Advances in High Energy Physics

Volume 2014, Article ID 689719, 11 pages

http://dx.doi.org/10.1155/2014/689719

## Cosmic Baryon Asymmetry in Different Neutrino Mass Models with Mixing Angles

Department of Physics, Tezpur University, Tezpur, Assam 784028, India

Received 31 July 2014; Revised 8 November 2014; Accepted 21 November 2014; Published 21 December 2014

Academic Editor: Filipe R. Joaquim

Copyright © 2014 Ng. K. Francis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We investigate the comparative studies of cosmological baryon asymmetry in different neutrino mass models with and without by considering the three-diagonal form of Dirac neutrino mass matrices and the three aspects of leptogenesis, unflavoured, flavoured, and nonthermal. We found that the estimations of any models with are consistent in all the three stages of calculations of leptogenesis and the results are better than the predictions of any models without which are consistent in a piecemeal manner with the observational data in all the three stages of leptogenesis calculations. For the normal hierarchy of Type-IA with charged lepton matrix, model with and without predicts inflaton mass required to produce the observed baryon asymmetry to be GeV and GeV, and the corresponding reheating temperatures are GeV and GeV respectively. These predictions are not in conflict with the gravitino problem which required the reheating temperature to be below GeV. And these values apply to the recent discovery of Higgs boson of mass 125 GeV. One can also have the right order of relic dark matter abundance only if the reheating temperature is bounded to below GeV.

#### 1. Introduction

Recent measurement of a moderately large value of the third mixing angle by reactor neutrino oscillation experiments around the world particularly by Daya Bay [1] and RENO [2] signifies an important breakthrough in establishing the standard three-flavour oscillation picture of neutrinos. Thereby, we will address the issues of the recent indication of nonmaximal 2-3 mixing by MINOS accelerator experiment [3] leading to determining the correct octant of and neutrino mass hierarchy. Furthermore, now, this has opened the door to study leptonic CP violation in a convincing manner, 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, [4], is one of the burning open issues in both particle physics and 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 [6–9]. 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 [9], 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 it; 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 [10]. One of the keys 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 nonzero .

Broadly the leptogenesis can be grouped into two groups: thermal with and without flavour effects and nonthermal leptogenesis. 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 [11–13] and provides several implications for the light neutrino mass spectrum [14, 15]. 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, GeV [16–19], which in turn put constraints on reheating temperature after inflation to be GeV. This will lead to an excessive gravitino production and conflicts with the observed data. In the postinflation era, these gravitinos 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 GeV [8, 20–24]. 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 GeV [25–29]. While thermal leptogenesis in SUSY SO with high seesaw scale easily satisfies the lower bound, the tension with the gravitino constraint is manifest.

According to Fukuyama et al. [30, 31], the nonthermal leptogenesis scenario in the framework of a minimal supersymmetric SO model with Type-I seesaw shows that the predicted inflaton mass needed to produce the observed baryon asymmetry of the universe is found to be GeV for the reheating temperature GeV and weak scale gravitino mass GeV without causing the gravitino problem. It also claims that even if these values are relaxed by one order of magnitude ( TeV, GeV), the result is still valid. In [32, 33] using the Closed-Time-Path approach, they performed a systematic leading order calculation of the relaxation rate of flavour correlations of left-handed standard model leptons; and for flavoured leptogenesis in the early universe they found the reheating temperature to be GeV to GeV. These values apply to the standard model with a Higgs-boson mass of 125 GeV [34]. 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 sterile right-handed neutrinos into a Higgs boson and a lepton. In [35] split neutrinos were introduced where there is one Dirac neutrino and two Majorana neutrinos with a slight departure from tribimaximal mixing (TBM), which explains the reactor angle , 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 experiments. In [36] too a deviation from TBM case was studied with model-independent discussion and the existing link between low- and high-energy parameters that connect to the parameters governing leptogenesis was analysed. However, in [37] exact TBM, , was considered with charged lepton and up-quark type and set to be zero; eventually their results differ from ours. We slightly modify the neutrino models in [37]; consequently the inputs parameters are different for zero but for nonzero our formalism is entirely different than the one done in [37]; besides we consider for detail analysis. Our work in this paper is consistent with the values given in [30–35].

Now, the theoretical framework supporting leptogenesis from low-energy phases has some other realistic testable predictions in view of nonzero . 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 . The current investigation is twofold. The first part deals with zero reactor mixing angle in different neutrino mass models within - symmetry [38–49], while in the second part we construct matrix from fitting of incorporating the nonzero third reactor angle along with the observed data and subsequently predict the baryon asymmetry of the universe (BAU). We must also mention that there are several works analysing the link between leptogenesis and low-energy data in more general scenarios. However, we have not come across in the literature where all the three categories of leptogenesis, that is, the thermal leptogenesis with or without flavour effects and nonthermal leptogenesis, are studied in a single paper. Take, for instance, some of the major players working on leptogenesis. Professor Wilfried Buchmuller works are mostly confined to standard unflavoured thermal leptogenesis by solving Boltzmann’s equation whereas Professor Steven Blanchet and Professor P. Di. Bari generally worked on flavoured effects in leptogenesis and lesser people work on nonthermal leptogenesis (cf. [30, 31]). But we attempt to study all the three aspects of leptogenesis in this paper, which makes our work apparently different from others on this account.

The detailed plan of the paper is as follows. In Section 2, methodology and classification of neutrino mass models for zero are presented. Section 3 gives an overview of leptogenesis. The numerical and analytic results for neutrino mass models without and with are given in Sections 4 and 5, respectively. We end with conclusions in Section 6.

#### 2. 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 without are given in Table 4. And can be related to the right-handed Majorana mass matrix and the Dirac mass matrix through the inversion seesaw mechanism: where In (2) are two integers depending on the type of Dirac mass matrix we choose. Since the texture of Yukawa matrix for Dirac neutrino is not known, we take the diagonal texture of 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 GUT models.

For computations of leptogenesis, we choose a basis where with real and positive eigenvalues. And the Dirac mass matrix in the prime basis transforms to , where is the complex matrix containing CP-violating Majorana phases and derived from . The values of and are chosen arbitrarily other than and 0. We then set the Wolfenstein parameter as and compute the three choices of in . In this prime basis the Dirac neutrino Yukawa coupling becomes and subsequently this value is used in the expression of CP asymmetry. The new Yukawa coupling matrix also becomes complex, and hence the term appearing in CP asymmetry parameter gives a nonzero contribution.

In the second part of this paper, we construct from matrix with value: where is the Pontecorvo-Maki-Nakagawa-Sakata parameterised matrix taken from the standard particle data group (PDG) [50], and the corresponding mixing angles are

A global analysis [51, 52] current best-fit data is used in the present analysis:

Neutrino oscillation data are insensitive to the low-energy individual neutrino masses. However, it can be measured in tritium beta decay [53] and neutrinoless double beta decay [54] and from the contribution of neutrinos to the energy density of the universe [55]. Very recent data from the Planck experiment have set an upper bound over the sum of all the neutrino mass eigenvalues of eV at C.L. [56]. But, oscillations experiments are capable of measuring the two independent mass-squared differences and only. This two flavours oscillation approach has been quite successful in measuring the solar and atmospheric neutrino parameters. In the future the neutrino experiments must involve probing the full three flavor effects, including the subleading ones proportional to . The is positive as is required to be positive by the observed energy dependence of the electron neutrino survival probability in solar neutrinos but is allowed to be either positive or negative by the present data. Hence, two patterns of neutrino masses are possible: called normal hierarchy (NH) where is positive and called inverted hierarchy (IH) where is negative. A third possibility, where the three masses are nearly quasi-degenerate with very tiny differences, , between them, also exists with two subcases of being positive or negative.

Leptonic CP violation (LCPV) can be established if CP-violating phase is shown to differ from 0 to . A detailed review on LCPV can be found in [57]. It was not possible to observe a signal for CP violation in the present data so far. Thus, can have any value in the range []. The Majorana phases and are free parameters. In the absence of constraints on the phases and , these have been given full variation between 0 and excluding these two extreme values.

#### 3. Leptogenesis

As pointed out above leptogenesis can be thermal or nonthermal; again thermal leptogenesis can be unflavoured (single flavoured) or flavoured which are all explained in the subsequent pages. In the simplest form of leptogenesis the heavy Majorana neutrinos are produced by thermal processes, which is therefore called the “thermal leptogenesis.” For our estimations of CP asymmetry parameter [6, 58, 59], we list here only the required equations for computations. However, interested reader can find more details in [60]. The low-energy neutrino physics is related to the high-energy leptogenesis physics through the seesaw mechanism. In (1), is the transpose of and is the inverse of . For the third generation Yukawa coupling unification, in SO grand unified theory, one obtains the heavy and light neutrino masses as GeV and eV respectively. Remarkably, the light neutrino mass is compatible with eV, as measured in atmospheric neutrino oscillations. This suggests that neutrino physics probes the mass scale of grand unification (GUT), although other interpretations of neutrino masses are possible as well. The heavy Majorana neutrinos have no gauge interactions. Hence, in the early universe, they can easily be out of thermal equilibrium. This makes the lightest () of the heavy right-handed Majorana neutrino an ideal candidate for baryogenesis, satisfying the third condition of Sarkarov, the deviation from thermal equilibrium. Assuming hierarchical heavy neutrino masses , the CP asymmetry generated due to CP-violating out-of-equilibrium decay of is given by where is the antilepton of lepton and is the Higgs doublets chiral supermultiplets. Consider where is the Yukawa coupling of the Dirac neutrino mass matrix in the diagonal basis of and GeV is the vev of the standard model. At high temperatures, between the critical temperature of the electroweak phase transition and a maximal temperature , these processes are believed to be in thermal equilibrium [9]. Although this important phenomenon is accepted by theorists as a correct explanation of baryogenesis via leptogenesis, it is yet to be tested experimentally. Therefore it is very fascinating that the corresponding phenomenon of chirality changing processes in strong interactions might be observed in heavy decay ion collisions at the LHC [61, 62]. The evolution of lepton number () and baryon number () is given by a set of coupled equations [63] by the electroweak sphaleron processes which violates () but conserves (). At temperature above the electroweak phase transition temperature , the baryon asymmetry can be expressed in terms of () number density as [64] where () asymmetry per unit entropy is just the negative of the ratio of lepton density and entropy (), since the baryon number is conserved in the right-handed Majorana neutrino decays. At , any primodial () will be washed out and (10) can be written as [64, 65] For standard model (SM) the number of fermion families , and the number of Higgs doublets ; and (11) reduces to The ratio of baryon to photon is not conserved due to variation of photon density per comoving volume [66] at different epoch of the expanding universe. However, for very slow baryon number nonconserving interactions, the ratio of baryon to entropy in a comoving volume is conserved. Considering the cosmic ray microwave background temperature K, we have . Here is a photon number density. And finally the observed baryon asymmetry of the universe [67, 68] for the case of standard model is calculated from

The efficiency or dilution factor describes the washout of the lepton asymmetry due to various lepton number violating processes, which mainly depends on the effective neutrino mass where is the electroweak vev; GeV. For , the washout factor can be well approximated by [69] We adopt a single expression for valid only for the given range of [69–73]. And the comparison of the effective neutrino mass with the equilibrium neutrino mass gives the information whether the system is weak or strong washout regime. For the weak washout regime we have and GeV whereas for the strong washout regime we have and GeV. However, the strong washout regime appears to be favoured by the present evidence for neutrino masses.

In the flavoured thermal leptogenesis [74–77], we look for enhancement in baryon asymmetry over the single flavour approximation and the equation for CP asymmetry in decay where becomes where and . The efficiency factor is given by . Here too eV and . This leads to the BAU:

For single flavour case, the second term in vanishes when summed over all flavours. Thus this leads to baryon symmetry: where and . The conditions of weak or strong washout regime for flavoured leptogenesis are the same as in the case of single favoured/unflavoured leptogenesis, however, with one difference that is the effective mass due to unflavoured leptogenesis while is the resultant effective mass due to contributions of three leptons (flavoured leptogenesis).

In nonthermal leptogenesis [78–83] the right-handed neutrinos with masses produced through the direct nonthermal decay of the inflaton interact only with leptons and Higgs through Yukawa couplings. The inflaton decay rate is given by [30] where is the mass of inflaton . The reheating temperature () after inflation is [84] and the produced baryon asymmetry of the universe can be calculated by the following relation [85]: where is related to in (23). From (23) the connection between and is expressed as

Two boundary conditions in nonthermal leptogenesis are and . The values of and for all neutrino mass models are also used in the calculation of theoretical bounds: and . Only those models which satisfy these constraints can survive in the nonthermal leptogenesis.

#### 4. Numerical Analysis and Results without

We first begin our numerical analysis for without given in the Appendix. The predicted parameters for , given in Table 1, are consistent with the global best-fit value. For computations of leptogenesis, we employ the well-known inversion seesaw mechanism as explained in Section 2. Finally the estimated BAU for both unflavoured and flavoured leptogenesis for without is tabulated in Table 2. As expected, we found that there is an enhancement in BAU in the case of flavoured leptogenesis compared to unflavoured . We also observe the sensitivity of BAU predictions on the choice of models without and all but the five models are favourable with good predictions (see Table 2). Streaming lining further, by taking the various constraints into consideration, quasi-degenerate Type-1A, 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 nonthermal leptogenesis.