Flavor symmetric origin of texture zeros in minimal inverse seesaw and impacts on leptogenesis

We study the prediction of maximal zeros of the Dirac mass matrix on neutrino phenomenology and baryon asymmetry of the universe (BAU) within the framework of an inverse seesaw ISS $(2,3)$. We try to find the origin of the allowed two zero textures of the Dirac mass matrix from $S_{4}$ flavor symmetry. ISS $(2,3)$ model contains two pairs of quasi-Dirac particles and one sterile state in the keV scale along with three active neutrinos. The decays of the quasi-Dirac pairs create lepton asymmetry that can be converted to baryon asymmetry of the universe by the sphaleron process. Thus, BAU can be explained in this framework through leptogenesis. We study BAU in all the two zero textures of the Dirac mass matrix. The viabilities of the textures within the framework have been verified with the latest cosmology data on BAU.

Baryon asymmetry of the universe (BAU) has been an appealing mystery in particle physics as well as cosmology. 1-3 . There are robust observational evidences of BAU that was produced after the Big Bang, but the origin is still an open question to the physics community. The quantitative measurement of the asymmetry can be expressed as the baryons to photon ratio where n B , nB and n γ are the baryon number density, anti baryon number density and photon density respectively. Another expression in terms of entropy can be obtained as, Current cosmological bounds for the baryon asymmetry are 4 , η B = (6.1 ± 0.18) × 10 −10 , Y B = (8.75 ± 0.23) × 10 −11 (3) Among several beyond standard model frameworks, the inverse seesaw mechanism can provide a way to explain BAU along with neutrino phenomenology at a very low scale compared to the conventional seesaw [5][6][7][8][9] . In inverse seesaw ISS (2,3), lepton asymmetry can be produced through out of equilibrium decay of the quasi-Dirac pairs present in the model [10][11][12][13] . The Yukawa couplings of the heavy neutrinos are the sole source of CP violation in this mechanism. The leptonic asymmetry gets resonantly enhanced as the RH neutrino mass spectrum has a certain amount of degeneracy comparable to their decay width in this model [14][15][16] . Leptogenesis can provide a highly nontrivial link between baryon asymmetry of the universe and neutrino masses and mixings. Again, the free parameters in the model can be reduced by implementing texture zeros to the mass matrices involved in the model [17][18][19][20][21][22][23][24][25][26][27] .
As mentioned in our earlier work 28 , there are 12 possible two zero textures in ISS(2,3) that can be divided into six classes.
In this paper, we try to find the flavor symmetric origin of the mass matrices involved in this model. After constructing the zero textures of M D , we perform a detailed analysis of baryon asymmetry created in all the textures. The viability of the different textures is identified by implementing astrophysical and cosmological bounds on baryon asymmetry of the universe. The paper is structured as follows. In section II, we have shown the S 4 flavor symmetric origin of possible texture zeros in the framework of ISS (2, 3) along with the description of the framework. In section III, we briefly discuss about leptogenesis in ISS (2,3). Section IV contains the results of the numerical analysis and discussion. Finally, in section V, we present our conclusion.

II. S 4 FLAVOR SYMMETRIC ORIGIN OF TEXTURE ZEROS IN DIRAC MASS MATRICES OF ISS(2,3)
As already mentioned, we have chosen a minimal inverse seesaw model for our study where two RH neutrinos instead of three are included along with the gauge singlet fermions 29 . The relevant Lagrangian and the expressions for the light neutrino mass matrix can be found in 29 . In ISS(2,3) model, the light neutrino mass matrix can be obtained from three mass matrices M D , M N , and µ. Therefore, the allowed number of zeros in the light neutrino mass matrix further constrains the structure of these three matrices 23 . In our study, we have considered maximal possible zeros in M N and µ 30 . The charge assignments of the particles in the model under the symmetry group S 4 are shown in table I 31 Field L l RNR H s φ φ φ s φ l χ χ S 4 3 1 3 1 2 1 1 1 1 3 2 3 1 1 1 1 1 3 2 3 1 SU(2) L 2 1 1 2 1 1 1 1 1 1 1 The Yukawa Lagrangian for the charged leptons and also for the neutrinos can be expressed as: As mentioned in our work 31  and µ and the charged lepton mass matrix m l as follows, Here, we define y s v s = p ,y 1r v r v r = f and y 2r v r v r = g. With these structures of M N and µ, the allowed two zero texture zeros of M D have already been shown in 28 . However, in this work, we find the flavor symmetric origin of these two zero textures of M D 32-35 . Class A1: The Dirac Lagrangian in this model can be written as, The cut-off scale Λ is needed to lower the mass dimension to 4. Here, we have also taken extra scalar φ and φ with SM Higgs. Now, with Thus the structure of M D can be written as, We assign, b = yv h1 − yv h1 , c = yv h1 + yv h1 , e = yv h1 + yv h1 ,h = yv h1 − yv h1 This structure of M 1 D along with M 2 D 28 lead to 1 − 0 texture of M ν with zero at (1, 2) position.
Class A2: If after SSB, the VEV acquired by the scalars as, Thus the structure of M D can be written as, These two structures of M 3 D along with M 4 D 28 lead to 1 − 0 texture of M ν with zero at (2, 3) position.
Class A3: If after SSB, the VEV acquired by the scalars as, Thus the structure of M D can be written as, We assign, These two structures of M 5 D along with M 6 Class B1: If after SSB, the VEV acquired by the scalars as, Thus the structure of M D can be written as, We assign, These two structures of M D 67 and M 8 D 28 lead to texture of M ν of the form: where in M 7 Class B2: If after SSB, the VEV acquired by the scalars as, Thus the structure of M D can be written as, We assign, The structure of M ν arising from M 9 D and M 10 D 28 as follows: where in M D1 , Class B3: If after SSB, the VEV acquired by the scalars as, Thus the structure of M D can be written as, We assign, M 11 D and M 12 D 28 give rise to the following texture of M ν : After constructing the texture zero structures, we have studied neutrino phenomenology and BAU in all the categories.

III. LEPTOGENESIS IN ISS(2,3) FRAMEWORK
As mentioned above, the model leads to two pairs of heavy quasi-Dirac particles and a sterile neutrino in keV scale [36][37][38] . Though there are no direct Majorana mass terms for the RH neutrinos, there is possibility of producing lepton asymmetry in this model 11 . The decays of the heavy quasi-Dirac particles create the lepton asymmetry which in turn converts into baryon asymmetry by a process known as sphaleron [39][40][41][42] . The decays of the two pairs in the model can produce lepton asymmetries. However, the lepton number violating scatterings of the lightest pair will wash out the lepton asymmetries produced by the heavier pair 43 . Therefore, in this work, we calculate the lepton asymmetry produced by the lightest quasi-Dirac pair only. The complex Dirac Yukawa couplings are the sole source of CP violation in producing lepton asymmetry 44 . The two quasi-Dirac RH neutrino pairs are N i, j with masses M i, j where M i (i = 1, 2...4) denotes the four heavy neutrino mass eigenvalues. The CP asymmetry produced by the decay of N i into any lepton flavor is given by 45,46 , where, h iα is the effective Yukawa coupling in the diagonal mass basis. f ν i j can be written as 47,48 , Here, the decay width of the heavy-neutrino N i is represented by Γ i . M i are the real and positive eigenvalues of the heavy neutrino mass matrix which are grouped into two quasi-Dirac pairs with the mass splitting of order µ kk (k = 1, 2). The Yukawa couplings in the flavor basis (y iα ) can be expressed in terms of the Yukawa couplings in the mass basis (h iα ) as shown in 11,49 . The calculations of BAU in this model can be found in 49 . Previously, we have studied leptogenesis in a particular S 4 flavor symmetric model. In this paper, we study the effect of different two zero textures of the Dirac mass matrix on leptogenesis.

IV. RESULTS OF NUMERICAL ANALAYSIS AND DISCUSSIONS
The light neutrino mass matrix obtained in all the categories is diagonalised by a unitary PMNS matrix as 50 , The diagonal mass matrix of the light neutrinos can be written as, M diag ν = diag(0, ∆m 2 solar , ∆m 2 sol + ∆m 2 atm ) for normal ordering (NO) and M diag ν = diag( ∆m 2 atm , ∆m 2 solar + ∆m 2 atm , 0) for inverted ordering (IO) 51 . For the numerical analysis, we have fixed f and g at 9 × 10 4 GeV and 13.5 × 10 4 GeV respectively. Then we have numerically evaluated the other model parameters using 44. Subsequently, we have calculated the parameters in the light neutrino mass matrix viz k 1 , k 2 , k 3 and k 4 . With the same set of parameters, numerically evaluated for the model, we have calculated the baryon asymmetry in all the six textures.
In the calculations, only the decay of the lightest quasi-Dirac pair (N 1 , s 1 ) is considered as the asymmetry generated by the heavy pair (N 2 , s 2 ) is washed out very rapidly. Thus the same set of model parameters that give correct neutrino phenomenology can also be used for the calculation of the baryon asymmetry of the universe. We also have calculated the sum of the light neutrino masses and have implemented the current cosmological bounds 52 . Fig 1 to fig 9 demonstrate the results of our numerical analysis for both normal and inverted ordering.  obtained from the model respectively for NO and IO for the texture A1. It has been observed that A1 leads to the BAU that is consistent with the latest cosmology data. In this texture, the parameter space exceeds the current bound on the sum of the light neutrino masses. However, a small space is available that agrees with both the experimental data on mass squared differences and the current cosmological bound.  obtained from the model respectively for the textures B1 and B2. It has been observed that for IO, the textures B1 and B2 do not yield the observed BAU. In the case of NO, texture B1 can lead to the correct BAU, But the amount of BAU obtained in B2 is quite less than the observed values. It is seen that the two textures are consistent with the latest cosmology data on the sum of the light neutrino masses and mass squared differences. But the texture B2 is astonishingly weak in their predictions of BAU.   The predictions of these textures on effective mass characterizing neutrinoless double beta decay have been shown in fig  9. It is observed that all these textures are in good agreement with the current experimental limits on the effective Majorana neutrino mass.

V. CONCLUSION
In this work, we have studied the effect of two zero textures of Dirac mass matrix M D on leptogenesis in the framework of inverse seesaw ISS (2,3). The maximum allowed zeros in the structure of M D with the structures of M N and µ is two. We find the flavor symmetric origin of the six classes with S 4 discrete flavor symmetry. The S 4 flavor symmetry is further augmented with Z 4 × Z 3 . The flavons get VEV after the spontaneous symmetry breaking and the different VEVs can lead to different two zero structures of M D . With the six allowed classes of two zero textures of M D , we have studied the impact of texture zeros on the baryon asymmetry of the universe. It has been found that the textures A1, A3, and B3 in NO give rise to the observed baryon asymmetry. The other three textures are not in agreement with the current cosmological limit on BAU. Though texture A2 fails in explaining BAU in NO, yet it can lead to the observed BAU in IO. One can obtain the required amount of BAU with the textures A1 and B3 in IO also. Again, the texture A3, which is in good agreement in NO, can also yield correct amount of BAU in IO. However, all the textures are in good agreement with the current cosmological limit on the sum of the light neutrino masses. In our previous work 28 , it is observed that the textures A2 and B1 are not suitable from the cosmology data on dark matter phenomenology. In this work also it has been observed that the textures A2 and B1 are not able to yield the observed baryon asymmetry. So, in ISS(2,3) framework the two textures A2 and B1 can be discarded from the latest cosmology data. However, these textures are in good agreement with the neutrino data. As can be seen from 28 , the texture B2 in NO can account for good dark matter phenomenology, but it fails to explain BAU as can be observed from our results. Also B2 in IO is weak in explaining both dark matter and BAU. It can also be concluded that the texture A1 in ISS(2,3) is in good agreement with the latest cosmology data on dark matter as well as BAU for both NO and IO. Similarly, the textures A3 and B3 in NO can give good DM phenomenology along with the correct value of observed BAU.