Resonant leptogenesis and TM$_1$ mixing in minimal Type-I seesaw model with S$_4$ symmetry

We present an S$_4$ flavour symmetric model within a minimal seesaw framework resulting in mass matrices that leads to TM$_1$ mixing. Minimal seesaw is realized by adding two right-handed neutrinos to the Standard Model. The model predicts Normal Hierarchy (NH) for neutrino masses. Using the constrained six-dimensional parameter space, we have evaluated the effective Majorana neutrino mass, which is the parameter of interest in neutrinoless double beta decay experiments. The possibility of explaining baryogenesis via resonant leptogenesis is also examined within the model. A non-zero, resonantly enhanced CP asymmetry generated from the decay of right-handed neutrinos at the TeV scale is studied, considering flavour effects. The evolution of lepton asymmetry is discussed by solving the set of Boltzmann equations numerically and obtain the value of baryon asymmetry to be $\lvert \eta_B \rvert = 6.3 \times 10^{-10}$.


I. Introduction
Neutrino oscillation experiments have determined that the mass of neutrinos are small but nonzero, and indicate flavour mixing [1][2][3][4][5][6]. Observations such as these pose a question about the origin of the tiny neutrino masses. The absence of the right-handed counterpart of the neutrinos within the standard model (SM) suggests that, unlike charged fermions, Dirac masses could not be the origin of neutrino masses. There are numerous frameworks beyond the standard model (BSM) that can explain the origin of neutrino masses, for instance, the seesaw mechanism [7][8][9], radiative seesaw mechanism [10], models based on extra dimensions [11,12], and other models.
The minimal seesaw mechanism, which is the extension of SM with two right-handed neutrinos, can explain the origin of neutrino masses as well as that of the Baryon Asymmetry of the Universe (BAU) through leptogenesis [13]. Here, the generation of baryon asymmetry involves conversion of lepton asymmetry, obtained from the CP-violating decay of the heavy right-handed neutrinos, via the sphaleron processes [14]. It has been reported in Ref. [15] that the mass scale O(10 9 ) for the right-handed neutrino is needed to explain the observed BAU. However, this scale can be lowered if we have nearly degenerate mass of right-handed neutrinos. Such a case leads to resonantly enhanced CP-violating effects, and sufficient lepton asymmetry to account for BAU can be generated at relatively low masses (TeV scale). Such a situation is termed Resonant leptogenesis [16].
TM 1 mixing has proved to be compatible with the global data on neutrino oscillations, in a sense that it includes non-zero θ 13 and agrees very well with the experiments on its prediction on the mixing angles θ 13 , θ 23 and the Dirac phase, δ CP . Over the years, many discrete symmetry-based studies have been done that gives rise to TM 1 mixing [24][25][26][27][28][29]. Herein, we propose a model based constructed using S 4 discrete symmetry within the framework of the minimal seesaw model. The resulting mass matrix leads to TM 1 mixing and can simultaneously explain BAU via resonant leptogenesis. The choice of right-handed neutrino Majorana mass matrix, M R , is such that the right-handed neutrinos have degenerate mass at dimension five-level and successful resonant leptogenesis is achieved by introducing the higher-order term. In other words, our work is based on the extension of the model presented in [30], such that it makes it suitable to study resonant leptogenesis within minimal seesaw scenario, and the orthogonality condition [23,25] allows us to realize TM 1 mixing in the leptonic sector.
This paper is structured as follows. In section 2, we have presented the S 4 flavour symmetric

II. Model Framework
The S 4 flavour symmetry has been widely used to explain the observed flavour mixing of neutrinos [21,[29][30][31][32][33][34][35][36][37]. S 4 group is a non-Abelian discrete group of permutations of 4 objects. It has 24 elements and 5 irreducible representations 1 1 , 1 2 , 2, 3 1 and 3 2 . The product rules and Clebsch-Gordon coefficients are presented in Appendix A. In this work, we have considered the extension of the standard model (SM) with a discrete non-abelian group S 4 . Also, a Z 3 ×Z 2 group is introduced to avoid specific unwanted couplings and achieve desired structures for the mass matrices. The fermion sector includes, in addition to the SM fermions, two right-handed neutrinos N 1 and N 2 .
Flavons ϕ l , φ l , ϕ ν , φ ν , χ, ψ, β and ρ forms the extension in the scalar sector. The charges carried by the various fields under different symmetry groups are presented in Table 1. Following the representations of the fields given in Table 1, we can write the invariant Yukawa Lagrangian where H is SM Higgs doublet andH = iσ 2 H * , σ 2 being the 2 nd Pauli matrices. The vacuum expectation values (vev) of the scalar fields are of the form [30] As for the vev of ψ we choose ψ = (0, −v ψ , v ψ ) following the orthogonality conditions ψ · φ l and ψ · φ ν . After electroweak and flavour symmetry breaking, we obtain the following structure for the charged lepton mass matrix The charged lepton sector of the model is similar to that of [30] and we similarly assume that the Froggatt-Nielsen mechanism explains the observed mass hierarchy of the charged leptons.
Similarly, using the vev presented in Eq.(4) for the neutrino sector, we obtain the Dirac and where with v H being the vev of the SM Higgs. Taking In the seesaw framework, the resultant light neutrino mass matrix is given by the well-known In charged-lepton diagonal basis, the neutrino mixing matrix, U ν , is the unitary matrix that diagonalizes the mass matrix in Eq. (10). The resulting U ν matrix, which is determined entirely from the neutrino sector, is where U TBM is the tri-bimaximal mixing (TBM) matrix, U 23 is a unitary matrix whose (1,2), (1,3), (2,1), (3,1) entries are vanishing and the resulting matrix, U TM 1 , has its 1 st column coinciding with that of TBM matrix.  Table 2: Neutrino oscillation parameters used to fit the model parameters.

The diagonalization equation thus reads
The light neutrino masses are given as, where s = 2a 2 + 3b 2 + 2c 2 and t = −24a 2 b 2 . It is evident from Eq.(13) that the model predicts normal hierarchy of light neutrino.
In the following sections, we have presented the numerical approaches and discussed baryogenesis via resonant leptogenesis, neutrinoless double beta decay within the context of our model.

III. Numerical Analysis
In To do so, we use the 3σ interval [38] for the neutrino oscillation parameters (θ 12 , θ 23 , θ 13 , ∆m 2 21 , ∆m 2 31 ) as presented in Table 2. A further constraint on the model parameters was applied on the sum of absolute neutrino masses i m i < 0.12 eV from the cosmological bound [39].
In our analysis, the three complex parameters of the model are treated as free parameters and are allowed to run over the following ranges: where φ a , φ b , φ c are the phases given by arg(a ), arg(b ), arg(c ), respectively. Using relation , with M = m ν m † ν and U is a unitary matrix, we numerically diagonalize the effective neutrino mass matrix m ν . The mixing angles, θ 23 , θ 13 are obtained using the relation As seen from Eqs. (1) and (2), TM 1 mixing gives correlations among the mixing angles and CP phases. These relations are assumed to calculate the observables θ 12 and δ CP . The points in the 6-dimensional parameter space which corresponds to the observables that satisfy the 3σ bound on neutrino oscillations are taken to be the allowed region and the best-fit values for the model parameters (|a |, |b |, |c |, φ a , φ b ,φ c ) correspond to the minimum of the following χ 2 function where λ model i is the i th observable predicted by the model, λ model i stands for the i th experimental best-fit value ( Table 2) and ∆λ i is the 1σ range of the i th observable.

IV. Resonant Leptogenesis
The mechanism of leptogenesis, first proposed by Fukugita and Yanagida [13], is one of the popular mechanisms that can explain the observed baryon asymmetry of the universe (BAU). In the simplest scenario of thermal leptogenesis with a hierarchical mass spectrum of right-handed neutrinos, there is a lower bound on the mass of the lightest right-handed neutrino, M 1 10 9 GeV [15]. Although one can lower this limit if their masses are nearly degenerate, the scenario is popularly known as resonant leptogenesis [16,40]. In such a situation, one-loop self-energy contribution is enhanced resonantly, and the flavour-dependent asymmetry produced from the decay of right-handed neutrino into lepton and Higgs is given by [41][42][43][44][45]: where β Λ 2 is a parameter that quantifies the tiny difference between masses required for leptogenesis 2 . The mass matrix in Eq.(18) is diagonalized using a (2 × 2) matrix of the form with real eigenvalues M 1 = M − and M 2 = M + . In the basis where the charged-lepton and Majorana mass matrix are diagonal, the dirac mass matrix (Eq.6) takes the form From this point onward, we will take Y ν = m D /v, which is relevant for calculating CP asymmetry that arises during the decay of right-handed neutrinos in out-of-equilibrium way. Taking the bestfit values of the model parameters obtained in the previous section, we solve the following coupled Boltzmann equations describing the evolution, with respect to z = M 1 /T , of RH neutrino density, N N i and lepton number density for three flavours, N αα corresponding to α = e, µ, τ [16,44].
The following equation gives the equilibrium number density of N i , with K 1,2 (z) being the modified Bessel function. The parameter, D i , sometimes called the decay parameter is defined as which gives the total decay rate with respect to Hubble rate and W ∆L=2 denotes the washout coming from ∆L = 2 scattering process 3 .
We take M 1 = 10 TeV and d = (M 2 − M 1 ) /M 1 10 −8 in order to estimate the value of BAU.
We made the calculations related to baryon asymmetry using the ULYSSES package [46].   Figure 5 shows the predicted values of | m ee | with respect to m i for the allowed region of parameter space. We have also shown the sensitivity reach of some experiments such as nEXO [47], KamLAND-Zen [48], NEXT [49], AMoRE-II [50]. It shows that | m ee | ranges from 2.6 meV to 3.6 meV and probing such small parameters by (0νββ) experiments would be quite difficult.

V. Conclusion
We have explored the S 4 symmetric flavour model in the context of minimal Type-I seesaw mechanism leading to TM 1 mixing pattern in the leptonic sector. In order to achieve TM 1 mixing, we extended the scalar sector further by adding a flavon ψ and its vev is chosen such that it follows the orthogonality conditions (i.e., ψ · φ l and ψ · φ ν ). The resulting effective neutrino mass matrix predicts NH for masses of the neutrinos and 0.0576 eV < m i < 0.0599 eV. An allowed region for the model parameters is derived numerically such that the predictions on the mixing angles, CP phase, and mass squared differences lie within the 3σ bound of current oscillation data. Among various points within the 6-dimensional parameter space, the best-fit value is obtained through chi-squared analysis. Using the obtained parameter space, we evaluated the effective Majorana neutrino mass, | m ee | and we found that it is relatively small, and difficult to probe at the 0νββ experiments.
Furthermore, we investigated baryogenesis via flavoured resonant leptogenesis. The righthanded neutrinos are degenerate at dimension 5 level, and hence a tiny splitting was generated by including higher dimension term. We have taken the splitting parameter, d 10 −8 and thus, obtained a non-zero, resonantly enhanced CP asymmetry from the out-of-equilibrium decay of right-handed Majorana neutrinos. The analysis of the evolution of particles and asymmetry is done by solving the Boltzmann equations. Here, the best-fit values for the model parameters is considered as inputs, and the Boltzmann equations are solved numerically to estimate baryon asymmetry. It was found that the predicted baryon asymmetry comes out to be |η B | ≈ 6.3×10 −10 .

A. S 4 group
The irreducible representations of S 4 follow the following Kronecker products, Now, we write the Clebsch-Gordon coefficients in particular basis [34] For 1-dimensional representations: For 2-dimensional representations: For 3-dimensional representations: where α i and β i denotes the elements of the first and second elements, respectively.