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Baryon Asymmetry in Neutrino Mass Models with and without {\theta}_13

We investigate the comparative studies of cosmological baryon asymmetry in different neutrino mass models with and without {\theta}_13 by considering the three diagonal form of Dirac neutrino mass matrices, down-quark (4,2), up-quark (8,4) and charged lepton (6,2). The predictions of any models with {\theta}_13 are consistent in all the three stages of leptogenesis calculations and the results are better than the predictions of any models without {\theta}_13 which are consistent in a piecemeal manner with the observational data. For the best model, the normal hierarchy Type-IA for charged lepton (6,2) without {\theta}_13, the predicted inflaton mass required to produce the observed baryon asymmetry is found to be 3.6x10 to the power 10 GeV corresponding to reheating temperature TR 4.5x10 to the power 6 GeV, while for the same model with {\theta}_13, the inflaton mass is 2.24x10 to the power 11 GeV, TR 4.865x10 to the power 6 GeV and weak scale gravitino mass m(2 divided by 3) 100 GeV without causing the gravitino problem. These values apply to the recent discovery of Higgs boson of mass 125 GeV. The relic abundance of gravitino is proportional to the reheating temperature of the thermal bath. One can have the right order of relic dark matter abundance only if the reheating temperature is bounded to below 10 to the power 7 GeV.


Introduction
Recent measurement of a moderately large value of the third mixing angle θ 13 by reactor neutrino oscillation experiments around the world particularly by Daya Bay (sin 2 θ 13 = 0.089±0.010(stat)±0.005(syst)) [1], and RENO (sin 2 θ 13 = 0.113±0.013(stat)±0.019(syst)) [2], signifies an important breakthrough in establishing the standard three flavor oscillation picture of neutrinos. Thereby, will address the issues of the recent indication of non-maximal 2-3 mixing by MINOS accelerator experiment [3] leading to determining the correct octant of θ 23 and neutrino mass hierarchy. Furthermore, now, this has opened the door to search CP violation in the lepton sector, which in turn has profound implications for our understanding of the matter-antimatter asymmetry of the Universe. In fact, ascertaining the origin of the cosmological baryon asymmetry,η B = (6.5 +0. 4 −0.5 ) × 10 −10 [4], is one of the burning open issues in both particle physics as well as in cosmology. The asymmetry must have been generated during the evolution of the Universe. However, it is possible to dynamically generate such asymmetry if three conditions, i) the existence of baryon number violating interactions, ii) C and CP violations and iii) the deviation from thermal equilibrium, are satisfied [5]. There are different mechanisms of baryogenesis, but leptogenesis [6] is attractive because of its simplicity and the connection to neutrino physics. Establishing a connection between the low energy neutrino mixing parameters and high energy leptogenesis parameters has received much attention in recent years in Refs. [6,7,8]. In leptogenesis, the first condition is satisfied by the Majorana nature of heavy neutrinos and the sphaleron effect in the standard model (SM) at the high temperature [8]], while the second condition is provided by their CP-violating decay. The deviation from thermal equilibrium is provided by the expansion of the Universe. Needless to say the departures from thermal equilibrium have been very important-without them, the past history of the Universe would be irrelevant, as the present state would be merely that of a system at 2.75 K, very uninteresting indeed [9]! One of the key to understanding the thermal history of the Universe is the estimation of cosmological baryon asymmetry from different neutrino mass models with the inclusion of the latest non-zero θ 13 .
Broadly the leptogenesis can be grouped into two: thermal with and without flavour effects and non-thermal. The simplest scenario, namely the standard thermal leptogenesis, requires nothing but the thermal excitation of heavy Majorana neutrinos which generate tiny neutrino masses via the seesaw mechanism [10] and provides several implications for the light neutrino mass spectrum [11]. And with heavy hierarchical right-handed neutrino spectrum, the CP-asymmetry and the mass of the lightest right-handed Majorana neutrino are correlated. In order to have the correct order of light neutrino mass-squared differences, there is a lower bound on the mass of the right-handed neutrino, M N ≥ 10 9 GeV [12], which in turn put constraints on reheating temperature after inflation to be T R ≥ 10 9 GeV. This will lead to an excessive gravitino production and conflicts with the observed data. In the post-inflation era, these gravitino are produced in a thermal bath due to annihilation or scattering processes of different standard particles. The relic abundance of gravitino is proportional to the reheating temperature of the thermal bath. One can have the right order of relic dark matter abundance only if the reheating temperature is bounded to below 10 7 GeV [13].On the other hand, big-bang nucleosynthesis in SUSY theories also sets a severe constraint on the gravitino mass and the reheating temperature leading to the upper bound T R ≥ 10 7 GeV [14]. While thermal leptogenesis in SUSY SO (10) with high see-saw scale easily satisfies the lower bound, the tension with the gravitino constraint is manifest.
The analysis done in Ref. [15], the non-thermal leptogenesis scenario in the framework of a minimal supersymmetric SO (10) model with Type-I see-saw shows that the predicted inflaton mass needed to produce the observed baryon asymmetry of the universe is found to be M φ∼ ∼ 5 × 10 11 GeV for the reheating temperature T R = 10 6 GeV and weak scale gravitino mass m 3/2 ∼ 100 GeV without causing the gravitino problem. It also claims that even if these values are relaxed by one order of magnitude (m 3/2 ≤ 10T eV, T R = 10 7 GeV ), the result is still valid. In Ref. [16] using the Closed-Time-Path approach, they performed a systematic leading order calculation of the relaxation rate of flavour correlations of lefthanded Standard Model leptons; and for flavoured Leptogenesis in the Early Universe found the reheating temperature to be T R = 10 7 GeV to 10 13 GeV. These values apply to the Standard Model with a Higgs-boson mass of 125 GeV [17]. The recent discovery of a Standard Model (SM) like Higgs boson provides further support for leptogenesis mechanism, where the asymmetry is generated by out-of-equilibrium decays of our conjecture heavy right-handed neutrinos into a Higgs boson and a lepton. In [18] split neutrinos was introduced where there is one Dirac neutrino and two Majorana with a slight departure from tribimaximal mixing (TBM), which explains the reactor angle ∼ θ 13 , , and tied intimately to the lepton asymmetry and can explain inflation, dark matter, neutrino masses and the baryon asymmetry, which can be further constrained by the searches of SUSY particles at the LHC, the right handed sneutrino, essentially the inflaton component as a dark matter candidate, and from the 0νββ experiments. In Ref. [19] too a deviation from TBM case was studied with model-independent and analyse the existing link between low and high-energy parameters that connect to the parameters governing leptogenesis. However, in Ref.
[20] exact TBM, tan 2 θ 12 = 0.50 was considered with charged-lepton and up-quark type and set θ 13 zero, eventually their results differs from us. We slightly modify the neutrino models in [20]; consequently the inputs parameters are different for zero θ 13 but for non-zero θ 13 our formalism is entirely different than the one done in Ref.
[20], besides we consider for tan 2 θ 12 = 0.45 for detail analysis. Our work in this paper is consistent with the values given in Refs. [15,16,17,18]. Now, the theoretical framework supporting leptogenesis from low-energy phases has some other realistic testable predictions in view of non-zero θ 13 . So the present paper is a modest attempt to compare the predictions of leptogenesis from low-energy CP-violating phases in different neutrino mass matrices with and without θ 13 . The current investigation is of two fold. The first part deals with zero reactors mixing angle in different neutrino mass models within µ − τ symmetry [21], while in the second part we construct a m LL matrix from fitting of U P MN S incorporating the non-zero third reactor angle (θ 13 ) along with the observed data and subsequently predict the baryon asymmetry of the Universe (BAU).
The detailed plan of the paper is as follows. In Section 2, methodology and classification of neutrino mass models for zero θ 13 is presented. Section 3, gives an overview of leptogenesis. The numerical and analytic results for neutrino mass models m LL without and with θ 13 are given in Sections 4 and 5 respectively. We end with summary and conclusions in Section 6 .

Methodology and classification of neutrino mass models
We begin with Type-I seesaw mechanism for estimation of BAU. The required left-handed light neutrino mass models m LL without θ 13 are shifted to Appendix A . Since the texture of Yukawa matrix for Dirac neutrino is not known, we take the diagonal texture of m LR to be of charged lepton mass matrix (6,2), up-quark type mass matrix (8,4), or down-quark type mass matrix (4,2), as allowed by SO(10) GUT models.
In the second part of this paper, we construct m LL from U P MN S matrix with θ 13 value.
where U P MN S is the PMNS parameterised matrix taken from the standard Particle Data Group (PDG) [22], and the corresponding mixing angles are: A global analysis [23] current best-fit data is used in the present analysis: sin 2 θ 12 = 0.312, sin 2 θ 23 = 0.42, sin 2 θ 13 = 0.025, Oscillation data are insensitive to the lowest neutrino mass. However, it can be measured in tritium beta decay [24], neutrinoless double beta decay [25] and from the contribution of neutrinos to the energy density of the universe [26]. Very recent data from the Planck experiment have set an upper bound over the sum of all the neutrino mass eigenvalues of 3 i=1 m i ≤ 0.23eV at 95% C.L. [27]. But, oscillations experiments are capable of measuring the two independent mass-squared differences between the three neutrino masses: ∆m 2 21 = m 2 2 − m 2 1 and ∆m 2 31 = m 2 3 − m 2 1 . This two flavour oscillation approach has been quite successful in measuring the solar and atmospheric neutrino parameters. In future the neutrino experiments must involve probing the full three flavor effects, including the subleading ones proportional to α = ∆m 2 21 /|∆m 2 31 |. The ∆m 2 21 is positive as is required to be positive by the observed energy dependence of the electron neutrino survival probability in solar neutrinos but ∆m 2 31 is allowed to be either positive or negative by the present data. Hence, two patterns of neutrino masses are possible: m 1 <m 2 ≪ m 3 called normal hierarchy (NH) where ∆m 2 31 is positive and m 3 ≪ m 1 <m 2 , called inverted hierarchy (IH) where ∆m 2 31 is negative. A third possibility, where the three masses are nearly quasi-degenerate with very tiny differences m 1 ≤ m 2 ∼ m 3 , between them, also exists with two sub-cases of ∆ 2 31 being positive or negative. Leptonic CP violation (LCPV) can be established if CP violating phase δ CP is shown to differ from 0 and 180 0 . It was not possible to observe a signal for CP violation in the data so far. Thus, δ CP can have any value in the range [−180 0 , 180 0 ]. The Majorana phase φ 1 and φ 2 are free parameters. In the absence of constraints on the phases φ 1 and φ 2 , these have been given full variation between 0 to 2π and excluding these two extreme values.

Leptogenesis
For our estimations of lepton asymmetry [28], we list here only the required equations for computations. Interested reader can find more details in Ref. [29]. According to Type-1 Seesaw mechanism [30] the light left-handed Majorana neutrino mass matrix m LL heavy right-handed (RH) Majorana neutrinos M RR , and the Dirac neutrino mass matrix m LR are related in a simple way Where m T LR is the transpose of Dirac neutrino mass matrix m LR and M −1 RR is the inverse of M RR . In unflavoured thermal leptogenesis, the lepton asymmetry generated due to CP-violating out-of-equilibrium decay of the lightest of the heavy right-handed Majorana neutrinos, is given by wherel L is the anti-lepton of lepton l L and φ is the Higgs doublets chiral supermultiplets.
where h = m LR /v is the Yukawa coupling of the Dirac neutrino mass matrix in the diagonal basis of M RR and v = 174 GeV is the vev of the standard model. And finally the observed baryon asymmetry of the Universe [31] is calculated from, The efficiency or dilution factor κ 1 describes the washout of the lepton asymmetry due to various lepton number violating processes, which mainly depends on the effective neutrino massm Where v is the electroweak vev, v = 174 GeV. For 10 −2 eV <m 1 < 10 −3 eV , the washout factor κ 1 can be well approximated by [31] κ 1 (m 1 ) = 0.
We adopt a single expression for κ 1 valid only for the given range ofm 1 [32,33].
In the flavoured thermal leptogenesis [34], we look for enhancement in baryon asymmetry over the single flavour approximation and the equation for lepton asymmetry in N 1 → l α φ decay where α = (e, µ, τ ), becomes The efficiency factor is given by κ = m * m αα . Here This leads to the BAU, For single flavour case, the second term in ǫ αα vanishes when summed over all flavours. Thus this leads to baryon symmetry, where ǫ 1 = α ǫ αα andm = αm αα . In non-thermal leptogenesis [35]the right-handed neutrinos N i (i = 1, 2, 3) with masses (M 1 , M 2 , M 3 ) produced through the direct non-thermal decay of the inflaton φ interact only with leptons and Higgs through Yukawa couplings. In supersymmetric models the superpotential that describes their interactions with leptons and Higgs is [36] where Y ia is the matrix for the Yukawa couplings, H U is the superfield of the Higgs doublet that couples to up-type quarks and L α (α = e, µ, τ ) is the superfield of the lepton doublet. Furthermore, for supersymmetric models the interaction between inflaton and right-handed neutrinos is described by the superpotential [37] where λ i are the Yukawa couplings for this type of interaction and S is a gauge singlet chiral superfield for the inflaton. With such a superpotential the inflaton decay rate Γ φ is given by [37] where M φ is the mass of inflaton φ. The reheating temperature (T R ) after inflation is [38], (17) and the produced baryon asymmetry of the universe can be calculated by the following relation [39], where s = 7.0n γ , is related to Y B = n B /S = 8.7 × 10 −11 in Eq. (18). From Eq.(18) the connection between T R and M φ is expressed as, Two more boundary conditions are: M φ > 2M 1 and T R ≤ 0.01M 1 . M 1 and ǫ for all neutrino mass models are used in the calculation of theoretical bounds: T min Only those models which satisfy these constraints can survive in the non-thermal leptogenesis.

Numerical analysis and results without θ 13
We first begin our numerical analysis for m LL without θ 13 given in Appendix A . The predicted parameters for tan 2 θ 12 = 0.45, given in Table-1 [40,41]]. And the Dirac mass matrix m LR = diag(λ m , λ n , 1)v in the prime basis transform to m LR → m ′ LR = m LR U R Q, where Q is the complex matrix containing CP-violating Majorana phases φ 1 and φ 2 derived from M RR . We then set the Wolfenstein parameter as λ = 0.3, and compute for the three choices of (m, n) in m LR as explained in Section 2 .
In this primebasis the Dirac neutrino Yukawa coupling becomes h = m ′ LR v enters in the expression of CP-asymmetry in Eq. (6). The new Yukawa coupling matrix h also becomes complex, and hence the term Im(h † h) 1j appearing in lepton asymmetry ǫ 1 gives a nonzero contribution. For φ 1 and φ 2 we choose some arbitrary values other than π/2 and 0. Finally the estimated BAU for both unflavoured η 3B leptogenesis for m LL without θ 13 are respectively tabulated in Table-2 . As expected, we found that there is an enhancement in BAU in the case of flavoured leptogenesis η 3B compare to unflavoured η 1B . We also observe the sensitivity of BAU predictions on the choice of models with zero all but the five models are favourable with good predictions [see Table- 2 ]. Streaming lining further, by taking the various constraints into consideration, QD-1A (6, 2) and NH-III (8,4) are competing with each other, which can be tested for discrimination in the next level-the non-thermal leptogenesis.
However, the predictions of thermal leptogenesis Table-2 and non-thermal leptogenesis  Table-3 Table 1: Predicted values of the solar and atmospheric neutrino mass-squared differences and mixing angles for tan 2 θ 12 = 0.45 there is a problem with neutrino mass models without θ 13 . In the next section, we study neutrino mass models with non-zero θ 13 and check the consistency of above predictions.

Numerical analysis and results with θ 13
In this section, we investigate the effects of inclusion of non-zero θ 13 [1] [2] on the cosmological baryon asymmetry in neutrino mass models. Unlike in Section 4 analysis, we don't use the particular form of neutrino mass matrices, but we have constructed the lightest neutrino mass matrix m LL using Eq. , and calculated out for normal and inverted hierarchy patterns. The mass eigenvalues m i (i=1,2,3) can also be taken from Ref. [29]. The positive and negative value of m 2 corresponds to Type-IA and Type-IB respectively. Once the matrix m LL is determined the procedure for subsequent calculations are same as in Section 4 .
Here, we give the result of only the best model due to inclusion of reactor mixing angle θ 13 in prediction of baryon asymmetry, reheating temperature and Inflaton mass (M φ ). Undoubtedly, for tan 2 θ 12 = 0.45, the best model is NH-IA (6,2) with: baryon asymmetry in unflavoured thermal leptogenesis B uf = 3.313 × 10 −12 ; single flavoured approximation B 1f = 8.844 × 10 −12 and full flavoured B 3f = 2.093 × 10 −11 . If we examine these values, we found that, expectedly, there is an enhancement is baryon asymmetry due to flavour effects. Similarly in non-thermal leptogenesis, we found that NH-IA is the best model and the predicted results are: Type (m,n)   (6,2) or up quark mass matrix (8,4), or down quark mass matrix (4,2) as explained in the text. IA (6,2) and III (8,4) appears to be the best models.

Summary and conclusions
We now summarise the main points. We have investigated the comparative studies of baryon asymmetry in different neutrino mass models (viz QDN, IH and NH) with and without θ 13 for tan 2 θ 12 =0.45, and found that models with θ 13 are better than models without θ 13 . We found that the predictions of any models with zero θ 13 are erratic or haphazard in spite of the fact that their predictions are consistent in a piecemeal manner with the observational data (see Tables 2 & 3) whereas the predictions of any models with nonzero θ 13 are consistent throughout the calculations. And among them, only the values of NH-IA (6,2) satisfied Davidson-Ibarra upper bound on the lightest RH neutrino CP asymmetry |ǫ 1 | ≤ 3.4 × 10 −7 and M 1 lies within the famous Ibarra-Davidson bound, i.e., M 1 > 4×10 8 GeV [44]. Neutrino mass models either with or without θ 13 , Type-IA for charged lepton matrix (6,2) in normal hierarchy appears to be the best if Y CMB B = 6.1 × 10 −10 is taken as the standard reference value, on the other hand if then charged lepton matrix (5,2) is not ruled out. We observed that unlike neutrino mass models with zero θ 13 , where µ predominates over e and τ contributions, for neutrino mass models with non-zero θ 13 , τ predominates over e and µ contributions. This implies the factor changes for neutrino mass models with and without θ 13 . When flavour dynamics is included the lower bound on the reheated temperature is relaxed by a factor ∼ 3 to 10 as in Ref. [45]. We also observe enhancement effects in flavoured leptogenesis [46] compared to non-flavoured leptogenesis by one order of magnitude as in Ref. [47]. Such predictions may also help in determining the unknown Dirac Phase δ in lepton sector, which we have not studied in the present paper. The overall analysis shows that normal hierarchical model appears to be the most favourable choice in nature. Further enhancement from brane world cosmology [48]