CP violations in a predictive A4 symmetry model

T. Phong Nguyen, ∗ L. T. Hue, D. T. Si, and T. T. Thuc Department of Physics, Can Tho University, 3/2 Street, Can Tho, Vietnam Institute for Research and Development, Duy Tan University, Da Nang City, Vietnam Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam† Department of Education and Training of Can Tho, 3/2 Street, Can Tho, Vietnam‡ Department of Education and Training of Ca Mau, 70 Phan Dinh Phung, 970000 Ca Mau, Vietnam§ Abstract We study a seesaw model with A4 flavor symmetry and the physics phenomenological consequences. After symmetry breaking, the model leads to the neutrino mixing matrix that satisfies


I. INTRODUCTION
The data of neutrino oscillation experiments definitely affirmed that neutrinos have mass and they are mixing. Based on neutrino experimental data, in 2002, P. F. Harrison et al. [1] proposed the structure of neutrino mixing matrix which named Tri-Bimaximal (TB).
According to this structure, the reactor mixing angle, θ 13 , is zero and the Dirac CP violating phase has no meaning. Subsequently, there have been a lot of efforts to build a simple model that leads to TB mixing pattern of leptons, and an interesting way seems to be the use of some discrete non-Abelian flavor groups added to the gauge group of the Standard Model (SM). There is a series of such models based on the symmetry groups A 4 [2,3], T [4], and S 4 [5]... These models are usually realized at some high energy scale Λ and the groups are spontaneously broken due to a set of scalar multriplets. Based on the most updated data of neutrino oscillation experiments, the values of neutrino mixing angles are given in [6] where the reactor mixing angle is relatively large, θ 13 ∼ 8 0 . As a result, the mentioned models are needed to re-examine for their consistence with recent experimental results. There are lots of efforts to generate the non-zero value of neutrino mixing angle θ 13 as well as leptogenesis in the frame work of A 4 model, see for examples, Ref. [7]. However, according to these works, the only inclusion of higher order corrections would not produce such large value of θ 13 . Then the leading orders of the original A 4 models are required, and there are several attempts built in this direction, see for example in Refs. [8][9][10]. Besides two triplet flavons as usual, tree singlet flavons were used in Ref. [8], and two singlet flavons ξ, ξ transform as 1, 1 of the A 4 in Refs. [9,10] in order to accommodate with the present neutrino data. However, leptogenesis have not studied in Ref. [9], whereas in Ref. [10], for having conventional leptogenesis, the authors considered the contribution of the next-to-leading order corrections to right handed neutrino (RHN) mass matrix in the suppersymmetry framework. In this work, we will study flavored leptogenesis with the help of renormalization group evolution of the Dirac Yukawa coupling matrix.
Besides the explanation of neutrino mixing structure, one has to find a way of generating neutrino tiny mass which is zero in the SM. And the seesaw mechanism [11] seems to be the most interesting and effective solution. The seesaw has another physics consequence which is called leptogenesis for the generation of the observed Baryon Asymmetry of the Universe (BAU) by the CP asymmetric decay of heavy RHNs [12]. If the BAU was generated by leptogenesis, then CP violation in the lepton sector must be existed. For Majorana neutrinos, there are one Dirac and two Majorana CP violating phases. One of the phases (or a combination of them) in principle can be measured by neutrinoless double beta (0ν2β) decay [13] experiments. Besides, the TB mixing structure forbids the low energy CP violation in neutrino oscillation, due to U e3 = 0, and also forbids the high energy CP violation in leptogenesis. Therefore, any observation of the leptonic CP violation, for instance in 0ν2β decay and J CP , can strengthen our believe in leptogenesis by demonstrating that CP is not a symmetry of the leptons.
In this work, we consider an expansion of the SM by the seesaw realization of an A 4 discrete symmetric model and it's phenomenological relating with J CP and leptogenesis.
Apart from two SM scalar doublets taking responsibility for spontaneously breaking of the A 4 and the SM gauge groups, this model contains additional SU (2) L scalar singlets, namely two singlets ξ , ξ transform as 1 , 1 and two triplets of the A 4 . If the RHN mass matrix's components resulting from the contributions of VEVs of two scalar singlets (of both SU (2) L and A 4 ) are exactly the same, then the model generates the TB pattern of lepton mixing matrix and hence leptogenesis does not work. We, therefore, study the case where those components are independent and we find the parameter space of the model that satisfies the low energy data and that the BAU is successful generated through flavored leptogenesis.
This work is organized as follows. In the next section, section II, we present the overview of the A 4 model with seesaw mechanism. We also discuss the low energy phenomena of the lepton sector in this section. Section III is devoted to study the leptogenesis. Our discussions and the summary of our work are given in the last section, section IV.

II. THE A 4 SYMMETRY MODEL WITH SEESAW MECHANISM
The non-Abelian A 4 is a group of even permutations of 4 objects and has 4!/2 = 12 elements. All properties of this group needed for model construction was given in [2]. This paper will work in the A 4 basis introduced by G. Altarelli and F. Feruglio, which is reviewed in the A. In this work, we promote the A 4 proposed in [3] with two Higgs singlets to accompany with seesaw mechanism. The model contains several SU (2) L ⊗ U (1) Y Higgs singlets, where two of them (ξ , ξ ) are A 4 singlets, while the remaining (φ S , φ T ) are triplets.
The SM lepton doublets are assigned to be three components of one A 4 triplet, while three right handed charged leptons e R , µ R , τ R are assumed to transform as three different singlets 1, 1 , 1 , respectively. The standard Higgs doublets h u and h d remain invariant under A 4 .
The particle content for leptons and scalars, their VEVs, and symmetry groups considered in the model are shown in Table I. Two more discrete symmetries Z 3 and Z 4 are included in order to get minimal and necessary Yukawa couplings. The Lagrangian for lepton sector which is invariant under all symmetries given in Table I is: where Λ is the cut-off scale of the model. After spontaneous symmetry breaking, the charged lepton mass matrix comes out diagonally with The neutrino sector gives rise to the following Dirac and Majorana neutrino mass matrices where Hereafter, the the complex parameters are distinguished by the tildes (Z = Ze iφ 1 ,X = Xe iφ 2 and X, Y, Z are real parameters). The active neutrino mass matrix is then obtained by the seesaw formula [11]: where The active neutrino mass matrix is diagonalized by U PMNS matrix as where U PMNS is the neutrino mixing matrix parameterized as U PMNS = U ν K [6]. The precise forms of U ν and K are where c ij = cos θ ij , s ij = sin θ ij (ij = 12, 23, 13); δ and β 1 , β 2 are Dirac and two Majorana CP violating phases, respectively. However, instead of diagonalizing m ν , we diagonalize the Hermitian matrix m † ν m ν to examine the structure of m ν , so that the two Majorana phases become irrelevant and we can easy obtain the mixing angles and phase appeared in U ν in terms of the parameters appeared in m ν [14].
Then the Hermitian matrix m † ν m ν is diagonalized as where Then, the straightforward calculation leads to the expressions for the masses and mixing parameters [15] with It is worth to study the other low energy quantities such as effective neutrino mass in neutrinoless double beta decay (0νββ) | m | and the Jarlskorg invariant J CP with the forms given in [6] as: J CP = 1 8 sin 2θ 12 sin 2θ 23 sin 2θ 13 cos θ 13 sin δ.
As can be seen from Eqs. (11,12,13,14), three neutrino masses, three mixing angles and a CP phase are presented in terms of five independent parameter p, ω, κ, φ 1 , φ 2 . At the present, we have five experimental results, which are taken as inputs in our numerical analysis given at 3σ by [6] for the normal hierarchy (NH) of active neutrino mass spectrum: ). Imposing the current experimental data on neutrino masses and mixing angles for the case NH of active neutrino masses into above relations and scanning all the parameter space p, κ, ω, φ 1 , φ 2 , we investigate how those parameters are constrained and estimate possible prediction for leptogenesis. The allowed parameter spaces (κ, ω) and (φ 1 , φ 2 ) constrained by the experimental data given in Eq. (17) are respectively plotted in the figures 1 and 2. Whereas, the global parameter p in the Dirac neutrino Yukawa coupling matrix Y ν can be roughly estimated by p 2 In the above numerical calculation, we have used the random value from zero to 2π for φ 1 , φ 2 , and tan β = 30 as the inputs.
The RHN mass scale is chosen as M 0 = 5 × 10 13 GeV. Here we note that M 0 and tan β are  The prediction of the effective mass | m | is plotted in figure 5 as a function of the lightest active neutrino mass, m 1 . Numerically, our prediction of | m | is turned out to be 0.002 eV ≤ | m | ≤ 0.023 eV. Notice that, the results from 0ν2β by KamLAND-Zen [16] and EXO-200 [17] indicate a limit on the effective neutrino mass parameter | m | as, | m | ≤ (0.14 − 0.28) eV at 90% CL. and | m | ≤ (0.19 − 0.45)V at 90% CL., respectively. Therefore, our result of | m | is expected to be measured by KamLAND-Zen and other 0ν2β decay experiments in their new phase which is taking data since mid-2017 [18].
The prediction of J CP as a function of Dirac CP violation phase δ is plotted in figure 6.
Once the exact value of δ is confirmed the exact value of J CP can be determined. As can be seen from Eq. (16), within the constraints of θ 13 and θ 12 given in Eq. (17), J CP is strongly depends on sin δ (or on the CP phase δ). Notice that, from Eqs. (5,10,13) we find that δ directly depends on two high energy phases φ 1 , φ 2 leading to the depend on φ 1 , φ 2 of the J CP parameter. To be consistent, we would like to note that the CP asymmetry generated by the decay of RHN (Eq. 20) also directly depends on the phases φ 1 , φ 2 (besides two Majorana phases). This makes the correlation between the BAU η B and J CP .

III. LEPTOGENESIS
Now we consider how leptogenesis can work in our scenario. First of all, we have to diagonalize the RHN mass matrix M R given in Eq. (3) in order to go to the mass basis of the RHNs: where V R = U * PMNS . The parameters in this matrix are determined in the previous section with an exception that the Majorana CP phases will be taken as 0 ≤ β 1,2 ≤ 180 0 . And in the mass basis of the RHNs, the Dirac neutrino Yukawa coupling matrix is modified to be We study the case of flavored leptogenesis, the CP asymmetry in the decay of RHN N i to lepton flavor l α (α = e, µ, τ ) is defined as [19]: where H = Y ν Y † ν , and M i denotes the RHN masses. The loop function f (x) containing vertex and self-energy corrections is given as: As can be seen from Eq. (20), Therefore, the CP asymmetry increases with the increasing of M 0 , whereas it does not depend on tan β for a large range (tan β ≥ 3).
Notice from Eq. (20) that, in the studied model model a nonvanishing CP asymmetry (19). Therefore, to have leptogenesis we need to induce a non vanishing H ij (i = j) at the leptogenesis scale. Indeed, this is possible by the RG (Renormalization Group) effects. The RG equation for the Dirac neutrino Yukawa coupling can be written as [20] where As the structure of M R changes with the evolution of the energy scale, the V R depends on the scale Λ too. The RG evolution of V R (t) can be written as where A is an anti-Hermitian matrix A † = −A due to the unitary of V R . The RG equation for Y ν in the basis of diagonal M R is then obtained as Finally, we obtain the RG equation for the Hermitian matrix H = Y ν Y † ν responsible for the leptogenesis as then, if we keep only the τ -Yukawa coupling contribution as the leading order, we derive the off-diagonal terms of H matrix as The flavored CP asymmetries ε α i then can be obtained. After the CP asymmetry in the decay of N i , ε α i , are calculated, the final value of η B can be calculated by solving the flavor dependent Boltzmann equations (BE). Those BEs describe the out-of-equilibrium processes such as the decay, inverse decay, and scattering involving the RHNs, as well as the nonperturbative sphaleron interaction. Besides the CP asymmetries ε α i , the final value of BAU also depends on the wash-out factors K α i which measure the effects of the inverse decay of Majorana neutrino N i into the lepton flavor α and scalars. The parameter K α i is defined as [21]: where Γ α i is the partial decay width of N i into the lepton flavors and Higgs scalars; H(M i ) is the Hubble parameter at temperature T = M i defined as H(M i ) (4π 3 g * /45) Due to the flavor effects, each CP asymmetry ε α i contributes differently to the final formula for the baryon asymmetry as [21,22], if the RHN mass is about M i ≤ (1 + tan 2 β) × 10 9 GeV where the µ and τ Yukawa couplings are in equilibrium and all the flavors are to be treated separately. And if (1 + tan 2 β) × 10 9 GeV ≤ M i ≤ (1+tan 2 β)×10 12 GeV where only the τ Yukawa coupling is in equilibrium and treated separately while the e and µ flavors are indistinguishable, then baryon asymmetry is obtained as: here In the Eqs. (28,29), the wash-out factors κ α i are defined as We study the case NH of active neutrino masses hence, as can be seen from Eq. (18), the lightest RHN mass is M 3 , therefore the BAU is mainly generated by the decay of the 3 rd generation of RHNs. The prediction of η B as a function of the lightest RHN mass, M 3 is shown in fig. 7. In this figure (and in figs. 8,9), the central value of the experimental data of BAU is η CM B B = 6.1 × 10 −10 [23], and 2 × 10 −10 ≤ η B ≤ 10 −9 is the phenomenologically can be seen in figure 9, for successful leptogenesis, the scale of RHN mass, M 0 , is required about 10 13 GeV. However, as can be seen in the figure 9, the RHN mass scale for successful leptogenesis is also constrained by J CP . Therefore, once the low energy CP violation J CP is precisely determined by future experiments then the value of M 0 for successful leptogenesis is well established. And vice versa, J CP is constrained by the current data of η B , for some fixed value of M 0 , we can pin down the value of J CP and hence the value of δ.

IV. CONCLUSION
We have studied the seesaw version of a A 4 flavor symmetry model with two Higgs singlets beside other scalars as usual A 4 models. The neutrino mixing angles predicted by the model come out satisfy the current experimental data at 3σ CL. We have also investigated how effective neutrino mass | m | associated with 0ν2β decay can be predicted as a function of the lightest active neutrino mass m 1 , and our prediction for | m | can be measured by the in running 0ν2β decay experiments. Besides, we have calculated the Jarlskog invariant parameter J CP as a function of Dirac CP violation phase δ. In the near future, if the value of δ is precisely determined then we can point out the exact values of J CP and δ. The flavored leptogenesis is investigated in detail in this work. We find that the RHN mass about 10 12 GeV is required to successfully generate BAU. We have found that there is a correlation between low CP violation parameter J CP and high CP violation in the decay of RHN. Our prediction for J CP and therefore for δ for some fixed parameters can be constrained by the current observation of BAU.

and F. Feruglio
The non-Abelian A 4 is a group of even permutations of 4 objects and has 4!/2 = 12 elements. The group is generated by two generators S and T satisfying the relations There are three one-dimensional irreducible representations of the group denoted as It is easy to check that there is no two-dimensional irreducible representation of this group.
The three-dimensional unitary representations of T and S are given by where T has been chosen to be diagonal. The multiplication rules for the singlet and triplet representations correspond to the above basis of two generators T, S are given as below 1 × 1 = 1, 1 × 1 = 1, 3 × 3 = 3 + 3 A + 1 + 1 + 1 .
Note that while 1 remains invariant under the exchange of the second and the third elements of a and b, 1 is symmetric under the exchange of the first and the second elements while 1 is symmetric under the exchange of the first and the third elements.

≡ (ab)
We will only focus only 3 since the 3 A terms are antisymmetric and hence can not be used for neutrino mass matrix. In the triplet 3, we can see that the first element has 2-3 exchange symmetry, the second element has 1-2 exchange symmetry while the third element earns 1-3 interchange symmetry.
Moreover, if c, c , c are singlets of the type 1, 1 , 1 , and a = (a 1 , a 2 , a 3 ) is a triplet, then the products ac, ac , ac are triplets explicitly given by (a 1 c, a 2 c, a 3 c), (a 3 c , a 1 c , a 2 c ), (a 2 c , a 3 c , a 1 c ), respectively.
For the one dimensional reps, it is easy to see these property because (ω 2 ) * = ω. For the 3-reps we can find a transformation U that changes 3 * into 3 or 3 * ∼ 3. This is similar to the case of SU (2) symmetry.