for LFV are expected in µ − e conversion experiments. Several projects will begin their operation in the near future, with sensitivities that improve the current bounds by several orders of magnitude. The experimental situation is shown in **Table 3**.

The conversion rate, normalized to the the muon capture rate Ŵcapt, is given by Kuno and Okada [479] and Arganda et al. [480]

$$\begin{split} \text{CR}(\mu \text{ } - \text{ e, Nucleus}) \\ &= \frac{p\_e \text{E}\_e \text{m}^3\_{\mu} \text{ } \text{ $G\_F^2$ } \text{ $\alpha^3$  } \text{ $Z^4\_{\text{eff}} \text{$ F^2\_P} \text{ $} } \text{$ F^2\_P \text{ }} \text{ $\alpha^2$  } \left\{ \left| (\text{Z} + \text{N}) \left( \text{g}^{\text{(0)}}\_{LV} + \text{g}^{\text{(0)}}\_{LS} \right) \right. \right. \\ &\left. + (\text{Z} - \text{N}) \left( \text{g}^{\text{(1)}}\_{LV} + \text{g}^{\text{(1)}}\_{LS} \right) \right|^2 + \left. \left| (\text{Z} + \text{N}) \left( \text{g}^{\text{(0)}}\_{RV} + \text{g}^{\text{(0)}}\_{RS} \right) \right. \right. \\ &\left. + (\text{Z} - \text{N}) \left( \text{g}^{\text{(1)}}\_{RV} + \text{g}^{\text{(1)}}\_{RS} \right) \right|^2 \text{ } . \end{split}$$

Z and N are the number of protons and neutrons in the nucleus and Zeff is the effective atomic charge [481]. G<sup>F</sup> is the Fermi constant, α is the electromagnetic fine structure constant, p<sup>e</sup> and E<sup>e</sup> are the momentum and energy of the electron, m<sup>µ</sup> is the muon mass and F<sup>p</sup> is the nuclear matrix element. g (0) XK and g (1) XK (with X = L, R and K = S,V) are effective couplings at the nucleon level. They can be written in terms of effective couplings at the

<sup>22</sup>We clarify that this is only true for the processes <sup>ℓ</sup><sup>α</sup> <sup>→</sup> <sup>ℓ</sup><sup>β</sup> <sup>ℓ</sup><sup>β</sup> <sup>ℓ</sup><sup>β</sup> and <sup>ℓ</sup><sup>α</sup> <sup>→</sup> <sup>ℓ</sup><sup>β</sup> <sup>ℓ</sup>δℓ<sup>δ</sup> (with β 6= δ). The process ℓ<sup>α</sup> → ℓ<sup>β</sup> ℓδℓ<sup>δ</sup> (with β 6= δ) does not receive contributions from penguin diagrams, but only from boxes.

<sup>23</sup>In some models, cancellations due to certain Yukawa textures can affect some decays (like µ → eγ ), but it is virtually impossible to cancel all radiative decays simultaneously.

<sup>24</sup>Interestingly, the authors of Chowdhury and Nasri [129] have shown that in variants of the scotogenic model with higher SU(2) representations the LFV rates become larger due to additive effects from the components of the large multiplets.

TABLE 3 | Current experimental bounds and future sensitivities for µ − e conversion in nuclei.


quark level as

$$\begin{aligned} \mathcal{g}\_{\text{XK}}^{(0)} &= \frac{1}{2} \sum\_{q=\mu,d,s} \left( \mathcal{g}\_{\text{XK}(q)} G\_K^{(q,p)} + \mathcal{g}\_{\text{XK}(q)} G\_K^{(q,n)} \right), \\ \mathcal{g}\_{\text{XK}}^{(1)} &= \frac{1}{2} \sum\_{q=\mu,d,s} \left( \mathcal{g}\_{\text{XK}(q)} G\_K^{(q,p)} - \mathcal{g}\_{\text{XK}(q)} G\_K^{(q,n)} \right). \end{aligned} \tag{62}$$

The numerical values of the relevant G<sup>K</sup> factors can be found in Kuno and Okada [479], Kosmas et al. [482] and Porod et al. [445]. For coherent µ − e conversion in nuclei, only scalar (S) and vector (V) couplings contribute and sizable contributions are expected only from the u, d,s quark flavors. The gXK(q) effective couplings can be written in terms of the Wilson coefficients in Equations (48), (53), and (54) as

$$\mathcal{g}\_{LV(q)} = \frac{\sqrt{2}}{G\_F} \left[ e^2 \mathcal{Q}\_q \left( \mathcal{K}\_1^L - \mathcal{K}\_2^R \right) - \frac{1}{2} \left( \mathcal{C}\_{\ell \ell pq}^{VLL} + \mathcal{C}\_{\ell \ell pq}^{VLR} \right) \right] \tag{63}$$

$$\left. \mathcal{g}\_{\mathcal{V}(q)} \right|\_{\text{=}} = \left. \mathcal{g}\_{\mathcal{V}(q)} \right|\_{\text{L} \to \text{R}} \tag{64}$$

$$\mathcal{g}\_{\rm LS(q)} = -\frac{\sqrt{2}}{G\_F} \frac{1}{2} \left( \mathcal{C}\_{\ell\ell q q}^{\rm SL} + \mathcal{C}\_{\ell\ell q q}^{\rm SLR} \right) \tag{65}$$

$$\left.g\_{\rm RS(q)}\right|\_{L\to R}|\_{L\to R}\,,\tag{66}$$

where Q<sup>q</sup> is the quark electric charge (Q<sup>d</sup> = −1/3, Q<sup>u</sup> = 2/3) and C IXK ℓℓqq = B K XY C K XY for d-quarks (u-quarks), with X = L, R and K = S, V.

Radiative neutrino mass models can also be probed by looking for µ − e conversion in nuclei. As already pointed out, the search for this LFV process is going to be intensified in the next few years and, in case no observation is made, it will soon become one of the most constraining observables for this type of models. Similarly to the leptonic LFV 3-body decays discussed above, the dipole coefficients K L,R 2 also enter the µ − e conversion rate, potentially dominating it. In this case, one can derive a simple relation [483]

$$\frac{\text{CR}(\mu - e, \text{Nucleus})}{\text{BR}(\mu \to e\gamma)} \approx \frac{f(Z, N)}{428} \,, \tag{67}$$

where f(Z, N) is a function of the nucleus ranging from 1.1 to 2.2 for the nuclei of interest. The reader is referred to de Gouvea and Vogel [484] and Crivellin et al. [485] for a discussion on the complementarity of µ → eγ and µ − e conversion in nuclei. One can easily depart from this dipole dominance scenario in radiative neutrino mass models due to the existence of sizable contributions to other LFV operators. For instance, non-dipole contributions have been shown to be potentially large in the scotogenic model in Toma and Vicente [472] and Vicente and Yaguna [460]. The dipole coefficients may also be reduced due to partial cancellations in non-minimal models, see for example Ahriche et al. [360, 361] and Rocha-Moran and Vicente [266]. Finally, as already pointed out in the case of ℓ<sup>α</sup> → ℓβℓδℓ<sup>δ</sup> decays, some radiative neutrino mass models contain new states that mediate LFV processes at tree level. For instance, in R-parity violating models with trilinear terms (discussed in section 5.5), the superpotential terms λ ′bLbQb<sup>d</sup> c induce µ − e conversion at tree level [486]. This easily breaks the expectation in Equation (67).

Finally, we point out that the experiments looking for µ → eee and µ−e conversion in nuclei will soon take the lead in the search for LFV. Therefore, even if dipole contributions turn out to be dominant in a given model, µ → eee and µ − e conversion in nuclei might become the most constraining LFV processes in the near future. Prospects illustrating this point for specific radiative neutrino mass models have been presented in Angel et al. [90], Vicente and Yaguna [460], and Klasen et al. [487].

### 4.2.4. h → ℓαℓ<sup>β</sup>

In many radiative neutrino mass models, there can also be contributions to lepton-flavor violating Higgs (HLFV) decays, like h → τ −µ +, τ −e + and their CP-conjugates. These same interactions, however, also generate LFV processes such as τ → µ(e) γ , as no symmetry can prevent the latter [488], which are subject to much stronger constraints. In the effective field theory with just the 125 GeV Higgs boson, HLFV decays involving the tau lepton can be sizable, and ATLAS and CMS constraints on its flavor violating couplings (shown in **Table 4**) are comparable or even stronger than those coming from lowenergy observables [489–491]. However, in UV models, specially in radiative neutrino mass models, the situation is generally the opposite.

The relevant gauge-invariant effective operators that generate HLFV are the Yukawa operator:

$$\mathcal{O}\_Y = \overline{L}e\_\mathbb{R}H(H^\dagger H) \,, \tag{68}$$

and derivative operators like

$$\mathcal{O}\_{D\_\*e\_\mathbb{R}} = (\overline{e\_\mathbb{R}}H^\dagger) \wr \mathcal{O}(e\_\mathbb{R}|H) \,, \tag{69}$$

or

$$\mathcal{O}\_{\rm D,L} = (\overline{\mathcal{L}}H) \dot{\imath} \, \not\!\!\!\!\!/ (H^{\dagger}L) \, , \tag{70}$$

plus their Hermitian conjugates. In Herrero-García et al. [491] all the possible tree-level realizations of these operators were outlined, some of which include particles that are present in radiative neutrino mass models, as we will see below. In **Figure 16**, we show some possible UV completions of operators OY, OD, <sup>L</sup> and OD, <sup>e</sup><sup>R</sup> . The authors concluded that only O<sup>Y</sup> can have sizable rates, and in particular only for UV completions that involve scalars, like in a type-III two-Higgs doublet model.

TABLE 4 | Experimental 95 % C.L. upper bounds on HLFV decays from ATLAS and CMS in the tau sector using the 13 TeV data sets.


After electroweak symmetry breaking the Yukawa operator gives rise to the interaction Lagrangian in Equation (50). For instance, the SL,<sup>R</sup> couplings are given by

$$\mathcal{S}\_L = \frac{\nu^2}{\sqrt{2}\Lambda^2} \mathcal{C}\_Y^\dagger + D\_f \quad , \quad \mathcal{S}\_R = \frac{\nu^2}{\sqrt{2}\Lambda^2} \mathcal{C}\_Y + D\_f \,, \tag{71}$$

where D<sup>f</sup> is the SM flavor-diagonal contribution, not relevant for the present discussion, and C<sup>Y</sup> is the Wilson coefficient of the O<sup>Y</sup> operator defined in Equation (68). Focusing on the contributions from the Yukawa operator, the branching ratio of the Higgs into a tau and a muon reads:

$$\text{BR}(h \to \tau \mu) = \frac{m\_h}{8\pi \Gamma\_h} \left(\frac{\nu^2}{\sqrt{2}\Lambda^2}\right)^2 \left( |\langle \mathbf{C}\_{\bar{Y}} \rangle\_{\text{\tau}\mu}|^2 + |\langle \mathbf{C}\_{\bar{Y}} \rangle\_{\mu\tau}|^2 \right). \tag{72}$$

Most radiative neutrino mass models generate HLFV at 1-loop order [491] <sup>25</sup>. For instance, the doubly-charged scalar singlet and the singly-charged scalar singlet of the Zee-Babu model (see section 5.1) generate respectively the derivative operators OD, <sup>e</sup><sup>R</sup> and OD, <sup>L</sup> at 1-loop order. The scotogenic model (see section 5.3) also generates HLFV at 1-loop order (OD, <sup>L</sup>).

We can estimate the loop-induced HLFV in radiative neutrino mass models. Denoting a generic Yukawa coupling of the fermions and scalars with the SM leptons as Y, and a scalar quartic coupling with the Higgs as λih, and taking into account that the amplitude of h → µτ involves a tau mass, one can estimate the dominant contribution to be [491]:

$$\text{BR}(h \to \mu \tau) \sim \text{BR}(h \to \tau \tau) \frac{\lambda\_{ih}^2}{(4\pi)^4} \left(\frac{\nu}{\text{TeV}}\right)^4 \left(\frac{Y}{M\_i/\text{TeV}}\right)^4. \tag{73}$$

where M is the largest mass in the loop. In all these models, in addition to the loop factor, there are in general limits from charged LFV processes, as usually all radiative neutrino mass models have charged particles that can generate ℓ<sup>α</sup> → ℓβγ . As <sup>τ</sup> <sup>→</sup> µγ typically gives the constraint <sup>Y</sup>/(M/TeV)<sup>4</sup> . <sup>O</sup>(0.01 <sup>−</sup> 1), we get:

$$\text{BR}(h \to \mu\tau) \lessapprox 10^{-8},\tag{74}$$

well below future experimental sensitivities. Thus, unless cancellations are invoked (which are difficult to achieve in all possible radiative decays), HLFV rates are very suppressed, well below future experimental sensitivities.

One class of models which can have large HLFV are those with another Higgs doublet such that both the SM and the new scalar doublet couple to the lepton doublets [488, 499–502]. In such scenarios, both Yukawa couplings cannot be diagonalized simultaneously, which leads to LFV Higgs interactions. One example is the Zee model discussed in section 5.1, which can have BR(h → µτ ) up to the percent level [91].

### 4.3. Anomalous Magnetic Moments and Electric Dipole Moments

The anomalous magnetic moments (AMMs) and electric dipole moments (EDMs) of the SM leptons receive new contributions in radiative neutrino mass models (see Raidal et al. [503] for a review on the topic). These are contained in the dipole coefficients that also contribute to the radiative ℓ<sup>α</sup> → ℓβγ decays, typically leading to tight correlations between these observables. Using the effective Lagrangian in Equation (48), the anomalous magnetic moment aα and the electric dipole moment d<sup>α</sup> of the charged lepton ℓ<sup>α</sup> are given by Raidal et al. [503]

$$a\_{\alpha} = m\_{\ell\_{\alpha}}^2 \operatorname{Re} \left( K\_2^L + K\_2^R \right), \quad \frac{d\_{\alpha}}{e} = \frac{1}{2} \operatorname{Im} \left( K\_2^R - K\_2^L \right). \tag{75}$$

The experimental values for the AMMs and EDMs of charged leptons are collected in **Table 5**. In particular the muon AMM received a lot of intention in recent years due to the discrepancy between the experimentally measured value given in **Table 5** and the SM prediction [504]

$$a\_{\mu}^{\text{SM}} = 116591803(1)(42)(26) \times 10^{-11} \tag{76}$$

with the errors due to electroweak, lowest-order, and higherorder hadronic contributions.

There are many examples of radiative neutrino mass models leading to sizable effects in these two observables. For some examples in the case of AMMs see for instance Dicus et al. [234], Babu and Julio [312],Nomura et al. [239], Nomura and Okada [299], Chiang et al. [206], and Lee et al. [505]. In some cases, the new contributions effects can help close the gap between the theory prediction and the experimental measurement of the muon AMM, although in other cases they increase the disagreement, depending on their sign. We refer to the recent review [506] for a guide regarding new physics contributions to the muon AMM.

Regarding lepton EDMs, some examples in radiative neutrino models are given in Borah and Dasgupta [397, 402], and Chiang et al. [206]. In this case one requires CP-violating new physics in the lepton sector, something that is easily accommodated in new Yukawa couplings.

### 4.4. Neutrinoless Double Beta Decay

One of the main experimental probes to test the Majorana/Dirac nature of neutrinos is neutrinoless double beta decay (0νββ), in which a nucleus (A, Z) decays into another nucleus (A, Z + 2) and two electrons [507]. In order to have sizable 0νββ rates, the nuclei should not have single beta decays. This is achieved with even-even nuclei which, thanks to the nuclear pairing force, are lighter than the odd-odd nucleus, making single beta decays kinematically forbidden. The current strongest experimental limits are obtained using <sup>136</sup>Xe by EXO-200 [508]

<sup>25</sup>Also in type-I seesaw (and inverse seesaw), and in the MSSM, HLFV is generated at 1-loop order [494–498].

TABLE 5 | Experimental values for AMMs and EDMs [504].


Both statistical and systematic uncertainties are given for the muon AMM aµ.

and KamLAND-Zen [509, 510] which yield lower bounds of the lifetime of 1.1 · <sup>10</sup><sup>25</sup> y and 1.9 · <sup>10</sup><sup>25</sup> y at 90 % C.L, respectively. Uncertainties in the nuclear matrix elements translate into uncertainties in the extracted values of |mee| (see Equation 6), whose current strongest upper limits are in the ballpark of ∼0.15 eV. For further details regarding the present and future experimental situations see Dell'Oro et al. [511].

The observation of 0νββ decay would imply that lepton number is violated by two units (1L = 2), and therefore that neutrinos are Majorana particles [512]. However, quantitatively, this contribution to neutrino masses occurs at 4-loop order and is therefore extremely suppressed, much lighter than the observed neutrino masses (see Duerr et al. [513] for a quantitative study of this statement). So, even if it is true that neutrinos will necessarily be Majorana if 0νββ is observed, the main contribution to their masses may no be necessarily related to 0νββ.

We will mainly focus in this section on radiative models which have new direct contributions to neutrinoless double beta decay beyond the standard ones mediated by the light Majorana neutrinos, which are indirect, as they are generated by the new particles at higher-loop order (via light neutrino masses). For general reviews on the subject the interested reader is referred to Rodejohann [514], Bilenky and Giunti [515] and Dell'Oro et al. [511].

In Päs et al. [516, 517] a general phenomenological formula for the process including both long and short-range interactions was given. The authors considered all possible Lorentz structures for the quarks involved in the process and the outgoing electrons. In del Aguila et al. [107] effective operators that involve gauge bosons were considered, such that there are new effective vertices of the W-boson and the electrons.

In **Figure 12** (reproduced from del Aguila et al. [349]) all possible contributions to 0νββ are shown, with the red dot representing the 1L = 2 vertex. **Figure 12A** shows the light neutrino contribution, while **Figure 12F** involves a dimension-9 effective operator. In Bonnet et al. [106] a systematic classification of possible UV models stemming from the dimension 9 operator was performed (**Figure 12F**). See also Helo et al. [182] for scalarmediated UV completions and its connection to neutrino masses. **Figures 12D,E** involve new vertices between quarks, leptons and gauge bosons. **Figures 12B,C** involve new vertices with just leptons and gauge bosons and no quarks. In del Aguila et al. [107] operators that involve gauge bosons were considered, such that there are new effective vertices of the W-boson and the electrons, as in **Figures 12B,C**. See section 3.4 for a discussion of the effective operators that generate the latter diagrams and their connection to neutrino masses. A systematic classification of UV models for all the dimension-7 operators was given in Helo et al. [108]. Many (if not all) of these particles can be present in radiative neutrino mass models.

We outline in the following two typical new contributions to 0νββ from radiative neutrino mass models:


Let us also mention that, in addition to 0νββ, there are also limits on other lepton number violating elements mαβ of the neutrino mass matrix in flavor basis (where the charged lepton mass matrix is diagonal), different from the mee (which equals mββ) one, stemming from meson decays, tau decays, e +p collider data among other processes [527]. Also indirect bounds using neutrino oscillations and the unitarity of the PMNS matrix can be set [528]. However, both the direct and indirect (even if much stronger than the direct) bounds obtained are typically very weak [527, 528]. µ −e + conversion also offers a possibility to test the me<sup>µ</sup> element, however typically the rates are not competitive with those of 0νββ, although of course they test a different element and flavor effects could be relevant. A study of the contributions from effective operators was performed in Berryman et al. [529], while a doubly-charged scalar was studied in detail in Geib and Merle [530].

Lepton number violation can also be searched for at colliders. This is specially interesting for channels that do not involve electrons, as it is necessarily the case for 0νββ. Those will be discussed in section 4.5. Also the connection of lepton number violation to the matter-antimatter asymmetry of the universe will be discussed in section 4.6.

### 4.5. Collider Searches

Radiative neutrino mass models generally have a much lower UV scale than the GUT scale, which makes them testable at either current or future colliders. The diversity of exotic particles and their interaction with the SM particles in radiative neutrino mass models leads to an extremely rich phenomenology at colliders. Processes pertaining to the Majorana nature of neutrino masses or LFV couplings between the exotic particles and the SM, i.e., processes violating lepton number and/or lepton flavor, are often chosen as signal regions in collider searches due to the low SM background26. Of course, there are searches for exotic particles in general if they are not too heavy and the couplings are sizable <sup>27</sup> . In the following, we sketch different search strategies at colliders, which often utilize the low SM background for LNV and LFV processes. We thus discuss LNV and LFV processes separately before discussing general searches for new particles, which rely on processes without any LNV/LFV.

#### 4.5.1. Lepton Number Violation

At the LHC, the most sought-after channel of LNV<sup>28</sup> are samesign leptons

$$
\!\!\!\!\!\!p\underline{p} \to \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X \,,\tag{77}
$$

where ℓ denotes e or µ, and X can be any number of jets, E miss T or other SM objects. The details of the production and the actual content of X are very model-dependent: typically heavy states are produced and decay to final states with same-sign dilepton due to their Majorana nature. We will take a doublycharged scalar as a simple example to illustrate the basics of this search strategy. A doubly-charged scalar φ ++ is an SU(2)<sup>L</sup> singlet with hypercharge Y = 2. They can be pair-produced via Drell-Yan process and subsequently decay to two same-sign dileptons. For large masses the photon-initiated process becomes important and leads to an enhancement [532]. Assuming the branching fraction of φ ++ → e +e + is 100%, the signature for pair-produced doubly-charged scalars is four electrons and thus ZZ production is the main SM background. To reduce the SM background, discriminating variables such as the samesign dilepton mass, the difference between the opposite sign dilepton mass and the Z boson mass, and the scalar sum of the lepton p<sup>T</sup> can be utilized. ATLAS [533] has excluded doublycharged SU(2)<sup>L</sup> singlet scalar with mass lower than 420 GeV at 95% CL with LHC Run 2 data. The improved limit can be extracted from the CMS search for doubly-charged component of an SU(2)<sup>L</sup> triplet [534]. In Sugiyama et al. [535], del Aguila and Chala [536, 537] and Kanemura et al. [538] studies of doublycharged scalars and how to discriminate the multiplet to which they belong were performed.

The sensitivities of 0νββ searches detailed in section 4.4 and the same-sign dilepton searches at the LHC can be compared in any specific model (see for example [539–541]). Specifically in Helo et al. [539] and Peng et al. [540] a simplified model with a scalar doublet S ∼ (1, 2, 1) and a Majorana fermion F, which has the same matter content as the scotogenic model, is adopted. In this model, the reach of tonne-scale 0νββ generally beats that of the LHC. In the parameter space region where the heavy particle masses are near the TeV scale, however, the two probes are complementary.

#### 4.5.2. Lepton Flavor Violation

As described in section 4.2, lepton flavor violating processes are commonly predicted in radiative neutrino mass models, which can also be probed at colliders. The actual production topology of the LFV processes varies from model to model. For example, in models with the leptoquark S<sup>1</sup> ∼ (3, 1, 1 ¯ /3), there are two possible decay channels, S<sup>1</sup> → ¯ν ¯b or S<sup>1</sup> → ℓ <sup>+</sup>t¯ [542]. The dilepton final states are produced from

$$pp \to S\_1^\* S\_1 \to b\nu \bar{t} \ell^+ \to \ell^+ \ell'^- b\bar{b} + X \,,$$

$$pp \to S\_1^\* S\_1 \to t \ell'^- \bar{t} \ell^+ \to \ell^+ \ell'^- b\bar{b} + X \,,\tag{78}$$

where X can represent E miss T , multiple jets and leptons, and the former contributes dominantly for normal ordering in the minimal model with two leptoquarks. SUSY stop searches in the dilepton final states have the same signatures and their collider bounds can be translated into that of the leptoquark. This has been done for the LHC 8 TeV run [543] and the limit was mS<sup>1</sup> & 600 GeV [214]. Note that this limit in LFV channel is stronger than lepton flavor conserving ones (mS<sup>1</sup> & 500 GeV) as the SM background is lower. The stop search has been updated for LHC

<sup>26</sup>Theoretically there is no SM background. Realistically, however, object misidentification, undetected particles and fake objects can result in similar final states at the detector level.

<sup>27</sup>Some of the exotic particles may also show up in tree-level neutrino mass models. The interested reader is referred to the recent review [531] for the collider tests of specific tree-level models.

<sup>28</sup>Strictly speaking the process is not necessarily LNV, because X may carry lepton number as well, for example in form of neutrinos. Currently the searches are limited to electrons and muons. However, τ -leptons may also be used to search for LNV.

Run 2 [544, 545], though a recast for leptoquarks in LFV dilepton final states still awaits further analysis.

Alternatively, LFV processes can also be studied in an independent manner. In the framework of effective operators with two flavor-diagonal quarks and two flavor-off-diagonal leptons, constraints from LHC searches for LFV final states are interpreted as lower limits on the UV cut-off scale [546]. Compared with the limits derived from low energy precision measurements [449, 546], LHC delivers less stringent limits for light quarks. For heavier quarks, however, competitive limits of 3UV & 600 − 800 GeV can already be set for operators with right-handed τ leptons using only LHC Run 1 data.

#### 4.5.3. Searches for New Particles

Radiative neutrino mass models may contain exotic particles such as vector-like quarks (VLQs), vector-like leptons (VLLs), scalar leptoquarks, singly- or doubly-charged scalars, colored octet fermions or scalars, and electroweak multiplets. Note that the examples here are far from complete and searches for each individual particle require their own dedicated discussion. In Cai et al. [214], LHC searches for exotic particles in UV complete models based on 1L = 2 dimension 7 operators are discussed systematically. Here we will only present a simple summary about a handful of new particles.

#### **Vector-like quarks**

We refer by VLQs to new SU(3)<sup>c</sup> triplets which mix with the SM quarks and Higgs via Yukawa couplings [547]. The VLQs include different SU(2) representations: two singlets T and B with hypercharge 2/3 and −1/3; three doublets (T, B), (X, T), and (B, Y) with hypercharge 1/6, 7/6, and −5/6; and two triplets (X, T, B), and (T, B, Y) with hypercharge 2/3 and −1/3. They can be pair produced at the LHC via gluon fusion and quarkantiquark annihilations. Single production is model-dependent and can be dominant for large vector-like quark masses and large mixings [547]. The mass splitting among the components of the fields is suppressed by the mixing angles between the SM quarks and the vector-like quarks, which in turn suppresses the decays between the component fields. Therefore, VLQs will dominantly decay to either a gauge boson or a Higgs plus a SM quark. Both ATLAS and CMS have performed searches for VLQs and have set lower limits on the VLQs masses up to 990 GeV at the 95% confidence level (CL) depending on the representations and the decay branching ratio [548–559].

#### **Vector-like leptons**

VLLs are the colorless version of VLQs. Similar to VLQs, VLLs mix with the SM leptons via Yukawa couplings with Higgs. Due to the absence of right-handed neutrinos, there are less VLLs: two singlets N and E with hypercharge 0 and 1; two doublets (N, E) and (E, D) with hypercharge 3/2 and 1/2; and triplets (P, N, E) and (N, E, D) with hypercharge 0 and 1, respectively. Detailed studies have been performed in Altmannshofer et al. [560], Falkowski et al. [561], Dermisek et al. [562], and Kumar and Martin [563]. Contrary to the colored VLQs, VLLs are dominantly pair produced at the LHC via Drell-Yan process as the phase space suppression is less significant in the parameter space of interest at the moment. They can also be singly produced in association with W, Z or H, which can be dominant if the pair production channel is phase space suppressed and sizable mixing parameters are assumed. Likewise VLLs decay either to a SM lepton and a boson, W or Z, or Higgs. So far there is no dedicated search for VLLs at colliders, though SUSY searches for sleptons or charginos can be used to derive bounds on VLLs (see Altmannshofer et al. [560] and Hamada et al.[564] for example).

#### **Leptoquarks**

Leptoquarks appear frequently in theories beyond the SM such as grand unified theories [565, 566]. As its name suggests, a leptoquark, which can be either a scalar or a vector [542], possesses both non-zero lepton and baryon numbers. Here we will focus on scalar leptoquarks. At hadron colliders, leptoquarks are primarily produced in pairs via gluon fusion and quark-antiquark annihilation. Each leptoquark subsequently decays to one quark and one charged or neutral lepton. Both ATLAS [567, 568] and CMS [569–571] have performed searches for leptoquarks in final states with two charged leptons plus multiple jets. Assuming 100% branching fraction of the leptoquark decay into a charged lepton and a quark, current searches at the LHC Run 2 with 13 TeV center of mass energy have excluded leptoquarks with masses less than 1,130 GeV [569], 1,165 GeV [570] and 900 GeV [571] at 95% CL for leptoquark couplings to the first, second and third generations respectively.

#### **Charged scalars singlets**

Singly- and doubly-charged scalars are introduced in various radiative neutrino mass models (see Babu and Julio [158], Zee [288], and Babu [289], for instance). As singlets under SU(3)<sup>c</sup> × SU(2)L, the singly (doubly) charged scalar can only couple to the lepton doublet (right-handed charged lepton) bilinear. So the doubly-charged scalar can only decay to a pair of charged leptons, which leads to LNV signature at colliders (see discussion in section 4.5.1 for details). As for the singly-charged scalar, it decays to a charged lepton and a neutrino whose LNV effects can not be detected at the LHC. Singly-charged scalars are mainly produced in pairs via the Drell-Yan pair process. They are searched for in final states with two leptons plus E miss T <sup>29</sup>. SUSY searches for sleptons and charginos at the LHC share the same signature as the singly-charged scalars. Thus, we can in principle recast the slepton search in The ATLAS Collaboration [574] and extract the limit for our singly-charged scalars. Note a slepton can also be produced via a W-boson, while singly-charged scalar only via a virtual photon.

#### **Higher-dimensional electroweak multiplet**

SU(2)<sup>L</sup> higher-dimensional representations can also be incorporated in radiative neutrino mass theories [126– 129, 362, 473, 575, 576]. While the mass splittings among the component fields for scalar multiplets can be generally

<sup>29</sup>Long-lived charged particles have been searched at the LHC using anomalously high ionization signal [572], also in the context of dark matter [573]. However, charged scalars in radiative neutrino mass models usually have sizable couplings to SM leptons and decay promptly.

large due to couplings to the SM Higgs, those for fermion multiplets are only generated radiatively and are typically <sup>∼</sup> <sup>O</sup>(100) MeV, with the neutral component being the lightest. This small mass splitting results in lifetimes <sup>∼</sup> <sup>O</sup>(0.1) ns. At the LHC, charged component field can be produced in pair via electroweak interaction and decay to the neutral component plus a very soft pion, which leads to a disappearing track signature. For a triplet with a lifetime of about 0.2 ns, the current LHC searches set the lower mass limit to be 430 GeV at 95% CL [577–579].

### 4.6. Generation of the Matter-Antimatter Asymmetry of the Universe

The matter-antimatter asymmetry of the universe has been inferred independently (and consistently) by big bang nucleosynthesis (BBN) predictions of light elements, and by the temperature anisotropies of the cosmic microwave background. In order to generate it, the Sakharov conditions need to be fulfilled [580]. There should be:


In the standard model, it is well-known that due to the chiral nature of weak interactions B+L is violated by sphaleron processes, while B−L is preserved [581]. Also C and CP are violated in the quark sector (in the CKM matrix), although the amount is too small to generate the required CP asymmetry. In the lepton sector (with massive neutrinos) CP can be violated, and there are in fact hints of δ ∼ −π/2 [20]. However, the measurement of the Higgs mass at 125 GeV implies that the phase transition is not strongly first-order, with no departure from thermal equilibrium. Therefore, the SM has to be extended to explain the matter-antimatter asymmetry which raises the question whether this new physics is related to neutrino masses or not.

When sphalerons are active and in thermal equilibrium, roughly at temperatures above the electroweak phase transition, B+L can be efficiently violated. Therefore, one natural option in models of Majorana neutrinos is that an asymmetry in lepton number is generated, which is converted by sphalerons into a baryon asymmetry. This is known as leptogenesis [25] (see Davidson et al. [582] for a review on the topic), the most popular example being the case of type-I seesaw, where the outof-equilibrium decays of the lightest of the heavy right-handed neutrinos into lepton and Higgs doublets and their conjugates, at a temperature equal or smaller than its mass, generate the lepton asymmetry due to CP-violating interactions.

The scotogenic model and its variants, see section 5.3, have been studied in detail regarding the generation of the baryon asymmetry from particle decays with TeV-scale masses. Ma [583] briefly discusses leptogenesis within the scotogenic model. This discussion is extended in Kashiwase and Suematsu [584, 585] and Racker [586] to include resonant leptogenesis. Resonant leptogenesis has also been studied in a gauge extension of the scotogenic model [224–226] in Kashiwase and Suematsu [252] and resonant baryogenesis in an extension with new colored states in Dev and Mohapatra [587]. Hambye et al. [588] and Babu and Ma [589] consider extensions of the scotogenic model by an additional charged or neutral scalar to achieve viable nonresonant leptogenesis. The baryon asymmetry can similarly be enhanced by producing the SM singlet fermions in the scotogenic model non-thermally beyond the usual thermal abundance [590]. Leptogenesis via decays of an inert Higgs doublet or a heavy Dirac fermion were studied in Lu and Gu [119, 154] in scotogenic-like models, respectively. In Chen and Law [127] leptogenesis was studied in a scotogenic-like model with fermionic 5-plets and a scalar 6-plet, via the decays of the second-lightest fermionic 5 plet. Baldes et al. [278] demonstrated the feasibility to generate the correct matter-antimatter asymmetry via leptogenesis in the model proposed in Ma [335]. It also showed that any pre-existing baryon asymmetry in the two models proposed in Ma [335] and Law and McDonald [277] is washed out at temperatures above the mass of their heaviest fields.

In radiative models with extra scalars coupled to the Higgs field, the phase transition can generally be stronger, as they contribute positively to the beta function of the Higgs and therefore, they help to stabilize the Higgs potential. Moreover, in these models there are typically extra sources of CP violation. These two ingredients allow the possibility of having electroweak baryogenesis. In particular, the strong first-order phase transition has been discussed using an effective potential in Bertolini et al. [591], and in Aoki et al. [592] for the model of Aoki et al. [370]. Also in the case of a supersymmetric radiative model in Kanemura et al. [202].

However, in general the new states can also destroy a preexisting asymmetry, irrespective of their production mechanism, as they violate necessarily lepton number by two units [593–596]. The new particles typically have gauge interactions, so that they are in thermal equilibrium at lower temperatures than those at which the asymmetry is generated (by high-scale baryogenesis or by leptogenesis, for instance30) potentially washing it out.

Some works have focused on the fact that if LNV is observed at the LHC, one could falsify leptogenesis, as the wash-out processes would be too large [597–599]. Similarly, observations of 0νββ rates beyond the one generated by the light neutrinos could impose constraints for the first family [600]. LFV processes could be used to extend it to all families. See Deppisch et al. [531] for further discussions about LNV processes in leptogenesis.

The limits on radiative models due to the requirement of not washing-out any pre-existing asymmetry are model-dependent. A more systematic way to go is to consider the LNV effective operators related to radiative models [81, 85, 87]. These operators lead to wash-out processes if they are in thermal equilibrium above the electroweak phase transition, and therefore their strength can be bounded by this requirement.

<sup>30</sup>In this last case, of course, the presence of low scale LNV can be regarded as being less motivated, as in principle there would already be an explanation for neutrino masses (at least for one neutrino).

### 4.7. A Possible Connection to Dark Matter Models

In many radiative neutrino mass models the generation of neutrino masses at tree-level is forbidden by a symmetry, G. This symmetry can be global or gauge, continuous or discrete (a typical example is a Z<sup>2</sup> parity), imposed or accidental (a byproduct of other symmetries in the model). If G is preserved after electroweak symmetry breaking, the lightest state transforming non-trivially under it, the so-called lightest charged particle (LCP), is completely stable and, in principle, could constitute the dark matter (DM) of the universe. This opens up an interesting connection between radiative neutrino masses and dark matter. DM may be produced via its coupling to neutrinos and thus the annihilation cross section is closely related to neutrino mass. This has been studied using an effective Lagrangian for light, MeV-scale, scalar DM [601] in a scotogenic-like model and for fermionic DM [460, 468–472] in the scotogenic model. A key signature of this close connection is a neutrino line from DM annihilation. The constraints from neutrino mass generation on the detectability of a neutrino line has been recently discussed in El Aisati et al. [602].

Based on the general classification of 1-loop models [100], the authors of Restrepo et al. [111] performed a systematic study for models compatible with DM stabilized by a discrete Z<sup>2</sup> symmetry. They focused on the topologies T1-x and T3. The topologies T4-2-i and T4-3-i require an additional symmetry to forbid the tree-level contribution and thus were not studied in Restrepo et al. [111]. A similar classification for 2-loop models has been presented in Simoes and Wegman [112] based on the possible 2-loop topologies discussed in Aristizabal Sierra et al. [102]. Symmetries forbidding tree and lower-order loop diagrams have been discussed in Farzan et al. [110]. In section 5.3 we discuss the prototype example of such models: the scotogenic model.

Besides dark matter being stabilized by a fundamental symmetry, it may be stable due to an accidental symmetry. For example, higher representations of SU(2)<sup>L</sup> cannot couple to the SM in a renormalizable theory, which leads to an accidental Z<sup>2</sup> symmetry at the renormalizable level. This has been dubbed minimal dark matter [385, 386]. After the initial proposal to connect the minimal dark matter paradigm and radiative neutrino mass generation [126], it has been conclusively demonstrated that the minimal dark matter paradigm cannot be realized in 1-loop neutrino mass models [473, 575, 576]. However, there is a viable variant of the KNT model at 3-loop order [362], which realizes the minimal dark matter paradigm without imposing any additional symmetry beyond the SM gauge symmetry.

Finally, the DM abundance in the universe may be explained by a light pseudo-Goldstone boson (pGB) associated with the spontaneous breaking of a global symmetry. It is commonly called Majoron in case the lepton number plays the role of the global symmetry. The possibility of pGB dark matter has been discussed in one of the models in Dasgupta et al. [219] which provides a pGB dark matter candidate after the breaking of a continuous U(1) symmetry to its Z<sup>2</sup> subgroup in addition to the LCP. Recently the authors of Ma et al. [603] proposed an extension of the Fileviez-Wise model [120] to incorporate a Majoron DM candidate which simultaneously solves the strong CP problem.

### 5. SELECTED EXAMPLES OF MODELS

In the following subsections, we list and discuss different benchmark models for neutrino mass that are qualitatively different. We start with the most well-studied models, which are the Zee model, discussed in section 5.1.1, that is the first 1-loop model for Majorana neutrino masses, and the Zee-Babu model, revisited in section 5.1.2, which is the first 2-loop model. In section 5.2 we discuss the first 3-loop model [355], which was proposed by Krauss, Nasri, and Trodden and is commonly called KNT-model, and its variants. It is also the first model with a stable dark matter candidate. The scotogenic model is discussed in section 5.3. It generates neutrino mass at 1-loop order and similarly to the KNT-model it features a stable dark matter candidate due to the imposed Z<sup>2</sup> symmetry. These are the most well-studied models in the literature. However, this preference is mostly due to the historic development (and also simplicity) and we are proposing a few other interesting benchmark models in the following subsections.

### 5.1. Models with Leptophillic Particles

There are only three different structures which violate lepton number (LN) by two units that can be constructed with SM fields [76]:

$$
\overline{\tilde{L}} \not\equiv (1, 3, -1) \,, \quad \overline{\tilde{L}} \not\sim (1, 1, -1) \,, \quad \overline{e\_{\mathbb{R}}^{\zeta}} \not\equiv (1, 1, -2) \, . \tag{79}
$$

The three different structures can couple respectively to a SU(2) triplet scalar with Y = 1 (we denote it by 1), a singly-charged SU(2) singlet scalar (we call it h +) and a doubly-charged SU(2) singlet scalar (we call it k ++).

In all cases, we could assign LN equal to −2 to the new fields so that such interactions preserve it. However, dimension-3 terms in the scalar potential will softly break LN, as there is no symmetry to prevent them. In the first case, the triplet can have in the potential the lepton-number violating term (with 1L = 2) with the SM Higgs doublet H

$$\mathcal{V}\_{\Delta} \subset \mu\_{\Delta} \tilde{H}^{\dagger} \Delta^{\dagger} H + \text{H.c.} \tag{80}$$

Then, after electroweak symmetry breaking, the triplet gets an induced VEV v<sup>T</sup> ≃ −µ1v 2 /m<sup>2</sup> 1 (strongly bounded by the T parameter to be . O(1) GeV), and neutrino masses are generated at tree-level via the type-II seesaw.

If only the singly-charged scalar h <sup>+</sup> is present, a 1L = 2 term can be constructed with two Higgs doublets, the SM Higgs H and an extra Higgs doublet 8

$$V\_{\rm Zee} \subset \mu\_{\rm Zee} \tilde{H}^{\dagger} \Phi (h^{+})^{\*} + \text{H.c.} \tag{81}$$

In this case, however, neutrino masses are not induced by the Higgs VEV at tree-level, but they are generated at 1-loop order. This is known as the Zee model [104, 105].

For the case of the doubly-charged scalar, one can construct the 1L = 2 term precisely with two singly-charged scalars h +

$$V\_{\rm ZB} \subset \mu\_{\rm ZB} \, h^+ h^+ (k^{++})^\* + \text{H.c.} \tag{82}$$

Notice that no other combination with SM fields exist, given the large electric charge of k ++. In this case, neutrino masses are generated at 2-loop order. This is known as the Zee-Babu model [76, 604].

These are the simplest radiative models. By using particles that couple to a lepton and a quark (leptoquarks), one can also have 1L = 2 interactions and generate neutrino masses at a different number of loops. In the following, we will discuss the Zee and Zee-Babu models.

#### 5.1.1. The Zee Model

In addition to the SM content with a Higgs scalar doublet H, the Zee model [104, 105] contains an extra Higgs scalar doublet 8 and a singly-charged scalar singlet h +, which is shown in **Table 6**. It is an example of the operator O<sup>2</sup> = L iL jL k e <sup>c</sup>H<sup>l</sup> ǫijǫkl. Several aspects of the phenomenology of the model have been studied in Petcov [605], Zee [288], Bertolini and Santamaria [606, 607], Yu et al. [608, 609], Frampton and Glashow [610], Jarlskog et al. [611], Ghosal et al. [451], Kanemura et al. [612], Balaji et al. [613], Koide [614], Brahmachari and Choubey [615], Frampton et al. [616], Assamagan et al. [617], He [156], Kanemura et al. [618], and Aristizabal Sierra and Restrepo [619]. While the Zee-Wolfenstein version where just the SM Higgs doublet couples to the leptons has been excluded by neutrino oscillation data [155, 156], the most general version of the Zee model in which both couple remains allowed [157] and has been recently studied in Herrero-García et al. [91] (see also Babu and Julio [158] and Aranda et al. [159] for a variant with a flavor-dependent Z<sup>4</sup> symmetry).

The Yukawa Lagrangian is

$$-\mathcal{L}\_L = \overline{L}\left(Y\_1^\dagger H + Y\_2^\dagger \Phi\right) \text{e}\mathbb{R} + \overline{\tilde{L}}f \, Lh^+ + \text{H.c.},\tag{83}$$

where L = (νL, eL) T and e<sup>R</sup> are the SU(2) lepton doublets and singlets, respectively, and L˜ ≡ iτ2L <sup>c</sup> <sup>=</sup> <sup>i</sup>τ2CL T with τ<sup>2</sup> being the second Pauli matrix. Due to Fermi statistics, f is an antisymmetric Yukawa matrix in flavor space, while Y<sup>1</sup> and Y<sup>2</sup>

TABLE 6 | Quantum numbers for new particles in the Zee model.


Assuming CP-invariance there are two CP-even neutral scalars (one of which is the 125 GeV Higgs boson, with mass mh, and the other is a heavy one with mass mH), one neutral CP-odd scalar with mass mA, and two charged-scalars of masses m<sup>h</sup> + 1,2 , whose mixing due to the trilinear term in Equation (81) is given by

$$s\_{2\psi} = \frac{\sqrt{2}\nu\mu\_{\text{Zee}}}{m\_{h\_2^+}^2 - m\_{h\_1^+}^2} \,. \tag{85}$$

Interestingly, µZee cannot be arbitrarily large, as it contributes at 1-loop level to the mass of the light Higgs. Demanding no fine-tuning, we can estimate |µZee| . 4π m<sup>h</sup> ≃ 1.5 TeV.

The Yukawa couplings of Equation (83), together with the term in the potential given in Equation (81), imply that lepton number is violated by the product m<sup>E</sup> (Y1v<sup>2</sup> − Y2v1)f µZee. Therefore, neutrino masses will be necessarily generated, in particular the lowest order contribution appears at 1-loop order, as shown diagram of **Figure 17**, where the charged scalars run in the loop. The neutrino mass matrix is given by:

$$\mathcal{M}\_{\nu} = A \left[ f \, m\_{\mathrm{E}}^{2} + m\_{\mathrm{E}}^{2} f^{T} - \frac{\nu}{\sqrt{2}} \, \text{(} f \, m\_{\mathrm{E}} \, Y\_{2} + Y\_{2}^{T} \, m\_{\mathrm{E}} f^{T} \text{)} \right] \ln \frac{m\_{h\_{2}^{+}}^{2}}{m\_{h\_{1}^{+}}^{2}},$$

$$A \equiv \frac{s\_{2\varphi} \, t\_{\beta}}{8 \sqrt{2} \pi^{2} \, \nu}, \tag{86}$$

with ϕ being the mixing angle for the charged scalars given in Equation (85). Therefore, in the Zee model, due to the loop and the chiral suppressions, the new physics scale can be light. From the form of the mass matrix it is clear that if one takes Y<sup>2</sup> → 0 (Zee-Wolfenstein model), the diagonal elements vanish, yielding neutrino mixing angles that are not compatible with observations.

Neglecting m<sup>e</sup> ≪ mµ, m<sup>τ</sup> and taking fe<sup>µ</sup> = 0, the following Majorana mass matrix is obtained

$$\mathcal{M}\_{\nu} = A \frac{m\_{\text{t}}v}{\sqrt{2}} \begin{pmatrix} -2f^{\text{cr}}Y\_{2}^{\text{cr}} & -f^{\text{cr}}Y\_{2}^{\text{cr}} - f^{\text{cr}}Y\_{2}^{\text{cr}} & \frac{\sqrt{2}\rho\_{\text{f}}m\_{\text{t}}}{\nu}f^{\text{cr}} - f^{\text{cr}}Y\_{2}^{\text{cr}} \\ -f^{\text{cr}}Y\_{2}^{\text{cr}} - f^{\text{ar}}Y\_{2}^{\text{cr}} & -2f^{\text{cr}}Y\_{2}^{\text{cr}} & \frac{\sqrt{2}\rho\_{\text{f}}m\_{\text{t}}}{\nu}f^{\text{cr}} - f^{\text{cr}}Y\_{2}^{\text{cr}} \\ \frac{\sqrt{2}\rho\_{\text{f}}m\_{\text{t}}}{\nu}f^{\text{cr}} - f^{\text{cr}}Y\_{2}^{\text{cr}} & \frac{\sqrt{2}\rho\_{\text{f}}m\_{\text{t}}}{\nu}f^{\text{cr}} - f^{\text{cr}}Y\_{2}^{\text{cr}} & 2\frac{m\_{\text{t}}}{m\_{\text{t}}}f^{\text{cr}}Y\_{2}^{\text{cr}} \end{pmatrix}. \tag{87}$$

are completely general complex Yukawa matrices. Furthermore, the charged-lepton mass matrix is given by

$$m\_{\rm E} = \frac{\nu}{\sqrt{2}} (c\_{\beta} Y\_1^{\dagger} + s\_{\beta} Y\_2^{\dagger}) \, , \tag{84}$$

where tan <sup>β</sup> <sup>=</sup> <sup>s</sup>β/c<sup>β</sup> <sup>=</sup> <sup>v</sup>2/v<sup>1</sup> with <sup>h</sup>H<sup>0</sup> i = <sup>v</sup><sup>1</sup> and <sup>h</sup>8<sup>0</sup> i = v<sup>2</sup> and v <sup>2</sup> <sup>=</sup> <sup>v</sup> 2 <sup>1</sup> + v 2 2 . Without loss of generality, one can work in the basis where m<sup>E</sup> is diagonal.

Notice that if the term proportional to the muon mass is neglected, one neutrino remains massless. In order to obtain correct mixing angles, we need both Y τµ 2 and Y τ e 2 different from zero [91, 491], as they enter in the 1-2 submatrix of Equation (87). This implies that LFV mediated by the scalars will be induced. In fact, in the model large LFV signals are generated, like τ → µγ and µ − e conversion in nuclei. Moreover, also a full numerical scan of the model performed in in Herrero-García et al. [91] showed that large LFV Higgs decays are possible, in particular

BR(h → τµ) can reach the percent level. BR(h → τ e) is roughly two-orders of magnitude smaller than BR(h → τµ). The singly-charged h also generates violations of universality, as it interferes constructively with the W boson, as well as nonstandard interactions, see section 4.1, which however are too small to be observed [91].

In Herrero-García et al. [91] it was also shown that the model is testable in next-generation experiments. While normal mass ordering (NO) provided a good fit, inverted mass ordering (IO) is disfavored, and if θ<sup>23</sup> happens to be in the second octant, then IO will be ruled-out. Notice also that the lightest neutrino is required to be massless for IO, as it has also been obtained in He and Majee [157]. Furthermore, future τ → µγ (µ−e conversion) will test most regions of the parameter space in NO (IO). Regarding direct searches at the LHC, the new scalars have to be below ∼2 TeV, which implies that they can be searched for similarly as in a two-Higgs doublet model (with an extra charged scalar that could be much heavier). Particularly, the charged scalars are searched for at colliders. See the discussion in section 4.5.

Let us mention that an interesting modification of the Zee model was proposed in Babu and Julio [158] (see also Aranda et al. [159]), where a Z<sup>4</sup> symmetry was imposed, being able to reduce significantly the number of parameters. In that case, among the predictions of the model, is that the spectrum should be inverted. Other flavor symmetries beyond Z<sup>4</sup> in this framework have been studied in Babu and Mohapatra [168, 169], Koide and Ghosal [170], Kitabayashi and Yasue, [171], Adhikary et al. [172], Fukuyama et al. [173], Aranda et al. [174, 175].

#### 5.1.2. The Zee-Babu Model

The Zee-Babu model contains, in addition to the SM, two SU(2) singlet scalar fields with electric charges one and two, denoted by h + and k ++ [76, 604] as shown in **Table 7**. It is a UV completion of the operator O<sup>9</sup> = L iL jL k e cL l e c ǫijǫkl. Several studies of its phenomenology exist in the literature [95, 435, 620–622].

The leptonic Yukawa Lagrangian reads:

$$\mathcal{L}\_L = \overline{L}\,Y^\dagger \,e\_\mathcal{R}\mathcal{H} + \overline{\overline{L}}f\mathcal{L}h^+ + \overline{e\_\mathcal{R}^\zeta} \,e\_\mathcal{R} \,k^{++} + \text{H.c.},\tag{88}$$

TABLE 7 | Quantum numbers for new particles in the Zee-Babu model.


where like in the Zee model, due to Fermi statistics, f is an antisymmetric matrix in flavor space. On the other hand, g is symmetric. Charged lepton masses are given by <sup>m</sup><sup>E</sup> <sup>=</sup> <sup>√</sup><sup>v</sup> 2 Y † , which be take to be diagonal without loss of generality.

Lepton number is violated by the simultaneous presence of the trilinear term µZB in Equation (82), together with mE, f , g. Note that the trilinear term cannot be arbitrarily large, as it contributes to the charged scalar masses at loop level, and can also lead to charge-breaking minima, if |µZB| is large compared to the charged scalar masses. For naturalness considerations we demand |µZB| ≪ 4π min(mh, m<sup>k</sup> ). See Nebot et al. [435] and Herrero-Garcia et al. [95] for detailed discussions.

As lepton number is not protected, neutrino masses are generated radiatively, in particular at 2-loop order, via the diagram of **Figure 18**. The mass matrix is approximately given by (see for instance McDonald and McKellar [286], Nebot et al. [435] and Herrero-Garcia et al. [95] for more details)

$$\mathcal{M}\_{\nu} \simeq \frac{\nu^2 \mu\_{\rm ZB}}{96\pi^2 M^2} f \, Y \, g^\dagger \, Y^T f^T,\tag{89}$$

where M is the heaviest mass of the loop, either that of the singly-charged singlet h + or of the doubly-charged singlet k ++. A prediction of the model is that, since f is a 3 × 3 antisymmetric matrix, det <sup>f</sup> <sup>=</sup> 0, and therefore detM<sup>ν</sup> <sup>=</sup> 0. Thus, at least one of the neutrinos is exactly massless at this order.

In the model, both NO and IO can be accommodated. The phenomenology of the singly-charged scalar is similar to that discussed in the Zee model, apart from the fact that in the Zee model the charged singlet mixes with the charged component of the doublet. Some of the most important predictions of the model are due to the presence of the doubly-charged scalar k ++. Firstly, k ++ mediates trilepton decays (ℓ<sup>i</sup> → ℓjℓkℓ<sup>l</sup> ) at tree-level which unlike, in the Zee model, are not suppressed by the small charged lepton masses, as well as radiative decays (ℓ<sup>i</sup> → ℓjγ ). Secondly, k ++ can be pair-produced at the LHC via Drell-Yan, decaying among other final states into same-sign leptons which yields a clean experimental signature. See the discussion in section 4.5.

### 5.2. KNT-Models

The first radiative neutrino mass model at 3-loop order is the KNT model [355] which has one fermionic singlet N and two singly-charged scalars S1,2 in addition to the SM particles. A discrete Z<sup>2</sup> symmetry is imposed, under which only S<sup>2</sup> and N are odd. We list the quantum numbers of the exotic particles in **Table 8**.

The Z<sup>2</sup> symmetry forbids the usual type-I seesaw contribution at tree-level. The relevant Lagrangian is expressed as

$$\mathcal{L} = f \text{ } L^T \text{Cir}\_2 L \text{S}\_1^\* + g \text{ } \overline{\text{N}^c} e \text{R} \text{S}\_2^\* + \frac{1}{2} M\_{\text{NN}} \text{N}^T \text{CN} + \text{H.c.} \tag{90}$$

$$1 + M\_{\text{S}1} \text{S}\_1 \text{S}\_1^\* + M\_{\text{S}2} \text{S}\_2 \text{S}\_2^\* + \frac{1}{4} \lambda\_{\text{S}} \text{(S}\_1 \text{S}\_2^\*\text{)}^2,\tag{91}$$

where the flavor indices of f and g are all suppressed. With this setup, neutrino masses are generated first at 3-loop order as shown in **Figure 19**. The neutrino mass matrix is then

$$(\mathcal{M}\_{\boldsymbol{\nu}})\_{ij} = \sum\_{\alpha\beta} \frac{\lambda\_{\boldsymbol{\mathcal{S}}}}{(4\pi^2)^3} \frac{m\_{\alpha}m\_{\beta}}{M\_{\boldsymbol{\mathcal{S}}\_2}} f\_{i\alpha} f\_{j\beta} g\_{\alpha}^\* g\_{\beta}^\* F \left(\frac{M\_N^2}{M\_{\boldsymbol{\mathcal{S}}\_2}^2}, \frac{M\_{\boldsymbol{\mathcal{S}}\_1}^2}{M\_{\boldsymbol{\mathcal{S}}\_2}^2}\right), \tag{92}$$

where the function F is defined in Ahriche and Nasri [358]. This matrix is, however, only rank one and thus can give exactly one non-zero neutrino mass. Adding more copies of N can increase the rank of the matrix. The phenomenology of this model including flavor physics, dark matter, Higgs decay, electroweak phase transition and collider searches is discussed in detail in Ahriche and Nasri [358].

This model is subject to constraints from LFV experiments such as µ → eγ which requires three copies of N for the neutrino mixing to be in agreement with the observations<sup>31</sup> . Meanwhile in order to be consistent with the measurements of muon anomalous magnetic moment and the 0νββ decay, strong constraints are imposed. For MS1,S<sup>2</sup> > 100 GeV, 10 <sup>−</sup><sup>5</sup> . <sup>g</sup>i1gi<sup>2</sup>  . 10 and 10−<sup>5</sup> . f13f<sup>23</sup>  . 1, it can satisfy all flavor constraints while reproducing the neutrino mixing data.

Assuming a mass hierarchy M<sup>N</sup> < MS<sup>2</sup> , the lightest fermion singlet is stable and serves as a good DM candidate. This is also the first radiative neutrino mass theory with a stable DM candidate running in the loop. If the DM relic density is saturated and all previously discussed constraints are satisfied, the DM mass cannot exceed 225 GeV while the lighter charged scalar S<sup>2</sup> cannot be heavier than 245 GeV. If the fermion singlets have very small mass splitting, DM coannihilation effects should be taken into account. With about 5% mass splitting, the DM relic density increases by 50%.

As discussed in section 4.5.3, the singly-charged scalars can be pair-produced at the LHC and subsequently decay to a pair of charged leptons and the fermion singlets which appear as missing transverse energy. This signature is exactly the same as the direct slepton pair production in SUSY theories. ATLAS has performed the search for sleptons in this channel with 36.1 fb−<sup>1</sup> data of

TABLE 8 | Quantum numbers for new particles in the original KNT model.


√ s = 13 TeV [574] and has ruled out slepton masses below ∼500 GeV in the non-compressed region. The actual constraint on MS<sup>2</sup> depends on the decay branching ratio of S<sup>2</sup> to different leptons and in principle will be substantially relaxed compared to the ATLAS search.

With the same topology, a lot of variations of the KNT model can be constructed. Chen et al. [356] discusses several possibilities to replace the electron with other SM fermions<sup>32</sup> or vector-like fermions. A similar model in which the electron is replaced by a fermion doublet with hypercharge 5/2 and S1,2 with doublycharged scalar is discussed in Okada and Yagyu [368]. The Z2-odd particles in this model form instead the outer loop.

### 5.3. The Scotogenic Model

The most popular model linking dark matter to the radiative generation of neutrino masses is the one proposed by E. Ma in 2006. We will refer to it as scotogenic model [113] <sup>33</sup>. In the scotogenic model, the SM particle content is extended with three singlet fermions, N<sup>i</sup> (i = 1, 2, 3), and one SU(2)<sup>L</sup> doublet, η, with hypercharge <sup>1</sup> 2 ,

$$
\eta = \begin{pmatrix} \eta^+ \\ \eta^0 \end{pmatrix}. \tag{93}
$$

This setup is supplemented with a Z<sup>2</sup> parity, under which the new states are odd and all the SM particles are even34. The newly-introduced particles with their respective charges of the scotogenic model are shown in **Table 9**. The gauge and discrete

<sup>31</sup> Less copies of N means less contribution to the neutrino mass matrix, which in turn generally leads to larger Yukawa couplings to generate the same neutrino mass scale and thus more likely to violate constraints from LFV processes.

<sup>32</sup>The authors of Chen et al. [356] also point out that up-quarks are not feasible due to gauge invariance.

<sup>33</sup>The scotogenic model has been extensively studied, sometimes referring to it with different names. For instance, some authors prefer the denomination radiative seesaw. In this review we will stick to the more popular name scotogenic model, which comes from the Greek word skotos (σ oτoς), darkness. scotogenic would then mean created from darkness.

<sup>34</sup>The Z<sup>2</sup> symmetry can obtained from the spontaneous breaking of an Abelian U(1) factor, see for instance Aristizabal Sierra et al. [623].

TABLE 9 | Quantum numbers of new particles in the scotogenic model.


symmetries of the model allow us to write the Lagrangian terms involving the fermion singlets

$$\mathcal{L}\_N = \frac{M\_N}{2} \overline{N^c} N + Y\_N \eta \,\overline{N}L + \text{H.c.} \tag{94}$$

We do not write the kinetic term for the fermion singlet as it takes the standard canonical form. Y<sup>N</sup> is an arbitrary 3 × 3 complex matrix, whereas the 3×3 Majorana mass matrix M<sup>N</sup> can be taken to be diagonal without loss of generality. We highlight that the usual neutrino Yukawa couplings with the SM Higgs doublet are not allowed due to the Z<sup>2</sup> symmetry. This is what prevents the light neutrinos from getting a non-zero mass at tree-level. The scalar potential of the model is given by

$$\begin{split} \mathcal{V} &= -m\_H^2 H^\dagger H + m\_\eta^2 \eta^\dagger \eta + \frac{\lambda\_1}{2} \left( H^\dagger H \right)^2 + \frac{\lambda\_2}{2} \left( \eta^\dagger \eta \right)^2 \\ &+ \lambda\_3 \left( H^\dagger H \right) \left( \eta^\dagger \eta \right) + \lambda\_4 \left( H^\dagger \eta \right) \left( \eta^\dagger H \right) \\ &+ \frac{\lambda\_5}{2} \left[ \left( H^\dagger \eta \right)^2 + \left( \eta^\dagger H \right)^2 \right]. \end{split} \tag{95}$$

Neutrino masses are induced at the 1-loop level via the diagram in **Figure 20**

$$\begin{split} (\mathcal{M}\_{\upsilon})\_{ij} &= \sum\_{k=1}^{3} \frac{Y\_{Nki} Y\_{Nkj}}{32\pi^{2}} M\_{Nk} \left[ \frac{m\_{R}^{2}}{m\_{R}^{2} - M\_{Nk}^{2}} \ln \left( \frac{m\_{R}^{2}}{M\_{Nk}^{2}} \right) \right. \\ &\quad - \frac{m\_{I}^{2}}{m\_{I}^{2} - M\_{Nk}^{2}} \ln \left( \frac{m\_{I}^{2}}{M\_{Nk}^{2}} \right) \Bigg], \end{split} \tag{96}$$

where the masses of the scalar η<sup>R</sup> and pseudo-scalar part η<sup>I</sup> of the neutral scalar η <sup>0</sup> <sup>=</sup> (η<sup>R</sup> <sup>+</sup> <sup>i</sup>ηI)/ √ 2 are given by

$$m\_{\mathbb{R},I}^2 = m\_{\eta}^2 + \frac{1}{2} \left(\lambda\_3 + \lambda\_4 \pm \lambda\_5\right) \nu^2 \tag{97}$$

with the electroweak VEV v = √ 2 H0 ≃ 246GeV. Neutrino mass vanishes in the limit of λ<sup>5</sup> = 0 and thus degenerate masses for the neutral scalars ηR,<sup>I</sup> , because it is possible to define a generalized lepton number which forbids a Majorana mass term.

In the scotogenic model, the Z<sup>2</sup> parity is assumed to be preserved after electroweak symmetry breaking. This will be so if hηi = 0. In this case, the lightest Z2-odd state (to be identified with the LCP defined in section 4.7) will be stable and, if neutral, will constitute a potentially good DM candidate. The LCP in the scotogenic model can be either a fermion or a scalar: the lightest singlet fermion N<sup>1</sup> or the lightest neutral η scalar (η<sup>R</sup> or ηI). As the neutrino Yukawa couplings are generally required to be small to satisfy LFV constraints, the DM phenomenology

for a scalar LCP is generally the same as in the inert doublet model [624, 625]. Recently it has been pointed out [626] that late decay of the lightest SM singlet fermion N<sup>1</sup> may repopulate the dark matter abundance and thus resurrect the intermediate dark matter mass window between mW, the mass of the W boson, and 550 GeV. In the case of a fermionic LCP, for which the annihilation cross section is governed by the neutrino Yukawa couplings, the connection of the dark matter abundance with neutrino masses leads to a very constrained scenario due to the bounds from lepton flavor violation [460, 468–472].

Many scotogenic variations have been proposed since the publication of the minimal model described above. All these models are characterized by neutrino masses being induced by new dark sector particles running in a loop [114–135, 138, 139, 141–143, 204, 261, 265]. One of them involves a global continuous dark symmetry, instead of a discrete dark symmetry [145], Hagedorn, (in prep). A gauge dark symmetry was considered in Yu [253] and a scale-invariant version presented in Ahriche et al. [245]. The collider [627–630] and dark matter [631–634] phenomenologies of different scotogenic variants have also been discussed in detail. Finally, we point out that the authors of Merle and Platscher [635] identified a potential problem in this family of models, since some parameter regions lead to the breaking of the Z<sup>2</sup> parity at high energies. This problem, how it can be escaped and its phenomenological implications have been explored in Merle et al. [636], Merle and Platscher [267], and Lindner et al. [637].

#### 5.4. Models with Leptoquarks

Leptoquarks are common ingredients of radiative neutrino mass models. For example neutrino mass can be generated at loop level by two leptoquarks which mix via a trilinear coupling to the SM Higgs boson [178–185]. Neutrino mass generation at 1 loop order with all possible leptoquarks has been systematically studied in in Aristizabal Sierra et al. [181]. At 1-loop order and especially at a higher-loop order, leptoquarks usually appear together with other exotic particles such as vector-like quarks and leptons, charged scalar singlets and electroweak multiplets [214]. We will review two models here, one at 1-loop and one at 2-loop order.

#### 5.4.1. A 1-Loop Model

Without introducing exotic fermions, the only possible topology that can contribute at 1-loop order to the Weinberg operator is T1-ii shown in **Figure 3** as we need the fermion arrow to flip only once. With this topology and leptoquarks as the only exotic particles, the only UV completion we can realize is depicted in **Figure 21**. The relevant scalar leptoquarks<sup>35</sup> are S1, S<sup>3</sup> and R˜ <sup>2</sup> with quantum numbers detailed in **Table 10**. The relevant Lagrangian reads

$$\begin{split} \Delta \mathcal{L} &= \mathcal{y}\_1 \overline{Q^c} L \mathcal{S}\_1 + \mathcal{y}\_3 \overline{Q^c} \mathcal{S}\_3 L + \tilde{\mathcal{y}}\_2 \bar{d} L \tilde{\mathcal{R}}\_2 + \lambda\_1 \mathcal{S}\_1^\* \tilde{\mathcal{R}}\_2^\dagger H \\ &+ \lambda\_3 \tilde{\mathcal{R}}\_2^\dagger \mathcal{S}\_3^\dagger H + \text{H.c.}, \end{split} \tag{98}$$

following the convention in in Doršner et al. [185] with all generation indices suppressed. Apparently only the leptoquark component fields with electric charge Q = − 1 3 can contribute. These leptoquarks, in the interaction basis (S1, S 1 3 3 , R˜ − 1 3 ∗ 2 ), will mix with each other through the λ1,3 terms in Equation (98)<sup>36</sup> . We will consider simplified scenarios where either S<sup>1</sup> or S<sup>3</sup> appears together with R˜ <sup>2</sup>. For the model with S1,3, the squaredmass matrix will be diagonalized with angle θ1,3 and the mass eigenvalues are m<sup>1</sup> and m2. So the neutrino mass matrix is expressed as [181, 185]

$$\mathcal{M}\_{\upsilon} \simeq \frac{3\sin 2\theta\_{1,3}}{32\pi^2} \ln \frac{m\_2^2}{m\_1^2} \left( \tilde{\nu}\_2^T \, M\_d \, \mathcal{Y}\_{1,3} + \mathcal{Y}\_{1,3}^T \, M\_d \, \tilde{\mathcal{Y}}\_2 \right) \,, \tag{99}$$

where M<sup>d</sup> = diag(md, m<sup>s</sup> , m<sup>b</sup> ) with md,s,<sup>b</sup> being the down, strange and bottom quark masses. Due to the hierarchy of down-type quark masses, the neutrino mass matrix will be approximately rank-2 with one nearly massless neutrino. Current neutrino oscillation data put lower bounds on the product of Yukawa couplings ranging from 10−<sup>12</sup> to 10−<sup>7</sup> for leptoquarks with TeV scale masses [181]. On the other hand, low energy precision experiments constrain the Yukawa couplings from above. For example, µ − e conversion in titanium bounds the first generation Yukawa couplings with

$$\left(\left(\hat{\mathbf{y}}\_{2}\right)\_{11}\left(\hat{\mathbf{y}}\_{2}\right)\_{21} < 2.6 \times 10^{-3}, \qquad \left(\mathbf{y}\_{3}\right)\_{11}\left(\mathbf{y}\_{3}\right)\_{21} < 1.7 \times 10^{-3}, \tag{100}$$

for 1 TeV leptoquark masses. Their decay branching fractions are dictated by the same couplings that determine the neutrino masses and mixings, which leads to a specific connection between the decay channels of the leptoquark and the neutrino mixings. Generally LFV decays with similar branching ratios to final states with muon and tau are expected in some leptoquark decays. This neutrino mass model can also be tested at colliders. The leptoquarks running in the loop can be created in pairs and decay to final states containing leptons plus jets with predicted branching ratios. We refer to section 4.5 for further details on searches of leptoquarks at colliders.

Päs and E. Schumacher [183] explored the possibility to explain the anomalous b → sll transitions with S<sup>3</sup> and R˜ 2.

TABLE 10 | Quantum numbers of leptoquarks.


Different texture of the Yukawa coupling matrices y<sup>3</sup> and y˜<sup>2</sup> were considered and leptoquark masses in the the range of 1 to 50 TeV can reproduce the neutrino masses and mixings in addition to R<sup>K</sup> [639].

#### 5.4.2. A 2-Loop Model

Based on the gauge-invariant effective operator <sup>O</sup>11<sup>b</sup> <sup>=</sup> LLQdcQd<sup>c</sup> , which violates lepton number by two units, a UV complete radiative neutrino mass model at 2-loop order containing leptoquark S<sup>1</sup> and fermion color octet f can be constructed [287]. We list their quantum numbers in **Table 11** for the convenience of the readers.

The general gauge invariant Lagrangian for the exotic particles is then expressed as

$$
\Delta \mathcal{L} = \left( \lambda^{LQ} \overline{L}^{\epsilon} Q S\_1 + \lambda^{d \overline{f}} \overline{d} f \, S\_1^\* + \lambda^{\epsilon u} \overline{\varepsilon}^{\epsilon} u \, S\_1 + \text{H.c.} \right) - \frac{1}{2} \, m\_f \overline{f}^{\epsilon} f,\tag{101}
$$

where generation indices for all parameters and fields are suppressed. We demand baryon number conservation to forbid the terms QQS ¯ <sup>1</sup> and ud¯ c S<sup>1</sup> which induce proton decay. With this setup, Majorana neutrino mass will be generated at 2-loop order as shown in **Figure 22**. Generally the contribution to the neutrino mass matrix is proportional to the down-type Yukawa coupling squared which is dominated by the third generation unless strong hierarchy in λ LQλ df exists. As a result, we can simplify the formula for the neutrino mass matrix to

$$(M\_{\nu})\_{ij} \simeq 4 \frac{m\_f m\_b^2 V\_{tb}^2}{(2\pi)^8} \sum\_{\alpha,\beta=1}^{N\_{\mathbb{S}\_1}} \left( \lambda\_{i3\alpha}^{LQ} \lambda\_{3\alpha}^{d\overline{f}} \right) \left( I\_{\alpha\beta} \right) \left( \lambda\_{j3\beta}^{LQ} \lambda\_{3\beta}^{d\overline{f}} \right), \quad \text{(102)}$$

with the CKM-matrix element Vtb and Iαβ as a function of m<sup>f</sup> and mS<sup>1</sup> whose exact form can be read from Angel et

<sup>35</sup>We follow the nomenclature in Doršner et al. [638] and Buchmuller et al. [542] for the names of the leptoquarks, where subscripts indicate dimension of the SU(2)<sup>L</sup> representations.

<sup>36</sup> Aristizabal Sierra et al. [181] considered the most general interactions with all possible leptoquarks and found in total four mass matrices for leptoquarks with electric charges Q = − 1 3 , − 2 3 , − 4 3 and − 5 3 .



al. [287]. The indices α and β label the leptoquark copies. This neutrino mass matrix is only rank one if there is only one leptoquark flavor assuming the dominance of the bottom-quark loop37. At least two leptoquarks are needed to fit to the current neutrino oscillation data in this model, where one neutrino mass eigenvalue is nearly vanishing. Among all flavor processes, µ − e conversion in nuclei, µ → eγ and µ → eee give the most stringent constraints.

The leptoquark S<sup>1</sup> can explain the recent anomalies observed in semileptonic B decays, i.e., the violation of lepton flavor universality (LFU) of RK(∗) [639] and RD(∗) [640–645]. In the parameter space with relatively large λ eu 32, the combination of leftand right-handed couplings induces scalar and tensor operators, which lift the chirality suppression of the semi-leptonic B-decay B → D (∗) ℓν and produce sizable effects in the LFU observables RD(∗) [313].

### 5.5. Supersymmetric Models with R-Parity Violation

Supersymmetric models with R-parity violation naturally lead to non-zero neutrino masses and mixings. These models have been regarded as very economical, since no new superfields besides those already present in the MSSM are required. Moreover, their phenomenology clearly departs from the standard phenomenology in the usual SUSY models, typically providing new experimental probes.

With the MSSM particle content, one can write the following superpotential, invariant under supersymmetry, as well as the gauge and Lorentz symmetries,

$$\mathcal{W} = \mathcal{W}^{\text{MSSM}} + \mathcal{W}^{\mathbb{R}\_p} \,. \tag{103}$$

Here WMSSM is the MSSM superpotential, whereas

$$\mathcal{W}^{\mathbb{R}\_p} = \frac{1}{2} \lambda\_{ijk} \widehat{L\_i} \widehat{L\_j} \widehat{e}^\epsilon\_k + \lambda'\_{ijk} \widehat{L\_i} \widehat{Q}\_j \widehat{d}^c\_k + \epsilon\_i \widehat{L\_i} \widehat{H}\_\mu + \frac{1}{2} \lambda''\_{ijk} \widehat{u}^c\_i \widehat{d}^c\_j \widehat{d}^c\_k. \tag{104}$$

The ǫ coupling has dimensions of mass, {i, j, k} denote flavor indices and gauge indices have been omitted for the sake of clarity. The first three terms in WR/<sup>p</sup> break lepton number (L) whereas the last one breaks baryon number (B). The nonobservation of processes violating these symmetries impose strong constraints on these parameters, which are required to be rather small [646]. Also importantly, their simultaneous presence would lead to proton decay, a process that has never been observed and whose rate has been constrained to increasingly small numbers along the years. For this reason, it is common to forbid the couplings in Equation (104) by introducing a discrete symmetry called R-parity. The R-parity of a particle is defined as

$$R\_{\mathcal{P}} = (-1)^{3(\mathcal{B}-\mathcal{L}) + 2s},\tag{105}$$

where s is the spin of the particle. With this definition, all SM particles have R<sup>p</sup> = +1 while their superpartners have R<sup>p</sup> = −1, and the four terms in WR/<sup>p</sup> are forbidden. Furthermore, as a side effect, the lightest supersymmetric particle (LSP) becomes stable and can be a dark matter candidate.

However, there is no fundamental reason to forbid all four couplings in WR/<sup>p</sup> . When R-parity is conserved both lepton and baryon numbers are conserved, but in order to prevent proton decay just one these two symmetries suffices. Furthermore, the breaking of R-parity by L-violating couplings generates non-zero neutrino masses, and thus constitutes a well-motivated scenario beyond the standard SUSY models. This scenario (with only Lviolating couplings) can be theoretically justified by replacing Rparity by a less restrictive symmetry, such as baryon triality [647].

We can distinguish two types of R-parity violating (RPV) neutrino mass models:


We now proceed to discuss some of the central features of these two types of leptonic RPV models, highlighting the most remarkable experimental predictions. Although in general one can have both types of leptonic RPV simultaneously, we will discuss them separately for the sake of clarity.

<sup>37</sup>The contributions of the strange and down quarks are suppressed by m<sup>2</sup> s,d /m<sup>2</sup> b and thus have been neglected in the discussion of Angel et al. [287].

#### Neutrino Masses with b-R/ <sup>p</sup>

Bilinear R-parity violation [177] is arguably the most economical supersymmetric scenario for neutrino masses. The bilinear <sup>ǫ</sup><sup>i</sup> <sup>=</sup> ǫe, ǫµ, ǫ<sup>τ</sup> terms in the superpotential come along with new B i ǫ = B e ǫ , B µ ǫ , B τ ǫ terms in the soft SUSY breaking potential. Therefore, the number of new parameters in b-R/<sup>p</sup> with respect to the MSSM is 6, without modifying its particle content, and they suffice to accommodate all neutrino oscillation data. For a comprehensive review on b-R/<sup>p</sup> see Hirsch and Valle [648].

The ǫ<sup>i</sup> couplings induce mixing between the neutrinos and the MSSM neutralinos. In the basis (ψ 0 ) <sup>T</sup> <sup>=</sup> (−iB˜ <sup>0</sup> , −iW˜ 0 3 , He0 d , He0 u , νe, νµ, ν<sup>τ</sup> ), the neutral fermion mass matrix M<sup>N</sup> is given by

$$\mathcal{M}\_N = \begin{pmatrix} \mathcal{M}\_{\mathcal{X}^0} \ m^T \\\\ m & 0 \end{pmatrix}. \tag{106}$$

Here Mχ<sup>0</sup> is the standard MSSM neutralino mass matrix and m ∝ ǫ is the matrix containing the neutrino-neutralino mixing. Assuming the hierarchy <sup>m</sup>≪Mχ<sup>0</sup> (naturally fulfilled if <sup>ǫ</sup>≪mW), one can diagonalize the mass matrix in Equation (106) in the seesaw approximation, <sup>m</sup><sup>ν</sup> = −<sup>m</sup> · <sup>M</sup>−<sup>1</sup> <sup>χ</sup><sup>0</sup> <sup>m</sup><sup>T</sup> , obtaining

$$m\_{\boldsymbol{\upsilon}} = \frac{M\_1 \boldsymbol{g}^2 + M\_2 \boldsymbol{g}'^2}{4 \operatorname{Det}(\mathcal{M}\_{\boldsymbol{\chi}^0})} \begin{pmatrix} \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}}^2 & \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \boldsymbol{\Lambda}\_{\boldsymbol{\mu}} & \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \\ \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \boldsymbol{\Lambda}\_{\boldsymbol{\mu}} & \boldsymbol{\Lambda}\_{\boldsymbol{\mu}}^2 & \boldsymbol{\Lambda}\_{\boldsymbol{\mu}} \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \\ \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} & \boldsymbol{\Lambda}\_{\boldsymbol{\mu}} \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}} & \boldsymbol{\Lambda}\_{\boldsymbol{\varepsilon}}^2 \end{pmatrix} \tag{107}$$

where 3<sup>i</sup> = µv<sup>i</sup> + vdǫ<sup>i</sup> are the so-called alignment parameters. Here M1,2 are the usual gaugino soft mass terms, µ is the Higgsino superpotential mass term, vd/ √ 2 is the H 0 d VEV and vi/ √ 2 are the sneutrino VEVs (induced by ǫ<sup>i</sup> 6= 0). The special (projective) form of mν implies that it is a rank 1 matrix, with only one non-zero eigenvalue, identified with the atmospheric mass scale. Furthermore, one can obtain two leptonic mixing angles in terms of the alignment parameters,

$$\tan \theta\_{13} = -\frac{\Lambda\_{\varepsilon}}{(\Lambda\_{\mu}^{2} + \Lambda\_{\varepsilon}^{2})^{\frac{1}{2}}} \quad , \quad \tan \theta\_{23} = -\frac{\Lambda\_{\mu}}{\Lambda\_{\varepsilon}} \, . \tag{108}$$

The generation of the solar mass scale, which is much smaller (1m<sup>2</sup> sol <sup>≪</sup> <sup>1</sup>m<sup>2</sup> atm), requires one to go beyond the tree-level approximation. This makes b-R/<sup>p</sup> a hybrid radiative neutrino mass model, since loop corrections are necessary in order to reconcile the model with the observations in neutrino oscillation experiments. An example of such loops is shown in **Figure 23**, where the bottom–sbottom diagrams are displayed. These are found to be the dominant contributions to the solar mass scale generation in most parts of the parameter space of the model. Other relevant contributions are given by the taustau and neutrino-sneutrino loops [649–651]. In all cases two R/<sup>p</sup> projections are required, hence leading to the generation of 1L = 2 Majorana masses for the light neutrinos.

The most important consequence of the breaking of R-parity at the LHC is that the LSP is no longer stable and decays. In fact, this is the only relevant change with respect to the standard MSSM phenomenology. Since the R/<sup>p</sup> couplings are constrained to be small, they do not affect the production cross-sections or the intermediate steps of the decay chains, and hence only the LSP decay is altered in an observable way. For instance, the smallness of the R/<sup>p</sup> couplings typically imply observable displaced vertices at the LHC, see for instance de Campos et al. [652]. Furthermore, in b-R/<sup>p</sup> there is a sharp correlation between the LSP decay and the mixing angles measured in neutrino oscillation experiments [653–656]. This connection allows to test the model at colliders. For instance, for a neutralino LSP one finds

$$\frac{\text{BR}(\tilde{\chi}\_1^0 \to W\mu)}{\text{BR}(\tilde{\chi}\_1^0 \to W\tau)} \cong \left(\frac{\Lambda\_\mu}{\Lambda\_\tau}\right)^2 = \tan^2 \theta\_{23} \cong 1\,. \tag{109}$$

A departure from this value would rule out the model completely. Interestingly, these correlations are also found in extended models which effectively lead to bilinear R/<sup>p</sup> [657–659].

#### Neutrino Masses with t-R/ <sup>p</sup>

Supersymmetry with trilinear R/<sup>p</sup> has many similarities with leptoquark models. Once the trilinear RPV interactions are allowed in the superpotential, the sfermions become scalar fields with lepton and/or baryon number violating interactions, defining properties of a leptoquark. For instance, the right sbottom ˜b<sup>R</sup> has the same quantum numbers as the leptoquark S<sup>1</sup> discussed in section 5.4.2 and the λ ′ coupling in Equation (104) originates a Yukawa interaction exactly like λ LQ in Equation (101)38. For this reason, neutrino mass generation takes place in analogous ways, t-R/<sup>p</sup> being a pure radiative model.

As already discussed, the breaking of R-parity leads to the decay of the LSP. This is the most distinctive signature of this family of models. However, in contrast to b-R/<sup>p</sup> , the large number of free parameters in t-R/<sup>p</sup> exclude the possibility of making definite predictions for the LSP decay. Nevertheless, one expects novel signatures at the LHC, typically with many leptons in the final states [661]. Other signatures, already mentioned in section 4.2, include LFV observables, see for instance de Gouvea et al. [486].

### 6. CONCLUSIONS AND OUTLOOK

The discovery of neutrino oscillations and its explanation in terms of massive neutrinos has been one of the most exciting discoveries in particle physics in recent years and a clear sign of lepton flavor violation and physics beyond the SM. Neutrino masses being the first discovery of physics beyond the SM may be related to the fact that the lowest-order effective operator, the Weinberg operator, generates Majorana neutrino masses. This may point to Majorana neutrinos and consequently lepton number violation introducing a new scale beyond the SM. The magnitude of this scale, and that of lepton flavor violation, are unknown.

<sup>38</sup>There are, however, additional couplings that supersymmetry forbids but would be allowed for general leptoquarks. Therefore, t-R/<sup>p</sup> can then be regarded as a constrained leptoquark scenario. See Deshpande and He [660] for a paper on t-R/<sup>p</sup> as a possible explanation for the B-meson anomalies that highlights the similarities between this setup and leptoquark models.

The sensitivity to many lepton flavor violating processes will be increased by 2-4 orders of magnitude in the next decade and thus test lepton flavor violation at scales of <sup>O</sup>(1 <sup>−</sup> 1, 000) TeV. In particular the expected improvement of up to 4 orders of magnitude for µ − e conversion and the decay µ → eee, but also other processes, will yield strong constraints on the parameter space of currently allowed models or even more excitingly lead to a discovery. Moreover, the LHC is directly probing the TeVscale and several possible options for colliders are discussed to probe even higher scales. These exciting experimental prospects, together with the simplicity of the explanation for the smallness of neutrino mass, are the main motivations to study radiative neutrino mass models.

Radiative neutrino mass models explain the lightness of neutrinos without introducing heavy scales. The main idea is that neutrino masses are absent at tree-level, being generated radiatively at 1- or higher-loop orders. This, together with the suppressions due to the possible presence of SM masses and/or extra Yukawa and quartic couplings, implies that the scale of these models may be in the range of <sup>O</sup>(1 <sup>−</sup> 100) TeV. This is also theoretically desirable, because all new particles are light and no hierarchy problem is introduced.

The plethora of neutrino mass models studied in the last decades is overwhelming, reaching the hundreds. We believe that at this point an ordering principle for the theory space is necessary to (i) help scientists outside the field to acquire an overview of the topic, (ii) cover the theory space and spot possible holes, (iii) try to draw generic phenomenological conclusions that can be looked for experimentally, and last but not least (iv) serve as reference for model-builders and phenomenologists.

One can choose to systematically classify the different possibilities and models in different complementary ways: in terms of (i) the effective operators they generate after integrating out the heavy particles at tree-level, (ii) the number of loops at which the Weinberg operator is generated, and (iii) the possible topologies within a particular loop order39. In the first case, the contribution of the matching to the Weinberg operator can be easily estimated, and possible UV completions can be outlined. The second option also sheds light on the scale of the new particles. Finally, the study of possible topologies, which have been analyzed up to 2-loop order, helps to systematically pin down neutrino mass models.

We presented selected examples of radiative neutrino mass models in section 5 which serve as benchmark models and discussed their main phenomenological implications such as lepton flavor-violating processes and direct production of the heavy particles at colliders. The phenomenology is generally very rich and quite model-dependent including extra contributions to neutrinoless double beta decay, electric dipole moments, anomalous magnetic moments, and meson decays. Furthermore, radiative neutrino mass models may solve the dark matter problem with a weakly-interacting massive particle running in the loop generating neutrino mass. Also, the new states can play a crucial role for the matter-antimatter asymmetry, although not necessarily in a positive way, and therefore extra bounds can be set on the lepton number violating interactions.

From our work, we have found that there are several interesting avenues that can be pursued in the future:


To conclude, it is interesting that there are many combinations of what one may call "aesthetically reasonable" particles—those that have SM multiplet assignments and hypercharges that are not too high—that couple to SM particles in such a way as to realize neutrino mass generation at loop level. Radiative mass generation, as well as being a reasonable hypothesis for explaining the smallness of neutrino masses, also provides many phenomenological signatures at relatively low new-physics scales. So, even if nature realizes the seesaw mechanism with heavy

<sup>39</sup>A fourth complementary classification in terms of particles can be done, which will appear in a future publication Cai, (in prep).

<sup>40</sup>All possible dimension-7 operators with SM fields and right-handed neutrinos have been listed in Bhattacharya and Wudka [86].

right-handed neutrinos, given the difficulty of testing such a paradigm, falsifying radiative models by means of studying in detail their phenomenology and actively searching for their signals seems the only way to strengthen the case of the former by reducing as much as possible the theory space. Not to mention all the useful insights learned on such a journey.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### REFERENCES


### ACKNOWLEDGMENTS

We acknowledge the use of the TikZ-Feynman package [663]. JHG acknowledges discussions with Arcadi Santamaria and Nuria Rius. This work was supported in part by the Australian Research Council through the ARC Centre of Excellence for Particle Physics at the Terascale (CoEPP) (CE110001104). A.V. acknowledges financial support from the "Juan de la Cierva" program (27-13-463B-731) funded by the Spanish MINECO as well as from the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064, SEV-2014-0398 and PROMETEOII/ 2014/084 (Generalitat Valenciana).


model with an extra flavor symmetry. Phys Rev. (2014) **D90**:016006. doi: 10.1103/PhysRevD.90.016006


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Cai, Herrero García, Schmidt, Vicente and Volkas. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## APPENDIX

### On the Relative Contribution of Operators

Oftentimes the effective 1L = 2 operators are discussed using a cutoff regularization scheme. In the following, however, we outline the relative contribution of the different 1L = 2 operators to neutrino mass using dimensional regularization with a momentum-independent renormalization scheme such as MS renormalization. Power counting in the SM effective theory establishes that the dominant contributions to neutrino mass are given by (i) the lowest-dimensional Weinberg-like operator O (n) <sup>1</sup> <sup>≡</sup> LLHH(H†H) <sup>n</sup> which is induced via matching at the new physics scale 3 and (ii) the contributions induced by mixing via renormalization group running of the operator O (n) 1 into the Weinberg operator or other lower-dimensional Weinberg-like operators.

Using naive dimensional analysis we discuss in more detail the relative contribution to neutrino mass from each operator in the SM effective field theory. Note that here we follow the matching and running from low energy scale to high energy scale. Below the electroweak scale effective operators that can contribute to neutrino masses should contain two neutrinos and possibly additional fields. Those additional fields have to be closed off and their contribution to neutrino masses vary: for photons and gluons, the contribution from the tadpole diagram vanishes; for fermions f , the contribution is proportional to a factor m<sup>3</sup> /16π <sup>2</sup>3<sup>3</sup> per fermion loop. Thus, the contribution of operators with additional fields to neutrino mass either vanishes or is generally suppressed. Matching at the electroweak scale may similarly include loops with electroweak gauge bosons or the top quark and lead to a suppression of the respective operator. Additional Higgs fields yield a factor v/3 each. Above the electroweak scale the operators generally mix. Higher-dimensional operators also mix into lower dimensional ones. For example although the operator O ′ <sup>1</sup> mixes into the operator O<sup>1</sup> via renormalization group running and thus it is an operator of lower dimension, its contribution to the Wilson coefficient is suppressed by a factor of order m2 H /16π <sup>2</sup>3<sup>2</sup> and therefore it is of the same order as the operator O ′ 1 . At the new physics scale the relative size of the Wilson coefficients is determined by the couplings and the loop level at which they are generated. The Wilson coefficient of the Weinberg-like operators at the new physics scale may be suppressed by a loop factor compared to other operators, but the other operators receive a further loop-factor suppression when matching onto the effective interactions at the electroweak scale or finally onto the neutrino mass term at a lower scale. The contributions of all operators to neutrino mass has at least the same loop-factor suppression as the leading Weinberg-like operator which is induced by matching at the new physics scale. Higher-dimensional Weinberg-like operators will induce the lower-dimensional ones via mixing when running the Wilson coefficients to the low scale, but the contribution of the induced operator is still of the same order as the original higherdimensional operator. In summary, an order of magnitude estimate of neutrino mass can be obtained from the leading Weinberg-like operator which is induced from matching at the new physics scale keeping in mind that its contribution to lowerdimensional Weinberg-like operators will be of a similar order of magnitude.

# Status of Neutrino Properties and Future Prospects—Cosmological and Astrophysical Constraints

Martina Gerbino<sup>1</sup> \* and Massimiliano Lattanzi <sup>2</sup> \*

<sup>1</sup> The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm, Sweden, 2 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Ferrara, Italy

Cosmological observations are a powerful probe of neutrino properties, and in particular of their mass. In this review, we first discuss the role of neutrinos in shaping the cosmological evolution at both the background and perturbation level, and describe their effects on cosmological observables such as the cosmic microwave background and the distribution of matter at large scale. We then present the state of the art concerning the constraints on neutrino masses from those observables, and also review the prospects for future experiments. We also briefly discuss the prospects for determining the neutrino hierarchy from cosmology, the complementarity with laboratory experiments, and the constraints on neutrino properties beyond their mass.

#### Edited by:

Diego Aristizabal Sierra, Federico Santa María Technical University, Chile

#### Reviewed by:

Sergio Pastor, Instituto de Física Corpuscular (IFIC), Spain Davide Meloni, Universitá degli Studi Roma Tre, Italy

> \*Correspondence: Martina Gerbino martina.gerbino@fysik.su.se Massimiliano Lattanzi lattanzi@fe.infn.it

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 22 September 2017 Accepted: 19 December 2017 Published: 06 February 2018

#### Citation:

Gerbino M and Lattanzi M (2018) Status of Neutrino Properties and Future Prospects—Cosmological and Astrophysical Constraints. Front. Phys. 5:70. doi: 10.3389/fphy.2017.00070 Keywords: neutrinos, cosmic microwave background, large scale structure, cosmology, neutrino mass

# 1. INTRODUCTION

Flavor oscillation experiments have by now firmly established that neutrinos have a mass. Current experiments measure with great accuracy the three mixing angles, as well as the two mass-squared splittings between the three active neutrinos. In the framework of the standard model (SM) of particle physics neutrinos are massless, and consequently do not mix, since it is not possible to build a mass term for them using the particle content of the SM. Therefore, flavor oscillations represent the only laboratory evidence for physics beyond the SM. Several unknowns in the neutrino sector still remain, confirming these particles as being the most elusive within the SM. In particular, the absolute scale of neutrino masses has yet to be determined. Moreover, the sign of the largest mass squared splitting, the one governing atmospheric transitions, is still unknown. This leaves open two possibilities for the neutrino mass ordering, corresponding to the two signs of the atmospheric splitting: the normal hierarchy, in which the atmospheric splitting is positive, and the inverted hierarchy, in which it is negative. Other unknowns are the value of a possible CP-violating phase in the neutrino mixing matrix, and the Dirac or Majorana nature of neutrinos.

There are different ways of measuring the absolute neutrino mass scale. One is to use kinematic effects, for example by measuring the energy spectrum of electrons produced in the β-decay of nuclei, looking for the distortions due to the finite neutrino mass. This approach has the advantage of being very robust and providing model-independent results, as it basically relies only on energy conservation. Present constraints on the effective mass of the electron neutrino mβ (an incoherent sum of the mass eigenvalues, weighted with the elements of the mixing matrix) are m<sup>β</sup> < 2.05 eV from the Troitsk [1] experiment, and m<sup>β</sup> < 2.3 eV from the Mainz [2] experiment, at the 95% CL. The KATRIN spectrometer [3], that will start its science run in 2018, is expected to improve the sensitivity by an order of magnitude. Another way to measure neutrino masses in the laboratory is to look for neutrinoless double β decay (0ν2β in short) of

**142**

nuclei, a rare process that is allowed only if neutrinos are Majorana particles [4]. The prospects for detection of neutrino mass with 0ν2β searches are very promising: current constraints for the effective Majorana mass of the electron neutrino mββ, a coherent sum of the mass eigenvalues, weighted with the elements of the mixing matrix, are in the mββ < 0.1 ÷ 0.4 eV ballpark (see section 9 for more details). There are a few shortcomings, however. First of all, there is some amount of model dependence: one has to assume that neutrinos are Majorana particles to start, and even if this is, in some sense, a natural and very appealing scenario from the theoretical point of view—as it could explain the smallness of neutrino masses [5–9]—we have at the moment no indication that this is really the case. Moreover, inferring the neutrino mass from a (non-)observation of 0ν2β requires the implicit assumption that the mass mechanism is the only contribution to the amplitude of the process, i.e., that no other physics beyond the SM that violates lepton number is at play. Another issue is that the amplitude of the process also depends on nuclear matrix elements, that are known only with limited accuracy, introducing an additional layer of uncertainty in the interpretation of experimental results. Finally, given that mββ is a coherent superposition of the mass eigenvalues, it could be that the values of the Majorana phases arrange to make mββ vanishingly small.

The third avenue to measure neutrino masses, and in fact the topic of this review, is to use cosmological observations. As we shall discuss in more detail in the following, the presence of a cosmic background of relic neutrinos (CνB) is a robust prediction of the standard cosmological model [10]. Even though a direct detection is extremely difficult and still lacking, (but experiments aiming at this are currently under development, like the PTOLEMY experiment [11]), nevertheless cosmological observations are in agreement with this prediction. The relic neutrinos affect the cosmological evolution, both at the background and perturbation level, so that cosmological observables can be used to constrain the neutrino properties, and in particular their mass (see e.g., [10, 12, 13] for excellent reviews on this topic). In fact, cosmology currently represents the most sensitive probe of neutrino masses. The observations of cosmic microwave background (CMB) anisotropies from the Planck satellite, without the addition of any external data, constrain the sum of neutrino masses already at the 0.6 eV level [14], which is basically the same as the KATRIN sensitivity. Combinations of different datasets yield even stronger limits, at the same level or better than the ones from 0ν2β searches, although a direct comparison is not immediate, due to the fact that different quantities are probed, and also due to the theoretical assumptions involved in the interpretation of both kinds of data. Future-generation experiments will likely have the capability to detect neutrino masses, and to disentangle the hierarchy, provided that systematics effects can be kept under control—and that our theoretical understanding of the Universe is correct, of course! Concerning this last point, the drawback of cosmological measurements of neutrino mass and other properties, is that they are somehow model dependent. Inferences from cosmological observations are made in the framework of a model—the so-called 3CDM model—, and of its simple extensions, that currently represents our best and simple description of the Universe that is compatible with observations. This model is based on General Relativity (with the assumption of a homogeneous and isotropic Universe at large scales) and on the SM of particle physics, with the addition of massive neutrinos, complemented with a mechanism for the generation of primordial perturbations, i.e., the inflationary paradigm. When cosmological data are interpreted in this framework, they point to the following picture: our Universe is spatially flat and is presently composed by baryons (∼5% of the total density), dark matter (∼25%), and an even more elusive component called dark energy (∼70%), that behaves like a cosmological constant, and is responsible for the present accelerated expansion, plus photons (a few parts in 10<sup>5</sup> ) and light neutrinos. The constraints from Planck cited above imply that, in the framework of the 3CDM model, neutrinos can contribute 1% to the present energy density at most. The structures that we observe today have evolved from adiabatic, nearly scale-invariant initial conditions. Even though this model is very successful, barring some intriguing but for the moment still mild (at the ∼2σ level) discrepancies between observational probes, this dependence should be borne in mind. On the other hand, such a healthy approach should not, in our opinion, be substituted with its contrary, i.e., a complete distrust toward cosmological constraints. A pragmatic approach to this problem is to test the robustness of our inferences concerning neutrino properties against different assumptions, by exploring extensions of the 3CDM model. This has been in fact done quite extensively in the literature, and we will take care, toward the end of the review, to report results obtained in extended models.

Another advantage of cosmological observations is that they are able to probe neutrino properties beyond their mass. A wellknown example is the effective number of neutrinos, basically a measure of the energy density in relativistic species in the early Universe, that is a powerful probe of a wide range of beyond the SM model physics (in fact, not necessarily related to neutrinos). For example, it could probe the existence of an additional, light sterile mass eigenstate, as well as the physics of neutrino decoupling, or the presence of lepton asymmetries generated in the early Universe. Cosmology can also be used to constrain the existence of non-standard neutrino interactions, possibly related to the mechanism of mass generation. Even though they are not the focus of this review, we will briefly touch some of these aspects in the final sections of the review.

Cosmological data have reached a very good level of maturity over the last decades. Measurements of the CMB anisotropies from the Planck satellite have put the tightest constraints ever on cosmological parameters from a single experiment [14], dramatically improving the constraints from the predecessor satellite WMAP [15]. From the ground, the Atacama Cosmology Telescope (ACT) polarization-sensitive receiver and the South Pole Telescope (SPT) have been measuring with incredible accuracy CMB anisotropies at the smallest scales in temperature and polarization [16, 17]. At degree and subdegree scales, the BICEP/Keck collaboration [18, 19] and the POLARBEAR telescope [20] are looking at the faint CMB "B-mode" signal, containing information about both the early stages of the Universe (primordial B-modes) and the late time evolution (lensing B-modes). The Cosmology Large Angular Scale Surveyor (CLASS) [21] is working at mapping the CMB polarization field over 70% of the sky. The SPIDER balloon [22] successfully completed its first flight and is in preparation for the second launch likely at the end of 2018. In addition to CMB data, complementary information can be obtained by looking at the large-scale structure of the Universe. The SDSS III-BOSS galaxy survey has recently released its last season of data [23]. Extended catalogs of galaxy clusters have been completed from several surveys (see e.g., [24] and references therein). In addition, weak lensing surveys (Canada-France-Hawaii Telescope Lensing Survey [25], Kilo-Degree Survey [26], Dark Energy Survey [27]) are mature enough to provide constraints on cosmological parameters that are competitive with those from other observables. They also allow to test the validity of the standard cosmological paradigm by comparing results obtained from high-redshift observables to those coming from measurements of the low-redshift universe.

The current scenario is just a taste of the constraining power of cosmological observables that will be available with the next generation of experiments, that will be taking measurements in the next decade. Future CMB missions—including Advanced ACTPol [28], SPT-3G [29], CMB Stage-IV [30], Simons Observatory<sup>1</sup> , Simons Array [31], CORE [32], LiteBIRD [33], PIXIE [34]—will test the Universe over a wide range of scales with unprecedented accuracy. The same accuracy will enable the reconstruction of the weak lensing signal from the CMB maps down to the smallest scales and with high sensitivity, providing an additional probe of the distribution and evolution of structures in the universe. On the other hand, the new generation of large-scale-structure surveys—including the Dark Energy Spectroscopic Instrument [35], the Large Synoptic Survey Telescope [36], Euclid [37], and the Wide Frequency InfraRed Spectroscopic Telescope [38]—will also probe the late-time universe with the ultimate goal of shedding light on the biggest unknown of our times, namely the nature of dark energy and dark matter.

The aim of this review is to provide the state of the art of the current knowledge of neutrino masses from cosmological probes and give an overview of future prospects. The review is organized as follows: in section 2, we outline the framework of this review, introducing some useful notation and briefly reviewing the basics of neutrino cosmology. Section 3 is devoted to discussing, from a broad perspective, cosmological effects induced by massive neutrinos. In section 4, we will describe in detail how the effects introduced in section 3 affect cosmological observables, such as the CMB anisotropies, large-scale structures and cosmological distances. Sections 5 and 6 present a detailed collection of the current and future limits on 6m<sup>ν</sup> from the measurements of the cosmological observables discussed in section 4, mostly derived in the context of the 3CDM cosmological model. Constraints derived in more extended scenarios are summarized in section 7. Section 8 briefly deals with the issue of whether cosmological probes are able to provide information not only on 6m<sup>ν</sup> , but also on its distribution among the mass eigenstates, i.e., about the neutrino hierarchy. In section 9, we will briefly go through the complementarity between cosmology and laboratory searches in the quest for constraining neutrino properties. Finally, section 10 offers a summary of the additional information about neutrino properties beyond their mass scale that we can extract from cosmological observables. We derive our conclusions in section 11. The impatient reader can access the summary of current and future limits from **Tables 1**–**4**.

### 2. NOTATION AND PRELIMINARIES

### 2.1. Basic Equations

Inferences from cosmological observations are made under the assumption that the Universe is homogeneous and isotropic, and as such it is well-described, in the context of general relativity, by a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Small deviations from homogeneity and isotropy are modeled as perturbations over of a FLRW background.

In a FLRW Universe, expansion is described by the Friedmann equation<sup>2</sup> for the Hubble parameter H:

$$H(a)^2 = \frac{8\pi G}{3}\rho(a) - \frac{K}{a^2},\tag{1}$$

where G is the gravitational constant, K parameterizes the spatial curvature<sup>3</sup> , a is the cosmic scale factor and the ρ is the total energy density. This is given by the sum of the energy densities of the various components of the cosmic fluid.

Considering cold dark matter (c), baryons (b), photons (γ ), dark energy (DE), and massive neutrinos (ν), and introducing the redshift 1 + z = a −1 , the Friedmann equation can be recast as:

$$\begin{aligned} H(z)^2 &= H\_0^2 \left[ \left( \Omega\_c + \Omega\_b \right) (1+z)^3 + \Omega\_Y (1+z)^4 + \\\\ &+ \Omega\_{DE} (1+z)^{3(1+w)} + \Omega\_k (1+z)^2 + \frac{\rho\_{v, \text{tot}}(z)}{\rho\_{\text{crit},0}} \right], \end{aligned} \tag{2}$$

where we have introduced the present value of the critical density required for flat spatial geometry <sup>ρ</sup>crit,0 <sup>≡</sup> <sup>3</sup>H<sup>2</sup> 0 /8πG (in general, we use a subscript 0 to denote quantities evaluated today), and the present-day density parameters <sup>i</sup> = ρi,0/ρcrit,0 (since we will be always referring to the density parameters today, we omit the subscript 0 in this case). The scalings with (1 + z) come from the fact that the energy densities of non-relativistic matter and radiation scale with a −3 and a −4 , respectively. For DE, in writing Equation (2) we have left open the possibility for an arbitrary (albeit constant) equation-of-state parameter w. In the case of neutrinos, since the parameter of their equation of state is not constant, we could not write a simple scaling with redshift, although this is possible in limiting regimes (see section 2.5). We use ρν,tot to denote the total neutrino density, i.e., summed over all mass eigenstates. Finally, we have defined a "curvature density

<sup>1</sup>https://simonsobservatory.org

<sup>2</sup>All throughout this review, we take <sup>c</sup> <sup>=</sup> <sup>h</sup>¯ <sup>=</sup> <sup>k</sup><sup>B</sup> <sup>=</sup> 1.

<sup>3</sup>We choose not to rescale <sup>K</sup> to make it equal to <sup>±</sup>1 for an open or closed Universe, so that we are left with the freedom to rescale the scale factor today a<sup>0</sup> to unity.

parameter" <sup>k</sup> = −K/H<sup>2</sup> 0 . From Equation (2) evaluated at z = 0 it is clear that the density parameters, including curvature, satisfy the constrain P <sup>i</sup> <sup>i</sup> = 1.

Let us also introduce some extra notation and jargon that will be useful in the following. We will use <sup>m</sup> to refer to the total density of non-relativistic matter today. Thus, this in general includes dark matter, baryons, and those neutrinos species that are heavy enough to be non-relativistic today. In such a way we have that <sup>m</sup> + DE = 1 in a flat Universe (or <sup>m</sup> + DE = 1−<sup>k</sup> in general), since the present density of photons and other relativistic species is negligible. Since many times we will have to consider the density of matter that is non-relativistic at all the redshifts that are probed by cosmological observables, i.e., dark matter and baryons but not neutrinos, we also introduce c+<sup>b</sup> , with obvious meaning. When we consider dark energy in the form of a cosmological constant (w = −1) we use <sup>3</sup> in place of DE to make this fact clear. Finally, we also use the physical density parameters ω<sup>i</sup> ≡ ih 2 , with h being the present value of the Hubble parameter in units of 100 km s−<sup>1</sup> Mpc−<sup>1</sup> .

As we shall discuss in more detail in the following, cosmological observables often carry the imprint of particular length scales, related to specific physical effects. For this reason we recall some definitions that will be useful in the following. The causal horizon r<sup>h</sup> at time t is defined as the distance traveled by a photon from the Big Bang (t = 0) until time t. This is given by:

$$r\_h(t) = \int\_0^t \frac{dt'}{a(t')} = \int\_{z(t)}^\infty \frac{dz'}{H(z')}.\tag{3}$$

Note that this is actually the comoving causal horizon; in the following, unless otherwise noted, we will always use comoving distances. We also note that the comoving horizon is equal to the conformal time η(t) (defined through dt = adη and η(t = 0) = 0). In a Friedmann Universe (i.e., one composed only by matter and radiation), the physical causal horizon is proportional, by a factor of order unity, to the Hubble length dH(t) ≡ H(t) −1 . For this reason, we shall sometimes indulge in the habit of calling the latter the Hubble horizon, even though this is, technically, a misnomer.

A related quantity is the sound horizon rs(t), i.e., the distance traveled in a certain time by an acoustic wave in the baryonphoton plasma. The expression for r<sup>s</sup> is very similar to the one for the causal horizon, just with the speed of light (equal to 1 in our units) replaced by the speed of sound c<sup>s</sup> in the plasma:

$$r\_s(t) = \int\_0^t \frac{c\_s(t')}{a(t')} dt' = \int\_{z(t)}^\infty \frac{c\_s(z')}{H(z')} dz'.\tag{4}$$

The speed of sound is given by c<sup>s</sup> = 1/ √ 3(1 + R), with R = (p<sup>b</sup> + ρ<sup>b</sup> )/(p<sup>γ</sup> + ρ<sup>γ</sup> ) being the baryon-to-photon momentum density ratio. When the baryon density is negligible relative to the photons, c<sup>s</sup> ≃ 1/ √ 3 and r<sup>s</sup> ≃ rh/ √ <sup>3</sup> <sup>=</sup> η/<sup>√</sup> 3.

Imprints on the cosmological observables of several physical processes usually depend on the value of those scales at some particular time. For example, the spacing of acoustic peaks in the CMB spectrum is reminiscent of the sound horizon at the time of hydrogen recombination; the suppression of small-scale matter fluctuations due to neutrino free-streaming is set by the causal horizon at the time neutrinos become non-relativistic; and so on. Moreover, since today we see those scales through their projection on the sky, what we observe is actually a combination of the scale itself and the distance to the object that we are observing. We find then useful also to recall some notions related to cosmological distances. The comoving distance χ between us and an object at redshift z is

$$\chi(z) = \int\_0^z \frac{dz'}{H(z')},\tag{5}$$

and this is also equal to η0−η(z). The comoving angular diameter distance dA(z) is given by

$$d\_A(z) = \frac{\sin\left(\sqrt{K}\chi\right)}{\sqrt{K}},\tag{6}$$

so that

$$d\_A(z) = \chi(z) = \int\_0^z \frac{dz'}{H(z')} \qquad \text{for } \Omega\_k = 0. \tag{7}$$

The angular size θ of an object is related to its comoving linear size λ through θ = λ/dA(z). This justifies the definition of an object of known linear size as a standard ruler for cosmology. In fact, knowing λ, we can use a measure of θ to get d<sup>A</sup> and make inferences on the cosmological parameters that determine its value through the integral in Equation (6).

Another measure of distance is given by the luminosity distance dL(z), that relates the observed flux F to the intrinsic luminosity L of an object at redshift z:

$$d\_L(z) \equiv \sqrt{\frac{L}{4\pi F}} = (1+z)d\_A(z). \tag{8}$$

Similarly to what happened for the angular diameter distance, this allows to use standard candles—objects of known intrinsic luminosity—as a mean to infer the values of cosmological parameters, after their flux has been measured.

### 2.2. Neutrino Mass Parameters

According to the standard theory of neutrino oscillations, the observed neutrino flavors ν<sup>α</sup> (α = e, µ, τ ) are quantum superpositions of three mass eigenstates ν<sup>i</sup> (i = 1, 2, 3):

$$\left| \boldsymbol{\nu}\_{\alpha} \right\rangle = \sum\_{i} U^{\*}\_{\alpha i} \left| \boldsymbol{\nu}\_{i} \right\rangle,\tag{9}$$

where U is the Pontecorvo-Maki-Nakagawa-Sasaka (PMNS) mixing matrix. The PMNS matrix is parameterized by three mixing angles θ12, θ13, θ23, and three CP-violating phases: one Dirac, δ, and two Majorana phases, α<sup>21</sup> and α31. The Majorana phases are non-zero only if neutrinos are Majorana particles. They do not affect oscillation phenomena, but enter lepton number-violating processes like 0ν2β decay. The actual form of the PMNS matrix is:

$$U = \begin{bmatrix} c\_{12}c\_{13} & s\_{12}c\_{13} & s\_{13}e^{-i\delta} \\ -s\_{12}c\_{23} - c\_{12}s\_{23}s\_{13}e^{i\delta} & c\_{12}c\_{23} - s\_{12}s\_{23}s\_{13}e^{i\delta} & s\_{23}c\_{13} \\\\ s\_{12}s\_{23} - c\_{12}c\_{23}s\_{13}e^{i\delta} & -c\_{12}s\_{23} - s\_{12}c\_{23}s\_{13}e^{i\delta} & c\_{23}c\_{13} \end{bmatrix}$$
 
$$\times \text{diag}\left(1, e^{i\alpha\_{21}/2}, e^{i\alpha\_{31}/2}\right),\tag{10}$$

where cij ≡ cos θij and sij ≡ sin θij.

In addition to the elements of the mixing matrix, the other parameters of the neutrino sector are the mass eigenvalues m<sup>i</sup> (i = 1, 2, 3). Oscillation experiments have measured with unprecedented accuracy the three mixing angles and the two mass squared differences relevant for the solar and atmospheric transitions, namely the solar splitting 1m<sup>2</sup> sol <sup>=</sup> <sup>1</sup>m<sup>2</sup> <sup>21</sup> <sup>≡</sup> <sup>m</sup><sup>2</sup> 2 − m2 <sup>1</sup> <sup>≃</sup> 7.6 <sup>×</sup> <sup>10</sup>−<sup>5</sup> eV<sup>2</sup> , and the atmospheric splitting 1m<sup>2</sup> atm = <sup>|</sup>1m<sup>2</sup> 31| ≡ |m<sup>2</sup> <sup>3</sup> <sup>−</sup> <sup>m</sup><sup>2</sup> 1 | ≃ 2.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> eV<sup>2</sup> (see e.g., [39–41] for a global fit of the neutrino mixing parameters and mass splittings). We know, because of matter effects in the Sun, that, of the two eigenstates involved, the one with the smaller mass has the largest electron fraction. By convention, we identify this with eigenstate "1," so that the solar splitting is positive. On the other hand, we do not know the sign of the atmospheric mass splitting, so this leaves open two possibilities: the normal hierarchy (NH), where 1m<sup>2</sup> <sup>31</sup> > 0 and m<sup>1</sup> < m<sup>2</sup> < m3, or the inverted hierarchy, where 1m<sup>2</sup> <sup>31</sup> < 0 and m<sup>3</sup> < m<sup>1</sup> < m2.

Oscillation experiments are unfortunately insensitive to the absolute scale of neutrino masses. In this review, we will mainly focus on cosmological observations as a probe of the absolute neutrino mass scale. To a very good approximation, cosmological observables are mainly sensitive to the sum of neutrino masses 6m<sup>ν</sup> , defined simply as

$$
\Sigma m\_{\upsilon} = \sum\_{i} m\_{i}.\tag{11}
$$

Absolute neutrino masses can also be probed by laboratory experiments. These will be reviewed in more detail in section 9, where their complementarity with cosmology will be also discussed. For the moment, we just recall the definition of the mass parameters probed by laboratory experiments. The effective (electron) neutrino mass mβ

$$m\_{\beta} = \left(\sum\_{i} |U\_{ei}|^2 \, m\_i^2\right)^{1/2},\tag{12}$$

can be constrained by kinematic measurements like those exploiting the β decay of nuclei. The effective Majorana mass of the electron neutrino mββ:

$$m\_{\beta\beta} = \left| \sum\_{i} U\_{ei}^{2} m\_{i} \right|,\tag{13}$$

can instead be probed by searching for 0ν2β decay.

#### 2.3. The Standard Cosmological Model

Our best description of the Universe is currently provided by the spatially flat 3CDM model with adiabatic, nearly scaleinvariant initial conditions for scalar perturbations. With the exception of some mild (at the ∼2σ level) discrepancies that will be discussed in the part devoted to observational limits, all the available data can be fit in this model, that in its simplest ("base") version is described by just six parameters. In the base 3CDM model, the Universe is spatially flat (<sup>k</sup> = 0), and the matter and radiation content is provided by cold dark matter, baryons, photons, and neutrinos, while dark energy is in the form of a cosmological constant (w = −1). The energy density of photons is fixed by measurements of the CMB temperature, while neutrinos are assumed to be very light, usually fixing the sum of the masses to 6m<sup>ν</sup> = 0.06 eV, the minimum value allowed by oscillation experiments. In this way, the energy density of neutrinos is also fixed at all stages of the cosmological evolution (see section 2.5). From Equation (2), and taking into account the flatness constraint, it is clear then that the background evolution in such a model is described by three parameters, for example<sup>4</sup> h, ω<sup>c</sup> , and ω<sup>b</sup> , with <sup>3</sup> given by 1 − m. The initial scalar fluctuations are adiabatic and have a power-law, nearly scale invariant, spectrum, that is thus parameterized by two parameters, an amplitude A<sup>s</sup> and a logarithimc slope n<sup>s</sup> − 1 (with n<sup>s</sup> = 1 thus corresponding to scale invariance). Finally, the optical depth to reionization τ parameterizes the ionization history of the Universe.

This simple, yet very successful, model can be extended in several ways. The extension that we will be most interested in, given the topic of this review, is a one-parameter extension in which the sum of neutrino masses is considered as a free parameter. We call this seven-parameter model 3CDM+6m<sup>ν</sup> . This is also in some sense the best-motivated extension of 3CDM, as we actually know from oscillation experiments that neutrinos have a mass, and from β-decay experiments that this can be as large as 2 eV. In addition to this minimal extension, we will also discuss how relaxing some of the assumptions of the 3CDM model affects estimates of the neutrino mass. Among the possibilities that we will consider, there are those of varying the curvature (<sup>k</sup> ), the equation-of-state parameter of dark energy (w), or the density of radiation in the early Universe (Neff, defined in section 2.5).

There are many relevant extensions to the 3CDM model that however we will not consider here (or just mention briefly). The most important one concerns the possibility of non-vanishing tensor perturbations, i.e., primordial gravitational waves, that, if detected, would provide a smoking gun for inflation. This scenario is parameterized through an additional parameter, the tensor-to-scalar ratio r. In the following, we will always assume r = 0. In any case, this assumption will not affect the estimates reported here, as the effect of finite neutrino mass and of tensor modes on the cosmological observables are quite distinct. Similarly, we will not consider the possibility of non-adiabatic

<sup>4</sup> In the analysis of CMB data, the angle subtended by the sound horizon at recombination is normally used in place of h, as it is measured directly by CMB observations, see section 4.1.

initial perturbations, nor of more complicated initial spectra for the scalar perturbations, including those with a possible running of the scalar spectral index, although we report a compilation of relevant references in section 7 for the reader's convenience.

### 2.4. Short Thermal History

Given that cosmological observables carry the imprint of different epochs in the history of the Universe, we find it useful to shortly recall some relevant events taking place during the expansion history, and their relation to the cosmological parameters. For our purposes, it is enough to start when the temperature of the Universe was T ∼ 1 MeV, i.e., around the time of Big Bang Nucleosynthesis (BBN) and neutrino decoupling. At these early times (<sup>z</sup> <sup>∼</sup> <sup>10</sup>10), since matter and radiation densities scale as (1 + z) 3 and (1 + z) 4 , respectively, the Universe is radiation-dominated.


decoupling of radiation from matter. This is the time at which the CMB radiation is emitted. After decoupling, the CMB photons undergo last interactions with residual free electrons. Finally, the CMB photons emerge from this last scattering surface and free-stream until the present time (with some caveats, see below). Most of the features that we observe in the CMB anisotropy pattern are created at this time. Given the current estimates of cosmological parameters, zrec ≃ 1, 090 [14]. Note that in fact the temperature at recombination is basically fixed by thermodynamics, so once the present CMB temperature is determined through observations, zrec = T(z = zrec)/T(z = 0) depends very weakly on the other cosmological parameters.

• Even if photons decoupled from matter shortly after recombination, the large photon-to-baryon ratio keeps baryons coupled to the photon bath for some time after that. The drag epoch zdrag is the time at which baryons stop feeling the photon drag. A good fit to numerical results in a CDM cosmology is given by Eisenstein and Hu [42]

$$z\_{\rm drag} = 1.291 \frac{(\omega\_{\rm c} + \omega\_{\rm b})^{0.251}}{1 + 0.659(\omega\_{\rm c} + \omega\_{\rm b})^{0.828}} [1 + b\_1(\omega\_{\rm c} + \omega\_{\rm b})^{b\_2}],$$

$$\begin{split} b\_1 &= 0.313(\omega\_{\rm c} + \omega\_{\rm b})^{-0.419} [1 + 0.607(\omega\_{\rm c} + \omega\_{\rm b})^{0.674}], \\ b\_2 &= 0.238(\omega\_{\rm c} + \omega\_{\rm b})^{0.223} \end{split} \tag{14}$$

Given the current estimates of cosmological parameters, zdrag ≃ 1,060 [14].


### 2.5. Evolution of Cosmic Neutrinos

In this section, we discuss the thermal history of cosmic neutrinos.

As anticipated above, in the early Universe neutrinos are kept in equilibrium with the cosmological plasma by weak interactions. The two competing factors that determine if equilibrium holds are the expansion rate, given by the Hubble parameter H(z), and the interaction rate Ŵ(z) = nhσvi, where n is the number density of particles, σ is the interaction cross section, and v is the velocity of particles (brackets indicate a thermal average). In fact, neutrino interactions become too weak to keep them in equilibrium once Ŵ < H. The left-hand

side of this inequality is set by the standard model of particle physics, as the interaction rate at a given temperature only depends on the cross-section for weak interactions, and thus, ultimately, on the value of the Fermi constant (σ<sup>w</sup> ∼ G 2 F T 2 ). The right-hand side is instead set through Equation (2) by the total radiation density (the only relevant component at such early times): <sup>H</sup><sup>2</sup> <sup>=</sup> (8πG/3)(ρ<sup>γ</sup> <sup>+</sup> <sup>ρ</sup><sup>ν</sup> ). In the framework of the minimal 3CDM model, once the present CMB temperature is measured, the radiation density at any given temperature is fixed. Thus the temperature of neutrino decoupling, defined through Ŵ(Tν,dec) = H(Tν,dec) does not depend on any free parameter in the theory. A quite straightforward calculation shows that Tν,dec ≃ 1 MeV [44].

While they are in equilibrium, the phase-space distribution f(p) of neutrinos is a Fermi-Dirac distribution<sup>5</sup> :

$$f(p,t) = \frac{1}{e^{p/T\_\upsilon(t)} + 1},\tag{15}$$

where it has been taken into account that at T & 1 MeV, the active neutrinos are certainly ultrarelativistic (i.e., T<sup>ν</sup> ≫ m<sup>ν</sup> ) and thus E(p) ≃ p. The distribution does not depend on the spatial coordinate xE, nor on the direction of momentum pˆ, due to the homogeneity and isotropy of the Universe. Before decoupling, the neutrino temperature Tν is the common temperature of all the species in the cosmological plasma, that we denote generically with T, so that T<sup>ν</sup> = T. We recall that the temperature of the plasma evolves according to g 1/3 <sup>∗</sup><sup>s</sup> aT = const., where g∗<sup>s</sup> counts the effective number of relativistic degrees of freedom that are relevant for entropy [44].

Since decoupling happens while neutrinos are ultrarelativistic, it can be shown that, as a consequence of the Liouville theorem, the shape of the distribution function is preserved by the expansion. In other words, the distribution function still has the form Equation (15), with an effective temperature Tν (z) (that for the sake of simplicity we will continue to refer to as the neutrino temperature) that scales like a −1 (i.e., aT = const). We stress that this means that, when computing integrals over the distribution function, one still neglects the mass term in the exponential of the Fermi-Dirac function, even at times when neutrinos are actually non-relativistic.

Shortly after neutrino decouple, electrons and positrons annihilate and transfer their entropy to the rest of the plasma, but not to neutrinos. In other words, while the neutrino temperature scales like a −1 , the photon temperature scales like a −1 g −1/3 ∗s , and thus decreases slightly more slowly during e +e − annihilation, when g∗<sup>s</sup> is decreasing. In fact, applying entropy conservation one finds that the ratio between the neutrino and photon temperatures after electron-positron annihilation is <sup>T</sup><sup>ν</sup> /<sup>T</sup> <sup>=</sup> (4/11)1/<sup>3</sup> . The photon temperature has been precisely determined by measuring the frequency spectrum of the CMB radiation: T<sup>0</sup> = (2.725 ± 0.002) K [45, 46], so that the present temperature of relic neutrinos should be Tν,0 ≃ 1.95 K ≃ 1.68 × 10−<sup>4</sup> eV.

The number density nν of a single neutrino species (including both neutrinos and their antiparticles) is thus given by:

$$m\_{\boldsymbol{\nu}}(T\_{\boldsymbol{\nu}}) = \frac{\mathcal{g}}{(2\pi)^3} \int \frac{d^3 \boldsymbol{p}}{\mathcal{e}^{\boldsymbol{p}/T\_{\boldsymbol{\nu}}} + 1} = \frac{3\zeta(3)}{4\pi^2} T\_{\boldsymbol{\nu}}^3,\tag{16}$$

where ζ (3) is the Riemann zeta function of 3, and in the last equality we have taken into account that g = 2 for neutrinos. This corresponds to a present-day density of roughly 113 particles/cm<sup>3</sup> .

The energy density of a single neutrino species is instead

$$\rho\_{\boldsymbol{\nu}}(T\_{\boldsymbol{\nu}}) = \frac{\mathcal{g}}{(2\pi)^3} \int \frac{\sqrt{p^2 + m^2}}{e^{p/T\_{\boldsymbol{\nu}}} + 1} d^3 p. \tag{17}$$

Thisis the quantity that appears, among other things, in the righthand side of the Friedmann equation (summed over all mass eigenstates). In the ultrarelativistic (T<sup>ν</sup> ≫ m) and non-relativistic (T<sup>ν</sup> ≪ m) limits, the energy density takes simple analytic forms:

$$\rho\_{\boldsymbol{\upnu}}(T\_{\boldsymbol{\upnu}}) = \begin{cases} \frac{7\pi^2}{120} T\_{\boldsymbol{\upnu}}^4 \text{ (UR)}\\\\ m\_{\boldsymbol{\upnu}} n\_{\boldsymbol{\upnu}} \quad \text{(NR)} \end{cases} \tag{18}$$

These scalings are consistent with the fact that one expects neutrinos to behave as pressureless matter, ρ<sup>ν</sup> ∝ (1 + z) 3 , in the non-relativistic regime, and as radiation, ρ<sup>ν</sup> ∝ (1 + z) 4 , in the ultrarelativistic regime.

Given that the present-day neutrino temperature is fixed by measurements of the CMB temperature and by considerations of entropy conservation, it is clear from the above formulas how the present energy density of neutrinos depends only on one free parameter, namely the sum of neutrino masses 6m<sup>ν</sup> defined in Equation (11). Introducing the total density parameter of massive neutrinos <sup>ν</sup> ≡ P i ρνi ,0/ρcrit,0, one easily finds from Equation (16):

$$
\Omega\_{\upsilon} h^2 = \frac{\Sigma m\_{\upsilon}}{93.14 \,\text{eV}}.\tag{19}
$$

where we have already included the effects of non-instantaneous neutrino decoupling, see below. In the instantaneous decoupling approximation, the quantity at denominator would be 94.2 eV.

On the other hand, the neutrino energy density in the early Universe only depends on the neutrino temperature, and thus it is completely fixed in the framework of the 3CDM model. Using the fact that for photons ρ<sup>γ</sup> = (π 2 /15)T 4 , together with the relationship between the photon and neutrino temperatures, one can write for the total density in relativistic species in the early Universe, after e +e − annihilation:

$$
\rho\_{\mathcal{Y}+\boldsymbol{\upsilon}} = \rho\_{\mathcal{Y}} \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N\_{\boldsymbol{\upsilon}} \right], \tag{20}
$$

where Nν is the number of neutrino families. In the framework of the standard model of particle physics, considering the active neutrinos, one has N<sup>ν</sup> = 3. However, the above formula slightly

<sup>5</sup>We are assuming a vanishing chemical potential for neutrinos and antineutrinos, i.e., a vanishing lepton asymmetry.

underestimates the total density at early times; the main reason is that neutrinos are still weakly coupled to the plasma when e +e − annihilation occurs, so that they share a small part of the entropy transfer. Moreover, finite temperature QED radiative corrections and flavor oscillations also play a role. This introduces nonthermal distortions at the subpercent level in the neutrino energy spectrum; the integrated effect is that at early times the combined energy densities of the three neutrino species are not exactly equal to 3ρ<sup>ν</sup> , with ρ<sup>ν</sup> given by the upper row of Equation (18), but instead are given by (3.046ρ<sup>ν</sup> ) [12, 47]. A recent improved calculation, including the full collision integrals for both the diagonal and off-diagonal elements of the neutrino density matrix, has refined this value to (3.045ρ<sup>ν</sup> ) [48]. It is then customary to introduce an effective number of neutrino families Neff and rewrite the energy density at early times as:

$$
\rho\_{\mathcal{Y}+\upsilon} = \rho\_{\mathcal{Y}} \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N\_{\text{eff}} \right]. \tag{21}
$$

In this review, we will consider Neff = 3.046 as the "standard" value of this parameter in the 3CDM model, and not the more precise value found in de Salas and Pastor [48], since most of the literature still makes use of the former value. This does not make any difference, however, from the practical point of view, given the sensitivity of present and next-generation instruments.

It is also customary to consider extensions of the minimal 3CDM model in which one allows for the presence of additional light species in the early Universe ("dark radiation"). In this kind of extension, the total radiation density of the Universe is still given by the right-hand side of Equation (21), where now however Neff has become a free parameter. In other words, Equation (21) becomes a definition for Neff, that is, just a way to express the total energy density in radiation. The effect on the expansion history of this additional radiation component can be taken into account by the substitution

$$
\Omega\_{\mathcal{Y}} \rightarrow \Omega\_{\mathcal{Y}} \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} \Delta N\_{\text{eff}} \right] \tag{22}
$$

in the rhs of the Hubble equation (2), with 1Neff ≡ Neff − 3.046. Note that this substitution fully captures the effect of the additional species only if this is exactly massless, and not just very light (as in the case of a light massive sterile neutrino, for example—see section 10).

It is often useful, to understand some of the effects that we will discuss in the following, to have a feeling for the time at which neutrinos of a given mass become non-relativistic, or, thinking the other way around, for the mass of a neutrino that becomes non-relativistic at a given redshift. The average momentum of neutrinos at a temperature T<sup>ν</sup> is hpi = 3.15T<sup>ν</sup> . We take as the moment of transition from the relativistic to the non-relativistic regime the time when hpi = m<sup>ν</sup> . Then, using the fact that <sup>T</sup><sup>ν</sup> (z) <sup>=</sup> (4/11)1/3T0(1 <sup>+</sup> <sup>z</sup>) <sup>=</sup> 1.68 <sup>×</sup> <sup>10</sup>−<sup>4</sup> (1 + z) eV, one has

$$1 + z\_{\rm nr} \cong 1,900 \left( \frac{m\_{\rm v}}{\rm eV} \right). \tag{23}$$

This relation can be used to show e.g., that neutrinos with mass m<sup>ν</sup> . 0.6 eV turn non-relativistic around or after recombination. In the following, when discussing the effect of neutrino masses on the CMB anisotropies, we will assume that this is the case. Note however that the actual statistical analyses from which bounds on neutrino masses are derived do not make such an assumption. We also note that, given the current measurements of the neutrino mass differences, only the lightest mass eigenstate can still be relativistic today. Thus at least two out of the three active neutrinos become non-relativistic before the present time.

We conclude this section with a clarification on the role of neutrinos in determining the redshift of matter-radiation equality. Given the present bounds on neutrino masses, we know that equality likely takes place when neutrino are relativistic. In fact, observations of the CMB anisotropies constrain zeq ≃ 3,400, so that neutrinos with mass m<sup>ν</sup> ≃ 1.8 eV, just below the current bound from tritium beta-decay, turn non-relativistic at equality. Thus, for masses sufficiently below the tritium bound, the total density of matter at those times is proportional to c+<sup>b</sup> . The radiation density is instead provided by photons and by the relativistic neutrinos (and as such does not depend on the neutrino mass), plus any other light species present in the early Universe. So the redshift of equivalence is given by

$$1 + z\_{\rm eq} = \frac{\Omega\_{\rm c} + \Omega\_{\rm b}}{\Omega\_{\rm \prime} \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N\_{\rm eff} \right]} = \frac{\alpha\_{\rm c} + \alpha\_{\rm b}}{\alpha\_{\rm \prime} \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N\_{\rm eff} \right]},\tag{24}$$

where the last equality makes it clear that, in the framework of the minimal 3CDM model, the redshift of equivalence only depends on the quantity ω<sup>c</sup> + ω<sup>b</sup> , since Neff is fixed and ω<sup>γ</sup> is determined through observations (it is basically the CMB energy density).

### 3. COSMOLOGICAL EFFECTS OF NEUTRINO MASSES

The impact of neutrino masses—and in general of neutrino properties—on the cosmological evolution can be divided in two broad categories: background effects, and perturbation effects. The former class refers to modifications in the expansion history, i.e., in changes to the evolution of the FLRW background. The latter class refers instead to modifications in the evolution of perturbations in the gravitational potentials and in the different components of the cosmological fluid. We shall now briefly review both classes; we refer the reader who is interested in a more detailed analysis to the excellent review by Lesgourgues and Pastor [13].

To start, we shall consider a spatially flat Universe, i.e., <sup>k</sup> = 0, in which dark energy is in the form of a cosmological constant (w = −1) and there are no extra radiation components (Neff = 3.046). Let us also consider a particular realization of this scenario, that we refer to as our reference model, in which the sum of neutrino masses is very small; for definiteness, we can think that 6m<sup>ν</sup> is equal to the minimum value allowed by oscillation measurements, 6m<sup>ν</sup> = 0.06 eV (see section 8 for further details). When needed, we will take the other parameters as fixed to their 3CDM best-fit values from Planck 2015 [14]. Our aim is to understand what happens when we change the value of 6m<sup>ν</sup> . Increasing the sum of neutrino masses 6m<sup>ν</sup> will increase ω<sup>ν</sup> = νh 2 according to Equation (19). Remember that the sum of the density parameters P <sup>i</sup> <sup>i</sup> = 1; this constraint can be recast in the form:

$$
\alpha\_{\mathcal{C}} + \alpha\_{\mathcal{b}} + \alpha\_{\Lambda} + \alpha\_{\mathcal{Y}} + \alpha\_{\mathcal{V}} + \alpha\_{\mathcal{k}} = h^2. \tag{25}
$$

Since ω<sup>γ</sup> is constrained by observations, and ω<sup>k</sup> is zero by assumption, we have four degrees of freedom that we can use to compensate for the change in ω<sup>ν</sup> , namely: increase h, or decrease any of ω<sup>c</sup> , ω<sup>b</sup> , or ω3. For the moment, for simplicity, we will not distinguish between baryons and cold dark matter, pretending that as non-relativistic components they have the same effect on cosmological observables. This is of course not the case, but we will come back to this later. Then we are left with three independent degrees of freedom that we can use to compensate for the change in ω<sup>ν</sup> : h, ωb+<sup>c</sup> , and ω3. We prefer to use <sup>3</sup> in place of ω3, so that in the end our parameter basis for this discussion will be h, ωc+<sup>b</sup> , <sup>3</sup> .

The first option, increasing the present value of the Hubble constant while keeping <sup>3</sup> and ωb+<sup>c</sup> constant has the effect of making the Hubble parameter at any given redshift after neutrinos become non-relativistic larger with respect to the reference model. This can be understood by looking at Equation (2), that we rewrite here in this particular case

$$H(z)^2 = H\_0^2 \left[ \left( \Omega\_\epsilon + \Omega\_b \right) (1+z)^3 + \Omega\_\mathcal{V} (1+z)^4 \right]$$

$$+ \left. \Omega\_\Lambda + \frac{\rho\_{v, \text{tot}}(z)}{\rho\_{\text{crit},0}} \right]. \tag{26}$$

With respect to the reference model, the first two terms in the RHS are unchanged, while the third increases because <sup>3</sup> is fixed but h is larger. The last term does not depend on h (because the factor H<sup>2</sup> 0 in front of the square brackets cancel the one in the critical density) but yet increases because ρ<sup>ν</sup> = 6mνn<sup>ν</sup> is larger as long as neutrinos are in the non-relativistic regime. On the other hand, before neutrino become non-relativistic, ρ<sup>ν</sup> is the same in the two models, and the change in the 3h 2 term is irrelevant, because the DE density is only important at very low redshifts. So we can conclude that at z ≫ znr, the two models share the same expansion history, while for z . znr the model with "large" neutrino mass is always expanding faster (larger H), or equivalently, is always younger, at those redshifts. In terms of the length scales and of the distance measures introduced in section 2.1, it is easily seen that the causal and sound horizons at both equality and recombination (as well as at the drag epoch) are unchanged, because the expansion history between z = ∞ and z ≃ znr is unchanged. On the other hand, distances between us and objects at any redshift—for example, the angular diameter distance to recombination—are always smaller than in the reference model, because H is always larger between z ≃ znr and z = 0. H increases with the extra neutrino density, so this effect increases with larger neutrino masses (and moreover, znr also gets larger for larger masses). Given this, we expect for example the angle subtended by the sound horizon at recombination, θ<sup>s</sup> = rs(zrec)/dA(zrec) to become larger when we increase 6m<sup>ν</sup> . We conclude this part of the discussion that in this case the redshift of equality zeq does not change, since ωb+<sup>c</sup> is being kept constant, and neutrinos contribute to the radiation density at early times (see discussion at the end of the previous section).

If we instead choose to pursue the second option, i.e., we keep h and <sup>3</sup> constant while lowering ωc+<sup>b</sup> , we are again changing the expansion history, but this time on a different range of redshifts. In fact, when neutrinos are non-relativistic, the RHS of Equation (26) is unchanged, because the changes in the presentday densities of neutrinos and non-relativistic matter perfectly compensate; this continues to hold as long as both densities scale as (1+z) 3 , i.e., roughly for z < znr. On the other hand, at z > znr the neutrino density is the same as in the reference model, while the matter density is smaller, so H(z) is smaller as well. Finally, when the Universe is radiation dominated, the two models share again the same expansion history. Then in this scenario we change the expansion history, decreasing H, for znr . z . zeq. The sound horizon at recombination increases, and so does the angular diameter distance, so one cannot immediately guess how their ratio varies. However, a direct numerical calculation shows that, starting from the Planck best-fit model, the net effect is to increase θ<sup>s</sup> , meaning that the sound horizon will subtend a larger angular scale on the sky when 6m<sup>ν</sup> increases. For what concerns instead the redshift of matter-radiation equality, it is immediate to see that it decreases proportionally to ωc+<sup>b</sup> , i.e., equality happens later in the model with larger 6m<sup>ν</sup> .

Finally, when <sup>3</sup> is decreased, the main effect is to delay the onset of acceleration and make the matter-dominated era last longer. This has some effect on the evolution of perturbations, as we shall see in the following. For what concerns the expansion history, since the model under consideration and the reference model only differ in the neutrino mass and in the DE density, they are identical when neutrinos are relativistic and DE is negligible, i.e., at z > znr. For z < znr, instead, starting as usual from Equation (26) one finds, with some little algebra, that H(z) is always larger in the model with smaller <sup>3</sup> and larger 6m<sup>ν</sup> . As in the previous case, both r<sup>s</sup> and d<sup>A</sup> at recombination vary in the same direction (decreasing in this case); the net effect is again that θ<sup>s</sup> becomes larger with 6m<sup>ν</sup> . Also, since the matter density at early times is not changing in this case, the redshift of equivalence is the same in the two models.

We now comment briefly about ω<sup>b</sup> . One could choose to modify ω<sup>b</sup> in place of ω<sup>c</sup> in order to compensate for the change in ω<sup>ν</sup> . From the point of view of the background expansion, both choices are equivalent, since the baryon and cold dark matter density only enter through their sum ωb+<sup>c</sup> in the RHS of Equation (26). However, changing the baryon density also produces some peculiar effects, mainly related to the fact that (i) it determines the BBN yields, and (ii) it affects the evolution of photon perturbations prior to recombination. Thus, the density of baryons is quite well constrained by the observed abundances of light elements and by the relative ratio between the heights of odd and even peaks in the CMB, (see section 4.1) and there is little room for changing it without spoiling the agreement with observations.

Let us now turn to discuss the effects on the evolution of perturbations. Given that we have observational access to the fluctuations in the radiation and matter fields, it is useful to discuss separately these two components. The photon perturbations are sensitive to time variations in the gravitational potentials along the line of sight from us up to the lastscattering surface; this is the so-called integrated Sachs-Wolfe (ISW) effect. The gravitational potentials are constant in a purely matter-dominated Universe, so that the observed ISW gets an early contribution right after recombination, when the radiation component is not yet negligible, and a late contribution, when the dark energy density begins to be important. Coming back to our previous discussion, it is clear to see how delaying the time of equality will increase the amount of early ISW, while anticipating dark energy domination will increase the late ISW, and viceversa. For what concerns matter inhomogeneities, a first effect is again related to the time of matter-radiation equality. Changing zeq affects the growth of perturbations, since most of the growth happens during the matter dominated era. Apart from that, a very peculiar effect is related to the clustering properties of neutrinos. In fact, while neutrinos are relativistic, they tend to free stream out of overdense regions, damping out all perturbations below the horizon scale. The net effect is that neutrino clustering is suppressed below a certain critical scale, the free-streaming scale, that corresponds to the size of the horizon at the time of the nonrelativistic transition. If the transition happens during matter domination, this is given by:

$$k\_{\rm fs} \simeq 0.018 \,\Omega\_m^{1/2} \left(\frac{m\_\nu}{1 \,\text{eV}}\right)^{1/2} h \text{Mpc}^{-1}.\tag{27}$$

On the contrary, above the free-streaming scale neutrinos cluster as dark matter and baryons do. Thus, increasing the neutrino mass and consequently the neutrino energy density will suppress small-scale matter fluctuations relative to the large scales. It will also make small-scale perturbations in the other components grow slower, since neutrino do not source the gravitational potentials at those scales. It should also be noted that the freestreaming scale depends itself on the neutrino mass—specifically, heavier neutrinos will become non-relativistic earlier and the free-streaming scale will be correspondingly smaller. Moreover, there is actually a free-streaming scale for each neutrino species, each depending on the individual neutrino mass. In principle one could think to go beyond observing just the small-scale suppression and try to access instead the scales around the nonrelativistic transition(s), in order to get more leverage on the mass and perhaps also on the mass splitting. We shall see however in the following that this is not the case.

The suppression of matter fluctuations due to neutrino free-streaming also affects the path of photons coming from distant sources, since those photons will be deflected by the gravitational potentials along the line of sight, resulting in a gravitational lensing effect. This is relevant for the CMB, as it modifies the anisotropy pattern by mixing photons that come from different directions. Another application of this effect, of particular importance for estimates of neutrino masses, is to use the distortions of the shape of distant galaxies due to lensing, to reconstruct the intervening matter distribution.

### 4. COSMOLOGICAL OBSERVABLES

In this section we review the various cosmological observables, and explain how the effects described in the previous section propagate to the observables.

### 4.1. CMB Anisotropies

The CMB consists of polarized photons that, for the most part, have been free-streaming from the time of recombination to the present time. The pattern of anisotropies in both temperature (i.e., intensity) and polarization thus encodes a wealth of information about the early Universe, down to z = zrec ≃ 1, 100. Moreover, given that the propagation of photons from decoupling to the present time is also affected by the cosmic environment, the CMB also has some sensitivity to physics at z < zrec. Two relevant examples for the topic under consideration are the CMB sensitivity to the redshift of reionization (because the CMB photons are re-scattered by the new population of free electrons) and to the integrated matter distribution along the line of sight (because clustering at low redshifts modifies the geodesics with respect to an unperturbed FLRW Universe, resulting in a gravitational lensing of the CMB, see next section). However, the CMB sensitivity to these processes is limited due to the fact that these are integrated effects.

The information in the CMB anisotropies is encoded in the power spectrum coefficients C TT ℓ , i.e., the coefficients of the expansion in Legendre polynomials of the two-point correlation function. In the case of the temperature angular fluctuations 1T(nˆ)/T:

$$\left\langle \frac{\Delta T(\hat{n})}{\overline{T}} \frac{\Delta T(\hat{n}')}{\overline{T}} \right\rangle = \sum\_{\ell=0}^{\infty} \frac{2\ell+1}{4\pi} C\_{\ell}^{TT} P\_{\ell}(\hat{n} \cdot \hat{n}'). \tag{28}$$

For Gaussian fluctuations, all the information contained in the anisotropies can be compressed without loss in the two-point function, or equivalently in its harmonic counterpart, the power spectrum. A similar expression holds for the polarization field and for its cross-correlation with temperature. In detail, the polarization field can be decomposed into two independent components, known as E− (parity-even and curl-free) and B− (parity-odd and divergence-free) modes. Given that, it is clear that we can build a total of six spectra C XY <sup>ℓ</sup> with X, Y = T, E, B; however, if parity is not violated in the early Universe, the TB and EB correlations are bound to vanish. Let us also recall that, in linear perturbation theory, B modes are not sourced by scalar fluctuations. Thus, in the framework of the standard inflationary paradigm, primordial B modes can only be sourced in the presence of tensor modes, i.e., gravitational waves.

The shape of the observed power spectra is the result of the processes taking place in the primordial plasma around the time of recombination. In brief, in the early Universe, standing, temporally coherent acoustic waves set in the coupled baryon-photon fluid, as a result of the opposite action of gravity and radiation pressure [49]. Once the photons decouple after hydrogen recombination, the waves are "frozen" and thus we observe a series of peaks and throughs in the temperature power spectrum, corresponding to oscillation modes that were caught at an extreme of compression or rarefaction (the peaks), or exactly in phase with the background (the throughs). The typical scale of the oscillations is set by the sound horizon at recombination rs(zrec), i.e., the distance traveled by an acoustic wave from some very early time until recombination, see Equation (4). The position of the first peak in the CMB spectrum is set by the value of this quantity and corresponds to a perturbation wavenumber that had exactly the time to fully compress once. The second peak corresponds to the mode with half the wavelength, that had exactly the time to go through one full cycle of compression and rarefaction, and so on. Thus, smaller scales (larger multipoles) than the first peak correspond to scales that could go beyond one full compression, while larger scales (smaller multipoles) did not have the time to do so. In fact, scales much above the sound horizon are effectively frozen to their initial conditions, provided by inflation. This picture is complicated a little bit by the presence of baryons, that shift the zero of the oscillations, introducing an asymmetry between even and odd peaks. Finally, the peak structure is further modulated by an exponential suppression, due to the Silk damping of photon perturbations (further related to the fact that the tight coupling approximation breaks down at very small scales). This description also holds for polarization pertubations, with some differences, like the fact that the polarization perturbations have opposite phase with respect to temperature perturbations.

As noted above, the large-scale temperature fluctuations, that have entered the horizon very late and did not have time to evolve, trace the power spectrum of primordial fluctuations, supposedly generated during inflation. On the contrary, since there are no primordial polarization fluctuations, but those are instead generated at the time of recombination and then again at the time of reionization, the polarization spectra at large scales are expected to vanish, with the exception of the so-called reionization peak.

We can now understand how the CMB power spectra are shaped by the cosmological parameters, in a minimal model with fixed neutrino mass. The overall amplitude and slope of the spectra are determined by A<sup>s</sup> and n<sup>s</sup> , since these set the initial conditions for the evolution of perturbations. The height of the first peak strongly depends on the redshift of equivalence zeq (that sets the enhancement in power due to the early ISW), while its position is determined by the angle θ<sup>s</sup> subtended by the sound horizon at recombination. As we have discussed before, zeq and θ<sup>s</sup> are in turn set by the values of the background densities and of the Hubble constant. The baryon density further affects the relative heights of odd and even peaks, and also the amount of damping at small scales, through its effect on the Silk scale. The ratio of the densities of matter and dark energy fixes the redshift of dark energy domination and the amount of enhancement of large-scale power due to the late ISW. Finally, the optical depth at reionization τ induces an overall power suppression, proportional to e −2τ , in all spectra, at all but the largest scales. This can be easily understood as the effect of the new scatterings effectively destroying the information about the fluctuation pattern at recombination, at the scales that are inside the horizon at reionization. Reionization also generates the large-scale peak in the polarization spectra, described above. Measuring the power spectra gives a precise determination of all these parameters: simplifying a little bit, the overall amplitude and slope give Ase −2τ and n<sup>s</sup> (the latter especially if we can measure a large range of scales), the ratio of the peak heights and the amount of small-scale damping fix ω<sup>b</sup> , while the position and height of the first peak fix θ<sup>s</sup> and zeq, and thus h and ωb+<sup>c</sup> . The polarization spectra further help in that they are sensitive to τ directly, allowing to break the A<sup>s</sup> − τ degeneracy, and that the peaks in polarization are sharper and thus allow, in principle, for a better determination of their position [50]. It is clear that adding one more degree of freedom to this picture, for example considering curvature, the equation of state parameter of dark energy, or the neutrino mass as a free parameter, will introduce parameter degeneracies and degrade the constraints.

Coming to massive neutrinos, as we have discussed in section 3, there is a combination of the following effects when 6m<sup>ν</sup> , and consequently ω<sup>ν</sup> , is increased, depending on how we are changing the other parameters to keep P <sup>i</sup> <sup>i</sup> = 1: (i) an increase in θ<sup>s</sup> ; (ii) a smaller zeq and thus a longer radiation-dominated era; (iii) a delay of the time of dark energy domination. These changes will in turn result in: (i) a shift towards the left of the position of the peaks; (ii) an increased height of the first peak, that is set by the amount of early ISW; (iii) less power at the largest scales, due to the smaller amount of late ISW. A more quantitative assessment of these effects can be obtained using a Boltzmann code, like CAMB [51] or CLASS [52], to get a theoretical prediction for the CMB power spectra in presence of massive neutrinos. These are shown in **Figure 1**. In the left panel we plot the unlensed CMB temperature power spectra for a reference model with <sup>6</sup>m<sup>ν</sup> <sup>=</sup> 0.06 eV (ω<sup>ν</sup> <sup>≃</sup> 6.4 <sup>×</sup> <sup>10</sup>−<sup>4</sup> ) (the other parameters are fixed to their best-fit values from Planck 2015) and for three models with 6m<sup>ν</sup> = 1.8 eV (ω<sup>ν</sup> ≃ 1.9 × 10−<sup>2</sup> ), where either h, ω<sup>c</sup> , or <sup>3</sup> are changed to keep P <sup>i</sup> <sup>i</sup> = 1. We consider three degenerate neutrinos with m<sup>ν</sup> = 0.6 eV each, so that they become non-relativistic around recombination. We also show the ratio between these spectra and the reference spectrum in the right panel of the same figure.

These imprints are in principle detectable in the CMB, especially the first two, since the position and height of the first peak are very well measured; much less so the redshift of DE domination, due to the large cosmic variance at small ℓ's. However, following the above discussion, it is quite easy to convince oneself that these effects can be pretty much canceled due to parameter degeneracies. In fact, simplifying again a little bit, in standard 3CDM we use the very precise determinations of the height and position of the first peak to determine θ<sup>s</sup> and zeq, and from them ωc+<sup>b</sup> and h. In an extension with massive neutrinos, we still have the same determination of θ<sup>s</sup> and zeq, but we have to use them to fix three parameters, namely ωc+<sup>b</sup> , h, and ω<sup>ν</sup> , so that the system is underdetermined. One could argue that the amount of late ISW, as measured by the largescale power, could be used to break this degeneracy, as it would provide a further constraint on the matter density (given that

vanishing by changing h (green), 3 (yellow, always below the green apart from the lowest ℓ's), or ωc (blue). The model in blue has a smaller zeq with respect to the reference; the models in yellow and green have a larger θs; in addition, the yellow model also has a smaller z3. (Bottom) Ratio between the models with 6m<sup>ν</sup> = 1.8 eV and the reference model.

the DE density is fixed by the flatness condition). Unfortunately, measurements of the large-scale CMB power are plagued by large uncertainties, due to cosmic variance, so they are of little help in solving this degeneracy. Given the experimental uncertainties, then, it is clear that, when trying to fit a theory to the data, there will be a strong degeneracy direction corresponding to models having the same θ<sup>s</sup> and zeq, and thus with identical predictions for the first peak, and slightly different values of z3, with very low statistical weight due to the large uncertainties in the corresponding region of the spectrum. In other words, the effects of neutrino masses will be effectively "buried" in the small-ℓ plateau, where experimental uncertainties are large. The situation is even worse in extended models, for example if we allow the spatial curvature or the equation of state of dark energy to vary [13]. In any case, the degeneracy between h and ωc+<sup>b</sup> is not completely exact, so that the unlensed CMB still has some degree of sensitivity to neutrinos that were relativistic at recombination. For example, the Planck 2013 temperature data, in combination with high-resolution observations from ACT and SPT, were able to constrain 6m<sup>ν</sup> < 1.1 eV after marginalizing over the effects of lensing [53].

#### 4.1.1. Secondary Anisotropies and the CMB Lensing

As observed above, in addition to the features that are generated at recombination, the so-called primary anisotropies, the CMB spectra also carry the imprint of effects that are generated along the line of sight. We have already given an example of one of these secondary anisotropies when we have mentioned the re-scattering of photons over free electrons at low redshift, that creates the distinctive "reionization bump" in the low-ℓ region of the polarization spectra. Another important secondary anisotropy is the gravitational lensing of the CMB (see [54, 55]): photon paths are distorted by the presence of matter inhomogeneities along the line of sight. In the context of General Relativity, the deflection angle α for a CMB photon is

$$\alpha = -2 \int\_0^{\chi\_\*} \mathrm{d}\chi \frac{f\_\mathsf{K}(\chi\_\* - \chi)}{f\_\mathsf{K}(\chi\_\*) f\_\mathsf{K}(\chi)} \nabla \Psi(\chi \mathbf{n}, \eta\_0 - \chi) \tag{29}$$

where χ<sup>∗</sup> is the comoving distance to the last scattering surface, fK(χ) is the angular-diameter distance (Equation 6) thought as a function of the comoving distance, 9 is the gravitational potential, η<sup>0</sup> − χ is the conformal time at which the photon was along the direction **n**. If we then define the lensing potential as

$$\phi(\hat{\mathbf{n}}) \equiv -2 \int\_0^{\chi\_\*} \mathrm{d}\chi \frac{f\_\mathbf{K}(\chi\_\* - \chi)}{f\_\mathbf{K}(\chi\_\*) f\_\mathbf{K}(\chi)} \Psi(\chi \mathbf{n}, \eta\_0 - \chi), \tag{30}$$

it is straightforward to see that the deflection angle is the gradient of the lensing potential, α = ∇φ. From the harmonic expansion of the lensing potential, we can build an angular power spectrum<sup>6</sup> as < φℓmφ ∗ ℓ ′m′ >≡ δℓℓ′δmm′C φφ ℓ . The lensing power spectrum C φφ ℓ is therefore proportional to the integral along the line of sight of the power spectrum of the gravitational potential P9, which in turn can be expressed in terms of the power spectrum of matter fluctuations P<sup>m</sup> (see the next section for its definition).

The net effect of lensing on the CMB is that photons coming from different directions are mixed, somehow "blurring" the anisotropy pattern. This effect is mainly sourced by inhomogeneities at z < 5 and has a typical angular scale of 2.5′ . In the power spectra, this translates in a several percent level smoothing of the primary peak structure (ℓ & 1,000), while the lensing effect becomes dominant at ℓ & 3,000. We stress that lensing only alters the spatial distribution of CMB fluctuations, while leaving the total variance unchanged. Lensing, being a non-linear effect, creates some amount of non-gaussianity in the anisotropy pattern. Thus, other than through its indirect effect on the temperature and polarization power spectra (i.e., on the two-point correlation functions), lensing can be detected and measured by looking at higher-order correlations, in particular at the four-point correlation function. In fact, in such a way it has been possible to directly measure the power spectrum C φφ ℓ of the lensing potential φ. Another consequence of the nonlinear nature of lensing is that it is able to source "spurious" B modes by converting some of the power in E polarization, thus effectively creating B polarization also in the absence of a primordial component of this kind. The latter effect represents an additional tool to enable the reconstruction of the lensing potential, especially for future CMB surveys. An alternative reconstruction technique is based on the possibility to crosscorrelate the CMB signal with tracers of large-scale structures, such as Cosmic Infrared Background (CIB) maps, therefore leading to an "external" reconstruction [56] (opposite to the "internal" reconstruction performed with the use of CMB-based only estimators [57, 58]).

The lensing power spectrum basically carries information about the integrated distribution of matter along the line of sight. Given the peculiar effect of neutrino free-streaming on the evolution of matter fluctuations, CMB lensing offers an important handle for estimates of neutrino masses. Since a larger neutrino mass implies a larger neutrino density and less clustering on small scales, because of neutrino free-streaming, the overall effect of larger neutrino masses is to decrease lensing. In the temperature and polarization power spectra, the result is that the peaks and throughs at high ℓ's are sharper. Concerning the shape of the lensing power spectrum, for light massive neutrinos the net effect is a rescaling of power at intermediate and small scales (see e.g., [59]). Thus, the lensing power spectrum is a powerful tool for constraining 6m<sup>ν</sup> and will probably drive even better constraints on 6m<sup>ν</sup> in the future. In fact, it is almost free from systematics coming from poorly understood astrophysical effects, it directly probes the (integral over the line of sight of the) distribution of the total matter fluctuations (as opposed to what galaxy surveys do, as we will see in the next section) at scales that are still in the linear regime.

Given a cosmological model, it is quite straightforward, using again CAMB or CLASS, to get a theoretical prediction for the lensing power spectrum, as well as for the lensing BB power spectrum. Note that non-linear corrections (see next section for further details) to the lensing potential are important in this case to get accurate large-scale BB spectrum coefficients [54]. Additional corrections that take into account modifications to the CMB photon emission angle due to lensing can further modify the large-scale lensing BB spectrum [60].

### 4.2. Large Scale Structures 4.2.1. Clustering

The clustering of matter at large scales is another powerful probe of cosmology. The clustering can be described in terms of the two-point correlation function, or, equivalently, of the power spectrum of matter density fluctuations:

$$\left\langle \delta\_m(\vec{k}, z)\delta\_m(\vec{k}', z) \right\rangle = P\_m(k, z)\delta^{(3)}\left(\vec{k} - \vec{k}'\right),\tag{31}$$

where δm( Ek, z) is the Fourier transform of the matter density perturbation at redshift z. Note that, contrarily to the CMB, that we are bound to observe at a single redshift (that of recombination), the matter power spectrum can, in principle, be measured at different times in the cosmic history, thus allowing for a tomographic analysis.

As for the CMB, the large-scale (small k's) part of the power spectrum traces the primordial fluctuations generated during inflation, while smaller scales reflect the processing taking place after a given perturbation wavenumber enters the horizon. A relevant distinction in this regard is whether a given mode enters the horizon before or after matter-radiation equality. Since subhorizon perturbations grow faster during

<sup>6</sup>We are assuming that the lensing field is isotropic.

matter domination, the matter power spectrum shows a turning point at a characteristic scale, corresponding to the horizon at zeq. Given that perturbations grow less efficiently also during DE domination, increasing z3 produces a suppression in the power spectrum. Also, increasing h will make the horizon at a given redshift smaller; so the mode k that is entering the horizon at that redshift will be larger.

Varying the sum of neutrino masses has some indirect effects on the shape of matter power spectrum, related to induced changes in background quantities, similarly to what happens for the CMB. As explained in section 3, increasing 6m<sup>ν</sup> while keeping the Universe flat has to be compensated by changing (a combination of) ωm, 3, or h. This will in turn result in a shift of the turning point and/or in a change in the global normalization of the spectrum. This can be seen in **Figure 2**, where we show the matter power spectra for the same models considered when discussing the background effects of neutrino masses on the CMB.

As it is for the CMB, these effects can be partly canceled due to parameter degeneracies. Neutrinos, however, have also a peculiar effect on the evolution of matter perturbations. This is due to the fact that neutrinos possess large thermal velocities for a considerable part of the cosmic history, so they can free-stream out of overdense regions, effectively canceling perturbations on small scales. In particular, one can define the free-streaming length at time t as the distance that neutrinos can travel from decoupling until t. The comoving free-streaming length reaches a maximum at the time of the non-relativistic transition. This corresponds to a critical wavenumber kfs, given in equation (27) for transitions happening during matter-domination, above which perturbations in the neutrino component are erased.

A first consequence of neutrino free-streaming is that, below the free-streaming scale, there is a smaller amount of matter that can cluster. This results in an overall suppression of the power spectrum at small scales, with respect to the neutrinoless case. Secondly, subhorizon perturbations in the non-relativistic (i.e., cold dark matter and baryons) components grow more slowly. In fact, while in a perfectly matter-dominated Universe, the gravitational potential is constant and the matter perturbation grows linearly with the scale factor, δ<sup>m</sup> ∝ a, in a mixed matterradiation Universe the gravitational potential decays slowly inside the horizon. Below the free-streaming scale, neutrinos effectively behave as radiation; then in the limit in which the neutrino fraction f<sup>ν</sup> = <sup>ν</sup> /<sup>m</sup> is small, one has for k ≫ kfs

$$(\delta\_m(k \gg k\_{\rm fs}) \propto a^{1 - (3/5)\ell\_\vee}, \tag{32}$$

while δ<sup>m</sup> ∝ a for k ≪ kfs. These two effects can be qualitatively understood as follows: if one considers a volume with linear size well below the free-streaming scale, this region will resemble a Universe with a smaller <sup>m</sup> and a larger radiation-to-matter fraction than the "actual" (i.e., averaged over a very large volume) values. This yields a smaller overall normalization of the spectrum, as well as a larger radiation damping; the two effects combine to damp the matter perturbations inside the region. So, looking again at the full power spectrum, the net effect is that, in the presence of free-streaming neutrinos, power at small-scales is suppressed with respect to the case of no neutrinos. At z = 0, the effect saturates at <sup>k</sup> <sup>≃</sup> <sup>1</sup> <sup>h</sup> Mpc−<sup>1</sup> , where a useful approximation is Pm(k, f<sup>ν</sup> )/Pm(k, f<sup>ν</sup> = 0) ≃ 1 − 8f<sup>ν</sup> [61].

It is useful to stress that since f<sup>ν</sup> is linear in 6m<sup>ν</sup> , we have the somehow counterintuitive result that the effects of free-streaming are more evident for heavier, and thus colder, neutrinos. The reason is simply that the asymptotic suppression of the spectrum depends only on the total energy density of neutrinos, as

this determines the different amount of non-relativistic matter between small and large scales.

Until now, we have somehow ignored the role of baryons in shaping the matter power spectrum. In fact, on scales that enter the horizon after zdrag, the baryons are effectively collisionless and behave exactly like cold dark matter. On the other hand, baryon perturbations at smaller scales, entering the horizon before zdrag exhibit acoustic oscillations due to the coupling with photons. This causes the appearance of an oscillatory structure in the matter power spectrum. These wiggles in Pm(k), that go under the name of baryon acoustic oscillations (BAO), have a characteristic frequency, related to the value of the sound horizon at zdrag. Thus they can serve as a standard ruler and can be used very effectively in order to constrain the expansion history.

In more detail, the acoustic oscillations that set up in the primordial Universe produce a sharp feature in the two-point correlation function of luminous matter at the scale of the sound horizon evaluated at the drag epoch, rs(zd) ≡ rd; this sharp feature translates in (damped) oscillations in the Fourier transform of the two-point correlation function, i.e., the power spectrum. Measuring the BAO feature at redshift z allows in principle to separately constrain the combination dA(z)/rd, for measurements in the transverse direction with respect to the line of sight, or rdH(z) for measurements along the line of sight. An isotropic analysis instead measures, approximately, the ratio between the combination

$$d\_V(z) = \left[\frac{z d\_A^2(z)}{H(z)}\right]^{1/3},\tag{33}$$

called the volume-averaged distance, and the sound horizon rd. Given that the value of the sound horizon is well constrained by CMB observations, measuring the BAO features, possibly at different redshifts, allows to directly constrain the expansion history, as probed by the evolution of the angular diameter distance dA(z) and of the Hubble function H(z), or of their average dV(z). In particular, it is straightforward to see that BAO measurements put tight constraints on the <sup>m</sup> − H0r<sup>d</sup> plane, along a different degeneracy direction that it is instead probed by CMB [62, 63]. Therefore, when estimating neutrino masses, the addition of BAO constraints to CMB data helps breaking the parameter degeneracies discussed in the previous section, yielding in general tighter constraints on this quantity.

The linear matter power spectrum for a given cosmological model can be computed using a Boltzmann solver. However, comparison with observations is complicated by the non-linear evolution of cosmic structures. Note that both CAMB and CLASS are able to handle non-linearities in the evolution of cosmological perturbations with the inclusion of nonlinear corrections from the Halofit model [64] calibrated over numerical simulations. In particular, for cosmological models with massive neutrinos, the preferred prescription is detailed in Bird et al. [65].

From the observational point of view, Pm(k, z) can be probed in different ways. In galaxy surveys, the 3-D spatial distribution of galaxies is measured, allowing to measure the two-point correlation function and to obtain an estimate of the power spectrum of galaxies P<sup>g</sup> (k, z). Since in this case one is measuring the distribution of luminous matter only, and not of all matter (including dark matter), this does not necessarily coincide with the quantity for which we have a theoretical prediction, i.e., Pm; in other words, galaxies are a biased tracer of the matter distribution. To take this into account, one relates the two quantities through a bias b(k, z):

$$P\_{\mathcal{S}}(k, z) = b^2(k, z) P\_m(k, z). \tag{34}$$

The bias is in general a function of both redshift and scale. If it is approximated as a scale-independent factor, then the presence of the bias only amounts to an overall rescaling of the matter power spectrum (at a given redshift). In this case, one marginalizes over the amplitude of the matter spectrum, effectively only using the information contained in its shape. A scale-independent bias is considered to be a safe approximation for the largest scales: as an example, for Luminous Red Galaxies sampled at an efficient redshift of 0.5 (roughly corresponding to the CMASS sample of the SDSS III-BOSS survey), a scaleindependent bias is a good approximation up to k . 0.2 h Mpc−<sup>1</sup> [66]. On the other hand, scale-dependent features are expected to appear on smaller scales. In this case, the bias can still be described using a few "nuisance" parameters, that are then marginalized over. In any case the exact functional form of the bias function, the range of scales considered, as well as prior assumptions on the bias parameters, are delicate issues that should be treated carefully. An additional complication arises from the fact that massive neutrinos themselves induce a scale-dependent feature in the bias parameter, due to the scaledependent growth of structures in cosmologies with massive neutrinos [67, 68].

It has to be mentioned that, at any given redshift, there exists a certain scale kNL below which the density contrast approaches the limit δ ∼ 1. In this regime, the evolution of cosmic structures cannot be completely captured by a linear theory of perturbations. The modeling of structures in the non-linear regime relies on numerical N-body simulations that must take into account the astrophysical and hydrodynamical processes at play at those scales. The level of complexity of Nbody simulations has been increasing over the years, so that the physical processes included in the simulations and the final results are much closer to the observations than they used to be at the beginning. Recent examples are given by the MassiveNuS [69] suite, based on the Gadget-2 code [70] modified to include the effects of massive neutrinos, the DEMNUni suite [71–73], the TianNu simulation [74–76], the BAHAMAS project [77], the gevolution simulations [78], and the nuCONCEPT simulations [79] (see also [80] for a method combining the particle and fluid descriptions)<sup>7</sup> . Nevertheless, the uncertainties related to the non-linear evolution of cosmological structures are still higher

<sup>7</sup>Prescriptions for the matter power spectrum in the non-linear regime are also provided by the Halofit model [65], the Coyote Universe emulator [81], the semianalytical approach of PINOCCHIO [82], and additional methods referenced in Rizzo et al. [82].

than those affecting the linear theory, therefore reducing the constraining power coming from the inclusion of those scales in cosmological analysis. In fact, the conservative choice of not including measurements at k > kNL is usually made when performing cosmological analyses. It is easy to understand that at higher redshifts, a wider range of scales is still in the linear regime.

Additional probes of P<sup>m</sup> are measurements of Lyman-α (Lyα) forests and 21-cm fluctuations (see e.g., [83, 84] for reviews). Although they are promising avenues since they can probe the matter distribution at higher redshifts and smaller scales than those usually accessible with typical galaxy samples, they still have to reach the level of maturity required to take full advantage of their constraining power. The observation of high-redshift (z ∼ 2) quasars and in particular the measurement of their flux provides a powerful tool for cosmological studies. Indeed, the absorption of the Lyα emission from quasars by the intervening intergalactic medium—an observational feature known as "Lyα forest"—constitutes a tracer of the total matter density field at higher redshifts and smaller scales than those usually probed by galaxy surveys. Similarly to what is done for galaxy samples, one can compute a correlation function of the measured flux variation, or equivalently its power spectrum PLyα. The latter is again proportional to the total P<sup>m</sup> via a bias parameter bLyα. The Lyα bias factor is in general different from the galaxy bias, as each tracer of the underlying total matter distribution exhibits its own characteristics. The Lyα forest is ideally a powerful cosmological tool, being able to access high redshifts. Therefore, at fixed scale k, the physics governing the Lyα spectrum is much closer to the linear regime than that related to the galaxy power spectrum. Furthermore, the redshift window probed by Lyα is complementary to that probed by traditional galaxy surveys, in a sense that at higher redshift the relative impact of dark energy on the cosmic inventory is much smaller. However, a reliable description of the astrophysics at play in the intergalactic medium is essential for deriving the theoretical model for the Lyα absorption features along the line of sight. This description heavily depends on hydrodynamical simulations that reproduce the behavior of baryonic gas and on poorly known details of the reionization history. In addition, uncertainties in the theory of non-linear physics of the intergalactic medium at small scales can play a non-negligible role.

Finally, another tracer of the total matter fluctuations is represented by fluctuations in the 21-cm signal. The 21-cm line is due to the forbidden transition of neutral hydrogen (HI) between the two hyperfine levels of the ground state (spin flip) of the hydrogen atom. The observational technique resides in the possibility to measure the brightness temperature relative to the CMB temperature. Fluctuations in the 21-cm brightness are related to fluctuations in HI (or equivalently to the fraction of free electrons xe), which in turn trace the matter fluctuations. Therefore, one can infer P<sup>m</sup> observationally by measuring the power spectrum of 21-cm fluctuations P21−cm. Apart from the technological challenges associated with the detection of the 21-cm signal, the main source of systematics come from the difficulties to separate the faint 21-cm signal from the much brighter foreground contamination, mostly due to synchrotron emission from our own galaxy.

#### 4.2.2. Cluster Abundances

The variation of the number of galaxy clusters of a certain mass M with redshift dN(z, M)/dz is also a valid source of information about the evolution of the late time Universe (see e.g., [85] for a review). The expected number of clusters to be observed in a given redshift window is an integral over the redshift bin of the quantity

$$\frac{\mathrm{dN}}{\mathrm{d}z} = \int \mathrm{d}\Omega \int \mathrm{d}M \hat{\chi} \, \frac{\mathrm{d}N}{\mathrm{d}M \mathrm{d}z \mathrm{d}\Omega} \tag{35}$$

where is the solid angle, χˆ is the so-called completeness of the survey (a measure of the probability that the survey will detect a cluster of a given mass M at a given redshift z) and <sup>d</sup><sup>N</sup> dM (z, M) is the mass function giving the number of clusters per unit volume. The latter can be predicted once a cosmological model has been specified. The quantity in Equation (35) is thus directly sensitive to the matter density <sup>m</sup> and to the current amplitude of matter overdensities, usually parametrized in terms of σ8, the variance of matter fluctuations within a sphere of 8 h <sup>−</sup>1Mpc. As a result, this probe can be highly beneficial for putting bounds on 6m<sup>ν</sup> .

Extended catalogs of galaxy clusters have been published in the last decade by the Atacama Cosmology Telescope (ACT) [86, 87], the South Pole Telescope (SPT) [88], and the Planck [24] collaborations. CMB experiments are in fact able to perform searches for galaxy clusters by looking for the thermal Sunyaev-Zeldovich (SZ) effect, the characteristic upward shift in frequency of the CMB signal induced by the inverse-Compton scattering of CMB photons off the hot gas in clusters. The redshift of cluster candidates is identified with follow-up observations, whereas their mass is usually inferred with X-ray observations or, more recently, calibrated through weak lensing. Regardless of how it is calibrated, the determination of the cluster mass is the largest source of uncertainty for the cluster count analysis, due to possibly imprecise assumptions about the dynamical state of the cluster and/or survey systematics. A common way to factorize the uncertainties related to the mass calibration is to introduce a mass bias parameter that relates the true cluster mass to the mass inferred with observations.

#### 4.2.3. Weak Lensing

The weak gravitational lensing effect is the deflection of the light emitted by a source galaxy caused by the foreground large-scale mass distribution (lens). The shape of the source galaxy therefore appears as distorted, i.e., it acquires an apparent ellipticity. The cosmic shear is the weak lensing effect of all the galaxies along the line of sight (see e.g., [89] for a review). Weak lensing surveys offer the possibility to directly test the distribution of intervening matter at low redshifts, thus providing a powerful tool to investigate the late-time evolution of the Universe. By correlating the apparent shapes of source galaxies at different redshifts, one can compute the shear field γ (nˆ, z) as a function of the angular position nˆ and redshift z. The shear field is usually decomposed in two components: the curl-free E-modes and the divergencefree B-modes. It can be shown that, in absence of systematics, the B-modes are expected to vanish, whereas the power spectrum of the E-modes is equivalent to the lensing power spectrum C φφ ℓ . The integrated lensing potential has been defined in Equation (30) for a source located at recombination. The corresponding expression for a source at a generic redshift z can be obtained simply by substituting χ<sup>∗</sup> with the comoving distance of the source.

Thus, the power spectrum of the lensing potential—which is due to intervening matter along the line of sight—is recovered from the measurements of the lensing-induced ellipticity of background galaxies; in a similar way, the lensing power spectrum is recovered from the redistribution of CMB photons due to the forming structures along the line of sight. As we have seen in section 4.1.1, the spectrum of the lensing potential is a function of the matter power spectrum integrated along the line of sight. Therefore, it carries information about the distribution and growth of structures, representing a powerful tool for constraining 6m<sup>ν</sup> . It should be mentioned that the observed shear signal γobs is a biased tracer of the true shear γtrue. This effect, mostly due to noise in the pixels when galaxy ellipticity is measured, is usually taken into account by introducing a multiplicative bias m that relates γtrue and γobs: γobs = (1 + m)γtrue + c, where c is the additional noise bias [90].

In addition, the shear signal can be cross-correlated with the angular distribution of foreground (lens) galaxies (the so-called galaxy-shear or galaxy-galaxy lensing cross-correlation). This cross-correlation is a powerful way to overcome the limitations induced in the galaxy-galaxy auto-correlation by the unknown galaxy bias. Indeed, the galaxy-galaxy lensing is basically a cross-correlation between the galaxy field and the total matter fluctuation field. Measurements of the galaxy-galaxy lensing cross spectrum can therefore help determine the form of the bias.

Cosmological constraints from weak lensing surveys are often summarized in terms of bounds on <sup>m</sup> and σ8. As an additional probe of the large-scale structure in the Universe, weak lensing can be profitably used to constrain 6m<sup>ν</sup> .

### 4.3. Supernovae Ia and Direct Measurements of the Hubble Constant

Measurements of the distance-redshift relation of Supernovae Ia (SNIa) have provided the compelling evidence of the accelerated Universe [91, 92]. SNIa are produced in binary stellar systems in which one of the stars is a white dwarf. Accreting matter from its companion, the white dwarf explodes once it reaches the Chandrasekhar mass limit. Therefore, SNIa are standard candles, because their absolute magnitude can be theoretically inferred from models of stellar evolution. A comparison between the absolute magnitude and the apparent luminosity yields an estimate of their luminosity distance dL(z). The expected value of d<sup>L</sup> in turn depends on the underlying cosmological model. The constraints coming from SNIa in the <sup>m</sup> − <sup>3</sup> plane are orthogonal to those obtained from CMB. As a result, the combination of the two probes is extremely efficient in breaking the degeneracy between the two parameters. For this reason, SNIa are very useful for constraining models of dark energy and/or arbitrary curvature. Nonetheless, constraints on 6m<sup>ν</sup> can benefit from the use of SNIa data, thanks to the improved bounds on m.

As already discussed, the effect of light massive neutrinos on the background evolution of the Universe can be also compensated by a change in the value of the Hubble constant H0. Therefore, it is clear that any direct measurement of H<sup>0</sup> can be highly beneficial for putting bounds on 6m<sup>ν</sup> . Direct measurements only rely on local distance indicators (i.e., redshift z≪1), therefore they are little or not-at-all sensitive to changes in the underlying cosmological model. In contrast, indirect estimates from high-redshift probes, such as primary CMB, can suffer from model dependency.

Direct measurements of H<sup>0</sup> are based on the geometric distance calibration of nearby Cepheids luminosity-period relation and the subsequent calibration of SNIa over Cepheids observed in the same SNIa galaxy hosts (see e.g., [93] and references therein). The goal is to connect the precise geometric distances measured in the nearby Universe (usually referred to as "anchors") with the distant SNIa magnitude-redshift relation in order to extract the estimate of H0. The main systematics are of course related to the calibration procedure. Further improvements on the precision of direct measurements of H<sup>0</sup> are expected to come once the precise parallaxes measurements from the Gaia satellite will be available [94].

Local measurements of H<sup>0</sup> are not directly sensitive to 6m<sup>ν</sup> . Besides, their results, in combination with cosmological probes, can break the degeneracy between cosmological parameters and improve constraints on 6m<sup>ν</sup> . The main example is in fact the possibility to break the strong (inverse) degeneracy between H<sup>0</sup> and 6m<sup>ν</sup> that affects CMB constraints.

Indirect estimates of H<sup>0</sup> can be obtained from CMB and BAO measurements. We have already seen in section 4 that the position and amplitude of the first acoustic peak in the CMB spectrum depends on H<sup>0</sup> in combination with other parameters. In addition, we shall mention that, once the BAO are calibrated with the precise determination of r<sup>d</sup> from CMB, measurements of dA/r<sup>d</sup> and Hr<sup>d</sup> (or dV/rd) yields bounds on H<sup>0</sup> that are competitive with CMB estimates and direct measurements.

We finally mention an additional independent measurement of H0. The gravitational wave (GW) signal emitted by merging compact objects in combination with the observation of an electromagnetic counterpart has been proposed as a standard siren [95, 96]. The GW waveform reconstruction allows for a determination of the luminosity distance to the source. Precise determinations of the source localization can lead to percent accuracy in the luminosity distance estimation. The observation of the electromagnetic counterpart of the GW event is then essential to determine the redshift to the source. The full combination of distance-redshift pair can finally be employed to constrain H0. In the absence of the detection of an electromagnetic counterpart, methods to infer the redshift of the source of the GW signal have been proposed (see e.g., [97]).

### 4.4. Summary of the Effects of Neutrino Masses

Before moving to report the current observational constraints, we find it useful to summarize the constraining power of different cosmological observables with respect to the neutrino mass. The discussion is somehow qualitative, also given the high-level complexity of the cosmological models. The purpose is also to underline the importance of combining different cosmological probes.

We start from the CMB. For the present discussion, it is useful to consider separately the information coming from the unlensed CMB (i.e., the primary CMB plus all the secondary effects with the exclusion of lensing) and that coming from the weak lensing of CMB photons. For what concerns the former, the sensitivity of the unlensed CMB to neutrino masses is somehow limited. This is mainly due to a geometrical degeneracy between h and ω<sup>ν</sup> thanks to which one can simultaneously change the two parameters (decreasing h and increasing ω<sup>ν</sup> ) to keep θ<sup>s</sup> constant, thus preserving the position of the first peak, with only limited changes to other parts of the spectrum (especially changes in the low-ℓ region, where the sensitivity is limited by cosmic variance, induced by variations in 3). The height of the first peak is preserved by keeping ω<sup>c</sup> fixed. Having access to the information contained in the CMB lensing, either through its effect on the temperature and polarization power spectra, or through a direct estimation of the lensing power spectrum, helps because 6m<sup>ν</sup> also affects the matter distribution and then the amplitude of the lensing potential at small scales. This helps breaking the degeneracy described above.

To illustrate this point, in the upper panel of **Figure 3** we show the parameter correlations derived by an analysis of the Planck observations of the temperature, over a wide range of scale, and large-scale polarization anisotropies. We remember that this dataset contains some information about lensing through the high-ℓ part of the temperature power spectrum. The negative degeneracy between 6m<sup>ν</sup> and H<sup>0</sup> is particularly evident. Given that ω<sup>c</sup> and ω<sup>b</sup> are both measured quite well from the CMB, this also translates into a strong degeneracy with <sup>m</sup> = (ω<sup>c</sup> + ω<sup>b</sup> )/h 2 and <sup>3</sup> = 1 − m. Among the other parameters, one can notice mild correlations with A<sup>s</sup> and τ . These are due to the small-scale effects related to the increased lensing in models with larger 6m<sup>ν</sup> . The overall amplitude of the spectrum Ase −2τ is very precisely determined by CMB observations. On the other hand, the lensing amplitude depends on A<sup>s</sup> but not on τ . So, the lensing amplitude can be kept constant by increasing both A<sup>s</sup> and ω<sup>ν</sup> . At this point τ has to be increased as well to preserve the scalar amplitude Ase −2τ .

Geometric measurements, like those coming from BAO, SNIa, or direct measurements of H0, greatly help solving the geometrical degeneracy between H<sup>0</sup> and 6m<sup>ν</sup> . This is evident in the lower panel of **Figure 3**, where we show parameter correlations from an analysis of the same dataset as above with the addition of BAO data, if one compares the (H0, 6m<sup>ν</sup> ) square with the corresponding square in the upper panel. Measurements of large scale structures, and especially those that are directly sensitive to the total matter distribution at small scales, are very helpful, in that on the one hand they allow to further constrain m, A<sup>s</sup> , and n<sup>s</sup> and thus reduce degeneracies with these parameters; on the other hand, they allow to probe the regime in which neutrino free-streaming is important. Finally, it is also clear that a precise measurement of τ from a CMB experiment that is sensitive to the large-scale polarization (meaning that it can access a large fraction of the sky) will be highly beneficial.

inclusion of BAO data helps reduce the degeneracy between parameters (see e.g., the correlation between 6mν and H0, 3); in a few cases, in fact, the inclusion of BAO reverts the degeneracy (see e.g., the correlation between 6mν and ns).

We have focused our attention to the 3CDM+6m<sup>ν</sup> model. In extended dark energy models (as well as modified gravity models), for example for arbitrary equations of state of the dark energy fluid, the degeneracy between 6m<sup>ν</sup> and <sup>3</sup> is amplified. Both massive neutrinos and dark energy-modified gravity affect the late time evolution of the Universe, so that the individual effects on cosmological observables (mostly structures) can be reciprocally canceled.

## 5. CURRENT OBSERVATIONAL CONSTRAINTS ON 6m<sup>ν</sup>

In this section we report current constraints on 6m<sup>ν</sup> from cosmological and astrophysical observations. These constraints are also summarized in **Table 1** for the reader's convenience. Unless otherwise stated, the results are obtained in the framework of a minimal one-parameter extension of the 3CDM model with varying neutrino mass, dubbed 3CDM+6m<sup>ν</sup> , in which the three mass eigenstates are degenerate (m<sup>i</sup> = 6m<sup>ν</sup> /3). Given the sensitivity of current experiments, the degenerate approximation is appropriate. See section 8 for a more detailed discussion on this point.

### 5.1. CMB

CMB observations are probably the most mature cosmological measurements. The frequency spectrum is known with great accuracy [46]. Measurements of the power spectrum of CMB anisotropies in temperature are cosmic-variance limited down to very small scales (ℓ ∼ 1, 500) and the quality of current CMB data in polarization is already good enough to tighten constraints on cosmological parameters [14, 16, 17, 19, 20]. The next generation of CMB experiments will further improve our knowledge of CMB polarization anisotropies [21, 30–33]. The main systematics involved in CMB measurements are due to foreground contamination (atmospheric, galactic, extragalactic), calibration uncertainties and spurious effects induced by an



Bounds given in this table are 95% CL.

BAO+FS for row 5 are from SDSS BOSS DR12 [23]. BAO data for rows no. 6–7 are from 6dFGS [100], WiggleZ [101], SDSS BOSS DR11 LOWZ and SDSS BOSS DR11 CMASS [102] (see [98] for details). FS for row no. 8 is from SDSS BOSS DR12 CMASS [103] (see [98] for details). BAO for row no. 9–10 are from 6dFGS [100], SDSS MGS [104], BOSS DR12 [23] (see [27] for details). BAO data for row no. 11 are from 6dFGS [100], SDSS MGS [104], BOSS LOWZ DR11 and BOSS CMASS DR11 [102] (see [14] for details). JLA for row no. 9–10 is the catalog of luminosity distance measurements from the Joint Lightcurve Analysis [105, 106]. WL for row no. 10 is the combination of galaxy, shear and galaxy-galaxy lensing spectra from DES Year1 [27]. SZ in row no. 11 is the SZ cluster count dataset from Ade et al. [24]. Lyα-FS in the last row is the Lyα power spectrum measurement from BOSS [107].

imprecise knowledge of the instrument (see e.g., [108–112] for a sample list of references).

The tightest constraints on 6m<sup>ν</sup> from a single experiment come from the measurements of the Planck satellite [14]. In the context of a one-parameter extension of the 3CDM cosmological model, the state of the art after the 2015 data release was as follows. The combination of the measurements of the CMB temperature anisotropies up to the multipole ℓ ≃ 2, 500 (hereafter, "Planck TT") and the large scale (ℓ < 30) polarization anisotropies (hereafter "lowP") leads to an upper bound of 6m<sup>ν</sup> < 0.72 eV at 95% CL. The inclusion of the small scale (ℓ ≥ 30) polarization measurements (which we globally label as "Planck TE,EE") provides a tighter upper bound of 6m<sup>ν</sup> < 0.49 eV at 95% CL. This latter bound should be regarded as less conservative, as a small level of residual systematics could still affect the small scale polarization data.

The Planck collaboration also provides the most significant measurements of the CMB lensing potential power spectrum for the multipole range 40 < L < 400 (labeled as "lensing") [113]. When this dataset is included in the analysis, the 95% CL constraints on 6m<sup>ν</sup> become: 6m<sup>ν</sup> < 0.68 eV for Planck TT+lowP+lensing and 6m<sup>ν</sup> < 0.59 eV for Planck TT,TE,EE+lowP+lensing [14]. When combining the lensing reconstruction data from Planck with the measurements of the CMB power spectra, it should be kept in mind that CMB power spectra as measured by Planck prefer a slightly higher lensing amplitude than that estimated with the lensing reconstruction. As a result, the bounds on 6m<sup>ν</sup> obtained by their combination have less weight for smaller values of 6m<sup>ν</sup> than the corresponding bounds obtained from CMB power spectra only. Nevertheless, higher values of 6m<sup>ν</sup> are still disfavored.

In 2016, new estimates of the reionization optical depth τ have been published by the Planck collaboration [43], obtained from the analysis of the high-frequency CMB maps, in 2015 still affected by unexplained systematics effects at large scales. The estimated 68% credible interval for τ coming from the EE−only low-ℓ data is τ = 0.055 ± 0.009. This estimate is lower than the corresponding interval obtained in 2015 from the analysis of the low-frequency maps (τ = 0.067 ± 0.023), though the two estimates are well in agreement with each other. The lower value of τ has an impact on the constraints on 6m<sup>ν</sup> , due to the degeneracy between the optical depth and the amplitude of primordial perturbations A<sup>s</sup> , as they together fix the normalization amplitude A<sup>s</sup> e −2τ . A lower τ implies a lower A<sup>S</sup> and thus a lower lensing amplitude, leaving less room for large values of 6m<sup>ν</sup> (that would further reduce lensing). If the "lowP" dataset is replaced by the new estimate of τ (labeled as "SimLow"), the 95% CL bounds improve as follows: 6m<sup>ν</sup> < 0.59 eV for Planck TT+SimLow and 6m<sup>ν</sup> < 0.34 eV for Planck TT,TE,EE+SimLow [43].

### 5.2. Large-Scale Structure Data

Although the CMB is an extremely powerful dataset, multiple degeneracies between cosmological parameters limit the constraining power on 6m<sup>ν</sup> from CMB only, as seen in section 4.4. Measurements of the large scale structures (LSS) can help solving these degeneracies. LSS surveys map the distribution and clustering properties of matter at later times (or equivalently at lower redshift) than those accessible with CMB data and are directly sensitive to cosmological parameters that CMB data can only constrain indirectly, such as the total matter abundance at late times (see e.g., [114] for a review). In this section, we gather constraints on 6m<sup>ν</sup> from different LSS probes alone and in combination with CMB data.

#### 5.2.1. Baryon Acoustic Oscillations and the Full Shape of the Matter Power Spectrum from the Clustering of Galaxies

BAO measurements, obtained by mapping the distribution of matter at relatively low redshifts (z < 3) if compared to the redshifts relevant for CMB, constrain the geometry of the expanding Universe, providing estimates of the comoving angular diameter distance dA(z) and the Hubble parameter H(z) at different redshifts (or an angle-averaged combination of the two parameters, <sup>d</sup>V(z) <sup>=</sup> [zd<sup>2</sup> A (z)/H(z)]1/<sup>3</sup> ). Therefore, BAO constrain cosmological parameters which are relevant for the late-time history of the Universe, helping break the degeneracy between those parameters and 6m<sup>ν</sup> .

BAO extraction techniques rely on the ability to localize the peak of the two-point correlation function of some tracer of the baryon density, or equivalently the locations of the acoustic peaks in the matter power spectrum, thus neglecting the information coming from the broad-band shape of the matter power spectrum itself. In principle, the full shape (FS) of the matter power spectrum is a valuable source of information about clustering properties of the different constituents of the Universe and their reciprocal interactions. In particular, full shape measurements of the power spectrum also provide estimates of the growth of structures at low redshifts through the anisotropies induced by the redshift-space distortions (RSD), usually encoded in the parameter f(z)σ8(z), where f(z) is the logarithmic growth rate and σ8(z) is the normalization amplitude of fluctuations at a given redshift in terms of rms fluctuations in a 8h <sup>−</sup><sup>1</sup> Mpc sphere.

In 2016, the final galaxy clustering data from the Baryon Oscillation Spectroscopic Survey (BOSS) were released, as part of the Sloan Digital Sky Survey (SDSS) III<sup>8</sup> . Joint consensus constraints on dA(z), H(z), and f(z)σ8(z) from BAO and FS measurements at three different effective redshifts (zeff = 0.38, 0.51, 0.61) are employed to derive constraints on 6m<sup>ν</sup> 9 in combination with Planck TT,TE,EE+ lowP [23]. The 95% upper bound is 6m<sup>ν</sup> < 0.16 eV. When relaxing the constraining power coming from CMB weak lensing (through the rescaling of the lensing potential with the lensing amplitude AL) and the RSD (through the rescaling of the f σ<sup>8</sup> parameter with the amplitude A<sup>f</sup> <sup>σ</sup><sup>8</sup> ), the bound degrades up to 6m<sup>ν</sup> < 0.25 eV.

When using the FS measurements, it has to be noted that the constraining power of this dataset is highly reduced if one considers that (1) the majority of information encoded in the FS usually comes from the small-scale region of the power spectrum, where the still imprecisely known non-linearities play a non-negligible role; (2) the exact shape and scale-dependence of the bias b between the observed galaxy clustering and the underlying total matter distribution is still debated. Therefore, it is useful to disentangle BAO and FS measurements, to gauge the relative importance of the two in constraining 6m<sup>ν</sup> . For a thorough comparison between the constraining power of the two datasets, we refer the reader to Vagnozzi et al. [98] (see also [116, 117] for analyses using older data), where the authors focus on recent BAO and FS measurements. Here, we summarize the conclusion of the paper: "The analysis method commonly adopted [for FS measurements] results in their constraining power still being less powerful than that of the extracted BAO signal."

### 5.2.2. Weak Lensing

The most recent weak lensing datasets have been released by the Kilo-Degree Survey (KiDS [26, 118]) and the Dark Energy Survey (DES [27, 119]). It is interesting to note that all of the aforementioned datasets provide results in terms of cosmological parameters which are slightly in tension with the corresponding estimates coming from CMB data (which we remind is a highredshift probe). In particular, the values of <sup>m</sup> and S<sup>8</sup> = σ8(m/0.3)0.5 inferred from weak lensing data are lower than the best fit obtained with CMB data. The significance of this tension is at ∼ 2σ level for KiDS and more than 1σ level for the 1-D marginalized constraints on <sup>m</sup> and S<sup>8</sup> for DES (even though a more careful measure of the consistency between the two datasets in the full parameter space provides "substantial" evidence for consistency, see Abbott et al. [27] for details).

Weak lensing data tend to favor higher values of 6m<sup>ν</sup> than those constrained by CMB power spectrum data. In fact, lower values of <sup>m</sup> and S<sup>8</sup> imply a reduced clustering amplitude, an effect that can be obtained by increasing the sum of neutrino masses. In Abbott et al. [27], the combination of DES shear, galaxy and galaxy-shear spectra with Planck TT+lowP and other cosmological datasets in agreement with CMB results (i.e., BAO from 6dFGS [100], SDSS DR<sup>7</sup> MGS [104], and BOSS DR12 [23], and luminosity distances from the Joint Lightcurve Analysis (JLA) of distant SNIa [105, 106]) yields an upper bound at 95% CL on the sum of the neutrino masses of 6m<sup>ν</sup> < 0.29 eV, almost 20% higher than the corresponding bound obtained dropping DES data (6m<sup>ν</sup> < 0.245 eV). Interestingly enough, the DES collaboration shows that a marginal improvement in the agreement between DES and Planck data is obtained when the sum of the neutrino masses is fixed to the minimal mass allowed by oscillation experiments 6m<sup>ν</sup> = 0.06 eV.

To conclude this section, we also report the upper bound on 6m<sup>ν</sup> obtained by weak lensing only data from the tomographic weak lensing power spectrum as measured by the KiDS collaboration [26]. They found 6m<sup>ν</sup> < 3.3 eV and 6m<sup>ν</sup> < 4.5 eV at 95% CL depending on the number of redshift bins retained in the analysis. These bounds are significantly broader

<sup>8</sup>Recently, the DES collaboration has reported a 4% measurement of the angular diameter distance from the distribution of galaxies to redshift z = 1 [115]. Cosmological constraints are derived in the 3CDM framework, with 6m<sup>ν</sup> fixed to the minimal value of 0.06 eV. Therefore, no bounds on 6m<sup>ν</sup> have been extracted from the BAO measurements from DES yet.

<sup>9</sup>Note that the authors follow the assumption that all the mass is carried by only one of three neutrino species, i.e., m<sup>1</sup> = 6m<sup>ν</sup> , m2,3 = 0 eV, instead of the more widely used fully-degenerate approximation of m<sup>i</sup> = 6m<sup>ν</sup> /3, i = 1, 2, 3 for each of the three neutrino species.

than the constraints coming from CMB only data. Nevertheless, they come from independent cosmological measurements and are still tighter than the constraints coming from kinematic measurements of β decay.

#### 5.2.3. Cluster Counts

An additional low-redshift observable is represented by measurements of the number of galaxy clusters as a function of their mass at different redshifts. Cluster number counts provide a tool to infer the present value of the matter density <sup>m</sup> and the clustering amplitude σ8, to be compared with the equivalent quantities probed at higher redshift by the primary CMB anisotropies.

Depending on the prior imposed on the mass bias, cluster counts tend to prefer lower values of <sup>m</sup> and σ<sup>8</sup> than the corresponding values obtained with primary CMB. The tension between the two datasets can be as high as 3.7σ for the lowest value of the mass bias as quantified by the Planck collaboration in 2015 [24]. Again, this preference for less power in the matter distribution favors higher values of the sum of the neutrino masses. Indeed, the Planck collaboration reports [24] an upper bound of 6m<sup>ν</sup> < 0.20 eV at 95% CL when Planck TT,TE,EE+lowP+BAO is combined with the SZ cluster count dataset (with a prior on the mass bias (1−b) = 0.780±0.092 from the gravitational shear measurements of the Canadian Cluster Comparison Project, CCCP [120]), to be compared with the corresponding 95% upper bound 6m<sup>ν</sup> < 0.17 eV without the SZ cluster count dataset [14].

Recently, Salvati et al. [121] updated constraints on cosmological parameters, including 6m<sup>ν</sup> , from the SZ clusters in the Planck SZ catalog, considering cluster count alone and in combination with the angular power spectrum of SZ sources. A comparison with bounds coming from primary CMB anisotropies is also performed. The combination of the two SZ probes (complemented with BAO measurements from [102] to fix the underlying cosmology) confirms the discrepancy in <sup>m</sup> and σ<sup>8</sup> at the level of 2.1σ and provides an independent upper limit on the sum of the neutrino masses of 6m<sup>ν</sup> < 1.47 eV at 95% CL. When combined with primary CMB, the bound reduces to 6m<sup>ν</sup> < 0.18 eV. This bound is slightly higher than 6m<sup>ν</sup> < 0.12 eV found by Vagnozzi et al. [98] in absence of SZ data, as we should expect due to the aforementioned tension between SZ and primary CMB estimates of matter density and power.

#### 5.2.4. Lyman-α Forests

Like all the datasets that probe the clustering of matter over cosmological distances, the Lyα power spectrum is sensitive to 6m<sup>ν</sup> primarily through the power suppression induced by massive neutrinos at small scales. The Lyα spectrum alone can constrain 6m<sup>ν</sup> at the level of 1 eV (see e.g., [107]). The constraining power of the Lyα spectrum is evident when it is combined with CMB data. In this case, the Lyα data are used for setting the overall normalization of the spectrum through their sensitivity to <sup>m</sup> and σ8, whereas the CMB fixes the underlying cosmological parameters and helps break degeneracies between m, σ8, and 6m<sup>ν</sup> . Recently, Yèche et al. [122] reported constraints on 6m<sup>ν</sup> from the combination of the one-dimensional (i.e., angle-averaged) Lyα power spectra from the SDSS III-BOSS collaboration and from the VLT/XSHOOTER legacy survey (XQ- 100). When the power spectra are used alone [complemented with a Gaussian prior on H<sup>0</sup> = (67.3 <sup>±</sup> 1.0) km s−<sup>1</sup> Mpc−<sup>1</sup> ], the authors obtain 6m<sup>ν</sup> < 0.8 eV at 95% CL. The bounds dramatically improve to 6m<sup>ν</sup> < 0.14 eV when CMB power spectrum data from Planck TT+lowP are added to the analysis. The tightest bound on 6m<sup>ν</sup> from Lyα power spectrum comes from Palanque-Delabrouille et al. [107], with 6m<sup>ν</sup> < 0.12 eV from Planck TT+lowP in combination with the Lyα flux power spectrum from BOSS-DR12. Interestingly enough, in both analyses, the limit set by Lyα+Planck TT+lowP does not further improve when the Lyα spectra are combined instead with the full set of CMB data from Planck, including small-scale CMB polarization (Planck TT,TE,EE+lowP), and with BAO data from 6dFGS, SDSS MGS, BOSS-DR11.

The BAO signal can be also extracted from the Lyα spectrum (see [123] for a pivotal study), providing estimates of the comoving angular diameter distance dA(z) and of the Hubble parameter H(z) at redshift z ≃ 2. Recently, the SDSS III-BOSS DR12 collaboration reported measurements of the BAO signal at z = 2.33 from Lyα forest [99]. The estimated values of d<sup>A</sup> and H are in agreement with a 3CDM model (even though a slight tension with Planck primary CMB is present), although their precision is smaller than the precision obtained with galaxyderived BAO measurements. Therefore, at present, the impact of Lyα-BAO data on simple extensions of the 3CDM model is minimal.

We conclude that it is a conservative choice to take the constraints coming from Lyα with some caution (a similar comment applies to constraints coming from aggressive analyses of the broadband shape of the matter power spectrum from galaxy surveys), until this probe will reach the level of maturity comparable with other traditional cosmological probes.

### 5.3. Local Measurements of the Hubble Constant and Supernovae Ia

The most recent estimate of the Hubble constant has been reported in Riess et al. [93]. The authors improved over their previous measurement of H<sup>0</sup> from 3.3 to 2.4% thanks to an increased sample of reliable SNIa in nearby galaxies calibrated over Cepheids. Their final estimate, based on the combination of three different anchors, is <sup>H</sup><sup>0</sup> <sup>=</sup> (73.24 <sup>±</sup> 1.74) km s−<sup>1</sup> Mpc−<sup>1</sup> , 3.2σ higher than the indirect estimate of H<sup>0</sup> from Planck TT+SimLow (3.4σ higher than Planck TT,TE,EE+SimLow) in the context of a 3CDM cosmology with 6m<sup>ν</sup> = 0.06 eV. Previous analyses from the same authors also pointed to a ∼ 2σ tension between direct measurements of H<sup>0</sup> and indirect estimate from primary CMB anisotropies from Planck (although see [124] for a re-analysis of the same dataset which slightly reduces the discrepancy to within 1σ agreement). A discussion about the possible reasons behind this discrepancy and ways to alleviate it invoking non-standard cosmological scenarios are beyond the scope of this work. We refer the reader to the dedicated works [62, 125, 126] for further reading.

Since the Hubble constant and the sum of neutrino masses are anti-correlated, given the tension between the two probes it is clear that the combination of direct measurements of H<sup>0</sup> with CMB data leads to a preference for smaller values of 6m<sup>ν</sup> with respect to CMB-only constraints. Indeed, several authors have pointed out the tight constraints on 6m<sup>ν</sup> for such a combination. As an example, Vagnozzi et al. [98] showed that constraints on 6m<sup>ν</sup> can be as tight as 6m<sup>ν</sup> < 0.148 eV at 95% CL when Planck TT+lowP+BAO are complemented with a Gaussian prior on H<sup>0</sup> equal to the estimate of the Hubble constant in Riess et al. [93], to be compared with 6m<sup>ν</sup> < 0.186 eV from Planck TT+lowP+BAO only. When lowP is replaced by a Gaussian prior on τ compatible with the new estimates from SimLow, these numbers change to 6m<sup>ν</sup> < 0.115 eV (6m<sup>ν</sup> < 0.151 eV) with

(without) the H<sup>0</sup> prior. For the sake of completeness, we shall also mention that independent estimates of H<sup>0</sup> from BAO measurements conducted by the SDSS III-BOSS DR12 collaboration [23] are in agreement with CMB estimates (see also [62] for a recent discussion). See also Abbott et al. [127] for an additional independent estimate of H<sup>0</sup> with a combination of clustering and weak lensing measurements from DES-Y1 with BAO and BBN data. A discussion about the combination of different measurements of H<sup>0</sup> from cosmological probes and local measurements is also reported in Abbott et al. [127], Vega-Ferrero et al. [128].

Finally, we report that a standard siren measurement of H<sup>0</sup> has been performed after the detection of the neutron star-neutron star merger GW170817 [129–131]. The Hubble constant has been constrained as <sup>H</sup><sup>0</sup> <sup>=</sup> 70.0+12.0 <sup>−</sup>8.0 km s−<sup>1</sup> Mpc−<sup>1</sup> at 68% CL. The accuracy of this determination is not comparable with the precise estimates of direct measurements and other cosmological constraints. However, the standard siren approach represents an additional independent estimate of H<sup>0</sup> and appears as a promising avenue as more GW events with electromagnetic counterparts are detected.

Concerning the inclusion of SNIa, the bounds from Planck TT+lowP improve from 6m<sup>ν</sup> < 0.72 eV to 6m<sup>ν</sup> < 0.33 eV at 95% CL when data from the Joint Lightcurve Analysis [105, 106] are included10. The most relevant systematics that affect SNIa measurements are related to the way in which SNIa light curves are standardized, with issues mostly arising from photometric calibrations and lightcurve fitting procedures.

### 6. CONSTRAINTS ON 6m<sup>ν</sup> FROM FUTURE SURVEYS

In this section, we will discuss the expected improvements in the constraints on 6m<sup>ν</sup> from the upcoming generation of CMB and LSS surveys. These constraints are also summarized in **Table 2** for the reader's convenience.



<sup>a</sup>The combination assumes a Gaussian prior on <sup>τ</sup> <sup>=</sup> 0.06 <sup>±</sup> 0.01 roughly corresponding to the new estimate from Aghanim et al. [43].

<sup>b</sup>The combination assumes <sup>σ</sup>(<sup>τ</sup> ) <sup>=</sup> 0.002 and noise level of 2.5µ<sup>K</sup> · arcmin.

<sup>c</sup>For a fiducial value <sup>6</sup>m<sup>ν</sup> <sup>=</sup>0 eV and marginalizing over dynamical dark energy, arbitrary curvature and Neff .

Unless otherwise stated, the sensitivity σ(6m<sup>ν</sup> ) is forecasted assuming a standard cosmological model with 6m<sup>ν</sup> = 0.06 eV. DESI refers to the simulated DESI-BAO dataset based on expected experimental performances [35] (see [30, 132] for details). FS refers to the use of the (simulated) measurements of the full shape of the matter power spectrum. The last line implies the use of CMB lensing, Euclid and WFIRST to calibrate the multiplicative bias in the shear measurements from LSST [38].

## 6.1. CMB Surveys: CORE and CMB Stage-IV

The tightest bounds on 6m<sup>ν</sup> from a single CMB experiment are those from the Planck satellite, reported in section 5.1. As already explained, this sensitivity mostly comes from the ability to (1) detect, at the level of CMB power spectrum, the smoothing effect of gravitational lensing of CMB photons, and, (2) directly reconstruct the lensing power spectrum itself. These effects arise at small angular scales (higher multipoles ℓ), therefore it is crucial to observe this region of the power spectrum with high accuracy in order to improve the sensitivity on 6m<sup>ν</sup> . Improved measurements of the polarization power spectra at all scales are also important to break degeneracies between cosmological parameters. The main example is the effect that a better estimate of the reionization optical depth τ from the large scale polarization spectrum has on 6m<sup>ν</sup> . Concerning the lensing power spectrum, this is internally reconstructed by the Planck collaboration with high statistical significance up to intermediate scales. However, the full power of this probe will be definitively unveiled when better measurements of polarization maps are available, enabling reconstruction from E-B estimators with lower variance and up to smaller scales [57].

A detailed summary of the expected sensitivity to cosmological parameters, including 6m<sup>ν</sup> , of all pre-2020 and post-2020 CMB missions can be found in Errard et al. [134]. As relevant examples, in this section we focus on two classes of future (post 2020) CMB experiments: a space mission and a ground based telescope.

<sup>10</sup>Bounds from the Planck Legacy Archive: https://wiki.cosmos.esa.int/ planckpla2015/index.php/Cosmological\_Parameters

Recently, a proposal for a future CMB space mission has been submitted to the European Space Agency (ESA) in response to a call for medium-size mission proposals (M5). The mission, named Cosmic ORigin Explorer (CORE), is designed to have 19 frequency channels in the range 60 − 600 GHz for simultaneously solving for CMB and foreground signals, angular resolution in the range 2′ − 18′ depending on the frequency channel and aggregate sensitivity of 2µK · arcmin [32] (for comparison, the Planck satellite has 9 frequency channels in the range 30 − 900 GHz, angular resolution in the range 5′ − 33′ and the most sensitive channel shows a temperature noise of 0.55µK · deg at 143 GHz [135]). This experimental setup would enable to constrain <sup>6</sup>m<sup>ν</sup> <sup>=</sup> (0.072+0.037 <sup>−</sup>0.051) eV at 68% CL assuming a 3CDM model with a fiducial value of the sum of the neutrino masses 6m<sup>ν</sup> = 0.06 eV, for the combination of CORE TT,TE,EE,PP (temperature and E-polarization auto and cross spectra and lensing power spectrum PP) [132]. This roughly corresponds to a sensitivity of σ(6m<sup>ν</sup> ) ∼ 0.044 eV (note that the target threshold for a 3σ detection in the minimal mass scenario is σ(6m<sup>ν</sup> ) = 0.020 eV; for comparison, a simulated Planck-like experiment could only put an upper limit of 6m<sup>ν</sup> < 0.315 eV at 68% CL for the same model). Other than to the capability of measuring with high precision the small scale polarization (also in order to reconstruct the lensing potential), part of this high sensitivity also comes from the improved limits that a science mission like CORE can put on τ : compared to Planck, CORE would achieve an almost cosmic-variance-limited (CVL) detection of the reionization optical depth [σCVL(τ ) ≃ 0.002].

A roadmap towards a Stage-IV (S4) generation of CMB ground-based experiments<sup>11</sup> has been also developing [30]. The goal is to set a definitive CMB experiment with ∼250,000 detectors surveying half of the sky, with angular resolution of 1 ′ −2 ′ and a sensitivity of 1µK · arcmin at 150 GHz. The greatest contaminant for a ground-based experiment is the atmospheric noise, which highly reduces the accessible frequencies for CMB observations to a total of four windows, roughly 35, 90, 150, and 250 GHz. The main advantages with respect to a spaceborne mission are a larger collecting area with an incredibly higher number of detectors (for a comparison, the CORE proposal accounts for a total of 2,100 detectors [32], the Planck satellite has 74 detectors [135]) and subsequent suppression of experimental noise. At large scales, the Stage-IV target is the recombination bump at ℓ > 20. The reduced sky fraction accessible from ground, foreground contaminations and atmospheric noise are the main issues that limit the possibility to target also the range ℓ < 20. Therefore, it is likely that S4 would be complemented by balloon-based and satellite-based measurements at the largest scales. As a result, forecasts for S4 relies on external measurements of τ . The sensitivity σ(6m<sup>ν</sup> ) of S<sup>4</sup> TT,TE,EE,PP complemented with a Gaussian prior on the optical depth of τ = 0.06 ± 0.01 (roughly corresponding to the latest estimate from Planck-HFI [43]) is in the range [0.073 − 0.110] eV, depending on the angular resolution and noise level, for fsky = 40% [30].

Neither of the two classes of future CMB mission proposals can achieve alone the necessary sensitivity to claim a detection of 6m<sup>ν</sup> = 0.06 eV at the 3-σ level. Nevertheless, we will see in the next section that the combination of future CMB missions with future galaxy surveys could possibly lead to the first detection of neutrino masses from cosmology.

### 6.2. Future LSS Surveys: DESI, Euclid, LSST, WFIRST

Improved performances from future galaxy surveys with respect to the current status can be achieved by mapping a larger volume of the sky, therefore increasing the number of samples observed and going deeper in redshift. In this section, we will briefly review the expected performances of the main Stage-IV LSS surveys.

The successor to SDSS III-BOSS survey will be the groundbased Dark Energy Spectroscopic Instrument<sup>12</sup> (DESI). It is designed to operate for 5 years and cover roughly a 14,000 deg<sup>2</sup> survey area. The extension in redshift is expected to be up to z = 1 for Luminous Red Galaxies (LRG), z = 1.7 for Emission Line Galaxies (ELG) and z = 3.5 for Lyα forests, for a total of over 20 million galaxy and quasar redshifts. With these numbers, DESI will improve over the BOSS survey by an order of magnitude in both volume covered and number of objects observed. It can achieve a 3.49% and 4.78% determination of the BAO signal across (dA/rd) and along (Hrd) the line-of-sight, respectively, at z = 1.85, and 16% and 9% determination of the same quantities at the highest redshift achievable with Lyα forest z = 3.55 [35]. Even in the most conservative scenario when DESI BAO only (i.e., without including information from the broadband shape of the matter power spectrum and Lyα forests) are combined with future CMB experiments, the sensitivity on 6m<sup>ν</sup> greatly improves. It goes down to σ(6m<sup>ν</sup> ) = 0.021 eV for CORE TT,TE,EE,PP+DESI BAO, forecasting a ∼ 3σ detection of 6m<sup>ν</sup> in the minimal mass scenario [132]. In the case of S4+DESI BAO [30], σ(6m<sup>ν</sup> ) is in the range [0.023 − 0.036] eV ( or [0.020 − 0.032] eV) with a prior of τ = 0.06 ± 0.01 (or τ = 0.060 ± 0.006, the expected sensitivity from Planck-HFI [136]) and fsky = 0.40, depending on the S4 angular resolution and noise level. For a 1′ resolution and a noise level lower than 2.5µK · arcmin, σ(6m<sup>ν</sup> ) could be further improved with a better measurement of τ down to the level of σ(6m<sup>ν</sup> ) < 0.015 eV, that would guarantee a > 4σ detection of 6m<sup>ν</sup> in the minimal mass scenario.

The DESI mission will be complementary to the science goals of the Large Synoptic Survey Telescope<sup>13</sup> (LSST), a Stage-IV ground-based optical telescope. The main science fields in which LSST will mostly operate are [36]: "Inventory of the Solar System, Mapping the Milky Way, Exploring the Transient Optical Sky, and Probing Dark Energy and Dark Matter". These goals will be achieved by surveying a <sup>∼</sup>30,000 deg<sup>2</sup> area (2/3 of which in a "deep-wide-fast" survey mode) over 10 years, in six bands (ugrizy), with incredible angular resolution (∼ 0.7′′), producing measurements of roughly 10 billion stars and galaxies. Thanks to its peculiar observational strategy, LSST will provide multiple probes of the late-time evolution of the Universe with

<sup>11</sup>https://cmb-s4.org

<sup>12</sup>http://desi.lbl.gov

<sup>13</sup>https://www.lsst.org

a single experiment, namely, weak lensing cosmic shear, BAO in the galaxy power spectrum, evolution of the mass function of galaxy clusters, and a compilation of SNIa redshift-distances. The expected sensitivity on 6m<sup>ν</sup> [36] is in the range σ(6m<sup>ν</sup> ) = [0.030 − 0.070] eV, depending on the fiducial value of 6m<sup>ν</sup> assumed when performing forecasts (6mfid <sup>ν</sup> = [0 − 0.66] eV). Larger fiducial values for the mass yield better sensitivity. These numbers include a marginalization over the uncertainties coming from an extended cosmological scenario, where a number of relativistic species different than 3.046, a non-zero curvature and a dynamical dark energy component are allowed. They also take into account the combination of the three-dimensional cosmic shear field as measured by a LSST-like survey with Planck-like CMB data and can be improved by a factor of 2 if either BAO or SNIa measurements are also considered, whereas a factor of √ 2 degradation could come from systematic effects. Interestingly enough, the observational strategy of LSST (large and deep survey) could provide the necessary sensitivity to explore the faint effects that the distinct neutrino mass eigenstates have on cosmological probes. This is a highly debated topic and we refer the reader to section 8 for related discussion.

Synergy between these large ground-based observatories and future space missions is expected. We consider here the ESA Euclid satellite<sup>14</sup> and the NASA Wide Field Infrared Survey Telescope<sup>15</sup> (WFIRST) as representative space-borne missions. Euclid will be a wide-field satellite that operates with imaging and spectroscopic instruments for 6 years and covers roughly 15,000 deg<sup>2</sup> in the optical and near-infrared bands, observing a billion galaxies and measuring ∼100 million galaxy redshifts [37]. The redshift depth will be up to z ∼ 2 for galaxy clustering and up to z ∼ 3 for cosmic shear. The combination of the galaxy power spectrum measured with Euclid and primary CMB from Planck is expected to give σ(6m<sup>ν</sup> ) = 0.04 eV; if instead the weak lensing dataset produced by Euclid is considered in combination with primary CMB, we expect σ(6m<sup>ν</sup> ) = 0.05 eV [133]. Both combinations provide a ∼ 1σ evidence in the minimal mass scenario. Some authors have also pointed out that weak lensing data as measured by Euclid could discriminate between the two neutrino hierarchies if the true value of 6m<sup>ν</sup> is small enough (i.e., far enough from the degenerate region of the neutrino mass spectrum), see [133] and references therein<sup>16</sup> .

WFIRST is an infrared telescope with a primary mirror as wide as the Hubble Space Telescope's primary (2.4 m) and will operate for 6 years [38]. The primary instrument on board, the Wide Field Instrument, will be able to operate both in imaging and spectroscopic mode, observing a billion galaxies. The instrumental characteristics of WFIRST will more than double the surface galaxy density measured by Euclid. With this setup, WFIRST will test the late expansion of the Universe with great accuracy employing supernovae, weak lensing, BAO, redshift space distortions (RSD), and clusters as probes. From the BAO and broadband measurements of the matter power spectrum, WFIRST in combination with a Stage-III CMB experiment could provide σ(6m<sup>ν</sup> ) < 0.03 eV [38].

We want to conclude this section by pointing out that the aforementioned missions will be extremely powerful if combined together. Indeed, they are quite complementary [137]. A significant example concerning the improvement of constraints on massive neutrinos is the combination of all the previously discussed surveys with the lensing reconstruction from CMB. The cross correlation of weak lensing (optical), CMB lensing power spectrum and galaxy clustering (spectroscopic) can highly reduce the systematics affecting each single probe, in particular the multiplicative bias in cosmic shear [138]. For example, a combination of WFIRST, Euclid, LSST, and CMB Stage-III can achieve σ(6m<sup>ν</sup> ) < 0.01 eV [38]. Another example is the calibration of the cluster mass for SZ cluster count analyses. This calibration can be performed through optical surveys such as LSST or through CMB lensing calibration, with comparable results. In Madhavacheril et al. [139], the authors show that lensing-calibrated SZ cluster counts can provide a detection of the minimal neutrino mass 6m<sup>ν</sup> at > 3σ level, also in extended cosmological scenarios.

#### 6.3. 21-cm Surveys

In this section, we will briefly comment about the possibility to use 21-cm survey data to constrain 6m<sup>ν</sup> . We refer the reader to the relevant papers for further readings. Measurements of the 21-cm signal such as those expected from the Square Kilometer Array<sup>17</sup> (SKA) and the Canadian Hydrogen Intensity Mapping Experiment<sup>18</sup> (CHIME) can shed light on the Epoch of Reionization, including a better determination of the reionization optical depth τ . In addition, they map the distribution of neutral hydrogen in the Universe, a tracer of the underlying matter distribution. Therefore, constraints on 6m<sup>ν</sup> can benefit from 21-cm measurements in two ways: by breaking the degeneracy between 6m<sup>ν</sup> and τ (see e.g., [140], where the authors report σ(6m<sup>ν</sup> ) = 0.012 eV for a combination of CORE+Euclid lensing and FS+ a prior on τ compatible with expectations from future 21-cm surveys); by detecting the effect of 6m<sup>ν</sup> on the evolution of matter perturbations (see e.g., [141–143]).

### 7. CONSTRAINTS ON 6m<sup>ν</sup> IN EXTENDED COSMOLOGICAL SCENARIOS

The constraints reported so far apply to the simple one-parameter extension of the standard cosmological model, 3CDM + 6m<sup>ν</sup> . When derived in the context of more complicated scenarios, such as models that allow arbitrary curvature and/or nonstandard dark energy models and/or modified gravity scenarios etc., constraints on 6m<sup>ν</sup> are expected in general to degrade (although tighter constraints on 6m<sup>ν</sup> can be also possible in particular extended scenarios) with respect to those obtained in a 3CDM + 6m<sup>ν</sup> cosmology. This effect is due to the multiple degeneracies arising between cosmological parameters

<sup>14</sup>https://www.euclid-ec.org

<sup>15</sup>https://wfirst.gsfc.nasa.gov

<sup>16</sup>Note that the specifics of the Euclid mission have changed since the time when [133] was published. The new specifics are not publicly available, however the Euclid collaboration is expected to release updated forecasts in the near future.

<sup>17</sup>http://skatelescope.org

<sup>18</sup>https://chime-experiment.ca

that describe the cosmological model under scrutiny. In other words, when more degrees of freedom are available—in terms of cosmological parameters that are not fixed by the model—, more variables can be tuned in order to adapt the theoretical model to the data. For example, CMB data measure with incredible accuracy the location (expressed by the angular size of the horizon at recombination θs) and amplitude (basically driven by the exact value of zeq) of the first acoustic peak. Therefore, we want to preserve this feature in any cosmological model. As explained before, h, m, and 6m<sup>ν</sup> can be varied together in order to do this. Adding other degrees of freedom, like curvature or evolving dark energy, allows for even more freedom, thus making the degeneracy worse. Of course, the addition of different cosmological data, which are usually sensitive to different combinations of the aforementioned parameters, is extremely helpful in tightening the constraints on 6m<sup>ν</sup> (and, in general, on any other cosmological parameter) in complex scenarios.

In more detail, constraints on the sum of neutrino masses are particularly sensitive to the so-called "geometric degeneracy." This term refers to the possibility of adjusting the parameters in order to keep constant the angle subtended by the sound horizon at last scattering, that controls the position of the first peak of the CMB anisotropy spectrum. The degeneracy is worsened in models with a varying curvature density <sup>k</sup> or parameter of the equation of state of dark energy w. Constraints on the expansion history, like those provided by BAO or by direct measurements of the Hubble constant, are particularly helpful in breaking the geometric degeneracy. In principle, one could also expect a degeneracy between the effective number of degrees of freedom Neff and 6m<sup>ν</sup> , but for a different reason: both parameters can be varied in order to keep constant the redshift of matter-radiation equality. However, this can be done only at the expense of changing the CMB damping scale (see section 10 for further details). High-resolution measurements of the CMB anisotropies are therefore a key to partially break the degeneracy. Finally, a non-standard relation between the matter density distribution and the lensing potential can be modeled by introducing a phenomenological parameter AL, which modulates the amplitude of the lensing signal [144]. Most of the current constraining power of CMB experiments on 6m<sup>ν</sup> comes from CMB lensing. Therefore, it is clear that in models with varying A<sup>L</sup> the limits on neutrino masses are strongly degraded. However, it should also be noted that A<sup>L</sup> is usually introduced as a proxy for instrumental systematics; if considered as an actual physical parameter, its value is fixed by general relativity to be A<sup>L</sup> = 1.

To make the discussion more quantitative, we see how this applies to the constraints obtained with present data and future data. In **Table 3**, we report a comparison of the constraints on 6m<sup>ν</sup> for some extensions of the 3CDM model. In the upper part of the table, we report constraints obtained from the PlanckTT+lowP+lensing+BAO dataset combination, described in section 5.1. These are taken from the full grid of results made available by the Planck collaboration<sup>19</sup> and have been



<sup>a</sup>From the Planck 2015 Explanatory Supplement Wiki.

Ω<sup>K</sup> is the curvature density parameter, w is the (constant) equation of state parameter for the dark energy, Neff is the number of relativistic species at recombination, A<sup>L</sup> is the phenomenological rescaling of the lensing power that smears the CMB power [144], YHe is the primordial Helium abundance, r is the tensor to scalar ratio. Upper section: 95% CL constraints from the full grid of results from the Planck collaboration (see text for details). BAO data are from 6dFGS, SDSS MGS, BOSS LOWZ DR11, and BOSS CMASS DR11 (see [14] for details). Lower section: Forecasted 68% CL constraints from Di Valentino et al. [132]. BAO refers to simulated data for DESI and Euclid surveys. The fiducial model adopted for the analysis is the following: 6m<sup>ν</sup> = 0.06 eV, Ω<sup>K</sup> = 0, w = −1, Neff = 3.046, YHe = 0.24, r = 0.

obtained with the same statistical techniques used for the 3CDM model. We see that the constraints are degraded by 30% in models with varying Neff, by 50% in models with varying <sup>K</sup> or w, and by 65% in models with varying AL. This information is also conveyed, for an easier visual comparison, in **Figure 4**, where we show the sum of neutrino masses as a function of the mass mlight of the lightest eigenstate. The green and red curves are for normal and inverted hierarchy, respectively. We show 95% constraints on 6m<sup>ν</sup> for different models and dataset combinations as horizontal lines. In the lower section of **Table 3** we instead report a similar comparison, based on the expected sensitivities of future CMB and LSS probes [132]. The pattern is very similar to that observed for present data, although it should be noted that the increased precision of future experiments will allow to further reduce the degeneracies. In particular, it is found that the constraints on 6m<sup>ν</sup> are degraded by ∼30% in models with varying <sup>K</sup> or w, and not degraded at all in models with varying Neff (models with varying A<sup>L</sup> have not been considered in Di Valentino et al. [132]).

The cases reported in **Table 3** hardly exhaust all the possible, well-motivated extensions to the 3CDM + 6m<sup>ν</sup> model. To make a few examples of more complicated extensions, without the aim of being complete, the interplay between inflationary parameters and the neutrino sector has been investigated in Gerbino et al. [145] and Di Valentino et al. [146]. In Di Valentino et al. [147–149] "extended parameter spaces" are considered, in which 12 parameters, including 6m<sup>ν</sup> , are varied simultaneously.

<sup>19</sup>The full grid can be downloaded from the Planck Legacy Archive.

Neutrino-dark matter interactions are discussed in Di Valentino et al. [150], while low-reheating scenarios are studied in de Salas et al. [151]. Finally, constraints on 6m<sup>ν</sup> in the context of cosmological models with time-varying dark energy are derived for example in Lorenz et al. [152] and Yang et al. [153]. Neutrino masses in interacting dark energy-dark matter models and in extended neutrino models (including neutrino viscosity, anisotropic stress and lepton asymmetry) have instead been considered in Kumar and Nunes [154] and in Nunes and Bonilla [155].

### 8. COSMOLOGY AND THE NEUTRINO MASS HIERARCHY

Cosmology is mostly sensitive to the total energy density in neutrinos, directly proportional to the sum of the neutrino masses 6m<sup>ν</sup> ≡ m<sup>1</sup> + m<sup>2</sup> + m3. We can express 6m<sup>ν</sup> in the two hierarchies as a function of the lightest eigenstate mlight (either m<sup>1</sup> or m3) and of the squared mass differences 1m<sup>2</sup> <sup>12</sup> and <sup>1</sup>m<sup>2</sup> 13:

$$\begin{split} \Sigma m\_{\nu}^{\mathrm{NH}} &= m\_{\mathrm{light}} + \sqrt{m\_{\mathrm{light}}^{2} + \Delta m\_{12}^{2}} \\ &+ \sqrt{m\_{\mathrm{light}}^{2} + |\Delta m\_{13}^{2}|} \end{split} \tag{36}$$

$$\begin{split} \Sigma m\_{\upsilon}^{IH} &= \, m\_{\text{light}} + \sqrt{m\_{\text{light}}^2 + |\Delta m\_{13}^2|} \\ &+ \sqrt{m\_{\text{light}}^2 + |\Delta m\_{13}^2| + \Delta m\_{12}^2} \end{split} \tag{37}$$

When stating that oscillation experiments are insensitive to the absolute mass scale, one refers to the fact that the value of mlight is not accessible with oscillation data. When mlight = 0 eV, one obtains 6mNH <sup>ν</sup> <sup>≃</sup> 0.06 eV and <sup>6</sup>mIH <sup>ν</sup> ≃ 0.1 eV. Therefore, for each hierarchy, a minimum mass scenario exists in which 6m<sup>ν</sup> 6= 0.

It has been a long-standing issue whether or not cosmological probes are sensitive to the neutrino mass hierarchy. In principle, we expect physical effects on cosmological observables due to the choice of the neutrino hierarchy. Individual neutrino species that carry a slightly different individual mass exhibit a slightly different free-streaming scale kfs: depending on their individual mass, neutrinos can finish suppressing the matter power at different epochs, leaving three distinct "kinks" in the matter power spectrum. As a consequence, the weak lensing effects on the CMB and on high redshift galaxies can be slightly affected by the choice of the hierarchy. In practice, all of these signatures are at the level of permille effects on the matter and CMB power spectra, well below the current sensitivity [156].

Given the current sensitivity (roughly 6m<sup>ν</sup> < 0.2 eV at 95% CL), it is then a legitimate assumption to approximate the mass spectrum as perfectly degenerate (m<sup>i</sup> = 6m<sup>ν</sup> /3) when performing analysis of cosmological data. Very recently, several authors investigated the possibility that such an approximation could fail reproducing the physical behavior of massive neutrinos when observed with the high sensitivity of future cosmological surveys [132, 145, 157, 158]. In addition, the issue of whether future surveys could unravel the unknown hierarchy has been addressed by several groups [98, 158–162]. We refer the reader to the relevant papers for a thorough discussion of these issues. Here, we summarize the main results: (1) the sensitivity of future experiments will not be enough to clearly separate the effects of different choices of the neutrino hierarchy, for a given value of 6m<sup>ν</sup> ; therefore the fully-degenerate approximation is still a viable way to model the neutrino mass spectrum in the context of cosmological analysis; (2) the possibility to clearly identify the neutrino hierarchy with future cosmological probes is related to the capability of measuring 6m<sup>ν</sup> < 0.1eV at high statistical significance, in order to exclude the IH scenario. It is clear that the possibility to do this strongly depends on the true value of 6m<sup>ν</sup> : the closer it is to 6mNH <sup>ν</sup> = 0.06 eV, the larger will be the statistical significance by which we can exclude IH. This is true independently of whether we approach the issue from a frequentist or Bayesian perspective. In the latter case, however, since a detection of the hierarchy would be driven by volume effects, this posits the question of what is the correct prior choice for 6m<sup>ν</sup> . The issue is extensively discussed in Gerbino et al. [159], Simpson et al. [163], Schwetz et al. [164], Caldwell et al. [165], Long et al. [166], and Hannestad and Tram [167].

### 9. COMPLEMENTARITY WITH LABORATORY SEARCHES

Cosmological observables are ideal probes of the neutrino absolute mass scale, though they are not the only probes available. In fact, laboratory avenues such as kinematic measurements in β-decay experiments (see e.g., [168]) and neutrino-less double-β decay (0ν2β) searches (see e.g., [169, 170]) provide complementary pieces of information to those brought by cosmology.

Kinematic measurements are carried on with β-decay experiments mostly involving <sup>3</sup>H. The shape of the decay spectrum close to the end point is sensitive to the (electron) neutrino mass and can be parametrized in terms of constraints on the electron neutrino effective mass<sup>20</sup> defined in Equation (12). The current best limits on mβ come from the Troitzk and Mainz experiments, with m<sup>β</sup> < 2.05 eV [1] and m<sup>β</sup> < 2.3 eV [2] at 95% CL. The new generation <sup>3</sup>H β-decay experiment KATRIN (Karlsruhe Tritium Neutrino21) is expected either to reach a sensitivity of m<sup>β</sup> < 0.2 eV at 90% CL, an order of magnitude improvement with respect to current sensitivities, or to detect the neutrino mass if it is higher than m<sup>β</sup> = 0.35 eV. Note that a detection of non-zero neutrino mass in KATRIN would imply 6m<sup>ν</sup> & 1 eV, and would then be in tension with the cosmological constraints obtained in the framework of the 3CDM model. This could point to the necessity of revising the standard cosmological model, although it should be noted that none of the simple oneparameter extensions reported in **Table 3** could accommodate for such a value.

Future improvements in kinematic measurements involve technological challenges, since KATRIN reaches the experimental limitations imposed to an experiment with spectrometers. Future prospects are represented by the possibility of calorimetric measurements of <sup>136</sup>Ho (HOLMES experiment [171]) and measurements of the <sup>3</sup>H decay spectrum via relativistic shift in the cyclotron frequency of the electrons emitted in the decay (Project8 experiment<sup>22</sup> [172]). Although the bounds coming from β-decay experiments are very loose compared to bounds from cosmology, nevertheless they are appealing for the reason that they represent model-independent constraints on the neutrino mass scale, only relying on kinematic measurements.

0ν2β decay is a rare process that is allowed only if neutrinos are Majorana particles. A detection of 0ν2β events thus would solve the issue related to the nature of neutrinos, whether they are Dirac or Majorana particles. Searches for 0ν2β directly probe the number of 0ν2β events, which is related to the half life T1/<sup>2</sup> of the isotope involved in the decay. The half life can be translated in limits on the Majorana mass mββ (defined in Equation 13) once a nuclear model has been specified. In practice, a bound on T1/<sup>2</sup> is reflected in a range of bounds on mββ, due to the large uncertainties associated with the exact modeling of the nuclear matrix elements. Additional complications are due to model dependencies: when translating bounds on T1/<sup>2</sup> to bounds on mββ, a mechanism responsible for the 0ν2β decay has to be specified. This is usually the exchange of light Majorana neutrinos, though alternative mechanisms could be responsible for the lepton number violation that not necessarily allow a direct connection between T1/<sup>2</sup> and mββ. Finally, it can be shown that in the case of NH, disruptive interference between mixing parameters could prevent a detection of 0ν2β

events, regardless of the neutrino nature and the lepton-number violation mechanism.

We report here some of the more recent limits on mββ from 0ν2β searches. Constraints are reported as a range of 90% CL upper limits, due to the uncertainty on the nuclear matrix elements. We also specify the isotope used in each experiment. The current bounds are mββ < 0.120 − 0.270 eV from Gerda Phase-II (76Ge) [173, 174], <sup>m</sup>ββ <sup>&</sup>lt; 0.061 <sup>−</sup> 0.165 eV from KamLAND-Zen [175] (136Xe), <sup>m</sup>ββ <sup>&</sup>lt; 0.147 <sup>−</sup> 0.398 eV from EXO-200 (136Xe) [176], <sup>m</sup>ββ <sup>&</sup>lt; 0.140 <sup>−</sup> 0.400 eV from CUORE ( <sup>130</sup>Te) [177]. The next generation 0ν2β experiments, such as LEGEND, SuperNEMO, CUPID, SNO+, KamLAND2-Zen, nEXO, NEXT, PANDAX-III, aims to cover the entire region of IH, reaching a 3σ discovery sensitivity for mββ of 20 meV or better, roughly an order of magnitude improvement with respect to the current limits (see [178] for a more detailed discussion and for a full list of references).

As outlined above, laboratory searches and cosmology are sensitive to different combinations of neutrino mixing parameters and individual masses. Therefore, it makes sense to compare their performances in terms of constraints on the neutrino mass scale. It is also beneficial to combine these different probes of the mass scale, in order to overcome the limitations of each single probe and increase the overall sensitivity to the neutrino masses [40, 165, 179]. This is possible because, once the elements of the mixing matrix are known, specifying one of three mass parameters among (mβ, mββ, 6m<sup>ν</sup> ), together with the solar and atmospheric mass splittings, uniquely determines the other two. Oscillation experiments measure precisely the values of the mixing angles and of the squared mass differences, with an ambiguity on the sign of 1m<sup>2</sup> <sup>31</sup>, so that these parameters can be simply fixed to their best-fit values, given the larger uncertainties on the absolute mass parameters. The value of the Dirac phase, on the other hand, is known with lesser precision, and the Majorana phases, relevant for the interpretation of 0ν2β searches, are not probed at all by oscillation experiments. However this ignorance can be folded into the analysis using standard statistical techniques. Finally, the relation between the mass parameters also depends on the mass hierarchy. This can be taken into account either by performing different analyses for NH and IH, or by marginalizing over the hierarchy itself (see e.g., [159]).

Combining the different probes of the absolute mass scale, with the support of oscillation results, leads to some interesting considerations. First of all, basically all of the information on the absolute mass scale comes from cosmology and 0ν2β searches. This confirms the naive expectation that can be made by comparing the sensitivity of the different probes. However, we recall again that the robust limits on mβ from kinematic experiments represent an invaluable test for the consistency of the more model-dependent constraints coming from cosmology and 0ν2β decay experiments. At the moment, cosmology still provides most of the information on the neutrino masses, although the sensitivity of 0ν2β experiments is rapidly approaching that of cosmological observations. A summary of the current limits is reported in Figure 3 of Gerbino et al. [159]. To better illustrate the complementarity of cosmology

<sup>20</sup>It has to be noticed that the observable which β-decay experiments are sensitive to is m<sup>2</sup> β , rather than mβ . Nevertheless, it is useful to quote constraints in terms of mβ to facilitate the comparison with results from other probes.

<sup>21</sup> https://www.katrin.kit.edu

<sup>22</sup>http://www.project8.org/index.html

FIGURE 5 | Majorana mass mββ of the electron neutrino as a function of the mass mlight of the lightest neutrino eigenstate, for normal (green) or inverted (red) hierarchy. The filled regions correspond to the uncertainty related to the CP-violating phases. The horizontal dashed lines show 95% current upper limits from 0ν2β searches. In particular, we show the tightest and loosest limits among those reported in the text, namely the most stringent from KamLAND-Zen (labeled "KamLAND-Zen, optimistic NME"), and the less stringent from CUORE (labeled "CUORE, pessimistic NME"). NME refers to the uncertainty related to the nuclear matrix elements. We also show vertical dashed lines corresponding to 95% upper limits on 6mν from cosmological observations, translated to upper limits on mlight using the information from oscillation experiments. In particular we show different model and dataset combinations, from right to left: PlanckTT+lowP in the 3CDM + 6m<sup>ν</sup> model, PlanckTT+lowP+BAO in the 3CDM + 6m<sup>ν</sup> + <sup>K</sup> model, PlanckTT+lowP+BAO in the 3CDM + 6m<sup>ν</sup> model. The vertical lines shown in the plot assume normal hierarchy, but the difference with the case of inverted hierarchy is very small on the scale of the plot.

and 0ν2β searches, we show in **Figure 5** how they constrain, together with oscillation experiments, the allowed space in the (mββ, mlight) plane. In more detail, we show the region in that plane that is singled out by oscillation experiments, for normal and inverted hierarchy. The width of the allowed regions traces the uncertainties on the CP-violating phases. We show current upper 95% bounds on mββ from 0ν2β searches as horizontal lines, and current 95% bounds on mlight from cosmology as vertical lines. These are translated from the bounds on 6m<sup>ν</sup> using information from oscillation experiments and assuming normal hierarchy. Assuming inverted hierarchy would however make a barely noticeable difference on the scale of the plot. It can be seen that in general cosmological observations are more constraining than 0ν2β searches.

In the future, however, one can expect that the constraining power of these two probes will be roughly equivalent. This can be seen in **Figure 6** where, similarly to **Figure 5**, we show the allowed space in the (mββ, mlight) plane for future cosmological and 0ν2β probes. As shown in Gerbino et al. [159], the constraining power of 0ν2β searches for 6m<sup>ν</sup> would also depend

FIGURE 6 | The same as Figure 5, but for future cosmological observations and 0ν2β experiments. Note that in this figure we show 95% upper limits for both mββ and mlight, assuming that the true values of both quantities are much smaller that the corresponding experimental sensitivities. The horizontal yellow band labeled "Future 0ν2β" is the union of the regions that contain the 95% upper limits for LEGEND 1K, CUPID, and nEXO, assuming 5 years of live time. The vertical dashed lines correspond to 95% upper limits on 6mν . From right to left: CORE TT, TE, EE, PP in the 3CDM + 6m<sup>ν</sup> model, CORE TT, TE, EE, PP + the DESI and EUCLID BAO in the 3CDM + 6m<sup>ν</sup> + <sup>K</sup> model, CORE TT, TE, EE, PP + the DESI and EUCLID BAO in the 3CDM + 6m<sup>ν</sup> model. The vertical lines shown in the plot assume normal hierarchy.

crucially on the possibility of reducing the uncertainty on the nuclear matrix elements for the 0ν2β isotopes. In fact, provided that neutrinos are Majorana particles and that the leading mechanism responsible for the decay is a mass mechanism, the combination of cosmological probes and 0ν2β measurements could not only lead to a detection of the mass scale, but could also solve the hierarchy dilemma and provide useful information about (at least one of) the Majorana phases [179–181].

## 10. CONSTRAINTS ON Neff

Until now, we have focused on the capability of cosmological observations to constrain neutrino masses. However, as noted in the introduction, cosmology is also a powerful probe of other neutrino properties. The main example is without any doubt the effective number of neutrino families (also called effective number of relativistic degrees of freedom) Neff, defined in Equation (21). As it is clear from its definition, Neff is simply a measure of the total cosmological density during the radiation-dominated era. More precisely, it represents the density in relativistic species, other than photons, normalized to the energy density of a massless neutrino that decouples well before electron-positron annihilation (that, we remember, is not actually the case). As explained in section 2.5, the standard framework, in which photons and active neutrinos are the only relativistic degrees

of freedom present, and neutrino interactions follow the SM of particle physics, predicts Neff = 3.046 after electron-positron annihilation [12, 47, 48].

Given its meaning, it is clear that a deviation from the expected value of Neff can hint to a broad class of effects—in fact, all those effects that change the density of light species in the early Universe. Those effects are not necessarily related to neutrino physics, as the definition of Neff in terms of the number of relativistic degrees of freedom suggests. For example, the existence of a Goldstone boson that decouples well before the QCD phase transition would appear as an increased number of degrees of freedom, with 1Neff ≡ Neff − 3.046 = 0.027 [182]. Speaking however about changes in Neff that are somehow related to neutrino physics, the most notable example is probably the existence of one (or more) additional, sterile light eigenstate, produced through some mechanism in the early Universe. In such a situation, one would have Neff > 3.046, as well as an additional contribution to 6m<sup>ν</sup> . Note that a light sterile neutrino would not necessarily contribute with 1Neff = 1, as it does not share the same temperature as the active neutrinos.

In this section we will focus on cosmological constraints on sterile neutrinos. However, for completeness, we mention a few other examples of scenarios in which 1Neff can possibly be different from zero. One is the presence of primordial lepton asymmetries, related to the presence of a non-vanishing chemical potential in the neutrino distribution function, Equation (15). Constraints on the allowed amount of lepton asymmetry, obtained taking into account the effect of neutrino oscillations, have been reported in Castorina et al. [183] using CMB and BBN data. Another possibility is the so-called low-reheating scenario [151, 184, 185], in which the latest reheating episode of the Universe happens just before BBN, at temperatures of the order of a few MeV, so that neutrinos do not have time to thermalize completely. In this case, one has 1Neff ≤ 0. Finally, non-standard interactions between neutrino and electrons can modify the time of neutrino decoupling [186], so that the entropy transfer from e +e <sup>−</sup> annihilation and Neff are different with respect to the standard picture. We note that the effects related to these new scenarios are often more complicated that just a change in Neff: for example, both in the case of lepton asymmetries and low reheating, the neutrino distribution function is changed in a non-trivial way, affecting also the other moments of the distribution (like the number density, the average velocity, etc.). Finally, to mention a possibility that is not related to changes in Neff, cosmology can also probe the free-streaming nature of neutrinos, for example by looking for the effects of non-standard interactions among neutrinos [187–190], or between neutrinos and dark matter [191–193].

Let us briefly recall how Neff is constrained by cosmological observations [194]. Increasing Neff will make the Universe expand faster (larger H) during the radiation-dominated era, and thus be younger at any given redshift. Then the comoving sound horizon at recombination will be smaller, going like 1/H, while the angular diameter distance to recombination stays constant, because H is unchanged after equality, so that θ<sup>s</sup> is smaller. Also, for fixed matter content, this will make the radiation-dominated era last longer. Recalling our discussion in section 4.1, the effect on the CMB spectrum is that the first peak is enhanced due to the larger early ISW, and all the peaks are moved to the right. However, as we have already learned, these effects can be canceled by acting on other parameters. There is however a more subtle and peculiar effect of Neff, that is related to the scale of Silk damping. The damping scale roughly scales as 1/ √ <sup>H</sup>, i.e., as <sup>√</sup> t, as expected for a random walk process. Then the ratio between the angle subtended by the sound horizon and that subtended by the damping length scales like H−<sup>1</sup> /H−1/<sup>2</sup> <sup>=</sup> <sup>H</sup>−1/<sup>2</sup> . Since θ<sup>s</sup> is fixed by the position of the first peak, this means that, when increasing Neff, the damping length is projected on larger angular scales, or, equivalently, that damping at a given scale is larger. In conclusion the net effect is to lower the damping tail of the CMB spectrum. This effect is difficult to mimic with other parameters, at least in the standard framework. The damping length also depends on the density of baryons, so in principle one could think of changing this to compensate for the effect of Neff; however, the baryon density is very well determined by the ratio of the heights of the first and second peak, so that it is in practice fixed. One possibility, in extended models, is to vary the fraction of primordial helium. Since the mean free path of photons depends on the number of free electrons, and helium recombines slightly before hydrogen, changing the helium-to-hydrogen ratio alters the Silk scale. However, this requires the assumption of nonstandard BBN, since, in the framework of standard BBN, the helium fraction is fixed by ω<sup>b</sup> and Neff themselves, so it is not a free parameter.

We first review constraints on Neff in a simple one-parameter extension of 3CDM, in which Neff is left free to vary, and the mass of active neutrinos is kept fixed to the minimum value allowed by oscillations. This case can be considered as the most agnostic, in some sense, in which one does not make any hypothesis on the new physics that is changing Neff (and thus on any other effects this new physics might produce). Moreover, one can think of these as limits for a very light (massless) sterile neutrino. Finally, constraining Neff is a robustness check for the standard 3CDM model. In fact, measuring Neff = 3.046 within the experimental uncertainty can be seen as a great success of the standard cosmological model. It can be regarded as an indirect detection of the CνB, or, at least, of some component who has the same density, within errors, as we would expect for the three active neutrinos23. From PlanckTT+lowP, one gets Neff = 3.13 ± 0.32; adding BAO gives Neff = 3.15 ± 0.23 [14]. Both measurements, with a precision of ∼ 10%, are in excellent agreement with the standard prediction. Moreover, according to these results, 1Neff = 1 is excluded at least at the 3σ level. Using also information about the full shape of the matter power spectrum, the BOSS collaboration finds Neff = 3.03 ± 0.18 [23]. We note that adding information from direct measurements of the Hubble constant results in larger values of Neff (Neff = 3.41± 0.22 from Planck TT,TE,EE+lowP+lensing+BAO+JLA+H0, see [93]); this is due to the tension with the value of H<sup>0</sup> that is inferred from the CMB, that is alleviated in models with

<sup>23</sup>The fact that, when probed, there is no hint for deviations from the freestreaming behavior should strengthen our belief that we are really observing the CνB.

larger Neff. The next generation of cosmological experiments will improve these constraints by roughly one order of magnitude, getting close to the theoretical threshold of 1Neff = 0.027 discussed at the beginning of this section, corresponding to a Goldstone boson decoupling before the QCD phase transition. Moreover, it will be possible to confirm the effects of noninstantaneous decoupling, since future sensitivities will allow to distinguish, at the 1-σ level, between Neff = 3 and Neff = 3.046. The combination of CORE TT,TE,EE,PP will put an upper bound at 68% CL of 1Neff < 0.040 on the presence of extra massless (<sup>m</sup> <sup>≪</sup> 0.01 eV) species<sup>24</sup> [132] in addition to the three active neutrino families. The CORE collaboration puts limits also on the scenario in which the three active neutrinos have a fixed temperature, but their energy density is rescaled as (Neff/3.046)3/<sup>4</sup> . This scenario can account for an enhanced neutrino density (if Neff > 3.046) and reduced neutrino density (if Neff < 3.046 as for example in the case of low-reheating scenarios). In this case, CORE TT,TE,EE,PP yields Neff = 3.045± 0.041. Forecasts from S4 show that, in order to get closer to the threshold of 1Neff = 0.027, a sensitivity of 1µK · arcmin and fsky > 50% are needed for a 1′ beam size [30]. Efficient delensing will help improve the limits on Neff: delensed spectra will have sharper acoustic peaks, allowing to constrain Neff not only through the impact on the Silk scale, but also through the phase shift in the acoustic peaks [195]. Finally, having access to a larger sky fraction—and therefore to a larger number of modes observed—will be beneficial for constraints on Neff [30]. We conclude this summary about future limits by noticing that the inclusion of LSS data, such as BAO measurements from DESI and Euclid, provides only little improvements with respect to CMB-only constraints (e.g., from CORE TT,TE,EE,PP+DESI BAO+Euclid BAO, 1Neff < 0.038 at 68% CL for extra massless species and Neff = 3.046±0.039 for three neutrinos with rescaled energy density [132]). For a summary of current and future limits on Neff, we refer to **Table 4**.

Let us now come to the case of a massive sterile neutrino. A sterile neutrino would contribute both to Neff and to ω<sup>ν</sup> . Its effect on the cosmological observables will thus be related to changes in these two quantities, as explained through this review. In fact, in principle, we should specify the full form of the distribution function of the sterile neutrino, and its effects could not be fully parameterized through Neff and ω<sup>ν</sup> . Fortunately, one has that, when the distribution function is proportional to a Fermi-Dirac distribution, all the effects on the perturbation evolution of a light fermion can be mapped into two parameters [196]: its energy density in the relativistic limit (and thus its contribution to Neff) and its energy density in the non-relativistic limit (and thus its density parameter, let us denote it with ω<sup>s</sup> to distinguish it from the active neutrinos). This covers several physically interesting cases, namely those of a sterile neutrino that either (i) has a thermal distribution with arbitrary temperature T<sup>s</sup> , or (ii) is distributed proportionally to the active neutrinos, but with a suppression factor χ<sup>s</sup> (this corresponds to the Dodelson-Widrow (DW) prediction for the non-resonant TABLE 4 | Constraints on Neff (at 68% CL) from different combinations of cosmological data.


<sup>a</sup>The constrain applies to the scenario of extra light relics in addition to the three massive neutrino families, i.e., Neff ≥ 3.046.

<sup>b</sup>The constrain applies to the scenario of three massive neutrinos with energy density rescaled by Neff , i.e., Neff can be either lower or greater than 3.046.

<sup>c</sup>The combination includes delensed CMB spectra and a Gaussian prior on the optical depth τ = 0.06 ± 0.01.

Upper part: current 68% CL constraints on Neff . BAO in row no. 2 are from 6dFGS [100], SDSS MGS [104], BOSS LOWZ DR11 and BOSS CMASS DR11 [102] (see [14] for details). BAO and FS (full shape measurements) in row no. 3 are from BOSS DR12 [23]. Lower part: forecasts for future cosmological surveys. Unless otherwise stated, the sensitivity on Neff is forecasted assuming a standard cosmological model with Neff = 3.046 and also marginalizing over 6m<sup>ν</sup> . DESI and Euclid BAO refer to the simulated BAO datasets based on expected experimental performances [35, 37] (see [132] for details).

production scenario [197]; see also Merle et al. [198]). Defining an effective mass meff s by mimicking Equation 19, i.e.,

$$m\_s^{\text{eff}} \equiv \mathfrak{B} \mathfrak{A}.14 \,\,\alpha\_s \,\,\text{eV},\tag{38}$$

the actual mass m<sup>s</sup> of the sterile is related to the effective parameters by:

$$m\_s = (T\_s/T\_v)^{-3} m\_s^{\text{eff}} = \Delta N\_{\text{eff}}^{-3/4} m\_s^{\text{eff}} \qquad \text{(thermal)} \tag{39}$$

$$m\_s = \chi\_s^{-1} m\_s^{\text{eff}} = \Delta N\_{\text{eff}}^{-1} m\_s^{\text{eff}} \qquad \text{(DW)}.\tag{40}$$

Planck data are consistent with no sterile neutrinos: the 95% allowed region in parameter space is Neff < 3.7, meff <sup>s</sup> < 0.52 eV from PlanckTT + lowP + lensing + BAO. However, it should be noted that they do not exclude a sterile neutrino, provided its contribution to the total energy density is small enough. A light sterile neutrino has been proposed as an explanation of the anomalies observed in short-baseline (SBL) experiments (see e.g., [199] and references therein). However, a sterile neutrino with the mass (m<sup>s</sup> ≃ 1 eV) and coupling required to explain reactor anomalies would rapidly thermalize in the early Universe (see e.g., [200, 201]) and lead to 1Neff = 1, strongly at variance with cosmological constraints (excluded at more than 99% confidence considering the above combination of Planck and BAO data). We conclude this section by quoting the forecasts for future cosmological probes. In the context of a 3CDM + 6m<sup>ν</sup> model with 6mfid <sup>ν</sup> <sup>=</sup> 0.06 eV and <sup>m</sup>fid <sup>s</sup> = 0 eV, the combination of CORE TT,TE,EE,PP with BAO measurements from DESI and Euclid will provide 1Neff < 0.054 and m<sup>s</sup> < 0.035 eV [132].

<sup>24</sup>This constraint has been obtained in the context of a <sup>3</sup>CDM <sup>+</sup>6m<sup>ν</sup> cosmology, with 6mfid <sup>ν</sup> = 0.06 eV.

### 11. SUMMARY

The absolute scale of neutrino masses is one of the main open questions in physics to date. Measuring the neutrino mass could shed light on the mechanism of mass generation, possibly related to new physics at a high energy scale. From the experimental point of view, neutrino masses can be probed in the laboratory, with β- and double β-decay experiments, and with cosmological observations. In fact, cosmology is at the moment the most sensitive probe of neutrino masses. Upper limits from cosmology on the sum of neutrino masses are possibly based on combinations of different observables. Results from the CMB alone can be regarded as very robust: these are of the order of 6m<sup>ν</sup> < 0.7 eV (95% CL). The addition of geometrical measurements, like those provided by BAO also very robust—brings down this limit to 6m<sup>ν</sup> < 0.2 eV (95% CL). More aggressive analyses can get the bound very close to the minimum value allowed by oscillation experiments in the case of inverted hierarchy, but are based on observations where control of systematics is more difficult and thus should be taken with caution. It should also be borne in mind that cosmological inferences of neutrino masses are somehow model dependent. In extended cosmological models, especially those involving non-vanishing spatial curvature or dark energy, the constraints on 6m<sup>ν</sup> are degraded, even though they still remain very competitive with those obtained from laboratory experiments. Combination of future CMB and LSS experiments could reach, if systematics are kept under control, a sensitivity of 15 meV in the first half of the next decade, allowing a 4σ detection of neutrino masses if the hierarchy is normal and the lightest eigenstate is massless. In that case, it will also be possible to exclude the inverted hierarchy scenario with a high statistical significance.

Present data are also compatible with the standard description of the neutrino sector, based on the standard model of particle physics. CMB measurements constrain the number of relativistic

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species at recombination to be Neff = 3.13 ± 0.32 at 68% CL. The inclusion of LSS data further tightens the constraints to Neff = 3.03 ± 0.18 at 68% CL. These results exclude the presence of an additional thermalized species at more than 3σ level. Cosmological data are also consistent with no sterile neutrinos. Thus no new physics in the neutrino sector is presently required to interpret cosmological data. The standard picture will be tested more thoroughly by future experiments, that will allow to probe to an unprecedented level the physics of neutrino decoupling. An example would be the possibility to constrain non-standard neutrino-electron interactions. Future cosmological probes will also possibly reach the sensitivity necessary to detect, at the 1 σ level, the increase in the number of degrees of freedom due to a Goldstone boson that decouples well before the QCD phase transition.

### AUTHOR CONTRIBUTIONS

Both authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

We thank Carmelita Carbone, Francesco Forastieri, Keir Rogers, Sunny Vagnozzi for useful comments about the manuscript. MG acknowledges support by the Vetenskapsrådet (Swedish 3493 Research Council) through contract No. 638-2013-8993 and the 3494 Oskar Klein Centre for Cosmoparticle Physics. ML acknowledges support from INFN through the InDark and Gruppo IV fundings, and from ASI through the Grant 2016-24-H.0 (COSMOS) and through the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2. We would like to thank the University of Ferrara and the Oskar Klein Centre for kind hospitality.


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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer SP declared a past co-authorship with one of the authors ML to the handling Editor.

Copyright © 2018 Gerbino and Lattanzi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Neutrino Oscillations and Non-standard Interactions

Yasaman Farzan<sup>1</sup> and Mariam Tórtola<sup>2</sup>

<sup>1</sup> School of Physics, Institute for Research in Fundamental Sciences, Tehran, Iran, <sup>2</sup> AHEP Group, Institut de Física Corpuscular—Universitat de València/CSIC, Paterna, Spain

\*

Current neutrino experiments are measuring the neutrino mixing parameters with an unprecedented accuracy. The upcoming generation of neutrino experiments will be sensitive to subdominant neutrino oscillation effects that can in principle give information on the yet-unknown neutrino parameters: the Dirac CP-violating phase in the PMNS mixing matrix, the neutrino mass ordering and the octant of θ23. Determining the exact values of neutrino mass and mixing parameters is crucial to test various neutrino models and flavor symmetries that are designed to predict these neutrino parameters. In the first part of this review, we summarize the current status of the neutrino oscillation parameter determination. We consider the most recent data from all solar neutrino experiments and the atmospheric neutrino data from Super-Kamiokande, IceCube, and ANTARES. We also implement the data from the reactor neutrino experiments KamLAND, Daya Bay, RENO, and Double Chooz as well as the long baseline neutrino data from MINOS, T2K, and NOνA. If in addition to the standard interactions, neutrinos have subdominant yet-unknown Non-Standard Interactions (NSI) with matter fields, extracting the values of these parameters will suffer from new degeneracies and ambiguities. We review such effects and formulate the conditions on the NSI parameters under which the precision measurement of neutrino oscillation parameters can be distorted. Like standard weak interactions, the non-standard interaction can be categorized into two groups: Charged Current (CC) NSI and Neutral Current (NC) NSI. Our focus will be mainly on neutral current NSI because it is possible to build a class of models that give rise to sizeable NC NSI with discernible effects on neutrino oscillation. These models are based on new U(1) gauge symmetry with a gauge boson of mass . 10 MeV. The UV complete model should be of course electroweak invariant which in general implies that along with neutrinos, charged fermions also acquire new interactions on which there are strong bounds. We enumerate the bounds that already exist on the electroweak symmetric models and demonstrate that it is possible to build viable models avoiding all these bounds. In the end, we review methods to test these models and suggest approaches to break the degeneracies in deriving neutrino mass parameters caused by NSI.

Keywords: neutrino oscillations, leptonic CP violation, non-standard neutrino interactions, neutrino masses, neutrino physics

#### Edited by:

Diego Aristizabal Sierra, Federico Santa María Technical University, Chile

#### Reviewed by:

Andre De Gouvea, Northwestern University, United States Antonio Palazzo, Università degli Studi di Bari Aldo Moro, Italy

> \*Correspondence: Mariam Tórtola mariam@ific.uv.es

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 26 October 2017 Accepted: 30 January 2018 Published: 27 February 2018

#### Citation:

Farzan Y and Tórtola M (2018) Neutrino Oscillations and Non-standard Interactions. Front. Phys. 6:10. doi: 10.3389/fphy.2018.00010

### 1. INTRODUCTION

In the framework of "old" electroweak theory, formulated by Glashow, Weinberg and Salam, lepton flavor is conserved and neutrinos are massless. As a result, a neutrino of flavor α (α ∈ {e,µ, τ }) created in charged current weak interactions in association with a charged lepton of flavor α will maintain its flavor. Various observations have however shown that the flavor of neutrinos change upon propagating long distances. Historically, solar neutrino anomaly (deficit of the solar neutrino flux relative to standard solar model predictions) [1] and atmospheric neutrino anomaly (deviation of the ratio of muon neutrino flux to the electron neutrino flux from 2 for atmospheric neutrinos that cross the Earth before reaching the detector) [2] were two main observations that showed the lepton flavor was violated in nature. This conclusion was further confirmed by observation of flavor violation of man-made neutrinos after propagating sizable distances in various reactor [3–5] and long baseline experiments [6, 7]. The established paradigm for flavor violation which impressively explain all these anomalies is the three neutrino mass and mixing scheme. According to this scheme, each neutrino flavor is a mixture of different mass eigenstates. As neutrinos propagate, each component mass eigenstate acquires a different phase so neutrino of definite flavor will convert to a mixture of different flavors; hence, lepton flavor violation takes place.

Within this scheme, the probability of conversion of ν<sup>α</sup> to ν<sup>β</sup> (as well as that of ν¯<sup>α</sup> to ν¯β) in vacuum or in matter with constant density has a oscillatory dependence on time or equivalently on the distance traveled by neutrinos<sup>1</sup> . For this reason, the phenomenon of flavor conversion in neutrino sector is generally known as neutrino oscillation. Neutrino flavor eigenstates are usually denoted by να. That is ν<sup>α</sup> is defined as a state which appears in W boson vertex along with charged lepton l<sup>α</sup> (α ∈ {e,µ, τ }). The latter corresponds to charged lepton mass eigenstates. Neutrino mass eigenstates are denoted by ν<sup>i</sup> with mass m<sup>i</sup> where i ∈ {1, 2, 3}. The flavor eigenstates are related to mass eigenstate by a 3 × 3 unitary matrix, U, known as PMNS after Pontecorvo-Maki-Nakagawa-Sakata: ν<sup>α</sup> = P <sup>i</sup> Uαiν<sup>i</sup> . The unitary mixing matrix can be decomposed as follows

$$U \equiv \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta\_{23} & \sin \theta\_{23} \\ 0 & -\sin \theta\_{23} & \cos \theta\_{23} \end{pmatrix} \begin{pmatrix} \cos \theta\_{13} & 0 & \sin \theta\_{13} e^{-i\delta} \\ 0 & 1 \\ -\sin \theta\_{13} e^{i\delta} & 0 & \cos \theta\_{13} \end{pmatrix}$$

$$\begin{pmatrix} \cos \theta\_{12} & \sin \theta\_{12} & 0 \\ -\sin \theta\_{12} & \cos \theta\_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix},\tag{1}$$

where the mixing angles θij are defined to be in the [0, π/2] range while the phase δ can vary in [0, 2π). In this way, the whole physical parameter space is covered. Historically, ν1, ν<sup>2</sup> and ν<sup>3</sup> have been defined according to their contribution to νe. In other words, they are ordered such that |Ue1| > |Ue2| > |Ue3| so ν<sup>1</sup> (ν3) provides largest (smallest) contribution to νe. Notice that with this definition, θ12, θ<sup>13</sup> ≤ π/4. It is then of course a meaningful question to ask which ν<sup>i</sup> is the lightest and which one is the heaviest; or equivalently, what is the sign of 1m<sup>2</sup> ij <sup>=</sup> <sup>m</sup><sup>2</sup> <sup>i</sup> <sup>−</sup>m<sup>2</sup> j . The answer to this question comes from observation. Time evolution of ultra-relativistic neutrino state is governed by the following Hamiltonian: Hvac + Hm, where the effective Hamiltonian in vacuum is given by

$$H\_{\rm vac} = U \cdot \text{Diag}\left(\frac{m\_1^2}{2E}, \frac{m\_2^2}{2E}, \frac{m\_3^2}{2E}\right) \cdot U^\dagger. \tag{2}$$

Within the Standard Model (SM) of particles, the effective Hamiltonian in matter H<sup>m</sup> in the framework of the medium in which neutrinos are propagating can be written as

$$H\_m = \begin{pmatrix} \sqrt{2}G\_FN\_\varepsilon - \frac{\sqrt{2}}{2}G\_FN\_n & 0 & 0\\ 0 & -\frac{\sqrt{2}}{2}G\_FN\_n & 0\\ 0 & 0 & -\frac{\sqrt{2}}{2}G\_FN\_n \end{pmatrix},\tag{3}$$

where it is assumed that the medium is electrically neutral (N<sup>e</sup> = Np), unpolarized and composed of non-relativistic particles. In vacuum, H<sup>m</sup> = 0 and we can write

$$P(\nu\_{\alpha} \rightarrow \nu\_{\beta}) = \left| \sum\_{ij} U\_{\alpha i} U\_{\beta j}^\* e^{i \Delta m\_{ij}^2 L / (2E)} \right|^2. \tag{4}$$

By adding or subtracting a matrix proportional to the identity I3×<sup>3</sup> to the Hamiltonian, neutrinos obtain an overall phase with no observable physical consequences. That is why neutrino oscillation probabilities (both in vacuum and in matter) are sensitive only to 1m<sup>2</sup> ij rather than to <sup>m</sup><sup>2</sup> i . As a result, it is impossible to derive the mass of the lightest neutrino from oscillation data alone. Similarly, neutrino oscillation pattern within the SM only depends on N<sup>e</sup> not on Nn. Similar arguments can be repeated for antineutrinos by replacing U with U ∗ (or equivalently δ → −δ) and replacing H<sup>m</sup> → −Hm. The phase δ, similarly to its counterpart in the CKM mixing matrix of quark sector, violates CP. Just like in the quark sector, CP violation in neutrino sector is given by the Jarlskog invariant: <sup>J</sup> <sup>=</sup> sin θ<sup>13</sup> cos<sup>2</sup> θ<sup>13</sup> sin θ<sup>12</sup> cos θ<sup>12</sup> cos θ<sup>23</sup> sin θ<sup>23</sup> sin δ.

As we will see in detail in section 2, the mixing angles θ12, θ<sup>13</sup> and θ<sup>23</sup> are derived from observations with remarkable precision. The mixing angle θ<sup>23</sup> has turned out to be close 45◦ but it is not clear within present uncertainties whether θ<sup>23</sup> < π/4 or θ<sup>23</sup> > π/4. This uncertainty is known as the octant degeneracy. The value of δ is also unknown for the time being, although experimental data start indicating a preferred value close to 3π/2. The absolute value and the sign of 1m<sup>2</sup> <sup>21</sup> are however determined. While <sup>|</sup>1m<sup>2</sup> <sup>31</sup><sup>|</sup> is measured, the sign of <sup>1</sup>m<sup>2</sup> <sup>31</sup> is not yet determined. If 1m<sup>2</sup> <sup>31</sup> > 0 (1m<sup>2</sup> <sup>31</sup> < 0), the scheme is called normal (inverted) ordering or normal (inverted) mass spectrum. The main goals of current and upcoming neutrino oscillation experiments are determining sgn(cos 2θ23), sgn(1m<sup>2</sup> <sup>31</sup>) and the value of the CP–violating phase δ.

<sup>1</sup>One should bear in mind that in a medium with varying density, such as the Sun interior, the conversion may not have an oscillatory behavior for a certain energy range. Likewise, the presence of strong matter effects may suppress the oscillatory behavior even in the case of constant density [8].

The neutrino oscillation program is entering a precision era, where the known parameters are being measured with an ever increasing accuracy. Next generation of long–baseline neutrino experiments will resolve the subdominant effects in oscillation data sensitive to the yet unknown oscillation parameters (e.g., δ). Of course, all these derivations are within 3 × 3 neutrino mass and mixing scheme under the assumption that neutrinos interact with matter only through the SM weak interactions (plus gravity which is too weak to be relevant). Allowing for Non-Standard Interaction (NSI) can change the whole picture. Non-Standard Interaction of neutrinos can be divided into two groups: Neutral Current (NC) NSI and Charged Current (CC) NSI. While the CC NSI of neutrinos with the matter fields (e, u, d) affects in general the production and detection of neutrinos, the NC NSI may affect the neutrino propagation in matter. As a result, both types of interaction may show up at various neutrino experiments. In recent years, the effects of both types of NSI on neutrino experiments have been extensively studied in the literature, formulating the lower limit on the values of couplings in order to have a resolvable impact on the oscillation pattern in upcoming experiments. On the other hand, non-standard interaction of neutrinos can crucially affect the interpretation of the experimental data in terms of the relevant neutrino mass parameters. Indeed, as it will be discussed in this work, the presence of NSI in the neutrino propagation may give rise, among other effects, to a degeneracy in the measurement of the solar mixing angle θ<sup>12</sup> [9–11]. Likewise, CC NSI at the production and detection of reactor antineutrinos can affect the very precise measurement of the mixing angle θ<sup>13</sup> in Daya Bay [12, 13]. Moreover, it has been shown that NSI can cause degeneracies in deriving the CP–violating phase δ [14–17], as well as the correct octant of the atmospheric mixing angle θ<sup>23</sup> [18] at current and future long–baseline neutrino experiments. Along this review, we will discuss possible ways to resolve the parameter degeneracies due to NSI, by exploiting the capabilities of some of the planned experiments such as the intermediate baseline reactor neutrino experiments JUNO and RENO50 [19].

Most of the analyses involving NSI in neutrino experiments parameterize such interactions in terms of effective four-Fermi couplings. However, one may ask whether it is possible to build viable renormalizable electroweak symmetric UV complete models that underlay this effective interaction with coupling large enough to be discernible at neutrino oscillation experiments. Generally speaking if the effective coupling comes from integrating out a new state (X) of mass m<sup>X</sup> and of coupling gX, we expect the strength of the effective four-Fermi interaction to be given by g 2 X /m<sup>2</sup> X . We should then justify why X has not been so far directly produced at labs. As far as NC NSI is concerned, two solutions exist: (i) X is too heavy; i.e., m<sup>X</sup> ≫ mEW. Recent bounds from the LHC imply m<sup>X</sup> > 4 − 5 TeV [20] which for even g<sup>X</sup> ∼ 1 implies g 2 X /m<sup>2</sup> <sup>X</sup> ≪ G<sup>F</sup> [21, 22]. Moreover, as shown in Friedland et al. [23], in the range 10 GeV < mZ′ < TeV, the monojet searches at the LHC constrain this ratio to values much smaller than 1. (ii) Second approach is to take m<sup>X</sup> ≪ mEW and g<sup>X</sup> ≪ 1 such that g 2 X /m<sup>2</sup> <sup>X</sup> ∼ GF. In this approach, the null result for direct production of X is justified with its very small coupling. In Farzan [24], Farzan and Heeck [25], and Farzan and Shoemaker [26], this approach has been evoked to build viable models for NC NSI with large effective couplings. For CC NSI, the intermediate state, being charged, cannot be light. That is, although its Yukawa couplings to neutrinos and matter fields can be set to arbitrarily small values, the gauge coupling to the photon is set by its charge so the production at LEP and other experiments cannot be avoided. We are not aware of any viable model that can lead to a sizable CC NSI. Interested reader may see Agarwalla et al. [13], Vanegas Forero [27], Bakhti and Khan [28], Khan and Tahir [29], Kopp et al. [30], Bellazzini et al. [31], Akhmedov et al. [32], Biggio and Blennow [33]. Notice that throughout this review, we focus on the interaction of neutrinos with matter fields. Das et al. [34] and Dighe and Sen [35] study the effects of non-standard self-interaction of neutrinos in supernova. de Salas et al. [36], Brdar et al. [37] discuss propagation of neutrino in presence of interaction with dark matter. The effect of NSI on the decoupling of neutrinos in the early Universe has been considered in Mangano et al. [38].

This review is organized as follows. In section 2, we review the different neutrino oscillation experiments and discuss how neutrino oscillation parameters within the standard three neutrino scheme can be derived. We then discuss the prospect of measuring yet unknown parameters: δ, sgn(1m<sup>2</sup> <sup>31</sup>) and sgn(cos 2θ23). In section 3, we discuss how NSI can affect this picture and review the bounds that the present neutrino data sets on the effective ǫ parameters. We then discuss the potential effects of NSI on future neutrino experiments and suggest strategies to solve the degeneracies. In section 4, we introduce models that can lead to effective NSI of interest and briefly discuss their potential effects on various observables. In section 5, we review methods suggested to test these models. Results will be summarized in section 6.

## 2. NEUTRINO OSCILLATIONS

In this section, we will present the current status of neutrino oscillation data in the standard three–neutrino framework. Most recent global neutrino fits to neutrino oscillations can be found in de Salas et al. [39], Capozzi et al. [40], and Esteban et al. [41]. Here we will focus on the results of de Salas et al. [39], commenting also on the comparison with the other two analysis. First, we will describe the different experiments entering in the global neutrino analysis, grouped in the solar, reactor, atmospheric and long– baseline sectors. For each of them we will also discuss their main contribution to the determination of the oscillation parameters.

#### 2.1. The Solar Neutrino Sector: (sin<sup>2</sup> θ12, 1m<sup>2</sup> 21)

Under the denomination of solar neutrino sector, one finds traditionally not only all the solar neutrino experiments, but also the reactor KamLAND experiment, sensitive to the same oscillation channel, under the assumption of CPT conservation. Solar neutrino analysis include the historical radiochemical experiments Homestake [42], Gallex/GNO [43], SAGE [44], sensitive only to the interaction rate of electron neutrinos, but not to their energy or arrival time to the detector. This more detailed information became available with the start-up of the real– time solar neutrino experiment Kamiokande [45], that confirmed the solar neutrino deficit already observed by the previous experiments. Its successor Super–Kamiokande, with a volume 10 times larger, has provided very precise observations in almost 20 years of operation. Super–Kamiokande is a Cherenkov detector that uses 50 kton of ultra pure water as target for solar neutrino interactions, that are detected through elastic neutrino-electron scattering. This process is sensitive to all neutrino flavors, with a larger cross section for ν<sup>e</sup> due to the extra contribution from the charged–current neutrino–electron interaction. The correlation between the incident neutrino and the recoil electron in the observed elastic scattering makes possible the reconstruction of the incoming neutrino energy and arrival direction. After its first three solar phases [46–48], Super-Kamiokande is already in its fourth phase, where a very low energy detection threshold of 3.5 MeV has been achieved [49]. Moreover, during this last period, Super–Kamiokande has reported a 3σ indication of Earth matter effects in the solar neutrino flux, with the following measured value of the day–night asymmetry [50, 51]

$$A\_{DN} = \frac{\Phi\_D - \Phi\_N}{(\Phi\_D + \Phi\_N)/2} = (-3.3 \pm 1.0 \text{ (stat)} \pm 0.5 \text{ (syst)}) \text{ \%} \dots \text{ (5)}$$

Likewise, they have reconstructed a neutrino survival probability consistent with the MSW prediction at 1σ [52, 53].

The Sudbury Neutrino Observatory (SNO) used a similar detection technique with 1 kton of pure heavy water as neutrino target. The use of the heavy water allowed the neutrino detection through three different processes: charged–current ν<sup>e</sup> interactions with the deuterons in the heavy water (CC), neutral–current ν<sup>α</sup> with the deuterons (NC), and as well as elastic scattering of all neutrino flavors with electrons (ES). The measurement of the neutrino rate for each of the three reactions allows the determination of the ν<sup>e</sup> flux and the total active ν<sup>α</sup> flux of <sup>8</sup>B neutrinos from the Sun. SNO took data during three phases, each of them characterized by a different way of detecting the neutrinos produced in the neutrino NC interaction with the heavy water [54, 55].

Apart from Super–Kamiokande, the only solar neutrino detector at work nowadays is the Borexino experiment. Borexino is a liquid–scintillator experiment sensitive to solar neutrinos through the elastic neutrino–electron scattering, with a design optimized to measure the lower energy part of the spectrum. During its first detection phase, Borexino has reported precise observations of the <sup>7</sup>Be solar neutrino flux, as well as the first direct observation of the mono-energetic pep solar neutrinos and the strongest upper bound on the CNO component of the solar neutrino flux [56]. Moreover, Borexino has also measured the solar <sup>8</sup>B rate with a very low energy threshold of 3 MeV [57] and it has also provided the first real–time observation of the very low energy pp neutrinos [58].

The simulation of the production and propagation of solar neutrinos requires the knowledge of the neutrino fluxes produced in the Sun's interior. This information is provided by the Standard Solar Model (SSM), originally built by Bahcall [59]. The more recent versions of the SSM offer at least two different versions according to the solar metallicity assumed [60, 61]. de Salas et al. [39] uses the low metallicity model while Esteban et al. [41] reports its main results for the high–metallicity model. For a discussion on the impact of the choice of a particular SSM over the neutrino oscillation analysis see for instance Esteban et al. [41] and Schwetz et al. [62].

KamLAND is a reactor neutrino experiment designed to probe the existence of neutrino oscillations in the so-called LMA region, with 1m<sup>2</sup> <sup>21</sup> <sup>∼</sup> <sup>10</sup>−<sup>5</sup> eV<sup>2</sup> . KamLAND detected reactor antineutrinos produced at an average distance of 180 km, providing the first evidence for the disappearance of neutrinos traveling to a detector from a power reactor [63]. In KamLAND, neutrinos are observed through the inverse beta decay process ν¯<sup>e</sup> + p → e <sup>+</sup> + n, with a delayed coincidence between the positron annihilation and the neutron capture in the medium that allows the efficient reduction of the background. The final data sample released by KamLAND contains a total live time of 2,135 days, with a total of 2,106 reactor antineutrino events observed to be compared with 2,879 ± 118 reactor antineutrino events plus 325.9 ± 26.1 background events expected in absence of neutrino oscillations [64].

**Figure 1** reports the allowed region in the sin<sup>2</sup> <sup>θ</sup><sup>12</sup> <sup>−</sup> <sup>1</sup>m<sup>2</sup> 21 plane from the analysis of all solar neutrino data (black lines), from the analysis of the KamLAND reactor experiment (blue lines) and from the combined analysis of solar + KamLAND data (colored regions). Here the value of the θ<sup>13</sup> has been marginalized following the most recent short–baseline reactor experiments which will be described in the next subsection. From the figure, one can see that the determination of θ<sup>12</sup> is mostly due to solar neutrino experiments, while the very

FIGURE 1 | Allowed regions at 90 and 99% C.L. from the analysis of solar data (black lines), KamLAND (blue lines) and the global fit (colored regions). θ13 has been marginalized according to the latest reactor measurements [39]. Triangle and circle respectively denote KamLAND and solar best fit. The global best fit is denoted by a star.

accurate measurement of 1m<sup>2</sup> <sup>21</sup> is obtained thanks to the spectral information from KamLAND. There is also a mild but noticeable tension between the preferred values of 1m<sup>2</sup> <sup>21</sup> by KamLAND and by solar experiments. While the first one shows a preference for 1m<sup>2</sup> <sup>21</sup> <sup>=</sup> 4.96 <sup>×</sup> <sup>10</sup>−<sup>5</sup> eV<sup>2</sup> , the combination of all solar experiments prefer a lower value: 1m<sup>2</sup> <sup>21</sup> <sup>=</sup> 7.6 <sup>×</sup> <sup>10</sup>−<sup>5</sup> eV<sup>2</sup> . This discrepancy appears at the 2σ level. As we will see in the next section, non–standard neutrino interactions have been proposed as a way to solve the tension between solar and KamLAND data.

The best fit point for the global analysis corresponds to:

$$
\sin^2 \theta\_{12} = 0.321^{+0.018}\_{-0.016}, \,\Delta m^2\_{21} = 7.56 \pm 0.19 \times 10^{-5} \text{eV}^2 \,. \tag{6}
$$

Maximal mixing is excluded at more than 7σ.

## 2.2. Short–Baseline Reactor Neutrino Experiments and θ<sup>13</sup>

Until recently, the mixing angle θ<sup>13</sup> was pretty much unknown. Indeed, the only available information about the reactor angle was an upper-bound obtained from the non-observation of antineutrino disappearance at the CHOOZ and Palo Verde reactor experiments [65, 66]: sin<sup>2</sup> θ<sup>13</sup> < 0.039 at 90% C.L. for 1m<sup>2</sup> <sup>31</sup> <sup>=</sup> 2.5×10−<sup>3</sup> eV<sup>2</sup> . Later on, the interplay between different data samples in the global neutrino oscillation analyses started showing some sensitivity to the reactor mixing angle θ13. In particular, from the combined analysis of solar and KamLAND neutrino data, a non-zero θ<sup>13</sup> value was preferred [62, 67, 68]. The non-trivial constraint on θ<sup>13</sup> mainly appeared as a result of the different correlation between sin<sup>2</sup> θ<sup>12</sup> and sin<sup>2</sup> θ<sup>13</sup> present in the solar and KamLAND neutrino data samples [69, 70]. Moreover, a value of θ<sup>13</sup> different from zero helped to reconcile the tension between the 1m<sup>2</sup> <sup>21</sup> best fit points for solar and KamLAND separately. Another piece of evidence for a non-zero value of θ<sup>13</sup> was obtained from the combination of atmospheric and long-baseline neutrino data [71, 72]. Due to a small tension between the preferred values of <sup>|</sup>1m<sup>3</sup> <sup>31</sup>| at θ<sup>13</sup> = 0 by MINOS experiment and Super-Kamiokande atmospheric neutrino data, the combined analysis of both experiments showed a preference for θ<sup>13</sup> > 0 [73–75].

Nevertheless, the precise determination of θ<sup>13</sup> was possible thanks to the new generation of reactor neutrino experiments, Daya Bay, RENO and Double Chooz. The main features of these new reactor experiments are, on one side, their increased reactor power compared to their predecessors and, on the other side, the use of several identical antineutrino detectors located at different distances from the reactor cores. Combining these two features results in an impressive increase on the number of detected events. Moreover, the observed event rate at the closest detectors is used to predict the expected number of events at the more distant detectors, without relying on the theoretical predictions of the antineutrino flux at reactors. Several years ago, in the period between 2011 and 2012, the three experiments found evidence for the disappearance of reactor antineutrinos over short distances, providing the first measurement of the angle θ<sup>13</sup> [3–5]. We will now briefly discuss the main details of each experiment as well as their latest results.

The Daya Bay reactor experiment [4] in China is a multi-detector and multi-core reactor experiment. Electron antineutrinos produced at six reactor cores with 2.9 GW thermal power are observed at eight antineutrino 20 ton Gadoliniumdoped liquid scintillator detectors, located at distances between 350 and 2,000 m from the cores. The latest data release from Daya Bay has reported the detection of more than 2.5 millions of reactor antineutrino events, after 1,230 days of data taking [76]. This enormous sample of antineutrino events, together with a significant reduction of systematical errors has made possible the most precise determination of the reactor mixing angle to date [76]

$$
\sin^2 2\theta\_{13} = 0.0841 \pm 0.0027 \text{ (stat.)} \pm 0.0019 \text{ (syst.)}.\tag{7}
$$

Likewise, the sensitivity to the effective mass splitting 1m<sup>2</sup> ee has been substantially improved<sup>2</sup> ,

$$|\Delta m\_{ee}^2| = 2.50 \pm 0.06 \text{ (stat.)} \pm 0.06 \text{ (syst.)} \times 10^{-3} \text{ eV}^2,\tag{8}$$

reaching the accuracy of the long–baseline accelerator experiments, originally designed to measure this parameter.

The RENO experiment [5] in South Korea consists of six aligned reactor cores, equally distributed over a distance of 1.3 km. Reactor antineutrinos are observed by two identical 16 ton Gadolinium-doped Liquid Scintillator detectors, located at approximately 300 (near) and 1,400 m (far detector) from the reactor array center. The RENO Collaboration has recently reported their 500 live days observation of the reactor neutrino spectrum [78, 79], showing an improved sensitivity to the atmospheric mass splitting, <sup>|</sup>1m<sup>2</sup> ee| = 2.62+0.24 <sup>−</sup>0.26 <sup>×</sup> <sup>10</sup>−<sup>3</sup> eV<sup>2</sup> . Their determination for θ<sup>13</sup> is consistent with the results of Daya Bay:

$$
\sin^2 2\theta\_{13} = 0.082 \pm 0.009 \text{ (stat.)} \pm 0.006 \text{ (syst.)}\tag{9}
$$

The Double Chooz experiment in France detects antineutrinos produced at two reactor cores in a near and far detectors located at distances of 0.4 and 1 km from the neutrino source, respectively [3]. The latest results presented by the Double Chooz collaboration correspond to a period of 818 days of data at far detector plus 258 days of observations with the near detector. From the spectral analysis of the multi-detector neutrino data, the following best fit value for θ<sup>13</sup> is obtained [80]

$$
\sin^2 2\theta\_{13} = 0.119 \pm 0.016 \,\text{(stat.} + \text{syst.)}.\tag{10}
$$

**Figure 2** illustrates the sensitivity to sin<sup>2</sup> θ<sup>13</sup> obtained from the analysis of reactor and global neutrino data for normal and inverted mass ordering. The black line corresponds to the result obtained from the combination of all the reactor neutrino data samples while the others correspond to the individual reactor data samples, as indicated. One can see from the figure that the more constraining results come from Daya Bay and RENO experiments, while Double Chooz shows a more limited sensitivity to θ13. Moreover, the global constraint on

<sup>2</sup>Parke [77] discusses the correct form of the definition of 1m<sup>2</sup> ee.

θ<sup>13</sup> is totally dominated by the Daya Bay measurements, with some contributions from RENO to its lower bound. Notice also that global analyses of neutrino data do not show relevant differences between the preferred value of θ<sup>13</sup> for normal or inverted mass ordering, as we will discuss later. For more details on the analysis of reactor data presented in **Figure 2**, see de Salas et al. [39].

#### 2.3. The Atmospheric Neutrino Sector: (sin<sup>2</sup> θ23, 1m<sup>2</sup> 31)

The atmospheric neutrino flux was originally studied as the main source of background for the nucleon-decay experiments [81– 83]. For several years, most of the dedicated experiments observed a deficit in the detected number of atmospheric neutrinos with respect to the predictions. The solution to this puzzling situation arrived in 1998, when the observation of the zenith angle dependence of the µ-like atmospheric neutrino data in Super-Kamiokande indicated an evidence for neutrino oscillations [2]. Some years later, the Super-Kamiokande Collaboration reported a L/E distribution of the atmospheric ν<sup>µ</sup> data sample characteristic of neutrino oscillations [84]. Super-Kamiokande has been taking data almost continuously since 1996, being now in its fourth phase. Super-Kamiokande is sensitive to the atmospheric neutrino flux in the range from 100 MeV to TeV. The observed neutrino events are classified in three types, fully contained, partially contained and upward-going muons, based on the topology of the event. The subsequent data releases by the Super-Kamiokande Collaboration have increased in complexity. Currently it is very complicated to analyze the latest results by independent groups [39, 41]. From the analysis of the latest Super-Kamiokande atmospheric data, the following best fit values have been obtained for the oscillation parameters [85]:

$$
\sin^2 \theta\_{23} = 0.587, \quad \Delta m\_{32}^2 = 2.5 \times 10^{-3} \text{eV}^2. \tag{11}
$$

Thus, a slight preference for θ<sup>23</sup> > π/4 is reported. Likewise, the normal mass ordering (i.e., 1m<sup>2</sup> <sup>31</sup> > 0) is preferred over the inverted one (i.e., 1m<sup>2</sup> <sup>31</sup> < 0).

In recent years, atmospheric neutrinos are also being detected by neutrino telescope experiments. IceCube and ANTARES, originally designed to detect higher energy neutrino fluxes, have reduced their energy threshold in such a way that they can measure the most energetic part of the atmospheric neutrino flux.

The ANTARES telescope [86], located under the Mediterranean Sea, observes atmospheric neutrinos with energies as low as 20 GeV. Neutrinos are detected via the Cherenkov light emitted after the neutrino interaction with the medium in the vicinity of the detector. In Adrian-Martinez et al. [87], the ANTARES Collaboration has analyzed the atmospheric neutrino data collected during a period of 863 days. Their results for the oscillation parameters are in good agreement with current world data. For the first time, ANTARES results have also been included in a global neutrino oscillation fit [39].

The IceCube DeepCore detector is a sub-array of the IceCube neutrino observatory, operating at the South Pole [88]. DeepCore was designed with a denser instrumentation compared to IceCube, with the goal of lowering the energy threshold for the detection of atmospheric muon neutrino events below 10 GeV. Neutrinos are identified trough the Cherenkov radiation emitted by the secondary particles produced after their interaction in the ice. The most recent data published by DeepCore correspond to a live time of 3 years [89]. A total of 5,174 atmospheric neutrino events were observed, compared to a total 6,830 events expected in absence of neutrino oscillations. The obtained best fit values for the atmospheric neutrino parameters sin<sup>2</sup> <sup>θ</sup><sup>23</sup> <sup>=</sup> 0.53+0.09 −0.12 and 1m<sup>2</sup> <sup>32</sup> <sup>=</sup> 2.72+0.19 <sup>−</sup>0.20 <sup>×</sup> <sup>10</sup>−<sup>3</sup> eV<sup>2</sup> are also compatible with the atmospheric results of the Super–Kamiokande experiment.

The left panel of **Figure 3** shows the allowed regions at 90 and 99% C.L. in the atmospheric neutrino oscillation parameters sin<sup>2</sup> θ<sup>23</sup> and 1m<sup>2</sup> <sup>31</sup> obtained from ANTARES, DeepCore and Super-Kamiokande phases I to III [39]. From the combination

FIGURE 3 | 90 and 99% C.L. allowed regions at the sin<sup>2</sup> <sup>θ</sup><sup>23</sup> <sup>−</sup> <sup>1</sup>m<sup>2</sup> <sup>31</sup> plane obtained from the atmospheric (Left) and long–baseline accelerator experiments (Right), see the text for details. Notice the different scale in the 1m<sup>2</sup> <sup>31</sup> parameter. Both plots correspond to the normal ordered neutrino mass spectrum. Figures adapted de Salas et al. [39].

one sees that DeepCore results start being competitive with the determination of the atmospheric oscillation parameters by long-baseline experiments. Indeed, a recent reanalysis of DeepCore atmospheric data [90] shows an improved sensitivity with respect to the region plotted in **Figure 3**. The sensitivity of ANTARES shown in **Figure 3** is not yet competitive with the other experiments. However, it is expected that the ANTARES collaboration will publish an updated analysis that will certainly improve their sensitivity to the atmospheric neutrino parameters.

### 2.4. Long–Baseline Accelerator Experiments

After the discovery of neutrino oscillations in the atmospheric neutrino flux, several long-baseline accelerator experiments were planned to confirm the oscillation phenomenon with a man– made neutrino source. The first two experiments trying to probe the ν<sup>µ</sup> disappearance oscillation channel in the same region of 1m<sup>2</sup> as explored by atmospheric neutrinos were K2K and MINOS. Their successors, T2K and NOνA are still at work today.

The KEK to Kamioka (K2K) experiment used a neutrino beam produced by a 12 GeV proton beam from the KEK proton synchrotron. The neutrino beam was detected by a near detector 300 m away from the proton target and by the Super– Kamiokande detector, at a distance of 250 km. The number of detected neutrino events, as well as the spectral distortion of the neutrino flux observed by K2K was fully consistent with the hypothesis of neutrino oscillation [91].

The Main Injector Neutrino Oscillation Search (MINOS) experiment observed neutrino oscillations from a beam produced by the NuMI (Neutrinos at Main Injector) beamline at Fermilab in an underground detector located at the Soudan Mine, in Minnesota, 735 km away. MINOS searched for oscillations in the disappearance (ν<sup>µ</sup> → νµ) and appearance channels (ν<sup>µ</sup> → νe), for neutrinos and antineutrinos as well. After a period of 9 years, the MINOS experiment collected a data sample corresponding to an exposure of 10.71 <sup>×</sup> <sup>10</sup><sup>20</sup> protons on target (POT) in the neutrino mode, and 3.36 <sup>×</sup> <sup>10</sup><sup>20</sup> POT in the antineutrino mode [92, 93]. The combined analysis of all MINOS data shows a slight preference for inverted mass ordering and θ<sup>23</sup> below maximal as well as a disfavored status for maximal mixing with 1χ<sup>2</sup> <sup>=</sup> 1.54 [94]. The allowed ranges for the atmospheric parameters from the joint analysis of all MINOS data are the following

$$\begin{aligned} \sin^2 \theta\_{23} &\in [0.35, 0.65] \text{ (90\% C.L.)},\\ |\Delta m\_{32}^2| &\in [2.28, 2.46] \times 10^{-3} \,\text{eV}^2 \text{ (1}\sigma\text{) for normal ordering} \\ |\sin^2 \theta\_{23} &\in [0.34, 0.67] \text{ (90\% C.L.)},\\ |\Delta m\_{32}^2| &\in [2.32, 2.53] \times 10^{-3} \,\text{eV}^2 \text{ (1}\sigma\text{) for inverted ordering.} \end{aligned} \tag{12}$$

The Tokai to Kamioka (T2K) experiment uses a neutrino beam consisting mainly of muon neutrinos, produced at the J-PARC accelerator facility and observed at a distance of 295 km and an off-axis angle of 2.5◦ by the Super-Kamiokande detector. The most recent results of the T2K collaboration for the neutrino and antineutrino channel have been published in Abe et al. [95, 96]. A separate analysis of the disappearance data in the neutrino and antineutrino channels has provided the determination of the best fit oscillation parameters for neutrinos and antineutrinos [96]. The obtained results are consistent so, no hint for CPT violation in the neutrino sector has been obtained [97, 98] 3 . In both cases, the preferred value of the atmospheric angle is compatible with maximal mixing. The combined analysis of the neutrino and

<sup>3</sup> See Barenboim et al. [99] for updated bounds on CPT violation from neutrino oscillation data.

antineutrino appearance and disappearance searches in T2K, that corresponds to a total sample of 7.482×10<sup>20</sup> POT in the neutrino mode, and 7.471 <sup>×</sup> <sup>10</sup><sup>20</sup> POT in the antineutrino mode, results in the best determination of the atmospheric oscillation parameters to date [95]

$$
\sin^2 \theta\_{23} = 0.532 \left( 0.534 \right), \left| \Delta m\_{32}^2 \right| = 2.545 \left( 2.510 \right) \times 10^{-3} \,\text{eV}^2 \,, \tag{14}
$$

for normal (inverted) mass ordering spectrum. Furthermore, thanks to the combination of neutrino and antineutrino data, T2K has already achieved a mild sensitivity to the CP violating phase, reducing the allowed 90% C.L. range of δ in radians to [−3.13, −0.39] for normal and [−2.09, −0.74] for inverted mass ordering [95].

In the NOνA (NuMI Off-Axis ν<sup>e</sup> Appearance) experiment, neutrinos produced at the Fermilab's NuMI beam are detected in Ash River, Minnesota, after traveling 810 km through the Earth. In the same way as the T2K experiment, the NOνA far detector is located slightly off the centerline of the neutrino beam coming from Fermilab. Thanks to this configuration, a large neutrino flux is obtained at energies close to 2 GeV, where the maximum of the muon to electron neutrino oscillations is expected. The most recent data release from the NOνA collaboration corresponds to an accumulated statistics of 6.05 <sup>×</sup> <sup>10</sup><sup>20</sup> POT in the neutrino run [100, 101]. For the muon antineutrino disappearance channel, 78 events have been observed, to be compared with 82 events expected for oscillation and 473 ± 30 events predicted under the no-oscillation hypothesis. The searches for ν<sup>µ</sup> → ν<sup>e</sup> transitions in the accelerator neutrino flux have reported the observation of 33 electron neutrino events, with an expected background of 8.2 ± 0.8 ν<sup>e</sup> events. The analysis of the NOνA Collaboration disfavors maximal values of θ<sup>23</sup> at the 2.6σ level [100]. On the other hand, from the analysis of the appearance channel it is found that the inverted mass ordering is disfavored at 0.46σ, due to the small number of event predicted for this ordering in comparison to the observed results [101]. Furthermore, the combination of appearance and disappearance NOνA data with the θ<sup>13</sup> measurement at the reactor experiments results disfavors the scenario with inverted neutrino mass ordering and θ<sup>23</sup> < π/2 at 93% C.L., regardless of the value of δ [101].

The right panel of **Figure 3** shows the 90 and 99% C.L. allowed region in the atmospheric neutrino oscillation parameters sin<sup>2</sup> θ<sup>23</sup> and 1m<sup>2</sup> <sup>31</sup> according to the MINOS, T2K and NOνA data for normal mass ordering [39]. Note the different scale for 1m<sup>2</sup> <sup>31</sup> with respect to the left panel. The three long–baseline experiments provide similar constraints on this parameter, while the constraint on θ<sup>23</sup> obtained from T2K is a bit stronger. One can also see some small differences between the preferred values of θ<sup>23</sup> by the three experiments. While T2K prefers maximal mixing, MINOS and NOνA show a slight preference for non-maximal θ23. In any case, these differences are not significant and the agreement among the three experiments is quite good. Although not shown here, the agreement for inverted mass ordering is a bit worse, since in that case the rejection of NOνA against maximal mixing is stronger, whereas the preference for θ<sup>23</sup> ∼ π/4 in T2K remains the same as for normal ordering.

### 2.5. Global Fit to Neutrino Oscillations

In the previous subsections, we have reviewed the different experimental neutrino data samples, discussing their dominant sensitivity to one or two oscillation parameters. However, every data sample offers subleading sensitivities to other parameters as well. Although the information they can provide about such parameters may be limited, in combination with the rest of data samples, relevant information can emerge. This constitutes the main philosophy behind global analyses of neutrino oscillation data: joint analyses trying to exploit the complementarity of the different experiments to improve our knowledge on the neutrino oscillation parameters. Here, we will show the results of a combined analysis of neutrino oscillation data in the framework of the three-flavor neutrino oscillation scheme.

**Figure 4** reports the 90, 95, and 99% C.L. allowed regions in the parameters sin<sup>2</sup> θ23, sin<sup>2</sup> <sup>θ</sup>13, <sup>|</sup>1m<sup>2</sup> <sup>31</sup>| and δ from the global fit in de Salas et al. [39] for normal and inverted mass ordering. For the allowed regions in the solar plane sin<sup>2</sup> <sup>θ</sup><sup>12</sup> <sup>−</sup> <sup>1</sup>m<sup>2</sup> 21, see **Figure 1**. The best fit points, along with the corresponding 1σ uncertainties and 90% C.L. ranges for each parameter, are quoted in **Table 1**. The relative uncertainties on the oscillation parameters at 1σ range from around 2% for the mass splittings to 7–10% (depending on the mass ordering) for sin<sup>2</sup> θ23. In case of the CP phase, the 1σ uncertainties are of the order of 15–20%. Note also that, at the 3σ level, the full range of δ is still allowed for normal ordering. For the case of inverted ordering, a third of the total range is now excluded at the 3σ level. These results are in good agreement with Capozzi et al. [40] and Esteban et al. [41].

Despite the remarkable sensitivity reached in the determination of most of the neutrino oscillation parameters, there are still three unknown parameters in the oscillation of standard three neutrino scheme: the octant of θ23, the value of the CP phase δ and the neutrino mass ordering. The current status of these still unknown parameters will be discussed next.

Let us now comment on the maximality/non-maximality and octant preference for the atmospheric mixing angle. So far, experimental neutrino data have not shown a conclusive preference for values of θ<sup>23</sup> smaller, equal or larger than π/4. Different experiments may show a limited preference for one of the choices, but for the moment all the results are consistent at the 3σ level. On the other hand, one finds that the available global analyses of neutrino data [39–41], using very similar data samples show slightly different results for the octant preference. For this particular case, one can find the origin of the possible discrepancies in the different treatment of the Super– Kamiokande atmospheric data. See the previous references for more details on the chosen approach at each work. The results in **Figure 4** and **Table 1**, corresponding to the analysis in de Salas et al. [39], show a preference for θ<sup>23</sup> in the first octant. This global best fit point corresponds to normal mass ordering, but a local minimum can also be found with θ<sup>23</sup> > π/4 and inverted mass ordering with a 1χ<sup>2</sup> <sup>=</sup> 4.3. In the same way, additional local minima can be found with θ<sup>23</sup> in the second octant and inverted mass spectrum and the other way around. All these possibilities are allowed at 90% C.L. as can be seen in the right panel of **Figure 4**. With current data, the status of the maximal atmospheric mixing is a bit delicate, being allowed only at 99%

TABLE 1 | Neutrino oscillation parameters summary determined from the global analysis in de Salas et al. [39]. The ranges for inverted ordering refer to the local minimum of this neutrino mass ordering.


<sup>a</sup>Local min. at sin<sup>2</sup> <sup>θ</sup><sup>23</sup> = 0.596 with 1χ<sup>2</sup> <sup>=</sup> 2.1 w.r.t. the global min.

<sup>b</sup>Local min. at sin<sup>2</sup> <sup>θ</sup><sup>23</sup> = 0.426 with 1χ<sup>2</sup> <sup>=</sup> 3.0 w.r.t. the global min. for IO.

C.L. However, this result may change after the implementation of the partially published data release of T2K [102] in the global fit.

In the same way, the current neutrino oscillation data do not offer a definitive determination for the neutrino mass ordering. Individual neutrino experiments show in general a limited sensitivity to the mass ordering, with the exception of the latest atmospheric data from Super-Kamiokande, that prefer normal mass ordering with a significance of 1χ<sup>2</sup> <sup>=</sup> 4.3. Note however that this data sample is not included in some of the global analyses of neutrino oscillations [39, 41]. The sensitivity to the mass ordering in the global analysis arises instead from the interplay of the different neutrino data, as a result of the existing correlations and tensions among the other neutrino parameters. Indeed, the three global analysis discussed in this review show a preference for normal mass ordering, although the significance may be different in each case, depending on the particular details of the specific global fit. In the work in de Salas et al. [39], discussed in a bit more details here, a preference for normal ordering over inverted is obtained, with a significance of 1χ<sup>2</sup> <sup>=</sup> 4.3. In any case, the results reported are not conclusive yet, and we will have to wait for the next generation of experiments devoted to this purpose (among others), such as DUNE [103], PINGU [104], ORCA [105], JUNO [106], or RENO-50 [107].

Finally, we comment on the sensitivity to the CP-violating phase δ. Prior to the publication of the antineutrino run data from T2K, combined analyses were already showing a weak preference for δ = 3π/2, while δ = π/2 was disfavored above the 2σ level [108–110]. This sensitivity, absent in all the individual data samples, emerged from the tension between the value of θ<sup>13</sup> measured at the reactor experiments and the preferred value of θ<sup>13</sup> for δ = π/2 in T2K. This scenario has changed after the release of T2K results from its antineutrino run and now the sensitivity to δ comes mainly from the combined analysis of the neutrino and antineutrino channel in T2K. The remaining experiments contribute only marginally to the determination of the CP–violating phase.

### 3. CURRENT BOUNDS ON NON–STANDARD INTERACTIONS

New neutrino interactions beyond the Standard Model are natural features in most neutrino mass models [111, 112]. As commented in the introduction, these Non–Standard Interactions (NSI) may be of Charged-Current (CC) or of Neutral-Current (NC) type. In the low energy regime, neutrino NSI with matter fields can be formulated in terms of the effective four-fermion Lagrangian terms as follows:

$$\mathcal{L}\_{\text{CC}-\text{NSI}} = -2\sqrt{2}G\_F \epsilon\_{\alpha\beta}^{\text{f}'\text{X}} \left( \bar{v}\_{\alpha} \gamma^{\mu} P\_L \ell\_{\beta} \right) \left( \bar{f}' \gamma\_{\mu} P\_{\text{X}} f \right), \tag{15}$$

$$\mathcal{L}\_{\text{NC}-\text{NSI}} = -2\sqrt{2}G\_F \epsilon^{\chi\chi}\_{\alpha\beta} \left(\bar{\upsilon}\_{\alpha}\gamma^{\mu}P\_L\upsilon\_{\beta}\right) \left(\bar{f}\gamma\_{\mu}P\_Xf\right). \tag{16}$$

where G<sup>F</sup> is the Fermi constant and P<sup>X</sup> denote the left and right chirality projection operators PR,<sup>L</sup> = (1 ± γ5)/2. The dimensionless coefficients ǫ ff ′X αβ and ǫ fX αβ quantify the strength of the NSI between leptons of α and β flavor and the matter field f ∈ {e, u, d} (for NC-NSI) and f 6= f ′ ∈ {u, d} (for CC-NSI). At the limit ǫ fX αβ → 0, we recover the standard interactions, while ǫαβ ∼ 1 corresponds to new interactions with strength comparable to that of SM weak interactions. If ǫαβ is non-zero for α 6= β, the NSIs violate lepton flavor. If ǫαα − ǫββ 6= 0, the lepton flavor universality is violated by NSI.

The presence of neutrino NSI may affect the neutrino production and detection at experiments as well as their propagation in a medium through modified matter effects [52, 53]. In the literature, it is common denoting the CC-NSI couplings as ǫ s αβ or ǫ d αβ since they may often affect the source (s) and detector (d) interactions at neutrino experiments. On the other hand, ǫ mf αβ is used to refer to the NC-NSI couplings with the fundamental fermion f generally affecting the neutrino propagation in matter (m). In this case, what is relevant for neutrino propagation in a medium is the vector part of interaction ǫ fV αβ = ǫ fL αβ + ǫ fR αβ. In fact, the neutrino propagation in a medium is sensitive to the following combinations<sup>4</sup>

$$
\epsilon\_{\alpha\beta} \equiv \epsilon\_{\alpha\beta}^{\epsilon V} + \frac{N\_{\mu}}{N\_{\varepsilon}} \epsilon\_{\alpha\beta}^{\mu V} + \frac{N\_{d}}{N\_{\varepsilon}} \epsilon\_{\alpha\beta}^{dV} \tag{17}
$$

so most of the bounds from oscillation experiments are presented in the literature in terms of ǫαβ rather than in terms of ǫ fV αβ. Inside the Sun, Nu/N<sup>e</sup> ≃ 2Nd/N<sup>e</sup> ≃ 1 [60] and inside the Earth, Nu/N<sup>e</sup> ≃ Nd/N<sup>e</sup> ≃ 3 [113]. When studying the effect of NSI at neutrino detection, there will be independent sensitivity for the left and right chirality coefficients ǫ fL αβ and ǫ fR αβ.

Although this kind of interactions has not been confirmed experimentally, their potential effects have been extensively studied in a large variety of physical scenarios. As a result, stringent bounds on their strength have been derived [11, 111, 112]. Moreover, it has been shown that NSI may interfere with neutrino oscillations in different contexts, giving rise to parameter degeneracies that can affect the robustness of the neutrino parameter determination. In this section, we will review these results.

### 3.1. NSI in Solar Experiments

NSI may affect the propagation of solar neutrinos within the Sun and the Earth as well as the detection, depending on the type of NSI considered. Before the confirmation of neutrino oscillation as the phenomenon responsible for the solar neutrino anomaly by the KamLAND experiment, NSI with massless neutrinos was also proposed as the mechanism behind this anomaly [114–118]. After KamLAND confirmed the phenomenon of mass–induced electron neutrino (antineutrino) oscillation, NSI was excluded as the main mechanism behind the solar neutrino oscillations, although its presence has been considered at subleading level in solar neutrino experiments, see for instance [9–11, 119, 120]. These analyses have found that a small amount of NSI, ǫ dV ee ≃ 0.3, is in better agreement with data than the standard solution at the level of 2σ. Palazzo [121] finds the best fit value to lie at ǫ dV <sup>e</sup><sup>τ</sup> − ǫ dV <sup>e</sup><sup>µ</sup> = 0.23. On one hand, this result is due to the non–observation of the upturn of the solar neutrino spectrum predicted by the standard LMA–MSW solution at around 3 MeV [9, 119, 121]. On the other hand, there exists a small tension between the preferred value of 1m<sup>2</sup> <sup>21</sup> by KamLAND and by solar experiments that can be eased by introducing NSI. More surprisingly, these studies revealed an alternative solution to the standard LMA–MSW, known as LMA-Dark or LMA-D solution [9–11], requiring NSI with strength ǫ dV τ τ − ǫ dV ee ≃ 1. The presence of this new degenerate solution to the solar neutrino anomaly, shown in **Figure 5**, can be understood in the framework of two-neutrino mixing as follows. Under this approximation (justified by the fact that sin<sup>2</sup> θ<sup>13</sup> ≪ 1), the two by two Hamiltonian matrix can be diagonalized with an effective mixing angle given by

$$\tan 2\theta\_{12}^{'''} = \frac{\sin 2\theta\_{12} (\Delta m\_{12}^2 / 2E)}{\cos 2\theta\_{12} (\Delta m\_{12}^2 / 2E) - (H\_m)\_{ee}}.\tag{18}$$

The splitting between two eigenvalues is given by

$$
\Delta^m = \left( \langle H\_m \rangle\_{\text{ee}}^2 + \left( \frac{\Delta m\_{21}^2}{2E} \right)^2 - \frac{\Delta m\_{21}^2}{E} \langle H\_m \rangle\_{\text{ee}} \cos 2\theta\_{12} \right)^{1/2} \,\, . \tag{19}
$$

Under the simultaneous transformations cos 2θ<sup>12</sup> → − cos 2θ<sup>12</sup> and (Hm)ee → −(Hm)ee, we find θ m <sup>12</sup> → π/2 − θ m <sup>12</sup> and <sup>1</sup><sup>m</sup> <sup>→</sup> <sup>1</sup><sup>m</sup> which means that the off-diagonal elements of the 2 × 2 Hamiltonian remains the same but the diagonal elements (the 11 and 22 elements) flip. That is, P(ν<sup>e</sup> → νe) changes to

<sup>4</sup>Note that some references use the definition <sup>ǫ</sup>αβ <sup>=</sup> P f Nf Nd ǫ fV αβ . It is very relevant to distinguish between both notations, since the reported bounds will be different by a factor of 3. For this reason, we prefer to quote directly the results in terms of the effective lagrangian coefficients ǫ fV αβ . In any case, for the analysis including Earth matter effects, the relation between the corresponding NSI couplings is straightforward.

P(ν<sup>c</sup> → νc) where ν<sup>c</sup> ≡ c23ν<sup>µ</sup> − s23ν<sup>τ</sup> is the combination that ν<sup>e</sup> converts to (that is hν<sup>c</sup> |ν3i = hν<sup>c</sup> |νei = 0). Since in two neutrino approximation, we can write P(ν<sup>e</sup> → νe) + P(ν<sup>e</sup> → νc) = 1, P(ν<sup>c</sup> → νe) + P(ν<sup>c</sup> → νc) = 1 and P(ν<sup>e</sup> → νc) = P(ν<sup>c</sup> → νe). We therefore conclude P(ν<sup>e</sup> → νe) = P(ν<sup>c</sup> → νc). As a result, under the transformation described above, P(ν<sup>e</sup> → νe) remains invariant. This transformation is not possible for the case of standard matter effects, where the value of ( √ Hm)ee is fixed to 2GFNe. However, if one considers the presence of neutrino NSI with the matter field f , the effective Hamiltonian in the medium is modified to:

$$\begin{aligned} H'\_m &= H\_m + H\_{NSI} = \sqrt{2} G\_{FN} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ &+ \sqrt{2} G\_F \sum\_f N\_f \begin{pmatrix} \epsilon\_{\epsilon\epsilon}^{f\prime} & \epsilon\_{\epsilon\mu}^{f\prime} & \epsilon\_{\epsilon\tau}^{f\prime} \\ \epsilon\_{\epsilon\mu}^{f\prime\*} & \epsilon\_{\mu\mu}^{f\prime} & \epsilon\_{\mu\tau}^{f\prime} \\ \epsilon\_{\epsilon\tau}^{f\prime\*} & \epsilon\_{\mu\tau}^{f\prime\*} & \epsilon\_{\tau\tau}^{f\prime} \end{pmatrix}. \end{aligned} \tag{20}$$

Allowing for sufficiently large values of the ǫ fV ee coupling in the effective Hamiltonian in matter H′ <sup>m</sup>, it is now possible to apply the transformation described above, obtaining a degenerate solution to the solar neutrino anomaly with cos 2θ<sup>12</sup> < 0. Notice that, letting cos 2θ<sup>12</sup> to change sign, we are violating the historical choice of keeping θ<sup>12</sup> in the first octant and, therefore, ν<sup>1</sup> will not be anymore the state giving the largest contribution to νe, but the lighter one between those two eigenstates that give the main contribution to νe. This change of definition is in fact equivalent to maintain the same convention regarding the allowed range for the mixing angle, but allowing 1m<sup>2</sup> <sup>21</sup> to be negative. Indeed, changing 1m<sup>2</sup> <sup>21</sup> → −1m<sup>2</sup> <sup>21</sup> instead of cos 2θ<sup>12</sup> → − cos 2θ12, we would find the same degeneracy. In other words, for a given Hm, solar neutrino data only determine the sign of the product 1m<sup>2</sup> <sup>21</sup> cos 2θ12, not the signs of 1m<sup>2</sup> <sup>21</sup> and cos 2θ<sup>12</sup> separately, and therefore there is a freedom in definition. Since the LMA-D solution was introduced in the literature keeping 1m<sup>2</sup> <sup>21</sup> positive while allowing θ<sup>12</sup> to vary in the range (0, π/2) [9] and this convention has become popular in the literature since then, we will use it along this review. Note also that the degeneracy found at the neutrino oscillation probability is exact only for a given composition of matter (i.e., for a given Nn/N<sup>p</sup> = Nn/Ne). The composition slightly varies across the Sun radius and of course is quite different for the Sun and the Earth. Because of this, the allowed regions in the neutrino oscillation parameter space for the LMA and LMA-D solutions are not completely degenerate. A small χ <sup>2</sup> difference between the best fit point of the LMA solution and the local minimum of LMA-Dark solution appears because the relevant data analyses take into account the varying composition of the Sun and the day-night asymmetry due to propagation in the Earth.

Unfortunately, the degeneracy between the LMA and LMA-Dark solutions could not be lifted by the KamLAND reactor experiment because KamLAND was not sensitive to the octant of the solar mixing angle due to the lack of matter effects. A possible way to solve this problem was proposed in Escrihuela et al. [10]. There, it was found that the combination of solar experiments, KamLAND and neutrino neutral–current scattering experiments, such as CHARM [122], may help to probe the LMA-D solution. The relevance of the degeneracy in the solar neutrino parameter determination has been explored recently in Coloma and Schwetz [123]. As discussed in this analysis, the ambiguity of LMA-D does not affect only the octant of the solar mixing angle but it also makes impossible the determination of the neutrino mass ordering at oscillation experiments. More recently, a global analysis of neutrino scattering and solar neutrino experiments was performed to further investigate the situation of the LMA-D solution [124]. Besides the accelerator experiment CHARM, the authors also considered the NuTeV experiment [125]. They found that the degenerate LMA-D solution may be lifted for NSI with down quarks, although it does not disappear for the case of neutrino NSI with up quarks. As discussed in that work, constraints from CHARM and NuTeV experiments can be however directly applied only for NSI with relatively heavy mediators. For the case of NSI mediated by lighter particles (above 10 MeV), constraints coming from coherent neutrino-nucleus scattering experiments may be used to resolve the degeneracy. Indeed, after the recent observation of such process at the COHERENT experiment [126], a combined analysis of neutrino oscillation data including the observed number of events in this experiment has excluded the LMA-D solution (for up and down quarks) at the 3σ level [127] 5 . One should, however, bear in mind that the analysis [127] assumes the mediator of interaction in Equation (16) is heavier than 50 MeV. As we shall see in sect V, for light mediator, their conclusion should be revised. Besides that, COHERENT data along with neutrino oscillation data has been used to improve the current bounds on the flavor–diagonal NSI parameters [127] 6 :

$$-0.09 < \epsilon\_{\rm tr}^{\mu V} < 0.38\,, \ -0.075 < \epsilon\_{\rm tr}^{dV} < 0.33\,, \tag{90\% C.L.}\,\text{.}\tag{21}$$

These limits on NC vector interactions of ν<sup>τ</sup> improve previous bounds by one order of magnitude [10, 11, 128]. For the flavor– changing NC NSI couplings, however, the improvement is much smaller<sup>7</sup> :

$$-0.073 < \epsilon\_{\epsilon\mu}^{\mu V} < 0.044, \ -0.07 < \epsilon\_{\epsilon\mu}^{dV} < 0.04 \quad \text{(90\% C.L.)}, \tag{22}$$

$$-0.15 < \epsilon\_{\epsilon\mu}^{\mu V} < 0.13, \ -0.13 < \epsilon\_{\epsilon\mu}^{dV} < 0.12 \quad \text{(90\% C.L.)}. \tag{23}$$

The spectrum of coherent elastic neutrino–nucleus scattering events at COHERENT has also been analyzed to constrain the amplitude of NSI in Liao and Marfatia [130].

Besides their impact on solar neutrino propagation, NSI can also affect the detection processes at solar neutrino experiments. In experiments like Super–Kamiokande and Borexino, for instance, the presence of NSI may modify the cross section of neutrino elastic scattering on electrons, used to observe solar neutrinos. Analyzing data from solar neutrino experiments, and in particular the effect of NSI on neutrino detection in Super-Kamiokande, in combination with KamLAND, Bolanos et al. [131] reported limits on the NSI parameters which are competitive and complementary to the ones obtained from laboratory experiments. For the case of ν<sup>e</sup> NSI interaction with electrons, the reported bounds (taking one parameter at a time) are:

$$-0.021 < \epsilon\_{\rm ee}^{\epsilon L} < 0.052, \quad -0.18 < \epsilon\_{\rm ee}^{\epsilon R} < 0.51 \quad \text{(90\% C.L.)}, \tag{24}$$

while for the case of ν<sup>τ</sup> NSI interaction with electrons, looser constraints are obtained:

$$-0.12 < \epsilon\_{\rm tt}^{eL} < 0.060, \quad -0.99 < \epsilon\_{\rm tt}^{eR} < 0.23 \quad \text{(90\% C.L.)}.\tag{25}$$

The sensitivity of the Borexino solar experiment to NSI has also been investigated in Berezhiani et al. [132] and Agarwalla et al. [133]. Using <sup>7</sup>Be neutrino data from Borexino Phase I, the following 90% C.L. bounds have been derived [133]

$$\begin{aligned} -0.046 &< \epsilon\_{\text{eff}}^{\varepsilon L} < 0.053, \qquad -0.21 < \epsilon\_{\text{eff}}^{\varepsilon R} < 0.16, \\ -0.23 &< \epsilon\_{\text{tt}}^{\varepsilon L} < 0.87, \qquad -0.98 < \epsilon\_{\text{tt}}^{\varepsilon R} < 0.73. \end{aligned} \tag{26}$$

As can be seen, the NSI constraints obtained from Borexino and the combined analysis of solar (mainly Super-Kamiokande) and KamLAND data are comparable. It is expected that future results from Borexino Phase II, as well as the combination of all solar data, including Borexino, plus KamLAND data would allow a significant improvement on the current knowledge of neutrino NSI with matter [134].

### 3.2. NSI in Atmospheric Neutrino Experiments

The impact of non-standard neutrino interactions on atmospheric neutrinos was originally considered in Fornengo et al. [135, 136] and Friedland et al. [137, 138]. Assuming a two–flavor neutrino system, it was shown [136] that the presence of large NSI couplings together with the standard mechanism of neutrino oscillation can spoil the excellent description of the atmospheric neutrino anomaly given by neutrino oscillations. Thus, quite strong bounds on the magnitude of the non–standard interactions were derived. Using atmospheric neutrino data from the first and second phase of the Super–Kamiokande experiment, the following constraints were obtained, under the two–flavor neutrino approach [139]:

$$|\epsilon\_{\mu\tau}^{dV}| < 0.011, \quad |\epsilon\_{\mu\mu}^{dV} - \epsilon\_{\tau\tau}^{dV}| < 0.049 \quad \text{(90\% C.L.)}.\tag{28}$$

However, Friedland et al. [137, 138] showed that a three–family analysis significantly relaxes the previous bounds in such a way that the values of the NSI couplings with quarks comparable to the standard neutral current couplings can be still compatible with the Super–Kamiokande atmospheric data. A more recent three–neutrino analysis of NSI in the atmospheric neutrino flux can be found in Gonzalez-Garcia et al. [140], where the following limits on the effective NSI couplings with electrons have been obtained:

$$-0.035 \ \text{(}-0.035\text{)} < \epsilon\_{\mu\text{t}}^{\epsilon V} < 0.018 \text{(}0.035\text{)},$$

$$|\epsilon\_{\text{rt}}^{\epsilon V} - \epsilon\_{\mu\mu}^{\epsilon V}| < 0.097 \text{(}0.11\text{)} \quad \text{(}90\% \text{ C.L.)} \quad \text{(}29\text{)}$$

for the case of real (complex) ǫ eV µτ coupling.

The IceCube extension to lower energies, DeepCore, has made possible the observation of atmospheric neutrinos down to 5 GeV with unprecedented statistics. Indeed, with only 3 years of data, DeepCore allows the determination of neutrino oscillation parameters with similar precision as the one obtained from the long–lived Super–Kamiokande or the long–baseline accelerator experiments [90]. Focusing now on its sensitivity to NSI, the idea of using IceCube data to constrain the µ − τ submatrix of ǫ was first proposed in Esmaili and Smirnov [141]. Using the most

<sup>5</sup>The analysis of atmospheric neutrino data performed in Coloma et al. [127] employs two simplifying assumptions. First, the solar mass splitting is neglected. Second, rather than taking the most general matter potential, it is assumed that two of eigenvalues of this matrix are degenerate. As a result, the derived constraints on the NSI couplings are more stringent than what we expect in the most general case. <sup>6</sup>Notice that since the beam at COHERENT does not contain ν<sup>τ</sup> , this experiment cannot directly probe ǫττ . The bounds on ǫττ come from combining the limits on ǫµµ and ǫee by COHERENT with the bounds on ǫττ − ǫee and ǫττ − ǫµµ from oscillation experiments.

<sup>7</sup>Note that the existing bounds on ǫ qV <sup>e</sup><sup>µ</sup> were revised in Biggio et al. [129] showing that previously derived loop bounds do not hold in general.

recent data release from DeepCore, the IceCube collaboration has reported the following constraints on the flavor–changing NSI coupling [142]:

$$-0.0067 < \epsilon\_{\mu\tau}^{dV} < 0.0081 \qquad \text{(90\% C.L.)}.\tag{30}$$

From a different data sample containing higher energy neutrino data from IceCube, the authors of Salvado et al. [143] have derived somewhat more restrictive bounds on the same NSI interactions:

$$-0.006 < \epsilon\_{\mu\tau}^{dV} < 0.0054 \qquad \text{(90\% C.L.)}.\tag{31}$$

Both results are fully compatible and constitute the current best limits on NSI in the ν<sup>µ</sup> − ν<sup>τ</sup> sector.

Future prospects on NSI searches in atmospheric neutrino experiments have been considered in the context of PINGU, the future project to further lower the energy threshold at the IceCube observatory. Choubey and Ohlsson [144] shows that, after three years of data taking in PINGU in the energy range between 2 and 100 GeV, the Super–Kamiokande constraints on the NSI couplings may be improved by one order of magnitude:

$$-0.0043 < \epsilon\_{\mu\tau}^{\varepsilon V} < 0.0047, \quad -0.03 < \epsilon\_{\tau\tau}^{\varepsilon V} < 0.017 \quad \text{(90\% C.L.)}. \tag{32}$$

Likewise, the impact of NSI interactions on atmospheric neutrinos on the future India-based Neutrino Observatory (INO) has been analyzed in Choubey et al. [145]. Besides discussing its constraining potential toward NSI, this work studies how the sensitivity to the neutrino mass hierarchy of INO, one of the main goals of the experiment, may change in the presence of NSI.

Notice that the above bounds have been derived from the study of the atmospheric neutrinos flux at neutrino telescopes. Gonzalez-Garcia et al. [146] discusses the effects of NSI on high energy astrophysical neutrinos detected by IceCube when they propagate through the Earth.

### 3.3. NSI in Reactor Experiments

Modern reactor neutrino experiments, like Daya Bay, RENO and Double Chooz, provide a very accurate determination of the reactor mixing angle θ<sup>13</sup> [78, 147, 148]. Being at the precision era of the neutrino parameter determination, it is imperative to investigate the robustness of this successful measurement in the presence of NSI. Leitner et al. [12], Agarwalla et al. [13], Ohlsson and Zhang [149], Girardi and Meloni [150], and Khan et al. [151] have addressed this point. In principle, short–baseline reactor experiments may be affected by the presence of new neutrino interactions in β and inverse-β decay processes, relevant for the production and detection of reactor antineutrinos [152]. The NSI parameters relevant for these experiments are the CC NSI couplings between up and down quarks, positrons and antineutrinos of flavor α, ǫ ud eα . Considering unitarity constraints on the CKM matrix as well as the non-observation of neutrino oscillations in the NOMAD experiment, one may find the following constraints on these CC NSI couplings [33]:

$$|\epsilon\_{\epsilon\alpha}^{\rm udV}| < 0.041 \,\, |\, |\epsilon\_{\epsilon\mu}^{\rm udL}| < 0.026 \,\, |\, |\epsilon\_{\epsilon\mu}^{\rm udR}| < 0.037 \,\, \text{(90\% C.L.)}.\tag{33}$$

Agarwalla et al. [13] explored the correlations between the NSI parameters and the reactor mixing angle determination, showing that the presence of NSI may lead to relatively large deviations in the measured value of θ<sup>13</sup> in Daya Bay, as it can be seen in **Figure 6**. Conversely, the total number of events observed in Daya Bay was used to constrain the corresponding NSI couplings under two assumptions: (i) perfect theoretical knowledge of the reactor neutrino flux in absence of NSI and (ii) assuming a conservative error on its total normalization. In the latter case, it was shown that assuming an uncertainty of 5% on the reactor flux can relax the bounds by one order of magnitude, obtaining the following conservative limits on the NSI strengths<sup>8</sup>

$$|\epsilon\_{\epsilon\epsilon}^{\mathrm{udP}}| < 0.015 \text{, } |\epsilon\_{\epsilon\mu}^{\mathrm{udP}}| < 0.18 \text{, } |\epsilon\_{\epsilon\pi}^{\mathrm{udP}}| < 0.18 \text{, } \text{(90\% C.L.)}, \text{ (34)}$$

with P = L,R,V,A. Note that these results improve the existing bounds on the ǫ ud ee coupling reported above. On the other hand, one finds that an improved knowledge of the standard absolute neutrino flux from nuclear reactors together with a larger data sample from Daya Bay will result in a more stringent bound on the other two couplings in the near future. Notice also that previous results have been obtained assuming that the NSI couplings at neutrino production and detection satisfy ǫ s αβ = ǫ d∗ αβ. In this case, the presence of NSI would only produce a shift in the oscillation amplitude without altering the L/E pattern of the oscillation probability, and therefore, the analysis of the total neutrino rate in Daya Bay provides enough information. The investigation of more exotic scenarios where ǫ s αβ 6= ǫ d∗ αβ will require the spectral analysis of the Daya Bay data [12].

NSI at future intermediate baseline reactor experiments like JUNO and RENO-50 (see for instance, Ohlsson et al. [153] and Khan et al. [151]) are discussed at section 5.

### 3.4. NSI in Long–Baseline Neutrino Experiments

Besides neutrino production and detection, NSI can also modify the neutrino propagation through the Earth in long– baseline accelerator experiments<sup>9</sup> . This effect will be larger for experiments with larger baselines such as MINOS or NOνA. Using their neutrino and antineutrino data sample, the MINOS Collaboration reported the following bounds on the flavorchanging NC NSI with electrons [154]:

$$-0.20 < \epsilon^{eV}\_{\mu\tau} < 0.07 \text{ (90\% C.L.)}.\tag{35}$$

MINOS appearance data were also used to constrain NSI interactions between the first and third family [155], although the reported bound, |ǫ eV eτ | < 3.0 (90% C.L.) does not improve the previous limits on that parameter [33].

Regarding the long–baseline experiment NOνA, the presence of NSI in the neutrino propagation has been proposed as a way to solve the mild tension between the measured values of the

<sup>8</sup>The NSI parameters probed in this kind of analysis obtain contributions from the (V±A) operators in Equation 15 so, taking one parameter at a time, the derived bounds apply to all the chiralities.

<sup>9</sup> See for instance Kopp et al. [152], where the impact of NSI on long–baseline experiment is analyzed in detail.

atmospheric mixing angle in T2K and NOνA [156]. Under this hypothesis, the deviation of the NOνA preferred value for θ<sup>23</sup> from maximal mixing would be explained through the NSImodified matter effects. The T2K experiment, with a shorter baseline, has a limited sensitivity to matter effects in the neutrino propagation so, its θ<sup>23</sup> measurement would be unaffected by NSI. Note, however, that the size of the NSI required to reproduce the observed results is of the same order as the standard neutrino interaction [to be more precise ǫe<sup>τ</sup> , (ǫττ − ǫµµ) ≃ (ǫττ − ǫee) ∼ O(1)].

The presence of NSI has also been considered to reconcile the measured value of θ<sup>13</sup> in reactor experiments and T2K [157]. In that case, it is suggested that CC-NSI in the neutrino production and detection processes may be responsible for the different values of the reactor mixing angle measured in Daya Bay and T2K.

Finally, it has been shown that long–baseline neutrino facilities can also suffer from degeneracies in the reconstruction of some parameters due to the existence of new neutrino interactions with matter. For instance, Forero and Huber [14] states that NC NSI may affect the sensitivity to the CP–violating phase δ in experiments like T2K and NOνA. According to this analysis, it would be possible confusing signals of NSI with a discovery of CP violation, even if CP is conserved in nature. This result is illustrated in **Figure 7**, where it is shown how the standard CP–violating scenario may be confused with an hybrid standard plus NSI CP–conserving scenario.

Future sensitivities to NSI as well as the presence of new degeneracies due to NSI in future long–baseline experiments such as DUNE, T2HK and T2HKK are analyzed in more detail in section 5. It is worth mentioning that CC-NSI, affecting the production and detection of neutrinos can show up also in short baseline experiments [158–160].

### 3.5. NSI in Non-oscillation Neutrino Experiments

Neutrino scattering experiments constitute a very precise tool toward the understanding of neutrino interactions with matter. Indeed, this kind of experiments has been often used to measure the electroweak mixing angle θ<sup>W</sup> [161]. Non–standard neutrino interactions may contribute significantly to the neutrino– electron elastic scattering cross section and therefore they cannot be ignored when studying this process. Barranco et al. [162] compiled most of the neutrino scattering experiments potentially modified by the presence of NSI, from the neutrino accelerator–based experiments LSND and CHARM to the short– baseline neutrino reactor experiments Irvine, Rovno and MUNU, including as well as the measurement of the process e +e <sup>−</sup> → ννγ at LEP. From a combined analysis of all experimental data, allowed ranges on the ǫ e αβ were obtained. Some of these results are among the current strongest constraints on NSI couplings, and are reported in **Table 2**. The antineutrino–electron scattering data collected by the TEXONO Collaboration has been also used to constrain the presence of neutrino NC NSI with electrons [163] as well as CC NSI at neutrino production and detection [164].

In order to constrain the NSI between neutrinos and quarks, one may use data from the neutrino–nucleus experiments NuTeV, CHARM and CDHS. From the combination of atmospheric and accelerator data from NuTeV, CHARM and CDHS, the following limits on the non–universal vectorial and axial NSI parameters were derived [165]:

$$|\epsilon\_{\mu\mu}^{dV}| < 0.042, \qquad -0.072 < \epsilon\_{\mu\mu}^{dA} < 0.057 \quad \text{(90\% C.L.)} \quad \text{(36)}$$

For the case of the flavor changing NSI couplings (with q = u, d)

$$|\epsilon^{qV}\_{\mu\text{tr}}| < 0.007, \qquad |\epsilon^{qA}\_{\mu\text{tr}}| < 0.039 \quad \text{(90\% C.L.)}.\tag{37}$$

Under this category we include also the first observation of coherent neutrino–nucleus scattering observed at the COHERENT experiment recently [126]. As discussed above, the COHERENT data have been used to constrain neutrino NSI with quarks in Coloma et al. [127] and Liao and Marfatia [130]. The combination of solar neutrino oscillation data with COHERENT has been exploited to investigate the status of the solar degenerate solution LMA-D.

TABLE 2 | Bounds on flavor diagonal NC NSI couplings.


<sup>a</sup>Bound adapted from ǫ eV ττ .

### 3.6. Summary of Current Bounds on NSI Parameters

Here we summarize the current constraints on the NSI couplings from different experiments discussed throughout this section. For more details about the assumptions considered in each case, we refer the reader to the previous subsections as well as to the original references where the constraints have been calculated. The limits summarized in **Tables 2**–**4** have been obtained assuming only one nonzero NSI coupling at a time.

**Table 2** contains the limits on the flavor diagonal NC NSI couplings between neutrinos and electrons ǫ eP αα and neutrinos and quarks ǫ qP αα, with P = L, R,V, A being the chirality index and q = u, d. The table indicates the origin of the reported bound as well as the reference where it has been obtained as well. Most of the limits have been derived from the combination of neutrino oscillation and detection or production experimental results. For instance, the joint analysis of atmospheric neutrino data and accelerator measurements in NuTeV, CHARM and CDHS [165], or solar and KamLAND data together with the recent bounds of COHERENT [127] <sup>10</sup>. In other cases the constraints reported in the table come just from one type of experiment, as the limits derived only from CHARM [128], TEXONO [163] or atmospheric data [140]. Note that, for the latter case, we have adapted the bound on ǫ eV ττ reported in Gonzalez-Garcia et al. [140] to the corresponding bound for quarks, ǫ qV ττ .

**Table 3** collects the limits of the flavor changing NC NSI couplings between neutrinos and electrons ǫ eP αβ and neutrinos and quarks ǫ qP αβ, with the same conventions indicated above for P and q. As discussed before, in this case most of the bounds also emerge from the complementarity of different types of experiments, as the combination of reactor and accelerator non-oscillation experiments in Barranco et al. [162]. On the other hand, the first analyses on NSI obtained from IceCube data [142, 143] offer very strong bounds on ǫ qV µτ . This last constraint has also been adapted to get the equivalent bound for NSI with electrons, ǫ eV µτ .

Finally, **Table 4** contains the limits on the neutrino CC NSI with quarks and electrons (semileptonic CC NSI) and the CC NSI with leptons only (purely-leptonic CC NSI) in terms of the couplings ǫ udP αβ and ǫ ll′P αβ , respectively. The former ones, have been discussed in the context of the neutrino production and detection in the Daya Bay reactor experiment, as analyzed in Agarwalla

<sup>10</sup>The bounds in Coloma et al. [127] assume mediator mass to be heavier than ∼50 MeV. As we shall discuss in the next section, these bounds do not apply for mediator mass lighter than ∼10 MeV.

TABLE 3 | Bounds on flavor changing NC NSI couplings.


<sup>a</sup>Bound adapted from ǫ qV µτ .

TABLE 4 | Bounds on CC NSI couplings.


et al. [13]. Previous bounds on this type of NSI have been derived using the negative searches for neutrino oscillations at short distances in the NOMAD experiment [166, 167], as reported in the table [33]. Constraints on leptonic CC NSI using the results of the KARMEN experiment [168] as well as the deviations of Fermi's constant G<sup>F</sup> in the presence of these interactions, have also been obtained in Biggio et al. [33]. We refer the reader to that work for further details on the derivation of these constraints.

### 4. VIABLE MODELS LEADING TO SIZEABLE NSI

As we saw in the previous section, neutral current NSI of neutrinos with matter fields can lead to observable effect on neutrino oscillation provided that the NSI parameters ǫαβ are large enough. As briefly discussed in the introduction, it is possible to build viable models for NSI by invoking an intermediate state of relatively light mass (∼10 MeV) which has escaped detection so far because of its very small coupling. In this chapter, we review the models that give rise to sizeable NSI through integrating out a new gauge boson Z ′ with a mass smaller than ∼100 MeV. We however note that an alternative model has been suggested [169] in which NSI are obtained from SU(2)<sup>L</sup> scalar doublet-singlet mixing. We shall not cover this possibility in the present review. The models described in this chapter introduce a new U(1)′ gauge interaction which is responsible for NSI between neutrinos and quarks.

In section 4.1, we describe the general features of the model gauging a linear combination of lepton flavors and Baryon number with a light O(10 MeV) gauge boson. We then outline general phenomenological consequences. We show how a simple economic model can be reconstructed to reproduce the NSI pattern that gives the best fit to neutrino data, solving the small tension between KamLAND and solar neutrino by explaining the suppression of the upturn in the low energy part of the solar neutrino spectrum. In section 4.2, we describe another model which can provide arbitrary flavor structure ǫ u αβ = ǫ d αβ (both lepton flavor violating and lepton flavor conserving) without introducing new interactions for charged leptons. In section 4.3, the impact of the recent results from the COHERENT experiment is outlined.

# 4.1. NSI from New U(1)′

In this section, we show how we can build a model based on U(1)′ × SU(2)<sup>L</sup> × U(1)<sup>Y</sup> gauge symmetry which gives rise to NSI for neutrinos. Notice that the NSI of interest for neutrino oscillation involves only neutrinos and quarks of first generation which make up the matter. However, to embed the scenario within a gauge symmetric theory free from anomalies, the interaction should involve other fermions.

Let us first concentrate on quark sector and discuss the various possibilities of U(1)′ charge assignment. Remember that, in the flavor basis by definition, the interaction of Wµ boson with quarks is diagonal: Wµ P<sup>3</sup> i=1 u¯iLγ <sup>µ</sup>diL, where i is the flavor index. To remain invariant under U(1)′ , uiL and diL should have the same values of U(1)′ charge. As discussed in sect II.A, the SNO experiment has measured the rate of neutral current interaction of solar neutrinos by Deuteron dissociation ν + D → ν + p + n. In general, a large contribution to neutral current interaction from new physics should have affected the rate measured by SNO but this process, being a Gamow-Teller transition, is only sensitive to the axial interaction. In order to maintain the SM prediction for the total neutrino flux measured at the SNO experiment via NC interactions, the coupling to (at least the first generation of) quarks should be non-chiral. Thus, the U(1)′ charges of u1L, u1R, d1<sup>L</sup> and d1<sup>R</sup> should be all equal. In principle, different generations of quarks can have different U(1)′ charges. Such a freedom opens up abundant possibilities for anomaly cancelation. However, if the coupling of the new gauge bosons to different quark generations is non-universal, in the quark mass basis, off-diagonal couplings of form Z ′ µ q¯iγ µqj |i6=<sup>j</sup> appear which can lead to q<sup>i</sup> → Z ′ q<sup>j</sup> with a rate enhanced by m<sup>3</sup> qi /m<sup>2</sup> <sup>Z</sup>′ due to longitudinal component of Z ′ . These bounds are discussed in great detail in Babu et al. [170]. To avoid these decays, we assume the quarks couple to Z ′ universally. In other words, the U(1)′ charges of quarks are taken to be proportional to baryon number, B. Yukawa couplings of quarks to the SM Higgs will then be automatically invariant under U(1)′ .

Let us now discuss the couplings of leptons to the new gauge boson. There are two possibilities: (1) U(1)′ charges are assigned to a combination of lepton numbers of different flavors. In this case, the U(1)′ charges of charged leptons and neutrinos will be equal. (2) Neutrinos couple to Z ′ through mixing with a new fermion with mass larger than mZ′ . In this case charged leptons do not couple to Z ′ at tree level. We shall return to the second case in section 4.2. In the present section, we focus on the first case. As discussed in Farzan and Shoemaker [26], it is possible to assign U(1)′ charge to linear combinations of leptons which do not even correspond to charged lepton mass eigenstates. However, let us for the time being study the charge assignment as follows

$$a\_{\varepsilon}L\_{\varepsilon} + a\_{\mu}L\_{\mu} + a\_{\varepsilon}L\_{\varepsilon} + B. \tag{38}$$

Denoting the new gauge coupling by g ′ , the coupling of each generation of leptons and quarks to Z ′ are, respectively, g ′aα and g ′ /3. There are strong bounds on the new couplings of the electrons. If a<sup>e</sup> 6= 0, Z ′ with a mass of ∼ 10 MeV will dominantly decay into e −e + so strong bounds from beam dump experiments apply. These bounds combined with supernova cooling study yield g ′a<sup>e</sup> <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>11</sup> (see Figure 4 of Harnik et al. [171].) On the other hand, for mZ′ < m<sup>π</sup> , the bound from π <sup>0</sup> <sup>→</sup> <sup>γ</sup> <sup>Z</sup> ′ is g ′ <sup>&</sup>lt; <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> [172] (see **Figure 8** which is taken from Farzan and Heeck [25]). These bounds are too stringent to lead to a discernible ǫee. We therefore set a<sup>e</sup> = 0 which means at tree level, neither electron nor ν<sup>e</sup> couple to Z ′ . With such charge assignment, we obtain

$$
\epsilon^{\mu}\_{\alpha a} = \epsilon^{d}\_{\alpha a} = \frac{g'^2 a\_{\alpha}}{6\sqrt{2}G\_F m\_{Z'}^2} \quad \text{and} \quad \epsilon^{\mu}\_{\alpha \beta} = 0|\_{\alpha \neq \beta}. \tag{39}
$$

Notice that, with this technique, we only obtain lepton flavor conserving NSI. For neutrino oscillation not only the absolute value of ǫαα − ǫββ but also its sign is important. In fact, neutrino oscillation data favor positive value of ǫee−ǫµµ ≃ ǫee−ǫττ ∼ 0.3. If a<sup>µ</sup> + a<sup>τ</sup> = −3, the anomalies cancel without any need for new generations of leptons and/or quarks. However, just like in B − L and L<sup>µ</sup> − L<sup>τ</sup> gauge theories, the presence of righthanded neutrinos is necessary to cancel the U(1)′ −U(1)′ −U(1)′ anomaly. Let us take a<sup>µ</sup> = a<sup>τ</sup> = −3/2 so that anomalies cancel; moreover, we obtain ǫµµ = ǫττ . We can then accommodate the best fit with

$$g' = 4 \times 10^{-5} \frac{m\_{Z'}}{10 \text{ MeV}} \left(\frac{\epsilon\_{ee} - \epsilon\_{\mu\mu}}{0.3}\right)^{1/2}.\tag{40}$$

For the LMA-Dark solution ǫee−ǫµµ < 0 is required, so the value of a<sup>µ</sup> ≃ a<sup>τ</sup> should be positive. As a result, more chiral fermions are needed to be added to cancel anomalies. We will return to this point later.

Since the U(1)′ charges of the left-handed and right-handed charged leptons are equal, their Yukawa coupling (and therefore their mass terms) preserve U(1)′ automatically. We should however consider the mass matrix of neutrinos with more care. While the flavor diagonal elements of neutrino mass matrix can be produced without any need for U(1)′ breaking, if aα is not universal, obtaining the neutrino mass mixing requires symmetry breaking. As mentioned above, right-handed neutrinos are also required to cancel anomalies. If the masses of neutrinos are of Dirac type, right-handed neutrinos will be as light as left-handed neutrinos. They can be produced in the early universe via U(1)′ coupling so, if they are light, they can contribute to the relativistic degrees of freedom. To solve both problems at one shot, we can invoke the seesaw mechanism. For simplicity, we take a<sup>µ</sup> = a<sup>τ</sup> so that the mixing between the second and third generation does not break U(1)′ . Generalization to a<sup>µ</sup> 6= a<sup>τ</sup> will be straightforward. Let us denote the right-handed neutrino of generation "i" by N<sup>i</sup> . Under U(1)′ ,

$$N\_1 \to N\_1, \ N\_2 \to e^{ia\_\mu \alpha} N\_2 \text{ and } N\_3 \to e^{ia\_\ell \alpha} N\_3 = e^{ia\_\mu \alpha} N\_3. \tag{41}$$

Dirac mass terms come from

$$\begin{aligned} \lambda\_1 N\_1^T H^c \boldsymbol{c} \boldsymbol{L}\_{\boldsymbol{\varepsilon}} + \lambda\_2 N\_2^T H^c \boldsymbol{c} \boldsymbol{L}\_{\mu} + \lambda\_3 N\_3^T H^c \boldsymbol{c} \boldsymbol{L}\_{\boldsymbol{\varepsilon}} + \lambda\_4 N\_2^T H^c \boldsymbol{c} \boldsymbol{L}\_{\boldsymbol{\varepsilon}} \\ + \lambda\_5 N\_3^T H^c \boldsymbol{c} \boldsymbol{L}\_{\mu} + \boldsymbol{H} \boldsymbol{c}. \end{aligned} \tag{42}$$

By changing the basis, either of λ<sup>4</sup> and λ<sup>5</sup> can be set to zero, but the nonzero one will mix the second and the third generations. Moreover, we add electroweak singlet scalars S<sup>1</sup> and S<sup>2</sup> with U(1)′ charges −2a<sup>µ</sup> and −aµ, respectively. We can then write the following potential

$$\begin{aligned} \left(M\_1 N\_1^T c N\_1 + S\_1 \{A\_2 N\_2^T c N\_2 + A\_3 N\_3^T c N\_3 + A\_{23} N\_2^T c N\_3\}\right) \\ + S\_2 \{B\_2 N\_1^T c N\_2 + B\_3 N\_1^T c N\_3\} + H.c \end{aligned} \tag{43}$$

Once S<sup>1</sup> and S<sup>2</sup> develop a vacuum expectation value (VEV), U(1)′ will be broken leading to the desired neutrino mass and mixing scheme. The VEVs of S<sup>1</sup> and S<sup>2</sup> induce a mass of

$$\text{g}'a\_{\mu}\text{(}4\text{\textdegree S\_1}\text{)}^2 + \text{\textdegree S\_2}\text{)}^2\text{)}^{1/2} \tag{44}$$

American Physical Society.

for the Z ′ boson. Taking g ′a<sup>µ</sup> <sup>∼</sup> <sup>10</sup>−<sup>5</sup> <sup>−</sup> <sup>10</sup>−<sup>4</sup> , we find that as long as <sup>h</sup>S1i ∼ hS2i ∼ 100 GeV(10−<sup>4</sup> /g ′aµ), the contribution to the Z ′ mass will be ∼10 MeV as desired. In case that more scalars charged under U(1)′ are added to the model (we shall see examples in section 4.2), the Z ′ mass receives further contributions.

For mZ′ < m<sup>π</sup> , the Z ′ can decay only to neutrino pair at tree level with a lifetime of

$$c\tau\_{Z'} \sim 10^{-9} \text{ km} \left(\frac{7 \times 10^{-5}}{\text{g}'}\right)^2 \left(\frac{10 \text{ MeV}}{m\_{Z'}}\right) \frac{1}{a\_{\mu}^2 + a\_{\pi}^2}.$$

As a result, Z ′ evades the bounds from the beam dump experiments. In the following, we go through possible experiments that can search for the Z ′ boson.

In the presence of new interactions, new decay modes for charged mesons open up: K <sup>+</sup> → l <sup>+</sup> + ν + Z ′ and π <sup>+</sup> → l <sup>+</sup> + ν + Z ′ . The typical upper bounds from meson decay are of order of O(0.001) [173–176] which are too weak to be relevant for our models; see **Figures 10**, **11**, which are taken from Bakhti and Farzan [173]. As shown in Bakhti and Farzan [173], the bound on the ν<sup>e</sup> coupling to the Z ′ boson can be dramatically improved by customized searches for three body decays (K <sup>+</sup> → e <sup>+</sup> + missing energy) and (π <sup>+</sup> → e <sup>+</sup> + missing energy).

In principle, Z ′ can kinetically mix with the hypercharge gauge boson which gives rise to Z ′ mixings both with the photon and the Z bosons. Even if we set the kinetic mixing to zero at tree level, it can be produced at loop level as long as there are particles charged under both U(1) gauge symmetries. Going to a basis where the kinetic terms of gauge bosons is canonical, the Z ′ boson obtains a coupling to the electron given by eǫ where ǫ is the kinetic mixing between Z ′ and the photon. This coupling can affect neutrino interaction with the electron on which there are strong bounds from solar experiments (mostly Super-Kamiokande and Borexino) [131, 133]. Kamada and Yu [177], setting the tree level kinetic mixing equal to zero, has calculated the kinetic mixing for the L<sup>µ</sup> − L<sup>τ</sup> models and has found it to be finite and of order of eg′ /8π 2 . The Borexino bound [133] can then be translated into g ′ e < <sup>∼</sup> <sup>10</sup>−<sup>4</sup> which can be readily satisfied for <sup>g</sup> ′ < 10−<sup>4</sup> . The loop contribution to the photon Z ′ mixing from a charged particle is very similar to its contribution to the vacuum polarization (photon field renormalization) replacing (qe) <sup>2</sup> with (qe)(aαg ′ ). In case of the L<sup>µ</sup> − L<sup>τ</sup> gauge symmetry, a<sup>e</sup> = 0, a<sup>µ</sup> = −a<sup>τ</sup> and since the electric charges of µ and τ are the same, the infinite parts of their contribution to the mixing cancels out. In general, we do not however expect such a cancelation and counter terms are therefore required. Once we open up the possibility of tree level kinetic mixing, the sum of tree level and loop level mixing can be set to arbitrarily small value satisfying any bound.

The above discussion on the Z ′ −γ kinetic mixing also applies to the Z ′−Z kinetic mixing. Here, we should also check the Z−Z ′ mass mixing [178]. It is straightforward to show that, since the Z ′ couplings are taken to be non-chiral, there is no contribution to the Z − Z ′ mass mixing at one loop level. If the model contains scalars that are charged both under electroweak and U(1)′ and develop VEV, mass mixing between Z and Z ′ appears even at tree level. In the minimal version of the model that is described above there is no such scalar but we shall come back to this point in section 4.2.

Decay of Z ′ to neutrino pairs can warm up the neutrino background during and right after the Big Bang Nucleosynthesis (BBN) era. The effect can be described by the contribution to the effective extra relativistic degrees of freedom 1Neff . As shown in Kamada and Yu [177], BBN bounds rule out mZ′ < 5 MeV. Of course, this lower bound on mZ′ applies only if the coupling is large enough to bring Z ′ to thermal equilibrium with neutrinos before they decouple from the plasma at T ∼ 1 MeV. That is, for 1 MeV < mZ′ < 5 MeV the coupling should be smaller than ∼3 <sup>×</sup> <sup>10</sup>−<sup>10</sup> [179].

NSI can leave its imprint on the flavor composition of supernova neutrino flux [34, 180, 181]. Moreover, Z ′ particle can be produced and decay back to neutrinos within the supernova core. This leads to a shortening of the mean free path of neutrinos inside the supernova core [177]. This, in turn, results in prolonging the duration of neutrino emission from supernova. To draw a quantitative conclusion and bound, a full simulation is required.

Once we introduce the new interaction for neutrinos, high energy neutrinos (or antineutrinos) traveling across the universe resonantly interact with cosmic background antineutrinos (or neutrinos) producing Z ′ which decays back to νν¯ pair with energies lower than that of initial neutrino (or antineutrino). This will result in a dip in the spectrum of high energy neutrinos. Taking the cosmic background neutrinos as nonrelativistic, we expect the position of the dip to be given by <sup>E</sup><sup>ν</sup> <sup>∼</sup> PeV(mZ′/10 MeV)<sup>2</sup> (0.05 eV/m<sup>ν</sup> ). The value is tantalizingly close to the observed (but by no means established) gap in the high energy IceCube data. Moreover, as shown in Kamada and Yu [177] with g ′ <sup>∼</sup> <sup>10</sup>−<sup>5</sup> <sup>−</sup> <sup>10</sup>−<sup>4</sup> (the range of interest to us), the optical depth is larger than one. Thus, this rather robust prediction can be eventually tested by looking for the dip in the high energy neutrino data.

The contribution from the Z ′ loop to (g−2)<sup>µ</sup> can be estimated as g ′2 /(8π) up to corrections of order O(m<sup>2</sup> <sup>Z</sup>′/m<sup>2</sup> µ ) ∼ 0.01. For g ′ < 10−<sup>4</sup> , the contribution is too small to explain the claimed discrepancy [182].

Let us now discuss the neutrino scattering experiments. The amplitude of the contribution from t-channel Z ′ exchange to neutrino quark scattering is suppressed relative to that from SM by a factor of m<sup>2</sup> <sup>Z</sup>′/(<sup>t</sup> <sup>−</sup> <sup>m</sup><sup>2</sup> <sup>Z</sup>′), where t is the Mandelstam variable. At CHARM and NuTeV experiments, the energy momentum exchange was about 10 GeV (<sup>t</sup> <sup>≫</sup> <sup>m</sup><sup>2</sup> <sup>Z</sup>′), so the new effects were suppressed. As a result, the bound found in Escrihuela et al. [10], Coloma et al. [124], and Davidson et al. [128] does not apply to the model with a light gauge boson. However, as discussed in Farzan and Heeck [25], Coloma et al. [124], and Dutta et al. [183], low energy scattering experiments can be sensitive to low mass gauge interactions. Three categories of scattering experiments have been studied in this regard: (1) Scattering of solar neutrino at direct dark matter search experiments [25, 183–185]. As shown in Farzan and Heeck [25], the upcoming Xenon based experiments such as LUX-Zeplin and the future Germanium based experiments such as superCDMS at SNOLAB can test most of the parameter space of our interest (see **Figure 9**, adapted

from Farzan and Heeck [25]). (2) As shown in detail in Coloma et al. [124] and Shoemaker [186], the running COHERENT experiment [187] is an ideal setup to probe NSI with a light mediator. At this experiment, low energy ν<sup>µ</sup> and ν<sup>e</sup> fluxes are produced via pion and muon decay at rest. The LMA-Dark solution can be entirely probed by this experiment [124]. The COHERENT experiment has recently released its preliminary results, ruling out a significant part of the parameter space. We shall discuss the new results in section 4.3. (3) Scattering of reactor ν¯<sup>e</sup> flux off nuclei can also probe NSI of the type we are interested in Wong [188], Aguilar-Arevalo et al. [189, 190], Agnolet et al. [191], Billard et al. [192], Lindner et al. [193], Barranco et al. [194], and Kerman et al. [195].

The Z ′ gauge boson coupled to ν and µ can contribute to the so-called neutrino trident production ν+A → ν+A+µ <sup>+</sup> +µ −, where A is a nucleus. The rate of such interaction was measured by the CCFR [196] and CHARM II [197] collaborations, and is found to be consistent with the SM prediction. This observation sets the bound g ′<sup>a</sup> <sup>&</sup>lt; <sup>9</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> for mZ′ ∼ 10 MeV [198, 199].

As we saw earlier, taking a<sup>µ</sup> = a<sup>τ</sup> = −3/2, the contributions from the field content of the SM to anomalies cancel out. We can then obtain any negative values of ǫµµ − ǫee = ǫττ − ǫee ∼ −1 by choosing g ′ <sup>∼</sup> <sup>10</sup>−<sup>4</sup> (|ǫµµ − ǫee|) 1/2 (see Equation 40). Let us now discuss if with this mechanism we can reconstruct a model that embeds the LMA-Dark solution with positive ǫµµ − ǫee = ǫττ − ǫee ∼ 1. The condition ǫµµ − ǫee = ǫττ − ǫee ∼ 1 can be satisfied if a<sup>e</sup> = 0, a<sup>µ</sup> = a<sup>τ</sup> > 0 and

$$g' \sim 10^{-4} \left(\frac{m\_Z}{10 \text{ MeV}}\right) \left(\frac{1}{a\_\mu}\right)^{1/2}.\tag{45}$$

We should however notice that with a<sup>µ</sup> = a<sup>τ</sup> > 0, the cancelation of U(1)′ − SU(2) − SU(2) and U(1)′ − U(1) − U(1) anomalies require new chiral fermions. A new generation of leptons with U(1)′ charge equal to −(3 + a<sup>µ</sup> + a<sup>τ</sup> ) can cancel the anomalies but in order for these new fermions to acquire masses large enough to escape bounds from direct production at colliders, their Yukawa couplings enter the non-perturbative regime. Similar argument holds if we add a new generation of quarks instead of leptons. Another option is to add a pair of new generations of leptons (or quarks) with opposite U(1)<sup>Y</sup> charges but equal U(1)′ charge of −(3 + a<sup>µ</sup> + a<sup>τ</sup> )/2. Let us denote the field content of the fourth generation with νR4, eR<sup>4</sup> and L4, and similarly that of the fifth generation with νR5, eR<sup>5</sup> and L5. As pointed out, the hypercharges of fourth and fifth generation are opposite so we can write Yukawa terms of type

$$Y\_1 \text{Se}\_{\text{R4}}^T \text{ce}\_{\text{5R}} + Y\_2 \text{SL}\_4^T \text{cL5} + \text{H.c.}, \ldots$$

where S is singlet of the electroweak symmetry group, SU(2) × U(1) with a U(1)′ charge of 3 + a<sup>µ</sup> + a<sup>τ</sup> . Even for Y<sup>1</sup> ∼ Y<sup>2</sup> ∼ 1, in order to obtain heavy enough mass, hSi should be of order of TeV. On the other hand, hSi contributes to Z ′ mass so

Masses of 4th and 5th generation . hSi

$$<5\text{ TeV}\frac{m\_Z}{10\text{ MeV}}\frac{2\times10^{-6}}{\text{g}'(3+a\_\mu+a\_t)}.$$

In other words, g ′ . 5 TeV M4,5 <sup>2</sup>×10−<sup>6</sup> 3+aµ+a<sup>τ</sup> mZ′ 10 MeV , where M4,5 are the typical masses of the fourth and fifth generation leptons. Inserting this in Equation (39), we find ǫµµ = ǫττ . 0.01.

In general, the cancelation of U(1)′ − SU(2) − SU(2) and U(1)′ − U(1) − U(1) anomalies requires new chiral fermions charged under U(1)′ and SU(2) × U(1)′ (or both). In the former case, we need new U(1)′ charged scalars whose VEV contribute to the Z ′ mass. The lower bounds on the masses of new particles set a lower bound on the VEV of new scalars which, in turn, can be translated into an upper bound on g ′ /mZ′ which leads to ǫµµ . 0.01. In the second case, large masses of the 4th and 5th generations requires non-perturbative Yukawa coupling to the Higgs. If masses of the new fermions could be about few hundred GeV, none of these obstacles would exist. Fortunately, there is a trick to relax the strong lower bounds from colliders on the masses of new particles. Let us suppose the charged particles are just slightly heavier than their neutral counterparts. Their decay modes can be then of type e − 4(5) → ν4(5)lν, ν4(5)q ′ q¯ with a final charged lepton or jet too soft to be detected at colliders. In this case, the new generation can be as light as few 100 GeV so their mass can come from a perturbative Yukawa coupling to the SM Higgs or new scalars charged under U(1)′ and VEV of ∼100 GeV opening up a hope for g ′ /mZ′ <sup>∼</sup> <sup>10</sup>−<sup>5</sup> MeV−<sup>1</sup> and therefore for ǫµµ = ǫττ ∼ 1.

### 4.2. A Model Both for LF Conserving and LFV NSI

As mentioned in section 4.1, the coupling of Z ′ to neutrinos can be achieved with two mechanisms: (i) The (ν<sup>α</sup> ℓLα) doublet is assigned a charge under U(1)′ , so neutrinos directly obtain a gauge coupling to the Z ′ boson. This route was discussed in section 4.1. (ii) Active neutrinos mix with a new fermion singlet under electroweak group symmetry but charged under new U(1)′ . In this section, we focus on the second route. Using the notation in Equation (38), in the present scenario one has a<sup>e</sup> = a<sup>µ</sup> = a<sup>τ</sup> = 0 so to cancel the U(1)′ − SU(2) − SU(2) and U(1)′ − U(1) − U(1) anomalies, we should add new fermions. As discussed in the previous section, in order to make these new fermions heavier than ∼ 1 TeV, we need new scalars charged under U(1)′ with a VEV of 1 TeV. To keep the contribution from the new VEV to Z ′ mass under control,

$$g' < 10^{-5} \frac{m\_{Z'}}{10 \text{ MeV}}.\tag{46}$$

Notice that this tentative bound is stronger than the bound from π → Z ′γ (see **Figure 8**). Let us introduce a new Dirac fermion 9 which is neutral under electroweak symmetry but charged under U(1)′ . Its U(1)′ charge denoted by a9 can be much larger than one. Since we take equal U(1)′ charges for 9<sup>L</sup> and 9R, no anomaly is induced by this Dirac fermion. Let us denote the mixing of 9 with neutrino of flavor α with κα. Such a mixing of course breaks U(1)′ . Mixing can be obtained in two ways:

• We add a sterile Dirac N (neutral both under electroweak and under U(1)′ ) and a scalar (S) to break U(1)′ . The U(1)′ charge of the S is taken to be equal to that of 9. We can then add terms like the following to the Lagrangian:

$$m\_{\Psi}\Psi\Psi + m\_{N}\bar{N}N + Y\_{\alpha}\bar{N}\_{R}H^{T}cL\_{\alpha} + \lambda\_{L}\mathbb{S}\bar{\Psi}\_{R}N\_{L}.\tag{47}$$

Notice that we were allowed to add a term of λRS9¯ <sup>L</sup>N<sup>R</sup> too, but this term is not relevant for our discussion. Taking YαhHi, λLhSi, m<sup>9</sup> ≪ mN, we can integrate out N and obtain

$$\kappa\_{\alpha} = \frac{Y\_{\alpha} \langle H \rangle \lambda\_L \langle S \rangle}{m\_N m\_{\Psi}}.\tag{48}$$

Since we take λLhSi < mN, in order to have sizeable κα, the mass of 9 cannot be much larger than YαhHi. On the other hand, Y<sup>α</sup> determines the new decay mode of H → νN which is observationally constrained [200]. We therefore find an upper bound on m<sup>9</sup> of few GeV. For example, taking m<sup>9</sup> = 2 GeV, m<sup>N</sup> = 20 GeV, YαhHi = 0.1 GeV, λ<sup>L</sup> = 1 and hSi ∼ 4 GeV, we obtain κ<sup>α</sup> ≃ 0.01. With such small Yα, the rate of the Higgs decay into N and ν will be as small as Ŵ(H → µµ) and therefore negligible. With g ′ < 10−<sup>4</sup> , the contribution from hSi to mZ′ will also be negligible.

• Another scenario which has been proposed in Farzan and Heeck [25] invokes a new Higgs doublet H′ which has U(1)′ charge equal to that of 9. The Yukawa coupling will be then equal to <sup>L</sup> = −<sup>P</sup> α yαL¯ <sup>α</sup>H′<sup>T</sup> c9 which leads to

$$\kappa\_a = \frac{\wp\_a \langle H' \rangle}{\sqrt{2}m\wp}$$

where tan β = hHi/hH′ i. The VEV of H′ can contribute to the Z ′ mass so we obtain

$$2\cos\beta \le 4 \times 10^{-5} (\frac{m\_{Z'}}{10 \text{ MeV}}) \frac{1}{\text{g}\,\omega}.\tag{49}$$

Thus, to obtain sizeable κ<sup>α</sup> (e.g., κ<sup>α</sup> > 0.03), we find

$$M\mu < \text{few GeV} \frac{m\_{Z'}}{10 \text{ MeV}} \frac{0.2}{\text{g}'a\mu} \frac{0.03}{\kappa\_{\alpha}}.\tag{50}$$

Moreover, the VEV of H′ can induce Z − Z ′ mixing on which there are strong bounds [201]. These bounds can be translated into cos β < 10−<sup>4</sup> (mZ′/10 MeV)(1/g9) which is slightly weaker than the bound in (49). The smallness of hH′ i, despite its relatively large mass, can be explained by adding a singlet scalar(s) of charge <sup>a</sup><sup>9</sup> with <sup>L</sup> = −µ<sup>S</sup> † <sup>1</sup>H†H′ which induces hH′ i = −µhS1i/(2M<sup>2</sup> <sup>H</sup>′). Taking <sup>h</sup>S1<sup>i</sup> <sup>µ</sup> <sup>≪</sup> <sup>M</sup><sup>2</sup> H′ , we find cos β ≪ 1. The components of H′ can be pair produced at colliders via electroweak interactions. They will then decay to 9 and leptons. In particular, the charged component H− can decay into charged lepton plus 9 which appears as missing energy. Its signature will be similar to that of a charged slepton [25]. According to the present bounds [25, 202], mH′ & 300 GeV.

Regardless of the mechanism behind the mixing between 9 and ν, it will lead to the coupling of Z ′ to active neutrinos as follows

$$\begin{pmatrix} g'a\_{\Psi}Z'\_{\mu} \left(\sum\_{\alpha,\beta} \kappa\_{\alpha}^\* \kappa\_{\beta} \bar{\upsilon}\_{\alpha} \gamma^{\mu} P\_L \upsilon\_{\beta} - \kappa\_{\alpha}^\* \bar{\upsilon}\_{\alpha} \gamma^{\mu} P\_L \Psi - \kappa\_{\alpha} \bar{\Psi} \gamma^{\mu} P\_L \upsilon\_{\alpha} \right), \tag{51}$$

which leads to ǫ u αβ = ǫ d αβ = g ′2a<sup>9</sup> κ ∗ α κβ 6 √ 2GFm<sup>2</sup> Z′ . Notice that if the mixing of 9 with more than one flavor is nonzero, we can have lepton flavor violating NSI with ǫ u(d) αα ǫ u(d) ββ = |ǫ u(d) αβ | 2 . If more than one 9 is added, we may label the mixing of ith 9 to ν<sup>α</sup> with κiα. The Schwartz inequality (P i κ ∗ iα κiβ) <sup>2</sup> < ( P i |κiα| 2 )(P i |κiβ| 2 ) then still applies

$$
\epsilon\_{aa}^{\mathfrak{u}(d)} \epsilon\_{\beta\beta}^{\mathfrak{u}(d)} > (\epsilon\_{\alpha\beta}^{\mathfrak{u}(d)})^2.
$$

Taking κi<sup>α</sup> = δiα, meaning that each 9<sup>i</sup> mixes with only one να, only diagonal elements of ǫαβ will be nonzero, preserving lepton flavors.

Notice that 9 in our model decays into Z ′ and ν and appears as missing energy. 9 should be heavier than MeV; otherwise, it can contribute to extra relativistic degrees of freedom in the early universe. Remember that we have found that m<sup>9</sup> < few GeV. The mixing of active neutrinos with 9 results in the violation of the unitarity of 3×3 PMNS matrix on which there are strong bounds [203–205]

$$\left|\kappa\_{\epsilon}\right|^{2} < 2.5 \times 10^{-3}, \ \left|\kappa\_{\mu}\right|^{2} < 4.4 \times 10^{-4} \quad \text{and} \quad$$

$$\left|\kappa\_{\mathfrak{t}}\right|^{2} < 5.6 \times 10^{-3} \text{ at } 2\sigma \tag{52}$$

which immediately give

$$|\kappa\_{\mu}\kappa\_{\varepsilon}| < 10^{-3}, \quad |\kappa\_{\mu}\kappa\_{\varepsilon}| < 1.6 \times 10^{-3} \quad \text{and}$$

$$|\kappa\_{\varepsilon}\kappa\_{\varepsilon}| < 3.7 \times 10^{-3} \quad \text{at } 2\sigma. \tag{53}$$

Under certain assumptions, Fernandez-Martinez et al. [203] also derives independent bound on κακ ∗ β |α6=<sup>β</sup> from lepton flavor violation (LFV) processes l − <sup>α</sup> → l − β γ , but these bounds are valid only for m<sup>9</sup> ≫ mW. For our case with m<sup>9</sup> ≪ mW, a GIM mechanism is at work and suppresses the contribution to l − <sup>α</sup> → l − β γ from ν − 9 mixing. In the case that the mixing comes from Yukawa coupling to H′ , because of the LFV induced by H′ and 9 coupling to more than one flavor, a new contribution to l − <sup>α</sup> → l − β γ appears. As shown in [25], from Br(τ → eγ ) < 3.3 <sup>×</sup> <sup>10</sup>−<sup>8</sup> , Br(<sup>τ</sup> <sup>→</sup> µγ ) <sup>&</sup>lt; 4.4 <sup>×</sup> <sup>10</sup>−<sup>8</sup> [161] and Br(µ → <sup>e</sup><sup>γ</sup> ) <sup>&</sup>lt; 4.2 <sup>×</sup> <sup>10</sup>−13[206], one can respectively derive <sup>|</sup>yey<sup>τ</sup> <sup>|</sup> <sup>&</sup>lt; 0.46(mH<sup>−</sup> /(400 GeV))<sup>2</sup> , <sup>|</sup>yµy<sup>τ</sup> <sup>|</sup> <sup>&</sup>lt; 0.53(mH<sup>−</sup> /(400 GeV))<sup>2</sup> and <sup>|</sup>yeyµ<sup>|</sup> <sup>&</sup>lt; <sup>7</sup>×10−<sup>4</sup> (mH<sup>−</sup> /(400 GeV))<sup>2</sup> . As demonstrated in Farzan and Heeck [25], except for ǫe<sup>µ</sup> which is strongly constrained by the bound from µ → eγ within the model described in Farzan and Heeck [25], all components of ǫαβ can be within the reach of current and upcoming long baseline neutrino experiments. If mixing is achieved via the mechanism described in Equation (48), no new bound from LFV rare decay applies and we can obtain all ǫαβ (including ǫeµ) of the order of

$$
\epsilon^{\mu}\_{\alpha\beta} = \epsilon^{d}\_{\alpha\beta} = \mathcal{g}' a\_{\Psi} \left( \frac{\mathcal{g}'}{10^{-5}} \right) \frac{\kappa^\*\_{\alpha} \kappa\_{\beta}}{10^{-3}} \left( \frac{10 \text{ MeV}}{m\_{Z'}} \right)^2. \tag{54}
$$

Notice that in this model, the coupling of Z ′ to neutrino pairs can be much larger than the coupling to quarks: |g ′a9κακβ| ≫ g ′ , see Equation (51). The bounds from meson decays on Z ′ coupling to neutrino pairs have been studied in Bakhti and Farzan [173]. The results are shown in **Figures 10**, **11**. The strongest bound for mZ′ ∼ 10 MeV is of order of 0.001, which can be readily satisfied if κακ<sup>β</sup> < 10−<sup>3</sup> . However, further data on [π <sup>+</sup> → e +(µ <sup>+</sup>) + missing energy] or on [K <sup>+</sup> → e +(µ <sup>+</sup>) + missing energy] can probe parts of the parameter space of interest to us.

### 4.3. Impact of Recent Results from COHERENT Experiment

Recently, the COHERENT experiment has released its first results confirming the SM prediction of elastic scattering of neutrinos

off nuclei at 6.7σ, studying the interaction of νµ, ν¯<sup>µ</sup> and ν<sup>e</sup> flux from Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory on a 14.6 kg CsI[Na] scintillator detector [126]. The preliminary results already set strong bounds on NSI.

Assuming the validity of the contact interaction approximation (i.e., assuming the mass of the mediator is heavier than ∼10 MeV), Coloma et al. [127] shows that the recent COHERENT data rules out LMA-Dark solution. Liao and Marfatia [130], taking a universal coupling of Z ′ to SM fermions finds that

$$\left(\left(\text{g}\_{\text{v}}\text{g}\_{q}\right)^{1/2} < 6 \times 10^{-5} \text{ for } \, m\_{Z'} < 30 \,\text{MeV} \text{ at } 2\sigma.\right.\tag{55}$$

Let us discuss how this bound can constrain our model(s) for NSI. Regardless of the details of the underlying theory, we can write

$$\left(g\_{\mathbb{V}}g\_{\mathbb{q}}\right)^{1/2} = 5.47 \times 10^{-5} \sqrt{\epsilon\_{a\beta}} \left(\frac{m\_Z}{10 \text{ MeV}}\right)^{\varepsilon}$$

where g<sup>q</sup> = gB/3 is the coupling of Z ′ boson to quarks and (g<sup>ν</sup> )αβ is its coupling to ν<sup>α</sup> and νβ. In the model described in section 4.1, (g<sup>ν</sup> )αβ = δαβaαg ′ and in the model of section 4.2, (g<sup>ν</sup> )αβ = g ′a9κακβ. Remember that, in order for ǫαβ to show up in neutrino oscillation experiments, it should be non-universal. For example, the LMA-Dark solution requires ǫµµ − ǫee = ǫτ τ − ǫee ∼ 1. Setting ǫee = 0 and ǫµµ 6= 0, we expect the bound on |ǫµµ| from the COHERENT experiment to be slightly weaker than that found in Liao and Marfatia [130] taking ǫµµ = ǫee 6= 0. Thus, the LMA-Dark solution still survives with the present COHERENT data but, further data from COHERENT as well as the data from upcoming reactor neutrino-nucleus coherent scattering experiments such as the setup described in Lindner et al. [193] can probe the most interesting part of the parameter space.

### 5. NSI AT UPCOMING LONG BASELINE EXPERIMENTS: T2HK, T2HKK, DUNE, JUNO, AND MOMENT

In recent years, rich literature has been developed on the possibility of detecting the NSI effects in upcoming long baseline neutrino experiments. In particular, degeneracies induced by the presence of NSI in the DUNE experiment have been scrutinized [15–18, 213–223]. In section 5.1, we review the effects of NSI at DUNE, T2HK and T2HKK experiments. In section 5.2, we show how intermediate baseline reactor experiments such as JUNO and RENO-50 can help to determine sign(cos 2θ12) and therefore test the LMA-Dark solution. In section 5.3, we show how the MOMENT experiment can help to determine the octant of θ<sup>23</sup> and the true value of δ despite the presence of NSI. Throughout this section, we set ǫµµ = 0 for definiteness and consistency with the majority of our references.

### 5.1. NSI at Upcoming Long Baseline Neutrino Experiments

Let us first briefly review the setups of the three upcoming stateof-the-art long baseline neutrino experiments which are designed to measure the yet unknown neutrino parameters with special focus on the Dirac CP-violating phase of the PMNS matrix.


Notice that at T2HK, both energy of the neutrino beam and the baseline are lower than those at DUNE. We therefore expect the DUNE experiment to be more sensitive to both standard and non-standard matter effects than T2HK and T2HKK. Although the baseline for the Korean detector of T2HKK is comparable to the DUNE baseline, the DUNE experiment will be more sensitive to matter effects than T2HKK, because the energy of the beam at T2HKK is lower. Detailed simulation confirms this expectation [221]. In the presence of NSI, new degeneracies will appear in long baseline neutrino experiments for determination of the value of δ, mass ordering and the octant of θ23. One of the famous degeneracies is the so called generalized mass ordering degeneracy [11, 13, 19, 123, 221]. The oscillation probability remains invariant under the following simultaneous transformations

$$
\theta\_{12} \to \frac{\pi}{2} - \theta\_{12}, \,\delta \to \pi - \delta, \,\Delta m\_{31}^2 \to -\Delta m\_{31}^2 + \Delta m\_{21}^2,
$$

$$
\text{and } V\_{\text{eff}} \to -\mathbb{S} \cdot V\_{\text{eff}}^\* \cdot \mathbb{S} \tag{56}
$$

where S = Diag(1, −1, −1) and (Veff )αβ = √ 2GFNe[(δα1δβ1) + ǫαβ] in which ǫαβ = P f ∈{e,u,d} (N<sup>f</sup> /Ne)ǫ f αβ depends on the composition of medium. Notice that the LMA-Dark solution with θ<sup>12</sup> > π/4 and ǫ <sup>f</sup> ∼ −1 [13] is related to the generalized mass ordering transformation from the standard LMA solution with ǫ = 0. For the Earth (with N<sup>p</sup> ≃ Nn), we can write Nu/N<sup>e</sup> ≃ Nd/N<sup>e</sup> = 3. Notice however that the transformation in Equation (56) does not depend on the beam energy or baseline. As a result, by carrying out long baseline neutrino experiments on Earth with different baseline and beam energy configurations, this degeneracy cannot be resolved. Resolving this degeneracy requires media with different Nn/N<sup>e</sup> composition. Notice that although the nuclear compositions of the Earth core and mantle are quite different, Nn/N<sup>e</sup> is uniformly close to 1 across the Earth radius [113]. However, Nn/N<sup>e</sup> in the Sun considerably differs from that in the Earth. Moreover, it varies from the Sun center (with Nn/N<sup>e</sup> ≃ 1/2) to its outer region (with Nn/N<sup>e</sup> ≃ 1/6) [60, 61]. As a result, the solar neutrino data can in principle help to solve this degeneracy. In fact, Gonzalez-Garcia and Maltoni [11] by analyzing solar data shows that the LMA solution with ǫ u ee ≃ 0.3 is slightly favored over the LMA-Dark solution. The global analysis of solar, atmospheric and (very) long baseline data can in principle help to solve degeneracies. For the time being, however, since the terrestrial experiments are not precise enough to resolve the effects of sign(1m<sup>2</sup> <sup>31</sup>) and/or sign(cos 2θ12), the generalized mass ordering degeneracy cannot be resolved.

At relatively low energy long baseline experiments such as T2HK and T2HKK for which the contribution to the oscillation probability from higher orders of <sup>O</sup>(Veff ǫ/|1m<sup>2</sup> <sup>31</sup>|) can be neglected, the appearance oscillation probability along the direction ǫeµ/ǫe<sup>τ</sup> = tan θ<sup>23</sup> will be equal to that for standard ǫ = 0 [221]. The DUNE experiment being sensitive to higher orders of (Veff ǫ/|1m<sup>2</sup> <sup>31</sup>|) can solve this degeneracy [221]. At the DUNE experiment, another degeneracy appears when ǫee and ǫ<sup>τ</sup> <sup>e</sup> are simultaneously turned on and the phase of ǫe<sup>τ</sup> is allowed to be nonzero. As shown in **Figure 12**, (which corresponds to Figure 4 of Coloma [215] and confirmed in Figure 10 of Liao et al. [221]), in the presence of cancelation due to the phase of

<sup>11</sup>KEK Preprint 2016-21 and ICRR-Report-701-2016-1, https://lib-extopc.kek.jp/ preprints/PDF/2016/1627/1627021.pdf.

ǫe<sup>τ</sup> for |ǫe<sup>τ</sup> | ∼ 0.2 − 0.3, |ǫee| as large as 2 cannot be disentangled from standard case with ǫee = ǫe<sup>τ</sup> = 0 at the DUNE experiment. However, this figure also demonstrates that when information on ǫ from already existent data is used as prior, the degeneracy can be considerably solved, ruling out the ǫee < 0 wing of solutions. That is because solar data rules out ǫee < 0 for θ<sup>12</sup> < π/4 [11]. NSI can induce degeneracies in deriving sign(cos 2θ23). In principle, even with ǫe<sup>µ</sup> (ǫe<sup>τ</sup> ) as small as O(0.01), the degeneracy due to the phase of ǫe<sup>µ</sup> (ǫe<sup>τ</sup> ) makes the determination of the octant of θ<sup>23</sup> problematic [18]. Because of the generalized mass ordering degeneracy, the presence of NSI can also jeopardize determination of sign(1m<sup>2</sup> <sup>31</sup>) [226].

manuscript. We would like to thank P. Coloma for sending the original figure.

In summary, NSI induces degeneracies that makes determination of the true value of δ at DUNE impossible at 3 σ C.L. The T2HKK experiment can considerably solve this degeneracy as demonstrated in **Figures 13**, **14** (corresponding to Figures 11 and 12 of Liao et al. [221]).

### 5.2. JUNO and RENO-50 Shedding Light on LMA-Dark

To determine the sign of 1m<sup>2</sup> <sup>31</sup>, two reactor neutrino experiments with baseline of ∼ 50 km are proposed: The JUNO experiment in China which is planned to start data taking in 2020 and RENO-50 which is going to be an upgrade of the RENO experiment in South Korea12. In this section, we show that for known mass ordering, these experiments can determine the octant of θ12. At reactor experiments, since the energy is low, <sup>|</sup>1m<sup>2</sup> <sup>31</sup>|/E ≫ √ 2GFNe. Thus, the matter effects can be neglected and the survival probability can be written as

$$\begin{aligned} P(\bar{\nu}\_{\epsilon} \rightarrow \bar{\nu}\_{\epsilon}) &= \left| |U\_{\epsilon 1}|^2 + |U\_{\epsilon 2}|^2 e^{i\Delta\_{11}} + |U\_{\epsilon 3}|^2 e^{i\Delta\_{31}} \right|^2 \\ &= \left| c\_{12}^2 c\_{13}^2 + s\_{12}^2 c\_{13}^2 e^{i\Delta\_{21}} + s\_{13}^2 e^{i\Delta\_{31}} \right|^2 \\ &= c\_{13}^4 \left( 1 - \sin^2 2\theta\_{12} \sin^2 \frac{\Delta\_{21}}{2} \right) + s\_{13}^4 \\ &+ 2s\_{13}^2 c\_{13}^2 \left[ \cos \Delta\_{31} (c\_{12}^2 + s\_{12}^2 \cos \Delta\_{21}) \right. \\ &+ s\_{12}^2 \sin \Delta\_{31} \sin \Delta\_{21} \right], \end{aligned} \tag{57}$$

where <sup>1</sup>ij <sup>=</sup> <sup>1</sup>m<sup>2</sup> ijL/(2E<sup>ν</sup> ) in which L is the baseline. Notice that the first parenthesis (which could be resolved at KamLAND) is only sensitive to sin<sup>2</sup> 2θ<sup>12</sup> and cannot therefore resolve the octant of θ12. The terms in the last parenthesis, however, are sensitive to the octant of θ12. To solve these terms two main challenges have to be overcome: (i) These terms are suppressed by s 2 <sup>13</sup> ∼ 0.02 so high statistics is required in order to resolve them. (ii) In the limit, 1<sup>12</sup> → 0, we can write P(ν¯<sup>e</sup> → ¯νe) = c 4 <sup>13</sup> + s 4 <sup>13</sup> + 2s 2 13c 2 <sup>13</sup> cos 1<sup>31</sup> so the sensitivity to θ<sup>12</sup> is lost. To determine θ<sup>12</sup> baseline should be large enough (i.e., L & 10 km). (iii) Condition 1<sup>12</sup> & 1 naturally implies 1<sup>13</sup> ≫ 1 so the terms sensitive to the octant of θ<sup>12</sup> (and sign of 1m<sup>2</sup> <sup>31</sup>) oscillate rapidly. To resolve these terms, the energy resolution and accuracy of reconstruction of the total energy scale must be high. Notice that reactor experiments such

<sup>12</sup>Joo [227] reports the current status of RENO-50.

2.5◦ off-axis angle vs. true value of δ. Normal mass ordering is assumed and taken to be unknown. The values of ǫee, ǫeµ and ǫeτ are allowed to vary. The parameters that are not shown are marginalized. This plot is taken from Liao et al. [221], published under the terms of the Creative Commons Attribution Noncommercial License and therefore no copyright permissions were required for its inclusion in this manuscript. We would like to thank D. Marfatia for sending the original figure.

as Daya Bay satisfy the first condition and resolve the terms proportional to s 2 <sup>13</sup>, but cannot overcome the second challenge because at these experiments, 1<sup>12</sup> ≪ 1. At KamLAND, 1<sup>12</sup> > 1 but the statistics was too low to resolve the s 2 <sup>13</sup> terms. JUNO and RENO-50, being designed to be sensitive to these terms to determine sign(1m<sup>2</sup> <sup>31</sup>), can overcome all these three challenges. The detectors at JUNO and RENO-50 experiments will employ liquid scintillator technique with an impressive energy resolution of

$$\frac{\delta E\_{\upsilon}}{E\_{\upsilon}} \simeq \mathfrak{Poly} \times \left(\frac{E\_{\upsilon}}{\text{MeV}}\right)^{1/2}$$

.

Moreover, the energy calibration error can be as low as 3%. Using the GLoBES software [228, 229], Bakhti and Farzan [19] shows how JUNO and RENO-50 experiments can test LMA-Dark solution with θ<sup>12</sup> > π 4 . Results are shown in **Figures 15**, **16**. The star denotes the true value of 1m<sup>2</sup> <sup>31</sup> and θ12. In **Figures 15**, **16**, normal and inverted mass orderings are respectively assumed. Ellipses show 3σ C.L. contours, after 5 years of data taking. As seen from these figures, these upcoming experiments will be able to determine <sup>|</sup>1m<sup>2</sup> <sup>31</sup>| with much better accuracy than the present global data analysis so no prior on <sup>|</sup>1m<sup>2</sup> <sup>31</sup>| is assumed. The uncertainties of other relevant

neutrino parameters are taken from [230] and are treated by pull-method.

manuscript. We would like to thank D. Marfatia for sending the original figure.

For JUNO experiment, the uncertainties in the flux normalization and the initial energy spectrum at the source are taken respectively equal to 5 and 3%. RENO-50 enjoys having a near detector (the detectors of present RENO) which can measure the flux with down to O(0.3%) uncertainty. To perform the analysis, the energy range of 1.8–8 MeV is divided to 350 bins of 17.7 keV size. The pull-method is applied by defining

$$\chi^{2} = \text{Min}|\_{\theta\_{\text{pull}}\mathcal{ax}\_{i}} \left[ \sum\_{i} \frac{[N\_{i}(\theta\_{0}, \bar{\theta}\_{pull}) - N\_{i}(\theta, \theta\_{pull})(1 + \alpha\_{i})]^{2}}{N\_{i}(\theta\_{0}, \bar{\theta}\_{pull})} \right]$$

$$+ \sum\_{i} \frac{\alpha\_{i}^{2}}{(\Delta \alpha\_{i})^{2}} + \frac{(\theta\_{pull} - \bar{\theta}\_{pull})^{2}}{(\Delta \theta\_{pull})^{2}} \right],\tag{58}$$

where N<sup>i</sup> is the number of events at bin i. α<sup>i</sup> is the pull parameter that accounts for the uncertainty in the initial spectrum at bin i. Pull parameters taking care of the other uncertainties are collectively denoted by θpull.

As seen from these figures, JUNO and RENO-50 can determine the octant of θ<sup>12</sup> for a given mass ordering. This result is relatively robust against varying the calibration error but as expected, is extremely sensitive to the energy resolution. Increasing the uncertainty in energy resolution from 3 to 3.5%, Bakhti and Farzan [19] finds that JUNO and RENO-50 cannot determine the octant at 3 σ C.L. after five years. As seen from the figures, JUNO and RENO-50 experiments cannot distinguish two solutions which are related to each other with

show degenerate solutions respectively with opposite octant, with opposite mass ordering and with both opposite octant and mass ordering. Plots are taken from Bakhti and Farzan [19], published under the terms of the Creative Commons Attribution Noncommercial License and therefore no copyright permissions were required for their inclusion in this manuscript.

<sup>θ</sup><sup>12</sup> <sup>↔</sup> π/<sup>2</sup> <sup>−</sup> <sup>θ</sup><sup>12</sup> and <sup>1</sup>m<sup>2</sup> <sup>31</sup> → −1m<sup>2</sup> <sup>31</sup> <sup>+</sup> <sup>1</sup>m<sup>2</sup> <sup>21</sup> which stems from the generalized mass ordering degeneracy that we discussed in section 5.1.

### 5.3. NSI at the MOMENT

The MOMENT experiment is a setup which has been proposed to measure the value of CP-violating phase, δ [231, 232]. MOMENT stands for MuOn-decay MEdium baseline NeuTrino beam. This experiment will be located in China. The neutrino beam in this experiment is provided by the muon decay. Beam can switch between muon decay (µ → eν¯eνµ) and antimuon decay (µ¯ → e <sup>+</sup>νeν¯µ). The energies of neutrinos will be relatively low with a maximum energy at 700 MeV and peak energy at 150 MeV. The detector is going to be Gd doped water Cherenkov with fiducial mass of 500 kton, located at a distance of 150 km from the source. The detection modes are

$$
\nu\_e + n \to p + e^- \quad \bar{\nu}\_\mu + n \to p + \mu^+ 
$$

and

$$
\bar{\nu}\_e + n \to p + e^+ \quad \nu\_\mu + n \to p + \mu^-.
$$

Gd at the detector can capture the final neutron so although the detector lacks magnetic field, it can distinguish between neutrino and antineutrino with Charge Identification (CI) of 80% [233]. The MOMENT experiment, with a baseline of 150 km and relatively low energy is not very sensitive to matter effects so it enjoys an ideal setup to determine δ and octant of θ<sup>23</sup> without ambiguity induced by degeneracies with NSI. The potential of this experiment for determining δ and the octant of θ<sup>23</sup> is studied in Bakhti and Farzan [234] using GLoBES [228, 229]. The unoscillated flux of each neutrino mode is taken to be 4.7 <sup>×</sup> <sup>10</sup>11m−<sup>2</sup> year−<sup>1</sup> and 5 years of data taking in each muon and antimuon modes is assumed. Uncertainties in flux normalization of ν¯<sup>e</sup> and ν<sup>µ</sup> modes are taken to be correlated and equal to 5% but the uncertainties of fluxes from muon and antimuon modes are uncorrelated.

One of the main sources of background is atmospheric neutrinos. Since the neutrino beam at the MOMENT experiment will be sent in bunches, this source of background can be dramatically reduced. Reduction of background is parameterized by Suppression Factor (SF). Results of Bakhti and Farzan [234] are shown in **Figure 17**. The assumed true value of δ and θ<sup>23</sup> are shown with a star. The mass ordering is taken to be normal and assumed to be known. All the appearance and disappearance modes are taken into account. In all these figures, the true values of ǫ are taken to be zero. In **Figures 17B–D**, pull method is

solution is assumed to be true. Panel (B) shows the true solution. Panels (A,C,D) show degenerate solutions respectively with opposite octant, with opposite mass ordering and with both opposite octant and mass ordering. Plots are taken from Bakhti and Farzan [19], published under the terms of the Creative Commons Attribution Noncommercial License and therefore no copyright permissions were required for their inclusion in this manuscript.

applied on ǫ = P f (N f /Ne)ǫ f , taking 1σ uncertainties on ǫ as follows [11]

$$|\epsilon\_{\varepsilon\mu}| < 0.16, \ |\epsilon\_{\varepsilon\tau}| < 0.26 \text{ and } |\epsilon\_{\mu\tau}| < 0.02 \tag{59}$$

and

−0.018 < ǫττ −ǫµµ < 0.054 and 0.35 < ǫee−ǫµµ < 0.93. (60)

Results shown in **Figures 17C,D** assume that T2K (NOνA) takes data in neutrino mode for 2 (3) years and in antineutrino mode for 6 (3) years. For more details on the assumptions, see Bakhti and Farzan [234]. As seen from **Figure 17B**, turning on NSI, NOνA and T2K (even combined) cannot establish CP-violation even at 1 σ C.L.: while the true value of δ is taken to be 270◦ (maximal CP-violation), δ = 0, 360◦ (CP-conserving) is within the 1 σ C.L. contour. At 3σ C.L., these experiments cannot determine the octant of θ23. Moreover, in the presence of NSI, these experiments cannot rule out the wrong octant even at 1 σ C.L. But, comparing **Figure 17A** and **Figure 17B**, we observe that turning on NSI within range (59,60) does not considerably reduce the power of MOMENT to measure the CP-violating phase and rule out the wrong octant solution. In this figure, SF is taken to be 0.1% which is rather an optimistic assumption. **Figure 18** shows that increasing SF up to 10%, the power of octant determination is significantly reduced but the determination of δ is not dramatically affected.

Similar result holds valid when instead of normal mass ordering, inverted mass ordering is taken (and again assumed that the ordering is known) [234]. Bakhti and Farzan [232] shows that the MOMENT experiment itself can determine the mass ordering. According to Bakhti and Farzan [234], as long as ǫ can vary in the range shown in Equations (59, 60), MOMENT maintains its power to determine the mass ordering. Of course, once we allow ǫ to vary in a wide range such that transformation in Equation (56) can be made, the power of mass ordering determination is lost due to the generalized mass ordering degeneracy.

### 6. SUMMARY

After multiple decades of experimental progress in the area of neutrino physics, the phenomenon of neutrino oscillations has been observed in a wide variety of experiments. The discovery of neutrino oscillations implies the existence of neutrino masses and therefore a need for an extension of the SM to include them. Many possibilities have been proposed so far, see for instance

King [235], Hirsch and Valle [236], Boucenna et al. [237], and Cai et al. [238]. Motivated by the two original anomalies in the solar and atmospheric neutrino sector, various other experiments have been proposed to search for neutrino oscillations in the solar and atmospheric neutrino flux as well as in man-made neutrino beams such as reactors or accelerators. The large amount of experimental data collected over more than 20 years has allowed a very precise determination of some of the parameters responsible for the oscillations. These include the solar mass splitting, 1m<sup>2</sup> 21, the absolute value of the atmospheric mass splitting, <sup>|</sup>1m<sup>2</sup> <sup>31</sup>|, as well as the solar (θ12) and reactor (θ13) mixing angles, measured with relative accuracies below 5%. Nevertheless, the current precision of atmospheric angle θ<sup>23</sup> and the CP phase δ is not at that level. The sign of cos(2θ23) or in other words the octant of θ<sup>23</sup> is yet unknown. Moreover, although there are some hints for CP phase (δ) to be close to 3π/2, its value is not yet established. The sign of 1m<sup>2</sup> <sup>31</sup> or equivalently the scheme of mass ordering (normal vs. inverted) is also still unknown. In section 2, we have discussed the most relevant experimental information used in the global fits of neutrino oscillations [39–41] to obtain precise measurements of the oscillation parameters, exploiting the complementarity of the different data sets. The main results of these analysis have also been commented, with an emphasis on the still unknown parameters.

Since their discovery, neutrinos have always surprised us by showing unexpected characteristics. In the dawn of the neutrino precision era, it is intriguing to ask whether neutrinos have new interactions beyond those expected within the standard model of particles. Such new interactions can give a signal in different neutrino oscillation as well as non-oscillation experiments. No evidence for the presence of NSI has been reported so far. As a consequence of these negative searches, upper bounds on the magnitude of the new interactions can be set. In section 3, we have discussed the constraints on the NSI interactions, parameterized in terms of the ǫαβ couplings introduced in Equations (15) and (16). The presence of NSI has been extensively analyzed in the literature, at the level of the production, detection and propagation of neutrinos in matter. The most restrictive limits on NSI are summarized in **Tables 2**–**4**.

In principle, adding any new particle which couples both to neutrinos and to quarks will induce Non-Standard Interaction (NSI) for neutrinos. However, it is very challenging to build an electroweak symmetric model that leads to large enough NSI to be discernible at neutrino oscillation experiments without violating various bounds. We have discussed a class of models in which the new particle responsible for NSI is a light U(1)′ gauge boson Z ′ with mass 5 MeV − few 10 MeV with a coupling of order of 10−<sup>5</sup> <sup>−</sup> <sup>10</sup>−<sup>4</sup> to quarks and neutrinos. Within this range of parameter space, the NSI effective coupling can be as large as the standard effective Fermi coupling, GF.

The total flux of solar neutrino has been measured by SNO experiment via dissociation of Deuteron through axial part of neutral current interaction and has been found to be consistent with the standard model prediction. To avoid a deviation from this prediction, the coupling of quarks to the new gauge boson is taken to be non-chiral with equal U(1)′ charges for left-handed and right-handed quarks. Moreover, the U(1)′ charges of up and down quarks are taken to be equal to make the charged current weak interaction term invariant under U(1)′ . The U(1)′ charges of quarks is taken to be universal; otherwise, in the mass basis of quarks, we would have off-diagonal interactions leading to huge q<sup>i</sup> → qjZ ′ rate enhanced by (mq<sup>i</sup> /mZ′) 2 . In summary, U(1)′ charges of quarks is taken to be proportional to their baryon number. We have discussed two different scenarios for U(1)′ charge assignment to leptons: (i) assigning U(1)′ charge to the SM fermion as aeL<sup>e</sup> + aµL<sup>µ</sup> + a<sup>τ</sup> L<sup>τ</sup> + B where L<sup>α</sup> denotes lepton flavor α and B denotes Baryon number. With this assignment, lepton flavor will be preserved and both charged leptons and neutrinos obtain lepton flavor conserving NSI. A particularly interesting scenario is a<sup>e</sup> = 0, a<sup>µ</sup> = a<sup>τ</sup> = −3/2 for which gauge symmetry anomalies automatically cancel without a need to add new specious. Choosing appropriate value of coupling [i.e., 4 <sup>×</sup> <sup>10</sup>−<sup>5</sup> (mZ′/10 MeV)], the best fit to solar neutrino data with ǫµµ − ǫee = ǫττ − ǫee = −0.3 can be reproduced. (ii) In the second scenario, the leptons are not charged under U(1)′ . A new Dirac fermion, denoted by 9, with a mass of 1 GeV which is singlet under SM gauge group but charged under U(1)′ is introduced which mixes with neutrinos. As a result, neutrinos obtain coupling to Z ′ through mixing with the new fermion but charged leptons do not couple to Z ′ at the tree level. If the new fermion mixes with more than one flavor, both LFV and LFC NSI will be induced. Within this scenario, new fermions are needed to cancel the gauge anomalies. We have discussed different possibilities. To give masses to these new fermions, new scalars charged under U(1)′ are required whose VEV also gives a significant contribution to the mass of Z ′ boson.

We have suggested two mechanisms for inducing a mixing between 9 and neutrinos: (1) Introducing a new Higgs doublet, H′ , with U(1)′ charge equal to that of 9 which couples to lefthanded lepton doublets and 9. H′ obtains a VEV of few MeV which induces mixing. (2) Introducing a sterile neutrino, N (singlet both under SM gauge group and U(1)′ ) and a new scalar singlet with a U(1)′ charge equal to that of 9 which couples to N and 9. Its VEV then induces the coupling.

Even though the mass of Z ′ particle is taken to be low (i.e., of order of solar neutrino energies and much smaller than the typical energies of atmospheric neutrinos or the energies of the neutrinos of long baseline experiments), the effect of new interaction on propagation of neutrinos in matter can be described by an effective four-Fermi Lagrangian integrating out Z ′ because at forward scattering of neutrinos off the background matter, the energy momentum transfer is zero. At high energy scattering experiments, such as NuTeV and CHARM, the energy momentum transfer, q 2 , is much higher than m<sup>2</sup> <sup>Z</sup>′ so the effective four-Fermi coupling loses its viability. The amplitude of new effects will be suppressed by a factor of ǫm<sup>2</sup> <sup>Z</sup>′/q <sup>2</sup> <sup>≪</sup> 1 relative to SM amplitude and will be negligible. Thus, unlike the case that the intermediate state responsible for NSI is heavy, these experiments cannot constrain ǫ ∼ 1. However, by studying scattering of low energy neutrinos (E<sup>ν</sup> ∼few 10 MeV) off matter, these models can be tested. The current COHERENT experiment and the upcoming CONUS experiment<sup>13</sup> are ideal set-ups to eventually test this model. An alternative way to test such models is to search for a dip in the energy spectrum of high energy cosmic neutrinos around few hundred TeV.

### AUTHOR CONTRIBUTIONS

MT has contributed mainly to review the current status of neutrino oscillations as well as the current status of bounds on non-standard neutrino interactions. YF has contributed mainly to the review of models giving rise to sizeable non-standard neutrino interactions and the review of non-standard interaction searches at upcoming experiments.

<sup>13</sup>https://indico.cern.ch/event/606690/contributions/2591545/attachments/ 1499330/2336272/Taup2017\_CONUS\_talk\_JHakenmueller.pdf.

### REFERENCES


### ACKNOWLEDGMENTS

We thank J. Jeeck, I. Shoemaker, A. Yu Smirnov, M. Lindner, M. M. Sheikh-Jabbari, and O. G. Miranda for useful discussions. MT is supported by a Ramón y Cajal contract (MINECO) and the Spanish grants FPA2014-58183-P and SEV-2014-0398 (MINECO) and PROMETEOII/2014/084 and GV2016-142 (Generalitat Valenciana). YF thanks MPIK in Heidelberg where a part of this work was done for their hospitality. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896 and No 690575. YF is also grateful to ICTP associate office and Iran National Science Foundation (INSF) for partial financial support under contract 94/saad/ 43287.


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Farzan and Tórtola. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Neutrino Masses and Leptogenesis in Left–Right Symmetric Models: A Review From a Model Building Perspective

#### Chandan Hati <sup>1</sup> \*, Sudhanwa Patra<sup>2</sup> , Prativa Pritimita<sup>3</sup> and Utpal Sarkar 1,2,3,4

<sup>1</sup> Laboratoire de Physique Corpusculaire, Centre National de la Recherche Scientifique/IN2P3 UMR 6533, Aubière, France, <sup>2</sup> Department of Physics, Indian Institute of Technology Bhilai, Chhattisgarh, India, <sup>3</sup> Center of Excellence in Theoretical and Mathematical Sciences, Siksha 'O' Anusandhan University, Bhubaneswar, India, <sup>4</sup> Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, India

#### Edited by:

Alexander Merle, Max Planck Institute for Physics (MPG), Germany

#### Reviewed by:

Eduardo Peinado, Instituto de Física, Universidad Nacional Autónoma de México, Mexico Bhupal Dev, Washington University in St. Louis, United States

#### \*Correspondence:

Chandan Hati chandan.hati@clermont.in2p3.fr

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 17 October 2017 Accepted: 14 February 2018 Published: 06 March 2018

#### Citation:

Hati C, Patra S, Pritimita P and Sarkar U (2018) Neutrino Masses and Leptogenesis in Left–Right Symmetric Models: A Review From a Model Building Perspective. Front. Phys. 6:19. doi: 10.3389/fphy.2018.00019 In this review, we present several variants of left–right symmetric models in the context of neutrino masses and leptogenesis. In particular, we discuss various low scale seesaw mechanisms like linear seesaw, inverse seesaw, extended seesaw and their implications to lepton number violating process like neutrinoless double beta decay. We also visit an alternative framework of left–right models with the inclusion of vector-like fermions to analyze the aspects of universal seesaw. The symmetry breaking of left–right symmetric model around few TeV scale predicts the existence of massive right-handed gauge bosons W<sup>R</sup> and Z<sup>R</sup> which might be detected at the LHC in near future. If such signals are detected at the LHC that can have severe implications for leptogenesis, a mechanism to explain the observed baryon asymmetry of the Universe. We review the implications of TeV scale left–right symmetry breaking for leptogenesis.

Keywords: left-right symmetry, neutrino mass, leptogenesis, neutrinoless double beta decay, low scale seesaw

## 1. INTRODUCTION

Although the Standard Model (SM) of particle physics is highly successful in explaining the low energy phenomenology of fundamental particles, the reasons to believe it is incomplete are not less. The most glaring of them all is the issue of neutrino mass which has been confirmed by the oscillation experiments. Some more unsolved puzzles like dark matter, dark energy, baryon asymmetry of the universe strongly suggest that SM is only an effective limit of a more fundamental theory of interactions. In addition to the fact that gravity is completely left out in the SM, the strong interaction is not unified with weak and electromagnetic interactions. In fact, even in the electroweak "unification" one still has two coupling constants, g and g ′ corresponding to SU(2)<sup>L</sup> and U(1)Y. Thus, one is tempted to seek for a more complete theory where the couplings g<sup>s</sup> , g, and g ′ unify at some higher energy scale giving a unified description of the fundamental interactions. Given that the ratio <sup>m</sup>Pl/m<sup>W</sup> is so large, where <sup>m</sup>Pl <sup>=</sup> 1.2 <sup>×</sup> <sup>10</sup><sup>19</sup> GeV is the Planck scale, another major issue in the SM is the infamous "hierarchy problem." The discovery of the Higgs boson with a mass around 125 GeV has the consequence that, if one assumes the Standard Model as an effective theory, then λ ∼ <sup>O</sup>(0.1) and µ <sup>2</sup> <sup>∼</sup> (O(100) GeV)<sup>2</sup> (including the effects of 2-loop corrections). The problem is that every particle that couples, directly or indirectly, to the Higgs field yields a correction to µ 2 resulting in an enormous quantum correction. For instance, let us consider a one-loop correction to µ 2 coming from a loop containing a Dirac fermion f with mass m<sup>f</sup> . If f couples to the Higgs boson via the coupling term (−λ<sup>f</sup> φ¯ f f), then the correction coming from the one-loop diagram is given by

$$
\Delta\mu^2 = \frac{\lambda\_f^2}{8\pi^2} \Lambda\_{\text{UV}}^2 + \cdots \text{ ,}\tag{1}
$$

where 3UV is the ultraviolet momentum cutoff and the ellipsis are the terms proportional to m<sup>2</sup> f , growing at most logarithmically with 3UV. Each of the quarks and leptons in the SM plays the role of f , and if 3UV is of the order of mPl, then the quantum correction to µ 2 is about 30 orders of magnitude larger than the required value of µ <sup>2</sup> <sup>=</sup> 92.9 GeV<sup>2</sup> . Since all the SM quarks, leptons, and gauge bosons obtain masses from hφi, the entire mass spectrum of the Standard Model is sensitive to 3UV. Thus, one expects some new physics between m<sup>W</sup> and mPl addressing this problem. There are also other questions such as why the fermion families have three generations; is there any higher symmetry that dictates different fermion masses even within each generation; in the CKM matrix the weak mixing angles and the CP violating phase are inputs of the theory, instead of being predicted by the SM. Finally, in the cosmic arena, the observed baryon asymmetry of the universe cannot be explained within the SM. Also there are no suitable candidates for dark matter and dark energy in the SM. These also point toward the existence of physics beyond the SM.

In this review, we study several variants of left–right symmetric models which is one of the most popular candidates for physics beyond the standard model. We will review the left– right symmetric models in the context of neutrino masses and leptogenesis. We will study various low scale seesaw mechanisms in the context of left–right symmetric models and their implications to lepton number violating process like neutrinoless double beta decay. We will also discuss an alternative framework of left–right models with the inclusion of vector-like fermions as proposed to analyze various aspects. Interestingly, the breaking of left–right symmetry around few TeV scale predicts the existence of massive right-handed gauge bosons W<sup>R</sup> and Z<sup>R</sup> in left–right symmetric models. These heavy gauge bosons might be detected at the LHC in near future. If such signals are detected at the LHC that can have conclusive implications for leptogenesis, a mechanism to explain the observed baryon asymmetry of the Universe. In this review we will also discuss the implications of such a detection of left–right symmetry breaking for leptogenesis in detail. Before closing this paragraph we would like to stress the fact that this review is far from comprehensive and covers only a limited variety of topics from the vast choices of LRSM-related scenarios. For example, a detailed discussion of the relevant collider phenomenology of the right-handed gauge bosons W<sup>R</sup> and Z<sup>R</sup> and the Higgs sector is beyond the scope of this review. Some relevant references for the LHC phenomenology of W<sup>R</sup> and heavy neutrinos are in Keung and Senjanovic [1], Nemevsek et al. [2], Das et al. [3], Chen et al. [4] and Mitra et al. [5] and for Higgs sector some relevant references are in Bambhaniya et al. [6], Dutta et al. [7], Dev et al. [8] and Mitra et al. [9].

The plan for rest of the review is as follows. In section 2 we briefly review the standard seesaw and radiative mechanisms for light neutrino mass generation. In section 3 we first introduce and then review the standard left–right symmetric theories and the implementation of different types of low scale seesaw implementations. In section 4 we review an alternative formulation of left–right symmetric theories which uses a universal seesaw to generate fermion masses. We also discuss the implications of this model for neutrinoless double beta decay in this case for the specific scenario of type II seesaw dominance. In section 5 we give a brief introduction to leptogenesis and review some of the standard leptogenesis scenarios associated with neutrino mass generation. In section 6 we review the situation of leptogenesis in left–right symmetric theories and the implications of a TeV scale left–right symmetry breaking for leptogenesis. Finally, in section 7 we make concluding remarks.

### 2. NEUTRINO MASSES

The atmospheric, solar and reactor neutrino experiments have established that the neutrinos have small non-zero masses which are predicted to be orders of magnitude smaller than the charged lepton masses. However, in the SM the left handed neutrinos νiL, i = e,µ, τ , transform as (1, 2, −1) under the gauge group SU(3)c×SU(2)L×U(1)Y. Consequently, one cannot write a gauge singlet Majorana mass term for the neutrinos. On the other hand, there are no right handed neutrinos in the SM which would allow a Dirac mass term. The simplest way around this problem is to add singlet right handed neutrinos νiR with the transformation (1, 1, 0) under the SM gauge group. Then one can straightaway write the Yukawa couplings giving Dirac mass to the neutrinos

$$-\mathcal{L}\_{\text{mass}} = \frac{1}{2} h\_{ij} \bar{\psi}\_{iL} \upsilon\_{jR} \phi\_\* \tag{2}$$

such that once φ acquires a VEV, the neutrinos get Dirac mass mDij = hijυ. Here ψiL stands for the SU(2)<sup>L</sup> lepton doublet. However, to explain the lightness of the neutrinos one needs to assume a very small Yukawa coupling for neutrinos in comparison to charged leptons and quarks. However, we do not have a theoretical understanding of why the Yukawa coupling should be so small. Moreover, the accidental B − L symmetry of the SM forbids Majorana masses for the neutrinos. One way out is to consider the dimension-5 effective lepton number violating operator [10–13] of the form

$$\mathcal{L}\_{\text{dim-5}} = \frac{(\nu\phi^0 - e\phi^+)^2}{\Lambda},\tag{3}$$

where 3 is the scale corresponding to some new extension of the SM violating lepton number. This dimension-5 term can induce small Majorana masses to the neutrinos after the eletroweak symmetry breaking

$$-\mathcal{L}\_{\text{mass}} = m\_{\upsilon} \upsilon\_{iL}^T \mathbf{C}^{-1} \upsilon\_{jL},\tag{4}$$

with m<sup>ν</sup> = υ 2 /3. Here, C is the charge conjugation matrix. Consequently, lepton number violating new physics at a high scale 3 would naturally explain the smallness of neutrino masses. In what follows, we discuss some of the popular mechanisms of realizing the same.

#### 2.1. Seesaw Mechanism: Type-I

The type-I seesaw mechanism<sup>1</sup> [14–20] is the simplest mechanism of obtaining tiny neutrino masses. In this mechanism, three singlet right handed neutrinos NiR are added to the SM; and one can write a Yukawa term similar to Equation (2) and a Majorana mass term for the right handed neutrinos since they are singlets under the SM gauge group. The relevant Lagrangian is given by

$$-\mathcal{L}\_{\text{type-I}} = h\_{i\alpha} \bar{\mathbf{N}}\_{iR} \phi l\_{\alpha L} \phi + \frac{1}{2} M\_{\dot{\imath}\dot{\jmath}} \mathbf{N}\_{iL\dot{\imath}L}^{c} \mathbf{C}^{-1} \mathbf{N}\_{\dot{\jmath}L}^{c} + \text{h.c.} \tag{5}$$

Note that, the Majorana mass term breaks the lepton number explicitly and since the right handed neutrinos are SM gauge singlets, there is no symmetry protecting Mij and it can be very large. Now after the symmetry breaking, combining the Dirac and Majorana mass matrices we can write

$$\begin{aligned} -2\mathcal{L}\_{\text{mass}} &= \, ^\text{D}\boldsymbol{\omega} \boldsymbol{\upsilon}\_{\alpha L}^{\text{T}} \, \mathrm{C}^{-1} \boldsymbol{N}\_{\mathrm{i}L}^{\text{c}} + \, ^\text{M}\boldsymbol{N}\_{\mathrm{i}L}^{\text{c}} \, ^\text{C} \mathrm{C}^{-1} \boldsymbol{N}\_{\mathrm{j}L}^{\text{c}} + \, \text{h.c.}\\ &= \, \left( \boldsymbol{\upsilon}\_{\alpha} \, \, \boldsymbol{N}\_{\mathrm{i}}^{\text{c}} \right)\_{L}^{\text{T}} \, \mathrm{C}^{-1} \begin{pmatrix} 0 & \boldsymbol{m}\_{\mathrm{D}\alpha i} \\ \boldsymbol{m}\_{\mathrm{D}\alpha i}^{\text{T}} & \boldsymbol{M}\_{\mathrm{i}} \end{pmatrix} \begin{pmatrix} \boldsymbol{\upsilon}\_{\alpha} \\ \boldsymbol{N}\_{\mathrm{i}}^{\text{c}} \end{pmatrix}\_{L} + \text{h.c.} , \text{ (6)} \end{aligned}$$

where mDα<sup>i</sup> = hDαiυ. Now assuming that the eigenvalues of m<sup>D</sup> are much less than those of M one can block diagonalize the mass matrix to obtain the light Majorana neutrinos with masses <sup>m</sup><sup>ν</sup> ij = −mDαiM−<sup>1</sup> <sup>i</sup> <sup>m</sup><sup>T</sup> Dαi and heavy neutrinos with mass m<sup>N</sup> = M<sup>i</sup> . Note that if any of the right handed neutrino mass eigenvalues (Mi) vanish then some of the left handed neutrinos will combine with the right handed neutrinos to form Dirac neutrinos. For n generations, if the rank of M is r, then there will be 2r Majorana neutrinos and n − r Dirac neutrinos. The type-I seesaw mechanism not only generates tiny neutrino masses, but also provides the necessary ingredients for explaining the baryon asymmetry of the universe via leptogenesis [21], which we will discuss in length in the next section.

#### 2.2. Seesaw Mechanism: Type-II

In type-II seesaw mechanism [18, 19, 22–28], the effective operator given in Equation (3) is realized by extending the SM to include an SU(2)<sup>L</sup> triplet Higgs ξ which transforms under the SM gauge group SU(3)<sup>c</sup> × SU(2)<sup>L</sup> × U(1)<sup>Y</sup> as (1, 3, 1). For simplicity we assume that there are no right handed neutrinos in this model and only one triplet scalar is present. The Yukawa couplings of the triplet Higgs with the left handed lepton doublet (ν<sup>i</sup> , li) are given by

$$-\mathcal{L}\_{\text{type-II}} = f\_{\text{ij}} \left[ \xi^0 \upsilon\_i \upsilon\_j + \xi^+ (\upsilon\_i l\_j + \upsilon\_j l\_i) / \sqrt{2} + \xi^{++} l\_i l\_j \right]. \tag{7}$$

Now a non-zero VEV acquired by ξ 0 (hξ 0 i = u) gives Majorana masses to the neutrinos. Note that u has to be less than a few GeV to not affect the electroweak ρ-parameter. The most general Higgs potential with a doublet and a triplet Higgs has the form

$$V = m\_{\phi}^{2} \phi^{\dagger} \phi + m\_{\xi}^{2} \xi^{\dagger} \xi + \frac{1}{2} \lambda\_{1} (\phi^{\dagger} \phi)^{2} + \frac{1}{2} \lambda\_{2} (\xi^{\dagger} \xi)^{2}$$

$$+ \lambda\_{3} (\phi^{\dagger} \phi)(\xi^{\dagger} \xi) + \lambda\_{4} \phi^{T} \xi^{\dagger} \phi. \tag{8}$$

We assume λ<sup>4</sup> 6= 0, which manifests explicit lepton number violation and the mass of the triplet Higgs M<sup>ξ</sup> ∼ λ<sup>4</sup> ≫ υ. The mass matrix of the scalars <sup>√</sup> 2 Imφ 0 and <sup>√</sup> 2 Imξ 0 is given by

$$
\mathcal{M}^2 = \begin{pmatrix} -4\lambda\_4\mu & 2\lambda\_4\upsilon \\ 2\lambda\_4\upsilon & -\lambda\_4\upsilon^2/\mu \end{pmatrix},\tag{9}
$$

which tells us that one combination of these fields remains massless, which becomes the longitudinal mode of the Z boson; while the other combination becomes massive with a mass of the order of triplet Higgs and hence the danger of Z decaying into Majorons <sup>2</sup> is absent in this model. The minimization of the scalar potential yields

$$
\mu = -\frac{\lambda\_4 \upsilon^2}{M\_\xi^2},
\tag{10}
$$

giving a seesaw mass to the left handed neutrinos

$$m\_{\upsilon\_{i\bar{j}}} = f\_{\bar{i}\bar{j}}\mu = -f\_{\bar{i}\bar{j}}\frac{\lambda\_4 \upsilon^2}{M\_{\bar{\xi}}^2}.\tag{11}$$

Note that in the left–right symmetric extension of the SM, which we will discuss in the next subsection, both type-I and type-II seesaw mechanisms are present together. The type-II seesaw mechanism can also provide a very attractive solution to leptogenesis, which we will discuss in the next section.

#### 2.3. Seesaw Mechanism: Type-III

In type-III seesaw mechanism [29, 30] the SM is extended to include SU(2)<sup>L</sup> triplet fermions to realize the effective operator given in Equation (3)<sup>3</sup> . The Yukawa interactions in Equation (5) are generalized straightforwardly to SU(2)<sup>L</sup> triplet fermions 6 with hypercharge Y = 0. The corresponding interaction Lagrangian is given by

$$-\mathcal{L}\_{\text{type-III}} = h\_{\Sigma \dot{a} \alpha} \bar{\Psi}\_{iL} \left(\vec{\Sigma}\_{\alpha} \cdot \vec{\mathfrak{r}}\right) \tilde{\phi} + \frac{1}{2} M\_{\Sigma a \beta} \vec{\Sigma}\_{\alpha}^{\epsilon T} C^{-1} \vec{\Sigma}\_{\beta}^{\epsilon} + h.c.\tag{12}$$

where α = 1, 2, 3. In exactly similar manner as in the case of type-I seesaw, one obtains for M<sup>6</sup> ≫ υ, the left handed neutrino mass

$$m\_{\upsilon\_{i\bar{j}}} = -\upsilon^2 h\_{\Sigma i\alpha} M\_{\Sigma \beta \alpha}^{-1} h\_{\Sigma j \beta}^T. \tag{13}$$

<sup>1</sup>The seesaw mechanisms generically require a new heavy scale (as compared to the electroweak scale) in the theory, inducing a small neutrino mass (millions of times smaller than the charged lepton masses). Hence the name "seesaw."

<sup>2</sup>Majorons correspond to Goldstone bosons associated with the spontaneous breaking of a global lepton number symmetry.

<sup>3</sup>Ma [30] established the nomenclature Types I, II, III, for the three and only three tree-level seesaw mechanisms.

### 2.4. Radiative Models of Neutrino Mass

Small neutrino masses can also be induced via radiative corrections. The advantage of these models is that without introducing a very large scale into the theory the smallness of the neutrino masses can be addressed. In fact, several of these models can explain naturally the smallness of the neutrino masses with only TeV scale new particles. Thus, new physics scale in these models can be as low as TeV, which can be probed in current and next generation colliders.

One realization of this idea is the so-called Zee model [31, 32], where one extends the SM to have two (or more) Higgs doublets φ<sup>1</sup> and φ2, and a scalar η + which transforms under the SM gauge group SU(3)<sup>c</sup> × SU(2)<sup>L</sup> × U(1)<sup>Y</sup> as (1, 1, 1). The lepton number violating Yukawa couplings are given by

$$\mathcal{L}\_{\text{Zee}} = f\_{\text{ij}} \psi\_{i\text{L}}^T \mathbf{C}^{-1} \psi\_{j\text{L}} \boldsymbol{\eta}^+ + \mu \varepsilon\_{ab} \phi\_a \phi\_b \boldsymbol{\eta}^- + \text{h.c.} \,, \tag{14}$$

where fij is antisymmetric in the family indices i, j and εab is the totally antisymmetric tensor. Now, the VEV of the SM Higgs doublet allows mixing between the singlet charged scalar and the charged component of the second Higgs doublet, resulting in a neutrino mass induced through the one-loop diagram showed in **Figure 1** (left). The antisymmetric couplings of η + with the leptons make the diagonal terms of the mass matrix vanish, with the non-diagonal entries given by

$$m\_{ij}^{\upsilon}(\mathbf{i} \neq j) = A f\_{ij} (m\_i^2 - m\_j^2) \,, \tag{15}$$

where i, j = e,µ, τ and A is a numerical constant. In the Zee model, if the second Higgs doublet is replaced by a doubly charged singlet scalar ζ ++, then one gets what is called Zee-Babu Model [33, 34]. In this model a Majorana neutrino mass can be obtained through a two loop diagram shown in **Figure 1**(right). In fact, there are several other radiative models of Majorana neutrino mass such as the Ma model [35] connecting the Majorana neutrino mass to dark matter at oneloop; Krauss-Nasri-Trodden model [36] and Aoki-Kanemura-Sato model [37] giving neutrino mass at the three loop level with a dark matter candidate in the loop; Gustafsson-No-Rivera model [38] involving a three loop diagram with a dark matter candidate and the W boson; and Kanemura– Sugiyama model [39] utilizing an extension of the Higgs triplet model. There are also models for radiative Dirac neutrino masses such as the Nasri-Moussa model [40] utilizing a softly broken symmetry; Gu-Sarkar model [41] with dark matter candidates in the loop; Kanemura-Matsui-Sugiyama model [42] utilizing an extension of the two Higgs doublet model; Bonilla-Ma-Peinado-Valle model where the Dirac neutrino masses are generated at two-loops with dark matter in the loop [43], etc.

### 3. LEFT–RIGHT SYMMETRIC THEORIES

The SM gauge group <sup>G</sup>SM ≡ SU(3)<sup>c</sup> × SU(2)<sup>L</sup> × U(1)<sup>Y</sup> explains the (V −A) structure of the weak interaction and parity violation, which is reflected by the trivial transformation of all right handed fields under SU(2)L. However, the origin of parity violation is not explained within the SM, and it is natural to seek an explanation for parity violation starting from a parity conserved theory at some higher energy scale. This motivated a left–right symmetric extension of the SM gauge theory, called the Left– Right Symmetric Model (LRSM) [44–49], in which the Standard Model gauge group is extended to

$$\mathcal{G}\_{LR} \equiv \mathrm{SU(3)}\_{\mathcal{C}} \times \mathrm{SU(2)}\_{L} \times \mathrm{SU(2)}\_{R} \times U(1)\_{B-L}$$

where B − L is the difference between baryon (B) and lepton (L) numbers. The left–right symmetric theory, initially proposed to explain the origin of parity violation in low-energy weak interactions has come a long way answering various other issues like small neutrino mass, dark matter as left by the Standard Model. Originally suggested by Pati-Salam, the model has been studied over and over because of its versatility and many alternative formulations of the model have also been proposed. The model stands on the foundation of a complete symmetry between left and right which means Parity is an explicit symmetry in it until spontaneous symmetry breaking occurs. As evident from the gauge group, the natural inclusion of a righthanded neutrino in it makes the issue of neutrino mass an easy affair to discuss. Three new gauge bosons namely W± R that are the heavier parity counterparts of W± L of the standard model and a Z ′ boson analogous to the Z boson also find place in the framework. LRSM breaks down to Standard Model gauge theory at low energy scales, SU(2)<sup>L</sup> × SU(2)<sup>R</sup> × U(1)B−<sup>L</sup> × SU(3)<sup>C</sup> −→ SU(2)<sup>L</sup> × U(1)<sup>Y</sup> × SU(3)C. It has been noticed that the choices of Higgs and their mass scales in the model offers rich phenomenology which can be verified at the current and planned experiments.

The basic framework and properties of Left–Right Symmetric Models are already discussed at length in various original works [46–48], thus we only intend to study here various seesaw mechanisms for the generation of neutrino mass and its implications to leptogenesis in various Left–Right Symmetric models.

A very brief sketch of the manifest left–right symmetric model is given here. The model is based on the gauge group,

$$\mathcal{G}\_{LR} \equiv \mathrm{SU(2)}\_{L} \times \mathrm{SU(2)}\_{R} \times U(1)\_{B-L} \times \mathrm{SU(3)}\_{C} \,. \tag{16}$$

The electric charge Q is difined as,

$$Q = T\_{3L} + T\_{3R} + \frac{B - L}{2} = T\_{3L} + Y. \tag{17}$$

Here, T3<sup>L</sup> and T3<sup>R</sup> are, respectively, the third components of isospin of the gauge groups SU(2)<sup>L</sup> and SU(2)R, and Y is the hypercharge. The particle spectrum of a generic LRSM can be sketched as,

$$\ell\_L = \binom{\upsilon L}{\mathcal{e}\_L} \sim \langle \mathbf{2}, \mathbf{1}, -1, \mathbf{1} \rangle, \; \ell\_R = \binom{\upsilon\_R}{\mathcal{e}\_R} \sim \langle \mathbf{1}, \mathbf{2}, -1, \mathbf{1} \rangle \langle \mathbf{1} \mathbf{3} \rangle$$

$$q\_L = \binom{\iota\_R}{d\_R} \sim \langle \mathbf{2}, \mathbf{1}, \frac{1}{3}, \mathbf{3} \rangle, \; q\_R = \binom{\iota\_R}{d\_R} \sim \langle \mathbf{1}, \mathbf{2}, \frac{1}{3}, \mathbf{3} \rangle. \; \text{(19)}$$

The spontaneous symmetry breaking of the gauge group which occurs in two steps gives masses to fermions including neutrinos. In the first step the gauge group SU(2)<sup>L</sup> × SU(2)<sup>R</sup> × U(1)B−<sup>L</sup> × SU(3)<sup>C</sup> breaks down to SU(2)<sup>L</sup> × U(1)<sup>Y</sup> × SU(3)<sup>C</sup> i.e., the SM gauge group. This gauge group then breaks down to U(1)em × SU(3)C. However these symmetry breakings totally depend upon the choices of Higgs that we consider in the framework and their mass scales. Thus, in this review we intend to discuss fermion masses emphasizing on neutrino mass in possible choices of symmetry breakings of LRSM.

### 3.1. LRSM With Bidoublet (B − L = 0) and Doublets (B − L = −1).

Here we use Higgs bidoublet 8 to implement the symmetry breaking of SM down to low energy theory leading to charged fermion masses. The symmetry breaking of LRSM to SM occurs via RH Higgs doublet H<sup>R</sup> (B − L = −1). We need the lefthanded counterpart H<sup>L</sup> to ensure left–right invariance. The fermions including usual quarks and leptons along with scalars are presented in **Table 1**.

The matrix structure of the scalar fields looks as follows,

$$\begin{aligned} \Phi &\equiv \begin{pmatrix} \phi\_1^0 & \phi\_2^+ \\ \phi\_1^- & \phi\_2^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1} \end{pmatrix}, \\\ H\_L &\equiv \begin{pmatrix} h\_L^+ \\ h\_L^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, -\mathbf{1}, \mathbf{1} \end{pmatrix}, \qquad H\_R \equiv \begin{pmatrix} h\_R^+ \\ h\_R^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, -\mathbf{1}, \mathbf{1} \end{pmatrix}. \end{aligned} \tag{20}$$



With usual quarks and leptons the Yukawa Lagrangian reads as,

$$-\mathcal{L}\_{Yuk} \supset \overline{q\_L} \left[ Y\_1 \Phi + Y\_2 \widetilde{\Phi} \right] q\_R + \overline{\ell\_L} \left[ Y\_3 \Phi + Y\_4 \widetilde{\Phi} \right] \ell\_R + \text{ h.c.} \tag{21}$$

where <sup>8</sup><sup>e</sup> <sup>=</sup> <sup>σ</sup>28∗σ<sup>2</sup> and <sup>σ</sup><sup>2</sup> is the second Pauli matrix. When the scalar bidoublet (8) takes non-zero VEV ,

$$
\langle \Phi \rangle = \begin{pmatrix} \upsilon 1 & 0 \\ 0 & \upsilon 2 \end{pmatrix}, \tag{22}
$$

it gives masses to quarks and charged leptons in the following manner,

$$\begin{aligned} M\_u &= Y\_1 \upsilon 1 + Y\_2 \upsilon\_2^\*, & M\_d &= Y\_1 \upsilon 2 + Y\_2 \upsilon\_1^\*, \\ M\_e &= Y\_3 \upsilon 2 + Y\_4 \upsilon\_1^\*. \end{aligned} \tag{23}$$

It also yields Dirac mass for light neutrinos as

$$M\_D^\upsilon \equiv M\_D = Y\_3 \upsilon\_1 + Y\_4 \upsilon\_2^\* \,. \tag{24}$$

The only role that the Higgs doublets play here is helping in the spontaneous symmetry breaking of LRSM to SM. It is also important to note that the breaking of SU(2)<sup>R</sup> by doublet Higgs leads to Dirac neutrinos.

### 3.2. LRSM With Bidoublet (B − L = 0) and Triplets (B − L = 2).

Along with the bidoublet 8, here we use triplets 1L, 1<sup>R</sup> for the spontaneous symmetry breakings.

$$\begin{aligned} \Phi &= \begin{pmatrix} \phi\_1^0 & \phi\_2^+ \\ \phi\_1^- & \phi\_2^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1} \end{pmatrix}, \\ \Delta\_L &= \begin{pmatrix} \delta\_L^+ / \sqrt{2} & \delta\_L^{++} \\ \delta\_L^0 & -\delta\_L^+ / \sqrt{2} \end{pmatrix} \sim \begin{pmatrix} \mathbf{3}, \mathbf{1}, \mathbf{2}, \mathbf{1} \end{pmatrix}, \\ \Delta\_R &= \begin{pmatrix} \delta\_R^+ / \sqrt{2} & \delta\_R^{++} \\ \delta\_R^0 & -\delta\_R^+ / \sqrt{2} \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{3}, \mathbf{2}, \mathbf{1} \end{pmatrix}, \end{aligned} \tag{25}$$

The particle content of the model is shown in **Table 2**. The Yukawa Lagrangian is given by

$$\begin{split}-\mathcal{L}\_{Yuk} &\supset \overline{q\_{L}} \Big[Y\_{1}\Phi + Y\_{2}\widetilde{\Phi}\Big]q\_{R} + \overline{\ell\_{L}} \Big[Y\_{3}\Phi + Y\_{4}\widetilde{\Phi}\Big]\ell\_{R} \\ &+ f\left[\overline{(\ell\_{L})^{c}}\ell\_{L}\Delta\_{L} + \overline{(\ell\_{R})^{c}}\ell\_{R}\Delta\_{R}\right] + \text{h.c.},\end{split} \tag{26}$$


TABLE 2 | LRSM representations of extended field content.

The scalar triplets 1<sup>L</sup> , 1<sup>R</sup> give Majorana masses to light lefthanded and heavy right-handed neutrinos. The neutral lepton mass matrix is given by

$$M\_{\upsilon} = \begin{pmatrix} M\_L \ M\_D \\ M\_D^T \ M\_R \end{pmatrix},\tag{27}$$

Here M<sup>L</sup> = fLh1Li = f υ<sup>L</sup> (M<sup>R</sup> = fRh1Ri = f υR) denoted as the Majorana mass matrix for left-handed (right-handed) neutrinos and M<sup>D</sup> = Y3υ<sup>1</sup> + Y4υ<sup>2</sup> is the Dirac neutrino mass matrix connecting light-heavy neutrinos. The complete diagonalization results type-I+II seesaw formula for light neutrinos as,

$$m\_{\upsilon} = M\_L - m\_D M\_R^{-1} m\_D^T = m\_{\upsilon}^H + m\_{\upsilon}^I,\tag{28}$$

#### 3.3. LRSM With Inverse Seesaw

In canonical seesaw mechanisms, the tiny mass of light neutrinos is explained with large value of seesaw scale thereby making it inaccessible to the ongoing collider experiments. On the other hand, the light neutrino masses may arise from low scale seesaw mechanisms like inverse seesaw [50, 51] where the seesaw scale can be probed at upcoming accelerators. The inverse seesaw mechanism in LRSM can be realized with the following particle content;

**Fermions**:

$$\begin{aligned} q\_{\mathbb{L}} &= \begin{pmatrix} u\_{L} \\ d\_{L} \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, \mathbf{1}/\mathbf{3}, \mathbf{3} \end{pmatrix}, \; q\_{\mathbb{R}} = \begin{pmatrix} u\_{\mathbb{R}} \\ d\_{\mathbb{R}} \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, \mathbf{1}/\mathbf{3}, \mathbf{3} \end{pmatrix}, \\\ \ell\_{L} &= \begin{pmatrix} \mathbb{v}\_{L} \\ \mathcal{e}\_{L} \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, -\mathbf{1}, \mathbf{1} \end{pmatrix}, \; \ell\_{\mathbb{R}} = \begin{pmatrix} \mathbb{v}\_{\mathbb{R}} \\ \mathcal{e}\_{\mathbb{R}} \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, -\mathbf{1}, \mathbf{1} \end{pmatrix}, \\\ \mathbb{S} &\sim \begin{pmatrix} \mathbf{1}, \mathbf{1}, \mathbf{1}, \mathbf{0} \end{pmatrix}, \end{aligned}$$

**Scalars**:

$$H\_L = \begin{pmatrix} h\_L^+ \\ h\_L^0 \end{pmatrix} \sim (\mathbf{2}, \mathbf{1}, \mathbf{1}, \mathbf{1}), \quad H\_R = \begin{pmatrix} h\_R^+ \\ h\_R^0 \end{pmatrix} \sim (\mathbf{1}, \mathbf{2}, \mathbf{1}, \mathbf{1})$$

$$\Phi = \begin{pmatrix} \phi\_1^0 & \phi\_2^+ \\ \phi\_1^- & \phi\_2^0 \end{pmatrix} \sim (\mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1}), \tag{29}$$

The fermion sector here comprises of the usual quarks and leptons plus one extra fermion singlet per generation. The scalar sector holds the doublets HL,<sup>R</sup> with B − L charge −1 and the bidoublet 8 with B − L charge 0. The Yukawa Langrangian for inverse seesaw mechanism is given by,

$$-\mathcal{L}\_{Yuk} = \overline{\ell\_L} \left[ Y\_3 \Phi + Y\_4 \widetilde{\Phi} \right] \ell\_R + F \overline{(\ell\_R)} H\_R \mathcal{S}\_L^\circ + \mu \overline{\mathcal{S}\_L^\circ} \mathcal{S}\_L + \text{h.c.} \tag{30}$$

$$\supset M\_D \overline{\nu\_L} N\_R + M \overline{N\_R} \mathcal{S}\_L + \mu\_S \overline{\mathcal{S}\_L^\circ} \mathcal{S}\_L + \text{h.c.} \tag{31}$$

After spontaneous symmetry breaking the resulting neutral

L

$$\mathbf{M} = \begin{pmatrix} 0 & M\_D & 0 \\\\ M\_D^T & 0 & M \\\\ 0 & M^T & \mu\_S \end{pmatrix}. \tag{32}$$

With the mass hierarchhy mD, M ≫ µS, the light neutrino mass formula is given by,

$$m\_{\boldsymbol{\vartheta}} = M\_{\boldsymbol{D}} (\boldsymbol{M}^T)^{-1} \boldsymbol{\mu} \boldsymbol{M}^{-1} \boldsymbol{M}\_{\boldsymbol{D}}^T. \tag{33}$$

#### 3.4. LRSM With Linear Seesaw

lepton mass matrix reads as follows,

Another interesting low scale seesaw type is linear seesaw mechanism [52, 53] which can be realized with the following particle content in a LRSM.

**Fermions**:

$$\begin{aligned} Q\_L &= \begin{pmatrix} \iota\_L \\ d\_L \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, \mathbf{1}/\mathbf{3}, \mathbf{3} \end{pmatrix}, \ Q\_R = \begin{pmatrix} \iota\_R \\ d\_R \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, \mathbf{1}/\mathbf{3}, \mathbf{3} \end{pmatrix}, \\\ \ell\_L &= \begin{pmatrix} \upsilon\_L \\ \varepsilon\_L \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, -\mathbf{1}, \mathbf{1} \end{pmatrix}, \ \ell\_R = \begin{pmatrix} \upsilon\_R \\ \varepsilon\_R \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, -\mathbf{1}, \mathbf{1} \end{pmatrix}, \\\ S &\sim \{\mathbf{1}, \mathbf{1}, \mathbf{0}, \mathbf{1}\}, \end{aligned}$$

**Scalars**:

$$\begin{aligned} H\_L &= \begin{pmatrix} h\_L^+ \\ h\_L^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{1}, \mathbf{1}, \mathbf{1} \end{pmatrix} & H\_R &= \begin{pmatrix} h\_R^+ \\ h\_R^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{1}, \mathbf{2}, \mathbf{1}, \mathbf{1} \end{pmatrix}, \\\ \Phi &= \begin{pmatrix} \phi\_1^0 & \phi\_2^+ \\ \phi\_1^- & \phi\_2^0 \end{pmatrix} \sim \begin{pmatrix} \mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1} \end{pmatrix}. \end{aligned}$$

The scalars take non-zero vev as follows:

$$
\langle \Phi \rangle = k\_1, k\_2, \quad \langle H\_L \rangle = \upsilon\_L, \quad \langle H\_R \rangle = \upsilon\_R,
$$

Let us write down the relevant Yukawa terms in the Lagrangian that contribute to the fermion masses:

$$\mathcal{L}\_{\text{Yuk}} = h\_{\ell} \overline{\ell\_R} \Phi \,\ell\_L + \overline{h\_{\ell}} \overline{\ell\_R} \,\widetilde{\Phi} \,\ell\_L + f\_R \,\overline{\mathbb{S}} \,\widetilde{H}\_R \,\ell\_R + f\_L \,\overline{\mathbb{S}} \,\widetilde{H}\_L \,\ell\_L$$

$$+ \mu\_S \overline{(\mathbb{S}\_L)^c} \mathbf{S}\_L + \text{h.c.},\tag{34}$$

where <sup>H</sup>e<sup>j</sup> <sup>=</sup> <sup>i</sup>τ2H<sup>∗</sup> <sup>j</sup> with <sup>j</sup> <sup>=</sup> <sup>L</sup>, <sup>R</sup> and <sup>8</sup><sup>e</sup> <sup>=</sup> <sup>τ</sup>28<sup>∗</sup> τ2. The singlet Majorana field S in Equation (34) is defined as

$$\mathcal{S} = \frac{\mathcal{S}\_L + (\mathcal{S}\_L)^c}{\sqrt{2}}.\tag{35}$$

resulting in the neutral lepton mass matrix

$$M\_N = \begin{pmatrix} 0 & h\ell\_l k\_1 + \widetilde{h}\ell\_k k\_2 & f\_l \upsilon\_L \\ h\_\ell^T k\_1 + \widetilde{h}\_\ell^T k\_2 & 0 & f\_R \upsilon\_R \\ f\_L^T \upsilon\_L & f\_R^T \upsilon\_R & \mu\_S \end{pmatrix}$$

$$= \begin{pmatrix} 0 & m\_D & m\_L \\ m\_D^T & 0 & M \\ m\_L^T & M^T & \mu\_S \end{pmatrix}. \tag{36}$$

The violation of lepton number by two units arises here through the combination m<sup>L</sup> and µS. As a result, assuming mL≪m<sup>D</sup> < M, one gets the light Majorana masses of the active neutrinos to be

$$\begin{aligned} m\_{\boldsymbol{\upsilon}} &= \left[ m\_{\boldsymbol{D}} \boldsymbol{M}^{\boldsymbol{T}^{-1}} \boldsymbol{m}\_{\boldsymbol{L}}^{\boldsymbol{T}} + m\_{\boldsymbol{L}} \boldsymbol{M}^{-1} \boldsymbol{m}\_{\boldsymbol{D}}^{\boldsymbol{T}} \right] \\ &= \left[ m\_{\boldsymbol{D}} \left( \boldsymbol{f}\_{\boldsymbol{L}} \boldsymbol{f}\_{\boldsymbol{R}}^{-1} \right)^{\boldsymbol{T}} + \left( \boldsymbol{f}\_{\boldsymbol{L}} \boldsymbol{f}\_{\boldsymbol{R}}^{-1} \right) \boldsymbol{m}\_{\boldsymbol{D}}^{\boldsymbol{T}} \right] \times \frac{\left( \mu\_{1} k\_{1} + \mu\_{2} k\_{2} \right)}{\boldsymbol{M}^{\boldsymbol{\prime}} \boldsymbol{\eta}\_{\boldsymbol{P}}} . \end{aligned} \tag{37}$$

The last line in Equation (37) follows from the fact that in left– right symmetric model where Parity and SU(2)<sup>R</sup> breaking occurs at different scales υ<sup>L</sup> is given by

$$\nu\_L \simeq -\frac{(\mu\_1 k\_1 + \mu\_2 k\_2)\nu\_R}{M'\eta\_P},\tag{38}$$

where µ1,µ<sup>2</sup> are the trilinear terms arising in the Higgs potential involving Higgs bidoublet and Higgs doublets, η<sup>P</sup> is the parity breaking scale and M′ is the SU(2)<sup>R</sup> breaking scale. From Equation (37) it is clear that the light neutrino mass is suppressed by the parity breaking scale η<sup>P</sup> ≃ M′ . The f<sup>L</sup> and f<sup>R</sup> are Majorana couplings, k1, k<sup>2</sup> being VEV of Higgs bidoublet while υL(υR) is the VEV of LH (RH) scalar doublet. The smallness of ν<sup>L</sup> thus ensures the smallness of the observed sub-eV scale neutrino masses. The SU(2)<sup>R</sup> × U(1)B−<sup>L</sup> breaking scale υ<sup>R</sup> can be as low as a few TeV. This is in contrast to the usual left–right symmetric model without D-parity, where the neutrino mass is suppressed by v<sup>R</sup> and hence cannot be brought to TeV scales easily [54].

In addition we get two heavy pseudo-Dirac states, whose masses are separated by the light neutrino mass, given by

$$
\widetilde{M} \approx \pm M + m\_{\vee} \,. \tag{39}
$$

In the above equation, the small masses of active neutrinos can arise through small values of mL/M. As a result of M around TeV and m<sup>D</sup> in the range of 100 GeV, sizable mixing between the light and heavy states arises, and the Pseudo-Dirac pair with mass M can be probed at colliders<sup>4</sup> .

#### 3.5. LRSM With Extended Seesaw

The LRSM here is extended with the addition of a neutral fermion S<sup>L</sup> per generation to the usual quarks and leptons.<sup>5</sup> The scalar sector consists of bidoublet 8 with B − L = 0, triplets 1<sup>L</sup> ⊕ 1<sup>R</sup>

with B − L = 2 and doublets H<sup>L</sup> ⊕ H<sup>R</sup> with B − L = −1. We call the model Extended LR model and thus the seesaw mechanism is called extended seesaw. **Table 3** shows the complete particle spectrum.

The leptonic Yukawa interaction terms can be written as,

$$-\mathcal{L}\_{Yuk} = \overline{\ell\_L} \left[ Y\_3 \Phi + Y\_4 \widetilde{\Phi} \right] \ell\_R + f \left[ \overline{(\ell\_L)^c} \ell\_L \Delta\_L + \overline{(\ell\_R)^c} \ell\_R \Delta\_R \right]$$

$$+ F \overline{(\ell\_R)} H\_R \underline{\ell\_L^c} + F' \overline{(\ell\_L)} H\_L \underline{\ell\_L} + \mu\_S \overline{\mathcal{S}\_L^c} \mathcal{S}\_L + \text{h.c.} \tag{40}$$

$$\supset M\_D \overline{\nu\_L} N\_R + M\_L \overline{\nu\_L^c} \nu\_L + M\_R \overline{\mathcal{N}\_R^c} N\_R$$

$$+ M \overline{\mathcal{N}\_R} \mathcal{S}\_L + \mu\_L \overline{\nu\_L^c} \mathcal{S}\_L + \mu\_S \overline{\mathcal{S}\_L^c} \mathcal{S}\_L \tag{41}$$

The neutral lepton mass matrix comes out to be;

$$\mathbb{M}\_{\boldsymbol{\nu}} = \begin{pmatrix} M\_L & M\_D \ \mu\_L \\ M\_R^T & M\_R \ M \\ \mu\_L^T & M^T \ \mu\_S \end{pmatrix},\tag{42}$$

in the basis νL, N c R , S<sup>L</sup> after spontaneous symmetry breaking. The individual elements of the matrix hold the following meaning; M<sup>D</sup> = Yh8i measures the light-heavy neutrino mixing and is usually called the Dirac neutrino mass matrix, M<sup>N</sup> = f υ<sup>R</sup> = f h1Ri (M<sup>L</sup> = f υ<sup>L</sup> = f h1Li) is the Majorana mass term for heavy (light) neutrinos, M = F hHRi is the N − S mixing matrix, µ<sup>L</sup> = F ′ hHLi stands for the small mass term connecting ν−S and µ<sup>S</sup> is the Majorana mass term for the singlet fermion SL. **Inverse Seesaw:-** In Equation (42), following the mass hierarchy M≫MD≫µ<sup>S</sup> and with the assumption that ML, MR,µ<sup>L</sup> → 0 one obtains the inverse seesaw mass formula for light neutrinos [59]

$$m\_{\upsilon} = \left(\frac{M\_D}{M}\right)\mu\_s \left(\frac{M\_D}{M}\right)^T.$$

Let us have a look at the model parameters of inverse seesaw framework and see how the light neutrino mass can be parametrized in terms of these.

$$\left(\frac{m\_{\upsilon}}{0.1\ \text{eV}}\right) = \left(\frac{M\_{\text{D}}}{100\ \text{GeV}}\right)^{2} \left(\frac{\mu\_{\text{s}}}{\text{keV}}\right) \left(\frac{M}{10^{4}\ \text{GeV}}\right)^{-2}.$$

TABLE 3 | LRSM representations of extended field content.


<sup>4</sup>Most often the linear seesaw is assumed to be realized with <sup>M</sup> <sup>≫</sup> <sup>m</sup><sup>D</sup> <sup>∼</sup> <sup>m</sup><sup>L</sup> <sup>∼</sup> 100 GeV, which results in the same expression for the mν . This would result in unobservable heavy fermions and negligible mixing.

<sup>5</sup>The discussion of extended seesaw mechanism can be found in Gavela et al. [55], Barry et al. [56], Zhang [57] and Dev and Pilaftsis[58].

Testable collider phenomenology can be expected in such a scenario because M lies at a few TeV scale which allows large left– right mixing. For an extension of such a scenario which allows large LNV and LFV one may refer the work [60].

**Linear Seesaw:-** Alternatively, in Equation (42), the assumption of ML, MR,µ<sup>S</sup> → 0 leads to the linear seesaw mass formula for light neutrinos given by Deppisch et al. [61]

$$
\mu m\_{\vee} = M\_D^T M^{-1} \mu\_L + \text{transpose} \,, \tag{43}
$$

whereas the heavy neutrinos form a pair of pseudo-Dirac states with masses

$$M\_{\pm} \approx \pm M + m\_{\vee}.\tag{44}$$

**Type-II Seesaw Dominance:-** On the other hand a type-II seesaw dominance can be realized with the assumption that µL,µ<sup>S</sup> → 0 in Equation (42).This allows large left–right mixing and thus leads to an interesting scenario.

A natural type-II seesaw dominance can be realized from the following Yukawa interactions

$$-\mathcal{L}\_{Yuk} = \overline{\ell\_L} \left[ Y\_3 \Phi + Y\_4 \widetilde{\Phi} \right] \ell\_R + f \left[ \overline{(\ell\_L)^c} \ell\_L \Delta\_L + \overline{(\ell\_R)^c} \ell\_R \Delta\_R \right] \tag{45}$$

$$+ F \left( \overline{\ell\_R} \right) H\_R \mathcal{S}\_L^\complement + \text{h.c.} \tag{45}$$

$$\supset M\_D \overline{\upsilon\_L} N\_R + M\_L \overline{\upsilon\_L^c} \upsilon\_L + M\_R \overline{N\_R^c} N\_R + M \overline{N\_R} \mathcal{S}\_L + \text{h.c.} \tag{46}$$

The gauge singlet mass term µSS <sup>c</sup>S does not appear in the above Lagrangian since we have considered this to be zero or negligbly small to suppress the generic inverse seesaw contribution involving µS. We have also assumed the induced VEV for H<sup>L</sup> to be zero, i.e., hHLi → 0.

Now the complete 9 × 9 mass matrix for the neutral fermions in flavor basis can be written as

$$\mathbb{M} = \begin{pmatrix} \begin{array}{cc} \nu\_L & \mathbb{S}\_L & N\_R^c \\ \hline \nu\_L & M\_L & 0 & M\_D \\ \mathbb{S}\_L & 0 & 0 & M \\ N\_R^c \begin{vmatrix} M\_D^T & M^T & M\_R \end{vmatrix} \end{vmatrix} . \tag{47}$$

The heaviest right-handed neutrinos can be integrated out following the standard formalism of seesaw mechanism. Using mass hierarchy M<sup>R</sup> > M > M<sup>D</sup> ≫ M<sup>L</sup> one obtains

$$\begin{aligned} \mathbb{M}' &= \begin{pmatrix} M\_L & 0 \\ 0 & 0 \end{pmatrix} - \begin{pmatrix} M\_D \\ M \end{pmatrix} M\_R^{-1} \begin{pmatrix} M\_D^T \ M^T \end{pmatrix} \\ &= \begin{pmatrix} M\_L - M\_D M\_R^{-1} M\_D^T & -M\_D M\_R^{-1} M^T \\ M M\_R^{-1} M\_D^T & -M M\_R^{-1} M^T \end{pmatrix}, \end{aligned} \tag{48}$$

where the intermediate block diagonalised neutrino states are modified as

$$\begin{aligned} \nu' &= \nu\_L - M\_D M\_R^{-1} N\_R^\epsilon, \\ S' &= \mathbb{S}\_L - M\_D M\_R^{-1} N\_R^\epsilon, \\ N' &= N\_R^\epsilon + (M\_R^{-1} M\_D^T)^\* \nu\_L + (M\_R^{-1} M^T)^\* \mathbb{S}\_L. \end{aligned} \tag{49}$$

The following transformation relates the intermediate block diagonalised neutrino states to the flavor eigenstates.

$$
\begin{pmatrix} \boldsymbol{\nu}' \\ \boldsymbol{S}' \\ \boldsymbol{N}' \end{pmatrix} = \begin{pmatrix} \mathbb{I} & \mathbb{O} & -M\_{D}\boldsymbol{M}\_{R}^{-1} \\ \mathbb{O} & \mathbb{I} & -M\boldsymbol{M}\_{R}^{-1} \\ \left(M\_{D}\boldsymbol{M}\_{R}^{-1}\right)^{\dagger} & \left(M\boldsymbol{M}\_{R}^{-1}\right)^{\dagger} & \mathbb{I} \end{pmatrix} \begin{pmatrix} \boldsymbol{\nu}\_{L} \\ \boldsymbol{S}\_{L} \\ \boldsymbol{N}\_{R}^{c} \end{pmatrix} \tag{50}
$$

In the mass matrix M′ the (2, 2) entry is larger than other entries in the limit M<sup>R</sup> > M > M<sup>D</sup> ≫ ML. The same procedure can be repeated in Equation (48) and S ′ can be integrated out. Now the mass formula for light neutrino is given by

$$\begin{aligned} m\_{\boldsymbol{\nu}} &= \left[M\_{\boldsymbol{L}} - M\_{\boldsymbol{D}}M\_{\boldsymbol{R}}^{-1}M\_{\boldsymbol{D}}^{T}\right] \\ &- \left(-M\_{\boldsymbol{D}}M\_{\boldsymbol{R}}^{-1}M^{T}\right)\left(-M M\_{\boldsymbol{R}}^{-1}M^{T}\right)^{-1}\left(-M M\_{\boldsymbol{R}}^{-1}M\_{\boldsymbol{D}}^{T}\right) \\ &= \left[M\_{\boldsymbol{L}} - M\_{\boldsymbol{D}}M\_{\boldsymbol{R}}^{-1}M\_{\boldsymbol{D}}^{T}\right] + M\_{\boldsymbol{D}}M\_{\boldsymbol{R}}^{-1}M\_{\boldsymbol{D}}^{T} \\ &= M\boldsymbol{L} = m\_{\boldsymbol{\nu}}^{\Pi}, \end{aligned} \tag{51}$$

and the physical block diagonalised states are

$$\begin{aligned} \hat{\boldsymbol{\nu}} &= \boldsymbol{\nu}\_{L} - \boldsymbol{M}\_{D} \boldsymbol{M}^{-1} \boldsymbol{S}\_{L} \\ \hat{\boldsymbol{S}} &= \boldsymbol{S}\_{L} - \boldsymbol{M} \boldsymbol{M}\_{R}^{-1} \boldsymbol{N}\_{R}^{c} + \left( \boldsymbol{M}\_{D} \boldsymbol{M}^{-1} \right)^{\dagger} \boldsymbol{S}\_{L} \end{aligned} \tag{52}$$

with the corresponding block diagonalised transformation as

$$
\begin{pmatrix} \hat{\imath} \\ \hat{\jmath} \end{pmatrix} = \begin{pmatrix} \mathbb{I} & -M\_D M^{-1} \\ (M M^{-1})^\dagger & \mathbb{I} \end{pmatrix} \begin{pmatrix} \boldsymbol{\nu}' \\ \boldsymbol{\mathcal{S}}' \end{pmatrix} \tag{53}
$$

Following this block diagonalization procedure the flavor eigenstates can be related to mass eigenstates through the following transformation

$$
\begin{pmatrix} \boldsymbol{\upsilon}\_{L} \\ \boldsymbol{\mathcal{S}}\_{L} \\ \boldsymbol{N}\_{R}^{\boldsymbol{\varepsilon}} \end{pmatrix} = \begin{pmatrix} \mathbb{I} & \boldsymbol{M}\_{D} \boldsymbol{M}^{-1} & \boldsymbol{M}\_{D} \boldsymbol{M}\_{R}^{-1} \\ (\boldsymbol{M}\_{D} \boldsymbol{M}^{-1})^{\dagger} & \mathbb{I} & \boldsymbol{M} \boldsymbol{M}\_{R}^{-1} \\ \mathbb{O} & -(\boldsymbol{M} \boldsymbol{M}\_{R}^{-1})^{\dagger} & \mathbb{I} \end{pmatrix} \begin{pmatrix} \boldsymbol{\upsilon}' \\ \boldsymbol{S}' \\ \boldsymbol{N}' \end{pmatrix} \tag{54}
$$

Finally, the physical masses can be obtained by diagonalising the final block diagonalised mass matrices by a 9 × 9 unitary matrix V9×9. The block diagonalised neutrino states can be expressed in terms of mass eigenstates as follows,

$$
\hat{\upsilon}\_{\alpha} = U\_{\upsilon\_{\alpha i}} \upsilon\_i, \quad \hat{\mathsf{S}}\_{\alpha} = U\_{\mathsf{S}\alpha i} \mathsf{S}\_i, \quad \hat{\mathsf{N}}\_{\alpha} = U\_{\mathsf{N}\alpha i} \mathsf{N}\_i. \tag{55}
$$

while the block diagonalised mass matrices for light lefthanded neutrinos, heavy right-handed neutrinos and extra sterile neutrinos are

$$\begin{aligned} m\_{\boldsymbol{\nu}} &= M\_L, \\ M\_N &\equiv M\_R = \frac{\nu\_R}{\nu\_L} M\_L, \\ M\_S &= -M M\_R^{-1} M^T. \end{aligned} \tag{56}$$

Further these mass matrices can be diagonalised by respective 3 × 3 unitarity matrices as,

$$\begin{aligned} m\_{\upsilon}^{\text{diag}} &= U\_{\upsilon}^{\dagger} m\_{\upsilon} U\_{\upsilon}^{\*} = \text{diag}\{m\_{1}, m\_{2}, m\_{3}\}, \\ M\_{S}^{\text{diag}} &= U\_{S}^{\dagger} M\_{S} U\_{S}^{\*} = \text{diag}\{M\_{S\_{1}}, M\_{S\_{2}}, M\_{S\_{3}}\}, \\ M\_{N}^{\text{diag}} &= U\_{N}^{\dagger} M\_{N} U\_{N}^{\*} = \text{diag}\{M\_{N\_{1}}, M\_{N\_{2}}, M\_{N\_{3}}\}. \end{aligned} \tag{57}$$

The complete block diagonalization results,

$$\begin{split} \widehat{\mathbb{M}} &= \mathsf{V}\_{g \times g}^{\dagger} \mathbb{M} \mathsf{V}\_{g \times g}^{\*} = (\mathbb{W} \cdot \mathbb{U})^{\dagger} \mathbb{M} \left( \mathbb{W} \cdot \mathbb{U} \right) \\ &= \text{diag} \{ m\_{1}, m\_{2}, m\_{3}; \ M\_{\mathrm{S}1}, M\_{\mathrm{S}2}, M\_{\mathrm{S}3}; M\_{\mathrm{N}1}, M\_{\mathrm{N}2}, M\_{\mathrm{N}3} \}, \end{split} \tag{58}$$

where W is the block diagonalised mixing matrix and U is the unitarity matrix given by,

$$
\mathbb{W} = \begin{pmatrix}
\mathbb{I} & M\_{D}M^{-1} & M\_{D}M\_{R}^{-1} \\
\{M\_{D}M^{-1}\}^{\dagger} & \mathbb{I} & M M\_{R}^{-1} \\
\mathbb{O} & -\{M M\_{R}^{-1}\}^{\dagger} & \mathbb{I}
\end{pmatrix},
$$

$$
\mathbb{U} = \begin{pmatrix}
U\_{\boldsymbol{\nu}} & \mathbb{O} & \mathbb{O} \\
\mathbb{O} & U\_{S} & \mathbb{O} \\
\mathbb{O} & \mathbb{O} & U\_{N}
\end{pmatrix}.
\tag{59}
$$

Thus, the complete 9×9 unitary mixing matrix diagonalizing the neutral leptons is as follows

$$\mathbb{V} = \mathbb{W} \cdot \mathbb{U} = \begin{pmatrix} U\_{\boldsymbol{\upsilon}} & M\_{\boldsymbol{D}} \boldsymbol{M}^{-1} U\_{\boldsymbol{S}} & M\_{\boldsymbol{D}} \boldsymbol{M}\_{\boldsymbol{R}}^{-1} U\_{\boldsymbol{N}} \\ (M\_{\boldsymbol{D}} \boldsymbol{M}^{-1})^{\dagger} U\_{\boldsymbol{\upsilon}} & U\_{\boldsymbol{S}} & M \boldsymbol{M}\_{\boldsymbol{R}}^{-1} U\_{\boldsymbol{N}} \\ \mathbb{O} & -(\boldsymbol{M} \boldsymbol{M}\_{\boldsymbol{R}}^{-1})^{\dagger} U\_{\boldsymbol{S}} & U\_{\boldsymbol{N}} \end{pmatrix} \tag{60}$$

**Expressing Masses and Mixing in terms of U**PMNS **and light neutrino masses:-** Usually, the light neutrino mass matrix is diagonalised by the UPMNS mixing matrix in the basis where the charged leptons are already diagonal i.e., m diag ν = U † PMNSmνU ∗ PMNS. The structure of the Dirac neutrino mass matrix M<sup>D</sup> which is a complex matrix in general can be considered to be the up-quark type in LRSM. Its origin can be motivated from a high scale Pati-Salam symmetry or SO(10) GUT. If we consider M to be diagonal and degenerate i.e., M = m<sup>S</sup> diag{1, 1, 1}, then the mass formulas for neutral leptons are given by

$$m\_{\upsilon} = M\_L = f\upsilon\_L = U\_{\text{PMNS}} m\_{\upsilon}^{\text{diag}} U\_{\text{PMNS}}^T,$$

$$M\_N \equiv M\_R = f\upsilon\_R = \frac{\upsilon\_R}{\upsilon\_L} M\_L = \frac{\upsilon\_R}{\upsilon\_L} U\_{\text{PMNS}} m\_{\upsilon}^{\text{diag}} U\_{\text{PMNS}}^T,$$

$$M\_S = -M M\_R^{-1} M^T = -m\_S^2 \left[\frac{\upsilon\_R}{\upsilon\_L} U\_{\text{PMNS}} m\_{\upsilon}^{\text{diag}} U\_{\text{PMNS}}^T\right]^{-1}, \text{(61)}$$

After some simplification the active LH neutrinos νL, active RH neutrinos N<sup>R</sup> and heavy sterile neutrinos S<sup>L</sup> in the flavor basis are related to their mass basis as

 νL SL N c R α = V νν V <sup>ν</sup><sup>S</sup> V νN V <sup>S</sup><sup>ν</sup> V SS V SN V <sup>N</sup><sup>ν</sup> V NS V NN αi νi Si Ni = UPMNS 1 mS MDU ∗ PMNS vL vR MDU −1 PMNSm diag. ν −1 1 mS M † <sup>D</sup>UPMNS U ∗ PMNS vL vR mSU −1 PMNSm diag. ν −1 O vL vR mSU −1 PMNSm diag. ν −1 UPMNS αi νi Si Ni (62)

### 4. ALTERNATIVE FORMULATION OF LEFT–RIGHT SYMMETRIC MODEL: UNIVERSAL SEESAW

Among the various alternative formulations of left–right symmetric model that have been proposed so far, the model which includes isosinglet vector like fermions looks more upgraded. The advantages of this alternative formulation over the manifest one are the following:


### 4.1. Left–Right Symmetry With Vector-Like Fermions and Universal Seesaw

The fermion content of this model includes the usual quarks and leptons,

$$\begin{aligned} q\_L &= \begin{pmatrix} \mu\_L \\ d\_L \end{pmatrix} \sim \{ \mathbf{2}, \mathbf{1}, \mathbf{1}/\mathbf{3}, \mathbf{3} \}, \ q\_R = \begin{pmatrix} \mu\_R \\ d\_R \end{pmatrix} \sim \{ \mathbf{1}, \mathbf{2}, \mathbf{1}/\mathbf{3}, \mathbf{3} \}, \\\ \ell\_L &= \begin{pmatrix} \nu\_L \\ \mathcal{e}\_L \end{pmatrix} \sim \{ \mathbf{2}, \mathbf{1}, -\mathbf{1}, \mathbf{1} \}, \ \ell\_R = \begin{pmatrix} \nu\_R \\ \mathcal{e}\_R \end{pmatrix} \sim \{ \mathbf{1}, \mathbf{2}, -\mathbf{1}, \mathbf{1} \}, \end{aligned}$$

and the additional vector-like quarks and charged leptons [62– 70]

$$U\_{L,R} \sim \{1, 1, 4/3, 3\}, \quad D\_{L,R} \sim \{1, 1, -2/3, 3\},$$

$$E\_{L,R} \sim \{1, 1, -2, 1\}.\tag{63}$$

To this new setup of left–right symmetric model, we add vector like neutral lepton in the fermion sector and a singlet scalar in the Higgs sector. The purpose behind the inclusion of vector like neutral lepton is to allow seesaw mechanism for light neutrinos leading to Dirac neutrino mass. Similarly, the scalar singlet is introduced to give consistent vacuum stability in the scalar sector. The particle content and the relevant transformations under the LRSM gauge group are shown in **Table 4**.

We now extend the standard LRSM framework having isosinglet vector-like copies of fermions with additional neutral vector like fermions [71–75]. This kind of a vector-like fermion spectrum is very naturally embedded in gauged flavor groups with left–right symmetry [76] or quark-lepton symmetric models [77].

The relevant Yukawa part of the Lagrangian is given by

$$\mathcal{L} = -\sum\_{X} \langle \lambda\_{\text{SXX}} \text{S\!\!X}X + M\_{\text{X}} \text{\!\!X}X \rangle - \langle \lambda\_{U}^{\text{L}} \tilde{H}\_{L} \overline{q}\_{L} \text{\!\!U}R + \lambda\_{U}^{\text{R}} \tilde{H}\_{R} \overline{q}\_{R} \text{\!\!U}L \rangle$$

$$\begin{split} + \quad \lambda\_{D}^{\text{L}} H\_{L} \overline{q}\_{L} D\_{R} + \lambda\_{D}^{\text{R}} H\_{R} \overline{q}\_{R} D\_{L} + \lambda\_{E}^{\text{L}} H\_{L} \overline{\ell}\_{L} E\_{R} + \lambda\_{E}^{\text{R}} H\_{R} \overline{\ell}\_{R} E\_{L} \\ + \quad \lambda\_{N}^{\text{L}} \tilde{H}\_{L} \overline{\ell}\_{L} N\_{R} + \lambda\_{N}^{\text{R}} \tilde{H}\_{R} \overline{\ell}\_{R} N\_{L} + \text{h.c.} \rangle, \end{split} \tag{64}$$

where the summation is over X = U, D, E, N and we suppress flavor and color indices on the fields and couplings. H˜ <sup>L</sup>,<sup>R</sup> denotes τ2H<sup>∗</sup> L,R , where τ<sup>2</sup> is the usual second Pauli matrix. We would like to stress that Parity Symmetry is present in order to distinguish between for instance N<sup>R</sup> and NL, otherwise extra terms in the Lagrangian Equation (64) would appear with the vector-like fermions Left and Right exchanged.

The LRSM gauge group breaks to the SM gauge group when HR(1, 2, −1) acquires a VEV and the SM gauge group breaks to U(1)EM when HL(2, 1, −1) acquires a VEV. However, parity can break either at TeV scale or at a much higher scale MP. For the latter case the Yukawa couplings can be different for right-type and left-type Yukawa terms (λ R X 6= λ L X ) because of the renormalization group running below MP. Consequently, we will distinguish the left and right handed couplings explicitly with the subscripts L and R. We use the VEV normalizations hHLi = (0, υL) T and hHRi = (0, υR) T . The scale of v<sup>R</sup> has to lie between at around a few TeV (depending on the right-handed gauge coupling) to suit the experimental searches for the heavy right-handed W<sup>R</sup> boson at colliders and at low energies.

Since the particle spectrum does not contain a bidoublet Higgs, Dirac mass terms for the SM fermions can not be written and the charged fermion mass matrices assume a seesaw structure. Alternatively, a Higgs bidoublet 8 can be introduced along with HL,R.

After symmetry breaking, the mass matrices for the fermions are given by

$$M\_{\boldsymbol{\nu}\boldsymbol{U}} = \begin{pmatrix} 0 & \lambda\_{\boldsymbol{U}}^{L}\boldsymbol{\upsilon}\_{\boldsymbol{U}}\\ \lambda\_{\boldsymbol{U}}^{R}\boldsymbol{\upsilon}\_{\boldsymbol{R}} & M\_{\boldsymbol{U}} \end{pmatrix},\\ M\_{\boldsymbol{d}\boldsymbol{D}} = \begin{pmatrix} 0 & \lambda\_{\boldsymbol{D}}^{L}\boldsymbol{\upsilon}\_{\boldsymbol{L}}\\ \lambda\_{\boldsymbol{D}}^{R}\boldsymbol{\upsilon}\_{\boldsymbol{R}} & M\_{\boldsymbol{D}} \end{pmatrix},$$



$$M\_{\epsilon E} = \begin{pmatrix} 0 & \lambda\_E^L \upsilon\_L \\ \lambda\_E^R \upsilon\_R & M\_E \end{pmatrix}, M\_{\upsilon N} = \begin{pmatrix} 0 & \lambda\_N^L \upsilon\_L \\ \lambda\_N^R \upsilon\_R & M\_N \end{pmatrix},\tag{65}$$

The mass eigenstates can be found by rotating the mass matrices via left and right orthogonal transformations O L,R (we assume all parameters to be real). For example, the up quark diagonalization yields O LT U · MuU · O R <sup>U</sup> = diag(mˆ <sup>u</sup>, Mˆ <sup>U</sup>). Up to leading order in λ L U vL, the resulting up-quark masses are

$$
\hat{M}\_U \approx \sqrt{{\mathcal{M}\_U^2 + (\lambda\_U^R \nu\_R)^2}}, \quad \hat{m}\_{\hat{u}} \approx \frac{(\lambda\_U^L \nu\_L)(\lambda\_U^R \nu\_R)}{\hat{M}\_U}, \tag{66}
$$

and the mixing angles θ L,R U parametrizing O L,R U ,

$$\tan(2\theta\_U^L) \approx \frac{2(\lambda\_U^L \nu\_L) M\_U}{M\_U^2 + (\lambda\_U^R \nu\_R)^2}, \tan(2\theta\_U^R) \approx \frac{2(\lambda\_U^R \nu\_R) M\_U}{M\_U^2 - (\lambda\_U^R \nu\_R)^2}. \tag{67}$$

The other fermion masses and mixings are given analogously. For an order of magnitude estimate one may approximate the phenomenologically interesting regime with the limit λ R U v<sup>R</sup> → M<sup>U</sup> in which case the mixing angles approach θ L <sup>U</sup> → ˆmu/Mˆ <sup>U</sup> and θ R <sup>U</sup> → π/4. This means that θ L U is negligible for all fermions but the top quark and its vector partner [72].

We here neglect the flavor structure of the Yukawa couplings λ L,R X and λSXX which will determine the observed quark and leptonic mixing. The hierarchy of SM fermion masses can be generated by either a hierarchy in the Yukawa couplings or in the masses of the of the vector like fermions.

As described above, the light neutrino masses are of Diractype as well, analogously given by

$$
\hat{m}\_{\upsilon} = \frac{\lambda\_N^L \lambda\_N^R \nu\_L \nu\_R}{M\_N},
\tag{68}
$$

It is natural to assume that M<sup>N</sup> ≫ vR, as the vector like N is a singlet under the model gauge group. In this case, the scenario predicts naturally light Dirac neutrinos [76].

### 4.2. Left–Right Symmetry With Vector-Like Fermions and Type-II Seesaw for Neutrino Masses

In **Table 5**, we present the field content of this model and their transformations under the LRSM gauge group.

We implement a scalar sector consisting of SU(2)L,<sup>R</sup> doublets and triplets, however the conventional scalar bidoublet is absent. We use the Higgs doublets to implement the left–right and the electroweak symmetry breaking: H<sup>R</sup> ≡ (h 0 R , h − R ) <sup>T</sup> <sup>≡</sup> [1, 2, <sup>−</sup>1, 1] breaks the left–right symmetry, while H<sup>L</sup> ≡ (h 0 L , h − L ) <sup>T</sup> <sup>≡</sup> [2, 1, −1, 1] breaks the electroweak symmetry once they acquire vacuum expectation values (VEVs),

$$
\langle H\_R \rangle = \begin{pmatrix} \frac{\nu\_R}{\sqrt{2}} \\ 0 \end{pmatrix}, \quad \langle H\_L \rangle = \begin{pmatrix} \frac{\nu\_L}{\sqrt{2}} \\ 0 \end{pmatrix}. \tag{69}
$$

Note that the present framework requires only doublet Higgs fields for spontaneous symmetry breaking. However, in the absence of a Higgs bidoublet, we use the vector-like new fermions to generate correct charged fermion masses through a universal seesaw mechanism. For the neutrinos we note that in the absence of a scalar bidoublet there is no Dirac mass term for light neutrinos and without scalar triplets no Majorana masses are generated either. To remedy this fact we introduce additional scalar triplets 1<sup>L</sup> and 1R,

$$
\Delta\_{L,R} = \begin{pmatrix}
\delta\_{L,R}^+ / \sqrt{2} & \delta\_{L,R}^{++} \\
\delta\_{L,R}^0 & -\delta\_{L,R}^+ / \sqrt{2}
\end{pmatrix},
\tag{70}
$$

which transform as 1<sup>L</sup> ≡ [3, 1, 2, 1] and 1<sup>R</sup> ≡ [1, 3, 2, 1], respectively. They generate Majorana masses for the light and heavy neutrinos although they are not essential in spontaneous symmetry breaking here. The particle content of the model is shown in **Table 5**. In the presence of the Higgs triplets, the manifestly Left–Right symmetric scalar potential has the form

$$\mathcal{L} = \left(D\_{\mu}H\_{L}\right)^{\dagger}D^{\mu}H\_{L} + \left(D\_{\mu}H\_{R}\right)^{\dagger}D^{\mu}H\_{R}$$

$$+ \left(D\_{\mu}\Delta\_{L}\right)^{\dagger}D^{\mu}\Delta\_{L} + \left(D\_{\mu}\Delta\_{R}\right)^{\dagger}D^{\mu}\Delta\_{R}$$

$$- \left(\mathcal{V}\left(H\_{L},H\_{R},\Delta\_{L},\Delta\_{R}\right)\right),\tag{71}$$

TABLE 5 | Field content of the LRSM with universal seesaw.


where the scalar potential is given by

<sup>V</sup> (HL, HR, 1L, 1R) = −µ 2 1 (H † <sup>L</sup>HL) − µ 2 2 (H † <sup>R</sup>HR) + λ1(H † <sup>L</sup>HL) <sup>2</sup> <sup>+</sup> <sup>λ</sup>2(<sup>H</sup> † <sup>R</sup>HR) <sup>2</sup> <sup>+</sup> <sup>β</sup>1(<sup>H</sup> † <sup>L</sup>HL)(H † <sup>R</sup>HR) − µ 2 <sup>3</sup>Tr(1 † <sup>L</sup>1L) − µ 2 <sup>4</sup>Tr(1 † <sup>R</sup>1R) + λ3Tr(1 † <sup>L</sup>1L) <sup>2</sup> <sup>+</sup> <sup>λ</sup>4Tr(<sup>1</sup> † <sup>R</sup>1R) 2 + β2Tr(1 † <sup>L</sup>1L)Tr(1 † <sup>R</sup>1R) + ρ1(Tr(1 † <sup>L</sup>1L)(H † <sup>R</sup>HR) + Tr(1 † <sup>R</sup>1R)(H † <sup>L</sup>HL)) + ρ<sup>2</sup> Tr(1 † <sup>L</sup>1L)(H † <sup>L</sup>HL) + Tr(1 † <sup>R</sup>1R)(H † <sup>R</sup>HR) + ρ<sup>3</sup> H † <sup>L</sup>1 † <sup>L</sup>1LH<sup>L</sup> + H † <sup>R</sup>1 † <sup>R</sup>1RH<sup>R</sup> + µ H T L iσ21LH<sup>L</sup> + H T R iσ21RH<sup>R</sup> + h.c. · · · . (72)

Assigning non-zero VEV to Higgs doublets H<sup>R</sup> and H<sup>L</sup> and triplets 1<sup>R</sup> and 1L,

$$
\langle H\_L^0 \rangle \equiv \upsilon\_L / \sqrt{2}, \quad \langle H\_R^0 \rangle \equiv \upsilon\_R / \sqrt{2},
$$

$$
\langle \Delta\_L^0 \rangle \equiv \mu\_L / \sqrt{2}, \quad \langle \Delta\_R^0 \rangle \equiv \mu\_R / \sqrt{2}.\tag{73}
$$

the scalar potential takes the form,

$$\begin{aligned} \mathcal{V}\left( \langle H\_{L} \rangle, \langle H\_{R}^{0} \rangle, \langle \Delta\_{L}^{0} \rangle, \langle \Delta\_{R}^{0} \rangle \right) &= \\ -\frac{1}{2}\mu\_{1}^{2}\upsilon\_{L}^{2} - \frac{1}{2}\mu\_{2}^{2}\upsilon\_{R}^{2} - \frac{1}{2}\mu\_{3}^{2}\mu\_{L}^{2} - \frac{1}{2}\mu\_{4}^{2}\mu\_{R}^{2} \\ +\frac{1}{4}\lambda\_{1}\upsilon\_{L}^{4} + \frac{1}{4}\lambda\_{2}\upsilon\_{R}^{4} + \frac{1}{4}\lambda\_{3}\mu\_{L}^{4} + \frac{1}{4}\lambda\_{4}\mu\_{R}^{4} \\ +\frac{1}{4}\rho\_{1}\upsilon\_{L}^{2}\upsilon\_{R}^{2} + \frac{1}{4}\rho\_{2}\upsilon\_{L}^{2}\mu\_{R}^{2} - \frac{1}{2\sqrt{2}}\mu\left(\upsilon\_{L}^{2}\mu\_{L} + \upsilon\_{R}^{2}\mu\_{R}\right) \\ +\frac{1}{4}\rho\_{1}\left(\upsilon\_{R}^{2}\mu\_{L}^{2} + \upsilon\_{L}^{2}\mu\_{R}^{2}\right) + \frac{1}{4}\rho\_{2}\left(\upsilon\_{L}^{2}\mu\_{L}^{2} + \upsilon\_{R}^{2}\mu\_{R}^{2}\right) + \cdots \end{aligned} \tag{74}$$

As non-zero VEV hH 0 R i = v<sup>R</sup> breaks LRSM to SM at high scale and hH 0 L i = v<sup>L</sup> breaks SM down to low energy at electroweak scale, we consider v<sup>L</sup> 6= vR. We chose the induced VEVs for scalar triplets much smaller than VEVs of Higgs doublets, i.e., uL, u<sup>R</sup> ≪ vL, vR.

One can approximately write down the Higgs triplets induced VEVs as follows,

$$
\mu\_L = \frac{\mu \nu\_L^2}{M\_{\delta\_L^0}^2}, \quad \mu\_R = \frac{\mu \nu\_R^2}{M\_{\delta\_R^0}^2}. \tag{75}
$$

#### 4.2.1. Fermion Masses via Universal Seesaw

As discussed earlier, in this scheme normal Dirac mass terms for the SM fermions are not allowed due to the absence of a bidoublet Higgs. However, in the presence of vector-like copies of quark and charged lepton gauge isosinglets, the charged fermion mass matrices can assume a seesaw structure. The Yukawa interaction Lagrangian in this model is given by

$$\begin{split} \mathcal{L} &= -\left. Y\_U^L H\_L \overline{Q}\_L U\_R + \left. Y\_U^R H\_R \overline{Q}\_R U\_L + \left. Y\_D^L \tilde{H}\_L \overline{Q}\_L D\_R \right| \right. \\ &\left. + \left. Y\_D^R \tilde{H}\_R \overline{Q}\_R D\_L + \left. Y\_E^L \tilde{H}\_L \overline{\ell}\_L E\_R + \left. Y\_E^R \tilde{H}\_R \overline{\ell}\_R E\_L \right| \right. \right. \\ &\left. + \frac{1}{2} f \left( \overline{\ell\_L^c} i \mathbf{r}\_2 \Delta\_L \ell\_L + \overline{\ell\_R^c} i \mathbf{r}\_2 \Delta\_R \ell\_R \right) \\ &\left. - M\_U \overline{U}\_L U\_R - M\_D \overline{D}\_L D\_R - M\_E \overline{E}\_L E\_R + \text{h.c.} \right. \end{split} \tag{76}$$

where we suppress the flavor and color indices on the fields and couplings. H˜ <sup>L</sup>,<sup>R</sup> denotes τ2H<sup>∗</sup> L,R , where τ<sup>2</sup> is the usual second Pauli matrix. Note that there is an ambiguity regarding the breaking of parity, which can either be broken spontaneously with the left– right symmetry at around the TeV scale or at a much higher scale independent of the left–right symmetry breaking. In the latter case, the Yukawa couplings corresponding to the righttype and left-type Yukawa terms can be different because of the renormalization group running below the parity breaking scale, Y R X 6= Y L X . Thus, while writing the Yukawa terms above we distinguish the left- and right-handed couplings explicitly with the subscripts L and R.

After spontaneous symmetry breaking we can write the mass matrices for the charged fermions as [73]

$$M\_{\mathcal{U}} = \begin{pmatrix} 0 & Y\_U^L \upsilon\_L \\ Y\_U^R \upsilon\_R & M\_U \end{pmatrix}, M\_{dD} = \begin{pmatrix} 0 & Y\_D^L \upsilon\_L \\ Y\_D^R \upsilon\_R & M\_D \end{pmatrix},$$

$$M\_{e\mathcal{E}} = \begin{pmatrix} 0 & Y\_E^L \upsilon\_L \\ Y\_E^R \upsilon\_R & M\_E \end{pmatrix}. \tag{77}$$

The corresponding generation of fermion masses is diagrammatically depicted in **Figure 3**. Note that we are interested in a scenario where the VEVs of the Higgs doublets are much larger than the VEVs of the Higgs triplets i.e., u<sup>L</sup> ≪ υL, u<sup>R</sup> ≪ υR. In the context of this work, we do not attempt to explain how the hierarchy between VEVs can be achieved.

Assuming all parameters to be real one can obtain the mass eigenstates by rotating the mass matrices via left and right orthogonal transformations OL,<sup>R</sup> . For example, up to leading order in Y L U vL, the SM and heavy vector partner up-quark masses are

$$m\_u \approx Y\_U^L Y\_U^R \frac{\upsilon\_L \upsilon\_R}{\hat{M}\_U}, \quad \hat{M}\_U \approx \sqrt{M\_U^2 + (Y\_U^R \upsilon\_R)^2}, \tag{78}$$

and the mixing angles θ L,R U in OL,<sup>R</sup> are determined as

$$\tan\left(2\theta\_U^{L,R}\right) \approx 2Y\_U^{L,R} \frac{\upsilon\_{L,R}M\_U}{M\_U^2 \pm (Y\_U^R \upsilon\_R)^2}.\tag{79}$$

The other fermion masses and mixing are obtained in an analogous manner. Note that here we have neglected the flavor structure of the Yukawa couplings Y L,R <sup>X</sup> which will determine the observed quark and charged lepton mixings. The hierarchy of SM fermion masses can be explained by assuming either a hierarchical structure of the Yukawa couplings or a hierarchical structure of the vector-like fermion masses.

#### 4.2.2. Neutrino Masses and Type II Seesaw Dominance

In the model under consideration there is no tree level Dirac mass term for the neutrinos due to the absence of a Higgs bidoublet. The scalar triplets acquire induced VEVs h1Li = u<sup>L</sup> and h1Ri = u<sup>R</sup> giving the neutral lepton mass matrix in the basis (νL, νR) given by

$$M\_{\mathbb{V}} = \begin{pmatrix} f\mu\_L & 0\\ 0 & f\mu\_R \end{pmatrix} \,. \tag{80}$$

Thus, the light and heavy neutrino masses are simply m<sup>ν</sup> = fu<sup>L</sup> ∝ M<sup>N</sup> = fuR. A Dirac mass term is generated at the two-loop level via the one-loop W boson mixing θ<sup>W</sup> (see the next section) and the exchange of a charged lepton. It is of the order m<sup>D</sup> . g 4 L /(16π 2 ) <sup>2</sup>mτmbmt/M<sup>2</sup> WR ≈ 0.1 eV for MW<sup>R</sup> ≈ 5 TeV. This is intriguingly of the order of the observed neutrino masses; as long

as the right-handed neutrinos are much heavier than the lefthanded neutrinos, the type-II seesaw dominance is preserved and the induced mixing mD/M<sup>N</sup> is negligible. The mixing between charged gauge bosons θ<sup>W</sup> ≈ g 2 L /(16π 2 )mbmt/M<sup>2</sup> WR is generated through the exchange of bottom and top quarks, and their vectorlike partners. This yields a very small mixing of the order θ<sup>W</sup> ≈ 10−<sup>7</sup> for TeV scale W<sup>R</sup> bosons.

Incorporating three fermion generations leads to the mixing matrices for the left- and right-handed matrices which we take to be equal

$$V\_N = V\_\upsilon \equiv U,\tag{81}$$

where U is the phenomenological PMNS mixing matrix. Thus, the unmeasured mixing matrix for the right-handed neutrinos is fully determined by the left-handed counterpart. The present framework gives a natural realization of type-II seesaw providing a direct relation between light and heavy neutrinos, M<sup>i</sup> ∝ m<sup>i</sup> , i.e., the heavy neutrino masses M<sup>i</sup> can be expressed in terms of the light neutrino masses m<sup>i</sup> as M<sup>i</sup> = mi(M3/m3), for a normal and M<sup>i</sup> = mi(M2/m2) for a inverse hierarchy of light and heavy neutrino masses.

### 4.3. Implication to Neutrinoless Double Beta Decay

As discussed earlier, there is no tree level Dirac neutrino mass term connecting light and heavy neutrinos. Consequently, the mixing between light and heavy neutrinos is vanishing at this order. Also, the mixing between the charged gauge bosons is vanishing at the tree level due to the absence of a scalar bidoublet.

The charged current interaction in the mass basis for the leptons is given by

$$\frac{\text{gL}}{\sqrt{2}}\sum\_{i=1}^{3}U\_{ei}\left(\overline{\ell\_L}\chi\_{\mu}\upsilon\_i\mathcal{W}\_L^{\mu} + \frac{\text{g}\_R}{\text{g}\_L}\overline{\ell\_R}\chi\_{\mu}\mathcal{N}\_i\mathcal{W}\_R^{\mu}\right) + \text{h.c.}\tag{82}$$

The charged current interaction for leptons leads to 0νββ decay via the exchange of light and heavy neutrinos. There are additional contributions to 0νββ decay due to doubly charged triplet scalar exchange. While the left-handed triplet exchange is suppressed because of its small induced VEV, the right-handed triplet can contribute sizeably to 0νββ decay.

Before numerical estimation, let us point out the mass relations between light and heavy neutrinos under natural type-II seesaw dominance. For a hierarchical pattern of light neutrinos the mass eigenvalues are given as m<sup>1</sup> < m<sup>2</sup> ≪ m3. The lightest neutrino mass eigenvalue is m<sup>1</sup> while the other mass eigenvalues are determined using the oscillation parameters as follows, m<sup>2</sup> 2 = m2 <sup>1</sup> <sup>+</sup> <sup>1</sup>m<sup>2</sup> sol, <sup>m</sup><sup>2</sup> <sup>3</sup> <sup>=</sup> <sup>m</sup><sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>1</sup>m<sup>2</sup> atm <sup>+</sup> <sup>1</sup>m<sup>2</sup> sol. On the other hand, for the inverted hierarchical pattern of the light neutrino masses m<sup>3</sup> ≪ m<sup>1</sup> ≈ m<sup>2</sup> where m<sup>3</sup> is the lightest mass eigenvalue while other mass eigenvalues are determined by m<sup>2</sup> <sup>1</sup> <sup>=</sup> <sup>m</sup><sup>2</sup> <sup>3</sup> <sup>+</sup> <sup>1</sup>m<sup>2</sup> atm, m2 <sup>2</sup> <sup>=</sup> <sup>m</sup><sup>2</sup> 3+1m<sup>2</sup> sol+1m<sup>2</sup> atm. The quasi-degenerate pattern of light neutrinos is m<sup>1</sup> ≈ m<sup>2</sup> ≈ m<sup>3</sup> ≫ q 1m<sup>2</sup> atm. In any case, the heavy neutrino masses are directly proportional to the light neutrino masses.

In the present analysis, we discuss 0νββ decay due to exchange of light neutrinos via left-handed currents, right-handed neutrinos via right handed currents as shown in **Figure 4**. 0νββ decay can also be induced by a right handed doubly charged scalar as shown in **Figure 5**<sup>6</sup> . The half-life for a given isotope for these contributions is given by

$$[[T\_{1/2}^{0\upsilon}]^{-1} = \mathcal{G}\_{01} \left( |\mathcal{M}\_{\upsilon} \eta\_{\upsilon}|^2 + |(\mathcal{M}\_{N}' \eta\_{N} + \mathcal{M}\_{N} \eta\_{\Delta})|^2 \right), \tag{83}$$

where G<sup>01</sup> corresponds to the standard 0νββ phase space factor, the M<sup>i</sup> correspond to the nuclear matrix elements for the different exchange processes and η<sup>i</sup> are dimensionless parameters determined below.

#### Light Neutrinos

The lepton number violating dimensionless particle physics parameter derived from 0νββ decay due to the standard mechanism via the exchange of light neutrinos is

$$
\eta\_{\upsilon} = \frac{1}{m\_{\text{e}}} \sum\_{i=1}^{3} U\_{\text{e}i}^{2} m\_{i} = \frac{m\_{\text{e}\text{e}}^{\upsilon}}{m\_{\text{e}}} \,. \tag{84}
$$

Here, m<sup>e</sup> is the electron mass and the effective 0νββ mass is explicitly given by

$$m\_{\rm ee}^{\upsilon} = \left| c\_{12}^{2} c\_{13}^{2} m\_{1} + s\_{12}^{2} c\_{13}^{2} m\_{2} e^{i\alpha} + s\_{13}^{2} m\_{3} e^{i\beta} \right| \,, \tag{85}$$

with the sine and cosine of the oscillation angles θ<sup>12</sup> and θ13, c<sup>12</sup> = cos θ12, etc. and the unconstrained Majorana phases 0 ≤ α, β < 2π.

#### Right-Handed Neutrinos

The contribution to 0νββ decay arising from the purely righthanded currents via the exchange of right-handed neutrinos generally results in the lepton number violating dimensionless particle physics parameter

$$\eta\_N = m\_p \left(\frac{g\_R}{g\_L}\right)^4 \left(\frac{M\_{W\_L}}{M\_{W\_R}}\right)^4 \sum\_{i=1}^3 \frac{U\_{ei}^2 M\_i}{|p|^2 + M\_i^2}.\tag{86}$$

The virtual neutrino momentum |p| is of the order of the nuclear Fermi scale, p ≈ 100 MeV. m<sup>p</sup> is the proton mass and for the manifest LRSM case we have g<sup>L</sup> = gR, or else the new contributions are rescaled by the ratio between these two couplings. We in general consider right-handed neutrinos that can be either heavy or light compared to nuclear Fermi scale.

<sup>6</sup>A detailed discussion of 0νββ decay within LRSMs can be found e.g., in Mohapatra and Senjanovic [19], Mohapatra and Vergados [78], Hirsch et al. [79], Tello et al. [80], Chakrabortty et al. [81], Patra [75], Awasthi et al. [60], Barry and Rodejohann [82], Bhupal Dev et al. [83], Ge et al. [84], Awasthi et al. [85], Huang and Lopez-Pavon [86], Bhupal Dev et al. [87], Borah and Dasgupta [88], Bambhaniya et al. [89], Gu [90], Borah and Dasgupta [91] and Awasthi et al. [92] and for an early study of the effects of light and heavy Majorana neutrinos in neutrinoless double beta decay see in Halprin et al. [93].

If the mass of the exchanged neutrino is much higher than its momentum, M<sup>i</sup> ≫ |p|, the propagator simplifies as

$$\frac{M\_i}{p^2 - M\_i^2} \approx -\frac{1}{M\_i},\tag{87}$$

and the effective parameter for right-handed neutrino exchange yields

$$\eta\_N = m\_p \left(\frac{\mathcal{g}\_R}{\mathcal{g}\_L}\right)^4 \left(\frac{M\_W}{M\_{W\_R}}\right)^4 \sum\_{i=1}^3 \frac{U\_{ei}^2}{M\_i} \propto \eta\_\nu(m\_i^{-1}),\tag{88}$$

where in the expression for η<sup>ν</sup> (m −1 i ) the individual neutrino masses are replaced by their inverse values. Such a contribution clearly becomes suppressed the smaller the right-handed neutrino masses are.

On the other hand, if the mass of the neutrino is much less than its typical momentum, M<sup>i</sup> ≪ |p|, the propagator simplifies in the same way as for the light neutrino exchange,

$$P\_R \frac{\mathfrak{p} + M\_i}{\mathfrak{p}^2 - M\_i^2} P\_R \approx \frac{M\_i}{\mathfrak{p}^2} \,, \tag{89}$$

because both currents are right-handed. As a result, the 0νββ decay contribution leads to the dimensionless parameter

$$\eta\_N = \frac{m\_p}{|\mathcal{P}|^2} \left(\frac{\mathcal{g}\_R}{\mathcal{g}\_L}\right)^4 \left(\frac{M\_W}{M\_{W\_R}}\right)^4 \sum\_{i=1}^3 U\_{ei}^2 M\_i \propto \eta\_\nu \,. \tag{90}$$

This is proportional to the standard parameter η<sup>ν</sup> but in the case of very light right-handed neutrinos, e.g., M<sup>i</sup> ≈ m<sup>i</sup> , the contribution becomes negligible because of the strong suppression with the heavy right-handed W boson mass.

In general, we consider right-handed neutrinos both lighter and heavier than 100 MeV and use (86) to calculate the contribution. In addition, the relevant nuclear matrix element changes; for M<sup>i</sup> ≫ 100 MeV it approaches <sup>M</sup>′ <sup>N</sup> → <sup>M</sup><sup>N</sup> whereas for M<sup>i</sup> ≪ 100 MeV it approaches <sup>M</sup>′ <sup>N</sup> → <sup>M</sup><sup>ν</sup> . For intermediate values, we use a simple smooth interpolation scheme within the regime 10 MeV – 1 GeV, which yields a sufficient accuracy for our purposes.

#### Right-Handed Triplet Scalar

Finally, the exchange of a doubly charged right-handed triplet scalar shown in **Figure 5** (where doubly charged left-handed triplet scalar contributes negligible and thus, neglected from the



present discussion) gives

$$\eta\_{\Delta} = \frac{m\_{\text{p}}}{M\_{\delta\_{R}^{--}}^{2}} \left(\frac{\text{g}\_{R}}{\text{g}\_{L}}\right)^{4} \left(\frac{M\_{W}}{M\_{W\_{R}}}\right)^{4} \sum\_{i=1}^{3} U\_{\varepsilon i}^{2} M\_{i} \propto \eta\_{\nu} \, . \tag{91}$$

This expression is also proportional to the standard η<sup>ν</sup> because the relevant coupling of the triplet scalar is proportional to the right-handed neutrino mass.

#### Numerical Estimate

In the following, we numerically estimate the half-life for 0νββ decay of the isotope <sup>136</sup>Xe as shown in **Figure 6**. We use the current values of masses and mixing parameters from neutrino oscillation data reported in the global fits taken from Gonzalez-Garcia et al. [94]. For the 0νββ phase space factors and nuclear matrix elements we use the values given in **Table 6**. In **Figure 6**, we show the dependence of the 0νββ decay half-life on the lightest neutrino mass, i.e., m<sup>1</sup> for normal and m<sup>3</sup> for inverse hierarchical neutrinos. The other model parameters are fixed as

$$\lg\_R = \lg\_\text{L}, M\_{W\_R} = M\_{\delta\_R^{--}} \approx 5 \text{ TeV}, M\_N^{\text{heaviest}} = 1 \text{ TeV} \, . \tag{92}$$

The lower limit on lightest neutrino mass is derived to be m<sup>&</sup>lt; ≈ 0.9 meV, 0.01 meV for NH and IH pattern of light neutrino masses respectively by saturating the KamLAND-Zen experimental bound.

As for the experimental constraints, we use the current best limits at 90% C.L., T 0ν 1/2 ( <sup>136</sup>Xe) <sup>&</sup>gt; 1.07×10<sup>26</sup> yr and <sup>T</sup> 0ν 1/2 ( <sup>76</sup>Ge) > 2.1 <sup>×</sup> <sup>10</sup><sup>25</sup> yr from KamLAND-Zen [96] and the GERDA Phase I [97], respectively. Representative for the sensitivity of future 0νββ experiments, we use the expected reach of the planned nEXO experiment, T 0ν 1/2 ( <sup>136</sup>Xe) <sup>≈</sup> 6.6 <sup>×</sup> <sup>10</sup><sup>27</sup> yr [98]. As for the other experimental probes on the neutrino mass scale, we use the future sensitivity of the KATRIN experiment on the effective single β decay mass m<sup>β</sup> ≈ 0.2 eV [99] and the current limit on the sum of neutrino masses from cosmological observations, 6im<sup>i</sup> . 0.7 eV [100].

For a better understanding of the interplay between the leftand right-handed neutrino mass scales, we show in **Figure 7** the 0νββ decay half-life as a function of the lightest neutrino mass and the heaviest neutrino mass for a normal (left) and inverse (right) neutrino mass hierarchy. The other model parameters are fixed, with right-handed gauge boson and doubly-charged scalar masses of 5 TeV. The oscillation parameters are at their best fit values and the Majorana phases are always chosen to yield the smallest rate at a given point, i.e., the longest half life. The nuclear matrix employed are at the lower end in **Table 6**. This altogether yields the longest, i.e., most pessimistic, prediction for the 0νββ

FIGURE 6 | 0νββ decay half-life as a function of the lightest neutrino mass in the case of normal hierarchical (NH) and inverse hierarchical (IH) light neutrinos in red and green bands respectively. We defined mlightest ≃ m<sup>i</sup> such that m<sup>1</sup> is the lightest neutrino mass for NH and m3 for IH pattern. The other parameters are fixed as MW<sup>R</sup> = 5 TeV, M<sup>δ</sup> −− R ≈ 5 TeV and the heaviest right-handed neutrino mass is 1 TeV. The gauge couplings are assumed universal, g<sup>L</sup> = gR, and the intermediate values for the nuclear matrix elements are used, <sup>M</sup><sup>ν</sup> <sup>=</sup> 4.5, <sup>M</sup><sup>N</sup> <sup>=</sup> 270. The bound on the sum of light neutrino masses from the KATRIN and Planck experiments are represented as vertical lines. The bound from KamLAND-Zen experiment is presented in horizontal line for Xenon isotope. The bands arise due to 3σ range of neutrino oscillation parameters and variation in the Majorana phases from 0 − 2π.

decay half-life. The red-shaded area is already excluded with a predicted half life of 10<sup>26</sup> yr or faster. As expected, this sets an upper limit on the lightest neutrino mass mlightest . 1 eV, but it also puts stringent constraints on the mass scale of the righthanded neutrinos. For an inverse hierarchy, the range 50 MeV . M<sup>2</sup> . 5 GeV is excluded whereas in the normal hierarchy case, large M<sup>3</sup> can be excluded if there is a strong hierarchy, m<sup>1</sup> → 0. This is due the large contribution of the lightest heavy neutrino N<sup>1</sup> in such a case.

### 5. LEPTOGENESIS

Cosmological observations (studies of the cosmic microwave background radiation, large scale structure data, the primordial abundances of light elements) indicate that our visible universe is dominated by matter and there is very little antimatter. The baryon asymmetry normalized to number density of photons (nγ ) can be extracted out of these observations, which gives

$$\eta(t = \text{present}) = \frac{n\_B - n\_{\bar{B}}}{n\_{\mathcal{I}}} \sim 10^{-10}.\tag{93}$$

The astrophysical observations suggest that at an early epoch before the big-bang nucleosynthesis this asymmetry was

generated. Thus, it is natural to seek an explanation for this asymmetry from the fundamental particle interactions within or beyond the SM of particle physics. There are three conditions, often called Sakharov's conditions [101], that must be met in order to generate a baryon asymmetry dynamically:

values of the nuclear matrix elements in Table 6 are employed. The other model parameters are chosen as g<sup>R</sup> = g<sup>L</sup> and MW<sup>R</sup>


In principle, the SM has all the ingredients to satisfy all three conditions.


However, in practice it turns out that only the first Sakharov condition is fulfilled in a satisfactory manner in the SM. The CP violation coming from the CKM phase is suppressed by a factor T 12 EW in the denominator, where TEW ∼ 100 GeV is the temperature during the electroweak phase transition. Consequently, the CP violation in the SM is too small to explain the observed baryon asymmetry of the universe. Furthermore, the electroweak phase transition is not first order; but just a smooth crossover.

= M<sup>1</sup> = 5 TeV.

Thus, to explain the baryon asymmetry of the universe one must go beyond the SM, either by introducing new sources of CP violation and a new kind of out-of-equilibrium situations (such as the out-of-equilibrium decay of some new heavy particles) or modifying the electroweak phase transition itself. One such alternative is leptogenesis. Leptogenesis is a mechanism where a lepton asymmetry is generated before the electroweak phase transition, which then gets converted to baryon asymmetry of the universe in the presence of sphaleron induced anomalous B + L violating processes, which converts any primordial L asymmetry, and hence B − L asymmetry, into a baryon asymmetry. A realization of leptogenesis via the decay of out-of-equilibrium heavy neutrinos transforming as singlets under the SM gauge group was proposed in Fukugita and Yanagida [21]. The Yukawa couplings provide the CP through interference between tree level and one-loop decay diagrams. The departure from thermal equilibrium occurs when the Yukawa interactions are sufficiently slow<sup>7</sup> . The lepton number violation in this scenario comes from

<sup>7</sup>The out of equilibrium condition can be understood as follows. In thermal equilibrium the expectation value of the baryon number can be written as hBi = Tr[Be−βH]/Tr[e−βH], where β is the inverse temperature. Since particles and anti particles have opposite baryon number, B is odd under C operation, while it is even under P and T operations. Thus, CPT conservation implies a vanishing total baryon number since B is odd and H is even under CPT, unless there is a non-vanishing chemical potential. Assuming a non-vanishing chemical potential implies that the above equation for the expectation value of the baryon number is no longer valid and the baryon number density departs from the equilibrium distribution. This is achieved when the interaction rate is very slow compared to the expansion rate of the universe.

the Majorana masses of the heavy neutrinos. The generated lepton asymmetry then gets partially converted to baryon asymmetry in the presence of sphaleron induced anomalous B+L violating interactions before the electroweak phase transition. In what follows, we will discuss the sphaleron processes and few of the most popular scenarios of leptogenesis in some detail to set the stage before discussing leptogenesis in LRSM scenarios in particular.

### 5.1. Anomalous B + L Violating Processes and Relating Baryon and Lepton Asymmetries

In the SM both B and L are accidental symmetries and at the tree level these symmetries are not violated. However, the chiral nature of weak interactions gives rise to equal global anomalies for B and L, giving a vanishing B − L anomaly, but a nonvanishing axial current corresponding to B + L, given by t Hooft [102, 103]

$$
\partial\_{\mu}j\_{(B+L)}^{\mu} = \frac{2N\_f}{8\pi} \left( \alpha\_2 \, W\_{\mu\nu}^a \, \tilde{W}^{a\mu\nu} - \alpha\_1 B\_{\mu\nu} \tilde{B}^{\mu\nu} \right), \tag{94}
$$

where W<sup>a</sup> µν and Bµν are the SU(2)<sup>L</sup> and U(1)<sup>Y</sup> field strength tensors and N<sup>f</sup> is the number of fermion generations. The corresponding B + L violation can obtained by integrating the divergence of the B + L current, which is related to the change in the topological charges of the gauge field

$$
\Delta(B+L) = \int d^4x \partial^\mu j\_\mu^{(B+L)} = 2N\_f \Delta N\_{\text{cs}},\tag{95}
$$

where Ncs = ±1, ±2, · · · corresponds to the topological charge of gauge fields, called the Chern-Simons number. In the SM there are three generations of fermions (N<sup>f</sup> = 3), leading to 1B = 1L = 3Ncs, thus the vacuum to vacuum transition changes B and L by multiples of 3 units. At the lowest order, one has the B + L violating effective operator

$$\mathcal{C}(B+L) = \prod\_{i=123} (q\_{Li}q\_{Li}q\_{Li}l\_{Li}),\tag{96}$$

which gives rise to 12-fermion sphaleron induced transitions, such as

$$|\text{vac}\rangle \rightarrow [\mu\_L \mu\_L d\_L e\_L^- + c\_L c\_L s\_L \mu\_L^- + t\_L t\_L b\_L \mathfrak{r}\_L^-].\tag{97}$$

At zero temperature the transition rate is suppressed by e <sup>−</sup>4π/α <sup>=</sup> O(10−165) [102, 103]. However, when the temperature is larger than the barrier height, this Boltzmann suppression disappears and B + L violating transitions can occur at a significant rate [104]. In the symmetric phase, when the temperature is grater than the electroweak phase transition temperature, T ≥ TEW, the transition rate per unit volume is [105–108]

$$\frac{\Gamma\_{B-L}}{V} \sim \alpha^5 \ln \alpha^{-1} T^4,\tag{98}$$

where <sup>α</sup> <sup>=</sup> <sup>α</sup>EM/ sin<sup>2</sup> θW, with αEM being the fine structure constant and θ<sup>W</sup> being the weak mixing angle.

An account of the B − L symmetry getting converted to a baryon asymmetry via an analysis of the chemical potential can be found in Khlebnikov and Shaposhnikov [109], Harvey and Turner [110] and Sarkar [111]. The baryon asymmetry in terms of the B − L number density can be written as

$$B(T > T\_{\rm EW}) = \frac{24 + 4m}{66 + 13m}(B - L),$$

$$B(T < T\_{\rm EW}) = \frac{32 + 4m}{98 + 13m}(B - L). \tag{99}$$

Thus, the primordial B − L asymmetry gets partially converted into a baron asymmetry of the universe after the electroweak phase transition.

### 5.2. Leptogenesis With Right Handed Neutrinos

In section 2, we have discussed how adding singlet right handed neutrinos NRi to the SM can generate tiny seesaw masses [14– 20] for light neutrinos. Beyond the generation of light neutrino masses, the interaction terms

$$\mathcal{L}\_{\text{int}} = h\_{\text{ail}} \bar{l}\_{\text{la}} \phi N\_{\text{Ri}} + M\_{\text{i}} \overline{(N\_{\text{Ri}})^c} N\_{\text{Ri}},\tag{100}$$

can also provide all the ingredients necessary for realizing leptogenesis. We will work on a basis where the right handed neutrino mass matrix is real and diagonal. Furthermore we assume a hierarchical mass spectrum for the right handed neutrinos M<sup>3</sup> > M<sup>2</sup> > M1. The Majorana mass term gives rise to lepton number violating decays of the right handed neutrinos

$$\begin{aligned} N\_{Ri} & \rightarrow & l\_{iL} + \bar{\phi}, \\ & \rightarrow & l\_{iL}{}^c + \phi, \end{aligned} \tag{101}$$

which can generate a lepton asymmetry if there is CP violation and the decay is out of equilibrium [21]. This lepton asymmetry (equivalently B − L asymmetry) then gets converted to baryon asymmetry in presence of anomalous B + L violating processes before the electroweak phase transition.

In the original proposal [21] and few subsequent works [112– 116], only the CP violation coming from interference of tree level and one-loop vertex diagrams, shown in **Figure 8**. was considered. This is somewhat analogous to the CP violation in Kphysics coming from the penguin diagram. The CP asymmetry parameter corresponding to the vertex type CP violation is given by

$$\begin{split} \varepsilon\_{\upsilon} & \equiv \frac{\Gamma(N \to l\phi^{\dagger}) - \Gamma(N \to l^{c}\phi)}{\Gamma(N \to l\phi^{\dagger}) + \Gamma(N \to l^{c}\phi)} \\ & = -\frac{1}{8\pi} \sum\_{i=2,3} \frac{\mathrm{Im}\left[\Sigma\_{\alpha}(h\_{\alpha1}^{\*}h\_{\alpha i})\Sigma\_{\beta}(h\_{\beta1}^{\*}h\_{\beta i})\right]}{\Sigma\_{\alpha}|h\_{\alpha1}|^{2}} f\_{\upsilon}\left(\frac{M\_{i}^{2}}{M\_{1}^{2}}\right), \text{(102))} \end{split} $$

where the loop function f<sup>v</sup> is defined by

$$f\_{\nu}(\mathbf{x}) = \sqrt{\mathbf{x}} \left[ 1 - (1+\varkappa) \ln \left( \frac{1+\varkappa}{\varkappa} \right) \right]. \tag{103}$$

In the limit M<sup>1</sup> ≪ M2, M<sup>3</sup> the asymmetry simplifies to

$$\varepsilon\_{\nu} \simeq -\frac{3}{16\pi} \sum\_{i=2,3} \frac{M\_1}{M\_i} \frac{\mathrm{Im}\left[\Sigma\_{\alpha}(h\_{\alpha1}^\* h\_{\alpha i}) \Sigma\_{\beta}(h\_{\beta1}^\* h\_{\beta i})\right]}{\Sigma\_{\alpha}|h\_{\alpha1}|^2}. \tag{104}$$

It was later pointed out in Flanz et al. [117] and Flanz et al. [118] and confirmed rigorously in Pilaftsis [119], Pilaftsis and Resonant [120], Roulet et al. [121], Buchmuller and Plumacher [122], Flanz and Paschos [123], Hambye et al. [124] and Pilaftsis and Underwood [125], that there is another source of CP violation coming from interference of tree level diagram with one-loop self-energy diagram shown in **Figure 9**. This CP violation is similar to the CP violation due to the box diagram, entering the mass matrix in K −K¯ mixing in K-physics. If the heavy neutrinos decay in equilibrium, the CP asymmetry coming from the selfenergy diagram due to one of the heavy neutrinos may cancel with the asymmetry from the decay of another heavy neutrino to preserve unitarity. However, in out-of-equilibrium decay of heavy neutrinos the number densities of the two heavy neutrinos differ during their decay and consequently, this cancellation is no longer present. This can be understood as the right handed neutrinos oscillating into antineutrinos of different generations, which under the condition Ŵ[particle → antiparticle] 6= Ŵ[antiparticle → particle], can create an asymmetry in right handed neutrinos before they decay. An elementary discussion regarding how the CP violation enters in Majorana mass matrix, which then generates a lepton asymmetry can be found in Sarkar [111] and Langacker et al. [126]. The basic idea is to treat the particles and the antiparticles independently. The CP eigenstates |Nii and |N c i i are no longer physical eigenstates, which evolves with time. Consequently, the physical states, which are admixtures of |Nii and |N c i i, can decay into both leptons and antileptons, giving rise to a CP violation. The CP asymmetry parameter coming from the interference of tree level and oneloop self-energy diagram is given by

$$\begin{split} \varepsilon\_{s} &= \frac{\Gamma(N \to l\phi^{\dagger} - N \to l^{c}\phi)}{\Gamma(N \to l\phi^{\dagger} + N \to l^{c}\phi)} \\ &= \frac{1}{8\pi} \sum\_{i=2,3} \frac{\mathrm{Im}\left[\Sigma\_{\alpha}(h\_{\alpha1}^{\*}h\_{\alpha i})\,\Sigma\_{\beta}(h\_{\beta1}^{\*}h\_{\beta i})\right]}{\Sigma\_{\alpha}|h\_{\alpha1}|^{2}} f\_{s}\left(\frac{M\_{i}^{2}}{M\_{1}^{2}}\right), (105) \end{split}$$

where the loop function f<sup>s</sup> is defined by

$$f\_s\left(\mathbf{x}\right) = \frac{\sqrt{\mathbf{x}}}{1 - \mathbf{x}}.\tag{106}$$

When the mass difference between the right handed neutrinos is very large compared to the width, <sup>M</sup><sup>1</sup> <sup>−</sup> <sup>M</sup><sup>2</sup> <sup>≫</sup> <sup>1</sup> 2 ŴN1,2 , the CP asymmetries coming from vertex and self-energy diagrams are comparable. However, when two right handed neutrinos are nearly degenerate, such that their mass difference is comparable to their width, then CP violation contribution coming from the self-energy diagram becomes very large (orders of magnitude larger than the CP asymmetry generated by the vertex type diagram). This is often referred to as the resonance effect.

To ensure that the lightest right handed neutrino decays outof-equilibrium so that an asymmetry is generated, the out-ofequilibrium condition given by

$$\frac{h\_{\alpha 1}}{16\pi} M\_1 < 1.66 \sqrt{\mathcal{g}\_\*} \frac{T^2}{m\_{\text{Pl}}} \qquad \text{at } T = M\_1. \tag{107}$$

must be satisfied, where g<sup>∗</sup> correspond to the effective number of relativistic degrees of freedom. This gives a lower bound mN<sup>1</sup> > 10<sup>8</sup> GeV [127]. Though this gives us a rough estimate, in an actual calculation of the asymmetry one solves the Boltzmann equation, which takes into account both lepton number violating as well as lepton number conserving processes mediated by heavy neutrinos. The Boltzmann equation governing lepton number asymmetry n<sup>L</sup> ≡ n<sup>l</sup> − n<sup>l</sup> <sup>c</sup>, is given by

$$\frac{d\boldsymbol{n}\_{L}}{dt} + 3\boldsymbol{H}\boldsymbol{n}\_{L} = (\boldsymbol{\varepsilon}\_{\boldsymbol{\nu}} + \boldsymbol{\varepsilon}\_{\boldsymbol{s}})\boldsymbol{\Gamma}\_{\boldsymbol{\psi}\_{1}}(\boldsymbol{n}\_{\psi\_{1}} - \boldsymbol{n}\_{\psi\_{1}}^{eq})$$

$$-\frac{\boldsymbol{n}\_{L}}{\boldsymbol{n}\_{\boldsymbol{\nu}}}\boldsymbol{n}\_{\psi\_{1}}^{eq}\boldsymbol{\Gamma}\_{\psi\_{1}} - 2\boldsymbol{n}\_{\boldsymbol{\nu}}\boldsymbol{n}\_{L}\langle\boldsymbol{\sigma}|\boldsymbol{\nu}\rangle,\qquad\text{(108)}$$

where Ŵψ<sup>1</sup> is the decay rate of the physical state |ψ1i, n eq ψ1 is the equilibrium number density of ψ<sup>1</sup> given by

$$m\_{\psi\_1}^{eq} = \begin{cases} \frac{\mathcal{S}\mathfrak{g}^{-1}}{\mathcal{S}^\eta} \frac{T}{T} \begin{cases} T \gg m\_{\psi\_1} \\ \frac{T}{T} \end{cases} \tag{109}$$

where s is the entropy density. The first term on the right hand side of Equation (108) corresponds to the CP violating contribution to the asymmetry and is the only term that generates asymmetry when ψ<sup>1</sup> decays out-of-equilibrium, while the second term corresponds to inverse decay of ψ1, and the last term corresponds to 2 ↔ 2 lepton number violating scattering process such as l + φ † <sup>↔</sup> <sup>l</sup> <sup>c</sup> <sup>+</sup> <sup>φ</sup>, with <sup>h</sup>σ|v|i being the thermally averaged cross section. The number density of ψ<sup>1</sup> is governed by the Boltzmann equation

$$\frac{d n\_{\psi\_1}}{dt} + 3H n\_{\psi\_1} = -\Gamma\_{\psi\_1} (n\_{\psi\_1} - n\_{\psi\_1}^{eq}).\tag{110}$$

One often defines a parameter K = Ŵψ<sup>1</sup> (T = mψ<sup>1</sup> )/H(T = mψ<sup>1</sup> ), where the Hubble rate H = 1.66g<sup>∗</sup> 1/2 (T 2 /MPl), which gives a measure of the deviation from thermal equilibrium. For K ≪ 1 one can find an approximate solution for Equation (108) given by

$$
\hbar n\_L = \frac{s}{\mathcal{g}\_\*} (\varepsilon\_\nu + \varepsilon\_s). \tag{111}
$$

The Yukawa couplings are constrained by the required amount of primordial lepton asymmetry required to generate the correct baryon asymmetry of the universe, while the lightest right handed neutrino mass is constrained from the out-ofequilibrium condition. In the resonant leptogenesis scenario, the CP violation is largely enhanced, making the constrains on Yukawa couplings relaxed. Consequently the scale of leptogenesis can be considerably lower, making it possible to realize a TeV scale leptogenesis, which can be put to test at the LHC [128, 129].

#### 5.3. Leptogenesis With Triplet Higgs

In section 2, we have discussed how small neutrino masses can be generated by adding triplet Higgs ξ<sup>a</sup> to the SM [22, 26–28, 130– 132]. The interactions of these triplet Higgs that are relevant for leptogenesis are given by

$$\mathcal{L}\_{\text{int}} = f\_{\hat{i}\hat{j}}\xi\,\psi\_{Li} = f\_{\hat{i}\hat{j}}^a \xi\_a^{+++} l\_{i\hat{l}} + \mu\_a \xi\_a^{\dagger} \phi \phi. \tag{1.12}$$

From these interactions we have the decay modes of the triplet Higgs

$$
\xi\_a^{++} \rightarrow \begin{cases} l\_i^+ l\_j^+ \\ \phi^+ \phi^+ \end{cases} \tag{113}
$$

The CP violation is obtained through the interference between the tree level and one-loop self-energy diagrams shown in **Figure 10**. There are no one-loop vertex diagrams in this case. One needs at least two ξ 's. To see how this works, we will follow the mass-matrix formalism [22], in which the diagonal tree-level mass matrix of ξ<sup>a</sup> is modified in the presence of interactions to

$$\frac{1}{2}\xi^{\dagger}\left(M\_{+}^{2}\right)\_{ab}\xi\_{b} + \frac{1}{2}\left(\xi\_{a}^{\*}\right)^{\dagger}\left(M\_{-}^{2}\right)\_{ab}\xi\_{b}^{\*},\tag{114}$$

where

$$M\_{\pm}^{2} = \begin{pmatrix} M\_{1}^{2} - i\Gamma\_{11}M\_{1} & -i\Gamma\_{12}^{\pm} \\ -i\Gamma\_{21}^{\pm}M\_{1} & M\_{2}^{2} - i\Gamma\_{22}M\_{2} \end{pmatrix},\tag{115}$$

with Ŵ + ab = Ŵab and Ŵ − ab = Ŵ ∗ ab. From the absorptive part of the one-loop diagram for ξ<sup>a</sup> → ξ<sup>b</sup> we obtain

$$
\Gamma\_{ab}M\_b = \frac{1}{8\pi} \left( \mu\_a \mu\_b^\* + M\_a M\_b \sum\_{k,l} f\_{kl}^{a\*} f\_{kl}^b \right). \tag{116}
$$

Assuming <sup>Ŵ</sup><sup>a</sup> <sup>≡</sup> <sup>Ŵ</sup>aa <sup>≪</sup> <sup>M</sup>a, the eigenvalues of <sup>M</sup><sup>2</sup> <sup>±</sup> are given by

$$
\lambda\_{1,2} = \frac{1}{2} (M\_1^2 + M\_2^2 \pm \sqrt{\mathcal{S}}),
\tag{117}
$$

FIGURE 9 | Tree level and one-loop self-energy diagrams contributing to the CP violation in models with right handed neutrinos.

where <sup>S</sup> <sup>=</sup> (M<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>M</sup><sup>2</sup> 2 ) <sup>2</sup> <sup>−</sup> <sup>4</sup>|Ŵ12M2<sup>|</sup> 2 and M<sup>1</sup> > M2. The physical states, which evolves with time, can be written as linear combinations of the CP eigenstates as

$$
\psi\_{1,2}^+ = a\_{1,2}^+ \xi\_1 + b\_{1,2}^+ \xi\_2 \ , \qquad \psi\_{1,2}^- = a\_{1,2}^- \xi\_1^\* + b\_{1,2}^- \xi\_2^\* \ , \tag{118}
$$

where a ± <sup>1</sup> = b ± <sup>2</sup> = 1/ q 1 + |C ± i | 2 , b ± <sup>1</sup> = C ± 1 / q 1 + |C ± i | 2 , a ± <sup>2</sup> = C ± 2 / q 1 + |C ± i | <sup>2</sup> with

$$C\_1^+ = -C\_2^- = \frac{-2i\Gamma\_{12}^\* M\_2}{M\_1^2 - M\_2^2 + \sqrt{\mathcal{S}}},$$

$$C\_1^- = -C\_2^+ = \frac{-2i\Gamma\_{12} M\_2}{M\_1^2 - M\_2^2 + \sqrt{\mathcal{S}}}.\tag{119}$$

The physical states ψ ± 1,2 evolve with time and decay into lepton and antilepton pairs. Assuming (M<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>M</sup><sup>2</sup> 2 ) <sup>2</sup> <sup>≫</sup> <sup>4</sup>|Ŵ12M2<sup>|</sup> 2 , the CP asymmetry is given by Ma [22]

$$\varepsilon\_{i} \simeq \frac{1}{8\pi^{2}(M\_{1}^{2} - M\_{2}^{2})^{2}} \sum\_{k,l} \operatorname{Im}\left(\mu\_{1}\mu\_{2}^{\*}f\_{kl}^{1}f\_{kl}^{2\*}\right) \left(\frac{M\_{i}}{\Gamma\_{i}}\right) . \tag{120}$$

For M<sup>1</sup> > M2, when the temperature drops below M1, ψ<sup>1</sup> decays away to create a lepton asymmetry. However, this asymmetry is washed out by lepton number violating interactions of ψ2; and the subsequent decay of ψ<sup>2</sup> at a temperature below M<sup>2</sup> sustains. The generated lepton asymmetry then gets converted to the baryon asymmetry in the presence of the sphaleron induced anomalous B+L violating processes before the electroweak phase transition. The approximate final baryon asymmetry is given by

$$\frac{m\_B}{s} \sim \frac{\varepsilon\_2}{3g\_\* K (\ln K)^{0.6}},\tag{121}$$

where K ≡ Ŵ2(T = M2)/H(T = M2) is the parameter measuring the deviation from thermal equilibrium when, H = 1.66g<sup>∗</sup> 1/2 (T 2 /MPl) is the Hubble rate, and g ∗ corresponds to the number of relativistic degrees of freedom.

In a more rigorous estimation of the baryon asymmetry, in addition to the decays and the inverse decays of triplet scalars, one needs to incorporate the gauge scatterings ψψ¯ ↔ FF¯, φφ¯,GG¯ (F corresponds to SM fermions and G corresponds to gauge bosons) and 1L = 2 scattering processes ll ↔ φ ∗φ ∗ and lφ ↔ ¯ lφ ∗ into the Boltzmann equation analysis of the asymmetry. Including these washout processes, one finds a lower limit on M<sup>ξ</sup> , M<sup>ξ</sup> & 10<sup>11</sup> GeV [133]. For a quasi-degenerate spectrum of scalar triplets the resonance effect can enhance the CP asymmetry by a large amount and a successful leptogenesis scenario can be attained for a much smaller value of triplet scalar mass. In Strumia [134] and Aristizabal Sierra et al. [135], an absolute bound of M<sup>ξ</sup> & 1.6 TeV is obtained for a successful resonant leptogenesis scenario with triplet Higgs.

### 6. LEPTOGENESIS IN LRSM

In Left–Right Symmetric Model (LRSM) [17, 19, 44–49] the left– right parity symmetry breaking implies the existence of a heavy right-handed charged gauge boson W± R . In this section, we will discuss the aspect that if W± R is detected at the LHC with a mass of a few TeV then it can have profound implications for leptogenesis. If indeed W± R is detected at the LHC then that will give rise to an excess in the dilepton + dijet channel as reported sometime back by the CMS collaboration. A signal of 2.8 σ level was reported in the mass bin 1.8 TeV < Mlljj < 2.1 TeV in the di-lepton + di-jet channel at the LHC by the CMS collaboration [136]. One of the popular interpretations of this signal was W± R decay in the framework of LRSM with g<sup>L</sup> 6= g<sup>R</sup> via an embedding of LRSM in SO(10) [137, 138]. Another popular interpretation was for the case g<sup>L</sup> = g<sup>R</sup> which utilized the CP phases and non-degenerate mass spectrum of the heavy neutrinos [139]. Around the same time the ATLAS collaboration had also reported a resonance signal decaying into a pair of SM gauge bosons. They found a local excess signal of 3.4σ (2.5σ global) in the WZ channel at around 2 TeV [140]. This signal was shown to be explained by a W<sup>R</sup> in LRSM framework for a coupling g<sup>R</sup> ∼ 0.4 in Brehmer et al. [141]. Some other notable work along this direction can be found in Dobrescu and Liu Bhupal [142], Dev and Mohapatra [143] and Das et al. [144]. However, these interesting signals were either washed out by more accumulated data or reduced significantly below their initial reported levels. Nevertheless such signals have intrigued several studies concerning the impact of a TeV scale W± R on leptogenesis.

As discussed earlier, the Higgs sector in one of the popular versions of LRSM consists of one bidoublet Higgs 8 and two triplet complex scalar fields 1L,R. The relevant gauge transformations are as follows

$$
\Phi \sim (\mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1}), \quad \Delta\_L \sim (\mathbf{3}, \mathbf{1}, \mathbf{2}, \mathbf{1}), \quad \Delta\_R \sim (\mathbf{1}, \mathbf{3}, \mathbf{2}, \mathbf{1}).\tag{122}
$$

Here one breaks the left–right symmetry in a spontaneous manner to reproduce the Standard Model. On the other hand the smallness of the neutrino masses is realized using the seesaw mechanism [14–20].

In another variant of LRSM one has only the doublet Higgs which are employed to break all the relevant symmetries. Here the Higgs sector consists of doublet scalars with the gauge transformations

$$\Phi \sim (\mathbf{2}, \mathbf{2}, \mathbf{0}, \mathbf{1}), \ H\_L \sim (\mathbf{2}, \mathbf{1}, \mathbf{1}, \mathbf{1}), \ H\_R \sim (\mathbf{1}, \mathbf{2}, \mathbf{1}, \mathbf{1}), \tag{123}$$

In addition there is one fermion gauge singlet S<sup>R</sup> ∼ (**1**, **1**, **0**, **1**). The Higgs doublet H<sup>R</sup> acquires a VEV to break the left–right symmetry which results in the mixing of S with right-handed neutrinos. This gives rise to a light Majorana neutrino and a heavy pseudo-Dirac neutrino or alternatively a pair of Majorana neutrinos.

Historically, in LRSM, the left–right symmetry was broken at a fairly high scale, M<sup>R</sup> > 10<sup>10</sup> GeV. This serves two purposes– firstly, the requirement of gauge coupling unification implies this scale to be high, and secondly, thermal leptogenesis in this scenario gives a comparable bound. To get around this problem one often introduces a parity odd scalar which is then given a large VEV. This is often called D-parity breaking. Hati et al. Neutrino Masses and Leptogenesis in LRSM

Consequently, one can have g<sup>L</sup> 6= g<sup>R</sup> even before the left– right symmetry breaking. This in turn allows the possibility of a gauge coupling unification with TeV scale WR. This is true for both triplet and doublet models of LRSM. Embedding the LRSM in an SO(10) GUT framework, the violation of D-parity [54] at a very high scale helps in explaining the CMS TeV scale W<sup>R</sup> signal for g<sup>R</sup> ≈ 0.6g<sup>L</sup> as shown in Deppisch et al. [137, 138].

#### 6.1. Can a TeV-Scale W<sup>±</sup> R at the LHC Falsify Leptogenesis?

For a TeV scale W± R , all leptogenesis scenarios may be broadly classified into two groups:


The following discussions hold for the LRSM variants with a Higgs sector consisting of triplet Higgs as well as a Higgs sector with exclusively doublet Higgs. We will often refer to these two broad classes of the LRSM mentioned above to discuss the lepton number violating washout processes and point out how all these possible scenarios of leptogenesis are falsifiable for a W<sup>R</sup> of TeV scale. In the case where high-scale leptogenesis happens at <sup>T</sup> <sup>&</sup>gt; <sup>10</sup><sup>9</sup> GeV, the low energy <sup>B</sup> <sup>−</sup> <sup>L</sup> breaking gives rise to gauge interactions which depletes all the baryon asymmetry very rapidly before the electroweak phase transition is over. Now, these same lepton number violating gauge interactions will significantly slow down the generation of the lepton asymmetry for resonant leptogenesis at around TeV scale. Consequently, it is not possible to generate the required baryon asymmetry of the universe for TeV scale W± R in this case.

For the case MN3<sup>R</sup> ≫ MN2<sup>R</sup> ≫ MN1<sup>R</sup> = MN<sup>R</sup> , severe constraints on the W± <sup>R</sup> mass for a successful scenario of highscale leptogenesis come from the SU(2)<sup>R</sup> gauge interactions as pointed out in Ma [145]. To have successful leptogenesis in the case MN<sup>R</sup> > MW<sup>R</sup> , the condition that the gauge scattering process e − <sup>R</sup> + W<sup>+</sup> <sup>R</sup> → N<sup>R</sup> → e + <sup>R</sup> + W<sup>−</sup> R goes out-of-equilibrium yields

$$M\_{N\_R} \underset{\sim}{\gtrless} 10^{16} \,\text{GeV} \tag{124}$$

with mW<sup>R</sup> /mN<sup>R</sup> & 0.1. For the scenario where MW<sup>R</sup> > MN<sup>R</sup> leptogenesis happens either at T > MW<sup>R</sup> after the breaking of B − L gauge symmetry or at T ≃ MN<sup>R</sup> , the out-of equilibrium condition for the scattering process e ± R e ± <sup>R</sup> → W<sup>±</sup> <sup>R</sup> W<sup>±</sup> R through N<sup>R</sup> exchange leads to the constraint

$$M\_{W\_R} \gtrsim 3 \times 10^6 \,\text{GeV} (M\_{N\_R}/10^2 \,\text{GeV})^{2/3}. \tag{125}$$

Thus, a W<sup>R</sup> with mass in the TeV range (in the case of a hierarchical neutrino mass spectrum) rules out the high-scale leptogenesis scenario. In Deppisch et al. [146, 147], neutrinoless double beta decay and the observation of the lepton number violating processes at the colliders were studied in the context of high-scale thermal leptogenesis. In Flanz et al. [117, 118], Pilaftsis [119], Roulet et al. [121], Buchmuller and Plumacher [122], Flanz and Paschos [123], Hambye et al. [124], Pilaftsis and Underwood [125] resonant leptogenesis has been discussed in the context of a considerably low mass WR. In Frere et al. [148] it was pointed out that one requires an absolute lower bound of 18 TeV on the W<sup>R</sup> mass in order to have successful low-scale leptogenesis with a quasi-degenerate right-handed neutrinos. Recently, it was found that just the correct lepton asymmetry can be obtained by utilizing relatively large Yukawa couplings, for W<sup>R</sup> mass scale higher than 13.1 TeV in Bhupal Dev et al. [149, 150]. Note that in Frere et al. [148] and Bhupal Dev et al. [150], the lepton number violating gauge scattering processes such as NRe<sup>R</sup> → ¯uRdR, NRu¯<sup>R</sup> → eRdR, NRd<sup>R</sup> → eRu<sup>R</sup> and NRN<sup>R</sup> → eRe¯<sup>R</sup> have been analyzed in detail. However, lepton number violating scattering processes with external W<sup>R</sup> were ignored because for a heavy W<sup>R</sup> there is a relative suppression of e −mWR /mNR in comparison to the processes where there is no W<sup>R</sup> in the external legs. Now, if indeed the mass of W<sup>R</sup> is around a few TeV, as was suggested by an excess signal reported by the CMS experiment then one has to take the latter processes seriously. In Dhuria et al. [151], it was pointed out that the lepton number violating washout processes (e ± R e ± <sup>R</sup> → W<sup>±</sup> <sup>R</sup> W<sup>±</sup> R and e ± <sup>R</sup> W<sup>∓</sup> <sup>R</sup> → e ∓ <sup>R</sup> W<sup>±</sup> R ) can be mediated via the doubly charged Higgs in the conventional LRSM. In Bhupal Dev et al. [149] it was shown that in a parityasymmetric type-I seesaw model with relatively small MN<sup>R</sup> one obtains a small contribution from this process which is expected for a large MW<sup>R</sup> /MN<sup>R</sup> . However, in this scenario some other relevant gauge scattering processes are efficient in washing out the lepton asymmetry. Including these washout processes one obtains a lower bound of 13.1 TeV on the W<sup>R</sup> mass [149]. Here we will mainly discuss 1 ++ R and N<sup>R</sup> mediated lepton number violating scattering processes in a much more general context to establish their importance as washout processes which can falsify the possibility of leptogenesis depending on W<sup>R</sup> mass [152]. One of the vertices in the 1 ++ <sup>R</sup> mediated process is gauge vertex while the other one is a Yukawa vertex. On the other hand for N<sup>R</sup> mediated lepton number violating scattering processes both the vertices are gauge vertices. Consequently, these lepton number violating scattering processes are very rapid as compared to the scattering processes involving only Yukawa vertices. It turns out that N<sup>R</sup> and 1 ++ <sup>R</sup> mediated scattering process e ± <sup>R</sup> W<sup>∓</sup> R → e ∓ <sup>R</sup> W<sup>±</sup> R does not go out of equilibrium till the electroweak phase transition if the mass of W<sup>R</sup> is around TeV scale. Consequently, these lepton number violating scattering processes continue to wash out or slow down the generation of lepton asymmetry<sup>8</sup> . In the scenario of LRSM involving only doublet Higgs in the Higgs sector the doubly charged Higgs is absent. Nevertheless, the N<sup>R</sup> mediated lepton number violating scattering processes will be present and will wash out the lepton asymmetry in such a scenario.

<sup>8</sup> In passing we would like to note that the other relevant lepton number violating scattering process is doubly phase space suppressed for a temperature below the W<sup>R</sup> mass scale. Consequently, we will neglect such a process for leptogenesis occurring at T . MW<sup>R</sup> .

In LRSM the right handed leptonic charged current interaction is given by

$$\mathcal{L}\_N = \frac{1}{2\sqrt{2}} \text{g}\_R J\_{R\mu} \mathcal{W}\_R^{-\mu} + h.c.\tag{126}$$

where JR<sup>µ</sup> = e¯Rγ<sup>µ</sup> (1 + γ5) NR. The relevant interactions of the right-handed Higgs triplet are given by

$$\mathcal{L}\_{\Delta\_{R}} \supset \left(D\_{R\mu}\vec{\Delta}\_{R}\right)^{\dagger}\left(D\_{R}^{\mu}\vec{\Delta}\_{R}\right),\tag{127}$$

where 1E R = 1 ++ R , 1 + R , 1<sup>0</sup> R . The covariant derivative is given by DR<sup>µ</sup> = ∂µ−ig<sup>R</sup> T j RA j Rµ −ig′Bµ, where A j Rµ and Bµ are gauge fields corresponding to SU(2)<sup>R</sup> and U(1)B−<sup>L</sup> gauge groups with the associated gauge couplings given by g<sup>R</sup> and g ′ , respectively. When the neutral Higgs field 1<sup>0</sup> R acquires a VEV <sup>h</sup>1<sup>0</sup> R i = √ 1 2 vR SU(2)R, the interaction between the gauge boson W<sup>R</sup> and the doubly charged Higgs is given by Doi [153]

$$\mathcal{L}\_{\Delta\_R} \supset \left(-\frac{\nu\_R}{\sqrt{2}}\right) g\_R^2 W\_{\mu R}^- W\_R^{-\mu} \Delta\_R^{+++} + h.c.\tag{128}$$

The Yukawa interaction between the lepton doublet ψeR = (NR, eR) T and the components of triplet Higgs 1E <sup>R</sup> are given by

$$\mathcal{L}\_Y = h\_{\rm ee}^R \overline{(\psi\_{eR})^c} \left( i \text{tr}\_2 \vec{\pi}. \vec{\Delta}\_R \right) \psi\_{eR} + h.c.,\tag{129}$$

where τ 's are the Pauli matrices. After the Higgs triplet field acquires a VEV, the relevant Yukawa coupling can be written as h R ee = MNR 2vR where MN<sup>R</sup> is the Majorana mass of NR.

The relevant Feynman diagrams for the lepton number violating processes induced by these interactions are depicted in **Figure 11**.

Using the interactions given in Equations (126)–(129), one can estimate the differential scattering cross section for the process e ∓ R (p)W± R (k) → e ± R (p ′ )W∓ R (k ′ ) to obtain [153]

$$\frac{d\sigma\_{\varepsilon\_{R}W\_{R}}^{\varepsilon\_{R}W\_{R}}}{dt} = \frac{1}{384\pi M\_{W\_{R}}^{4}\left(s - M\_{W\_{R}}^{2}\right)^{2}}\Lambda\_{\varepsilon\_{R}W\_{R}}^{\varepsilon\_{R}W\_{R}}(s, t, \mu), \tag{130}$$

where

$$\left. \Lambda\_{\varepsilon\_{R}W\_{R}}^{\varepsilon\_{R}W\_{R}}(\mathbf{s},t,\boldsymbol{\mu}) = \left. \Lambda\_{\varepsilon\_{R}W\_{R}}^{\varepsilon\_{R}W\_{R}}(\mathbf{s},t,\boldsymbol{\mu}) \right|\_{N\_{R}} + \left. \Lambda\_{\varepsilon\_{R}W\_{R}}^{\varepsilon\_{R}W\_{R}}(\mathbf{s},t,\boldsymbol{\mu}) \right|\_{\Delta\_{R}^{++}} \tag{131}$$

and

$$\left.\Lambda\_{e\_{R}W\_{R}^{R}}^{\epsilon\_{R}W\_{R}}(s,t,u)\right|\_{N\_{R}} = \mathcal{g}\_{R}^{4}\left\{-t\left|M\_{N\_{R}}\left(\frac{s}{s-M\_{N\_{R}}^{2}}+\frac{u}{u-M\_{N\_{R}}^{2}}\right)\right|^{2}\right.$$

$$-4M\_{W\_{R}}^{2}\left(su-M\_{W\_{R}}^{4}\right)(s-u)^{2}\left|\frac{M\_{N\_{R}}}{\left(s-M\_{N\_{R}}^{2}\right)\left(u-M\_{N\_{R}}^{2}\right)}\right|^{2}$$

$$-4M\_{W\_{R}}^{4}t\left(\left|\frac{m\_{N\_{R}}}{\left(s-M\_{N\_{R}}^{2}\right)}\right|^{2}+\left|\frac{M\_{N\_{R}}}{\left(u-M\_{N\_{R}}^{2}\right)}\right|^{2}\right)\right\},\tag{132}$$

$$\begin{split} \left. \Lambda\_{\mathfrak{c}\_{R}^{W\_{R}}}^{\mathfrak{c}\_{R},W\_{R}}(s,t,u) \right|\_{\Delta\_{R}^{\pm+}} &= 4g\_{R}^{4}(-t) \left\{ \frac{(s+u)^{2} + 8M\_{W\_{R}}^{4}}{\left(t - M\_{\Delta\_{R}}^{2}\right)^{2}} \left| M\_{N\_{R}} \right|^{2} \\ &+ \frac{(s+u)}{t - M\_{\Delta\_{R}}^{2}} \left| M\_{N\_{R}} \right|^{2} \left(\frac{s}{s - M\_{N\_{R}}^{2}} + \frac{u}{u - M\_{N\_{R}}^{2}}\right) \\ &+ \frac{4M\_{W\_{R}}^{4}}{t - M\_{\Delta\_{R}}^{2}} \left| M\_{N\_{R}} \right|^{2} \left(\frac{1}{s - M\_{N\_{R}}^{2}} + \frac{1}{u - M\_{N\_{R}}^{2}}\right) \right\}, \end{split} \tag{133}$$

where we neglect any mixing between W<sup>L</sup> and WR. On the righthand side of Equation (133), the first term corresponds to the Higgs exchange. The last two terms are due to the interference between Higgs and N<sup>R</sup> exchange. The Mandelstam variables s = p + k 2 , t = p − p ′ 2 and u = p − k ′ 2 are related by the scattering angle θ as follows

$$\left(\begin{matrix}st\\s\mu-M\_{W\_R}^4\end{matrix}\right) = -\frac{1}{2}\left(s-M\_{W\_R}^2\right)^2 \left(1\mp\cos\theta\right). \tag{134}$$

The differential scattering cross section for the process e ± R (p)e ± R (p ′ ) → W<sup>±</sup> R (k)W± R (k ′ ) is given by Doi [153]

$$\frac{d\sigma\_{W\_R W\_R}^{e\_{R\ell R}}}{dt} = \frac{1}{512\pi M\_{W\_R}^4 s^2} \Lambda\_{W\_R W\_R}^{e\_R e\_R} \text{(s, t, u)},\tag{135}$$

where

$$\left. \Lambda\_{W\_R W\_R}^{\epsilon\_R \epsilon\_R} \{ s, t, \mu \} \right| = \left. \Lambda\_{W\_R W\_R}^{\epsilon\_R \epsilon\_R} \{ s, t, \mu \} \right|\_{N\_R} + \left. \Lambda\_{W\_R W\_R}^{\epsilon\_R \epsilon\_R} \{ s, t, \mu \} \right|\_{\Delta\_R^{++}}.\tag{1.36}$$

The expressions for 3 eReR WRWR (s, t, u) can be obtained after interchanging s ↔ t in 3 eRWR eRWR (s, t, u): 3 eReR WRWR (t,s, u)= −3 eRWR eRWR (s, t, u). The Mandelstem variables t = p − k 2 and u = p − k ′ 2 can be written in terms of s = p + p ′ 2 and scattering angle θ as follows

$$\chi\left(\frac{t}{\mu}\right) = -\frac{s}{2}\left(1 - \frac{2M\_{W\_R}^2}{s}\right)\left\{1 \mp \sqrt{1 - \left(\frac{2M\_{W\_R}^2}{s - 2M\_{W\_R}^2}\right)^2}\cos\theta\right\}.\tag{137}$$

### 6.1.1. Wash Out of Lepton Asymmetry for T > MW<sup>R</sup>

During the period when the temperature is such that v<sup>R</sup> > T > MW<sup>R</sup> , the lepton number violating washout processes are very rapid in the absence any suppression. To have a quantitative estimate of the strength of these scattering processes in depleting the lepton asymmetry one can estimate the parameter defined as

$$K \equiv \frac{n \langle \sigma | \nu |}{H},\tag{138}$$

for both the processes during v<sup>R</sup> > T > MW<sup>R</sup> , where n corresponds to the number density of relativistic species and is given by n = 2 × 3ζ (3) <sup>4</sup>π<sup>2</sup> T 3 . H corresponds to the Hubble

rate H ≃ 1.7g 1/2 <sup>∗</sup> T 2 /MPl, where g<sup>∗</sup> ∼ 100 corresponds to the relativistic degrees of freedom. The thermally averaged cross section is denoted by hσ|υ|i. To choose a rough estimate of vR, let us compare the situation with the Standard Model, where we have hφi = <sup>√</sup> vL 2 where v<sup>L</sup> = 246 GeV, and MW<sup>L</sup> ∼ 80 GeV. Now in case of LRSM <sup>h</sup>1<sup>0</sup> R i = √ υR 2 breaks the left–right symmetry and <sup>M</sup>W<sup>R</sup> <sup>=</sup> <sup>g</sup>RυR. Taking <sup>g</sup><sup>R</sup> <sup>∼</sup> <sup>g</sup>L, we have <sup>h</sup>φ<sup>i</sup> MWL = <sup>h</sup>1<sup>0</sup> R i MWR ≈ 3.

Making use of the differential scattering cross sections in Equations (130) and (135), we plot the behavior of K as a

function of temperature in **Figure 12**. The plot corresponds to a temperature range 3MW<sup>R</sup> > T > MW<sup>R</sup> and right handed charged gauge boson mass MW<sup>R</sup> = 3.5 TeV.

In **Figure 12**, the large values of K for both the processes indicates that high wash out efficiency of these scattering processes for T & MW<sup>R</sup> . For the LRSM variant with its Higgs sector consisting of only doublet Higgs the doubly charged Higgs mediated channels are absent for these processes and the right handed neutrino will mediate these processes, which will washout the lepton asymmetry for T & MW<sup>R</sup> .

### 6.1.2. Wash Out of Asymmetry for T < MW<sup>R</sup>

During the period when the temperature is such that T < MW<sup>R</sup> , the process e ± <sup>R</sup> W<sup>∓</sup> <sup>R</sup> → e ∓ <sup>R</sup> W<sup>±</sup> R is of more importance. Let us now estimate a lower bound on T below T = MW<sup>R</sup> till which this process stays in equilibrium and continues to deplete lepton asymmetry. The scattering rate can be written as<sup>9</sup> <sup>Ŵ</sup> = ¯nhσvreli. For T < MW<sup>R</sup> the Boltzmann suppression of the scattering rate stems from the number density n¯ = g TMWR 2π 3/<sup>2</sup> exp − MWR T . Now, the scattering process stays in thermal equilibrium when the condition Ŵ > H is satisfied.

In **Figure 13** we show the temperature until which the scattering process e ± <sup>R</sup> W<sup>∓</sup> <sup>R</sup> → e ∓ <sup>R</sup> W<sup>±</sup> R stays in equilibrium as a function of MW<sup>R</sup> for three values of M1<sup>R</sup> and taking MN<sup>R</sup> < <sup>∼</sup> <sup>M</sup>W<sup>R</sup> and vrel = 1. We have taken the lowest value of M1<sup>R</sup> to be 500 GeV to be consistent with the recent collider limits on the doubly charged Higgs mass [154]. The plot shows that unless MW<sup>R</sup> is significantly heavier than a few TeV, the process e ± <sup>R</sup> W<sup>∓</sup> R → e ∓ <sup>R</sup> W<sup>±</sup> <sup>R</sup> will continue to be in equilibrium till a temperature similar to the electroweak phase transition. Consequently, this process will continue to washout or slow down the generation of lepton asymmetry until the electroweak phase transition. In the LRSM variant with doublet Higgs, the heavy neutrinos will mediated lepton number violating scattering processes will washout or slow down the generation of lepton asymmetry until the electroweak phase transition. Thus, the lower limit on the W<sup>R</sup> mass for a successful leptogenesis scenario is significantly higher a few TeV. This was also confirmed by explicitly solving the relevant Boltzmann equations in Bhupal Dev et al. [149, 150].

### 7. CONCLUDING REMARKS

We have reviewed the standard left–right symmetric theories and the implementation of different types of low scale seesaw mechanisms in the context of neutrino masses. We have also discussed a left–right symmetric model with additional vectorlike fermions in order to simultaneously explain the charged fermion and Majorana neutrino masses. In this model the quark and charged lepton masses and mixings are realized via a universal seesaw mechanism while spontaneous symmetry breaking is achieved with two doublet Higgs fields with non-zero

### REFERENCES


B − L charge, we have introduced scalar triplets with small induced VEVs such that they give Majorana masses to light as well as heavy neutrinos. We have also discussed how the Majorana nature of these neutrinos leads to 0νββ decay. Interestingly, the right-handed currents play an important role in discriminating between the mass hierarchy as well as the absolute scale of light neutrinos. To summarize the situation for leptogenesis, in the high-scale leptogenesis scenario (T > <sup>∼</sup> <sup>M</sup>W<sup>R</sup> ), in all the variants of LRSM the lepton number violating processes e ± R e ± <sup>R</sup> → W<sup>±</sup> <sup>R</sup> W<sup>±</sup> R and e ± <sup>R</sup> W<sup>∓</sup> <sup>R</sup> → e ∓ <sup>R</sup> W<sup>±</sup> R are highly efficient in washing out the lepton asymmetry. In the case of resonant leptogenesis scenario at around TeV scale we found that the latter process stays in equilibrium until the electroweak phase transition, making the generation of lepton asymmetry for T < MW<sup>R</sup> significantly weaker. Thus, if the LHC discovers a TeV scale W± R then one needs to look for some post-electroweak phase transition mechanism to explain the baryon asymmetry of the Universe. To this end the observation of the neutronantineutron oscillation [155, 156] or (B − L) violating proton decay [157] will play a guiding role in confirming such scenarios. Complementing these results, the low-energy subgroups of the superstring motivated E<sup>6</sup> model have also been explored which can also give rise to left–right symmetric gauge structures but with a number of additional exotic particles as compared to the conventional LRSM. Interestingly, one of the low-energy supersymmetric subgroups of E6, also known as the Alternative Left–Right Symmetric Model, gives a model alternative to successfully realize high-scale leptogenesis in the absence of the dangerous gauge washout processes [158]. The vector-like fermions added to the minimal framework of LRSM to realize a universal seesaw can pave new ways to realize baryogenesis as discussed in Deppisch et al. [73].

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

The authors would like to thank Frank F. Deppisch for many helpful discussions and encouragement. The authors are also thankful to both the reviewers for many valuable suggestions and CH and US are thankful to Mansi Dhuria and Raghavan Rangarajan for several wonderful collaborations which have resulted in some of the interesting findings covered in this review. The work of US is supported partly by the JC Bose National Fellowship grant under DST, India.

4. Chen CY, Dev PSB, Mohapatra RN. Probing heavy-light neutrino mixing in left-right seesaw models at the LHC. Phys Rev D (2013) **88**:033014. arXiv:1306.2342. doi: 10.1103/PhysRevD.88.0 33014

<sup>9</sup>We ignore any finite temperature effects to simplify the analysis.

<sup>3.</sup> Das SP, Deppisch FF, Kittel O, Valle JWF. Heavy neutrinos and lepton flavour violation in left-right symmetric models at the LHC. Phys Rev D (2012) **86**:055006. 055006, arXiv:1206.0256. doi: 10.1103/PhysRevD.86.0 55006


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Hati, Patra, Pritimita and Sarkar. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Lepton Number Violation: Seesaw Models and Their Collider Tests

Yi Cai <sup>1</sup> , Tao Han2,3, Tong Li 4,5 \* and Richard Ruiz <sup>6</sup>

<sup>1</sup> School of Physics, Sun Yat-sen University, Guangzhou, China, <sup>2</sup> Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, United States, <sup>3</sup> Department of Physics, Tsinghua University, and Collaborative Innovation Center of Quantum Matter, Beijing, China, <sup>4</sup> School of Physics, Nankai University, Tianjin, China, <sup>5</sup> ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics and Astronomy, Monash University, Melbourne, VIC, Australia, <sup>6</sup> Department of Physics, Institute for Particle Physics Phenomenology, Durham University, Durham, United Kingdom

The Majorana nature of neutrinos is strongly motivated from the theoretical and phenomenological point of view. A plethora of neutrino mass models, known collectively as Seesaw models, exist that could generate both a viable neutrino mass spectrum and mixing pattern. They can also lead to rich, new phenomenology, including lepton number non-conservation as well as new particles, that may be observable at collider experiments. It is therefore vital to search for such new phenomena and the mass scale associated with neutrino mass generation at high energy colliders. In this review, we consider a number of representative Seesaw scenarios as phenomenological benchmarks, including the characteristic Type I, II, and III Seesaw mechanisms, their extensions and hybridizations, as well as radiative constructions. We present new and updated predictions for analyses featuring lepton number violation and expected coverage in the theory parameter space at current and future colliders. We emphasize new production and decay channels, their phenomenological relevance and treatment across different facilities in e +e −, e −p, and pp collisions, as well as the available Monte Carlo tools available for studying Seesaw partners in collider environments.

Keywords: lepton number violation, neutrino mass models, collider physics, seesaw mechanisms, Majorana neutrinos

ARXIV EPRINT: 1711.02180

# 1. INTRODUCTION

Neutrino flavor oscillation experiments from astrophysical and terrestrial sources provide overwhelming evidence that neutrinos have small but nonzero masses. Current observations paint a picture consistent with a mixing structure parameterized by the 3×3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1–3] with at least two massive neutrinos. This is contrary to the Standard Model of particle physics (SM) [4], which allows three massless neutrinos and hence no flavor oscillations. Consequently, to accommodate these observations, the SM must [5] be extended to a more complete theory by new degrees of freedom.

One could of course introduce right-handed (RH) neutrino states (νR) and construct Dirac mass terms, m<sup>D</sup> νLνR, in the same fashion as for all the other elementary fermions in the SM. However, in this minimal construction, the new states do not carry any SM gauge charges, and thus these "sterile neutrinos" have the capacity to be Majorana fermions [6]. The most significant consequence of this would be the existence of the RH Majorana mass term MR(νR) <sup>c</sup>ν<sup>R</sup> and the explicit violation of lepton number (L). In light of this prospect, a grand frontier opens for theoretical model-building with rich and new

#### Edited by:

Frank Franz Deppisch, University College London, United Kingdom

#### Reviewed by:

Oliver Fischer, Karlsruher Institut für Technologie, Germany Miha Nemevsek, Jožef Stefan Institute (IJS), Slovenia

\*Correspondence:

Tong Li nklitong@hotmail.com

#### Specialty section:

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

Received: 20 November 2017 Accepted: 16 April 2018 Published: 09 May 2018

#### Citation:

Cai Y, Han T, Li T and Ruiz R (2018) Lepton Number Violation: Seesaw Models and Their Collider Tests. Front. Phys. 6:40. doi: 10.3389/fphy.2018.00040

**240**

phenomenology at the collider energy scales, which we will review in this article.

Generically, if we integrate out the new states, presumably much heavier than the electroweak (EW) scale, the new physics may be parameterized at leading order through the dimension-5 lepton number violating operator [7], the so-called "Weinberg operator,"

$$\mathcal{L}\_5 = \frac{\alpha}{\Lambda} \text{ (LH)}\\\text{(LH)} \xrightarrow{\text{EWSB}} \mathcal{L}\_5 \Rightarrow \frac{\alpha \nu\_0^2}{2\Lambda} \overline{(\nu\_L)^\epsilon} \text{ } \nu\_L,\tag{1.1}$$

where L and H are, respectively, the SM left-handed (LH) lepton doublet and Higgs doublet, with vacuum expectation value (vev) v<sup>0</sup> ≈ 246 GeV. After electroweak (EW) symmetry breaking (EWSB), L<sup>5</sup> generates a Majorana mass term for neutrinos. One significance of Equation (1.1) is the fact that its ultraviolet (UV) completions are severely restricted. For example: extending the SM field content minimally, i.e., by only a single SM multiplet, permits only three [5] tree-level completions of Equation (1.1), a set of constructions famously known as the Type I [8–14], Type II [14–18], and Type III [19] Seesaw mechanisms. These minimal mechanisms can be summarized with the following:

**Minimal Type I Seesaw** [8–14]: In the minimal Type I Seesaw, one hypothesizes the existence of a right-handed (RH) neutrino νR, which transforms as a singlet, i.e., (1, 1, 0), under the SM gauge group SU(3)c⊗SU(2)L⊗U(1)Y, that possesses a RH Majorana mass Mν<sup>R</sup> and interacts with a single generation of SM leptons through a Yukawa coupling yν . After mass mixing and assuming Mν<sup>R</sup> ≫ y<sup>ν</sup> v0, the light neutrino mass eigenvalue m<sup>ν</sup> is given by m<sup>ν</sup> ∼ y 2 ν v 2 0 /Mν<sup>R</sup> , If y<sup>ν</sup> ≃ 1, to obtain a light neutrino mass of order an eV, Mν<sup>R</sup> is required to be of order 10<sup>14</sup> <sup>−</sup> <sup>10</sup><sup>15</sup> GeV. Mν<sup>R</sup> can be made much lower though by balancing against a correspondingly lower yν .

**Minimal Type II Seesaw** [14–18]: The minimal Type II Seesaw features the introduction of a Higgs field 1 with mass M<sup>1</sup> in a triplet representation of SU(2)<sup>L</sup> and transforms as (1, 3, 2) under the SM gauge group. In this mechanism, light neutrino masses are given by LH Majorana masses m<sup>ν</sup> ≈ Y<sup>ν</sup> v1, where v1 is the vev of the neutral component of the new scalar triplet and Yν is the corresponding Yukawa coupling. Due to mixing between the SM Higgs doublet and the new scalar triplet via a dimensionful parameter µ, EWSB leads to a relation v<sup>1</sup> ∼ µv 2 0 /M<sup>2</sup> 1 . In this case the new scale 3 is replaced by M<sup>2</sup> 1 /µ. With <sup>Y</sup><sup>ν</sup> <sup>≈</sup> 1 and <sup>µ</sup> <sup>∼</sup> <sup>M</sup>1, the scale is also 10<sup>14</sup> <sup>−</sup> <sup>10</sup><sup>15</sup> GeV. Again, M<sup>1</sup> can be of TeV scale if Y<sup>ν</sup> is small or µ≪M1. It is noteworthy that in the Type II Seesaw, no RH neutrinos are needed to explain the observed neutrino masses and mixing.

**Minimal Type III Seesaw** [19]: The minimal Type III Seesaw is similar to the other two cases in that one introduces the fermionic multiplet 6<sup>L</sup> that is a triplet (adjoint representation) under SU(2)<sup>L</sup> and transforms as (1, 3, 0) under the SM gauge group. The resulting mass matrix for neutrinos has the same form as in Type I Seesaw, but in addition features heavy leptons that are electrically charged. The new physics scale 3 in Equation (1.1) is replaced by the mass of the leptons 6L, which can also be as low as a TeV if balanced with a small Yukawa coupling.

However, to fully reproduce oscillation data, at least two of the three known neutrinos need nonzero masses. This requires a nontrivial Yukawa coupling matrix for neutrinos if appealing to any of the aforementioned Seesaws mechanisms, and, if invoking the Type I or III Seesaws, extending the SM by at least two generations of multiplets [20], which need not be in the same SM gauge representation. In light of this, one sees that Weinberg's assumption of a high-scale Seesaw [7] is not necessary to generate tiny neutrino masses in connection with lepton (L) number violation. For example: the so-called Inverse [21–24] or Linear [25, 26] variants of the Type I and III Seesaw models, their generic extensions as well as hybridizations, i.e., the combination of two or more Seesaw mechanisms, can naturally lead to mass scales associated with neutrino mass-generation accessible at present-day experiments, and in particular, collider experiments. A qualitative feature of these low-scale Seesaws is that light neutrino masses are proportional to the scale of L violation, as opposed to inversely related as in high-scale Seesaws [27].

The Weinberg operator in Equation (1.1) is the lowest order and simplest parameterization of neutrino mass generation using only the SM particle spectrum and its gauge symmetries. Beyond its tree-level realizations, however, neutrino Majorana masses may alternatively be generated radiatively. Suppression by loop factors may provide a partial explanation for the smallness of neutrino masses and again allow much lower mass scales associated with neutrino mass-generation. The first of such models was proposed at one-loop in Zee [28] and Hall and Suzuki [29], at two-loop order in Cheng and Li [16], Zee [30], and Babu [31], and at three-loop order in Krauss et al. [32]. A key feature of radiative neutrino mass models is the absence of tree-level contributions to neutrino masses either because there the necessary particles, such as SM singlet fermion as in Type I Seesaw, are not present or because relevant couplings are forbidden by additional symmetries. Consequently, it is necessary that the new field multiplets run in the loops that generate neutrino masses.

As observing lepton number violation would imply the existence of Majorana masses for neutrinos [33–35], confirming the existence of this new mass scale would, in addition, verify the presence of a Seesaw mechanism. To this end, there have been on-going efforts in several directions, most notably the neutrinoless double beta (0νββ)-decay experiments, both current [36–39] and upcoming [40–42], as well as proposed general purpose fixed-target facilities [43, 44]. Complementary to this are on-going searches for lepton number violating processes at collider experiments, which focus broadly on rare meson decays [45–47], heavy neutral fermions in Type I-like models [48–52], heavy bosons in Type II-like models [53– 55], heavy charged leptons in Type III-like models [56–58], and lepton number violating contact interactions [59, 60]. Furthermore, accurate measurements of the PMNS matrix elements and stringent limits on the neutrino masses themselves provide crucial information and knowledge of lepton flavor mixing that could shed light on the construction of Seesaw models.

In this context, we present a review of searches for lepton number violation at current and future collider experiments. Along with the current bounds from the experiments at LEP, Belle, LHCb and ATLAS/CMS at 8 and 13 TeV, we present studies for the 13 and 14 TeV LHC. Where available, we also include results for a future 100 TeV hadron collider, an ep collider (LHeC), and a future high-energy e +e − collider. We consider a number of tree- and loop-level Seesaw models, including, as phenomenological benchmarks, the canonical Type I, II, and III Seesaw mechanisms, their extensions and hybridizations, and radiative Seesaw formulations in pp, ep, and ee collisions. We note that the classification of collider signatures based on the canonical Seesaws is actually highly suitable, as the same underlying extended and hybrid Seesaw mechanism can be molded to produce wildly varying collider predictions.

We do not attempt to cover the full aspects of UV-complete models for each type. This review is only limited to a selective, but representative, presentation of tests of Seesaw models at collider experiments. For complementary reviews, we refer readers to Gluza [61], Barger et al. [62], Mohapatra and Smirnov [63], Rodejohann [64], Chen and Huang [65], Atre et al. [66], Deppisch et al. [67] and references therein.

This review is organized according to the following: In section 2 we first show the PMNS matrix and summarize the mixing and mass-difference parameters from neutrino oscillation data. With those constraints, we also show the allowed mass spectra for the three massive neutrino scheme. Our presentation is agnostic, phenomenological, and categorized according to collider signature, i.e., according to the presence of Majorana neutrinos (Type I) as in section 3, doubly charged scalars (Type II) as in section 4, new heavy charged/neutral leptons (Type III) as in section 5, and new Higgs, diquarks and leptoquarks in section 6. Particular focus is given to state-ofthe-art computations, newly available Monte Carlo tools, and new collider signatures that offer expanded coverage of Seesaw parameter spaces at current and future colliders. Finally in section 7 we summarize our main results.

### 2. NEUTRINO MASS AND OSCILLATION PARAMETERS

In order to provide a general guidance for model construction and collider searches, we first summarize the neutrino mass and mixing parameters in light of oscillation data. Neutrino mixing can be parameterized by the PMNS matrix [1– 3] as

$$\begin{split} U\_{\rm PMNS} &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & c\_{23} & s\_{23} \\ 0 & -s\_{23} & c\_{23} \end{pmatrix} \begin{pmatrix} c\_{13} & 0 & e^{-i\delta} s\_{13} \\ 0 & 1 & 0 \\ -e^{i\delta} s\_{13} & 0 & c\_{13} \end{pmatrix} \begin{pmatrix} c\_{12} & s\_{12} & 0 \\ -s\_{12} & c\_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ \text{diag}(e^{i\Phi\_{1}/2}, 1, e^{i\Phi\_{2}/2}) & & \tag{2.1} \\ &= \begin{pmatrix} c\_{12}c\_{13} & c\_{13}s\_{12} & e^{-i\delta} s\_{13} \\ -c\_{12}s\_{13}s\_{23}e^{i\delta} & -c\_{23}s\_{12} & c\_{12}s\_{13}s\_{23} & c\_{13}s\_{23} \\ s\_{12}s\_{23} - e^{i\delta} c\_{12}c\_{23}s\_{13} & -c\_{23}s\_{12}s\_{13}e^{i\delta} - c\_{12}s\_{23} & c\_{13}c\_{23} \end{pmatrix} \\ &\times \text{diag}(e^{i\Phi\_{1}/2}, 1, e^{i\Phi\_{2}/2}), \tag{2.2} \end{split}$$

where sij ≡ sin θij, cij ≡ cos θij, 0 ≤ θij ≤ π/2, and 0 ≤ δ, 8<sup>i</sup> ≤ 2π, with δ being the Dirac CP phase and 8<sup>i</sup> the Majorana phases. While the PMNS is a well-defined 3×3 unitary matrix, throughout this review, we use the term generically to describe the 3 × 3 active-light mixing that may not, in general, be unitary.

The neutrino mixing matrix is very different from the quarksector Cabbibo-Kobayashi-Maskawa (CKM) matrix, in that most of the PMNS mixing angles are large whereas CKM angles are small-to-negligible. In recent years, several reactor experiments, such as Daya Bay [68], Double Chooz [69], and RENO [70] have reported non-zero measurements of θ<sup>13</sup> by searching for the disappearance of anti-electron neutrinos. Among these reactor experiments, Daya Bay gives the most conclusive result with sin<sup>2</sup> 2θ<sup>13</sup> ≈ 0.084 or θ<sup>13</sup> ≈ 8.4◦ [71, 72], the smallest entry of the PMNS matrix. More recently, there have been reports on indications of a non-zero Dirac CP phase, with δ ≈ 3π/2 [73–75]. However, it cannot presently be excluded that evidence for such a large Dirac phase may instead be evidence for sterile neutrinos or new neutral currents [76– 79].

Neutrino oscillation experiments can help to extract the size of the mass-squared splitting between three neutrino mass eigenstates. The sign of 1m<sup>2</sup> <sup>31</sup> <sup>=</sup> <sup>m</sup><sup>2</sup> <sup>3</sup> <sup>−</sup> <sup>m</sup><sup>2</sup> 1 , however, still remains unknown at this time. It can be either positive, commonly referred as the Normal Hierarchy (NH), or negative and referred to as the Inverted Hierarchy (IH). The terms Normal Ordering (NO) and Inverted Ordering (IO) are also often used in the literature in lieu of NH and IH, respectively. Taking into account the reactor data from the antineutrino disappearance experiments mentioned above together with other disappearance and appearance measurement, the latest global fit of the neutrino masses and mixing parameters from the NuFit collaboration [72], are listed in **Table 1** for NH (left) and IH (center). The tightest constraint on the sum of neutrino masses comes from cosmological data. Combining Planck+WMAP+highL+BAO data, this yields at 95% confidence level (CL) [80]

$$\sum\_{i=1}^{3} m\_i < \text{ 0.230 eV.} \tag{2.3}$$

Given this and the measured neutrino mass splittings, we show in **Figure 1** the three active neutrino mass spectra as a function of the lowest neutrino mass in (a) NH and (b) IH. With the potential sensitivity of the sum of neutrino masses being close to 0.1 eV in the near future (5–7 years) [81], upcoming cosmological probes will not be able to settle the issue of the neutrino mass hierarchy. However, the improved measurement ∼ 0.01 eV over a longer term (7 − 15 years) [81, 82] would be sensitive to determine the absolute mass scale of a heavier neutrino spectrum. In addition, there are multiple proposed experiments aiming to determine the neutrino mass hierarchy. The Deep Underground Neutrino Experiment (DUNE) will detect neutrino beams from the Long Baseline Neutrino Facility (LBNF), and probes the CP-phase and the mass hierarchy. With a baseline of 1,300 km, DUNE is able to determine the mass hierarchy with at least 5σ significance [83]. The Jiangmen Underground Neutrino Observatory (JUNO) plans to precisely measure the reactor electron antineutrinos and improve the accuracy of 1m<sup>2</sup> <sup>21</sup>, 1m<sup>2</sup> <sup>32</sup> and sin<sup>2</sup> θ<sup>12</sup> to 1% level [84]. The TABLE 1 | Three-neutrino oscillation fit based as obtained by the NuFit collaboration, taken from Esteban et al. [72], where 1m<sup>2</sup> 3ℓ <sup>=</sup> <sup>1</sup>m<sup>2</sup> <sup>31</sup> > 0 for NO (or NH) and 1m<sup>2</sup> 3ℓ <sup>=</sup> <sup>1</sup>m<sup>2</sup> <sup>32</sup> < 0 for IO (or IH).


Hyper-Kamiokande (Hyper-K) experiment as an update of T2K can measure the precision of δ to be 7◦ − 21◦ and reach 3 (5)σ significance of mass hierarchy determination for 5 (10) years exposure [85]. Finally, the Karlsruhe Tritium Neutrino experiment (KATRIN) as a tritium β decay experiment aims to measure the effective electron-neutrino mass with the sensitivity of sub-eV [86].

### 3. THE TYPE I SEESAW AND LEPTON NUMBER VIOLATION AT COLLIDERS

We begin our presentation of collider searches for lepton number violation in the context of Type I Seesaw models. After describing the canonical Type I mechanism [8–12] and its phenomenological decoupling at collider scales in section 3.1.1, we discuss various representative, lowscale models that incorporate the Type I mechanism and its extensions. We then present collider searches for lepton number violation mediated by Majorana neutrinos (N), which is the characteristic feature of Type I-based scenarios, in section 3.2. This is further categorized according to associated phenomena of increasing complexity: N production via massive Abelian gauge bosons is reviewed in section 3.2.4, via massive non-Abelian gauge bosons in section 3.2.5, and via dimension-six operators in section 3.2.6.

#### 3.1. Type I Seesaw Models

#### 3.1.1. The Canonical Type I Seesaw Mechanism

In the canonical Type I Seesaw mechanism one hypothesizes a single RH neutral leptonic state, N<sup>R</sup> ∼ (1, 1, 0), in addition to the SM matter content. However, reproducing neutrino oscillation data requires more degrees of freedom. Therefore, for our purposes, we assume i = 1, ... , 3 LH states and j = 1, ... , n RH states. Following the notation of Atre et al. [66] and Han et al. [87], the full theory is

$$\mathcal{L}\_{\text{Type I}} = \mathcal{L}\_{\text{SM}} + \mathcal{L}\_{N \text{ Kim}} + \mathcal{L}\_{N},\tag{3.1}$$

where LSM is the SM Lagrangian, L<sup>N</sup> Kin is NR's kinetic term, and its interactions and mass are

$$\mathcal{L}\_N = -\overline{L}\,Y\_\nu^D \,\tilde{H}\,N\_R - \frac{1}{2} \overline{(N^c)\_L} \,M\_R \,N\_R + \text{ H.c.} \tag{3.2}$$

L and H are the SM LH lepton and Higgs doublets, respectively, and H˜ = iσ2H<sup>∗</sup> . Once H settles on the vev hHi = v0/ √ 2, neutrinos acquire Dirac masses m<sup>D</sup> = Y D ν v0/ √ 2 and we have

$$\begin{split} \mathcal{L}\_{N} &\ni -\frac{1}{2} \left( \overline{\upsilon\_{L}} \, m\_{D} \, N\_{R} + \, \overline{\langle \mathcal{N}^{c} \rangle\_{L}} \, m\_{D}^{T} \left( \nu^{c} \right)\_{R} + \, \overline{\langle \mathcal{N}^{c} \rangle\_{L}} \, M\_{R} \, N\_{R} \right) \\ &+ \quad \text{H.c.} \end{split} \tag{3.3}$$

After introducing a unitary transformation into m (m′ ) light (heavy) mass eigenstates,

$$
\begin{pmatrix} \nu \\ N^c \end{pmatrix}\_L = \mathbb{N} \begin{pmatrix} \nu\_m \\ N^c\_{m'} \end{pmatrix}\_L, \quad \mathbb{N} = \begin{pmatrix} U & V \\ X & Y \end{pmatrix}, \tag{3.4}
$$

one obtains the diagonalized mass matrix for neutrinos

$$\mathbb{N}^{\dagger} \begin{pmatrix} 0 & m\_D \\ m\_D^T & M \end{pmatrix} \mathbb{N}^\* = \begin{pmatrix} m\_\vee & 0 \\ 0 & M\_N \end{pmatrix},\tag{3.5}$$

with mass eigenvalues m<sup>ν</sup> = diag(m1, m2, m3) and M<sup>N</sup> = diag(M1, · · · , Mm′). In the limit m<sup>D</sup> ≪ MR, the light (m<sup>ν</sup> ) and heavy (MN) neutrino masses are

$$m\_{\upsilon} \approx -m\_{D}M\_{R}^{-1}m\_{D}^{T} \quad \text{and} \quad M\_{N} \approx M\_{R}. \tag{3.6}$$

The mixing elements typically scale like

$$UU^{\dagger} \approx I - m\_{\upsilon}M\_{N}^{-1}, \quad VV^{\dagger} \approx m\_{\upsilon}M\_{N}^{-1},\tag{3.7}$$

with the unitarity condition UU† <sup>+</sup> VV† <sup>=</sup> <sup>I</sup>. With another matrix Uℓ diagonalizing the charged lepton mass matrix, we have the approximate neutrino mass mixing matrix UPMNS and the matrix VℓN, which transits heavy neutrinos to charged leptons, and are given by

$$U\_{\ell}^{\dagger}U \equiv \begin{array}{c} U\_{\ell}^{\dagger}U \equiv \ U\_{\text{PMNS}}, \quad U\_{\ell}^{\dagger}V \equiv V\_{\ell N}, \quad \text{and} \\ U\_{\text{PMNS}}U\_{\text{PMNS}}^{\dagger} + \; V\_{\ell N}V\_{\ell N}^{\dagger} = I. \end{array} \tag{3.8}$$

The decomposition of active neutrino states into a general number of massive eigenstates is then given by Atre et al. [66] and Han et al. [87], ν<sup>ℓ</sup> = P<sup>3</sup> <sup>m</sup>=<sup>1</sup> <sup>U</sup>ℓmν<sup>m</sup> <sup>+</sup> P<sup>n</sup> <sup>m</sup>′=<sup>1</sup> <sup>V</sup>ℓm′<sup>N</sup> c m′ . From this, the SM EW boson couplings to heavy mass eigenstates (in the mixed mass-flavor basis) are

$$\begin{split} \mathcal{L}\_{\text{Int.}} &= -\left. \frac{\mathcal{g}}{\sqrt{2}} W\_{\mu}^{+} \sum\_{\ell=\epsilon}^{\mathsf{r}} \left( \sum\_{m=1}^{3} \overline{\nu\_{m}} \, U\_{\ell m}^{\*} + \sum\_{m'=1}^{n} \overline{N\_{m'}^{c}} \, V\_{\ell N\_{m'}}^{\*} \right) \chi^{\mu} P\_{L} \ell^{-} \\ &- \frac{\mathcal{g}}{2 \cos \theta\_{W}} Z\_{\mu} \sum\_{\ell=\epsilon}^{\mathsf{r}} \left( \sum\_{m=1}^{3} \overline{\nu\_{m}} \, U\_{\ell m}^{\*} + \sum\_{m'=1}^{n} \overline{N\_{m'}^{c}} \, V\_{\ell N\_{m'}}^{\*} \right) \chi^{\mu} P\_{L} \nu\_{\ell} \\ &- \frac{\mathcal{g}}{2 M\_{W}} h \sum\_{\ell=\epsilon}^{\mathsf{r}} \sum\_{m'=1}^{n} m\_{N\_{m'}} \overline{N\_{m'}^{c}} \, V\_{\ell N\_{m'}}^{\*} P\_{L} \nu\_{\ell} + \text{H.c.} \end{split} \tag{3.9}$$

There is a particular utility of using this mixed mass-flavor basis in collider searches for heavy neutrinos. Empirically, |VℓNm′ | . 10−<sup>2</sup> [88–91], which means pair production of Nm′ via EW processes is suppressed by |VℓNm′ | <sup>2</sup> . 10−<sup>4</sup> relative to single production of Nm′ . Moreover, in collider processes involving ν<sup>m</sup> − Nm′ vertices, one sums over ν<sup>m</sup> either because it is an internal particle or an undetected external state. This summation effectively undoes the decomposition of one neutrino interaction state for neutral current vertices, resulting in the basis above. In phenomenological analyses, it is common practice to consider in only the lightest heavy neutrino mass eigenstate, i.e., Nm′=4, to reduce the effective number of independent model parameters. In such cases, the mass eigenstate is denoted simply as N and one reports sensitivity on the associated mixing element, labeled correspondingly as |VℓN| or |Vℓ4|, which are equivalent to <sup>|</sup>VℓNm′=<sup>4</sup> |. Throughout this text, the |VℓN| notation is adopted where possible.

From Equation (3.5), an important relation among neutrino masses can be derived. Namely, that

$$\left(U\_{\rm PMNS}^\* m\_\nu U\_{\rm PMNS}^\dagger + V\_{\ell N}^\* M\_N V\_{\ell N}^\dagger = 0\right). \tag{3.10}$$

Here the masses and mixing of the light neutrinos in the first term are measurable from the oscillation experiments, and the second term contains the masses and mixing of the new heavy neutrinos. We now consider a simple case: degenerate heavy neutrinos with mass <sup>M</sup><sup>N</sup> <sup>=</sup> diag(M1, · · · , <sup>M</sup>m′) <sup>=</sup> <sup>M</sup>NIm′ . Using this assumption, we obtain from Equation (3.10),

$$M\_N \sum\_N (V\_{\ell N}^\*)^2 = \langle U\_{\text{PMNS}}^\* m\_\nu \, U\_{\text{PMNS}}^\dagger \rangle\_{\ell \ell} \,. \tag{3.11}$$

Using the oscillation data in **Table 1** as inputs<sup>1</sup> , we display in **Figure 2** the normalized mixing of each lepton flavor in this scenario<sup>2</sup> . Interestingly, one can see the characteristic features:

$$\sum\_{N} |V\_{\varepsilon N}|^2 \ll \sum\_{N} |V\_{\mu N}|^2, \sum\_{N} |V\_{\varepsilon N}|^2 \qquad \text{for NH, (3.12)}$$

$$\sum\_{N} |V\_{\varepsilon N}|^2 \ge \sum\_{N} |V\_{\mu N}|^2, \sum\_{N} |V\_{\varepsilon N}|^2 \qquad \text{for IH.} \tag{3.13}$$

<sup>1</sup>This is done for simplicity since UPMNS in **Table 1** is unitary whereas here it is not; for more details, see Esteban et al. [72], and Parke and Ross-Lonergan [92].

<sup>2</sup>P N (V ∗ ℓN ) <sup>2</sup> <sup>=</sup> P N |Vℓ<sup>N</sup> | <sup>2</sup> only when all phases on the right-hand side of Equation (3.11) vanish [93].

As shown in **Figure 3**, a corresponding pattern also emerges in the branching fraction<sup>3</sup> of the degenerate neutrinos decaying into charged leptons plus a W boson,

$$\text{BR}(\mu^{\pm}W^{\mp}), \text{BR}(\mathfrak{r}^{\pm}W^{\mp}) \sim \text{ (20-30)} \% \gg \text{BR}(\mathfrak{e}^{\pm}W^{\mp})$$

$$\sim \text{ (3-4)} \% \qquad \text{for NH}, \qquad \text{(3.14)}$$

$$\text{BR}(\mathfrak{e}^{\pm}W^{\mp}) \sim 27\% > \text{BR}(\mu^{\pm}W^{\mp}), \text{BR}(\mathfrak{r}^{\pm}W^{\mp})$$

$$\sim \text{ (10-20)} \% \qquad \text{for IIH}, \qquad \text{(3.15)}$$

with BR(ℓ <sup>±</sup>W∓) = BR(N<sup>i</sup> → ℓ <sup>+</sup>W<sup>−</sup> + ℓ −W+). These patterns show a rather general feature that ratios of Seesaw partner observables, e.g., cross sections and branching fractions, encode information on light neutrinos, such as their mass hierarchy [93, 94]. Hence, one can distinguish between competing light neutrino mass and mixing patterns with high energy observables.

More generally, the Vℓ<sup>N</sup> in Equation (3.10) can be formally solved in terms of an arbitrary orthogonal complex matrix , known as the Casas-Ibarra parametrization [95], using the ansatz

$$V\_{\ell N} = U\_{\text{PMNS}} \, m\_{\nu}^{1/2} \Omega M\_N^{-1/2},\tag{3.16}$$

with the orthogonality condition <sup>T</sup> <sup>=</sup> <sup>I</sup>. For the simplest incarnation with a unity matrix = I, |VℓNm′ | 2 are proportional to one and only one light neutrino mass, and thus the branching ratio of Nm′ → ℓ ±W∓ for each lepton flavor is independent of neutrino mass and universal for both NH and IH [93]. Nevertheless, one can still differentiate between the three heavy neutrinos according to the decay rates to their leading decay channels. As shown in **Figure 4** for = I, one sees

$$\text{BR}(e^{\pm}W^{\mp}) \sim 40\% > \text{BR}(\mu^{\pm}W^{\mp}), \text{BR}(\tau^{\pm}W^{\mp})$$

$$\epsilon \quad \text{(\$\tau\$ 15)} \\ \alpha \quad \text{(\$\tau\$-}10^{\prime}\text{ or \$\tau^{\pm}\$-}M\$)}$$

$$\sim (4-15)\% \text{ for } N\_1,\tag{3.17}$$

$$\text{BR}(e^{\pm}W^{\mp}) \sim 20\% \approx \text{BR}(\mu^{\pm}W^{\mp}) \approx \text{BR}(\tau^{\pm}W^{\mp})$$

$$
\sim (10-30)\% \text{ for } N\_2,\tag{3.18}
$$

$$\begin{split} \text{BR}(\mu^{\pm}W^{\mp}), \text{BR}(\mathfrak{r}^{\pm}W^{\mp}) &\sim \text{ (15-40)}\% \gg \text{BR}(\mathfrak{e}^{\pm}W^{\mp})\\ &\sim 1\% \text{ for } N\_{3}. \end{split} \tag{3.19}$$

A realistic Dirac mass matrix can be quite arbitrary with three complex angles parameterizing the orthogonal matrix . However, this arbitrariness of the Dirac mass matrix is not a universal feature of Seesaw models; the neutrino Yukawa matrix in the Type II Seesaw, for example, is much more constrained.

Beyond this, **Figure 2** also shows another general feature of minimal, high-scale Seesaw constructions, namely that the active-sterile mixing |VℓN| is vanishingly small. For a heavy neutrino mass of M<sup>N</sup> ∼ 100 GeV, Equation (3.11) implies |VℓN| <sup>2</sup> <sup>∼</sup> <sup>10</sup>−14−10−12. This leads to the well-known decoupling of observable lepton number violation in the minimal, high-scale Type I Seesaw scenario at colliders experiments [27, 96, 97]. For low-scale Type I Seesaws, such decoupling of observable lepton number violation also occurs: Due to the allowed arbitrariness of the matrix in Equation (3.16), it is possible to construct and M<sup>N</sup> with particular entry patterns or symmetry structures,

 $\Gamma^3$  $\text{Where } \text{BR}(A \to X) \equiv \Gamma(A \to X) / \sum\_Y \Gamma(A \to Y) \text{ for partial width } \Gamma(A \to Y).$ 

also known as textures in the literature, such that Vℓ<sup>N</sup> is nonzero but mν vanishes. Light neutrino masses can then be generated as perturbations from these textures. In Moffat et al. [27] it was proved that such delicate (and potentially fine-tuned [98–100]) constructions result in small neutrino masses being proportional to small L-violating parameters, instead of being inversely proportional as in the high-scale case. Subsequently, in low-scale Seesaw scenarios that assume only fermionic gauge singlets, tiny neutrino masses is equivalent to an approximate conservation of lepton number, and leads to the suppression of observable L violation in high energy processes. Hence, any observation of lepton number violation (and Seesaw partners in general) at collider experiments implies a much richer neutrino massgeneration scheme than just the canonical, high-scale Type I Seesaw.

#### 3.1.2. Type I+II Hybrid Seesaw Mechanism

While the discovery of lepton number violation in, say, 0νββ or hadron collisions would imply the Majorana nature of neutrinos [33–35], it would be less clear which mechanism or mechanisms are driving light neutrino masses to their sub-eV values. This is because in the most general case neutrinos possess both LH and RH Majorana masses in addition to Dirac masses. In such hybrid Seesaw models, two or more "canonical" tree- and loop-level mechanisms are combined and, so to speak, may give rise to phenomenology that is greater than the sum of its parts.

A well-studied hybrid model is the Type I+II Seesaw mechanism, wherein the light neutrino mass matrix Mν , when MDM−<sup>1</sup> <sup>R</sup> ≪ 1, is given by Chen et al. [101], Akhmedov and Frigerio [102], Akhmedov et al. [103], Chao et al. [104, 105], Gu et al. [106], and Chao et al. [107]

$$M\_{\nu}^{light} = M\_L - M\_D M\_R^{-1} M\_D^T. \tag{3.20}$$

Here, the Dirac and Majorana mass terms, MD, MR, have their respective origins according to the Type I model, whereas M<sup>L</sup> originates from the Type II mechanism; see section 4 for details. In this scenario, sub-eV neutrino masses can arise not only from parametrically small Type I and II masses but additionally from an incomplete cancellation of the two terms [102–104]. While a significant or even moderate cancellation requires a high-degree of fine tuning and is radiatively instable [107], this situation cannot theoretically be ruled out a priori. For a onegeneration mechanism, the relative minus sign in Equation (3.20) is paramount for such a cancellation; however, in a multigeneration scheme, it is not as crucial as M<sup>D</sup> is, in general, complex and can even absorb the sign through a phase rotation. Moreover, this fine-tuning scenario is a caveat of the aforementioned decoupling of L-violation in a minimal Type I Seesaw from LHC phenomenology [27, 96, 97]. As we will discuss shortly, regardless of its providence, if such a situation were to be realized in nature, then vibrant and rich collider signatures emerges.

#### 3.1.3. Type I Seesaw in U(1)<sup>X</sup> Gauge Extensions of the Standard Model

Another manner in which the decoupling of heavy Majorana neutrinos N from collider experiments can be avoided is

through the introduction of new gauge symmetries, under which N is charged. One such example is the well-studied U(1)<sup>X</sup> Abelian gauge extension of the SM [108–112], where U(1)<sup>X</sup> is a linear combination of U(1)<sup>Y</sup> and U(1)B−<sup>L</sup> after the spontaneous breaking of electroweak symmetry and B − L (baryon minus lepton number) symmetries. In this class of models, RH neutrinos are introduced to cancel gauge anomalies and realize a Type I Seesaw mechanism.

Generally, such a theory can be described by modifying the SM covariant derivatives by Salvioni et al. [113]

$$D\_{\mu} \ni \text{ig}\_1 Y B\_{\mu} \quad \rightarrow \quad D\_{\mu} \ni \text{ig}\_1 Y B\_{\mu} + i(\tilde{\emptyset}Y + \mathcal{g}\_1' Y\_{BL}) B\_{\mu}', \text{(3.21)}$$

where Bµ(Y) and B ′ µ (YBL) are the gauge fields (quantum numbers) of U(1)<sup>Y</sup> and U(1)B−L, respectively. The most economical extension with vanishing mixing between U(1)<sup>Y</sup> and U(1)B−L, i.e., , U(1)<sup>X</sup> = U(1)B−<sup>L</sup> and g˜ = 0 in Equation (3.21), introduces three RH neutrinos and a new complex scalar S that are all charged under the new gauge group but remain singlets under the SM symmetries [114–116]. In this extension one can then construct the neutrino Yukawa interactions

$$\mathcal{L}\_I^Y = -\bar{L}\_L \, Y\_\nu^D \, \tilde{H} \, N\_R - \frac{1}{2} Y\_\nu^M \, \overline{\langle N^c \rangle\_L} \, N\_R \, \mathbb{S} + \text{ H.c.} \tag{3.22}$$

Once the Higgs S acquires the vacuum expectation value hSi = vS/ √ 2, B − L is broken, spontaneously generating the RH Majorana mass matrix M<sup>N</sup> = Y M ν vS/ √ 2 from Equation (3.22).

It isinteresting to note that the scalar vev provides a dynamical mechanism for the heavy, RH Majorana mass generation, i.e., a Type I Seesaw via a Type II mechanism; see section 4 for more details. The Seesaw formula and the mixing between the SM charged leptons and heavy neutrinos here are exactly the same as those in the canonical Type I Seesaw. The mass of neutral gauge field B ′ µ , MZ′ = MZB−<sup>L</sup> = 2gBLvS, is generated from S' kinetic term, DµS † (D <sup>µ</sup>S) with <sup>D</sup>µ<sup>S</sup> <sup>=</sup> <sup>∂</sup>µ<sup>S</sup> <sup>+</sup> <sup>i</sup>2gBL<sup>B</sup> ′ µ S. Note that in the minimal model, gBL = g ′ 1 . As in other extended scalar scenarios, the quadratic term H†HS† S in the scalar potential

results in the SM Higgs H and S interaction states mixing into two CP-even mass eigenstates, H<sup>1</sup> and H2.

### 3.1.4. Type I+II Hybrid Seesaw in Left-Right Symmetric Model

As discussed in section 3.1.2, it may be the case the light neutrino masses result from the interplay of multiple Seesaw mechanisms. For example: the Type I+II hybrid mechanism with light neutrino masses given by Equation (3.20). It is also worth observing two facts: First, in the absence of Majorana masses, the minimum fermionic field content for a Type I+II Seesaw automatically obeys an accidental global U(1)B−<sup>L</sup> symmetry. Second, with three RH neutrinos, all fermions can be sorted into either SU(2)<sup>L</sup> doublets (as in the SM) or SU(2)<sup>R</sup> doublets, its RH analog. As the hallmark of the Type II model (see section 4) is the spontaneous generation of LH Majorana masses from a scalar SU(2)<sup>L</sup> triplet 1L, it is conceivable that RH neutrino Majorana masses could also be generated spontaneously, but from a scalar SU(2)<sup>R</sup> triplet 1R. (This is similar to the spontaneous breaking of U(1)B−<sup>L</sup> in section 3.1.3.) This realization of the Type I+II Seesaw is known as the Left-Right Symmetric Model (LRSM) [117–121], and remains one of the best-motivated and well-studied extensions of the SM. For recent, dedicated reviews, see Mohapatra and Smirnov [63], Duka et al. [122], and Senjanovic´ [123].

The high energy symmetries of the LRSM is based on the extended gauge group

$$\mathcal{G}\_{\rm LRSM} = \mathrm{SU(3)}\_{\rm c} \otimes \mathrm{SU(2)}\_{\rm L} \otimes \mathrm{SU(2)}\_{\rm R} \otimes \mathrm{U(1)}\_{\rm B-L},\tag{3.23}$$

or its embeddings, and conjectures that elementary states, in the UV limit, participate in LH and RH chiral currents with equal strength. While the original formulation of model supposes a generalized parity <sup>P</sup><sup>X</sup> <sup>=</sup> <sup>P</sup> that enforces an exchange symmetry between fields charged under SU(2)<sup>L</sup> and SU(2)R, it is also possible to achieve this symmetry via a generalized charge conjugation <sup>P</sup><sup>X</sup> <sup>=</sup> <sup>C</sup> [124]. For fermionic and scalar multiplets QL,<sup>R</sup> and 8, the exchange relationships are [124],

$$\mathcal{P}: \begin{cases} Q\_L \leftrightarrow Q\_R \\ \Phi \leftrightarrow \Phi^\dagger \end{cases}, \quad \text{and} \quad \mathcal{C}: \begin{cases} Q\_L \leftrightarrow \begin{pmatrix} Q\_R \end{pmatrix}^\varepsilon \\ \Phi \leftrightarrow \Phi^T \end{cases},$$

$$\text{where} \quad (Q\_R)^\varepsilon = C \mathcal{V}^0 Q\_R^\*. \tag{3.24}$$

A non-trivial, low-energy consequence of these complementary formulations of the LRSM is the relationship between the LH CKM matrix in the SM, V L ij , and its RH analog, V R ij . For generalized conjugation, one has |V R ij | = |V L ij|, whereas |V R ij | ≈ |V L ij| + <sup>O</sup>(mb/mt) for generalized parity [124–128]. Moreover, LR parity also establishes a connection between the Dirac and Majorana masses in the leptonic sector [129, 130]. Under generalized parity, for example, the Dirac (Y D 1,2) and Majorana (YL,R) Yukawa matrices must satisfy [130],

$$Y\_{1,2}^{D} = Y\_{1,2}^{D\dagger} \quad \text{and} \quad Y\_L = Y\_R. \tag{3.25}$$

Such relationships in the LRSM remove the arbitrariness of neutrino Dirac mass matrices, as discussed in section 3.1.1, and permits one to calculate , even for nonzero 1<sup>L</sup> vev [129, 131]. However, the potential cancellation between Type I and II Seesaw masses in Equation 3.20 still remains.

In addition to the canonical formulation of the LRSM are several alternatives. For example: It is possible to instead generate LH and RH Majorana neutrino masses radiatively in the absence of triplet scalars [132, 133]. One can gauge baryon number and lepton number independently, which, for an anomaly-free theory, gives rise to vector-like leptons and a Type III Seesaw mechanism [134, 135] (see section 5), as well as embed the model into an R-parity-violating Supersymmetric framework [136, 137].

Despite the large scalar sector of the LRSM (two complex triplets and one complex bidoublet), and hence a litany of neutral and charged Higgses, the symmetry structure in Equation (3.23) confines the number in independent degrees of freedom to 18 [122, 138]. These consist of three mass scales µ1,...,3, 14 dimensionless couplings λ1,...,4, ρ1,...,4, α1,...,3, β1,...,3, and one CPviolating phase, δ2. For further discussions on the spontaneous breakdown of CP in LR scenarios, see also Senjanovic [ ´ 121], Basecq et al. [139], and Kiers et al. [140]. With explicit CP conservation, the minimization conditions on the scalar potential give rise to the so-called LRSM vev Seesaw relationship [138],

$$\nu\_{\rm L} = \frac{\beta\_2 k\_1^2 + \beta\_1 k\_1 k\_2 + \beta\_3 k\_2^2}{(2\rho\_1 - \rho\_3)\nu\_{\rm R}},\tag{3.26}$$

where, vL,<sup>R</sup> and k1,2 are the vevs of 1L,<sup>R</sup> and the Higgs bidoublet 8, respectively, with v 2 <sup>L</sup> ≪ k 2 <sup>1</sup> + k 2 <sup>2</sup> <sup>≈</sup> (246 GeV)<sup>2</sup> <sup>≪</sup> <sup>v</sup>R.

In the LRSM, the bidoublet 8 fulfills the role of the SM Higgs to generate the known Dirac masses of elementary fermions and permits a neutral scalar h<sup>i</sup> with mass mh<sup>i</sup> ≈ 125 GeV and SM-like couplings. In the absence of egregious fine-tuning, i.e., ρ<sup>3</sup> 6≈ 2ρ1, Equation (3.26) suggests that v<sup>L</sup> in the LRSM is inherently small because, in addition to k1, k<sup>2</sup> ≪ vR, custodial symmetry is respected (up to hypercharge corrections) when all β<sup>i</sup> are identically zero [141]. Consistent application of such naturalness arguments reveals a lower bound on the scalar potential parameters [141],

$$\begin{split} \rho\_{1,2,4} &> \frac{\mathcal{g}\_R^2}{4} \left( \frac{m\_{\text{FCNH}}}{M\_{W\_R}} \right)^2, \\ \rho\_3 &> \mathcal{g}\_R^2 \left( \frac{m\_{\text{FCNH}}}{M\_{W\_R}} \right)^2 + 2\rho\_1 \sim 6\rho\_1, \\ \alpha\_{1,\dots,3} &> \mathcal{g}\_R^2 \left( \frac{m\_{\text{FCNH}}}{M\_{W\_R}} \right)^2, \\ \mu\_{1,2}^2 &> \{m\_{\text{FCNH}}\}^2, \quad \mu\_3^2 > \frac{1}{2} (m\_{\text{FCNH}})^2, \end{split} \tag{3.28}$$

where MW<sup>R</sup> and g<sup>R</sup> are the mass and coupling of the W<sup>±</sup> R gauge boson associated with SU(2)R, and mFCNH is the mass scale of the LRSM scalar sector participating in flavor-changing neutral transitions. Present searches for neutron EDMs [125, 126, 142, 143] and FCNCs [143–147] require mFCNH > 10 − 20 TeV at 90% CL. Subsequently, in the absence of FCNC-suppressing mechanisms, ρ<sup>i</sup> > 1 for LHC-scale WR. Thus, discovering LRSM at the LHC may suggest a strongly coupled scalar sector. Conversely, for ρ<sup>i</sup> < 1 and mFCNH ∼ 15 (20) TeV, one finds MW<sup>R</sup> & 10 (12) TeV, scales that are within the reach of future hadron colliders [141, 148, 149]. For more detailed discussions on the perturbativity and stability of the LRSM scalar section, see Mitra et al. [141], Maiezza et al. [146], Bertolini et al. [150–152], Mohapatra and Zhang [153], and Maiezza and Senjanovic [ ´ 154] and references therein.

After 1<sup>R</sup> acquires a vev and LR symmetry is broken spontaneously, the neutral component of SU(2)R, i.e., W<sup>3</sup> R , and the U(1)B−<sup>L</sup> boson, i.e., XB−L, mix into the massive eigenstate Z ′ LRSM (sometimes labeled ZR) and the orthogonal, massless vector boson B. B is recognized as the gauge field associated with weak hypercharge in the SM, the generators of which are built from the remnants of SU(2)<sup>R</sup> and U(1)B−L. The relation between electric charge Q, weak left/right isospin T 3 L/R , baryon minus lepton number B-L, and weak hypercharge Y is given by

$$Q = T\_L^3 + T\_R^3 + \frac{\{B - L\}}{2} \equiv T\_L^3 + \frac{Y}{2}, \quad \text{with} \quad Y = 2T\_R^3 + (B - L). \tag{3.29}$$

This in turn implies that the remaining components of SU(2)R, W<sup>1</sup> R and W<sup>2</sup> R , combine into the state W± <sup>R</sup> with electric charge Q <sup>W</sup><sup>R</sup> = ±1 and mass <sup>M</sup>W<sup>R</sup> <sup>=</sup> <sup>g</sup>RvR/ √ 2. After EWSB, it is possible for the massive W<sup>R</sup> and W<sup>L</sup> gauge fields to mix, with the mixing angle ξLR given by tan 2ξLR = 2k1k2/(v 2 <sup>R</sup> − v 2 L ) . 2v 2 SM/v 2 R . Neutral meson mass splittings [124, 147, 155–158] coupled with improved lattice calculations, e.g., [159, 160], Weak CPV [124, 158, 161], EDMs [124–126, 158], and CP violation in the electron EDM [129], are particularly sensitive to this mixing, implying the competitive bound of MW<sup>R</sup> & 3 TeV at 95% CL [147]. This forces W<sup>L</sup> − W<sup>R</sup> mixing to be, tan 2ξLR/2 ≈ ξLR . M<sup>2</sup> W/M<sup>2</sup> WR <sup>&</sup>lt; <sup>7</sup> <sup>−</sup> 7.5 <sup>×</sup> <sup>10</sup>−<sup>4</sup> . A similar conclusion can be reached on Z − Z ′ LRSM mixing. Subsequently, the light and heavy mass eigenstates of LRSM gauge bosons, W± 1 , W± 2 , Z1, Z2, where MV<sup>1</sup> < MV<sup>2</sup> , are closely aligned with their gauge states. In other words, to a very good approximation, W<sup>1</sup> ≈ WSM, Z<sup>1</sup> ≈ ZSM, W<sup>2</sup> ≈ W<sup>R</sup> and Z ′ ≈ Z ′ LRSM (or sometimes Z ′ ≈ ZR). The mass relation between the LR gauge bosons is MZ<sup>R</sup> = p 2 cos<sup>2</sup> θW/ cos 2θWMW<sup>R</sup> ≈ (1.7) × MW<sup>R</sup> , and implies that bounds on one mass results in indirect bounds on the second mass; see, for example, Lindner et al. [162].

#### 3.1.5. Heavy Neutrino Effective Field Theory

It is possible that the coupling of TeV-scale Majorana neutrinos to the SM sector is dominated by new states with masses that are hierarchically larger than the heavy neutrino mass or the reach of present-day collider experiments. For example: Scalar SU(2)<sup>R</sup> triplets in the Left-Right Symmetric Model may acquire vevs O(10) TeV, resulting in new gauge bosons that are kinematically accessible at the LHC but, due to <sup>O</sup>(10−<sup>3</sup> <sup>−</sup> 10−<sup>2</sup> ) triplet Yukawa couplings, give rise to EW-scale RH Majorana neutrino masses. In such a pathological but realistic scenario, the LHC phenomenology appears as a canonical Type I Seesaw mechanism despite originating from a different Seesaw mechanism [163]. While it is generally accepted that such mimicry can occur among Seesaws, few explicit examples exist in the literature and further investigation is encouraged.

For such situations, it is possible to parameterize the effects of super-heavy degrees of freedom using the Heavy Neutrino Effective Field Theory (NEFT) framework [164]. NEFT is an extension of the usual SM Effective Field Theory (SMEFT) [165– 168], whereby instead of augmenting the SM Lagrangian with higher dimension operators one starts from the Type I Seesaw Lagrangian in Equation (3.1) and builds operators using that field content. Including all SU(3) ⊗ SU(2)<sup>L</sup> ⊗ U(1)Y-invariant, operators of mass dimension d > 4, the NEFT Lagrangian before EWSB is given by

$$\mathcal{L}\_{\text{NFET}} = \mathcal{L}\_{\text{Type I}} + \sum\_{d=5} \sum\_{i} \frac{\alpha\_i^{(d)}}{\Lambda^{(d-4)}} \mathcal{O}\_i^{(d)}.\tag{3.30}$$

Here, O (d) i are dimension d, Lorentz and gauge invariant permutations of Type I fields, and α (d) <sup>i</sup> ≪ 4π are the corresponding Wilson coefficients. The list of O (d) i are known explicitly for d = 5 [169, 170], 6 [164, 170], and 7 [170–172], and can be built for larger d following [173–175].

After EWSB, fermions should then be decomposed into their mass eigenstates via quark and lepton mixing. For example: among the <sup>d</sup> <sup>=</sup> 6, four-fermion contact operations <sup>O</sup> (6) i that contribute to heavy N production in hadron colliders (see Equation 3.33) in the interaction/gauge basis are [164]

$$\mathcal{O}\_V^{(6)} = \left(\overline{d}\boldsymbol{\gamma}^\mu P\_R \boldsymbol{\mu}\right) \left(\overline{e}\boldsymbol{\gamma}\_\mu P\_R \boldsymbol{N}\_R\right) \quad \text{and}$$

$$\mathcal{O}\_{S3}^{(6)} = \left(\overline{Q}\boldsymbol{\gamma}^\mu P\_R \boldsymbol{N}\_R\right) \boldsymbol{\varepsilon} \left(\overline{L}\boldsymbol{\gamma}\_\mu P\_R d\right). \tag{3.31}$$

In terms of light (νm) and heavy (Nm′) mass eigenstates and using Equation (3.4), one can generically [66, 87] decompose the heavy neutrino interaction state N<sup>ℓ</sup> as N<sup>ℓ</sup> = P<sup>3</sup> <sup>m</sup>=<sup>1</sup> <sup>X</sup>ℓm<sup>ν</sup> c <sup>P</sup> <sup>m</sup> <sup>+</sup> n <sup>m</sup>′=<sup>1</sup> <sup>Y</sup>ℓNm′ <sup>N</sup>m′ , with <sup>|</sup>YℓNm′ | of order the elements of UPMNS. Inserting this into the preceding operators gives quantities in terms of leptonic mass eigenstates:

$$\begin{split} \mathcal{O}\_{V}^{(6)} &= \sum\_{m=1}^{3} \left( \overline{d} \gamma^{\mu} P\_{R} \mu \right) \left( \overline{\ell} \gamma\_{\mu} P\_{R} \, X\_{\ell m} \, \upsilon\_{m}^{c} \right) \\ &+ \sum\_{m'=1} \left( \overline{d} \gamma^{\mu} P\_{R} \mu \right) \left( \overline{\ell} \gamma\_{\mu} P\_{R} \, \, Y\_{\ell N\_{m'}} \, N\_{m'} \right), \quad \text{and} \\ \mathcal{O}\_{S}^{(6)} &= \sum\_{m=1}^{3} \left( \overline{\mathbb{Q}} \gamma^{\mu} P\_{R} \, X\_{\ell m} \upsilon\_{m}^{c} \right) \left( \overline{\ell} \gamma\_{\mu} P\_{R} d \right) \\ &+ \sum\_{m'=1} \left( \overline{\mathbb{Q}} \gamma^{\mu} P\_{R} \, Y\_{\ell N\_{m'}} N\_{m'} \right) \left( \overline{\ell} \gamma\_{\mu} P\_{R} d \right). \end{split} \tag{3.32}$$

After EWSB, a similar decomposition for quarks gauge states in terms of CKM matrix elements and mass eigenstates should be applied. For more information on such decompositions, see, e.g., Ruiz [163] and references therein. It should be noted that after integrating out the heavy N field, the marginal operators at d > 5 generated from the Type I Lagrangian are not the same operators generated by integrating the analogous Seesaw partner in the Type II and III scenarios [176, 177].

#### 3.2. Heavy Neutrinos at Colliders

The connection between low-scale Seesaw models and colliders is made no clearer than in searches for heavy neutrinos, both Majorana and (pseudo-)Dirac, in the context of Type Ibased scenarios. While extensive, the topic's body of literature is still progressing in several directions. This is particularly true for the development of collider signatures, Monte Carlo tools, and high-order perturbative corrections. Together, these advancements greatly improve sensitivity to neutrinos and their mixing structures at collider experiments.

We now review the various searches for L-violating collider processes facilitated by Majorana neutrinos N. We start with low-mass (section 3.2.1) and high-mass (sections 3.2.2 and 3.2.3) neutrinos in the context of Type I-based hybrid scenarios, before moving onto Abelian (section 3.2.4) and non-Abelian (section 3.2.5) gauge extensions, and finally the semi-model independent NEFT framework (section 3.2.6). Lepton number violating collider processes involving pseudo-Dirac neutrinos are, by construction, suppressed [178–181]. Thus, a discussion of their phenomenology is outside the scope of this review and we refer readers to thorough reviews such as Ibarra et al. [94], Weiland [182], and Antusch et al. [183].

#### 3.2.1. Low-Mass Heavy Neutrinos at pp and ee Colliders

For Majorana neutrinos below the M<sup>W</sup> mass scale, lepton number violating processes may manifest in numerous way, including rare decays of mesons, baryons, µ and τ leptons, and even SM electroweak bosons. Specifically, one may discover L violation in three-body meson decays to lighter mesons M± 1 → M∓ 2 ℓ ± 1 ℓ ± 2 [66, 184–199], such as that shown in **Figure 5A**; fourbody meson decays to lighter mesons M± <sup>1</sup> → M<sup>∓</sup> <sup>2</sup> <sup>M</sup><sup>0</sup> 3 ℓ ± 1 ℓ ± 2 [195, 196, 200–202]; four-body meson decays to leptons M<sup>±</sup> → ℓ ± 1 ℓ ± 1 ℓ ∓ 2 ν [192, 193, 202–204]; five-body meson decays [202]; four-body baryon decays to mesons, B → Mℓ ± 1 ℓ ± 2 [205]; threebody τ decay to mesons, τ <sup>±</sup> → ℓ ∓M± <sup>1</sup> M<sup>±</sup> 2 [195, 206, 207]; four-body τ decays to mesons, τ <sup>±</sup> → ℓ ± 1 ℓ ± <sup>1</sup> M∓ν [195, 206, 208– 210]; four-body W boson decays, W<sup>±</sup> → ℓ ± 1 ℓ ± 1 ℓ ∓ 2 ν [211– 215]; Higgs boson decays, h → NN → ℓ ± 1 ℓ ± <sup>2</sup> + X [216– 219]. and even top quark decays, t → bW+∗ → bℓ + <sup>1</sup> N → bℓ + 1 ℓ ± 2 qq ′ [7, 211, 220, 221]. The W boson case is notable as azimuthal and polar distributions [87] or exploiting endpoint kinematics [214] can differentiate between L conservation and non-conservation. Of the various collider searches for GeVscale N, great complementarity is afforded by B-factories. As shown in **Figure 5B**, an analysis of Belle I [45] and LHCb Run I [46, 47] searches for L-violating final states from meson decays excluded [222] |VµN| <sup>2</sup> & <sup>3</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> for M<sup>N</sup> = 1 − 5 GeV. Along these same lines, the observability of displaced decays of heavy neutrinos [217, 223–227] and so-called "neutrino-antineutrino oscillations" [228–231] (in analogy to <sup>B</sup>−<sup>B</sup> oscillations) and have also been discussed.

Indirectly, the presence of heavy Majorana neutrinos can appear in precision EW measurements as deviations from lepton flavor unitarity and universality, and is ideally suited for e +e − colliders [88–91, 183, 232, 233], such as the International Linear Collider (ILC) [234, 235], Circular e −e + Collider (CepC) [236], and Future Circular Collider-ee (FCC-ee) [232]. An especially famous example of this is the number of active, light neutrino flavors Nν , which can be inferred from the Z boson's invisible width Ŵ Z Inv. At lepton colliders, Ŵ Z Inv can be determined in two different ways: The first is from line-shape measurements of the Z resonance as a function of √ s, and is measured to be N Line <sup>ν</sup> = 2.9840 ± 0.0082 [237]. The second is from searches for invisible Z decays, i.e., , e +e <sup>−</sup> → Zγ , and is found to be N Inv ν = 2.92±0.05 [238]. Provocatively, both measurements deviate from the SM prediction of N SM <sup>ν</sup> = 3 at the 2σ level. It is unclear if deviations from N SM ν are the result of experimental uncertainty or indicate the presence of, for example, RH neutrinos [224, 239]. Nonetheless, a future Z-pole machine can potentially clarify this discrepancy [224]. For investigations into EW constraints on heavy neutrinos, see del Águila et al. [88], Antusch and Fischer [89], de Gouvêa and Kobach [90], and Fernandez-Martinez et al. [91].

of <sup>N</sup> mass after <sup>L</sup> <sup>=</sup> 3 fb−<sup>1</sup> at 7-8 TeV LHC [222].

#### 3.2.2. High-Mass Heavy Neutrinos at pp Colliders

Collider searches for heavy Majorana neutrinos with masses above M<sup>W</sup> have long been of interest to the community [240– 243], with exceptionally notable works appearing in the early 1990s [96, 244–247] and late-2000s [66, 97, 248–253]. In the past decade, among the biggest advancements in Seesaw phenomenology is the treatment of collider signatures for such hefty N in Type I-based models. While coupled to concurrent developments in Monte Carlo simulation packages, the progression has been driven by attempts to reconcile conflicting reports of heavy neutrino production cross sections for the LHC. This was at last resolved in Alva et al. [254] and Degrande et al. [255], wherein new, infrared- and collinear- (IRC-)safe definitions for inclusive and semi-inclusive<sup>4</sup> production channels were introduced. The significance of such collider signatures is that they are well-defined at all orders in α<sup>s</sup> , and hence correspond to physical observables. We now summarize this extensive body of literature, emphasizing recent results.

For Majorana neutrinos with M<sup>N</sup> > MW, the most extensively studied [66, 105, 183, 230, 240, 241, 246, 248–253, 256] collider production mechanism is the L-violating, charged current (CC) Drell-Yan (DY) process [240], shown in **Figure 6A**, and given by

$$q\_1 \ \overline{q}\_2 \to W^{\pm \*} \to N \ \ell\_1^{\pm}, \quad \text{with} \quad N \to \ell\_2^{\pm} \\
W^{\mp} \to \ell\_2^{\pm} q\_1' \ \overline{q'}\_2. \tag{3.33}$$

A comparison of **Figure 6A** to the meson decay diagram of **Figure 5A** immediately reveals that Equation (3.33) is the former's high momentum transfer completion. Subsequently, much of the aforementioned kinematical properties related to L-violating meson decays also hold for the CC DY channel [87, 257]. Among the earliest studies are those likewise focusing on neutral current (NC) DY production [241, 242, 245–247], again shown in **Figure 6A**, and given by

$$q \ \overline{q} \to Z^\* \to N \stackrel{(-)}{\nu\_\ell}, \tag{3.34}$$

as well as the gluon fusion mechanism [242, 245], shown in **Figure 6B**, and given by

$$\text{gg g} \to Z^\*/h^\* \to N\stackrel{(-)}{\nu\_\ell}.\tag{3.35}$$

Interestingly, despite gluon fusion being formally an O(α 2 s ) correction to Equation (3.34), it is non-interfering, separately gauge invariant, and the subject of renewed interest [255, 258, 259]. Moreover, in accordance to the Goldstone Equivalence Theorem [260, 261], the ggZ∗ contribution has been shown [258, 259] to be as large as the ggh∗ contribution, and therefore should not be neglected. Pair production of N via s-channel scattering [242, 246], e.g., gg → NN, or weak boson scattering [244, 247, 248], e.g., W±W<sup>∓</sup> → NN, have also been discussed, but are relatively suppressed compared to single production by an additional mixing factor of |VℓNm′ | <sup>2</sup> . 10−<sup>4</sup> .

A recent, noteworthy development is the interest in semiinclusive and exclusive production of heavy neutrinos at hadron colliders, i.e., , N production in association with jets. In particular, several studies have investigated the semi-inclusive, photoninitiated vector boson fusion (VBF) process [247, 254, 255, 262], shown in **Figure 6C**, and given by

$$q \; \text{y} \to \text{N} \; \ell^{\pm} \; q', \tag{3.36}$$

and its deeply inelastic, O(α) radiative correction [247, 254, 255, 262–266],

$$q\_1 \ q\_2 \xrightarrow{W\mathcal{Y} + W\mathcal{Z} \to N\ell^{\pm}} N \; \ell^{\pm} \; q\_1' \; q\_2'.\tag{3.37}$$

At O(α 4 ) (here we do not distinguish between α and αW), the full, gauge invariant set of diagrams, which includes the sub-leading W±Z → Nℓ ± scattering, is given in **Figure 7**.

Treatment of the VBF channel is somewhat subtle in that it receives contributions from collinear QED radiation off the proton [262], collinear QED radiation off initial-states quarks [254], and QED radiation in the deeply inelastic/high momentum transfer limit [247]. For example: In the top line

<sup>4</sup> A note on terminology: High-p<sup>T</sup> hadron collider observables, e.g., fiducial distributions, are inherently inclusive with respect to jets with arbitrarily low pT. In this sense, we refer to hadronic-level processes with a fixed multiplicity of jets satisfying kinematical requirements (and with an arbitrary number of additional jets that do not) as exclusive, e.g., pp → W<sup>±</sup> + 3j + X; those with a minimum multiplicity meeting these requirements are labeled semi-inclusive, e.g., pp → W±+ ≥ 3j + X; and those with an arbitrary number of jets are labeled inclusive, e.g., pp → W<sup>±</sup> + X. Due to DGLAP-evolution, exclusive, partonic amplitudes convolved with PDFs are semi-inclusive at the hadronic level.

drawn using JaxoDraw [267].

of diagrams in **Figure 7**, one sees that in the collinear limit of the q<sup>2</sup> → γ ∗ q ′ 2 splitting, the virtual γ ∗ goes on-shell and the splitting factorizes into a photon parton distribution function (PDF), recovering the process in Equation (3.36) [254, 255]. As these sub-channels are different kinematic limits of the same process, care is needed when combining channels so as to not double count regions of phase space. While ingredients to the VBF channel have been known for some time, consistent schemes to combine/match the processes are more recent [254, 255]. Moreover, for inclusive studies, Degrande et al. [255] showed that the use of Equation (3.36) in conjunction with a γ -PDF containing both elastic and inelastic contributions [268] can reproduce the fully matched calculation of Ref. [254] within the O(20%) uncertainty resulting from missing NLO in QED terms. Neglecting the collinear q<sup>2</sup> → γ ∗ q ′ 2 splitting accounts for the unphysical cross sections reported in Deppisch et al. [67] and Dev et al. [262] . Presently, recommended PDF sets containing such γ -PDFs include: MMHT QED (no available lhaid) [268, 269], NNPDF 3.1+LUXqed (lhaid=324900) [270], LUXqed17+PDF4LHC15 (lhaid=82200) [271, 272], and CT14 QED Inclusive (lhaid = 13300) [273]. Qualitatively, the MMHT [268] and LUXqed [271, 272] treatments of photon PDFs are the most rigorous. In analogy to the gluon fusion and

NC DY, Equation (3.36) (and hence Equation 3.37) is a noninterfering, O(α) correction to the CC DY process. Thus, the CC DY and VBF channels can be summed coherently.

In addition to these channels, the semi-inclusive, associated n-jet production mode,

$$p\,p \to W^\* + \begin{array}{c} \neg nj + X \ \rightarrow \quad N\,\ell^{\pm} + \; \ge nj + X, \quad \text{for} \quad n \in \mathbb{N}, \tag{3.38}$$

has also appeared in the recent literature [255, 262, 274]. As with VBF, much care is needed to correctly model Equation (3.38). As reported in Degrande et al. [255] and Ruiz [275], the production of heavy leptons in association with QCD jets is nuanced due to the presence of additional t-channel propagators that can lead to artificially large cross sections if matrix element poles are not sufficiently regulated. (It is not enough to simply remove the divergences with phase space cuts). After phase space integration, these propagators give rise to logarithmic dependence on the various process scales. Generically Ruiz [275] and Collins et al. [276], the cross section for heavy lepton and jets in Equation (3.38) scales as:

$$\sigma(\not\!p p \to N\ell^{\pm} + \eta j + X) \sim \sum\_{k=1}^{n} \alpha\_s^k(Q^2) \log^{(2k-1)}\left(\frac{Q^2}{q\_T^2}\right), \text{ (3.39)}$$

Here, Q ∼ M<sup>N</sup> is the scale of the hard scattering process, q<sup>T</sup> = p |EqT| 2 , and qE<sup>T</sup> ≡ P<sup>n</sup> k pE j T,k , is the (Nℓ)-system's transverse momentum, which recoils against the vector sum of all jet pET. It is clear for a fixed M<sup>N</sup> that too low jet p<sup>T</sup> cuts can lead to too small q<sup>T</sup> and cause numerically large (collinear) logarithms such that log(M<sup>2</sup> N /q 2 T ) ≫ 1/αs(Q), spoiling the perturbative convergence of Equation (3.39). Similarly, for a fixed qT, arbitrarily large M<sup>N</sup> can again spoil perturbative convergence. As noted in Alva et al. [254] and Degrande et al. [255], neglecting this fact has led to conflicting predictions in several studies on heavy neutrino production in pp collisions.

It is possible [255], however, to tune p<sup>T</sup> cuts on jets with varying M<sup>N</sup> to enforce the validity of Equation (3.39). Within the Collins-Soper-Sterman (CSS) resummation formalism [276], Equation (3.39) is only trustworthy when αs(Q 2 ) is perturbative and q<sup>T</sup> ∼ Q, i.e.,

$$
\log(Q/\Lambda\_{\rm QCD}) \gg 1 \quad \text{and} \quad \alpha\_{\rm s}(Q)\log^2(Q^2/q\_T^2) \lesssim 1. \tag{3.40}
$$

Noting that at 1-loop αs(Q) can be expressed by 1/αs(Q) ≈ (β0/2π) log(Q/3QCD), and setting Q = MN, one can invert the second CSS condition and obtain a consistency relationship [255]:

$$qr = |\vec{q}r| = \left| \sum\_{k=1}^{n} \vec{p}\_{T,k}^{j} \right| \gtrsim M\_N \times e^{-(1/2)\sqrt{(\beta\_0/2\pi)\log(M\_N/\Lambda\_{\text{QCD}})}}.\tag{3.41}$$

This stipulate a minimum q<sup>T</sup> needed for semi-inclusive processes like Equation (3.39) to be valid in perturbation theory. When q<sup>T</sup> of the (Nℓ)-system is dominated by a single, hard radiation, Equation (3.41) is consequential: In this approximation, q<sup>T</sup> ≈ |pE j <sup>T</sup>,1| and Equation (3.41) suggests a scale-dependent, minimum jet p<sup>T</sup> cut to ensure that specifically the semi-inclusive pp → Nℓ+ ≥ 1j + X cross section is well-defined in perturbation theory. Numerically, this is sizable: for M<sup>N</sup> = 30 (300) [3000] GeV, one requires that |pE j <sup>T</sup>,1| & 9 (65) [540] GeV, or alternatively |pE j <sup>T</sup>,1| & 0.3 (0.22) [0.18] × MN, and indicates that naïve application of fiducial p j T cuts for the LHC do not readily apply for √ s = 27-100 TeV scenarios, where one can probe much larger MN. The perturbative stability of this approach is demonstrated by the (roughly) flat K-factor of K NLO <sup>≈</sup> 1.2 for the semi-inclusive pp <sup>→</sup> <sup>N</sup><sup>ℓ</sup> <sup>±</sup> + 1j process, shown in the lower panel of **Figure 8A**. Hence, the artificially large N production cross sections reported in Deppisch et al. [67], Dev et al. [262], and Das et al. [274] can be attributed to a loss of perturbative control over their calculation, not the presence of an enhancement mechanism. Upon the appropriate replacement of MN, Equation (3.41) holds for other color-singlet processes [255], including mono-jet searches, and is consistent with explicit p<sup>T</sup> resummations of high-mass lepton [275] and slepton [277, 278] production.

A characteristic of heavy neutrino production cross sections is that the active-sterile mixing, |VℓN|, factorizes out of the partonic and hadronic scattering expressions. Exploiting this one can define [248] a "bare" cross section σ0, given by

$$
\sigma\_0(\mathfrak{pp} \to N + X) \equiv \sigma(\mathfrak{pp} \to N + X) / |V\_{\ell N}|^2. \tag{3.42}
$$

Assuming resonant production of N, a similar expression can be extracted at the N decay level,

$$\sigma\_0(\not p \to \ell\_1^{\pm} \ell\_2^{\pm} + X) \equiv \sigma(\not p \to \ell\_1^{\pm} \ell\_2^{\pm} + X) / \mathcal{S}\_{\ell\_1 \ell\_2},$$

$$\mathcal{S}\_{\ell\_1 \ell\_2} = \frac{|V\_{\ell\_1 N}|^2 |V\_{\ell\_2 N}|^2}{\sum\_{\ell=\pm}^{\mathfrak{r}} |V\_{\ell N}|^2}. \tag{3.43}$$

These definitions, which hold at all orders in α<sup>s</sup> [255, 275], allow one to make cross section predictions and comparisons independent of a particular flavor model, including those that largely conserve lepton number, such as the inverse and linear Seesaws. It also allows for a straightforward reinterpretation of limits on collider cross sections as limits on Sℓ1ℓ<sup>2</sup> , or |VℓN| with additional but generic assumptions. An exception to this factorizablity is the case of nearly degenerate neutrinos with total widths that are comparable to their mass splitting [228, 249, 279, 280].

**Figure 8** shows a comparison of the leading, single N hadronic production cross sections, divided by active-heavy mixing |VℓN| 2 , as a function of (a) heavy neutrino mass M<sup>N</sup> at √ <sup>s</sup> <sup>=</sup> 14 [255] and (b) collider energy <sup>√</sup> s up to 100 TeV for M<sup>N</sup> = 500, 1000 GeV [259]. The various accuracies reported reflect the maturity of modern Seesaw calculations. Presently, state-of-the-art predictions for single N production modes are automated up to NLO+PS in QCD for the Drell-Yan and VBF channels [255, 281], amongst others, and known up to N3LL(threshold) for the gluon fusion channel [259]. With Monte Carlo packages, predictions are available at LO with multi-leg merging (MLM) [251, 255, 282, 283] as well as up to NLO with parton shower matching and merging [255, 283]. The NLO accurate [284], HeavyNnlo universal FeynRules object (UFO) [285] model file is available from Degrande et al. [255, 283]. Model files built using FeynRules [285– 287] construct and evaluate L-violating currents following the Feynman rules convention of Denner et al. [288]. A brief comment is needed regarding choosing MLM+PS or NLO+PS computations: To produce MLM Monte Carlo samples, one must sum semi-inclusive channels with successively higher leg multiplicities in accordance with Equations (3.39)–(3.41) and correct for phase space double-counting. However, such MLM samples are formally LO in O(αs) because of missing virtual corrections. NLO+PS is formally more accurate, under better perturbative control, and thus is recommended for modeling heavy N at colliders. Such computations are possible with modern, general-purpose event generators, such as Herwig [289], MadGraph5\_aMC@NLO [290], and Sherpa [291].

At the 13 and 14 TeV LHC, heavy N production is dominated by charged-current mechanisms for phenomenologically relevant mass scales, i.e., M<sup>N</sup> . 700 GeV [254]. At more energetic colliders, however, the growth in the gluon-gluon luminosity increases the gg → Nν cross section faster than the CC DY channel. In particular, at √ s = 20 − 30 TeV, neutralcurrent mechanisms surpass charged-current modes for heavy

N production with M<sup>N</sup> = 500 − 1000 GeV [259]. As seen in the sub-panel of **Figure 8A**, NLO in QCD contributions only modify inclusive, DY-type cross section normalizations by +20 to +30% and VBF negligibly, indicating that the prescriptions of Degrande et al. [255] are sufficient to ensure perturbative control over a wide-range of scales. One should emphasize that while VBF normalizations do not appreciably change under QCD corrections [292], VBF kinematics do change considerably [255, 293–295]. The numerical impact, however, is observable-dependent and can be large if new kinematic channels are opened at higher orders of α<sup>s</sup> . In comparison to this, the sub-panel of **Figure 8B** shows that QCD corrections to gluon fusion are huge (+150 to +200%), but convergent and consistent with SM Higgs, heavy Higgs, and heavy pseudoscalar production [296–298]; for additional details, see Ruiz et al. [259].

With these computational advancements, considerable collider sensitivity to L-violating processes in the Type I Seesaw has been reached. In **Figure 9** is the expected sensitivity to active-sterile neutrino mixing via the combined CC DY+VBF channels and in same-sign µ ±µ <sup>±</sup> <sup>+</sup> <sup>X</sup> final-state. With <sup>L</sup> <sup>=</sup> <sup>1</sup> ab−<sup>1</sup> of data for <sup>M</sup><sup>N</sup> <sup>&</sup>gt; <sup>M</sup><sup>W</sup> at <sup>√</sup> s = 14 (100) TeV, one can exclude at 2σ Sµµ ≈ |VµN| <sup>2</sup> & 10−<sup>4</sup> (10−<sup>5</sup> ) [254]. This is assuming the 2013 Snowmass benchmark detector configuration for √ s = 100 TeV [299]. Sensitivity to the e ±e ± and e ±µ ± channels is comparable, up to detector (in)efficiencies for electrons and muons. As shown in **Figure 10**, with <sup>L</sup> <sup>≈</sup> <sup>20</sup> fb−<sup>1</sup> at 8 TeV, the ATLAS and CMS experiments have excluded at 95% CLs |VℓN| <sup>2</sup> & <sup>10</sup>−<sup>3</sup> <sup>−</sup> <sup>10</sup>−<sup>1</sup> for M<sup>N</sup> = 100 − 450 GeV [48–52]. For heavier MN, quarks from the on-shell W boson decay can form a single jet instead of the usual two-jet configuration. In such cases, well-known "fat jet" techniques can be used [300, 301]. Upon discovery of L-violating processes involving heavy neutrinos, among the most pressing quantities to measure are N's chiral couplings to other fields [87, 257], its flavor structure [129, 228, 230, 256], and a potential determination if the signal is actually made of multiple, nearly degenerate N [105, 229].

#### 3.2.3. High-Mass Heavy Neutrinos at ep Colliders

Complementary to searches for L violation in pp collisions are the prospects for heavy N production at ep deeply inelastic scattering (DIS) colliders [183, 302–309], such as proposed Large Hadron-electron Collider (LHeC) [310], or a µp analog [304]. As shown in **Figure 10**, DIS production of Majorana neutrinos can occur in multiple ways, including (a) W exchange and (b) Wγ fusion. For treatment of initial-state photons from electron beams, see Frixione et al. [311]. Search strategies for Majorana neutrinos at DIS experiments typically rely on production via the former since eγ → NW associated production can suffer from large phase space suppression, especially at lower beam energies. On the other hand, at higher beam energies, the latter process can provide additional polarization information on N and its decays [183].

At DIS facilities, one usually searches for L violation by requiring that N decays to a charged lepton of opposite sign from the original beam configuration, i.e.,

$$\ell\_1^{\pm} \not{q}\_i \to N \not{q}\_f, \quad \text{with} \quad N \to \ell\_2^{\mp} \not{W}^{\pm} \to \ell\_2^{\mp} \not{q} \overline{\not{q}}', \tag{3.44}$$

which is only possible of N is Majorana and is relatively free of SM backgrounds: As in the pp case, the existence of a highp<sup>T</sup> charged lepton without accompanying MET (at the partonic level) greatly reduces SM backgrounds. At the hadronic level, this translates to requiring one charged lepton and three highp<sup>T</sup> jets: two that arise from the decay of N, which scale as p j <sup>T</sup> ∼ MN/4, and the third from the W exchange, which scales as p j <sup>T</sup> ∼ MW/2. However, it was recently noted [312] that tagging this third jet is not necessary to reconstruct and identify the heavy neutrino, and that a more inclusive search may prove more sensitive. Although Equation (3.44) represents the so-called "golden channel," searches for N → Z/h + ν decays, but

FIGURE 9 | At 14 TeV and as a function of <sup>M</sup>N, (A) the 2<sup>σ</sup> sensitivity to <sup>S</sup>ℓℓ′ for the pp <sup>→</sup> <sup>µ</sup> ±µ <sup>±</sup> + X process. (B) The required luminosity for a 3 (dash-circle) and 5σ (dash-star) discovery in the same channel (C,D) Same as (A,B) but for 100 TeV [254].

which do not manifestly violate lepton number, have also been proposed [308].

While the lower beam energies translate to a lower mass reach for MN, large luminosity targets and relative cleaner hadronic environment result in a better sensitivity than the LHC to smaller active-sterile mixing for smaller neutrino Majorana masses. In **Figure 11**, one sees the expected 90% CL active-sterile mixing |θ| 2 (or |VℓN| 2 ) sensitivity assuming (c) ep configuration with E<sup>e</sup> = 150 GeV and (d) µp configuration with E<sup>µ</sup> = 2 TeV. For <sup>L</sup> <sup>∼</sup> <sup>O</sup>(100) fb−<sup>1</sup> , one can probe |VℓN| <sup>2</sup> <sup>∼</sup> <sup>10</sup>−<sup>5</sup> <sup>−</sup> <sup>10</sup>−<sup>3</sup> for M<sup>N</sup> = 250 − 750 GeV [304].

#### 3.2.4. Heavy Neutrinos and U(1)<sup>X</sup> Gauge Extensions at Colliders

Due to the small mixing between the heavy neutrinos and the SM leptons in minimal Type I Seesaw scenarios, typically of the order |VℓN| <sup>2</sup> <sup>∼</sup> <sup>O</sup>(m<sup>ν</sup> /MN), the predicted rates for collider-scale lepton number violation is prohibitively small. With a new gauge

FIGURE 11 | Born diagrams for DIS heavy neutrino (N) production via (A) W-exchange and (B) Wγ fusion. 90% CL active-sterile mixing |θ| (or |VℓN| ) sensitivity vs. integrated luminosity at DIS experiment assuming (C) ep configuration with E<sup>e</sup> = 150 GeV and (D) µp configuration with E<sup>µ</sup> = 2 TeV; red (blue) [black] line in (C,D) correspond to M<sup>N</sup> = 250 (500) [750] GeV, whereas the solid/dotted lines are the sensitivities with/without cuts [304].

interaction, say, from U(1)B−L, the gauge boson Z ′ = ZBL can be produced copiously in pp and pp¯ collisions via gauge interactions in quark annihilation [113, 313–319] and at Linear Colliders in e +e − annihilation [317, 320–322],

$$q\bar{q} \to Z' \to \text{NN} \quad \text{and} \quad e^+e^- \to Z' \to \text{NN}.\tag{3.45}$$

ZBL's subsequent decay to a pair of heavy Majorana neutrinos may lead to a large sample of events without involving the suppression from a small active-sterile mixing angles [93, 323–330]. As a function of MZBL , **Figure 12A** shows the NLO+NLL(Thresh.) pp → ZBL → ℓ +ℓ − production and decay rate for √ s = 13 TeV and representative values of coupling gBL. As a function of Majorana neutrino mass MN<sup>1</sup> , **Figure 12B** shows the LO pp → ZBL → NN production and decay rate for √ s = 14 TeV and 100 TeV and representative MZBL . As N is Majorana, the mixing-induced decays modes N → ℓ <sup>±</sup>W∓, νZ, νh open for MN<sup>1</sup> > MW, MZ, Mh, respectively. Taking these into account, followed by the leptonic and/or hadronic decays of W, Z and h, the detectable signatures include the lepton number violating, same-sign dileptons, NN → ℓ ±ℓ <sup>±</sup>W∓W<sup>∓</sup> → ℓ ±ℓ <sup>±</sup> + nj [93, 301]; final states with three charged leptons, ℓ ±ℓ ±ℓ <sup>∓</sup> + nj+MET [325, 330, 331]; and fourcharged lepton, ℓ ±ℓ ±ℓ ∓ℓ <sup>∓</sup>+MET [324, 332]. Assuming only third generation fermions charged under B − L symmetry, HL-LHC can probe Z ′ mass up to 2.2 TeV and heavy neutrino mass in the range of 0.2 − 1.1 TeV as shown in **Figure 13** [301].

For super-heavy ZBL, e.g., MZBL & 5 TeV ≫MN, one should note that at the 13 TeV LHC, a nontrivial contribution of the total pp → ZBL → NN cross section comes from the kinematical threshold region, where the (NN) system's invariant mass is near mNN ∼ 2M<sup>N</sup> and Z ∗ BL is far off-shell. This implies that the L-violating process pp → NN → ℓ ±ℓ <sup>±</sup> + nj can still proceed despite ZBL being kinematically inaccessible [163]. For more details, see section 3.2.6. Additionally, for such heavy ZBL that are resonantly produced, the emergent N are highly boosted with Lorentz factors of γ ∼ MZBL /2MN. For M<sup>N</sup> ≪ MZBL , this leads to highly collimated decay products, with separations scaling as 1R ∼ 2/γ ∼ 4MN/MZBL , and eventually the formation of lepton jets [225, 333], i.e., collimated clusters of light, charged leptons and electromagnetic radiation, and neutrino jets [141, 301, 312, 334], i.e., collimated clusters of electromagnetic and hadronic activity from decays of high-p<sup>T</sup> heavy neutrinos.

Leading Order-accurate Monte Carlo simulations for treelevel processes involving Z ′ bosons and heavy neutrinos in U(1)<sup>X</sup> theories are possible using the SM+B-L FeynRules UFO model [325, 335, 336]. At NLO+PS accuracy, Monte Carlo simulations can be performed using the Effective LRSM at NLO in QCD UFO model [312, 337], and, for light, long-lived neutrinos and arbitrary Z ′ boson couplings, the SM + W' and Z' at NLO in QCD UFO model [338, 339].

In B − L models, heavy neutrino pairs can also be produced through the gluon fusion process mediated by the two H<sup>1</sup> and H<sup>2</sup> [330, 340–342], and given by

$$\text{gg} \to H\_1, H\_2 \to \text{NN}.\tag{3.46}$$

For long-lived heavy neutrinos with M<sup>N</sup> . 200 GeV, this process becomes important compared to the channel mediated by Z ′ . **Figure 14A** shows that for MH<sup>2</sup> < 500 GeV, M<sup>N</sup> < 200 GeV, and MZ′ = 5 TeV, the cross section σ(pp → H<sup>2</sup> → NN) can be above 1 fb at the √ s = 13 TeV LHC. For M<sup>N</sup> < 60 GeV, decays of the SM-like Higgs H<sup>1</sup> also contributes to neutrino pair

production. Summing over the contributions via H<sup>1</sup> and H<sup>2</sup> the total cross section can reach about 700 fb for MH<sup>2</sup> < 150 GeV as shown in **Figure 14B**.

s = 13 TeV and M<sup>N</sup> < M<sup>W</sup> [330].

Owing to this extensive phenomenology, collider experiments are broadly sensitive to Z ′ bosons from U(1)BL gauge theories. For example: Searches at LEP-II have set the lower bound

MZ′/gBL & 6 TeV [314]. For more generic Z ′ (including Z<sup>R</sup> in LRSM models), comparable limits from combined LEP+EW precision data have been derived in del Águila et al. [345, 346]. Direct searches for a Z ′ with SM-like couplings to fermions exclude MZ′ < 2.9 TeV at 95% CLs by ATLAS [347] and CMS [348] at √ s = 8 TeV. ZBL gauge bosons with the benchmark coupling g<sup>1</sup> ′ = gBL are stringently constrained by searches for dilepton resonances at the LHC, with MZ′ . 2.1 − 3.75 TeV excluded at 95% CLs for gBL = 0.15 − 0.95, as seen in **Figure 12A** [343]. Searches for Z ′ decays to dijets at the LHC have exclude MZ′ < 1.5 − 3.5 TeV for gBL = 0.07 − 0.27 [349, 350]. **Figure 15A** shows that ATLAS excludes <sup>M</sup>Z′ <sup>&</sup>lt; 4.5 TeV at <sup>√</sup> s = 13 TeV. Further constraints are given in the plane of coupling strength γ ′ <sup>=</sup> <sup>g</sup>BL/g<sup>Z</sup> vs. <sup>M</sup>Z′ by ATLAS at <sup>√</sup> s = 13 TeV with 36.1 fb−<sup>1</sup> [344] as shown in the lower curve of **Figure 15B**. For √ <sup>s</sup> <sup>=</sup> 27 TeV, early projections show that with <sup>L</sup> <sup>=</sup> 1 (3) ab−<sup>1</sup> , MZ′ . 19 (20) TeV can be probed in the dijet channel [351].

#### 3.2.5. Heavy Neutrinos and the Left-Right Symmetric Model at Colliders

In addition to the broad triplet scalar phenomenology discussed later in section 4.2, the LRSM predicts at low scales massive W± R and Z<sup>R</sup> gauge bosons that couple appreciably to SM fields as well as to heavy Majorana neutrinos N. The existence of these exotic states leads to a rich collider phenomenology that we now address, focusing, of course, on lepton number violating final states. The collider phenomenology for Z<sup>R</sup> searches is very comparable to that for Z ′ gauge bosons in U(1)<sup>X</sup> theories [93, 323–330], and thus we refer readers to section 3.2.4 for more generic collider phenomenology.

In the LRSM, for M<sup>N</sup> < MW<sup>R</sup> or M<sup>N</sup> < MZ<sup>R</sup> /2, the most remarkable collider processes are the single and pair production of heavy Majorana neutrinos N through resonant charged and neutral SU(2)<sup>R</sup> currents,

$$q\overline{q'} \to W\_R^{\pm} \to N\_{\vec{l}} \ell^{\pm} \quad \text{and} \quad q\overline{q'} \to Z\_R \to N\_{\vec{l}} N\_{\vec{l}}.\tag{3.47}$$

As first observed in Keung and Senjanovic [ ´ 240], N<sup>i</sup> can decay into L-violating final-states, giving rise to the collider signatures,

$$\begin{array}{rcl} pp \rightarrow \; W\_{\mathbb{R}}^{\pm} \rightarrow \; N\_{i} \; \ell^{\pm} \rightarrow \; \ell\_{1}^{\pm} \; \ell\_{2}^{\pm} + \eta j \quad \text{and} \\ pp \rightarrow \; Z\_{\mathbb{R}} \rightarrow \; N\_{i} \; N\_{j} \rightarrow \; \ell\_{1}^{\pm} \; \ell\_{2}^{\pm} + \eta j. \end{array} \tag{3.48}$$

In the minimal/manifest LRSM, the decay of N<sup>i</sup> proceeds primarily via off-shell three-body right-handed currents, as shown in **Figure 16A**, due to mixing suppression to lefthanded currents. In a generic LRSM scenario, the naïve mixing suppression of |VℓN| <sup>2</sup> <sup>∼</sup> <sup>O</sup>(m<sup>ν</sup> /MN) is not guaranteed due to the interplay between the Types I and II Seesaws, e.g., as in Anamiati et al. [228] and Das et al. [230]. (However, heavy-light neutrino mixing in the LRSM is much less free than in pure Type I scenarios due to constraints on Dirac and RH masses from LR parity; see section 3.1.4 for more details). Subsequently, if |VℓN| is not too far from present bounds (see e.g., [91]), then decays of N<sup>i</sup> to on-shell EW bosons, as shown in **Figure 16B**, can occur with rates comparable to decays via off-shell W∗ R [87]. The inverse process [352], i.e., N<sup>i</sup> production via off-shell EW currents and decay via off-shell RH currents as well as vector boson scattering involving t-channel W<sup>R</sup> and Z<sup>R</sup> bosons [353] are in theory also possible but insatiably phase space-suppressed. For M<sup>N</sup> > MW<sup>R</sup> , MZ<sup>R</sup> , resonant N production via off-shell SU(2)<sup>R</sup> currents is also possible, and is analogous to the production through off-shell, SU(2)<sup>L</sup> currents in Equations (3.33)–(3.34). As MW<sup>R</sup> , MZ<sup>R</sup> are bound to be above a few-to-several TeV, the relevant collider phenomenology is largely the same as when M<sup>N</sup> < MW<sup>R</sup> , MZ<sup>R</sup> [144], and hence will not be individually discussed.

Aside from the mere possibility of L violation, what makes these channels so exceptional, if they exist, are their production rates. Up to symmetry-breaking corrections, the RH gauge coupling is g<sup>R</sup> ≈ g<sup>L</sup> ≈ 0.65, which is not a small number. In **Figure 17**, we show for √ s = 13 and 100 TeV the production rate for resonant W<sup>R</sup> at various accuracies as a function of mass [141]; rates for Z<sup>R</sup> are marginally smaller due to slight coupling

(dash-dot) with 1σ PDF uncertainty (shaded); Lower: NLO (dash) and NLO+NNLL (dash-dot) K-factors and PDF uncertainties [141].

suppression. As in other Seesaw scenarios, much recent progress has gone into advancing the precision of integrated and differential predictions for the LRSM: The inclusive production of W<sup>R</sup> and Z<sup>R</sup> are now known up to NLO+NNLL(Thresh) [141], automated at NLO+NLL(Thresh+kT) [354, 355], automated at NNLO [356, 357], and differentially has been automated at NLO with parton shower matching for Monte Carlo simulations [312]. For √ τ<sup>0</sup> = MWR/Z<sup>R</sup> / √ s & 0.3, threshold corrections become as large as (N)NLO corrections, which span roughly +20% to +30%, and have an important impact cross section normalizations [141, 358]. For example: The inclusive W<sup>R</sup> cross section at LO (NLO+NNLL) for MW<sup>R</sup> = 5 TeV is σ ∼ 0.7 (1.7) fb. After <sup>L</sup> <sup>=</sup> 1 ab−<sup>1</sup> and assuming a combined branchingdetection efficiency-selection acceptance of BR×<sup>ε</sup> <sup>×</sup> <sup>A</sup> <sup>=</sup> 2%, the number of observed events is N ∼ 14 (34). For simple Gaussian statistics with a zero background hypothesis, this is the difference between a 6σ "discovery" and 4σ "evidence". Clearly, the HL-LHC program is much more sensitive to ultra-high-mass resonances than previously argued.

For the collider processes in Equation (3.48), such estimations of branching, acceptance/selection, and background rates resemble actual rates: see, e.g., [87, 141, 240, 352, 353, 359–361]. For MW<sup>R</sup> , MZ<sup>R</sup> ≫ MN, one finds generically that BR(W<sup>R</sup> → ℓ <sup>±</sup>Ni) <sup>∼</sup> <sup>1</sup>/(1 <sup>+</sup> <sup>3</sup>Nc) <sup>∼</sup> <sup>O</sup>(10%), BR(Z<sup>R</sup> <sup>→</sup> <sup>N</sup>iNj) <sup>∼</sup> O(10%), and, for the lightest heavy N<sup>i</sup> in this limit, BR(N<sup>1</sup> → ℓ <sup>±</sup>X) <sup>∼</sup> <sup>O</sup>(100%). Trigger rates for multi-TeV, stable charged leptons (e,µ) at ATLAS and CMS exceed 80–95%, but conversely, the momentum resolution for such energetic muons begins to degrade severely; for additional information, see Aad et al. [52], Collaboration [362], Khachatryan [363, 364] and references therein. As in searches for Majorana neutrinos in the previous Type I-based scenarios, the final-states in Equation (3.48) possess same-sign, high-p<sup>T</sup> charged leptons without accompanying MET at the partonic level [240, 248, 359]. For the LRSM, this is particularly distinct since the kinematics of the signal process scale with the TeV-scale W<sup>R</sup> and Z<sup>R</sup> masses. Accordingly, top quark and EW background processes that can mimic the fiducial collider definition correspondingly must carry multi-TeV system invariant masses, and are inherently more phase space suppressed than the signal processes at the LHC [359]. Consequently, so long as M<sup>N</sup> . MW<sup>R</sup> , MZ<sup>R</sup> ≪ √ s, schannel production of W<sup>R</sup> and Z<sup>R</sup> remains the most promising

mechanism for discovering L violation in the LRSM at hadron colliders. In **Figure 18** we show the discovery potential at 14 TeV LHC of W<sup>R</sup> and N in (a) the minimal LRSM as in **Figure 16A** after <sup>L</sup> <sup>=</sup> 30 fb−<sup>1</sup> [360] and (b) the agnostic mixing scenario as in **Figure 16B** [87]. Final-states involving τ leptons are also possible, but inherently suffer from the difficult signal event reconstruction and larger backgrounds due to partonic-level MET induced by τ decays [365].

Unfortunately, direct searches at the √ s = 7/8 TeV LHC via the DY channels have yielded no evidence for lepton number violating processes mediated by W<sup>R</sup> and Z<sup>R</sup> gauge bosons from the LRSM [52, 300, 363, 366]. As shown in **Figure 19**, searches for WR/Z<sup>R</sup> in the e ±e <sup>±</sup> + nj and µ ±µ <sup>±</sup> + nj final state have excluded, approximately, MWR/Z<sup>R</sup> . 1.5 − 2.5 TeV and M<sup>N</sup> . 2 TeV. However, sensitivity to the e ±e <sup>±</sup> + nj greatly diminishes for M<sup>N</sup> ≪ MWR/Z<sup>R</sup> .

Interestingly, for M<sup>N</sup> ≪ MW<sup>R</sup> , MZ<sup>R</sup> , decays of N become highly boosted and its decay products, i.e., ℓ ± 2 qq ′ , become highly collimated. In such cases, the isolation criterion for electrons (and some muons) in detector experiments fail, particularly when √ r = MN/MW<sup>R</sup> < 0.1 [52, 87, 141, 359]. Instead of requiring the identification of two well-isolated charged leptons for the processes given in Equation (3.48), one can instead consider the N-decay system as a single, high-p<sup>T</sup> neutrino jet [141, 312]. The hadronic-level collider signature is then

pp → W<sup>R</sup> → ℓ <sup>±</sup> N → ℓ <sup>±</sup> jN, (3.49)

where the neutrino jet j<sup>N</sup> is comprised of three "partons", (ℓ2, q, q ′ ), with an invariant mass of m<sup>j</sup> ∼ MN. (Neutrino jets are distinct from so-called "lepton jets" [225], which are built from collimated charged leptons and largely absent of hadrons). This alternative topology for M<sup>N</sup> ≪ MW<sup>R</sup> recovers the lost sensitivity of the same-sign dilepton final state, as seen in **Figure 20**. Inevitably, for N masses below the EW scale, rare L-violating decay modes also of SM particles open. In particular, for M<sup>N</sup> below the top quark mass m<sup>t</sup> , one has the rare decay mode, t → bW+∗ <sup>R</sup> → bℓ + <sup>1</sup> N → bℓ + 1 ℓ ± 2 qq ′ [220]. Such processes, however, can be especially difficult to distinguish from rare SM processes, e.g., t → Wbℓ +ℓ − [367], particularly due to the large jet combinatorics.

For too small MN/MW<sup>R</sup> ratio, the lifetime for N, which scales as <sup>τ</sup><sup>N</sup> <sup>∼</sup> <sup>M</sup><sup>4</sup> WR /M<sup>5</sup> N , can become quite long. In such instances, the decays of N are no longer prompt and searches for pp → W<sup>R</sup> → Nℓ map onto searches for Sequential Standard Model W′ bosons [338, 368]. Likewise, searches for L-violating top quark decays become searches for RH currents in t → bℓp<sup>T</sup> decays. For intermediate lifetimes, displaced vertex searches become relevant [223, 228, 230, 334, 369].

Another recent avenue of exploration is the reassessment for resonant production of W<sup>R</sup> and Z<sup>R</sup> in Equation (3.48). In the limit where MW<sup>R</sup> & √ s but M<sup>N</sup> ≪ √ s, resonant production of N, and hence a lepton number violating final state, is still possible despite W<sup>R</sup> being kinematically inaccessible [163]. In such cases, N is produced near mass threshold with p N <sup>T</sup> ∼ M<sup>N</sup> instead of the usual p N <sup>T</sup> ∼ MW<sup>R</sup> /2. The same-sign leptons discovery channel is then kinematically and topologically identical to Type I Seesaw searches, and hence is actively searched for at the LHC, despite this kinematic regime not being well-studied in the literature. Reinterpretation of observed and expected sensitivities at the 14 and 100 TeV LHC are shown in **Figure 21**. One sees that with the anticipated cache of LHC data, MW<sup>R</sup> . 9 TeV can be excluded for M<sup>N</sup> . 1 TeV.

In addition to the aforementioned DY and VBF channels, there has been recent attention [312, 353, 370, 371] given to the production of LRSM scalar and vector bosons in association with heavy flavor quarks, e.g.,

$$\text{gg}^{(-)} \stackrel{(-)}{\rightarrow} \stackrel{(-)}{t}W\_R^{\pm} \text{ or } \stackrel{(-)}{t}H\_R^{\pm} \quad \text{and} \quad \text{gg} \rightarrow t\overline{t}Z\_R \text{ or } t\overline{t}H\_R^0. \tag{3.50}$$

As in the SM, such processes are critical in measuring the couplings of gauge bosons to quarks as well as determining

heavy flavor PDFs. However, also as in the SM, care is needed in calculating the rates of these processes when M<sup>R</sup> ≫ m<sup>b</sup> , m<sup>t</sup> . Here, M<sup>R</sup> is generically the mass of the RH scalar or vector boson. As discussed just after Equation (3.38), it has been noted recently in Mattelaer et al. [312] that such associated processes possess logarithmic dependence on the outgoing top quarks' kinematics, i.e., that the inclusive cross section scales as σ ∼ α k s log2k−<sup>1</sup> M<sup>2</sup> R /(m<sup>2</sup> <sup>t</sup> + p t 2 T ) . Subsequently, for M<sup>R</sup> & 1 − 2 TeV, these logarithms grow numerically large since log<sup>2</sup> (M<sup>2</sup> R /m<sup>2</sup> t ) & 1/α<sup>s</sup> and can spoil the perturbativity convergence of fixed order predictions. For example, the (N)NLO K-factor of K (N)NLO & 1.6 <sup>−</sup> 2.0 claimed in Dev et al. [353] indicate a loss of perturbative control, not an enhancement, and leads to a significant overestimation of their cross sections. As in the case of EW boson production in association with heavy flavors [372, 373], the correct treatment requires either a matching/subtraction scheme with top quark PDFs to remove double counting of phase space configurations [374, 375] or kinematic requirements on the associated top quarks/heavy quark jets, e.g., Equation (3.41) [255].

In all of these various estimates for discovery potential, it is important to also keep in mind what can be learned from observing L violation and LR symmetry at the LHC or a future collider, including ep machines [312, 376–382]. Primary goals post-discovery include: determination of W<sup>R</sup> and Z<sup>R</sup> chiral coupling to fermions [87, 129, 383], which can be quantified for quarks and leptons independently [87], determination of the leptonic and quark mixing [129, 130, 228, 230, 384–387], as well as potential CP violation [228, 230, 386–388]. We emphasize that the discovery of TeV-scale LRSM could have profound implications on high-scale baryo- and leptogenesis [10, 389– 392] as well as searches for 0νββ [129, 162, 385, 393, 394]. The latter instance is particularly noteworthy as the relationship between mee ν and mν<sup>1</sup> in the LRSM is different because of the new mediating fields [385].

We finish this section by noting our many omissions, in particular: supersymmetric extensions of the LRSM, e.g., Frank and Saif [395], and Demir et al. [396]; embeddings into larger internal symmetry structures, e.g., Goh and Krenke [361] and Appelquist and Shrock [397]; as well as generic extensions with additional vector-like or mirror quarks, e.g., Goh and Krenke [361], and de Almeida et al. [398]. While each of these extensions have their phenomenological uniquenesses, their collider signatures are broadly indistinguishable from the minimal LRSM scenario. With regard to Type I-based Seesaws in extra dimensional frameworks, it is worthwhile to note that it has recently [399–401] been observed that in warped fivedimensional models, a more careful organization of Kaluza-Klein states and basis decomposition results in an inverse Seesaw mechanism as opposed to a canonical Type I-like Seesaw

, MN) parameter space for various collider configurations via direct and indirect searches in the µ

mechanism, as conventionally believed. Again, this leads to greatly suppressed L violation at collider experiments.

#### 3.2.6. Heavy Neutrino Effective Field Theory at Colliders

As discussed in section 3.1.5, the production and decay of Majorana neutrinos in colliders may occur through contact interactions if mediating degrees of freedom are much heavier than the hard scattering process scale. Such scenarios have recently become a popular topic [163, 171, 172, 218, 305, 403– 406], in part because of the considerable sensitivity afforded by collider experiments. This is particularly true for Lviolating final-states in pp collisions, which naturally have small experimental backgrounds. As shown in **Figure 22**, for various operators, searches for L-violating process pp → Nℓ ± 1 → ℓ ± 1 ℓ ± <sup>2</sup> + X by the ATLAS and CMS experiments have set wide

expected 95% CL<sup>s</sup> sensitivities to the (MW<sup>R</sup>

±µ

± final state [163].

limits on the effective mass scale of 3 > 1−5 TeV for M<sup>N</sup> = 100 GeV−4.5 TeV [59, 163, 403]. Projections for <sup>√</sup> s = 14 (100) TeV after <sup>L</sup> <sup>=</sup> 1 (10) ab−<sup>1</sup> show that 3 . 9 (40) TeV can be achieved [163]. These search strategies are also applicable for the more general situation where L violation is mediated entirely via SMEFT operators [176, 177] as introduced in section 3.1.5.

### 4. THE TYPE II SEESAW AND LEPTON NUMBER VIOLATION AT COLLIDERS

In this section we review lepton number violating collider signatures associated with the Type II Seesaw mechanism [14– 18, 407] and its extensions. The Type II model is unique among the original tree-level realizations of the Weinberg operator in that lepton number is spontaneously broken; in the original formulations of the Type I and III Seesaws, lepton number violation is explicit by means of a Majorana mass allowed by gauge invariance. In section 4.1, we summarize the main highlights of the canonical Type II Seesaw and other Type IIbased scenarios. We then review in section 4.2 collider searches for lepton number violation mediated by exotically charged scalars (H±, H±±), which is the characteristic feature of Type II-based scenarios.

### 4.1. Type II Seesaw Models

In the Type II mechanism [14–18, 407], tiny neutrino masses arise through the Yukawa interaction,

$$
\Delta \mathcal{L}\_{\text{II}}^{m} = -\overline{L^{c}} \, Y\_{\text{v}} \, i\sigma\_{2} \, \Delta\_{L} L + \text{ H.c.},\tag{4.1}
$$

between the SM LH lepton doublet L, its charge conjugate, and an SU(2)<sup>L</sup> scalar triplet (adjoint representation) 1<sup>L</sup> with mass M1 and Yukawa coupling Yν . More precisely, the new scalar transforms as (1, 3, 1) under the full SM gauge symmetry and possesses lepton number L = −2, thereby ensuring that Equation (4.1) conserves lepton number before EWSB. Due to its hypercharge and L assignments, 1<sup>L</sup> does not couple to quarks at tree-level. It does, however, couple to the SM Higgs doublet, particularly through the doublet-triplet mixing operator

$$
\Delta \mathcal{L}\_{H\Delta\_L} \ni \mu H^T \operatorname{ i} \sigma\_2 \,\Delta\_L^\dagger H + \text{ H.c.}\tag{4.2}
$$

The importance of this term is that after minimizing the full Type II scalar potential VType II, 1<sup>L</sup> acquires a small vev v<sup>1</sup> that in turn induces a LH Majorana mass for SM neutrinos, given by

$$M\_{\boldsymbol{\nu}} = \sqrt{2}Y\_{\boldsymbol{\nu}}\boldsymbol{\nu}\_{\Delta} \quad \text{with} \quad \boldsymbol{\nu}\_{\Delta} = \langle \Delta\_L \rangle = \frac{\mu \nu\_0^2}{\sqrt{2}M\_{\Delta}^2}. \tag{4.3}$$

In the above, v<sup>0</sup> = √ 2hHi is the vev of the SM Higgs and v 2 <sup>0</sup> + v 2 <sup>1</sup> = ( √ 2GF) <sup>−</sup><sup>1</sup> <sup>≈</sup> (246 GeV)<sup>2</sup> . As a result of B−L being spontaneously broken by 1L, tiny 0.1 eV neutrino masses follow from the combination of three scales: µ, v0, and M1. In addition, after EWSB, there are seven physical Higgses, including the singly and doubly electrically charged H± and H±± with masses MH±,H±± ∼ M1. As v<sup>1</sup> contributes to EWSB at tree-level, and hence the EW ρ/T-parameter, v<sup>1</sup> is constrained by precision EW observables, with present limits placing v<sup>1</sup> . O(1 GeV) [408–416]. The impact of triplet scalars on the naturalness of the SM-like Higgs at 125 GeV has also been studied [412, 417, 418]. The simultaneous sensitivity of Mν to collider, neutrino mass measurement, and neutrino oscillation experiments is one of the clearest examples of their complementarity and necessity to understanding neutrinos physics.

For SM-like Yukawas <sup>Y</sup><sup>ν</sup> <sup>∼</sup> <sup>10</sup>−<sup>6</sup> <sup>−</sup> 1, one finds that <sup>v</sup><sup>1</sup> <sup>∼</sup> 0.1 eV − 100 keV are needed in order to reproduce 0.1 eV neutrino masses. Subsequently, for µ ∼ M1, then M<sup>1</sup> ∼ µ ∼ <sup>10</sup><sup>8</sup> <sup>−</sup> <sup>10</sup><sup>14</sup> GeV, and for <sup>µ</sup> <sup>∼</sup> <sup>v</sup>0, then <sup>M</sup><sup>1</sup> <sup>∼</sup> <sup>10</sup><sup>5</sup> <sup>−</sup> <sup>10</sup><sup>8</sup> GeV. In either case, these scales are too high for present-day experiments. However, as nonzero µ is associated with both lepton number and custodial symmetry non-conservation, one may expect it to be small [121] and natural, in the t'Hooft sense [419]. Imposing technical naturalness can have dramatic impact on LHC phenomenology: for example, if µ ∼ 1 MeV (keV), then <sup>M</sup><sup>1</sup> <sup>∼</sup> <sup>10</sup><sup>2</sup> <sup>−</sup> <sup>10</sup><sup>5</sup> (10<sup>1</sup> <sup>−</sup> <sup>10</sup><sup>4</sup> ) GeV, scales well within the LHC's energy budget. Moreover, this also indicates that proposed future hadron collider experiments [148, 149] will be sensitive to MeV-to-GeV values of the scalar-doublet mixing parameter µ, independent of precision Higgs coupling measurements, which are presently at the 10% level [420]. Assuming Higgs coupling deviations of O(µ/Mh), this implies the weak 7/8 TeV LHC limit of µ . O(10 GeV). While not yet competitive with constraints from EW precision data, improvements on Higgs coupling measurements will be greatly improved over the LHC's lifetime.

After decomposition of leptons into their mass eigenstates, the Yukawa interactions of the singly and doubly charged Higgses are

$$\begin{aligned} \, \_\nu \nu\_L^T \, ^\alpha \Gamma + \, H^+ \, ^t \ell\_L, \quad : \quad \Gamma \_+ = \cos \theta + \, \frac{m\_\nu^{\text{diag}}}{\nu\_\Delta} \, ^t U\_{\text{PMNS}}^\dagger, \; \theta \approx \frac{\sqrt{2} \nu\_\Delta}{\nu\_0}, \end{aligned} \tag{4.4}$$

$$\begin{aligned} \ell\_L^T \, \text{C} \, \Gamma\_{++} \, H^{++} \, \ell\_L \quad : \quad \Gamma\_{++} = \frac{M\_\upsilon}{\sqrt{2} \nu\_\Delta} = U\_{\text{PMNS}}^\* \frac{m\_\upsilon^{\text{diag}}}{\sqrt{2} \, \nu\_\Delta} \, U\_{\text{PMNS}}^\dagger. \end{aligned} \tag{4.5}$$

The constrained neutrino mass matrix M<sup>ν</sup> = √ 2v1Ŵ++ and squared Yukawa coupling Y i <sup>+</sup> ≡ P j |Ŵ ji +| 2 v 2 <sup>1</sup> with vanishing Majorana phases are shown in **Figures 23**, **24** respectively. The results reveal the following mass and Yukawa patterns:

$$M\_{\upsilon}^{22}, M\_{\upsilon}^{33} \gg M\_{\upsilon}^{11}; \quad Y\_{+}^{2}, Y\_{+}^{3} \gg Y\_{+}^{1} \qquad \text{for NH;} \tag{4.6}$$

$$M^{11}\_{\upsilon} \gg M^{22}\_{\upsilon}, M^{33}\_{\upsilon}; \quad Y^1\_+ \gg Y^2\_+, Y^3\_+ \qquad \text{for IH.} \tag{4.7}$$

Below <sup>v</sup><sup>1</sup> <sup>≈</sup> <sup>10</sup>−<sup>4</sup> GeV, the doubly charged Higgs <sup>H</sup>±± decays dominantly to same-sign lepton pairs. For vanishing Majorana phases 8<sup>1</sup> = 8<sup>2</sup> = 0, we show in **Figures 25**, **26** the branching fraction of the decays into same-flavor and differentflavor leptonic final states, respectively. Relations among the branching fractions of the lepton number violating Higgs decays of both the singly- and doubly-charged Higgs in the NH and IH, with vanishing Majorana phases, are summarized in **Table 2**.

The impact of Majorana phases can be substantial in doubly charged Higgs decays [421, 422]. In the case of the IH, a large cancellation among the relevant channels occurs due to the phase at 8<sup>1</sup> = π. As a result, in this scenario, the dominant channels swap from H++ → e +e +, µ +τ <sup>+</sup> when 8<sup>1</sup> ≈ 0 to H++ → e +µ +, e +τ <sup>+</sup> when 8<sup>1</sup> ≈ π, as shown in **Figure 27**. Therefore this qualitative change can be made use of to extract the value of the Majorana phase 81. In the NH case, however, the dependence of the decay branching fractions on the phase is rather weak because of the lack of a subtle cancellation [408].

The Type II mechanism can be embedded in a number of extended gauge scenarios, for example the LRSM as discussed in section 3.1.4, as well as GUTs, such as (331) theories [423–426] and the extensions of minimal SU(5) [427]. For (331) models, one finds the presence of bileptons [428, 429], i.e., gauge bosons with L = ±2 charges and hence Q = ±2 electric charges. In a realistic extension of the Georgi-Glashow model, a scalar 15 dimensional representation is added [430] and the scalar triplet stays in the **15** representation together with scalar leptoquark 8 ∼ (3, 2, 1/6). The SU(5) symmetry thus indicates that the couplings of the leptoquark to matter gain the same Yukawas Yν responsible for neutrino mass matrix [431]. Extensions with vector-like leptons in nontrivial SU(2)<sup>L</sup> representations are also possible [432]. Unsurprisingly, the phenomenology [423, 425, 433–435] and direct search constraints [433, 434] for L-violating, doubly charged vector bosons are similar to L-violating, doubly charged scalar bosons, which we now discuss.

### 4.2. Triplet Higgs Scalars at Colliders

#### 4.2.1. Triplet Higgs Scalars and the Type II Seesaw at Colliders

If kinematically accessible, the canonical and well-studied [145, 408, 436, 437] triplet scalars production channels at hadron colliders are the neutral and charged current DY processes, given by

$$pp \to \gamma^\*/Z^\* \to H^{++}H^{--}, \quad pp \to W^{\pm \*} \to H^{\pm \pm}H^{\mp}, \tag{4.8}$$

and shown in **Figure 28A**. Unlike Type I models, scalars in the Type II Seesaw couple to EW bosons directly via gauge couplings. Subsequently, their production rates are sizable and can be predicted as a function of mass without additional input. In **Figure 29** we show the LO pair production cross section of triplet scalars via the (a) neutral and (b) charged current DY process at √ s = 14 and 100 TeV. NLO in QCD corrections to these processes are well-known [438] and span K NLO <sup>=</sup> σ NLO/σ LO <sup>=</sup> 1.1 <sup>−</sup> 1.3 away from boundaries of collider phase space; moreover, due to the color-structure of DY-like processes, inclusive kinematics of very heavy scalar triplets are Born-like and thus naïve normalization of kinematics by K NLO gives reliable estimates of both NLO- and NLO+PS-accurate results [275, 338]. For MH±± = 1 TeV, one finds that the LO pair production rates can reach <sup>σ</sup> <sup>∼</sup> 0.1 (10) fb at <sup>√</sup> s = 14 (100) TeV, indicating O(10<sup>2</sup> ) (O(10<sup>4</sup> )) of events with the ab−<sup>1</sup> -scale data sets expected at the respective collider program.

In addition to the DY channels are: single production of charged Higgses via weak boson scatter, as shown in **Figure 28B** and investigated in Han et al. [410], and Chen et al. [439]; charged Higgs pair production via γγ scattering, as shown in **Figure 28C**, studied in Dutta et al. [409], Han et al. [440], Drees et al. [441], Bambhaniya et al. [442], and Babu and Jana [443], and computed at √ s = 14 TeV [440] in **Figure 29C**; as well as pair production through weak boson scattering, as studied in Dutta et al. [409] and Bambhaniya et al. [442] and computed for the 14 TeV LHC [409] in **Figure 29D**. As in the case of Wγ scattering in heavy N production in section 3, there is renewed interest [442] in the γγ -mechanisms due to the new availability of photon PDFs that include both elastic and (deeply) inelastic contributions, e.g., NNPDF 2.3 and 3.0 QED PDF sets [444, 445]. However, care should be taken in drawing conclusions based on these specific PDF sets due to the (presently) large γ -PDF uncertainty, particularly at large Bjorken-x where this can reach greater than 100% [444]. For example: As shown in **Figure 29C**, γγ production is unambiguously sub-leading to the DY mechanism and only contributes about 10% despite recent claims to the contrary [443, 446]. The collinear behavior and the

FIGURE 23 | Constraints on the diagonal (A,B) and off-diagonal (C,D) elements of the neutrino mass matrix M<sup>ν</sup> ≡ √ <sup>2</sup>v1Ŵ++ vs. the lowest neutrino mass for NH (A,C) and IH (B,D) when 8<sup>1</sup> = 0 and 8<sup>2</sup> = 0.

FIGURE 25 | Scatter plots for the H ++ decay branching fractions to the flavor-diagonal like-sign dileptons vs. the lowest neutrino mass for NH (A) and IH (B) with 8<sup>1</sup> = 8<sup>2</sup> = 0.

TABLE 2 | Relations among the branching fractions of the lepton number violating Higgs decays for the neutrino mass patterns of NH and IH, with vanishing Majorana phases.

#### Relations


factorization scale dependence of the incoming photons must be treated with great care. As more data is collected and γ -PDF methodology further matures, one anticipates these uncertainties to greatly shrink; for further discussions of γ -PDFs, see Alva et al. [254], Degrande et al. [255], Martin and Ryskin [268], Harland-Lang et al. [269], Manohar et al. [271, 272]. For a list of recommended γ -PDFs, see the discussion just above Equation (3.38).

Similar to the γγ channel, production of triplet scalars from gluon fusion is sub-leading with respect to DY due to multiple vanishing contributions [258, 447] and despite an expectedly large QCD correction of K <sup>N</sup>3LL <sup>=</sup> <sup>σ</sup> <sup>N</sup>3LL/σ LO <sup>∼</sup> 2.5 <sup>−</sup> 3 [259]. If triplet scalar couplings to the SM-like Higgs are not too small and if sufficiently light, then such scalars may appear in pairs as rare decays of the 125 GeV scalar boson [448]. Likewise, if neutral triplet scalars mix appreciably with the SM-like Higgs, then single production via gluon fusion is also possible [448]; one should note that in such cases, the QCD K-factors calculated in Ruiz et al. [259] are applicable.

A noteworthy direction of progress in searches for triplet scalars at colliders are the implementation of exotically charged

FIGURE 27 | Scatter plots of the same (A) and different (B) flavor leptonic branching fractions for the H ++ decay vs. the Majorana phase 8<sup>1</sup> for the IH with m<sup>3</sup> = 0 and 8<sup>2</sup> ∈ (0, 2π).

scalars into FeynRules model files. In particular, lepton number violating scalars are available in the LNV-Scalars [449, 450] model file as well as in a full implementation of LRSM at LO accuracy [451, 452]; the Georgi-Machacek model [453] is also available at NLO in QCD accuracy [293, 454]. These permit simulation of triplet scalar production in inclusive ℓℓ/ℓp/pp collisions using modern, general-purpose event generators, such as Herwig [289], MadGraph5\_aMC@NLO [290], and Sherpa [291].

Due to the unknown Yukawa structure in Equation (4.1), the decays of the triplet scalars to SM states are much more ambiguous than their production. Subsequently, branching rates of H<sup>±</sup> → ℓ <sup>±</sup>ν and H±± → ℓ ± 1 ℓ ± 2 are often taken as phenomenological parameters in analyses and experimental searches. When taking such a model-agnostic approach, it may be necessary to also consider the lifetimes of scalar triplets: In a pure Type II scenario, for MH±± < 270 GeV and sub-MeV values of the triplet vev vL, the proper decay length of H±± can exceed 10 µm [410]. As a result, exotically charged triplet scalars may manifest at collider experiments in searches for long-lived, multi-charged particles such as Aad et al. [455, 456], Collaboration [457], and Barrie et al. [458].

For prompt decays of triplet scalars, the discovery potential at hadron colliders is quantified in **Figure 30**. In particular, following the analysis of Fileviez Pérez et al. [408], **Figures 30A,B** show event contours in the BR(H++ → µ +µ <sup>+</sup>) vs. MH±± plane after <sup>L</sup> <sup>=</sup> 300 (3000) fb−<sup>1</sup> of data at <sup>√</sup> s = 14 TeV and 100 TeV, respectively. At the 2σ level, one finds the sensitivity to doubly charged Higgs is about MH±± = 0.75 (1.1) TeV at 14 TeV and MH±± = 2 (3.5) TeV at 100 TeV. In **Figures 30C,D**, one similarly has the signal significance σ = S/ √ S + B after <sup>L</sup> <sup>=</sup> <sup>1</sup> and 3 ab−<sup>1</sup> at the 14 TeV LHC for VBF production of doubly charged Higgs pairs and their decays to e ±µ ± and τ ±τ ±

final-states, respectively [409]. Upon the fortuitous discovery of a doubly charged scalar, however, will require also observing other charged scalars to determine its precise weak isospin and hypercharge quantum numbers [145, 449, 459].

In light of such sensitivity at hadron colliders, it is unsurprising then that null results from searches at the 7/8/13 TeV LHC [54, 55, 460, 461] have placed stringent constraints on EW-scale triplet scalar masses, assuming benchmark branching rates. As seen in **Figure 31**, results from the ATLAS experiment in searches for doubly charged Higgs pairs decaying to leptons, after collecting <sup>L</sup> <sup>=</sup> 36 fb−<sup>1</sup> of data at 13 TeV, have ruled out MH±± > 600 − 900 GeV at 95% CLs in both the (a) single-flavor and (b) mixed light-lepton final states [460]. Comparable limits have been reached by the CMS experiment [461].

At future e −e + colliders, triplet scalars can appear in t-channel exchanges, inducing charged lepton flavor violation (cLFV) and forward-backward asymmetries [462]; in three-body decays of taus that are absent of light-neutrinos in the final state, i.e., τ <sup>±</sup> → ℓ <sup>∓</sup>H±±∗ → ℓ ∓µ ±µ ± [463]; and, of course, in pairs via s-channel gauge currents [464]. In the event of such observations, the nontrivial conversion of an e −e + beam into an e −e −/e −µ −/µ−µ − facility could provide complimentary information on scalar triplet Yukawa couplings by means of the "inverse" 0νββ processes, ℓ − i ℓ − <sup>j</sup> → W<sup>−</sup> L/RW<sup>−</sup> L/R [465–467].

#### 4.2.2. Triplet Higgs Scalars and the Left-Right Symmetric Model at Colliders

Turning to scalars in the LRSM, as introduced in section 3.1.4, it was recently observed [368, 448] that in a certain class of neutrino mass models, decays of the SM-like Higgs boson h(125 GeV) to heavy neutrino pairs, h → NN, may occur much more readily than previously thought. The significance of this reaction is one's ability to confirm neutrino masses are generated, in part, through EWSB. It would also indicate sensitivity to the scalar sector responsible for generating RH Majorana masses. Interactions between SM particles and N typically proceed through heavylight neutrino mixing, |VℓN|, which, is a numerically small quantity. As h → NN involves two N, the issue is compounded and usually renders the decay rate prohibitively small in a pure Type I scenario. For <sup>H</sup> ∈ {H<sup>0</sup> , H±, H±±} predicted in Type I+II Seesaws, and in particular the LRSM, the situation is more interesting: it may be that h(125 GeV) and the RH neutral scalars mix sufficiently that decays to relatively light (2M<sup>N</sup> < 125 GeV) heavy neutrino pairs are possible [368]. This is allowed as H

can couple appreciable to N and the mixing between H<sup>0</sup> and h is much less constrained. Subsequently, the naïve neutrino mixing suppression is avoided by exploiting that h → NN decays can proceed instead through <sup>H</sup><sup>0</sup> <sup>−</sup> <sup>h</sup> mixing. In a similar vein, it may be possible for h to decay to triplet pairs and subsequently to N or same-sign charged leptons, or for single H<sup>0</sup> production to proceed directly [448]. Such processes are shown diagrammatically in **Figure 32**. As a result, the L-violating Higgs decays,

$$\begin{aligned} h(125\,\text{GeV}) &\rightarrow \ N\,N \rightarrow W\_R^{\pm\ast}W\_R^{\pm\ast}\ell\_1^{\mp}\ell\_2^{\mp} \rightarrow \ell\_1^{\mp}\ell\_2^{\mp} + \eta j, \text{ (4.9)}\\ h(125\,\text{GeV}) &\rightarrow \ H^0\,H^0 \rightarrow 4N \rightarrow \ell\_1^{\pm}\,\ell\_2^{\mp}\,\ell\_3^{\pm}\,\ell\_4^{\mp} + \eta j, \text{ (4.10)}\\ h(125\,\text{GeV}) &\rightarrow \ H^{++}\,H^{--} &\rightarrow \ell\_1^{\pm}\,\ell\_2^{\pm}\,\ell\_3^{\mp}\,\ell\_4^{\mp}, \end{aligned} \tag{4.11}$$

are not only possible, but also provide complementary coverage of low-mass N scenarios that are outside the reach of 0νββ experiments and direct searches for W<sup>R</sup> at colliders. The sensitivity of such modes are summarized in **Figure 33** [368, 448]. The associated production channels,

$$pp \to H^{0, \pm \pm} \; W\_R^{\mp} \quad \text{and} \quad pp \to H^0 Z\_R,\tag{4.12}$$

are also possible. However, as in the SM, these channels are schannel and phase space suppressed, which lead to prohibitively small cross sections in light of present mass limits [145].

Lastly, one should note that the search for such Higgs decays is not limited to hadron colliders. As presently designed future lepton colliders are aimed at operating as Higgs factories, searches for such L-violating Higgs decays [468–470] at such facilities represent an attractive discovery prospect. In this context, a relatively understudied topic is the possible manifestation of Seesaw in precision measurements of the known SM-like Higgs boson [216, 368, 471]. Some related studies also exist in the literature such as for generic pheno [440, 440, 449]; for little Higgs [410, 472]; and for decay ratios and mixing patterns of exotically charged Higgs [473, 474].

### 5. THE TYPE III SEESAW AND LEPTON NUMBER VIOLATION AT COLLIDERS

We now turn to collider searches for lepton number violation in the context of the Type III Seesaw mechanism [19] as well

FIGURE 31 | ATLAS 95% CLs exclusion at 13 TeV after <sup>L</sup> <sup>=</sup> 36 fb−<sup>1</sup> on <sup>σ</sup>(pp <sup>→</sup> <sup>H</sup> ++H −−) for various representative branching rates to SM charged leptons in the (A) pure e ±e ±, (B) pure µ ±µ ±, (C) pure e ±µ ±, and (D) mixed final-states [460].

as its embedding in GUTs and other SM extensions. In some sense, the Type III model is the fermionic version of the Type II scenario, namely that Seesaw partner fermions couple to the SM via both weak gauge and Yukawa couplings. Subsequently, much of the Type III collider phenomenology resembles that of Type I-based models. However, quantitatively, the presence of gauge couplings lead to a very different outlook and level of sensitivity. We now summarize the main highlights of the canonical Type III Seesaw (section 5.1.1), Type III-based models (section 5.1.2), and then review their L-violating collider phenomenology (section 5.2). As with the previous Seesaw scenarios, a discussion of cLFV is outside the scope of this review. For recent summaries on cLFV in the Type III Seesaw, see Abada et al. [176, 475], Eboli

et al. [476], and Agostinho et al. [477] and references therein.

# 5.1. Type III Seesaw Models

#### 5.1.1. The Canonical Type III Seesaw Mechanism

In addition to the SM field content, the Type III Seesaw [19] consists of SU(2)<sup>L</sup> triplet (adjoint) leptons,

$$
\Sigma\_L = \Sigma\_L^a \sigma^a = \begin{pmatrix} \Sigma\_L^0 / \sqrt{2} & \Sigma\_L^+ \\ \Sigma\_L^- & -\Sigma\_L^0 / \sqrt{2} \end{pmatrix},
$$

$$
\Sigma\_L^\pm = \frac{\Sigma\_L^1 \mp i \Sigma\_L^2}{\sqrt{2}}, \quad \Sigma\_L^0 = \Sigma\_L^3. \tag{5.1}
$$

which transform as (1, 3, 0) under the SM gauge group. Here 6 ± L have U(1)EM charges Q = ±1, and the σ a for a = 1, ... , 3, are the usual Pauli SU(2) matrices. The RH conjugate fields are related by

$$
\Sigma\_R^\epsilon = \begin{pmatrix}
\Sigma\_R^{0c}/\sqrt{2} & \Sigma\_R^{-\epsilon} \\
\Sigma^{+\epsilon} & -\Sigma\_R^{0c}/\sqrt{2}
\end{pmatrix}, \quad \text{for} \quad \psi\_R^\epsilon \equiv (\psi^\epsilon)\_R = (\psi\_L)^\epsilon. \tag{5.2}
$$

The Type III Lagrangian is given by the sum of the SM Lagrangian, the triplet's kinetic and mass terms,

$$\mathcal{L}\_T = \frac{1}{2} \operatorname{Tr} \left[ \overline{\Sigma\_L} i \, \mathcal{J} \Sigma\_L \right] - \left( \frac{M\_\Sigma}{2} \overline{\Sigma\_L^0} \Sigma\_R^{0c} + M\_\Sigma \overline{\Sigma\_L^-} \Sigma\_R^{+c} + \text{H.c.} \right), \tag{5.3}$$

and the triplet's Yukawa coupling to the SM LH lepton (L) and Higgs (H) doublet fields,

$$\mathcal{L}\_Y = -Y\_{\Sigma}\overline{L}\,\Sigma\_R^\epsilon \, i\sigma^2 H^\* + \text{H.c.}\tag{5.4}$$

From Equation (5.4), one can deduce the emergence of a Yukawa coupling between the charged SM leptons and the charged triplet leptons. This, in turn, induces a mass mixing among charged leptons that is similar to doublet-singlet and doublettriplet neutrino mass mixing, and represents one of the more remarkable features of the Type III mechanism. The impact of EW fermion triplets on the SM Higgs, naturalness in the context of the Type III Seesaw has been discussed in Gogoladze et al. [478], He et al. [479], and Gogoladze et al. [480].

After expanding Equations (5.3)–(5.4), the relevant charged lepton and neutrino mass terms are [481]

$$\begin{split} \mathcal{L}\_{\text{III}}^{m} &= -\left(\overline{l\_{\text{R}}} \ \overline{\Psi\_{\text{R}}}\right) \begin{pmatrix} m\_{l} & 0 \\ Y\_{\Sigma} \nu\_{0} & M\_{\Sigma} \end{pmatrix} \begin{pmatrix} l\_{L} \\ \Psi\_{L} \end{pmatrix} \\ &- \left(\overline{\nu\_{L}^{c}} \ \overline{\Sigma\_{L}^{0c}}\right) \begin{pmatrix} 0 & Y\_{\Sigma}^{T} \nu\_{0}/2\sqrt{2} \\ Y\_{\Sigma} \nu\_{0}/2\sqrt{2} & M\_{\Sigma}/2 \end{pmatrix} \begin{pmatrix} \nu\_{L} \\ \Sigma\_{L}^{0} \end{pmatrix} + \text{H.c.}, \end{split} \tag{5.5}$$

with 9<sup>L</sup> ≡ 6 − L , 9<sup>R</sup> ≡ 6 +c L , and 9 = 9L+9R. After introducing unitarity matrices to transit light doublet and heavy triplet lepton fields as below

$$\begin{pmatrix} l\_{L,R} \\ \Psi\_{L,R} \end{pmatrix} = \ U\_{L,R} \begin{pmatrix} l\_{mL,R} \\ \Psi\_{mL,R} \end{pmatrix}, \quad \begin{pmatrix} \nu\_{L} \\ \Sigma\_{L}^{0} \end{pmatrix} = U\_{0} \begin{pmatrix} \nu\_{mL} \\ \Sigma\_{mL}^{0} \end{pmatrix}, \tag{5.6}$$

$$U\_{L} \equiv \begin{pmatrix} U\_{L\mathcal{U}} & U\_{L\mathcal{V}\Psi} \\ U\_{L\Psi I} & U\_{L\Psi\Psi} \end{pmatrix}, \quad U\_{R} \equiv \begin{pmatrix} U\_{R\mathcal{U}} & U\_{R\mathcal{U}\Psi} \\ U\_{R\Psi I} & U\_{R\Psi\Psi} \end{pmatrix},$$

$$U\_{0} \equiv \begin{pmatrix} U\_{0\nu v} & U\_{0\nu\Sigma} \\ U\_{0\Sigma v} & U\_{0\Sigma\Sigma} \end{pmatrix}, \tag{5.7}$$

one obtains the diagonal mass matrices and mass eigenvalues for neutrinos and charged leptons,

$$\text{diag}(\mathcal{N}) = \boldsymbol{U}\_0^\dagger \begin{pmatrix} 0 & \boldsymbol{Y}\_\Sigma^\dagger \boldsymbol{\nu}\_0 / \sqrt{2} \\ \boldsymbol{Y}\_\Sigma^\ast \boldsymbol{\nu}\_0 / \sqrt{2} & \boldsymbol{M}\_\Sigma^\ast \end{pmatrix} \boldsymbol{U}\_0^\ast = \begin{pmatrix} \boldsymbol{m}\_\upsilon^{\text{diag}} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{M}\_N^{\text{diag}} \end{pmatrix},\tag{5.8}$$

$$\text{diag}(\mathcal{E}) = \boldsymbol{U}\_L^\dagger \begin{pmatrix} \boldsymbol{m}\_l^\dagger & \boldsymbol{Y}\_\Sigma^\dagger \boldsymbol{\nu}\_0 \\ \mathbf{0} & \boldsymbol{M}\_\Sigma^\dagger \end{pmatrix} \boldsymbol{U}\_R = \begin{pmatrix} \boldsymbol{m}\_l^{\text{diag}} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{M}\_E^{\text{diag}} \end{pmatrix}.\tag{5.9}$$

The light neutrino mass eigenstates are denoted by ν<sup>j</sup> for j = 1, ... , 3; whereas the heavy neutral and charged leptons are respectively given by N<sup>j</sup> ′ and E ± k ′ . In the literature, N and E ± are often denoted as T 0 , T <sup>±</sup> or 6<sup>0</sup> , 6±. However, there is no standard convention as to what set of symbols are used to denote gauge and mass eigenstates. Where possible, we follow the convention of Arhrib et al. [482] and generically denote tripletdoublet mixing by Y<sup>T</sup> and εT. This means that in the mass basis, triplet gauge states are given by

$$\Psi^{\pm} = Y\_T E^{\pm} + \sqrt{2} \varepsilon\_T \,\ell^{\pm} \qquad \text{and} \qquad \Psi^0 = Y\_T N + \varepsilon\_T \,\nu\_m,$$

$$\text{with } |Y\_T| \sim \mathcal{O}(1) \quad \text{and} \qquad |\varepsilon\_T| \sim \frac{Y\_\Sigma \nu\_0}{\sqrt{2} M\_\Sigma} \ll 1. \text{(5.10)}$$

The resulting interaction Lagrangian, in the mass eigenbasis then contains [482]

$$\begin{aligned} \mathcal{L}\_{\text{Type III}}^{\text{Mass Basis}} & \Rightarrow -\,\,\overline{E\_{\text{k}}} \left( eY\_{T} A\_{\mu} \gamma^{\mu} + g \cos \theta\_{W} Y\_{T} Z\_{\mu} \gamma^{\mu} \right) E\_{\text{k}}^{-} \\ & \qquad -\,\, gY\_{T} \overline{E\_{\text{k}'}} W\_{\mu}^{-} \gamma^{\mu} N\_{\text{f}} \\ & \qquad -\,\, \frac{e}{2 \varepsilon\_{\text{w}} \varepsilon\_{\text{w}}} Z\_{\mu} \left( \varepsilon\_{T} \overline{N\_{\text{f}'}} \gamma^{\mu} P\_{R} \upsilon\_{\text{f}} + \sqrt{2} \varepsilon\_{T} \overline{E\_{\text{k}'}} \gamma^{\mu} P\_{R} \ell\_{\text{k}}^{-} \right) \\ & \qquad -\,\, \frac{e}{\varepsilon\_{\text{w}}} W\_{\mu}^{+} \Big( \varepsilon\_{T} \overline{\upsilon\_{\text{j}}} \gamma^{\mu} P\_{L} E\_{\text{k}'}^{-} + \frac{1}{\sqrt{2}} \varepsilon\_{T} \overline{N\_{\text{f}'}} \gamma^{\mu} P\_{R} \ell\_{\text{k}}^{-} \Big) + \text{H.c.} \end{aligned} \tag{5.111}$$

From this, one sees a second key feature of the Type III Seesaw, that gauge interactions between heavy lepton pairs proceeds largely through pure vector currents with axial-vector deviations (not shown) suppressed by O(ε 2 T ) at the Lagrangian level. This follows from the triplet fermions vector-like nature. Similarly, the mixing-suppressed gauge couplings between heavy and light leptons proceeds through SM-like currents.

Explicitly, the light and heavy neutrino mass eigenvalues are

$$m\_{\upsilon} \approx \frac{Y\_{\Sigma}^2 \nu\_0^2}{2M\_{\Sigma}}, \quad M\_N \approx M\_{\Sigma}, \tag{5.12}$$

and for the charged leptons are

$$m\_l - m\_l \frac{Y\_\Sigma^2 \nu\_0^2}{2M\_\Sigma^2} \approx m\_l, \quad M\_E \approx M\_\Sigma. \tag{5.13}$$

This slight deviation in the light, charged leptons' mass eigenvalues implies a similar variation in the anticipated Higgs coupling to the same charged leptons. At tree-level, the heavy leptons N and E <sup>±</sup> are degenerate in mass, a relic of SU(2)<sup>L</sup> gauge invariance. However, after EWSB, and for M<sup>6</sup> & 100 GeV, radiative corrections split this degeneracy by Arhrib et al. [482],

$$\begin{split} \Delta M\_{T} & \equiv M\_{E} - M\_{N} = \frac{\alpha\_{W}}{2\pi} \frac{M\_{W}^{2}}{M\_{\Sigma}} \left[ f\left(\frac{M\_{\Sigma}}{M\_{Z}}\right) - f\left(\frac{M\_{\Sigma}}{M\_{W}}\right) \right] \\ & \approx 160 \text{ MeV}, \\ \text{where} \quad f\left(\mathbf{y}\right) &= \frac{1}{4\mathbf{y}^{2}} \log \mathbf{y}^{2} - \left(1 + \frac{1}{2\mathbf{y}^{2}}\right) \sqrt{4\mathbf{y}^{2} - 1} \arctan \\ & \sqrt{4\mathbf{y}^{2} - 1}, \end{split} \tag{5.15}$$

and opens the E <sup>±</sup> → Nπ ± decay mode. Beyond this are the heavy lepton decays to EW bosons and light leptons that proceed through doublet-triplet lepton mixing. The mixings are governed by the elements in the unitary matrices UL,<sup>R</sup> and U0. Expanding UL,<sup>R</sup> and U<sup>0</sup> up to order Y 2 6 v 2 0M−<sup>2</sup> 6 , one gets the following results [475, 483]

$$\begin{split} & U\_{\rm III} = 1 - \epsilon \, \,, \quad U\_{\rm II\Psi} = Y\_{\Sigma}^{\dagger} M\_{\Sigma}^{-1} \nu\_{0} \,, \qquad U\_{\rm L\Psi l} = -M\_{\Sigma}^{-1} Y\_{\Sigma} \nu\_{0} \,, \\ & U\_{\rm L\Psi\Psi} = 1 - \epsilon \, \, \/ \,, \\ & U\_{\rm R\Psi l} = 1 \, \,, \quad U\_{\rm R\Psi\Psi} = m\_{l} Y\_{\Sigma}^{\dagger} M\_{\Sigma}^{-2} \nu\_{0} \, \,, \, U\_{\rm R\Psi l} = -M\_{\Sigma}^{-2} Y\_{\Sigma} m\_{l} \nu\_{0} \, \,, \\ & U\_{\rm R\Psi\Psi} = 1 \, \, \,, \\ & U\_{0\nu\nu} = (1 - \epsilon/2) U\_{\rm PM\text{MS}} \,, \, \, U\_{0\nu\Sigma} = Y\_{\Sigma}^{\dagger} M\_{\Sigma}^{-1} \nu\_{0} / \sqrt{2} \, \,, \, \, U\_{0\Sigma\nu} = 1 \\ & - M\_{\Sigma}^{-1} Y\_{\Sigma} U\_{0\nu\nu} \nu\_{0} / \sqrt{2} \, \,, \\ & U\_{0\Sigma\Sigma} = 1 - \epsilon'/2 \, \,, \, \epsilon = Y\_{\Sigma}^{\dagger} M\_{\Sigma}^{-2} Y\_{\Sigma} \nu\_{0}^{2} / 2 \, \,, \\ & \epsilon' = M\_{\Sigma}^{-1} Y\_{\Sigma} Y\_{\Sigma}^{\dagger} M\_{\Sigma}^{-1} \nu\_{0}^{2} / 2 \, . \end{split}$$

To the order of Y6v0M−<sup>1</sup> 6 , the mixing between the SM charged leptons and triplet leptons, i.e., Vℓ<sup>N</sup> = −Y † 6 v0M−<sup>1</sup> 6 / √ 2, follows the same relation as Equation (3.10) in the Type I Seesaw [481] and the couplings in the interactions in Equation (5.11) are all given by Vℓ<sup>N</sup> [326, 481].

Hence, the partial widths for both the heavy charged lepton and heavy neutrino are proportional to |VℓN| 2 . For M<sup>E</sup> ≈ M<sup>N</sup> ≫ MW, MZ, Mh, the partial widths behave like [252, 326]

$$\begin{split} \frac{1}{2}\Gamma(\boldsymbol{N}\rightarrow\sum\_{\ell}\ell^{+}\boldsymbol{W}^{-}+\ell^{-}\boldsymbol{W}^{+}) &\approx \Gamma(\boldsymbol{N}\rightarrow\sum\_{\boldsymbol{\nu}}\boldsymbol{\nu}\mathbf{Z}+\bar{\boldsymbol{\nu}}\mathbf{Z}) \\ &\approx \Gamma(\boldsymbol{N}\rightarrow\sum\_{\boldsymbol{\nu}}\boldsymbol{\nu}\boldsymbol{h}+\bar{\boldsymbol{\nu}}\boldsymbol{h}) \\ \approx \frac{1}{2}\Gamma(\boldsymbol{E}^{\pm}\rightarrow\sum\_{\boldsymbol{\nu}}\binom{\boldsymbol{\nu}}{\boldsymbol{\nu}}\boldsymbol{W}^{\pm}) &\approx \Gamma(\boldsymbol{E}^{\pm}\rightarrow\sum\_{\ell}\ell^{\pm}\mathbf{Z}) \\ &\approx \Gamma(\boldsymbol{E}^{\pm}\rightarrow\sum\_{\ell}\ell^{\pm}\boldsymbol{h}) \\ \approx \frac{G\_{F}}{8\sqrt{2}\pi}\sum\_{\ell}|\boldsymbol{V}\_{\ell\boldsymbol{N}}|^{2}\boldsymbol{M}\_{\boldsymbol{\Sigma}}^{3}. \end{split}$$

Thus the heavy lepton branching ratios exhibit asymptotic behavior consistent with the Goldstone Equivalence Theorem [260, 261], and are given by the relations [252, 326, 482, 484],

$$\frac{1}{2}\text{BR}(N \to \sum\_{\ell} \ell^{+}W^{-} + \ell^{-}W^{+}) \approx \text{BR}(N \to \sum\_{\nu} \nu Z + \bar{\nu}Z)$$

$$\approx \text{BR}(N \to \sum\_{\nu} \nu h + \bar{\nu}h)$$

$$\approx \frac{1}{2}\text{BR}(E^{\pm} \to \sum\_{\nu} \binom{-}{\nu}W^{\pm}) \approx \text{BR}(E^{\pm} \to \sum\_{\ell} \ell^{\pm}Z)$$

$$\approx \text{BR}(E^{\pm} \to \sum\_{\ell} \ell^{\pm}h) \approx \frac{1}{4}.\tag{5.17}$$

As displayed in **Figure 34** by Franceschini et al. [484], as the triplet mass grows, this asymptotic behavior can be seen explicitly in the triplet lepton partial widths.

#### 5.1.2. Type I+III Hybrid Seesaw in Grand Unified and Extended Gauge Theory

One plausible possibility to rescue the minimal grand unified theory, i.e., SU(5), is to introduce an adjoint 24<sup>F</sup> fermion multiplet in addition to the original 10<sup>F</sup> and 5¯ <sup>F</sup> fermionic representations [5, 485]. As the 24<sup>F</sup> contains both singlet and triplet fermions in this non-supersymmetric SU(5), the SM gauge couplings unify and neutrino masses can generated through a hybridization of the Types I and III Seesaw mechanisms. The Yukawa interactions and Majorana masses in this Type I+III Seesaw read [482]

$$
\Delta \mathcal{L}\_{\text{I+III}}^Y = Y\_S LHS + Y\_T LHT - \frac{M\_\text{S}}{2} \text{SS} - \frac{M\_T}{2} TT + \text{H.c., (5.18)}
$$

where S and T = T <sup>−</sup>+T + √ 2 , T <sup>−</sup>−T + i √ 2 , T 0 are the fermionic singlet and triplet fields, respectively, with masses M<sup>S</sup> and MT. In the limit that MS, M<sup>T</sup> ≫ YSv0, YTv0, the light neutrino masses are then given by the sum of the individual Type I and III contributions

$$m\_{\boldsymbol{\nu}} = - (Y\_{\mathcal{S}} \boldsymbol{\nu}\_0 / \sqrt{2})^2 M\_{\mathcal{S}}^{-1} - (Y\_T \boldsymbol{\nu}\_0 / \sqrt{2})^2 M\_T^{-1},\qquad \text{(5.19)}$$

The most remarkable prediction of this SU(5) theory is that the unification constraint and the stability of proton require the triplet mass to be small: M<sup>T</sup> . 1 TeV [485, 486]. Thus, in SU(5) scenarios, the triplet leptons of this Type I+III Seesaw are within the LHC's kinematic reach and can be tested via L-violating collider signatures [5, 487–491].

Other GUT models that can accommodate the Type III Seesaw and potentially lead to collider-scale L-violation include variations of SO(10) [492] theories. It is also possible to embed the Type III scenario into extended gauge sectors, including Left-Right Symmetric theories [134, 135, 493, 494], which also represents a Type I+II+III hybrid Seesaw hat trick. Additionally, Type III-based hybrid Seesaws can be triggered via fermions in other SU(2)L×U(1)<sup>Y</sup> representations [495–498], The collider phenomenology in many of these cases is very comparable to that of the Type I and II Seesaws, as discussed in sections 3 and 4, or the more traditional Type III scenario, which we now discuss.

### 5.2. Heavy Charged Leptons and Neutrinos at Colliders

#### 5.2.1. Heavy Charged Leptons and Neutrinos at pp Colliders

Due to the presence of both gauge and Yukawa couplings to SM fields, the collider phenomenology for triplet leptons is exceedingly rich. In hadron collisions, for example, pairs of heavy triplet leptons are produced dominantly via charged and neutral Drell-Yan (DY) currents, given by

$$q\bar{q}' \to W^{\*\pm} \to T^{\pm}T^0 \text{and} q\bar{q} \to \gamma^\*/Z^\* \to T^+T^-, \quad \text{(5.20)}$$

and shown in **Figure 35A**. For the DY process, the total cross section is now known up to NLO and differentially at NLO+LL in p<sup>T</sup> resummation [275]. As function of mass, the Nℓ ± (singlet) as well as T +T − and T ±T 0 (triplet) DY production cross sections at √ s = 14 and 100 TeV are displayed in **Figure 36A**. While the three rates are naïvely comparable, one should assign a mixing factor of |VℓN| <sup>2</sup> . 10−<sup>2</sup> to the singlet production since it proceeds through active-sterile neutrino mixing, i.e., Yukawa couplings, whereas triplet lepton pair production proceeds through gauge couplings. Heavy triplet leptons can also be produced singly in the association with light leptons and neutrinos,

$$q\bar{q}' \to W^{\*\pm} \to T^{\pm}\nu,\ T^0\ell^{\pm} \text{ and } q\bar{q} \to \nu^\*/Z^\* \to T^{\pm}\ell^{\mp}. \text{(5.21)}$$

As single production modes are proportional to the small [88] doublet-triplet mixing, denoted by |VℓT|, these processes suffer from the same small signal rates at colliders as does singlet production in Type I-based Seesaws (see section 3.1.1). However, as heavy-light lepton vertices also posses axial-vector contributions, new production channels are present, such as through the gluon fusion mechanism [242, 245, 258, 259], shown in **Figure 35B** and given by

$$\text{gg} \to Z^\*/h^\* \to T^{\pm} \ell^{\mp}.\tag{5.22}$$

It is noteworthy that the partonic expression for gluon fusion channels gg → Z ∗ /h <sup>∗</sup> → T ±ℓ ∓ is equal to the Type I analog gg → Nν<sup>ℓ</sup> [258], and hence its QCD corrections [259], but that

FIGURE 36 | (A) As a function of mass, the Nℓ ± (singlet) as well as T +T − and T ±T 0 (triplet) DY production cross sections at √ s = 14 and 100 TeV. (B) As a function of collider energy √ s, the T +T − and T ±ℓ <sup>∓</sup> (assuming benchmark |Vℓ<sup>T</sup> | <sup>2</sup> <sup>=</sup> <sup>10</sup>−<sup>2</sup> ) production cross sections via various production mechanisms.

heavy triplet pair production through gluon fusion, i.e., gg → TT, is zero since their couplings to weak bosons are vector-like, and hence vanish according to Furry's Theorem [242, 245, 447]. For √ <sup>s</sup> <sup>=</sup> <sup>7</sup> <sup>−</sup> 100 TeV, the N3LL(Threshold) corrections to the Born rates span +160% to +260% [259]. Hence, for singly produced triplet leptons, the gluon fusion mechanism is dominant over the DY channel for √ s & 20 − 25 TeV, over a wide range of EW- and TeV-scale triplet masses [258, 259]. More exotic production channels also exist, such as the γγ → T +T − VBF channel, shown in **Figure 35C**, as well as permutations involving W and Z. However, their contributions are sub-leading due to coupling and phase space suppression.

For representative heavy lepton masses of M<sup>T</sup> = 500 GeV and 1 TeV as well as doublet-triplet mixing of |VℓT| <sup>2</sup> <sup>=</sup> <sup>10</sup>−<sup>2</sup> , we display in **Figure 36B** the pp → T +T − and T ±ℓ ∓ production cross sections via various hadronic production mechanisms as a function of collider energy √ s. In the figure, the dominance of pair production over single production is unambiguous. Interestingly, considering that the triplet mass splitting is <sup>1</sup>M<sup>T</sup> <sup>∼</sup> <sup>O</sup>(200) MeV as stated above, one should not expect to discover the neutral current single production mode without also observing the charged channel almost simultaneously. Hence, despite sharing much common phenomenology, experimentally differentiating a Type I scenario from a Type III (or I+III) scenario is straightforward.

Leading order-accurate Monte Carlo simulations for treelevel processes involving Type III leptons are possible with the Type III Seesaw FeynRules UFO model [475, 499, 500], as well as a Minimal Lepton Flavor Violation variant MLFV Type III Seesaw [476, 477, 501]. The models can be ported into modern, general-purpose event generators, such at Herwig [289], MadGraph5\_aMC@NLO [290], and Sherpa [291].

Hadron collider tests of the Type III Seesaw can be categorized according to the final-state lepton multiplicities, which include: the L-violating, same-sign dilepton and jets final state, ℓ ± 1 ℓ ± 2 + nj [252, 326, 481, 482, 484, 485, 499, 502]; the four-lepton final state, ℓ ± 1 ℓ ± 2 ℓ ∓ 3 ℓ ∓ <sup>4</sup> + nj [252, 326, 481, 484, 499]; other charged lepton multiplicities [252, 326, 484, 499, 503]; and also displaced charged lepton vertices [484, 504]. Other "displaced" signatures, include triplet lepton decays to displaced Higgs bosons [505]. Direct searches for Type III Seesaw partners at the √ s = 7/8 TeV [56, 57, 506] and √ s = 13 TeV [58, 507, 508] LHC have yet to show evidence of heavy leptons. As shown in **Figure 37A**, triplet masses below M<sup>T</sup> . 800 GeV have been excluded at 95% CLs [508]. **Figure 37B** displays the discovery potential of

denote heavy neutral lepton mass in GeV) [490].

triplet leptons at high-luminosity 100 TeV collider. One can discover triplet lepton as heavy as 4 (6.5) TeV with 300 (3000) fb−<sup>1</sup> integrated luminosity. The absence of triplet leptons in multi-lepton final states can also be interpreted as a constrain on doublet-triplet neutrino mixing. In **Figures 37C,D**, one sees the exclusion contours of doublet-triplet neutrino mixing in |VµN| − <sup>|</sup>VeN<sup>|</sup> and <sup>|</sup>VτN|−|VeN<sup>|</sup> spaces after <sup>L</sup> <sup>=</sup> 4.9 fb−<sup>1</sup> of data at CMS (labels denote heavy neutral lepton mass in GeV) [490].

### 5.2.2. Heavy Charged Leptons and Neutrinos at ee and ep Colliders

The triplet leptons can also be produced at the leptonic colliders like the ILC and the Compact Linear Collider (CLIC) [482, 509], and the electron-hadron collider like LHeC [309]. Besides the similar s-channels as hadron colliders, at e +e − colliders, the triplet lepton single and pair productions can also happen in t-channel via the exchange of h, W, or Z boson. Triplet leptons can also lead to anomalous pair production of SM weak bosons [470]. Assuming M<sup>6</sup> = 500 GeV and VeN = 0.05, the cross sections of triplet lepton single and pair productions are shown in **Figure 38A**. For the single production at 1 TeV e +e − collider, the triplet lepton with mass up to about 950-980 GeV can be reached with 300 fb−<sup>1</sup> . To discover the heavy charged lepton through e +e <sup>−</sup> <sup>→</sup> <sup>6</sup>+6<sup>−</sup> production at <sup>√</sup> s = 2 TeV, the luminosity as low (high) as 60 (480) fb−<sup>1</sup> is needed as shown in **Figure 38B**.

## 6. RADIATIVE NEUTRINO MASS MODELS AND LEPTON NUMBER VIOLATION AT COLLIDERS

A common feature of the Seesaw mechanisms discussed in the previous sessions is that they are all tree-level, UV completion of the dimension-5 Weinberg operator in of Equation (1.1). Though economical and elegant, these models often imply subtle balancing between a Seesaw mass scale at TeV or below and small Yukawa couplings, in the hope for them to be observable in the current and near future experiments. In an altogether different paradigm, it may be the case that small neutrino masses are instead generated radiatively. In radiative neutrino mass models, loop and (heavy) mass factors can contribute to the suppression of light neutrino masses and partly explain their smallness. A key feature of radiative neutrino mass models is that the Weinberg

operator is not generated at tree-level: For some models, this may be because the particles required to generate tree-level masses, i.e., SM singlet fermions in Type I, triplet scalars in Type II, or triplet leptons in Type III, do not exist in the theory. For others, it may be the case that the required couplings are forbidden by new symmetries. Whatever the case, it is necessary that the new field multiplets run in the loops to generate neutrino masses.

At one-loop, such models were first proposed in Zee [28] and Hall and Suzuki [29], at two-loop in Cheng and Li [16], Zee [30], and Babu [31], and more recently at three-loop order in Krauss et al. [32]. Besides these early works, a plethora of radiative mass models exist due to the relative ease with which unique loop topologies can be constructed at a given loop order, as well as the feasibility to accommodate loop contributions from various exotic particles, including leptoquarks, vector-like leptons and quarks, electrically charged scalars, and EW multiplets. For a recent, comprehensive review, see Cai et al. [510].

However, the diversity of the exotic particles and interactions in radiative neutrino mass models make it neither feasible nor pragmatic to develop a simple and unique strategy to test these theories at colliders. Although some effort has been made to advance approaches to collider tests of radiative neutrino mass models more systematically [511, 512], it remains largely modeldependent. As a comprehensive summary of the literature for radiative neutrino mass models and their collider study is beyond the scope of this review, in this section, we focus on a small number of representative models with distinctive L-violating collider signatures.

It is worth pointing out that some popular radiative neutrino mass models do not predict clear lepton number violation at collider scales. A prime example are the Scotogenic models [513], a class of one-loop radiative neutrino mass scenario with a discrete Z<sup>2</sup> symmetry. Scotogenic models typically contain three SM singlet fermions N<sup>i</sup> with Majorana masses and are odd under the Z2, whereas SM fields are even. The discrete symmetry forbids the mixing between the SM neutrinos and N<sup>i</sup> that one needs to trigger the Type I and III Seesaw mechanisms. As a result, collider strategies to search for lepton number violation mediated by heavy Majorana neutrinos as presented in section 3 are not applicable to the Scotogenic model. Instead, collider tests of Scotogenic models include, for example, searches for the additional EW scalars [514–517] that facilitate lepton number conserving processes. Subsequently, we avoid further discussing radiative models without collider-scale lepton number violation.

Like in the previous sections, we first present in section 6.1 an overview of representative radiative models. Then, in section 6.2, we review collider searches for lepton number violation associated with radiative neutrino mass models.

### 6.1. Selected Radiative Neutrino Mass Models

#### 6.1.1. The Zee-Babu Model

The first radiative scenario we consider is the well-known Zee-Babu model, a two-loop radiative neutrino mass model proposed independently by Zee [30] and Babu [31]. In the model, the SM field content is extended by including one singly-charged scalar (h ±) and one doubly-charged scalar (k ±±). Both scalars are singlets under SU(3)<sup>c</sup> ×SU(2)L, leading to the lepton number violating interaction Lagrangian

$$
\Delta \mathcal{L} = \bar{L}Y^\dagger e\_\mathsf{R} H + \bar{\tilde{L}} f \mathcal{L} h^+ + \overline{e\_\mathsf{R}^c} g e\_\mathsf{R} k^{++} + \mu\_{\mathsf{Z}\mathsf{B}} h^+ h^+ k^{--} + \text{H.c.} \tag{6.1}
$$

where L (H) is the SM LH lepton (Higgs) doublet. The 3 × 3 Yukawa coupling matrices f and g are anti-symmetric and symmetric, respectively. The trilinear coupling µZB contributes to the masses of the charged scalars at the loop level. For large values of (µZB/mh<sup>±</sup> ) or (µZB/mk±± ), where mh±,k±± are the masses of h ± and k ±±, the scalar potential may have QEDbreaking minima. This can be avoided by imposing the condition |µZB| ≪ 4π min(mh, m<sup>k</sup> ).

The combined presence of Y, f , g and µZB collectively break lepton number and lead to the generation of a small Majorana neutrino mass. At lowest order, neutrino masses in the Zee-Babu model arise at two-loop order, as depicted in **Figure 39A**. The resulting neutrino mass matrix scales as

$$\mathcal{M}\_{\boldsymbol{\nu}} \simeq \left(\frac{\boldsymbol{\nu}^{2}\mu\_{\rm ZB}}{96\pi^{2}M^{2}}\right)f\mathbf{Y}\mathbf{g}^{\dagger}\boldsymbol{Y}^{T}\boldsymbol{f}^{T},\tag{6.2}$$

where M = max(mh<sup>±</sup> , mk±± ) is the heaviest mass in the loop. Since f is antisymmetric, the determinant of the neutrino mass matrix vanishes, detM<sup>ν</sup> <sup>=</sup> 0. Therefore the Zee-Babu models yields at least one exactly massless neutrino. An important consequence is that the heaviest neutrino mass is determined by the atmospheric mass difference, which can be estimated as

$$m\_{\upsilon} \approx 6.6 \times 10^{-3} f^2 g \left(\frac{m\_{\tau}^2}{M}\right) \approx 0.05 \,\text{eV} \,\text{,}\tag{6.3}$$

where m<sup>τ</sup> ≈ 1.778 GeV is the tau lepton mass. This implies the product f 2 g can not be arbitrarily small, e.g., for M ∼ 100 GeV, one finds g 2 f & 10−<sup>7</sup> . Subsequently, the parameter space of the Zee-Babu model is constrained by both neutrino oscillation data, low-energy experiments such as decays mediated k ±± at tree level, and high-energy searches for direct pair production of k ±±.

The study of h ± is mostly similar to that of the singlycharged scalar in the Zee model [28], although the lepton number violating effects are not experimentally observable due to the missing information carried away by the light (Majorana) neutrino in the decay product. The doubly-charged scalar k ±± can decay to a pair of same-sign leptons, which manifestly violates lepton number by 1L = ±2, with a partial decay width given by

$$
\Gamma(k^{\pm \pm} \to \ell\_a^{\pm} \ell\_b^{\pm}) = \frac{\left| \mathbf{g}\_{ab} \right|^2}{4\pi \left( 1 + \delta\_{ab} \right)} m\_k \,\,\,\,\tag{6.4}
$$

If mk±± > 2mh<sup>±</sup> , then the k ±± → h ±h ± decay mode opens with a partial decay width of

$$\Gamma(k^{\pm\pm} \to h^{\pm}h^{\pm}) = \frac{m\_{k^{\pm\pm}}}{8\pi} \left(\frac{\mu\_{ZB}}{m\_{k^{\pm\pm}}}\right)^2 \sqrt{1 - \frac{4m\_{h^{\pm}}^2}{m\_{k^{\pm\pm}}^2}}\,. \tag{6.5}$$

Doubly-charged scalars, appear in many other radiative neutrino mass models, including the three-loop Cocktail Model [518], whose eponymous mass-generating diagram is shown in the right panel of **Figure 39**. The doubly-charged scalar couples to the SM lepton doublet and a singly-charged scalar in the same manner as in the Zee-Babu model, and thus again is similar to a Type II scenario. Radiative Type II Seesaw model [519] that generates neutrino mass at one-loop order contains an SU(2)<sup>L</sup> triplet scalar and thus also has similar LHC phenomenology as the tree-level Type II Seesaw mechanism [520].

#### 6.1.2. The Colored Zee-Babu Model With Leptoquark

In a particularly interesting variant of the Zee-Babu model, proposed in Kohda et al. [521], all particles in the neutrino massloop are charged under QCD. As shown in **Figure 40**, the lepton doublet in the loop of the Zee-Babu model is replaced with down-type quark while the singly- and doubly-charged scalars are replaced with a leptoquark S − 1 3 LQ and a diquark S − 2 3 DQ . Under the SM gauge group, the leptoquark and diquark quantum numbers are

$$S\_{LQ}^{-\frac{1}{3}} \quad : \quad \text{(3, 1, } -\frac{1}{3}) \quad \text{and} \quad S\_{DQ}^{-\frac{2}{3}} \quad : \quad \text{(6, 1, } -\frac{2}{3}) \, . \tag{6.6}$$

The decay of the diquark S − 2 3 DQ is analogous to that of the doublycharged scalar k ±± in that it can decay to a pair of samesign down-type quarks or a pair of same-sign leptoquarks, if kinematically allowed.

For the models mentioned above, we will only review the collider study with the characteristics different from the tree-level Seesaws in the following.

$$S\_{LQ}^{-\frac{1}{2}} \stackrel{\mu}{\underset{\nu}{\longrightarrow}} \begin{array}{c} \mu \\ \nu \\ \nu \\ \nu\_{\ell L} \end{array} \stackrel{\nu}{\underset{\nu}{S\_{LQ}^{-\frac{1}{2}}}} \sim S\_{LQ}^{\frac{1}{2}}$$

$$\nu\_{\ell L} \frac{\int (d\_{iL})\_{\otimes}^{c} (d\_{iR})\_{\dagger}^{c} \, d\_{jR} \otimes \, d\_{jL}}{(Y\_{L})\_{\ell i} \, \, m\_{d\_i} \, (Y\_s)\_{ij} \, \, m\_{d\_j} \, (Y\_L^T)\_{j\ell'}} \, (\nu\_{\ell' L})^c$$

FIGURE 40 | Feynman diagram for the generation of neutrino masses at two-loop order in the colored Zee-Babu model [521].

### 6.2. Radiative Neutrino Mass Models at Colliders

#### 6.2.1. Doubly-Charged Scalar at the LHC

As mentioned above, the Zee-Babu model contains two singlet charged scalars, h ± and k ±±. Moreover, due to the presence of the doubly-charged scalar decay mode to two same-sign leptons k ±± → ℓ ±ℓ <sup>±</sup> via the coupling µZB, collider searches for L-violating effects in the context of the Zee-Babu model are centered on k ±± and its decays.

Like the triplet Higgs in Type II Seesaw, the doubly-charged scalar k ±± can be pair produced via the Drell-Yan process at the LHC if kinematically accessible and is given by

$$\text{pp} \to \text{y}^\*/\text{Z}^\* \to \text{k}^{++}\text{k}^{--}.\tag{6.7}$$

This is the same process as shown in **Figure 28A**. However, an important distinction is that while H±± in the Type II Seesaw is an SU(2)<sup>L</sup> triplet, the k ±± here is a singlet. As this quantumnumber assignment leads to different Z boson couplings, and hence different production cross section at colliders, it is a differentiating characteristic of the model. Note the γγ fusion processes, shown in **Figure 28**, also applies to k ++k −− pair production and leads to the same production cross section.

Since the collider signal for pair produced k ±± is the same as H±± in the Type II Seesaw, the search for doubly-charged scalar can be easily performed for both cases as shown in **Figure 31**. Obviously the constraint on the singlet is less stringent due to the absence of weak isospin interactions. With 36.1 fb−<sup>1</sup> data at 13 TeV, ATLAS has excluded k ±± mass lower than 656-761 GeV for BR(k ±± → e ±e <sup>±</sup>) + BR(k ±± → µ ±µ <sup>±</sup>) = 1 at 95% CLs [460].

Low energy LFV experiments, especially µ → eγ , impose very stringent constraints on the parameter space of the Zee-Babu model. The MEG experiment [522, 523] has placed an upper bound on the decay branching ratio BR(<sup>µ</sup> <sup>→</sup> <sup>e</sup><sup>γ</sup> ) <sup>&</sup>lt; 4.2×10−<sup>13</sup> , which can be roughly translated as [524]

$$\left|f\_{13}^{\*}f\_{23}\right|^2 \frac{m\_{k^{\pm\pm}}^2}{m\_{h^{\pm}}^2} + 16 \left|\sum g\_{1k}^{\*}g\_{i2}\right|^2 < 1.2 \times 10^{-6} \left(\frac{m\_k}{\text{TeV}}\right)^4. \tag{6.8}$$

To satisfy LFV constraints, the doubly- and singly-charged scalar masses are pushed well above TeV, with mk±± > 1.3 (1.9) TeV and mh<sup>±</sup> > 1.3 (2.0) TeV for the NH (IH), assuming µZB = min(mk±± , mh<sup>±</sup> ). This can be very easily relaxed, however, by choosing larger µZB and balancing smaller Yukawa couplings to generate the right neutrino mass spectrum.

A recent study has projected the sensitivities of the LHC with large luminosities by scaling the cross section bound by 1/ √ L for two benchmark scenarios: one for NH and one for IH [525]. The projected sensitivities are shown in **Figure 41** for model parameters consistant with neutrino oscillation data. Note that these benchmarks are chosen to have µZB = 5 min(mk±± , mh<sup>±</sup> ) such that the constraints from flavor experiments such as µ → eγ are much less stringent at the price of a more fine-tuned the scalar potential. We can see that the NH benchmark is less constrained than the IH one when mk±± < 2mh<sup>±</sup> because k ±± has a smaller branching ratio to leptons.

#### 6.2.2. Leptoquark at the LHC

In the colored Zee-Babu model, L-violating signals can be observed in events with pair produced leptoquarks S − 1 3 LQ via s-channel diquark S − 2 3 DQ , shown in **Figure 42**, and given by,

$$\mathcal{pp} \to \mathcal{S}\_{DQ}^{-\frac{2}{3}\*} \to \mathcal{S}\_{LQ}^{-\frac{1}{3}} \mathcal{S}\_{LQ}^{-\frac{1}{3}} \to \mathcal{u}\ell^{-}\mathcal{u}\ell^{\prime -}.\tag{6.9}$$

One benchmark has been briefly studied in Kohda et al. [521]. For leptoquark mass of 1 TeV and diquark mass of 4 TeV, a benchmark consistent with neutrino oscillation data and low energy experiments, the L-violating process in Equation (6.9) can proceed with an LHC cross section of 0.18 fb at √ s = 14 TeV. So far, no dedicated collider study for this model. In general, however, one can recast either ATLAS or CMS search for heavy neutrinos, such as Aad et al. [52] and Khachatryan et al. [363], to derive the limit on the model parameter space.

Lepton number violating collider processes, pp → ℓ ±ℓ <sup>±</sup> + nj, involving charged scalars, leptoquarks and diquarks have also been studied for the LHC in Peng et al. [394], Helo et al. [526, 527]. Example diagrams are shown in **Figure 43**. Even though these studies are performed without a concrete neutrino mass model, they possess the most important ingredient of Majorana neutrino mass models: L violation by two units. Therefore radiative neutrino mass models can be constructed from the relevant matter content. Some processes, however, are realized with a SM singlet fermion (for example the left panel of **Figure 43**), which implies the existence of a tree-level Seesaw. Other processes without SM singlet fermions, SU(2)<sup>L</sup> triplet scalars, or triplet fermions, such as the one on the right panel of **Figure 43**, can be realized in a radiative neutrino mass model. Detailed kinematical analyses for resonant mass reconstruction would help to sort out the underlying dynamics.

#### 6.2.3. Correlation With Lepton Flavor Violation

In radiative neutrino mass models the breaking of lepton number generally needs the simultaneous presence of multiple couplings. For example, in the Zee-Babu model, Y, f , g and µZB together break lepton number. The observation of pair produced k ±± itself is insufficient to declare L violation. In order to establish L violation in the theory and thus probe the Majorana nature of the neutrinos, the couplings of h ± to SM leptons and to k ±± have to be studied at the same time. For the colored Zee-Babu model, the L violation process shown in **Figure 42** involves all couplings except the SM Yukawa necessary to break the lepton number. Note, however, the cross section for this process is proportional to the product of couplings and suppressed by the heavy exotic masses which both contributes to the smallness of the neutrino masses. Thus the cross section for this processes must be kinematically suppressed. For radiative neutrino mass models with dark matter candidates, probing lepton number violation at colliders alone is generally much more difficult as the dark matter candidate appears as missing transverse energy just as neutrinos. Overall, the study of L-violation of radiative neutrino mass models can be performed either with the combination of different processes that test different subsets of

FIGURE 41 | Projection of sensitivities at the LHC in the mk±± -mh<sup>±</sup> plane: (A) the NH benchmark with g11,22 = 0.1, g12,13,33 = 0.001, f12,13 = 0.01 and f<sup>23</sup> = 0.02; (B) the IH benchmark with g11,23 = 0.1, g12,22,13,33 = 0.0001, f<sup>12</sup> = −f<sup>13</sup> = 0.1 and f<sup>23</sup> = 0.01. For both benchmarks, the trilinear coupling is chosen to be <sup>µ</sup>ZB <sup>=</sup> <sup>5</sup> min(mk±± , <sup>m</sup>h<sup>±</sup> ). The gray shaded region in the left panel is excluded by low energy experiments. The green and orange regions are excluded by future experiments with an integrated luminosity of 70 fb−<sup>1</sup> and 3 ab−<sup>1</sup> respectively [525].

the couplings or in a single process that involves all couplings at once whose production cross section is generally suppressed.

On the contrary, radiative neutrino mass models contain LFV couplings and exotic particles that can be tested much easier than L violation stated above. The search strategies for LFV couplings and new particles vary from model to model. It is definitely impossible to cover all and they are also not the focus of this review. Thus we will take a few simple examples to illustrate the searches.

The leading LFV signals can be produced in a radiative neutrino mass model from the QCD pair production of the leptoquark S − 1 3 LQ with its suitable subsequent decays such as

$$pp \to \ S\_{LQ}^{+\frac{1}{3}} \mathcal{S}\_{LQ}^{-\frac{1}{3}} \to \ \ \overline{t}\ell^{+}t\ell'^{-} \tag{6.10}$$

where S + 1 3 LQ = S − 1 3 LQ <sup>∗</sup> and the top quarks decay hadronically. Note that the leptoquark pair can also decay to ¯bν<sup>ℓ</sup> tℓ ′− or channels can result in final states ℓ +ℓ ′−X, inclusive flavour offdiagonal charged lepton pair accompanied by missing transverse energy, jets etc., if the quarks decay to appropriate leptons. The same final states have been used to search for stop in SUSY theories and thus the results for stop searches at the LHC can be translated to that of the leptoquark S − 1 3 LQ , m(S − 1 3 LQ ) & 600 GeV [511] based on the ATLAS stop search at √ s = 8 TeV [528] 5 . No recast of stop search has been performed for 13 TeV run yet. Besides leptoquarks, radiative neutrino mass models also comprise exotic particles such as vector-like quarks, vectorlike leptons, charged scalar singlets (both singly- and doublycharged) and higher-dimensional EW multiplets. For example, disappearing tracks can be used to search for higher-dimensional EW multiplet fermions whose mass splitting between the neutral and the singly-charged component is around 100 MeV. The current LHC searches have set a lower mass limit of 430 GeV at 95% CL for a triplet fermion with a lifetime of about 0.2 ns [534–536]. We refer the readers to the section about collider tests of radiative neutrino mass model in Cai et al. [510] and the references therein for details.

We want to stress, however, that even though L violation in the radiative models is more complicated and challenging to search for in collider experiments, their observation is essential and conclusive to establish the Majorana nature of neutrinos. So once we find signals in either LFV processes or new particles searches, we should search for L violation in specific radiative neutrino mass models that give these LFV processes or contain these new particles, in order to ultimately test the generation of neutrino masses.

### 7. SUMMARY AND CONCLUSIONS

Exploring the origin of neutrinos' tiny masses, their large mixing, and their Dirac or Majorana nature are among the most pressing

<sup>¯</sup>bν<sup>ℓ</sup> bν¯<sup>ℓ</sup> ′ , where the LFV effects are not easy to disentangle at colliders due to the invisible neutrinos. However, these decay

<sup>5</sup>There are many dedicated leptoquark searches at the LHC [529–533]. However, the leptoquarks searched only couple to one generation of fermions at a time and thus generate no LFV signals.

issues in particle physics today. If one or more neutrino Seesaw mechanisms are realized in nature, it would be ultimately important to identify the new scales responsible for generating neutrino masses. Neutrino oscillation experiments, however, may not provide such information, and thus complementary pathways, such as collider experiments, are vital to understanding the nature of neutrinos. Observing lepton number violation at collider experiments would be a conclusive verdict for the existence of neutrino Majorana masses, but also direct evidence of a mass scale qualitatively distinct from those in the SM.

In this context, we have reviewed tests of low-scale neutrino mass models at pp, ep, and ee colliders, focusing particularly on searches for lepton number (L) violation: We begin with summarizing present neutrino oscillation and cosmology data and their impact on the light neutrino mass spectra in section 2. We then consider several representative scenarios as phenomenological benchmarks, including the characteristic Type I Seesaw in section 3, the Type II Seesaw in section 4, the Type III in section 5, radiative constructions in section 6, as well as extensions and hybridizations of these scenarios. We summarize the current status of experimental signatures featuring L violation, and present anticipated coverage in the theory parameter space at current and future colliders. We emphasize new production and decay channels, their phenomenological relevance and treatment across different collider facilities. We also summarize available Monte Carlo tools available for studying Seesaw partners in collider environments.

The Type I Seesaw is characterized by new right-handed, SM gauge singlet neutrinos, known also as "sterile neutrinos," which mix with left-handed neutrinos via mass diagonalization. As this mixing scales with light neutrino masses and elements of the PMNS matrix, heavy neutrino decays to charged leptons may exhibit some predictable patterns if one adopts some simplifying assumptions for the mixing matrix, as shown for example in **Figures 3**, **4**, that are correlated with neutrino oscillation data. The canonical high-scale Type I model, however, predicts tiny active-sterile mixing, with |VℓN| <sup>2</sup> <sup>∼</sup> <sup>m</sup><sup>ν</sup> /MN, and thus that heavy N decouple from collider experiments. Subsequently, observing lepton number violation in collider experiments, as discussed in section 3.2, implies a much richer neutrino mass-generation scheme than just the canonical, high-scale Type I Seesaw. In exploring the phenomenological parameter space, the 14 TeV LHC (and potential 100 TeV successor) and <sup>L</sup> <sup>=</sup> 1 ab−<sup>1</sup> integrated luminosity could reach at least 2σ sensitivity for heavy neutrino masses of M<sup>N</sup> . 500 GeV (1 TeV) with a mixing |VℓN| <sup>2</sup> . 10−<sup>3</sup> , as seen in **Figure 9**. If N is charged under another gauge group that also couples to the SM, as in B-L or LR gauge extensions, then the discovery limit may be extended to M<sup>N</sup> ∼ MZ′ , MW<sup>R</sup> , when kinematically accessible; see sections 3.2.4 and 3.2.5.

The Type II Seesaw is characterized by heavy SU(2)<sup>L</sup> triplet scalars, which result in new singly- and doubly-charged Higgs bosons. They can be copiously produced in pairs via SM electroweak gauge interactions if kinematically accessible at collider energies, and search for the doubly-charged Higgs bosons via the same-sign dilepton channel H±± → ℓ ±ℓ ± is an on-going effort at the LHC. Current direct searches at 13 TeV bound triplet scalar masses to be above (roughly) 800 GeV. With anticipated LHC luminosity and energy upgrades, one can expect for the search to go beyond a TeV. Furthermore, if neutrino masses are dominantly from triplet Yukawa couplings, then the patterns of the neutrino mixing and mass relations from the oscillation experiments will correlate with the decays of the triplet Higgs bosons to charged leptons, as seen from the branching fraction predictions in **Figures 25**, **26** and in **Table 2**. Since a Higgs triplet naturally exists in certain extensions beyond the SM, such as in Little Higgs theory, the LRSM, and GUT theories, the search for such signals may prove beneficial as discussed in section 4.2.2.

The Type III Seesaw is characterized by heavy SU(2)<sup>L</sup> triplet leptons, which result in vector-like, charged and neutral leptons. Such multiplets can be realized in realistic GUT theories in hybridization with heavy singlet neutrinos from a Type I Seesaw. Drell-Yan pair production of heavy charged leptons at hadron colliders is sizable as it is governed by the SM gauge interactions. They can decay to the SM leptons plus EW bosons, leading to same-sign dilepton events. Direct searches for promptly decaying triplet leptons at the LHC set a lower bound on the triplet mass scale of around 800 GeV. A future 100 TeV pp collider can extend the mass reach to at least several TeV, as seen in **Figure 37**.

Finally, neutrino masses can also be generated radiatively, which provides an attractive explanation for the smallness of neutrino masses with a plausibly low mass scale. Among the large collection of radiative neutrino mass models, the Zee-Babu model contains a doubly-charged SU(2)<sup>L</sup> singlet scalar with collider signal akin to the doubly-charged Higgs in the Type II Seesaw. ATLAS has excluded k ±± mass below 660 − 760 GeV assuming the benchmark decay rate P <sup>ℓ</sup>i=e,<sup>µ</sup> BR(<sup>k</sup> ±± → ℓ ± 1 ℓ ± 2 ) = 1. The high luminosity LHC is sensitive up to about a TeV for both k ±± and its companion scalar h ± in the Zee-Babu model with constraints from neutrino oscillation data and other low energy experiments. For the colored variant of the Zee-Babu model, a pair of same-sign leptoquark can be produced via an s-channel diquark at the LHC. Their subsequent decay lead to the lepton number violating same-sign dilepton plus jets final state, which still await dedicated studies.

As a final remark, viable low-scale neutrino mass models often generate a rich flavor structure in the charged lepton sector that predict lepton flavor-violating transitions. Such processes are typically much more easily observable than lepton number violating processes, in part due to their larger production and decay rates, and should be searched for in both high- and lowenergy experiments.

### AUTHOR CONTRIBUTIONS

The conception and content scope of this review was mainly designed by TH. TL edited the chapter on neutrino masses; TL and RR edited the chapters on the Type I, II, III seesaw models and their collider tests; YC edited the chapter on radiative neutrino mass models. TH drafted the concluding section. TH and RR revised the manuscript critically for important, intellectual content and expressions. Most of the materials compiled in the review came from the early publications from the collective works by the authors.

### REFERENCES


### ACKNOWLEDGMENTS

We thank Michael A. Schmidt for useful discussions. Past and present members of the IPPP are thanked for discussions. The work of TH was supported in part by the U.S. Department of Energy under grant No. DE-FG02- 95ER40896 and in part by the PITT PACC. The work of TL was supported in part by the Australian Research Council Centre of Excellence for Particle Physics at the Tera-scale. The work of RR was funded in part by the UK Science and Technology Facilities Council (STFC), the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements No 690575 (InvisiblesPlus RISE) and No 674896 (InvisiblesPlus RISE). Support for the open accessibility of this work is provided by the Research Councils UK, external grant number ST/G000905/1.

Republication of the various figures is granted under the terms of the Creative Commons Attribution Licenses, American Physical Society PRD License Nos.: 4234250091794, 423425077 4584, 4234260138469, 4234270420615, 4234270625995, 423427 0758210, 4234280275772, 4234280455112, 4234291212539, 423 4300360477, 4234301170893, 4234310175148, and PRL License No. 4234301302410.


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Cai, Han, Li and Ruiz. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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