Effect of sterile neutrino on low energy processes in minimal extended seesaw with $\Delta(96)$ symmetry and $\text{TM}_{1}$ mixing

We study the effect of sterile neutrino on some low scale processes in the framework of minimal extended seesaw (MES). MES is the extension of the seesaw mechanism with the addition of sterile neutrino of intermediate mass. The MES model in this work is based on $\Delta(96)\times C_{2}\times C_{3}$ flavor symmetry. The structures of mass matrices in the framework lead to $TM_{1}$ mixing with $\mu \text{-}\tau$ symmetry. The model predicts maximal value of Dirac CP phase. We carry out our analysis to study the new physics contributions from the sterile neutrino to different charged lepton flavor violation (cLFV) processes involving muon and tau leptons as well as neutrinoless double beta decay (0$\nu\beta\beta$). The model predicts normal ordering (NO) of neutrino masses and we perform the numerical analysis considering normal ordering (NO) only. We find that sterile neutrino mass in GeV range can lead to cLFV processes that are within the reach of current and planned experiments. The GeV scale sterile neurtrino in our model is consistent with the current limits on the effective neutrino mass set by $0\nu\beta\beta$ experiments.


I. INTRODUCTION
The observed neutrino oscillation phenomenon, the origin of the idea behind the massive nature of neutrinos, has been one of the most appealing evidences to expect physics beyond standard model (BSM). Neutrino oscillation probabilities are dependent on the three mixing angles,the neutrino mass-squared differences (∆m 2 21 , ∆m 2 31 ) and the Dirac CP phase (δ CP ). Though there are precise measurements of the mixing angles and mass squared differences, yet there are no conclusive remarks on (δ CP ) or the mass ordering of the neutrinos. NOνA [1] and T2K [2] experiments have recently provided hint towards the CP violation in Dirac neutrino matrix. Again, another important unsolved issue is the mass ordering of the neutrinos whether it is normal (m 1 < m 2 < m 3 ) or inverted (m 3 < m 1 < m 2 ). There are some other open questions in particle physics as well as cosmology such as CP violation in lepton sector, baryon asymmetry of the universe and particle nature of dark matter. Motivated by these shortcomings, different beyond standard model (BSM) theories [3] are pursued in different experiments.
Many searches for new physics beyond standard model are going on in different experiments. Charged lepton flavor violating (cLFV) processes can provide a way to search for new physics beyond standard model. cLFV processes are heavily suppressed in the standard model . However, the well established neutrino oscillation phenomenon give a signal towards the flavor violation in the charged lepton sector also. There are present and planned experiments to search for lepton flavor violating radiative decay (l i −→ l j γ) [4] and also three body decays (l i −→ l j l k l k ) [5]. The present and future experimental constraints on cLFV processes can be found in table I and II . In this work, we study the transition among the three charged leptons. However, the transitions of muon such as µ − e, N , µ −→ eee, µ −→ eγ [6,7] and recently proposed µ − e − −→ e − e − [8] are extensively analyzed as the parent particle is substantially available in the cosmic radiation as well as in dedicated accelerators [9]. Many other challenging cLFV processes are those which involve the third family of leptons (taus) as it opens many flavor violating channels. Among these τ −→ eγ, τ −→ µγ, τ −→ 3e and τ −→ 3µ are significant. The processes involving taus also open up many channels involving hadrons in the final state such as τ −→ lπ 0 , τ −→ lπ + π − [9,10] .    [7,13] There are various theoretical models which are the extension of SM that can account for cLFV processes [14][15][16]. These models usually introduce new particle fields to act as a source of flavor violation. The models with heavy sterile neutrinos can provide prominent contributions to cLFV processes. There are many theoretical motivations as well as experimental background for the existence of sterile neutrinos. The anomalies of the LSND [17] and MiniBooNE [18] results provide a hint towards the presence of one or two sterile neutrino states. Again from the theoretical point of view, the addition of sterile fermions into the standard model can explain the neutrino mass and also mixing [19]. Moreover, sterile neutrino can account for many cosmological observations like dark matter [20][21][22][23][24] and baryon asymmetry of universe (BAU) [25,26]. Furthermore, their mixing with the active neutrinos can contribute to certain non-oscillation processes like neutrino-less double decay (0νββ) amplitude or in beta decay spectra in the KATRIN experiment [27,28]. To study the effect of sterile neutrino on low scale processes, we have chosen minimal extended seesaw (MES) framework augmented with ∆(96) flavor symmetry. In MES framework, three right-handed neutrinos and one additional gauge singlet field S are added to the SM particle content [29,30]. The extra sterile state may have significant contribution to cLFV processes and 0νββ depending on its mass and mixing with the active neutrinos in the model. In  [29,32,33]. Thus the Lagrangian in this MES model can be obtained as [29], Subsequently, the mass matrix arising from the Lagrangian in Eq.(1) in the basis (ν L , N c , S c ) can be written as, Since the right-handed neutrinos are much heavier than the electroweak scale as in case of type-I seesaw, they should be decoupled at low scales. Effectively, the full 7 × 7 matrix can be block diagonalised into a 4 × 4 neutrino mass matrix as follows [29], Assuming M S > M D , the active neutrino mass matrix of Eq.(3) takes the form as, The sterile neutrino mass can be obtained as, The charged lepton mass matrix in general can be diagonalised using unitary matrices U L and U R as follows , Again, we obtain the light neutrino masses using unitary matrix U ν as, The 4 × 4 neutrino mixing matrix in MES model using U L and U ν can be obtained as [34] , The matrix U L R governs the active-sterile mixing in which R can be expressed as, and, Finally, the 3 × 3 lepton mixing matrix (PMNS) can be written as [34], Thus PMNS matrix can be obtained by multiplying the diagonalising matrix of charged lepton mixing matrix and that of the effective seesaw matrix. U L is identity matrix in the framework where charged lepton mass matrix is diagonal.

B. T M 1 Mixing
Trimaximal (T M 1 ) mixing is a mixing ansatz that preserves the first column of tribimaximal mixing U T BM and mixes its second and third columns. It is a perturbation to TBM mixing and we can write the mixing matrix as [31,35,36], Comparing above mixing matrix in Eq. (14) with the standard PMNS mixing matrix, one can obtain the three mixing angles in terms of θ as follows [37]: The Jarlskog's rephasing invariant J CP can be written in terms of the elements of the mixing matrix as, One can write the expression for the CP phase in the T M 1 scenario as, For a given θ 13 (θ), the T M 1 mixing with µ-τ symmetry leads to maximal CP violation.
In this work, we have used ∆(96) flavor symmetry [38][39][40][41] giving rise to unique textures of the mass matrices involved in the MES model. For a brief discussion about properties of ∆(96) ,its character table and tensor product rules please refer Appendix A. ∆(96) symmetry is further augmented by C 2 ans C 3 discrete flavor symmetries to get rid of some unwanted interactions. The particle assignments in the model are shown in table III.
In our MES model, the lepton doublets of the SM and the SM gauge singlets transform as triplets 3 i and3 i of ∆(96) respectively. The sterile neutrino and the three right-handed charged leptons transform as singlets under this symmetry group. We introduce flavons . These fields are also assigned various charges under the C 2 and C 3 group which can be found in table III The Yukawa Lagrangian for the charged leptons and also for the neutrinos can be expressed as: L M D represents Dirac neutrino Lagrangian given as, The neutrino Majorana mass term L M can be expressed as, The interactions between the sterile and the right handed neutrinos are involved in L M S .
L M L is the Lagrangian for the charged leptons which can be written as After Spontaneous Symmetry Breaking (SSB) , the scalar fields acquire VEV's which are assigned as:

D. The Mass Matrices involved in the Model
The textures of the mass matrices involved in MES model can be obtained using flavon alignments defined with residual symmetries under our flavor group. With these flavon alignments mentioned above, we obtain the charged-lepton and the neutrino mass matrices.
In the charged lepton sector,L which couples to l R (l = e, µ, τ ) through the flavon φ µ and φ τ . Using the VEV's of the flavons and the Higgs in the Lagrangian given by Eq.(25) ,the charged lepton mass matrix can be written as, The charged lepton mass matrix M C is diagonalised using the unitary matrix U L given as, U L is referred to as the 3 × 3 trimaximal matrix (TM) or the magic matrix.
and we obtain the masses of the charged leptons as, It is seen from Eq.(30) that the mass scale of electron is suppressed by an additional factor 1 Λ compared to tau or muon mass similar to Froggatt-Nielsen mechanism of obtaining the mass hierarchy.
Again, from Eq. (22), we obtain the Dirac neutrino mass matrix as, Denoting Here also, we denote y M v M = m R and y M i v M i y M v M = r 2 and rewrite the above matrix as, m R has the dimension of mass at the scale of flavon VEV and r 2 is dimensionless.
Finally, we obtain the mass matrix representing the coupling between right handed neutrinos and sterile neutrino as, or we can rewrite it as, where, m S = y S v S has the dimension of mass.
The light neutrino mass matrix in the framework of MES arising from the mass matrices in Eqs. (32,34,36) can be written using Eq.(3) as: where, The effective seesaw mass matrix in Eq.(37) can be diagonalised in two steps using the unitary matrix U BM and U θ as, or one may write, The matrix U θ and the bimaximal matrix U BM in the Eq.(41) are given by, Comparing Eq. (41) with Eq. (7), we can write the neutrino mixing matrix U ν as, Therefore,using Eq.(12),the PMNS matrix in this model can be expressed as, Here, U L U BM is the tri-bimaximal (TBM) mixing matrix,U T BM . The multiplication of U T BM and U θ mixes the 2nd and the 3rd columns of U T BM resulting in T M 1 mixing matrix U T M 1 .
Our construction of M ν given in Eq.(37) leading to T M 1 mixing implies that m 1 = 0, which rules out inverted hierarchy. Using this in Eq. (42) and comparing with Eq.(37), we can find the expressions for model parameters K 1 ,K 2 and K 3 in terms of the parameters θ, m 2 and m 3 as,

E. Sterile Neutrino Mass and Mixing in the Model
Apart from the active neutrinos, the mass and mixing of the sterile neutrino present in the model play crucial role in cLFV processes which will be discussed in the next section. As mentioned above, the sterile neutrino mass can be obtained using Eq.(5) and we can write the mass term for sterile neutrino as, The active-sterile mixing using Eq.(9) and Eq.(10) can be obtained as, In the above Eqs. (49,50,51,52), m D ,m R ,r 1 and r 2 are the model parameters.

A. Processes involving Muonic atoms
Many on-going experiments like MECO, SINDRUM II [7],COMET [42] are involved in searching for µ − e conversion with different targets. The observable characterizing this process is defined as , These experiments are running with different targets like Titanium (Ti), Lead (Pb), Gold (Au) Aluminum (Al) and give bounds for different targets. There are also some planned future experiments like the second phase of COMET experiment, Mu2e [43] to improve the sensitivity to this cLFV process.
There are several theoretical models to account for such rare LFV processes. As explained in [44], in the extension of standard model with one heavy sterile neutrino, such processes originate from one-loop diagrams involving active and sterile neutrinos with non zero mixing angles. In the MES model,the conversion ratio can be written as [44], In the above expression, G F ,s ω ,Γ cap (Z) are Fermi constant, sine of weak mixing angle and capture rate of the nucleus respectively. Here, α = e 2 4π andF µe q are form factors given as, Here, Q q represents the quark electric charge which is 2 3 and − 1 3 for up and down quark respectively. The weak isospin I 3 q is 1 2 and − 1 2 for up and down quark respectively. The numerical values of V (p) ,V (n) and D in [15]. In the small limit of masses (x j = m 2 νj m 2 W 1), the form factors can be written as [44], There may be flavour violating non-radiative decay of µ − into three electrons (µ −→ eee) [45]. Mu3e experiment running at PSI aims at finding the signatures of this type of decay [46]. The branching ratio of this decay process can be written as, In the above equation, the form factors can be obtained from Eq.(56) to Eq.(59).
The MEG experiment [47] aims at investigating LFV process µ −→ eγ and there are many planned projects in search for this kind of decay. In the framework of minimal extended seesaw, the heavy neutrinos can cause µ −→ eγ decay. The branching ratio of the process can be given as, In the above equation,the total decay width of muon (Γ µ ) is obtained as, Another possible cLFV process is the decay of a bound µ − in a muonic atom into a pair of electrons (µ − e − −→ e − e − ) proposed by [8]. This particular decay process offers several advantages over three body decay processes from the experimental point of view. There are different classes of extension of SM which can show a contribution to such processes. In this model with one extra sterile state, the effective Lagrangian describing this process contains long range interactions and local interaction terms. The branching ratio of such process in muonic atoms,with an atomic number Z can be expressed as, Here, τ µ represents the lifetime of free muon and the lifetimeτ µ depends on specific elements.
In our analysis, we have considered Al and Au in which value ofτ µ are 8.64×10 −7 and 7.26× 10 −8 respectively. This decay process would possibly be probed in the COMET collaboration.
As suggested in many literature, we have used the future sensitivity of CR(µ − e, N ) to constrain such decay process.
There are many flavor violating channels open for tau lepton decays. Search for such decays involving taus is also challenging. Theoretical models which predict cLFV in the muon indicate a violation in the tau sector also. However, the amplitude of the process involving tau channel is enhanced by several order of magnitude in comparison to muon decays. Experiments like BaBar [48] and Belle [49] provide limits to cLFV decays involving tau leptons. In this work, we have investigated three processes involving tau leptons τ −→ eγ, τ −→ µγ and τ −→ eee. The branching ratios of these mentioned process can be written as [15], In the above equations, Γ τ represents the total width of tau leptons with experimental value Γ τ = 2.1581 × 10 −12 GeV [15].
where, the composite form factors F τ e γ ,G τ e γ ,F τ e Z and F τ eee Box can be defined as follows:

IV. NEUTRINOLESS DOUBLE BETA DECAY (0νββ)
The presence of sterile neutrinos in addition to the standard model particles may lead to new contributions to lepton number violating interactions like neutrinoless double beta decay(0νββ) [50][51][52]. We have studied the contributions of the sterile state to the effective electron neutrino majorana mass m ββ [27,53]. The most stringent bounds on the effective mass by provided by KamLAND-ZEN experiment [54].
The amplitude of these processes depends upon the neutrino mixing matrix elements and the neutrino masses. The decay width of the process is proportional to the effective electron neutrino majorana mass m ββ which is in the case of standard contribution i.e. in the absence of any sterile neutrino is given as , The above equation is modified with the addition of sterile fermions and is given by [27] where, m 4 and U e4 represent the mass and mixing of the sterile neutrino to the electron neutrino respectively. V.

RESULTS OF NUMERICAL ANALYSIS AND DISCUSSIONS
It is evident from the above discussion that the neutrino mass matrix in Eq. (37) contains three model parameters K 1 ,K 2 ,K 3 . We can express the experimentally measured six oscillation parameters ∆m 2 21 , ∆m 2 31 , sin 2 θ 12 , sin 2 θ 23 , sin 2 θ 13 , δ CP in terms of these model parameters. Hence, the three model parameters can be evaluated by comparing with the three oscillation parameters in 3σ range as given in table IV and then constrain the other parameters. These parameters K 1 , K 2 , K 3 in turn are related to m D ,m R ,r 1 and r 2 as given in Eqs. (38,39,40)   T M 1 with µ-τ symmetry fixes the atmospheric mixing angle θ 23 be π 4 i.e maximal atmospheric mixing angle. We have seen the predictions of the model on Jarlskog parameter J CP and the Dirac CP phase δ CP by evaluating these two parameters using Eq. (19). The model predicts maximal δ CP which is consistent with the current global fit. We have also  KamLAND-ZEN experiment [54] and m i from latest Planck data [56].
Apart from studying active neutrino phenomenology, we have calculated different observables related to the different cLFV processes with the numerically evaluated model parameters. All the masses and mixing in the model are dependent on the model parameters which are highly constrained from the neutrino oscillation data. The masses and mixing of the active as well as sterile neutrinos in turn are related to the observables of different cLFV processes and also 0νββ process as mentioned above. Hence, the same set of model parameters which are supposed to produce correct neutrino phenomenology can also be used to estimate the observables of different low energy processes. Thus this model is constrained from these processes also. The motivation is to see if the neutrino mass matrix that can explain the neutrino phenomenology can also provide sufficient parameter space for other low energy observables 0νββ, cLFV etc. We also correlate the sterile neutrino mass with 0νββ and cLFV processes to see the impact of sterile neutrino.
The effective mass (m ββ ) characterizing 0νββ process along with the presence of heavy sterile neutrino is calculated using Eq.(73). We have seen the sterile neutrino contribution to process µ − e − −→ e − e − in the model.     The estimation of the model on baryon asymmetry of the universe (BAU) can also be studied in future.