Predictive $S_4$ flavon model with $\text{TM}_1$ mixing and baryogenesis through leptogenesis

We use $S_4$ discrete group to construct a neutrino flavour model which leads to $TM_1$ mixing and is consistent with the neutrino oscillation data. Using the model's constrained parameter space, we predict the values of Dirac $CP$ phase and the light neutrino mass as $-1<\sin \delta<-0.9$ and $1.7<m_1 (\text{meV})<5.5$ respectively. We thoroughly examine the usefulness of this model in explaining the observed baryon asymmetry of the Universe. Near-maximal breaking of CP symmetry (arising due to the $\text{TM}_1$ constraint) helps us in generating adequate baryon asymmetry through leptogenesis. We study the evolution of the asymmetry (generated due to the decay of the heavy Majorana neutrinos) starting from the primordial Universe in two different ways (i)explicitly solving network of Boltzmann equations, (ii) using approximate analytic solution and we have shown the extent of their equivalence. Nearly accurate analytical fits are used thereafter to evaluate baryon asymmetry for the whole parameter space allowed by $3\sigma$ global fit of oscillation data and to impose a constraint on the yet unbounded mass scale parameter of Dirac neutrino mass matrix. Furthermore, significant contribution of $N_2$ decay in the context of flavoured leptogenesis is also estimated.


Introduction
The study of the nature and the properties of neutrinos is key to extending our understanding of particle physics beyond the Standard Model (SM). The phenomenon of neutrino oscillations [1,2] is one of the handles to learn about neutrinos. The story of neutrino oscillations started with the discovery of solar neutrino deficit by the Homestake experiment [3] in 1970s, which was also reported by a number of experiments in the following decades. A way to resolve the problem was through the neutrino oscillations under which electron neutrinos produced in the sun changed into other flavours causing the deficit. In early 2000s, the SNO experiment [4] gave the irrefutable confirmation of oscillations by showing that even though the flux of electron neutrinos showed the deficit, it disappeared when the total flux of all neutrino flavours was taken into account.
The theoretical foundation for neutrino oscillations was laid in 1957 by Bruno Pontecorvo [1] who showed that if neutrinos had mass, they would oscillate from one flavour to another. As we know today, the SM fermions exist in three families or flavours. In the quark sector as well as in the neutrino sector, the flavour mixing arises as a consequence of the flavour eigenstates being a superposition of the mass eigenstates. In the neutrino sector, this superposition is described in terms of a unitary matrix called the PMNS matrix [5,6]. The PMNS matrix, is parameterized by three mixing angles (θ 12 , θ 23 and θ 13 ), a Dirac CP phase (δ) and two Majorana phases [7,8]. Neutrino oscillation probabilities depend on the three mixing angles and the Dirac phase along with the neutrino mass-squared differences, ∆m 2 21 and ∆m 2 31 . Starting with the SNO experiment, the last couple of decades saw tremendous progress in detecting various oscillation modes by a large number of solar, atmospheric and reactor neutrino experiments and measuring the three mixing angles and the two mass-squared differences. To have a clear idea (qualitatively as well as quantitatively) about the neutrino oscillation phenomenon, precision measurements of the above mentioned six parameters are necessary. It has to be remembered that the Majorana phases (α 1 , α 2 ) do not show up in the ordinary neutrino flavour oscillation probabilities. Their effect is reflected only in neutrino-antineutrino oscillations [9,10] which have not been observed yet. While we have quite robust as well as precise results on mixing angles and mass squared differences, the ambiguity in Dirac CP phase (δ) and mass ordering (normal: m 1 < m 2 < m 3 or inverted: m 3 < m 1 < m 2 ) of the light neutrinos still persists. Recently, significant advancement has been made in measuring this δ by NOνA [11,12] and T2K [13,14] experiments which seems to shed light on the yet unsolved issue of CP violation in the Dirac neutrino mass matrix. Determination of the octant of θ 23 is also one of the major goals of the ongoing experiments. A global analysis [15] including the latest data coming out from the experiments MINOS [16], T2K [13,14], NOνA [11,12], Daya Bay [17], RENO [18], Double CHooz [19] shows that the Inverted mass ordering (IO) is disfavoured with a ∆χ 2 = 4.7 (9.3 with SK atm) and that for θ 23 , the second octant is preferred (first octant disfavoured with ∆χ 2 = 4.5). The best fit value of δ has been found to be 215 • (normal ordering) whereas the CP conserving case (δ = 180 • ) is disfavoured with ∆χ 2 = 1.5. Although this global analysis [15] is quite rigorous and points towards normal ordering (NO), second octant for θ 23 and non-zero Dirac type CP violation, more precise results are expected in the coming years. If the neutrinos are Majorana particles, it would result in double beta decays without the emission of neutrinos. Several experimental searches (EXO-200 [20], GERDA [21], KamLAND-Zen [22], CUORE [23], CUPID [24]) seek to detect and observe such neutrinoless double beta decays (0νββ). These experiments have not yet detected 0νββ, but they set upper bounds on the effective Majorana neutrino mass, m ββ .
Cosmological observations have already proven the overwhelming predominance of matter over antimatter in the Universe. To the best of our knowledge any trace of appreciable amount of antimatter hasn't been found yet. The baryon asymmetry observed in the present day Universe is assumed to be generated dynamically from a baryon symmetric state, rather than taking it as an initial condition (the asymmetry would have been highly fine tuned in that case). More importantly, any primordial asymmetry would have been diluted very much during the inflationary epoch. Therefore, to generate the baryon asymmetry the other approach is followed, i.e the asymmetry is generated dynamically starting from a baryon symmetric era which is known as baryogenesis [25][26][27]. Among the various alternatives of baryogenesis (Affleck Dyne baryogenesis [28,29], GUT baryogenesis [30,31]) the widely accepted and compatible to our present model is baryogenesis through leptogenesis [32][33][34][35][36][37][38][39][40][41].
Here the asymmetry is created in the leptonic sector first, later which is converted into baryon asymmetry through sphaleron process [42]. There are three very important conditions known as Sakharov conditions [43] which are inevitable for successful leptogenesis to take place. They are (i) Baryon number violation, (ii) C and CP violation, (iii) departure from thermal equilibrium. Here, the baryon number is violated through lepton number violation which is again ensured by the presence of the Majorana mass term. The Majorana mass matrix involves the couplings among the Right-Handed (RH) neutrinos which are theorised to be present in addition to the SM fermions. The RH neutrinos are SM gauge singlets and assumed to be much heavier than than the SM fermions. The Dirac neutrino mass matrix (couplings between the RH neutrinos and the left-handed SM neutrinos) is complex in general which acts as the source of CP violation. The out of equilibrium condition is satisfied naturally at some epoch of evolution of the Universe when the temperature of the thermal bath falls below the mass of the decaying particle. So, the CP violating (and L violating) out of equilibrium decays of heavy RH neutrinos to SM Higgs and leptons give rise to lepton asymmetry, the evolution of which with temperature (from the scale of RH neutrino decay to the present epoch) is tackled by the set of Boltzmann equations. The sphaleron induced (B + L) violating [44,45] processes are in thermal equilibrium between the temperature range from 10 2 GeV to 10 12 GeV. During this process the change in the baryon number and the lepton number is the same, i.e ∆B = ∆L = N f (N f is the number of fermion generations). So the sphalerons violate (B + L) keeping (B − L) 1 conserved and as a result, a fraction of the L (or B − L) asymmetry is converted into baryon asymmetry. This asymmetry is expressed by η B (or Y B ) which is the measure of excess of baryons over antibaryons scaled by photon density (n γ ) (or comoving entropy density (s)), i.e.
where n B (nB) are the number density of baryons (antibaryons) and the subscript zero denotes the value of the corresponding asymmetry parameter [46] at the present epoch.
In the present work, we aim to build a model which simultaneously tackles the issues of the construction of the mass matrices in the lepton sector as well as the generation of the baryon asymmetry through leptogenesis. The model should be consistent with the observed neutrino mass-squared differences and the mixing. Unlike the case of the quarks, the neutrino mixing angles are quite large. Early measurements of solar and atmospheric oscillation probabilities have pointed towards sin 2 θ 12 ≈ 1 3 and sin 2 θ 23 ≈ 1 2 . This led to the tri-bimaximal mixing (TBM) [47] which theorized |ν 2 = 1 Even though the observation of non-zero reactor angle in 2012 ruled out TBM, often model builders use it as a starting point. A large number of ansatze which modify TBM, by preserving some of its features or symmetries while breaking some others, have been proposed. One of the most promising among these is the TM 1 [48][49][50][51] mixing which preserves the first column of TBM and leads to constraints among the mixing angles and the CP phase. A number of papers [52][53][54][55][56][57][58][59][60][61][62][63][64] had been published in which the TM 1 mixing arises as a result of an underlying discrete symmetry. In our paper, we construct a flavon model based on the S 4 discrete symmetry. Flavons are scalar fields and they transform as multiplets under the discrete group. They acquire Vacuum Expectation Values (VEVs) through spontaneous symmetry breaking. These VEVs form the building blocks of the lepton mass matrices. The structure of the discrete group as well as the residual symmetries of the VEVs result in specific textures of the mass matrices. The mass matrices constructed in our model lead to TM 1 mixing and we use them in our study of the leptogenesis.
The plan of the paper is as follows. In Section 2, we first describe TM 1 mixing and briefly outline the properties of the flavour group considered, i.e. S 4 . The SM fermions as well as the theorised flavon fields are assigned as multiplets under S 4 . The Lagrangian is constructed and the flavons are given specific VEVs. Then we construct various mass matrices and extract the lepton masses along with generating the TM 1 mixing. In Section 3, we start by expressing our mass matrices in the standard basis employed in the leptogenesis calculations. We then move on to describe the basic techniques of these calculations such as the CP asymmetry parameters and the Boltzmann equations. We discuss two versions of approximate analytic solutions of the Boltzmann equations. We also discuss the phenomenon of flavour decoherence and the significance of next to lightest RH neutrino (N 2 ) in the context of our model. In Section 4, the neutrino oscillation data are fitted with the model parameters and their allowed ranges are found. The neutrino masses and the CP phase are predicted from the model constraints. The baryon asymmetry parameter is calculated in both the unflavoured and the τ -flavoured regimes. This calculation is carried out by numerically solving the Boltzmann equations and also using the analytic approximations. This analysis is used to further constrain the parameter space. Finally, our results are summarised in Section 5.

The Flavon Model
2.1 TM 1 mixing TM 1 mixing can be expressed as a perturbation on the tri-bimaximal mixing, using two parameters 1 and 2 . By comparing the above matrix with the standard PMNS mixing (PDG convention) matrix, Eq. (1.1), we obtain the parameters 1 and 2 in terms of the mixing angles θ 13 and θ 23 , The reactor mixing angle is parameterized by 1 , since we have θ 13 → 0 when 1 → 0. Since the reactor mixing angle is observed to be non-zero, 1 should not vanish. On the other hand, 2 parameterizes the breaking of µ-τ symmetry [65][66][67][68], since 2 = 0 leads to the µ and the τ rows of the mixing matrix, Eq. (2.1), to become equal. Conservation of µ-τ symmetry implies θ 23 = π 4 , as is evident by the vanishing of Eq. (2.3). The latest experimental results point towards non-maximal atmospheric mixing and indicate the breaking of µ-τ symmetry and thus a non-vanishing 2 . In the TM 1 mixing, the solar angle can be obtained in terms of the reactor angle, The Jarlskog's CP violation parameter [69][70][71] can be expressed in terms of the elements of |U P M N S | 2 using where P αi = |U P M N S αi | 2 [72]. Substituting (2.1) in (2.5) we obtain In terms of the mixing angles and the CP phase, we have J CP = sin δ sin θ 13 sin θ 12 sin θ 23 cos 2 θ 13 cos θ 12 cos θ 23 . From the above expression, we can see that when θ 23 = π 4 we obtain sin 2 δ = 1, i.e. δ = ± π 2 . Therefore, for TM 1 mixing µ-τ symmetry implies maximum CP violation.
The TM 1 mixing obtained in this paper has non-vanishing values of both 1 and 2 . Therefore, we have non-zero mixing angle, breaking of µ-τ symmetry (non-maximal atmospheric mixing) and non-maximal CP violation.

S 4 group
We construct the model in the framework of the discrete group S 4 which has been studied extensively in the literature [54-56, 60-62, 73, 74]. Here we briefly mention the essential features of this group in the context of model building. S 4 is the rotational symmetry group of the cube. S 4 has 24 elements which fall under four conjugacy classes. Its conjugacy classes and irreducible representations are listed in Table 1.
(1) (12)(34) (12) (1234) (123) S 4 group can be generated using The above matrices represent a convenient basis for the triplet (3) of S 4 . The tensor product of two triplets (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) leads to (2.14) For the triplet (3 ), the generators are given by S = S, T = T and U = −U . Table 2 provides the field content of our model. The three families of the left-handed-weakisospin lepton doublets, L, and the three right-handed heavy neutrinos, N , form triplets (3) under S 4 . The flavons φ C , η D , φ D , η M and φ M , are scalar fields and are gauge invariants. They transform as singlets (η D , η M ) and triplets, 3 (φ C ) and 3 (φ D , φ M ), under S 4 . The C 6 group is introduced to prevent unwanted couplings. It also helps in determining the vacuum alignment of φ C . Given the field assignments, Table 2, we write the Lagrangian,

The Model Lagrangian
where y i with i = τ, µ, e, D 1 , D 3 , M 1 , M 3 are the coupling constants. () 3 and () 3 denote the symmetric and the antisymmetric tensor products which transform as 3 and 3, Eqs. (2.13, 2.14). Note that φ C couples in the charged-lepton mass term, η D , φ D couple in the neutrino Dirac mass term and η M , φ M couple in the neutrino Majorana mass term. We assume that the coupling constants in the model, y i , are real numbers, i.e. we do not introduce CP violation explicitly. Rather, CP is broken spontaneously by the flavon vacuum alignments. Because of the complex C 6 assignments, Table 2, the flavons φ C , η M , φ M have complex degrees of freedom. In contrast, η D and φ D are real fields. The scalar fields acquire VEVs through Spontaneous Symmetry Breaking (SSB). The Higgs VEV is the familiar (0, v) where v ≈ 170 GeV. For the flavon fields, we assign the following VEVs, The alignments of φ C , φ D and φ M in their respective triplet flavour spaces are fully determined by their residual symmetries which we will discuss in the next section. We also obtain the charged-lepton and the neutrino mass matrices in terms of the coupling constants and the VEVs of the scalar fields. The residual symmetries of the flavon VEVs manifest as the symmetries of the mass matrices.

Charged-lepton mass matrix
In the charged-lepton sector,L which transforms as 3 couples with the flavon triplet φ C which also transforms as 3. Under C 6 ,Lτ R ,Lµ R andLe R transform asω, ω, and 1 respectively. Since φ C transforms as ω under C 6 , φ C , φ * C and (φ * C φ C ) 3 couple toLτ R ,Lµ R andLe R respectively. Note that the quadratic term, (φ * C φ C ) 3 , needs to be included because neither φ C nor φ * C can couple in the electron sector. The vacuum alignment of this term is calculated by taking the antisymmetric product of φ * C = v C (1,ω, ω) and φ C = v C (1, ω,ω) using Eq. (2.14), Substituting the Higgs VEV, H = (0, v), and the flavon VEVs, we obtain the charged-lepton mass term,l A charged-lepton mass matrix of the same form as Eq. (2.25), was recently obtained in Refs. [64,75]. M C , has the cyclic symmetry generated by the group element T , Eq. (2.9), (2.26) T has eigenvectors (1, 1, 1), (1, ω,ω), (1,ω, ω) corresponding to the eigenvalues 1, ω,ω. Therefore,ωT generates the residual symmetry of φ C , The C 3 group generated byωT , i.e. {ωT, ωT 2 , I} is a subgroup of our flavour group, S 4 ×C 6 . φ C can be uniquely defined (upto a scale factor) as the alignment that uniquely breaks the flavour group into the above mentioned C 3 subgroup. The cyclic symmetry of M C , Eq. (2.26), is in fact the consequence of the C 3 residual symmetry of φ C .
Using the unitary matrix, we diagonalize the charged-lepton mass matrix, are the charged-lepton masses. Compared to the muon and the tau masses, the electron mass is suppressed by an additional factor of v C Λ . This is similar to the Froggatt-Nielsen Mechanism of obtaining the mass hierarchy. The matrix V , Eq. (2.28), is often referred to as the 3 × 3 trimaximal matrix or the magic matrix. The absolute value of every element of this matrix is equal to 1 √ 3 . The middle column of the magic matrix coincides with the middle column of the TBM mixing matrix and often the magic matrix plays an important role in neutrino model building.

Neutrino Dirac mass matrix
Substituting the Higgs VEV, H = (0, v), and the flavon VEVs, we obtain the neutrino Dirac mass term,ν The three right-handed neutrino components in Eq.
M w has the dimension of mass and is at the scale of the SM fermion masses while k is dimensionless and is of the order of one. Both M w and k are real numbers because the flavons, η D , φ D , are real fields and the coupling constants are assumed to be real. Consider the group elements, They generate the C 2 and the C 3 groups, {T U T 2 , I} and {S T , (S T ) 2 , I}, respectively. They correspond to the following residual symmetries of φ D : These residual symmetries uniquely define the alignment of φ D . It can be shown that the group elements given in Eq. (2.37), when taken together, generate the dihedral group D 6 , which forms a subgroup of S 4 . In other words, φ D breaks S 4 into one of its D 6 subgroups which in turn uniquely defines φ D . Note that the residual symmetries of the VEV, Eqs. (2.38), manifests as the symmetries of the mass matrix, Eq. (2.35), as well,

Neutrino Majorana mass matrix
Substituting the flavon VEVs we obtain the neutrino Majorana mass term, where is the Majorana mass matrix. We rewrite M M as Since the flavons η M and φ M are complex fields, the values of v M 1 and v M 3 can be complex.
Therefore, M f and z are complex numbers in general. M f has the dimension of mass and is at the scale of flavon VEV. We assume that this scale is quite high ≈ 10 12 GeV. This leads to the suppression of the light neutrino masses through the Type-1 [76][77][78][79] seesaw mechanism. The parameter z is dimensionless and is of the order of one.
Consider the group elements, They generate the C 2 groups, {T 2 U T , I} and {T 2 S T , I} respectively. When taken together, they generate the C 2 × C 2 group, This group represents the residual symmetries of φ M , The second C 2 symmetry in Eqs. (2.49) is responsible for the off-diagonal zeros in M M .

Effective seesaw mass matrix
The effective seesaw mass matrix is given by where In the previous discussion, we have shown that the Dirac mass matrix and the Majorana mass matrix break the original S 4 group into the D 6 group, Eq. (2.39), and the C 2 × C 2 group, Eq. (2.47), respectively. Besides the identity, the only common element in Eq. (2.39) and Eq.
This element generates the C 2 group, {T 2 U S T , I}. Hence, the seesaw mass matrix, Eq. (2.51), constructed from both the Dirac and the Majorana mass matrices possesses only the C 2 residual symmetry, Using the (1, 3)-bimaximal unitary matrix, we block diagonalize the seesaw mass matrix, This matrix is diagonalized using a unitary matrix with vanishing entries in (12), (13), (21), (31) positions. We call this unitary matrix U 23 . Therefore, the complete diagonalization of the seesaw mass matrix is given by where m 1 , m 2 , m 3 are the light neutrino masses. The neutrino mixing matrix, U PMNS , is obtained as the product of the unitary matrices that diagonalize the charged-lepton mass matrix, V , Eq. (2.29), and the effective seesaw mass matrix, U BM U 23 , Eq. (2.58), (2.59) V U BM is nothing but the tri-bimaximal mixing matrix, U TBM . The multiplication of U TBM with U 23 mixes the 2 nd and the 3 rd columns of U TBM giving rise to U TM1 , Eq. (2.1).
Having discussed the S 4 -symmetry-motivated neutrino mass model followed by the diagonalization of the light neutrino mass matrix, now it is evident that this model is capable of generating large mixing. Whether the numerical values of these observables (as predicted by our model) are consistent with the latest global fit data of neutrino oscillations will be explored in Section 4. The existence of the heavy RH neutrino and the presence of its Majorana type mass term ensure lepton number violation. The complex mass matrix in the neutrino Dirac sector 2 acts as the source of CP violation which is a necessary ingredient of the asymmetry generation. The detailed theoretical framework to account for this asymmetry and its evolution to the present day since the primordial era of Universe is presented in the following section. The problem of finding the final asymmetry has been dealt with rigorous kinetic equations (known as the Boltzmann equations) as well as with some useful analytical approximations. The usefulness of our model in the simultaneous explanation of the oscillation data and the observed baryon asymmetry will be examined numerically in Section 4.

Baryogenesis through Leptogenesis
As discussed earlier, we know that one of the convenient and effective ways to generate light neutrino mass is through Type-I seesaw mechanism where three right handed heavy neutrinos (N i ) are added to the SM. These physical right handed neutrinos with definite mass can decay both to a charged lepton with a charged scalar and a light neutrino with a neutral scalar. Due to the Majorana character of N i , conjugate process is also possible. If out-of-equilibrium decay of N i in conjugate process occur at different rate from the actual process, net lepton asymmetry will be generated. As a first step to study the leptogenesis we have to compute the CP asymmetry [80] parameter which depends upon the structure of the Dirac neutrino mass matrix and hence, depends upon the specific model under consideration. The leptogenesis phenomenon may be flavour dependent or independent according to the temperature regime. Accordingly, the CP asymmetries too may be flavour dependent. The CP asymmetries are then plugged into the Boltzmann equations to get the final value of the lepton asymmetry (flavour independent/dependent) which is further converted into baryon asymmetry by sphaleron process. Both flavour dependent and independent phenomena are explored in detail in the following subsections.

Standard basis for leptogenesis calculation
In the standard basis utilized for leptogenesis calculations, the charged-lepton mass matrix, Eq. (2.25), as well as the Majorana mass matrix for the neutrinos, Eq. (2.44) should be diagonal. In our model, both these matrices are non-diagonal. To diagonalize the chargedlepton mass matrix, we use the transformation, where M 3 , M 2 , M 1 are the heavy neutrino masses and U p is the unitary diagonal matrix used to remove the phases from the masses, The diagonalization, Eq. (3.2), corresponds to the following transformation of the RH neutrino fields: In the basis in which both the charged-lepton mass matrix and the neutrino Majorana mass matrix are diagonal (hereafter referred to as the "standard basis"), the neutrino Dirac mass matrix, Eq. (2.34), becomes

CP asymmetry parameter
In the present work, the smallness of the mass of active neutrinos is achieved through the well known Type-I seesaw mechanism where the fermion sector of the SM is extended in a minimal fashion through the addition of three heavy right-chiral neutrinos (N i ) which are singlets under the SM gauge group. CP violating out-of-equilibrium decays of these neutrinos act as the source of lepton asymmetry. From the neutrino Dirac mass term, y αiLα N iH , it is clear that the decay product of the right handed neutrino is either the lefthanded neutrino and the neutral scalar or charged lepton and the charged scalar. The basic quantity we are interested in is the flavour dependent CP asymmetry parameter [38,81] which is expressed as where Γ denotes the decay width. Basically, CP asymmetry is a measure of the difference in decay widths of N i in a process and its conjugate process. At the tree level, these two are the same 3 giving rise to vanishing CP asymmetry. Therefore, we have to investigate with higher order terms to obtain non-zero CP asymmetry. Taking into account the one loop vertex and self energy diagrams, it is found that non-zero CP asymmetry arises due to the interference between the tree level and the one loop diagrams. In the standard basis, the most general expression (keeping up to the fourth order of the Yukawa couplings) of the flavour dependent CP asymmetry parameter comes out to be where It is worthwhile to mention that depending upon the temperature regime at which leptogenesis takes place, the lepton flavours may be fully distinguishable, partly distinguishable or indistinguishable. The flavours can not be treated separately when the leptogenesis process occurs above a temperature T > 10 12 GeV. The CP asymmetry parameter will also be flavour independent accordingly. If leptogenesis occurs at a lower temperature T ∼ M 1 (M 1 mass of the lightest right-handed neutrino) then there are two possibilities: for T < 10 9 GeV all three (e, µ, τ ) flavours are separately active 4 , for 10 9 < T (GeV) < 10 12 only τ flavour can be identified separately while e and µ act indistinguishably 5 . For T > 10 12 GeV the flavour summed CP asymmetry parameter is given by For the model under consideration, it is straight forward to find the expression of the unflavoured CP asymmetry parameters. The main ingredient H in Eq. (3.9) can be represented in terms of the parameters of the mass matrix and the mixing matrix as from which the unflavoured CP asymmetry parameters are computed to be Now, to calculate the flavoured CP asymmetry parameters we need individual elements of the Dirac neutrino mass matrix in the standard basis. Unlike the unflavoured case, here both the diagonalization matrices (of the charged-lepton mass matrix and the neutrino Majorana mass matrix) contribute to the CP asymmetry parameters. The exact analytic expressions of the CP asymmetry parameters are too cumbersome to write here. We calculate them numerically and use in the appropriate formulas to calculate the final baryon asymmetry parameter. However, different elements of M D are spelt out in Appendix A. 4 In this regime we need three CP asymmetry parameters ε e i , ε µ i , ε τ i for each generation of RH neutrino. 5 Here we need two CP asymmetry parameters ε 2 i = ε e i + ε µ i and ε τ i for each generation of RH neutrino.

Boltzmann equations for leptogenesis
A particle species being coupled or decoupled with the thermal bath, depends roughly on the rate of interaction of the particle 6 (Γ) and the Hubble parameter (H), which can be expressed explicitly as [82]Γ H (coupled),Γ H (decoupled) . It is interesting to find out the phase space distribution of the particle species near the epoch of decoupling. The microscopic evolution of particle's phase space distribution (f (p µ , x µ )) is governed by the Boltzmann equation (BE) which is written as [83] whereL 7 is the Liouville operator and C is the collision operator which takes into account all such interactions that change the number density of the particle species under consideration in the thermal bath. In simple words, through this equation the evolution of the number density of a particle species can be tracked from very high temperature (early epoch) down to present temperature. Our present model has ample scope for the generation of lepton asymmetry through CP violating decay of heavy Majorana neutrinos in the early Universe.
Since we need to know the lepton asymmetry (to calculate the baryon asymmetry and compare it with the observed value) in the present epoch, we have to solve the Boltzmann equation for lepton number density which in turn depends on the instantaneous value of the RH neutrino density. Thus the required set of classical kinetic equations [81] (which can be derived from Eq. (3.12)) for the RH neutrino density and the lepton number density are given by

14)
6 PreciselyΓ is the interaction rate per particle for the reaction responsible for keeping the species under consideration in thermal equilibrium. 7 The covariant form of the Liouville operator isL = p α ∂ ∂x α − Γ α βγ p β p γ ∂ ∂p α where Γ α βγ is the Christoffel symbol. respectively, where z = Mass of lightest RH neutrino temperature = M 1 T , η a (z) = number density of paticle species a photon density = n a (z) n γ (z) , H is the Hubble parameter, Γs (detailed expression can be found in Ref. [38,81]) are the different decay and scattering cross sections scaled by photon density. Following Maxwell Boltzmann distribution, the number density of a particle species a of mass m a with g a internal degrees of freedom is given by from which the equilibrium density is obtained by setting the chemical potential to be zero as where K 2 is the modified Bessel function of 2nd kind with order 2. It is to be noted that the first superscript D(S) on Γ denotes decay (scattering) and the other superscript designates whether the interaction is Yukawa mediated (Y ) or gauge boson (G) mediated. Different D i parameters are defined as (3.20) The above set of Boltzmann equations, Eqs. (3.13, 3.14), consider asymmetry generated by all the three generations of RH neutrinos, but they are valid in a temperature regime (T > 10 12 GeV) where the lepton flavours are indistinguishable. If the RH neutrinos are strongly hierarchical (M 1 M 2 , M 3 ), the asymmetry generated by heavier RH neutrinos is completely washed out due to N 1 interactions and the summation in the Boltzmann equation is required no more. We need quantities involving first generation only. In comparatively lower energies lepton flavours are partially distinguishable (10 9 < T ( GeV) < 10 12 ) or fully distinguishable (T < 10 9 GeV). Accordingly the flavour effects have to be introduced in the Boltzmann equations (3.13, 3.14). After incorporating the lepton flavour index in the suitable places, the modified Boltzmann equation for lepton number density is presented as whereas that of the RH neutrino number density remains unaltered since it does not involve lepton flavour index.
can be estimated using Eq.(3.18), Eq.(3.20) by introducing the flavour index α on different Γs. It is to be noted that instead of η parameter we can use an equivalent parameter Y (particle number density/entropy density) to express the abundance of a particle species at any instant of evolution. In this type of representation, the quantity associated with the lepton asymmetry is denoted as Y α which is related to η α L as Y α = (n γ s −1 )η α L . It is well known that n γ s −1 = 1/{1.8g * s (T )}, where g * s (T ) is the effective number of massless degrees of freedom [82] at temperature T . Although g * s is a function of temperature, for sufficiently high temperature (T > 10 12 ) GeV it becomes practically constants and its value is 112 if we include the contribution of the three RH neutrinos. The lepton asymmetry created by the decay of the heavy Majorana neutrinos (before sphaleron processes set in) gets converted into baryon asymmetry by the action by sphalerons. One interesting aspect of this sphaleron interactions is that both baryon number (B) and lepton number (L) are violated in this process keeping the difference (B − L) (for flavoured regime this is (B/3 − L α ) = ∆ α ) conserved. We now define a new asymmetry parameter Y ∆α which is related to the Y α through an asymmetry coupling matrix A as Y α = β A αβ Y ∆ β . The Boltzmann equations describing the evolution of flavour asymmetry parameters can be represented in terms of Y ∆α as We are now in a position to estimate the final baryon asymmetry for which we have to solve the set of Boltzmann equations for flavour asymmetry (Eq.(3.22)) and RH neutrino density (Eq.(3.13)) 8 simultaneously upto a large enough value of z for which the asymmetry freezes to certain constant value.

Baryon asymmetry in different regimes
The asymmetry creation takes place mostly around a temperature scale of the order of lightest RH neutrino mass (M 1 ). Accordingly, there are three different regimes of leptogenesis [84,85]. The final value of baryon asymmetry (Y B = n B /s) is calculated by summing over the flavour asymmetry parameters (obtained by solving the Boltzmann equations) followed by multiplication with the sphaleron conversion factor, i.e., The equivalent η B parameter is connected to Y B as η B = s nγ 0 Y B = 7.0394Y B where the zero subscript denote its value at the present epoch. The final baryon asymmetry parameter is given by

An approach towards analytic approximation of Boltzmann equations
Lets start with the set of Boltzmann equations in the unflavoured regime, i.e., It has already been shown in the existing literature that for a strong washout scenario, if the active neutrinos follow hierarchical pattern, the ∆L = 2 terms and the scattering terms can be safely neglected. Thus we consider a simplified picture where only decays and inverse decays are taken into account, i.e.
Now using Eq.(3.18), D i (z) can also be expressed in the form where r i = M 2 i /M 2 1 , Γ N i is the decay width and K i is the decay parameter corresponding to the ith right handed neutrino. In Eq.(3.31) the washout term gets maximum contribution from the inverse decay which is given by where η in B−L is the pre-existing asymmetry. If it is assumed that there is no such kind of pre-existing asymmetry, then the final value of (B − L) asymmetry is simply obtained as where the final efficiency factor κ f i [86,87] is given by It is well known that in the strong washout regime for hierarchical RH neutrino mass spectrum (M 1 M 2 M 3 ) only the asymmetry produced by the lightest RH neutrino survives. Using the approximation dz , the analytic expression of final efficiency factor (z → ∞) corresponding to the lightest RH neutrino is obtained as where z B is such a value of the independent variable (z ) around which the integrand of Eq. (3.36) receives maximum contribution and for strong washout its analytical expression is given by [86,87] z B (K 1 ) 2 + 4K 0.13 Now, if the masses of the RH neutrinos are close to each other 9 , we can not neglect the contribution of the next to lightest RH neutrino towards final asymmetry. Under the assumption that asymmetry production or washout by N 2 is not affected by that of N 1 , the efficiency factor corresponding to N 2 is given by a simple analytic expression as The final value of the baryon asymmetry parameter η B is obtained from η f B−L after multiplication by the sphaleron conversion factor (a sph ) 10 and division by a dilution factor f [88] (which is estimated assuming standard thermal history of the Universe as Broadly, we can say that these two results match exactly with each other in the strong washout regime for hierarchical RH neutrinos. In the strong washout regime, an excellent fit to the efficiency factor for any value of δ 12 has been found [87] to be In the present work, we will first show that the lagrangian parameters constrained by the 3σ oscillation data mostly favour strong washout regime. Then for a bench mark point (we choose it to be the best fit point, i.e. the set of points having least χ 2 ), we find η B using 9 In the present problem our neutrino mass model predicts closeness of mass two RH neutrinos where as third one is order of magnitude higher than these two. So in this case it is sufficient to deal with two generations of RH neutrinos 10 a sph stands for the fraction of (B − L) asymmetry being converted into baryon asymmetry through sphalerons process 11 A detailed analysis dealing with this issue has been carried out in Ref [87].
the analytical formulas (both κ ∞ i and κ fit ) and also using the direct numerical solution of Boltzmann equations (Eq.(3.30), Eq.(3.31) ) to show the accuracy of the analytical fit. Upon showing satisfactory accuracy of the analytical formulas, we then proceed to find the baryon asymmetry for each and every point of the 3σ parameter space using the analytical formulas.

Lepton flavour decoherence and flavoured leptogenesis
It has been pointed out earlier that depending upon the mass of the lightest RH neutrino we can get unflavoured, τ -flavoured or fully flavoured leptogenesis. The distinguishability of lepton flavours in different temperature regimes can be explained by flavour decoherence [86,87,89,90] phenomenon 12 .
When M 1 > 10 12 GeV, the RH neutrino decays producing lepton doublets (|l i ) which are nothing but a coherent superposition of corresponding flavour states, i.e where (i = 1, 2, 3) stands for generation index of RH neutrinos and (α = e, µ, τ ) are lepton flavour indices. The coefficients are given by Since at this temperature regime there doesn't exist any fast interaction that can break the coherence between the flavour states before it inverse decays into N 1 , the net asymmetry is produced along |l i in which we can not differentiate between flavours.
To carry out the analysis in the flavoured regime we define branching ratio to the individual lepton flavours as (3.45) The flavoured decay parameter is given by where m * 10 −3 eV is the equilibrium neutrino mass. The flavoured efficiency factor is given by (3.47) 12 An exhaustive discussion on the flavour decoherence issue is presented in Ref [86,87].
The analytical form of the flavoured efficiency factor [86,87,89] is obtained (using the same arguments as done in the unflavoured case) as where A αα is the asymmetry coupling matrix (Eq.(3.23) or Eq. (3.25) ).
For the intermediate mass regime (10 9 < M 1 (GeV) < 10 12 ), the τ lepton Yukawa interactions reach equilibrium, i.e they become faster than the inverse decay process. Thus coherence between the lepton flavours (generated due to decay of N 1 ) is broken projecting a portion of the lepton flavours along τ , whereas the other portion being projected in a plane perpendicular to τ 13 (which is actually a coherent superposition of e and µ). When M i < 10 9 GeV, µ lepton interaction rate becomes faster than the inverse decay process thereby breaking the coherence of e and µ in τ ⊥ direction. Therefore in this situation, flavour decoherence is completely achieved allowing us to track asymmetry along e, µ and τ directions separately (which is termed as the fully flavoured leptogenesis). We now discuss about the analytical formula to find the baryon asymmetry in the τ -flavoured (or two flavoured) regime. It is to be mentioned that in the present case it is sufficient to consider asymmetry generated by the lightest and the next to lightest RH neutrinos (since the third one is orders of magnitude higher than these two). Now, in the τ -flavoured regime when the partial flavour decoherence achieved, lepton doublet states |l 1 and |l 2 projected in τ ⊥ plane are expressed by as linear superposition of |e and |µ states as The asymmetry produced by N 2 along τ direction is washed out directly by N 1 interactions along τ through the exponential suppression factor exp(−3πK 1τ /8). So, the total asymmetry generated by N 1 and N 2 along τ direction is given by But this direct suppression of asymmetry doesn't hold for the asymmetry produced by N 2 along τ ⊥ direction. To account for the washout of N 2 generated asymmetry along τ ⊥ direction due to N 1 inverse decay, we have to calculate the probability of finding |l τ ⊥ 2 state along |l τ ⊥ 1 , which is given by (3.52) 13 A diagrammatic representation of these flavour projections is given in [87].
Therefore, the total asymmetry produced along τ ⊥ direction due to the decay of N 1 and N 2 (after proper inclusion of the washout effects) is given by The asymmetry produced by N 2 [91,92] along a direction which is again perpendicular to τ ⊥ 1 (being completely orthogonal to |l τ ⊥ 1 ) totally escapes the washout due to N 1 inverse decay. Thus, this asymmetry along |l τ ⊥ 1⊥ survives as a pure N 2 contribution which is mathematically expressed as Therefore in the flavour space, the asymmetry is assumed to be distributed along three direction, τ , τ ⊥ 1 , τ ⊥ 1τ ⊥ from which the total final asymmetry is computed as (3.55) The pure N 2 contribution (N ∆ τ ⊥

1⊥
) constitutes a significant part of the total (B − L) asymmetry since it is not suppressed by any washout factor (which will be demonstrated clearly in the numerical analysis). This contribution is generally overlooked in the straightforward solution of flavoured Boltzmann equations considering three generations of RH neutrinos.
To account for this asymmetry, we have to solve the Boltzmann equations for N 2 separately in which the source term gets contribution from ε 2τ ⊥ ( or ε e+µ 2 ) only, whereas the washout term contains N 2 interactions only (washout due to N 1 inverse decay has to be neglected completely).

Analysis of numerical results
The main objective of this numerical analysis is to find the allowed parameter space constrained by 3σ range [15] of neutrino oscillation observables (mixing angles: θ 12 , θ 23 , θ 13 , mass squared differences: ∆m 2 21 , ∆m 2 31/32 ) as well as the bound [46] on baryon asymmetry. Therefore, the whole analysis can be regarded as a two step process where in the first step we constrain the parameters with the 3σ range of oscillation data and thereafter the admissible parameter space gets a second round of restriction from the range of the observed baryon asymmetry.

Fitting with neutrino oscillation data
Along with the overall scale (M 2 w /M f ), the seesaw mass matrix (2.51) is parameterized by k and z (z is a complex number), i.e. we have a total of four real degrees of freedom. We fit these parameters with the experimental data on the neutrino mixing angles and the masssquared differences. In this fit, the number of experimental degrees of freedom is six, i.e. the three mixing angles (θ 12 , θ 23 , θ 13 ), the Dirac CP phase (δ) and the mass-squared differences ∆m 2 21 , ∆m 2 31 . As we have stated in Section 2, the TM 1 mixing has only two degrees of freedom, 1 and 2 . Since the oscillation mixing matrix is parametrised by four observables (θ 12 , θ 23 , θ 13 , δ), the TM 1 scenario leads two constraints among the mixing angles and the CP phase, Eqs. (2.4, 2.8). It is straightforward to see that the solar mixing angle (θ 12 ) given in Eq. (2.4) is smaller that the TBM value. We note that the global fit favours such a smaller θ 12 . The second constraint, Eq. (2.8), is also consistent with the global fit data. In TM 1 , a deviation from maximal atmospheric mixing (θ 23 = π 4 ) leads to a deviation from maximal CP violation (δ = ± π 2 ). The data shows a slight preference for non-maximal θ 23 as well as an indication towards large negative CP phase (δ ≈ − π 2 ) which are consistent with TM 1 . The fact that the data naturally satisfies the two TM 1 constraints helps us to fit the six observables with the four model parameters.
We scan the parameter space (having four real degrees of freedom -M 2 w /M f , k, Re(z), Im(z) and generate the neutrino masses and the mixing observables. The parameter values which correspond to the observables lying outside their experimental 3σ ranges are omitted. For the allowed parameter values, we perform a chi-squared goodness of fit analysis. We evaluate where (Obs i ) model are the observables generated using the model parameters, (Obs i ) expt are the experimental best fit values taken from [15] and σ i are the corresponding 1σ errors. The summation is made over the six observables: sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 , sin 2 δ, ∆m 2 21 and ∆m 2 31 . By minimising χ 2 , we obtain the best fit values of the model parameters and also the corresponding values of the observables. The allowed parameter space for the dimensionless parameters k and z is given in Figure 1. The parameter k generates non-zero reactor angle. When k = 0, we obtain -25 -   [93][94][95][96]. The model predicts 60.3 meV < m i < 66.5 meV which is consistent with the current cosmological bounds. In Figure 3, we show the predicted range of i m i against that of sin δ. We note that the model parameter space contains points corresponding to the inverted mass ordering of the light neutrinos also. However, we have omitted these points because the global oscillation analysis disfavours inverted ordering. Also, the inverted ordering in our model leads to i m i > 150 meV which is disfavoured by the cosmological bound obtained under certain assumptions [93][94][95][96]. The effective Majorana mass applicable in neutrinoless double-beta decays is given by m ββ = i U 2 ei m i where U ei are the elements of the first row of the U PMNS matrix. From the allowed parameter space of the model, we calculate the effective mass and obtain 2.89 meV < m ββ < 8.02 meV. This range is well below the bounds set by the current 0νββ experiments [20][21][22][23][24].

Constraining by baryon asymmetry bound
It is clear from the discussion of the previous section that, upon imposing the 3σ experimental bound of neutrino oscillation data, the scale factor sitting outside the effective light neutrino mass matrix (Eq.(2.51)) is constrained to vary within a certain range. This factor contains M 2 w in the numerator and M f in the denominator. The overall multiplicative factor to the Dirac neutrino mass matrix, i.e. M w , which consists of a Yukawa type coupling (y D 1 ) with the Higgs VEV and the flavon VEV reduced by the scale Λ, is not an experimentally known quantity. However, we expect that M w is around the typical mass scale of a fermion i.e. it can be assigned values ranging from the top quark mass (∼ 170 GeV) down to the electron mass (0.5 MeV) (which are of course the two extremes). Therefore for a fixed value of the ratio (M 2 w /M f ), we can get different values of M f (which is the scale of the RH neutrino mass) by varying M w 14 . This allows us to vary the RH neutrino masses over a wide range which thereby opens up the possibility to study Leptogenesis in different regimes. Let us first proceed to analyze the numerical results of unflavoured Leptogenesis and then examine whether we can put any constrain on M w (which was unbounded by oscillation data) by the baryon asymmetry bound.

Unflavoured regime
It can be understood from the analytical formula of baryon asymmetry (Eq. (3.40)) that the sign of the final baryon asymmetry will depend on the sign of the CP asymmetry parameter (since the efficiency factor κ is always positive). The expression of unflavoured CP asymmetry parameter (Eq. (3.11)) shows that its sign is completely determined by the phase of z. Therefore the multiplicative factor M 2 w /M f of the M ss matrix has no role in determining the sign of ε i (or equivalently Y B ). Therefore using the constraint Y B > 0 we can further constrain the 3D parameter space which is already constrained by the 3σ global fit of oscillation data. The new parameter space after constraining by the condition of positive baryon asymmetry is shown in Figure 4. It is clear from Figure 4  the parameter space has been reduced by a considerable amount with respect to that of Figure 1. The sign of Y B is determined by the set {k, Re(z), Im(z), M 2 w /M f }, whereas the magnitude is controlled by the scale factor M w . All the allowed points shown in Figure 4 has the potential to generate Y B within the experimentally allowed range. For a definite set of values of {k, Re(z), Im(z), M 2 w /M f }, M w can be constrained by the bound on Y B (8.522 < Y B × 10 11 < 9.375 [46]). This exercise can be repeated for all the allowed points shown in Figure 4 and for each such set a bound on M w can be obtained with the imposition of baryon asymmetry bound. Now, Y B can be calculated either by direct numerical solution of the Boltzmann equations or by using the analytical formulas of efficiency factors (κ f i , κ fit ). In Figure 5, we show the 3σ allowed range of the decay parameters corresponding to the lightest (N 1 ) and next-to-lightest (N 2 ) RH neutrinos. It shows their preference towards strong washout regime.
Using the set of values of {k, Re(z), Im(z), M 2 w /M f } corresponding to least χ 2 (which is sometimes referred to as the best fit point) as the bench mark point and keeping M w fixed at a certain value, we solve the set of Boltzmann equations numerically and show the variation of Y B with z in Figure 6. The value of Y B at high z (or equivalently at low temperature) where it doesn't change with z anymore is the final value of baryon asymmetry. We have repeated the same procedure for different values of M w keeping the  other lagrangian parameters fixed at the best fit value and shown the variation of final Y B with M w graphically in Figure 7 where the left panel is generated by direct numerical solution of Boltzmann equations whereas the plot of the right panel is generated using analytic approximation κ f i . In this plot, we draw two lines parallel to the M w axis which represents experimental upper and lower bounds on Y B respectively. The corresponding M w coordinates where these straight lines intersects the Y B −M w curve represent the upper and the lower bounds on M w respectively. To get an idea how precisely the analytical formulas can reproduce the results of the actual Boltzmann solution, we show in tabular (Table  3) Table 3 that analytical formulas produce results very close to the actual solution of Boltzmann equations. The error, if we use κ fit , is even less than 10%. Thus we can readily use the analytical formulas to scan the whole parameter space ( Figure 4) and estimate the bound on M w for all the points belonging to the parameter space constrained by 3σ oscillation data with positive Y B bound. In Figure 8, we show graphically the variation of M w with the allowed range of the Lagrangian parameters. The spread in M w for each value of the variable along the abscissa is the corresponding bound on M w imposed by the Y B bound.

τ -flavoured regime
It has been noticed during the analysis of unflavoured regime that (specially from the Table 3) the experimentally observed value of baryon asymmetry is obtained for such value of M 1 (lightest right handed neutrino) which lies at the lower edge of the unflavoured regime. So, there is a possibility of generating adequate asymmetry in the τ -flavoured regime (since washout in flavoured leptogenesis is less than that of unflavoured case). In this context we want to remind the reader about the pure N 2 contribution (asymmetry generated by N 2 decay along τ ⊥ ⊥ ). Since this asymmetry does not suffer washout of lightest RH neutrino    Figure 8. Allowed M w constrained by the baryon asymmetry bound for the whole 3σ parameter space of oscillation data. (Y B has been calculated using κ f i . The result will not vary much with that of κ fit ).
inverse decay it has a significant contribution to the final asymmetry (its typical value is comparable to that of lightest RH neutrino). It has been examined that the CP asymmetry (ε ⊥ 2τ ⊥ ) associated to this contribution is always negative throughout the whole parameter space which ascertain that this asymmetry will always have a positive contribution towards final baryon asymmetry. without taking into account the pure N 2 contribution and then using it in the formula for total Y B . The corresponding restricted parameter spaces are shown in the left and right panels of Figure 9. The parameter space without pure N 2 contribution is very much similar to its unflavoured counterpart, whereas the inclusion of pure N 2 contribution results in a larger parameter space. This happens because of the remarkable positive pure N 2 contribution which drives the Y B value to the positive side and thus a few more points appear in the parameter space of the right panel (which was absent in the left panel). The effect of pure N 2 contribution is shown vividly in Figure 10 where we show the variation of Y B with z (while the set {k, Re(z), Im(z), M 2 w /M f } is kept fixed at its best fit value and a definite value of M w is used such that the final Y B freezes to the experimentally admissible range). Y B with or without pure N 2 contribution is denoted by dashed or solid lines respectively. Addition of the new contribution results in a three fold gain in the final asymmetry. It is clear from the plot that, for M w = 2.1 GeV the final baryon asymmetry generated by the usual components of asymmetry falls short of the experimental lower limit, whereas the inclusion of the new contribution easily drives this value within the admissible range. We show a tabular (Table 4)  w /M f } is fixed at the best fit (i.e. the point corresponding to least χ 2 ). The % errors of the analytical approximations in comparison to the actual solution of Boltzmann equations are also shown in the same table.
It is found that the error is always less than 10% which allows us to scan the whole 3σ parameter space using analytical formulas instead of solving the chain of Boltzmann   equations. The allowed range of M w obtained by imposing the experimental bound on Y B is depicted in Figure 11. It is clearly visible from the plots that use of the new asymmetry component shifts this required M w (to generate Y B within the experimental range) to a lower value. This analysis is carried out for a fixed set (corresponding to least χ 2 ) of values of the Lagrangian parameters {k, Re(z), Im(z), M 2 w /M f } picked from 3σ parameter space. For a fixed value of the ratio M 2 w /M f , M f increases linearly with square of M w . Thus the bound on M w dictates a restriction on M 1 (the lightest RH neutrino) too. So, we can say that the inclusion of the new asymmetry component helps us to get the required baryon asymmetry at a lower RH neutrino mass. In the present work, although we are successful in getting the flavoured leptogenesis even excluding the pure N 2 contribution, situations may arise where adequate asymmetry in the flavoured regime can be obtained only if we consider this new contribution.
The final Y B for each and every point belonging to the whole 3σ parameter space has been calculated using the κ f iα formula. In Figure 12, we show 15 the graphical representation of the variation of M w with k, Re(z), Im(z). The spread in M w for a definite value of the abscissa signifies the bound on M w obtained due to the imposition of baryon asymmetry bound.

Concluding summary
We have investigated a flavon model based on Standard Model with S 4 discrete symmetry group adhering to Type-I seesaw mechanism. Our model leads to TM 1 mixing through the incorporation of appropriate flavon fields. In our model neutrino oscillation phenomenology is described using four parameters. We carry out a chi-squared analysis fitting these parameters with the three mixing angles, the CP phase and the two mass-squared differences. The fact that the T M 1 mixing has two inbuilt constraints which are consistent with  Figure 12. The allowed range of M w constrained by the baryon asymmetry bound for the whole 3σ parameter space of oscillation data. Y B is calculated using κ f iα and pure N 2 contribution is also taken into account.
the data enables us to successfully carry out this fit. One of these constraints leads to near-maximal breaking of the CP symmetry, −1 < sin δ < −0.9. The specific texture of the seesaw mass matrix in the model results in the prediction of the light neutrino masses. We show that their values are consistent with the m i and m ββ bound. Furthermore, we have studied baryogenesis via leptogenesis in two different ways and have also shown their equivalence. Lagrangian parameters already constrained by the 3σ limit of oscillation data are used in the calculations of leptogenesis. The estimation of final baryon asymmetry has been carried out by solving the network of coupled Boltzmann equations as well as using appropriate analytical fits. Equivalence between these two methods has been shown clearly with corresponding numerical results. Moreover we have also shown the substantial contribution from N 2 leptogenesis in the context of flavoured leptogenesis.