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The Physics Associated with Neutrino Masses

Original Research ARTICLE

Front. Phys., 01 November 2017 | https://doi.org/10.3389/fphy.2017.00053

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 Z2×Z2 symmetry is imposed. Due to the exact Z2×Z2 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.

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 Δm212=7.50-0.17+0.19×10-5 eV2 and Δm312=2.524-0.040+0.039 (-2.514-0.041+0.038)×10-3eV2  for normal (inverted) mass hierarchy. The cosmological data, on the other hand, gives the upper bound of the sum of the neutrino masses as Σjmνj < 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 [36] 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(10−6).

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 TeV-scale 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 Z2 symmetry. The right-handed neutrino is odd under the Z2 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., [1146] 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 Nk and the inert doublet scalar field η=(η+,ηR0+iηI0)T. The neutrino masses are generated at the one-loop level, in which the Yukawa interactions YikνLiηNk and the scalar interaction (λ5/2)(Hη)2 contribute to the neutrino mass generation. The mass matrix is expressed as

(Mv)ij=kYikνYjkνMk32π2[mηR02mηR02Mk2ln (mηR0Mk)2              mηI02mηI02Mk2ln(mηI0Mk)2],

where Mk is the Majorana mass of the k-th generation of right-handed neutrino, mηR0 and mηI0 are the mass of the ηR0 and ηI0, respectively. In this model, we have two scenarios with respect to the DM candidate; the lightest right-handed neutrino N1 or the lighter Z2-odd neutral scalar field (ηR0 or ηI0). 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 Z2 symmetry as in the above models. However, if the DM stabilizing symmetry is larger than Z2: ZN (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 [4146]. 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 Z2×Z2 symmetry a set of stable particles can exist. Introducing two new scalar fields, the λ5 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

n˙χ+3Hnχ=σχχXXv(nχ2n¯χ2),    (1)

where nχ is the DM number density, n̄χ is the equilibrium number density and σχχXXv 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*1/2T2/MPL, where g* 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 Yχ = nχ/s and the inverse temperature x = m/T. Here s is the entropy density s=(2π2/45)g*T3 and m is the mass of the DM particle. Using the replacement of dx/dt = H|T = m/x, we obtain

dYχdx=0.264 g*1/2mMPLx2 σχχXXv(YχYχY¯χY¯χ).    (2)

The thermally averaged cross section σχχXXv of O(10−9) GeV with a DM mass of 100 GeV gives Yχ10-12, which is consistent with the observed value of the relic abundance Ωh2 ≃ 0.12 [88].

In the multicomponent DM system three types of processes enter in the Boltzmann equations1:

χiχiXX    (standard annihilation),    (3)
χiχiχjχj    (DM conversion),    (4)
χiχjχkX    (semi-annihilation).    (5)

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 χi with mass mi are

dYidx=0.264 g*1/2μMPLx2{σχiχiXXv(YiYiY¯iY¯i)           +i>jσχiχiχjχjv​​(YiYiYjYjY¯jY¯jY¯iY¯i)           j>iσχjχjχiχiv(YjYjYiYiY¯iY¯iY¯jY¯j)             +j,kσχiχiχkXijkv(YiYjYkY¯kY¯iY¯j)           j,kσχjχkχkXijkv(YjYkYiY¯iY¯jY¯k)}.    (6)
FIGURE 1
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Figure 1. Standard annihilation (Left), DM conversion (Middle) and semi-annihilation (Right).

Here x = μ/T and 1/μ=(imi-1) 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 Z2×Z2 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 Z2×Z2 symmetry. We refer to them as “model A” and “model B.” Owing to the Z2×Z2 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 Nk, an SU(2)L doublet scalar η, and two SM singlet scalars χ and ϕ. Note that the lepton number L′ of N is zero. The Z2×Z2×L -invariant Yukawa sector and Majorana mass term for N can be described by

LY=YijeHLilRjc+YikνLiϵηNk12MkNkNk+h.c. ,    (7)

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

Vλ=λ1(HH)2+λ2(ηη)2+λ3(HH)(ηη)+λ4(Hη)(ηH)       +γ1χ4+γ2(HH)χ2+γ3(ηη)χ2+γ4|ϕ|4+γ5(HH)|ϕ|2       +γ6(ηη)|ϕ|2+γ7χ2|ϕ|2+κ2[(Hη)χϕ+h.c.] ,    (8)
Vm=m12HH+m22ηη+12m32χ2+m42|ϕ|2+12m52[ϕ2       +(ϕ*)2] .    (9)
TABLE 1
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Table 1. The matter content and the corresponding quantum numbers of the model A.

The Z2×Z2 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 “λ5 term,” (λ5/2)(Hη)2, is also forbidden by L′. A small λ5 of the original model of Ma [11] is “natural” according to 't Hooft [100], because the absence of λ5 implies an enhancement of symmetry. In fact, if λ5 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 λ5 dynamically, such that the λ5 term becomes calculable.

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

H=(H+(vh+h+iG)/2) , η=(η+(ηR0+iηI0)/2) ,ϕ=(ϕR+iϕI)/2 ,    (10)

where vh is the vacuum expectation value. The tree-level masses of the scalars are given by

mh2=2λ1vh2 ,    (11)
mη±2=m22+λ3vh2/2 ,    (12)
mηR02=mηI02=m22+(λ3+λ4)vh2/2 ,    (13)
mϕR2=m42+m52+γ5vh2,    (14)
mϕR2=m42-m52+γ5vh2,    (15)
mχ2=m32+γ2vh2.    (16)

Although the tree-level mass of ηR0 is the same as that of ηI0 as shown in Equation (13), the degeneracy is lifted at the one-loop level via the effective λ5 term:

λ5eff~κ2128π2m52mϕR2mχ2[1mχ2mϕR2mχ2lnmϕR2mχ2]          for m5mϕR .    (17)

This correction is embedded into the two-loop diagram to generate the neutrino mass (see Figure 2). The 3 × 3 neutrino mass matrix Mν can be given by

(Mν)ij=(116π2)2κ2vh216               kYikνYjkνMk 0dx x(x+mη02)2(x+Mk2)              01dzln[zmχ2+(1z)mϕI2+z(1z)xzmχ2+(1z)mϕR2+z(1z)x],    (18)

where we have assumed that mη0=mηR0mηI0.

FIGURE 2
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Figure 2. Two-loop radiative neutrino mass of the model A.

Using λ5eff given in Equation (17), the neutrino mass matrix can be approximated as

(Mν)ij~λ5effvh232π2kYikνYjkνMkmη02Mk2[1Mk2mη02Mk2lnmη02Mk2].    (19)

We see from Equations (17) and (19) that the neutrino mass matrix Mν is proportional to |Yνκ|2m52. When mχ, mϕR, mη0, Mk~O(102) GeV, for instance, implies that |Yνκ|m5~O(10-1) GeV to obtain the neutrino mass scale of O(0.1) eV. With the same set of the parameter values we find that λ5eff~10-6, where the smallness λ5eff is a consequence of the radiative generation of this coupling. As we will see, the product |Yνκ| enters into the semi-annihilation of DM particles which produces monochromatic neutrinos, while the upper bound of |Yν| follows from the μ constraint.

3.1. Multicomponent Dark Matter System

In the model A there are three type of dark matter candidates ; N1 (the lightest among Nk's) or ηR0 (or ηI0) with (Z2,Z2)=(-,+), χ with (Z2,Z2)=(+,-) and ϕR (or ϕI) with (Z2,Z2)=(-,-). For (Z2,Z2)=(-,+) there are two candidates. In the following discussions we assume that N1 is a DM candidate [40]. The other possibility, ηR0-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:

Standard annihilation: N1N1XX,  ϕRϕRXX,                                    χχXX,     (20)
DM conversion: ϕRϕRχχ,     (21)
Semi-annihilation: N1ϕRχν,  χN1ϕRν,                               ϕRχN1ν.    (22)

Here we have assumed mϕR > mχ and mϕR + mχ < M2, 3. Moreover, since the mass difference between ϕR and ϕI is controlled by the lepton-number breaking mass m5, which is assumed to be much smaller than mϕR. Then mϕR and mϕI are practically degenerate and the contribution of ϕI 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 35. Since the reaction rate of the conversion between ϕR and ϕI can reach chemical equilibrium during the decoupling of DMs, we can sum up the number densities of ϕR and ϕI and compute the relic abundance of ΩϕRh2 [40].

FIGURE 3
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Figure 3. The diagrams for the standard annihilation processes.

FIGURE 4
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Figure 4. The diagrams for the DM conversion processes.

FIGURE 5
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Figure 5. The diagrams for semi-annihilation process.

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

σieff=σi(Ωih2Ωtotalh2) .    (23)

In our model, only χ and ϕR scatter with the nucleus, and the right-handed neutrino N1 does not interact with nucleus at tree level. So we can neglect the N1 contribution at the lowest order in perturbation theory. The effective cross sections of ϕR and χ are expressed as

σχeff=σχ(Ωχh2Ωtotalh2),   σϕReff=σϕR(ΩϕRh2Ωtotalh2),    (24)

where σχ and σϕR are the spin independent cross sections and given by

σχ=1π(γ2f^mNmχmh2)2(mNmχmN+mχ)2 ,    (25)
σϕR=1π((γ5/2)f^mNmϕRmh2)2(mNmϕRmN+mϕR)2 .    (26)

Here f^~0.3 is the usual nucleonic matrix element [101] and mN 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 one-component 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 ≳1 TeV [104106]. In the model A with the multicomponent DM system, the constrains on the cross sections off the nucleon for χ and ϕR are also relatively severe. As a benchmark we take the mass of χ as mχ = mh/2 while vary the mass of ϕR in the following analysis2.

In the original one-loop neutrino mass model in Ma [11], the relic density of N1 tends to be larger than the observational value [48]. The additional contributions coming from the semi-annihilation can enhance the annihilation rate for N1 so that the N1 DM contribution to the total relic abundance can be suppressed. This situation is realized for M1 > mϕR, mχ as can be seen later.

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

mχ=62 GeV,  M1=300 GeV,    (27)
mηR0=mηI0=mη+=mχ+mϕR10 GeV,    (28)
mϕI=mϕR+5 GeV,    (29)
γ2=0.004,  γ5=0.05,  γ7=0.17,    (30)
κ=0.4,  Yijν=0.01.    (31)

The masses of heavier right-handed neutrinos are M2 = M3 = 1 TeV. The mass differences between mηR0 and the sum of mχ and mϕR 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 [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 Ωχh2, ΩϕRh2, and ΩN1h2 and the total relic abundance Ωtotalh2(=Ωχh2+ΩϕRh2+ΩN1h2) as a function of mϕR for the benchmark set. The horizontal dashed line stands for the observed value Ωobsh2~0.12. It is shown that the relic abundance of the χ is Ωχ ≃ Ωobs/2. When ϕR 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 mϕR ≃ 280 GeV.

FIGURE 6
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Figure 6. Relic abundances Ωχh2, ΩϕRh2, and ΩN1h2 and the total relic abundance Ωtotalh2 as a function of mϕR. The relevant masses and couplings are taken as in Equations (27–31). The horizontal dashed line stands for the observed value Ωobsh2~0.12.

The left panel in Figure 7 shows the contour plot for the mϕR − γ5 plane where the total relic density of DM can be made consistent with the observed value Ωobsh2~0.12. We take two values, 10 GeV (black line) and 1 GeV (red line), for the mass difference between mηR0 and mχ+mϕR in Equation (28). The other parameters are taken as the same in Equations (27–31). We can see the scalar coupling γ5 increases drastically as mϕR increases for mϕR ≳ 290 GeV. It is because the relic density of the N1 DM, ΩN1h2, becomes significant for mϕR ≳ 290 GeV, so that ΩϕRh2 should be drastically suppressed. The scalar couplings of DM particles with the SM Higgs boson, γ2 and γ5, and the DM masses are constrained by the DM direct detection experiments. For the χ DM, the effective cross section off nucleon σχeff in Equation (24) is σχeff~10-47 cm2 for the benchmark set. It is an order of magnitude smaller than the current experimental bound. For the ϕR DM, the right panel in Figure 7 shows the relation between mϕR and the effective cross section σϕReff for (mχ+mϕR)-mηR0=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 γ5 becomes large for mϕR ≳ 290 GeV and then the cross sections off the nucleon σϕR becomes large, the effective cross section σϕReff decreases for mϕRM1(= 300 GeV), since the abundance of ϕR decreases. For the case of (mχ+mϕR)-mηR0= 10 GeV, it can be seen that the mass region 288 GeV ≲ mϕR 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 ϕR for the case of (mχ+mϕR)-mηR0= 1 GeV. This is because the relic abundance of ϕR 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 M1 = 500 GeV and γ7 = 0.28. >From the right panel in Figure 8, we see that 485 (490) GeV ≲ mϕR ≲ 510 (502) GeV is excluded by the direct detection experiments for (mχ+mϕR)-mηR0= 10 (1) GeV.

FIGURE 7
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Figure 7. (Left) Contour plot for the total relic density Ωtotalh2~0.12. (Right) The relation between the mϕ 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 mηR0 and mχ+mϕR.

FIGURE 8
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Figure 8. The same as Figure 7 but for M1 = 500 GeV and γ7 = 0.28.

Before we go to discuss indirect detection, we summarize the parameter space, in which a correct value of the total relic DM abundance Ωtotalh2 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 γ2 (the Higgs portal coupling) also has to be close to 0.004 for an adequate amount of Ωχ. However, as for mϕR and γ5, there exist a certain allowed region. The allowed region in the mϕR − γ5 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 ϕR increases, but how the allowed range in the mϕR5 plane emerges remains the same. If we take the larger γ7, e.g., γ7 = 0.28, in Figure 7, the allowed region for mϕR becomes narrower as 295 GeV ≲ mϕR ≲ 300 GeV. The smaller mϕR (≲ 295 GeV) is excluded by Ωtotal < Ωobs due to the larger DM conversion i.e., the larger annihilation process of ϕRϕR → 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 [110120] 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 ni 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. [119], and Jungman et al. [120]

n˙i=CiCA(iiSM)ni2      mi>mjCA(iijj)ni2CA(ijkν)ninj ,    (32)

where i, j, k = χ, ϕR, N1 and Ci is the capture rates in the Sun:

Cχ2.5×1018s1f(mχ)(f^0.3)2(γ20.004)2(60 GeVmχ)2               (125 GeVmh)4(Ωχh2Ωtotalh2),    (33)
CϕR6.2×1017s1f(mϕR)(f^0.3)2(γ50.02)2                 (300 GeVmϕR)2(125 GeVmh)4(ΩϕRh2Ωtotalh2),    (34)
CN1=0 ,    (35)

and CA's are the annihilation rate:

CA(ij)=σ(ij)vVij ,Vij=5.7×1027(100 GeVμij)3/2cm3 .    (36)

Here f(mi) depends on the form factor of the nucleus, elemental abundance, kinematic suppression of the capture rate, etc., varying O(0.01-1) depending on the DM mass [118120]. Vij is an effective volume of the Sun with μij = 2mimj/(mi+mj) in the non-relativistic limit. In the Equation (32) we have neglected the DM production processes such as jjii and jkiX because the kinetic energy of the produced particle i is much larger than that corresponding to the escape velocity from the Sun, i.e., ~ 103 km/s [114, 121, 122]. Consequently, the number of the right-hand neutrino DM cannot increase in the Sun, and hence the semi-annihilation process, ϕRχ → N1ν, is the only neutrino production process3, where its reaction rate as a function of t is given by Γ(ϕRχ → ; t) = CARχ → N1ν)nϕR(t)nχ(t).

Figure 9 shows the mϕR dependence of the neutrino production rate today Γ(ϕRχ → ; t0), where t0=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 mϕRM1 where the relic abundance of N1 dominates that of ϕR, the neutrino production rate decreases since the capture rate of the ϕR becomes small. As we can see from Figure 5 a resonance effect for the s-channel annihilation process can be achieved if mηR0mϕR+mχ. Then the smaller neutrino mass difference mηR0-(mϕR+mχ) gives the larger neutrino production rate. For the case of mηR0-(mϕR+mχ)= 1 GeV, the rate Γ(ϕRχ → ; t0) reaches about 1018 s−1 at mϕR ≃ 290 GeV for M1 = 300 GeV and 4 × 1017 s−1 at mϕR ≃ 490 GeV for M1 = 500 GeV, respectively.

FIGURE 9
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Figure 9. The neutrino production rate Γ(ν) = Γ(ϕRχ → ; t0) in the Sun against the ϕR DM mass for M1 = 300 GeV (Left) and 500 GeV (Right). The parameter space, as well as the meaning of colors of the lines in the left and right panel, are the same as in Figures 7, 8, respectively. The hatched region is excluded by perturbativity. Arrows indicate the excluded regions by the direct detection experiments.

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 W+W is 1.13 × 1021 (2.04 × 1020) s−1 and that into τ+τ is 5.99 × 1020 (7.96 × 1019) s−1, which is at least 102 times larger than the rate Γ(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 [1036, 3948]. Resent cosmological observations with increasing accuracy [88, 127129] 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 H0 and in the density variance σ8 in spheres of radius 8h − 1 Mpc: The Planck values of 1/H0 and σ8 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 [131139] 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 ΔNeff = 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 Z2×Z2×L -invariant Yukawa sector (the quark sector is suppressed) is described by the Lagrangian

LY=YijeHLilRjc+YijνLiϵηNj+YijχNiχjϕ12Mχkχkχk+h.c.,    (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 Yije have only diagonal elements. The most general renormalizable form of the Z2×Z2×L-invariant scalar potential is given by

Vλ=λ1(HH)2+λ2(ηη)2+λ3(HH)(ηη)+λ4(Hη)(ηH)        +λ52[(Hη)2+h.c.]        +γ2(HH)|ϕ|2+γ3(ηη)|ϕ|2+γ4|ϕ|4,    (38)

and the mass terms are

Vm=m12HH+m22ηη+m32|ϕ|2m422[ϕ2+(ϕ*)2] ,    (39)

where the m4 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 Z2×Z2 is unbroken, the scalar fields above do not mix with other, so that their tree-level masses can be simply expressed:

mη±2=m22+λ3vh2/2 ,    (40)
mηR02=m22+(λ3+λ4+λ5)vh2/2 ,    (41)
mηI02=m22+(λ3+λ4λ5)vh2/2 ,    (42)
mϕR2=m32m42+γ2vh2/2 ,    (43)
mϕI2=m32+m42+γ2vh2/2 .    (44)

The two-loop diagram for the neutrino mass is shown in Figure 10. Because of the soft breaking of the dimension two operator ϕ2, the propagator between ϕ and ϕ can exist, generating the mass of N:

(MN)ij=132π2k(Yikχ)*(Yikχ)*Mχk[mϕϕR2mϕϕR2Mχk2ln(mϕϕRMχk)2                mϕI2mϕI2Mχk2ln(mϕIMχk)2] .    (45)

The 3 × 3 two-loop neutrino mass matrix Mν is given by

(Mν)ij=1(32π2)2l,kYilνYjmν(Ylkχ)*(Ymkχ)*Mχk(mηI02mηR02)                 0dx 1(x+mηR02)2(x+mηI02)                01dzln[zMχk2+(1z)mϕI2+z(1z)xzMχk2+(1z)mϕR2+z(1z)x].    (46)

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

(Mν)ij~132π2kYikνYjkνMklnmηR02mηI02 , Yjkν=YjlνUlkN,    (47)

where UN is the unitary matrix diagonalizing the mass matrix (MN)ij with the eigenvalues Mk and the mass eigenstates Nk, and we have used the fact that MkmηR0mηI0. In the following discussions we choose the theory parameters so as to be consistent with the global fit [1]:

Δm212=7.500.17+0.19×105 eV2,    Δm312=2.5240.040+0.039 (2.5140.041+0.038)×103 eV2,sin2θ12=0.306±0.012, sin2θ23=0.4410.021+0.027 (0.5870.024+0.020),sin2θ13=0.02166±0.00075 (0.02179±0.00076),    (48)

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

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

FIGURE 10
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Figure 10. Two-loop radiative neutrino mass of the model B. The upper cross means the soft breaking mass term m42, which should indicate that there are ϕR and ϕI 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.

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 ΔNeff 4. Without los of generality we may assume it is N1 with mass ≲ 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 N2 and N3 decay above the decoupling temperature TNdec of N1. Their decay widths are given by

 Γ(N2(3)N1νν¯)+Γ(N2(3)N¯1νν¯) =13072π3M2(3)5mη04 i,j|Yi2(3)ν|2|Yj1ν|2 ,    (49)

where we have used mη0=mηR0mηI0 and neglected the mass of N1 and νLs. Therefore, N2 and N3 can decay above TNdec if

 Γ(N2(3)N1νν¯)+Γ(N2(3)N¯1νν¯) H(TNdec)    (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 N1 at the time of the photon decoupling, where we denote the decoupling temperature of γ, νL and N1 by Tγ0,Tνdec and TNdec, respectively. Further, Tν0 (TN0) stands for the temperature of νL (N1) at the decoupling of γ. The most important fact is that the entropy per comoving volume is conserved, so that sa3 is constant, where s is the entropy density, and a is the scale factor. The effective number Neff follows from ρr(Tγ0)=(π2/15)(1+(7/8)(4/11)4/3Neff)Tγ04 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]

Neff=3.046+(g*s(Tνdec)g*s(TNdec))4/3    (51)

for Nν = 3, where ρr is the energy density of relativistic species. Since g*s(Tνdec)=(11/2)+(7/4)Nν=10.75, we need to compute the decoupling temperature TNdec to obtain g*s(TNdec) and hence Neff. For 0.05 ≲ ΔNeff ≲ 0.38 [139] we find 101g*s(TNdec)22 and also TNdec165 MeV to obtain g*s(TNdec)30 (which gives ΔNeff = 0.25). To estimate TNdec, we compute the annihilation rate ΓN(T) of N1 at T, which is given by

ΓN(T)=nN(T) [σN1N1νLνLv(T)+σN1N¯1νLν¯Lv(T)]              =π59ζ(3)(7120)2i,j|Yi1ν|2|Yj1ν|2T5(mη0)4 ,    (52)

where ζ(3) ≃ 1.202…  and nN(T) is the number density of N1. Then we calculate TNdec from ΓN(TNdec)=H(TNdec), which can be rewritten as5

(TNdec164.2 MeV)3(29.9g*s(TNdec))12=(mη0200 GeV0.0409Yν)4    (53)

with (Yν)2=i|Yi1ν|2.

It turns out that M2, 3 ~ O(10) GeV to obtain ΔNeff ~ 0.25 while satisfying M1 ≲ 0.24 eV. To see this, we first find that

(mη02i|Yi1v|2)~2.4×107 GeV2 ,    (54)

which follows from Equation (53) for ΔNeff ~ 0.25. Further we can estimate a part of Equation (49) from the neutrino mass Equation (47) with Mν ~ 0.05 eV:

M2(3)mη02 i|Yi2(3)ν|2~2.7×1015|λ5|1 GeV1 ,    (55)

where we have used mηR02-mηI02λ5vh2. Then using Equation (50) with T ≃ 165 MeV (which corresponds to ΔNeff ≃ 0.25), we obtain

M2,317|λ5|1/4 GeV .    (56)

Note that this is an order of magnitude estimate, and indeed M2(3) can not be smaller than 10 GeV to satisfy ΔNeff ≲ 0.38.

Since we require that M1 ≲ 0.24 eV, there exists a huge hierarchy in the right-handed neutrino mass. This has a consequence on the Yukawa coupling matrix Yν: To obtain realistic neutrino masses with the mixing parameters given in Equation (48),

|Yi1ν|  |Yi2(3)ν|    (57)

has to be satisfied. Note that only |Yi1ν| enters into the thermally averaged annihilation cross section of N1, as we can see from Equation (52). Because of ΔNeff ≲ 0.38, on the other hand, |Yi1ν| can not be made arbitrarily large. The hierarchy (Equation 57) has effects on the LFV radiative decays of the type liljγ, so that the LFV decays and ΔNeff are related, as we will see below. In the limit mjmi, where mi and mj stand for the mass of li and lj, respectively, the ratio of the partial decay width B^(liljγ)=Γ(liljγ)/Γ(liνieν̄e) can be written as Ma and Raidal [51]

B^(liljγ)=(α768πGF2)|k(Yikν)*Yjkν|2mη±4.    (58)

Here mη± and Yikv are defined in Equations (40) and (47), respectively, and the current upper bounds on the branching fraction of these processes [107, 148] require

μeγ:|k(Y2kν)*Y1kν|2.5×104(mη±220GeV)2,    (59)
τμγ:|k(Y3kν)*Y2kν|8.1×102(mη±220GeV)2,    (60)
τeγ:|k(Y3kν)*Y1kν|7.0×102(mη±220GeV)2.    (61)

From Equation (59) we find that Y31v is not constrained by the stringent constraint from μ, which will be crucial in obtaining a realistic Neff without having any contradiction with Equations (59–61). Furthermore, if Y31v is large compared with others and the hierarchy (Equation 57) is satisfied, the ratio R=B^(τμγ)B^(τeγ)/B^(μeγ) is ~ |Y31ν|2, and from the same reason ΔNeff depends mostly on Y31v. A benchmark set of the input parameters is given by

Yijv=(0.03822.510×1053.349×1050.001291.183×1061.081×1040.01547.723×1059.334×105),    (62)
M1=0.147 eV,   M2=M3=9.55 GeV,    (63)
mη±=220 GeV,   mηR0=200 GeV,   mηI0=207 GeV,    (64)

which yields

sin2θ12=0.305,  sin2θ23=0.441,  sin2θ13=0.0213,    (65)
Δm212=7.50×105 GeV2,  Δm312=0.00248 GeV2,    (66)
B^(μeγ)=2.30×1014, B^(τμγ)=3.75×1015,                       B^(τeγ)=3.31×1012,    (67)

where we have assumed that Y31v 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 × 10−21 (1.78 × 10−20) GeV for N2 (N3), where the rhs is H = 2.10 × 10−20 GeV with TNdec=166.8 MeV and g*s(TNdec)=30.83, and ΔNeff = 0.245.

In Figure 11 we plot R1/2 against ΔNeff with mη±=240 GeV and mηR0=220 GeV, where we have varied mηI0 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 M1 < 0.24 eV, (Equations 50 and 59–61) are satisfied. If ΔNeff and R1/2 would depend on Y31v only, we would obtain a line in the ΔNeff-R1/2 plane. The Y11v and Y21v dependence in R1/2 cancels, but this is not the case for ΔNeff. 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 ΔNeff ≲ 0.1 is absent. The main reason is that we have assumed that M2, M3 ≲ 16 GeV. This has also a consequence on the difference between mηR02 and mηI02, 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 mηR02 and mηI02 can not be made arbitrarily large, because it requires a smaller M2, 3, which due to H(T)∝T2 in turn implies that the decoupling temperature TNdec has to decrease to satisfy the constraint [Equation (50)]. A smaller TNdec, on the other hand, means a larger ΔNeff which is constrained to be below 0.38. This is why mηI0 is varied only in a small interval in Figure 11.

FIGURE 11
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Figure 11. R1/2 against ΔNeff with mη±=240 GeV and mηR0=220 GeV, where mηI0 is varied between 221 and 227 GeV, and R=B^(τμγ) B^(τeγ)/B^(μeγ).

Since the current upper bound on B(μeγ)B^(μeγ) is 4.2 × 10−13 [107], the model B predicts

[B(τμγ)B(τeγ)]1/2[B^(τμγ)0.17 B^(τeγ)0.18]1/2                                             1.2×1010 ,    (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 ηR depends on i,j|Yij|2, which is approximately i|Yi1|2 (we assume that ηR is the lightest among ηs). Therefore, ΔNeff is basically a function of the decay width. In Figure 12 we show ΔNeff against ΓηR/mηR0, the decay width of ηR0 over mηR0 , where we have used the same parameters as for Figure 11. ηR0 decays almost 100 percent into neutrinos and dark radiation N1, which is invisible. In contrast to this, η+ can decay into a charged lepton and N1, and the decay width over mη± is the same as ΓηR/mηR0. ΓηR should be compared with the decay width for η+W+* ηR,I0ff̄N1ν, which is ~ 10-8mη± 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 N1 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 Yi1v. In the parameter space we have scanned we cannot make any definite conclusion on the difference.

FIGURE 12
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Figure 12. ΔNeff against ΓηR/mηR0, where we have used the same parameters as for Figure 11.

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), ηR,I0 can not be DM candidates, because they decay into N and ν. So, DM candidates are χ and the lightest component of ϕ6. In the case that ηs are lighter than ϕR and the lightest component of ϕ (which is assumed to be ϕR) is DM, a correct relic abundance ΩϕRh2=0.1204±0.0027 [88] can be easily obtained, because γ3 for the scalar coupling (ηη)|ϕ|2 is an unconstrained parameter so far. So, in the following discussion we assume that ϕR is DM.

Because of the Higgs portal coupling γ2, the direct detection of ϕR is possible. The current experimental bound of XENON1T [103] of the spin-independent cross section σSI off the nucleon requires |γ2| ≲ 0.05 ~ 0.14 formϕR = 250 ~ 500 GeV. Since γ2 is allowed only below an upper bound (which depends on the DM mass mϕR), γ3 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, vσ(ϕRϕRη+η-,ηR0ηR0,ηI0ηI0) [110120]. If a pair of ϕRs annihilates into ηR0ηR0 and also ηI0ηI0, a pair of νL and ν̄L 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]

Γ(ϕRϕRηR0ηR0,ηI0ηI0;t)=Γ(ϕRϕRη0η0;t)=12CϕRCA(η0η0)CA(η+η)+CA(η0η0)+CA(XX)tanh2[t(CA(η+η)+CA(η0η0)+CA(XX))CϕR],    (69)

where CϕR is the capture rate in the Sun,

CϕR1.4×1020f(mϕR)(f^0.3)2(γ20.1)2(200 GeVmϕR)2            (125 GeVmh)4 ,    (70)

and CA is given by

CA()=(σϕRϕRv5.7×1027cm3)(mϕR100 GeV)3/2 s1            with =η+η, η0η0, and XX .    (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(η0η0)=CA(η+η-). In Figure 13 we plot the annihilation rate Γ(ϕRϕRη0η0;t0) today (t0=1.45×1017 s) against σSI for mϕR = 250 and 500 GeV with mη fixed at 230 GeV and 0.117<ΩϕRh2<0.123. The vertical dashed lines are the XENON1T upper bound on σSI [103]. The peak of Γ(ϕRϕRη0η0;t0) for mϕR = 250 (500) GeV appears at σSI=4.2 (4.7)×10-46 cm2 and is ≃ 1.7 (0.7) × 1018 s−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].

FIGURE 13
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Figure 13. The pair-annihilation rate of ϕR into ηR0ηR0 and ηI0ηI0 in the Sun, Γ(η0η0)=Γ(ϕRϕRη0η0;t0), against σSI for mϕR = 250 (black) and 500 (red) GeV, where mη is fixed at 230 GeV (all ηs have the same mass) and 0.117<ΩϕRh2<0.123. The black (red) vertical dashed line is the XENON1T [103] upper bound on σSI for mϕR = 250 (black) and 500 (red) GeV.

5. Conclusion

We have discussed the extensions of the Ma model by imposing a larger unbroken symmetry Z2×Z2. 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 “λ5 coupling,” which is O(10-6) to obtain small neutrino masses for Yν ~ 0.01, is radiatively generated. Consequently, the neutrino masses are generated at the two-loop level, where the unbroken Z2×Z2 symmetry acts to forbid the generation of the one-loop mass. Such larger unbroken symmetry implies that the model involves a multi-component 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 Z2×Z2 there exist at least two stable DM particles; a dark radiation N1 with a mass of O(1) eV and the other one, DM, is the real part of ϕ. The dark radiation contributes to ΔNeff <1 such that the 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 M2,3TNdecO(100) MeVMN1 O(1) eV, we are able to relate to the ratio of the lepton flavor violating decays to ΔNeff. 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 ΔNeff as a function of the decay width of ηR0 (which is assumed to be lightest among ηs). It decays 100 percent into left- and right-handed 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.

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.

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).

Footnotes

1. ^Semi-annihilation processes also exist in one-component DM systems when DM is a Z3 charged particle [89, 90, 91, 92, 93, 94, 95] or a vector boson [96, 97, 98, 99].

2. ^Two singlet scalar DM scenario in Z2×Z2 model has been explored in detail in Bhattacharya et al. [65].

3. ^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].

4. ^Within a similar framework of radiative seesaw mechanism, the lightest right-handed 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−L is regarded as dark radiation.

5. ^We use the relation between T and g*s given in Husdal [147] to solve Equation (53) for TNdec.

6. ^Both together can not be DM, because the heavier one decays into N1+lighter one.

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Keywords: neutrino mass, dark matter, radiative seesaw mechanism, flavor physics, dark radiation

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

Received: 04 August 2017; Accepted: 10 October 2017;
Published: 01 November 2017.

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

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.

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