Flavored Leptogenesis and Neutrino Mass with $A_4$ Symmetry

We propose a minimal $A_4$ flavor symmetric model, assisted by $Z_2 \times Z_3$ symmetry, which can naturally takes care of the appropriate lepton mixing and neutrino masses via Type-I seesaw. It turns out that the framework, originated due to a specific flavor structure, favors the normal hierarchy of light neutrinos and simultaneously narrows down the range of Dirac CP violating phase. It predicts an interesting correlation between the atmospheric mixing angle and the Dirac CP phase too. While the flavor structure indicates an exact degeneracy of the right handed neutrino masses, renormalization group running of the same from a high scale is shown to make it quasi-degenerate and a successful flavor leptogenesis takes place within the allowed parameter space obtained from neutrino phenomenology.


Introduction
Over the last couple of decades, we have witnessed remarkable success in neutrino experiments [1][2][3][4][5][6][7][8][9] indicating that neutrinos are indeed massive. Furthermore, mixing parameters have been measured with great precision. In fact, two of the three mixing angles namely solar (θ 12 ) and atmospheric (θ 23 ) ones are found to be large while the other one, reactor (θ 13 ) mixing angle, is relatively small. Such a finding clearly shows the distinctive feature associated to the lepton sector in contrast to the quark one where all the three mixing angles are measured to be small. To have a deeper understanding of it, one needs to investigate the origin of the neutrino mass by looking at the neutrino mass matrix as well as the charged lepton sector from a symmetry perspective.
Among the various discrete groups, A 4 turns out to be the most economical one 1 . It is a group of even permutations of four objects having three inequivalent one-dimensional representations (1, 1 and 1 ) as well as one three-dimensional representation (3). Interestingly, the three generations (or flavors) of right-handed charged lepton singlets can naturally fit into these three inequivalent one-dimensional representations of A 4 while the three SM lepton doublets can be accommodated into the triplet representation of A 4 [28][29][30]. Works along this direction [28,29,31,32] showed that type-I seesaw model with A 4 flavor symmetry in general leads to a typical tri-bimaximal (TBM) lepton mixing (sin 2 θ 12 = 1/3, sin 2 θ 23 = 1/2 and θ 13 = 0) pattern [33,34] in presence of SM singlet (though charged under A 4 ) flavon fields. Though such TBM pattern received a great deal of attention due to its close proximity with experimental observation prior to 2012, it fails to accommodate the recent observation of small, but non-zero θ 13 [8,35,36]. Subsequently, modifications over models based on A 4 (and other discrete groups) are suggested to accommodate non-zero θ 13 either by considering additional flavon fields or including corrections to vaccuum alignments of the flavons [37,38] or considering contributions to additional mixing from the charged lepton sector [39].
In this work, we particularly focus on a framework where a non-trivial contribution to lepton mixing is originated from charged lepton sector. We do not consider any additional flavon field apart from those ones incorporated in the original Altarelli-Feruglio (AF) model [28]. While the RHN mass matrix turns out to be diagonal as a result of the flavor symmetry imposed, the structure of the charged lepton mass matrix becomes such that it can be diagonalized by a complex 'magic' matrix [29]. Interestingly, an antisymmetric contribution to the Dirac neutrino mass matrix, originated from the product of two A 4 triplets, plays a crucial role in generating non-zero θ 13 [40][41][42][43] in our model which was overlooked in an earlier attempt [44]. In doing the analysis, we find the atmospheric mixing angle θ 23 ≤ 45 • i.e. to lie in lower octant (LO). We also note that only normal hierarchy (NH) of light neutrino masses are allowed in this model. This turns out to be another salient feature of our construction. These predictions can be tested in ongoing and future neutrino experiments as ambiguities are still present in determining octant for θ 23 as well as hierarchies of light neutrino masses.
Additionally, we also discuss the aspects of leptogenesis [45][46][47][48][49] from the CP-violating decays of RHNs in this A 4 based type-I seesaw scenario in line with observations [50][51][52][53][54][55][56][57]. In doing so, since the involvement of the neutrino Yukawa matrix in the charged-lepton mass diagonal basis is necessary, the specific flavor symmetric construction of it is expected to play an important role. In fact, due to this symmetry, exactly degenerate heavy RHNs result at tree level, thereby indicating the breaking of the perturbative field theory involved in CP asymmetry generation [58]. Following [44], we are able to show that running of the parameters involved in the neutrino sector from the flavor symmetry breaking scale to the RHN mass scale actually eliminates such exact degeneracies and as a result, leptogenesis can indeed be possible. The present study of matter-antimatter asymmetry generation via leptogenesis taking into account the effect of running however differs from that of [44] by two aspects. Firstly, we use less number of flavon fields and secondly, we present a detailed analysis of flavored leptogenesis by solving the relevant Boltzmann equations.
The rest of the paper is organized as follows. In Section 2 we present detail structure of the model including the analysis of the mixing matrices involved. Section 3 deals with phenomenology of neutrino mixing. Constrains and predictions on neutrino parameters (including neutrinoless double beta decay) involved are presented here. In Section 4 we perform a detailed study on leptogenesis solving flavored Boltzmann equations. Finally in Section 5 we summarize the results and make final conclusion.

Structure of The Model
To realize the canonical type-I seesaw mechanism, we first consider an extension of the SM by including three singlet RHN fields (N R ). Additionally, three flavon fields namely Φ, Ψ, ϕ and a discrete symmetry A 4 × Z 2 × Z 3 are also incorporated to probe the typical flavor structure involved in the lepton sector. Note that same fields content was also present in the original AF [18] construction. Here N R and the flavon fields Φ, Ψ transform as triplet, whereas ϕ transforms as a singlet under A 4 . A judicious choice of additional Z 2 × Z 3 symmetry assists the leptonic mass matrices to take specific forms and hence forbid several unwanted contributions. In Table 1, we present transformation properties of all the relevant SM fields, N R and flavons involved in the analysis.
The relevant effective Lagrangian involving charged leptons and neutrinos can be written as where y ,ν i=1,2,3 are the respective coupling constants, M is the mass parameter of RHNs and Λ is the cut-off scale of the theory. In the first line of Eq. (2.1), terms in the first parentheses represent products of two A 4 triplets forming a one-dimensional representation which further contract with 1, 1 and 1 of A 4 , corresponding to e R , µ R and τ R respectively, to make a true singlet under A 4 . On the other hand, in the second line of Eq. (2.1), the subscripts s, a correspond to symmetric and anti-symmetric parts of triplet products in the S diagonal basis of A 4 . The essential multiplication rules of the A 4 group elements are elaborated in appendix A.
From Table 1 it is evident that the tree level contribution to charged lepton Yukawa interaction,¯ Hα R (with α = e, µ, τ ), gets forbidden. Instead, such interactions are effectively generated once the flavon Φ gets a vacuum expectation value (vev) via the dimension-5 operators (present in first line of Eq. (2.1)). Similarly in the neutrino sector, the renormalizable Dirac Yukawa coupling is forbidden as the lepton doublet is charged under Z 3 whereas both N R and H transform trivially under it. However such effective Yukawa coupling is generated from dimension-5 operators involving flavons Ψ and ϕ, after they obtain vevs. Presence of Z 2 symmetry is important in identifying Φ from Ψ (both being A 4 triplet) so that they contribute to the charged lepton and Dirac neutrino Yukawa couplings differently.
The flavon fields break the flavor symmetry A 4 ×Z 3 ×Z 2 when they acquire vevs along 2 as a result of which the part of the Lagrangian contributing to the charged lepton sector can be written as Using the above Lagrangian one obtains the charged lepton mass matrix after the electroweak symmetry breaking as where v= 174 GeV stands for the vev of the SM Higgs.
In a similar way, the Lagrangian for neutrino sector after breaking of the flavor symmetries can be written as This yields the corresponding Dirac and Majorana mass matrices as
with f ν i = v Ψ Λ y ν i , i = 1, 2, 3. Let us now discuss the diagonalization of the charged lepton and neutrino mass matrices so as to obtain the lepton mixing matrix. First we note that the charged lepton mass matrix given in Eq. (2.4) can be diagonalized by a bi-unitary transformation where I 3×3 is a 3 × 3 identity matrix and where ω (= e 2iπ/3 ) is the cube root of unity. From Eq. (2.7), it is evident that the righthanded Majorana neutrino mass matrix M R is diagonal having degenerate mass eigenvalues (M ) to start with. On the other hand, f ν 1 and f ν 2 appearing in Eq. (2.6) are the symmetric and antisymmetric contributions to the Dirac neutrino Yukawa respectively, originated as products of two A 4 triplets and N R which further contract with Φ (see the product rules Eq. (A.5) and (A.6). This antisymmetric part plays an instrumental role 3 in realizing correct neutrino oscillation data.
Here it is worth mentioning that in the vanishing limit of f ν 2 → 0, (keeping the structure of the charged lepton and Majorana mass matrix intact) one can reproduce the TBM mixing as discussed in [44].
The effective light neutrino mass 4 matrix can be obtained within the type-I seesaw framework as where the structure of M R is given in Eq. (2.7). Now, from Eq. (2.6) and (2.8), in the basis where the charged leptons are diagonal, the Dirac neutrino mass matrix in that basis can be written as, Therefore, substituting Eq. (2.11) in the type-I seesaw formula given by Eq. (2.10) one obtains the light neutrino mass matrix as 3 Earlier the role of such antisymmetric contributions was analyzed in the context of Dirac neutrinos [40][41][42][43]. 4 With the symmetries mentioned in Table 1 in principle, there will be a contribution to the effective light neutrino mass via a dim-6 operator given by y ef f Λ 2 ( H HΦ). However, in the limit vΦ > M , this additional contribution can be neglected compared to the dominant type-I contribution considered here.

Clearly, to get the mass eigenvalues of light neutrinos we
Though Y ν is in general a complex matrix, Y ν Y ν T being a complex symmetric matrix can be diagonalized by an orthogonal transformation (in the (1, 3) plane) through the relation where the rotation matrix U 13 (parametrised by angle θ and phase ψ) is given by (2.16) The complex eigenvalues are given by Now substituting Eq. (2.15) in Eq. (2.13), we get , representative of three complex light neutrino mass eigenvalues.
In order to extract the real and positive light neutrino mass eigenvalues, we choose the following representations of the parameters f ν 1,2,3 (= |f ν 1,2,3 |e iφ 1,2,3 and φ 1,2,3 are the three phases associated) as where are the redefined parameters used for the rest of our analysis. Now we are in a position to define the rotation angle θ and phase ψ of U 13 matrix (see Eq. (2.16)) as: Similarly, the real and positive light neutrino masses can also be expressed in terms of χ 1,2 and γ 1,2 after we extract the phases from the complex eigenvalues. To proceed, note that Eq. (2.21) can be rewritten as Here U p stands for a diagonal phase matrix given by U p = diag(1, e iβ 21 /2 , e iβ 31 /2 ), and the real positive light neutrino masses are given by: where o r , o i , n r and n i can be written in terms of the associated parameters (χ 1 , χ 2 , γ 1 and γ 2 ) in our model as The phases β 21 (31) involved in U p are given by, Therefore using Eqs. (2.9), (2.16) and (2.27), the final form of the mixing matrix U which diagonalises the effective light neutrino mass matrix (in the charged lepton diagonal basis) can now be written as  Table 2: neutrino oscillation data obtained from NuFIT [64] for NH scenario of light neutrino mass.
U is therefore the lepton mixing matrix, called the Pontecorvo-Maki-Nakagawa-Sakata (U P M N S ) matrix, the standard form of which is given by [63], where c ij = cos θ ij and s ij = sin θ ij , and δ is the CP violating Dirac phase. Also, U m = diag(1, e iα 21 /2 , e iα 31 /2 ) is a phase matrix which contains two Majorana phases α 21 and α 31 . Comparing above two matrices given in Eq. (2.37) and (2.38) we get the correlation between the neutrino mixing angles (and Dirac CP phase) appearing in U P M N S and the model parameters as [40] | s 13 | 2 = 1 + sin 2θ cos ψ 3 , tan δ = sin θ sin ψ cos θ + sin θ cos ψ , (2.39)  .25), all the mixing angles (θ 13 , θ 12 , θ 23 ) and the Dirac CP phase (δ) involved in the lepton mixing matrix U P M N S are finally determined by the model parameters χ 1 , χ 2 , γ 1 and γ 2 . Hence, using the 3σ allowed ranges of the three mixing angles (θ 13 , θ 12 , θ 23 ) from neutrino oscillation data 5 presented in Table 2, we can restrict parameter space for χ 1,2 and γ 1,2 . This parameter space of the current set-up can be further constrained using the 3σ allowed ranges of the mass-squared differences (see Table 2). For that purpose, we introduce a dimensionless quantity r, defined as the ratio of solar to atmospheric mass squared difference for normal hierarchy, i.e., r = Substituting o r,i , n r,i from Eq. (2.31)-(2.34) into Eq. (3.1), we note that r now becomes function of χ 1 , χ 2 , γ 1 and γ 2 . Apart from the satisfaction of r value obtained from the ratio of the best fit values of mass-squared differences, we must satisfy both the individual masssquared differences, ∆m 2 21 and ∆m 2 31 , independently within their 3σ allowed ranges using Eqs. 2.28-2.30. There also exists a cosmological upper bound on sum of the light neutrinos masses as i m i ≤ 0.11 eV [66,67] which will also constrain the parameter space. Note that in order to evaluate i m i , we need to get an estimate of the pre-factor |f ν 2 3 |v 2 /M (see Eqs. (2.28)-(2.30)) which can be obtained by using the relation with the known value of ∆m 2 21 from current global analysis [64]. Equipped with all these, we provide a range of the allowed parameter space of our model in Fig. 1. In the left panel, we first indicate the correlation between two of the parameters χ 1 − χ 2 while the same for γ 1 − γ 2 is shown in the right panel, indicated by the light blue points. The corresponding values of the parameters (light blue points) satisfy the 3σ allowed ranges of the lepton mixing angles, θ 13 , θ 12 , θ 23 . In obtaining these points, we varied parameters within a large range. For example, χ 1,2 are varied from 0 to 2 while γ 1,2 are considered within their full range: 0-360 • . Once we also incorporate the constraints following from the mass-squared differences as well as the one on the sum of the light neutrino masses, the entire allowed parameter space is reduced to a smaller region indicated by the dark blue patch on the left panel (in χ 1 − χ 2 plane) and four cornered patches (red, magenta, brown and purple) on the right panel (in γ 1 − γ 2 plane).
From Fig. 1, we find 0.584 χ 1 1.462 whereas the ratio of the magnitudes of the antisymmetric contribution to the diagonal one (in view of Eq. (2.6)) falls in a range: 0.470 χ 2 0.145. Turning into the right panel, we find that γ 1 and γ 2 both are pushed toward four cornered regions represented by red, magenta, brown and purple patches respectively. Here we find that for 0 It is important to note that so far the analysis presented here is applicable only for normal hierarchy of light neutrino mass. In the present setup, due to the special flavor structure of the model an inverted hierarchy of light neutrino mass spectrum however can not be accommodated. This is an interesting prediction that will undergo tests in several ongoing and near-future experiments.

Implications for light neutrino masses and low energy phase
From the previous part of the analysis, we have an understanding on the allowed regions for the χ 1 , χ 2 , γ 1 and γ 2 which satisfy all the constrains in the form of mass square differences, mixing angles and sum of the light neutrino masses. Hence, we are now in a position to study the implications of this allowed parameter space toward the predictions involving sum of the light neutrino masses, and phases. We already have correlation between the Dirac CP phase δ and the atmospheric mixing angle θ 23 as seen from Eqs.   favors θ 23 to be below maximal mixing, i.e. θ 23 < 45 • . These are the salient features of our proposal. It is pertinent to also shed light on the effective neutrino mass parameter, m ββ , involved in the half life of neutrinoless double beta decay in our set-up, which is given by [63] m ββ = |m 1 c 2 12 c 2 13 + m 2 s 2 12 c 2 13 e iα 21 + m 3 s 2 13 e i(α 31 −2δ) |.

(3.3)
Note that for the normal hierarchy of light neutrino masses, one can write m 2 = (m 2 1 + ∆m 2 21 ), and m 3 = (m 2 1 + ∆m 2 31 ). Recall also that we have already elaborated on our finding for lightest neutrino masses m 1 (see Fig. 2b), and δ (see Fig. 2a) in the last subsections corresponding to the allowed parameter space of {χ 1 , χ 2 , γ 1 , γ 2 } from Fig. 1. Using the same, we could also estimate the respective allowed ranges of Majorana phases α 21 and α 31 via Eq. (2.36) (as α 21 = β 21 and α 31 = β 31 ) and in turn we can evaluate m ββ as function of m 1 (substituting m 2 and m 3 in Eq. (3.3)). With the allowed ranges for χ 1 , χ 2 , γ 1 and γ 2 satisfying all the neutrino data inclusive of the cosmological mass bounds (i.e. corresponding to the dark blue patch of left panel, and four cornered patches of right panel of Fig. 1), we therefore plot m ββ as a function of lightest neutrino masses m 1 for normal hierarchy as presented in Fig. 3 by the red patch. The background light red patch indicates the allowed region in general when mixing angles, mass squared differences along with δ are allowed to vary within their 3σ range. Hence from this m ββ vs m 1 plot (red patch), we notice that for m 1 within the range (0.001-0.027) eV (allowed in our set-up as per Fig. 2b), the effective mass parameter is predicted to be: 0.002 m ββ 0.021 eV. This prediction lies well within the limits on m ββ by combined analysis of GERDA and KamLAND-Zen experiments denoted by the light blue shade. The horizontal brown and blue dashed lines stand for future sensitivity by the LEGEND and nEXO experiments.

Lepton flavor violation
Due to the existence of active-sterile neutrino mixing, the possibility of rare lepton flavor violating processes should arise in our framework. Out of all the processes, contribution to µ → eγ is the most important one as it is significantly constrained. In the weak basis, i.e. where charged and RHN mass matrix is diagonal, the branching ratio of the same process can be written as [68,69]: where α = e 2 /4π is the fine structure constant, M W stands for W ± mass, R = m D M −1 R is the mixing matrix representing active-sterile mixing, M i is the mass of RHN mass eigenstates N i and F(x) = x(1−6x+3x 2 +2x 3 −6x 2 ln x)

2(1−x) 4
, with x = M i /M W . The current upper bound on the branching ratio of the µ → eγ is found to be BR(µ → eγ) 4.2 × 10 −13 (at 90% C.L.) [63]. In our analysis, with the allowed ranges for χ 1 , χ 2 , γ 1 and γ 2 (obtained from Fig. 1) and M i in the TeV scale, the contribution towards the branching ratio for µ → eγ turns out to be insignificant (O(10 −35 )) compared to the experimental limit.

Leptogenesis
The presence of RHNs in the seesaw realization of light neutrino mass provides an opportunity to study leptogenesis from the CP-violating out-of-equilibrium decay of RHNs into lepton and Higgs doublets in the early universe [45,49,70]. The lepton asymmetry created is expected to be converted to a baryon asymmetry via the sphaleron process [71,72]. In the previous part of our analysis, we have found that the phenomenology of the neutrino sector is mainly dictated by four parameters i.e χ 1 , χ 2 , γ 1 , and γ 2 which in turn determine most of the observables in the neutrino sector. However we also notice the presence of the prefactor |f ν 2 3 |v 2 /M associated to the light neutrino mass eigenvalues as given in Eq. (2.28)-(2.30). Using Eq. (3.2), though this prefactor can be evaluated, we can't have specific estimate for the degenerate mass of the RHNs (M ) as f ν 3 remains undetermined. To have a more concrete picture, we provide a plot for |f ν 2 3 |v 2 /M against one of the parameters, χ 1 , in Fig. 4 obtained using the correlation with other parameters fixed by neutrino oscillation and cosmological data. Hence barring the ambiguity in determining f ν 3 apart from a conservative limit |f ν 3 | < O(1), M is seen to be anywhere from a very large value (say 10 14−15 GeV) to a low one (say TeV). Furthermore, the RHNs are exactly degenerate in our framework. Hence unless we break this exact degeneracy, no CP asymmetry can be generated [58] . Below we proceed to discuss leptogenesis mechanism in the present framework keeping in mind that we need to remove the exact degeneracy of RHN masses and study of flavored leptogenesis becomes essential (as M can be below 10 12 GeV).

Generation of mass splitting and CP asymmetry
The CP asymmetry parameter generated as a result of the interference between the tree and one loop level decay amplitudes of RHN N i decaying into a lepton doublet with specific flavor l α and Higgs (H) is defined by : Considering the exact mass degeneracy is lifted by some mechanism (will be discussed soon), the general expression for such asymmetry can be written as [73,74] : where Y ν (≡ V † Y ν in our case, see Eq. (2.11)) is the neutrino Yukawa matrix in charge lepton diagonal basis, H and the loop factor f (x ij ) are given by where M i are the masses of the RHNs after the degeneracy is removed. This is applicable for both hierarchical as well as quasi-degenerate mass spectrum of RHNs [73]. For the hierarchical RHNs, one neglects H 2 jj 64π 2 compared to (1 − x ij ) 2 while the entire expression of Eq. (4.2) can be used for quasi-degenerate case inclusive of resonance situation [74,75]. Below we discuss the mass splittings induced by the running of the heavy RHNs.

Lifting the mass degeneracy
The exact mass degeneracy of heavy Majorana neutrinos is the result of the flavor symmetry imposed in our construction. To remove this degeneracy, here we adopt the renormalization group effects into consideration [76,77]. Considering the discrete A 4 × Z 3 × Z 2 symmetry breaking scale close to the GUT scale ∼ Λ (the cut-off scale introduced in Eq. (2.1)), we determine the running of the RHN mass matrix M R and Dirac neutrino Yukawa matrix Y ν from GUT scale to seesaw scale M (assuming M < Λ). Using renormalisation group equations, the evolution of the RHN mass matrix M (= diag(M 1 , M 2 , M 3 )) and Dirac neutrino Yukawa matrix Y ν (in charged lepton Y diagonal basis) at one-loop can be written as [76][77][78] with where Y u,d are the up-quark and down-quark Yukawa matrices respectively, g 1,2 are the gauge couplings and I 3 is the identity matrix of order 3 × 3. Here the matrix R is anti-hermitian defined by [77] is the degeneracy parameter for the RHN masses and t = 1 16π 2 ln Λ M . Now as the RHNs are exactly degenerate at scale Λ, the right hand side (first term) of Eq. (4.8) becomes singular unless we impose Re(H ij ) = 0. Note that, in our construction, H 12 and H 23 are already zero due to the flavor symmetry imposed. Hence the above condition should be exercised only to realize Re(H 13 ) = 0 in our case which can be materialized if we choose to useỸ ν , obtained by performing an orthogonal rotation (by a matrix O say) on Dirac Yukawa matrix Y ν as, having the rotation angle Θ determined by the relation . (4.10) In obtaining the rightmost expression above, we employ Eqs. (2.6), (2.22), (2.23) in Eq. (4.3). This flexibility in usingỸ ν prevails due the following reason. Note that, if we rotate the Y ν in this manner, the neutrino Yukawa Lagrangian gets modified to: We can now redefine N R by:Ñ R = O T N R , i.e. if we rotate RHN fields by O T , RHN mass term will not change as N C R M R N R =Ñ C R M RÑR due to the orthogonal property of O matrix.
The Eqs. (4.5) and (4.6) can now be rewritten in terms ofH = O T HO andỸ ν by using the above relations. The form ofH can be obtained bỹ where ∆ ≡ tan Θ Re (H 13 ). As seen from the Eq. (4.5) (with right hand side written in terms ofH now), we find that a mass splitting generated at a scale (M ) as thanks to the effect of running. Using Eq. (4.6), we also get a off-diagonal contribution (H R ij , i = j) toH [77],H ν 3j t; (i = j) (4.14) Figure 5: Variation of mass splitting δ M ij with respect to scale M for the benchmark points BP1, BP2, BP3 and BP4 respectively. whileH M ii =H ii . As mentioned earlier, the seesaw scale M remains undetermined even after applying neutrino mass and mixing constraints, we have shown in Fig. 5 how such splitting δ M 12 varies with the degenerate RHN mass M due to running corresponding to benchmark points: BP1, BP2, BP3 and BP4 allowed by the neutrino data. We find that below M 10 12 GeV, δ M ij become smaller than O(10 −4 ) implying that the masses of the three RHNs fall in the quasi-degenerate category [73]. Such a a small splitting, although crucial for generation of CP asymmetry, won't alter our findings of the neutrino section. Note that the estimated splitting does not correspond to the requirement of resonant leptogenesis. We are now in a position to evaluate the CP asymmetry generated at scale M , as discussed below.

Estimating CP asymmetry
Starting with exact degeneracy of RHN masses, we have shown that the running of involved parameters from a typical high scale to the scale of the heavy neutrino masses leads to a quasi-degenerate spectrum of RHNs. Hence we can now estimate the CP asymmetry created at a scale M by using Eq. (4.2) while replacing H byH M and δ ij by δ M ij in view of our discussion above. Furthermore, it can be shown that maximum contribution to CP asymmetry comes from self energy diagram [48,79,80]. Therefore, the asymmetry expression of Eq. (4.2) gets modified to Now, using Eqs. (4.12) to (4.14) and employing them in Eq. (4.15), we estimate for the cp asymmetry parameter for the heavy RHNs decaying into various flavors which will be useful to evaluate the final lepton asymmetry taking the flavor effects into account. Since all the entities of Eq. (4.15) are function of set of parameters {χ 1 , χ 2 , γ 1 , γ 2 } and M , we can make use of the allowed parameter space from neutrino phenomenology (refer to Fig. 1) and finally calculate the CP asymmetries produced from all three RHN decays (i = 1, 2, 3) to different flavors of lepton doublets and Higgs. For representation purpose, in Fig. 6, we depict the variation of individual flavor components of CP asymmetry with respect to χ 1 at three different RHN mass scales: M =  It is found that maximum asymmetry falls in the ballpark of | i=1,3 | max ∼ 6 × 10 −7 whereas (| 2 |) max remains subdominant. At T = 10 11 GeV (and above 10 8 GeV), tau Yukawa comes to equilibrium, so effectively the scenario with M = 10 11 GeV becomes a two flavor scenario (τ and another orthogonal direction, say a) and the corresponding CP asymmetries are marked by: i=2,3 | max ∼ 2 × 10 −6 (middle panel of Fig. 6) and | τ,a 1 | max becomes relatively small. We also estimate CP asymmetry at M = 10 8 GeV(bottom panel of Fig. 6). At this temperature (or scale), all Yukawa couplings are in equilibrium and hence contributions to CP asymmetries from all the three flavors, { e i , µ i , τ i }, become important. We find | τ i=2,3 | max ∼ 3 × 10 −6 and | τ 1 | max < | τ i=2,3 | max . An analogous pattern is observed for µ i . CP asymmetry along electron flavor is shown in the third plot of the bottom panel of Fig.  6 and is found to be |( e i=2,3 ) max | ∼ 1.5 × 10 −6 , |( e 1 ) max | ∼ 5 × 10 −7 . With these various flavor dependent CP asymmetries, we can now proceed for evaluation of baryon asymmetry by solving the Boltzmann equations as illustrated below.

Solution of Boltzmann equation
It is worth mentioning that while estimating the final lepton asymmetry, one needs to take care of decays and inverse decays of heavy RHNs as well as various scattering processes. As stated earlier, we consider the contributions of all three RHNs having M i 10 12 GeV. Hence flavor effects have to be considered [81] as with the mass equivalent temperature regime T ∼ 10 12 GeV, decay rate of τ (Γ τ ∼ 5 × 10 −3 y 2 τ T ) [82,83] becomes comparable to the Hubble expansion rate. Below this temperature, the relation becomes Γ τ > H indicative of the start of equilibrium era for τ Yukawa interactions and τ lepton doublet becomes distinguishable. In a similar way, for the temperature regime 10 8 GeV T 10 11 GeV, muon Yukawa interaction comes to equilibrium (and both µ and τ flavors of lepton doublets are distinguishable henceforth) and finally below T 10 8 GeV, e Yukawa interaction are in equilibrium.
In our analysis, therefore, we include these flavor effects into consideration while constructing the Boltzmann equations. We work in a most general setup for leptogenesis, where all three RHNs are contributing to the asymmetry due to their quasi degenerate spectrum of masses. As standard, the produced lepton doublets from the RHN decay needs to be appropriately projected to flavor states in the three above mentioned temperature regimes differently where the related respective lepton asymmetries are characterized by the C matrices (C H stands for that of Higgs) [84,85]. For example, when only the τ Yukawa interaction is in equilibrium (10 11 GeV T 10 13 GeV), effectively the scenario becomes a two flavor case (as the flavor space is spanned by τ and another orthogonal direction) and so C l is a matrix of order 2 × 2. For a further smaller temperature, the situation comprises of three effective flavors and so C is of 3 × 3. Below we write down the relevant Boltzmann equations to study the time evolution of the lepton-number asymmetries (for a system of three RHNs) as [84][85][86] sHz where z = M i /T and α = e, µ, τ . In the above, Y ∆α(N i ) = n ∆α(N i ) /s denotes the density of ∆ α = B 3 − L α (relevant heavy neutrino) with respect to the entropy s, Y eq 's are the  Fig. 1).
respective number densities while in thermal equilibrium. Here, total decay rate density of N i is given by where Γ i is the total decay rate of N i at tree level and written as 19) and γ N i s , γ N i t (both are Higgs mediated scattering process with change in lepton number ∆L = 1), γ (1) N i N j , γ (2) N i N j (both are neutrino pair annihilation process) are the reaction rate densities for the scattering processes: [47,87]. Here in Eq. (4.18), K 1 (z) and K 2 (z) are the modified Bessel functions.
With all the ingredients at hand, we first substitute the evaluated CP asymmetry (from Eq. (4.15)) in Eq. (4.17) and proceed for solving the coupled Boltzmann equations  Fig. 1). Here the horizontal patch (light greenish-blue) indicates the observed value of baryon asymmetry [67].
in order to find out the final lepton asymmetry as well as final baryon asymmetry. In doing so, we divide the temperature range into three zones so as to take care of the flavor effects as discussed before while taking into account the ∆L = 1 processes (and ignoring ∆L = 2 processes). We have considered different benchmark values for RHN degenerate mass M (splittings are automatically taken cared by running in terms of other parameters): M = 10 9 , 10 6 , 10 5 GeV so that the effects of flavor can be visible. These benchmark values of M are so chosen that they can produce requisite amount of baryon asymmetry corresponding to a specific choice of parameters: {χ 1 , χ 2 , γ 1 , γ 2 }.
In Fig. 7, we present our findings in terms of estimate of the evolution of the B − L asymmetry (denoted by red dotted line) as well as B asymmetry (denoted by Magenta solid line) for specific choices of the parameters {χ 1 , χ 2 , γ 1 , γ 2 } which correctly produce neutrino data as discussed in Section 3. Fig. 7a, 7b,7c,7d represent the benchmark points BP1, BP2, BP3, and BP4 respectively from the allowed cornered patches of γ 1 and γ 2 plot of Fig. 1. Asymmetries of individual flavors are also drawn in these figures.
While solving the Boltzmann equations, we have assumed that initially the abundance of all the RHNs was very less and they were out of equilibrium. Then due to annihilation of bath particles it gets produced and comes to equilibrium. Around M T ∼ 1, the production rate and decay rate of the RHN become almost equal and afterward the decay rate dominates over the production rate and hence it's abundance starts to fall. The correct baryon asymmetry can be produced with M 10 6 GeV for BP1, M 10 5 GeV for BP2 and BP4, M 10 9 GeV for BP3 region respectively. For these individual sets of parameters, we have checked the variation of final baryon asymmetry, Y B , with respect to mass of M as shown in Fig. 8. From this Fig. 8, we also see that final Y B is increasing with the decreasing of M . There seems to be two discontinuities for each such plot. For example, with blue-dotted line, these are observed at or around M = 10 11 GeV and at M = 10 6 GeV. These are indicative of the eras where different flavors of lepton doublets enter in (or exit from) equilibrium and the Boltzmann equations get modified.

Conclusion
In this analysis, we present an economical, predictive flavor symmetric setup based on A 4 × Z 3 × Z 2 discrete group to explain neutrino masses, mixing via type-I seesaw mechanism while matter-antimatter asymmetry is also addressed via leptogenesis. In the original AF model, TBM mixing scheme was realized introducing three flavon fields. With similar fields content, here we show that correct neutrino mixing and mass-squared differences are originated from non-trivial structure of the neutrino Dirac Yukawa coupling and diagonal RHN mass matrix, thanks to the contribution from the charged lepton sector too. In particular, the antisymmetric contribution in the Dirac Yukawa coupling plays an instrumental role in generating the non-zero θ 13 . Using the current experimental observation on neutrino oscillation and other cosmological limits, we find the allowed parameter space for parameters χ 1 , χ 2 , γ 1 , γ 2 which in turn not only restricts some of the observables associated to neutrinos like Dirac CP phase, neutrino-less double beta decay, lepton flavor violating decays, estimation of Majorana phases etc. but also are helpful in determining the matter-antimatter asymmetry of the universe. More specifically, we find that this model is highly predictive in nature. Only normal mass hierarchies are found to be allowed in the current setup. Interestingly the atmospheric mixing angle θ 23 lies in the lower octant while the leptonic Dirac CP phase falls within the range 33 • (213 • ) δ 80 • (260 • ) and 100 • (280 • ) δ 147 • (327 • ). Apart from these predictions for absolute neutrino mass and effective mass parameter appearing the neutrino-less double beta decay have also been made. The model also predicts an interesting correlation between the atmospheric mixing angle θ 23 and the Dirac CP phase which is a feature of the specific flavor symmetry considered here. At high scale, owing to the symmetry of the model, the heavy RHNs are found to be exactly degenerate apparently forbidding the generation of baryon asymmetry via leptogenesis. However, this is accomplished here elegantly by considering the renormalization group effects into the picture. A tiny mass splitting produced as a result of running from a high scale (GUT scale) to the scale of RHN mass opens the room for leptogenesis. We have incorporated the flavor effects in leptogenesis as our working regime of RHN mass falls near or below 10 9 GeV. Finally, we figure out that the parameter space allowed by the neutrino data in fact is good enough to generate sufficient amount of baryon asymmetry of the universe with RHN mass as low as 10 5 GeV.
of the work has been completed. AD would like to thank Rishav Roshan and Dibyendu Nanda for fruitful discussions.