Implications of residual CP symmetry for leptogenesis in a model with two right-handed neutrinos
We analyze the interplay between leptogenesis and residual symmetry in the framework of two right-handed neutrino model. Working in the flavor basis, we show that all the leptogenesis CP asymmetries are vanishing for the case of two residual CP transformations or a cyclic residual flavor symmetry in the neutrino sector. If a single remnant CP transformation is preserved in the neutrino sector, the lepton mixing matrix is determined up to a real orthogonal matrix multiplied from the right side. The -matrix is found to depend on only one real parameter, it can take three viable forms, and each entry is either real or purely imaginary. The baryon asymmetry is generated entirely by the CP violating phases in the mixing matrix in this scenario. We perform a comprehensive study for the flavor group and CP symmetry which are broken to a single remnant CP transformation in the neutrino sector and an abelian subgroup in the charged lepton sector. The results for lepton flavor mixing and leptogenesis are presented.
A large amount of experiments with solar, atmospheric, reactor and accelerator neutrinos have provided compelling evidence for oscillations of neutrinos caused by nonzero neutrino masses and neutrino mixing [Kajita:2016cak, McDonald:2016ixn, nobel_NPB]. Both three flavor neutrino and antineutrino oscillations can be described by three lepton mixing angles , and , one leptonic Dirac CP violating phase , and two independent mass-squared splittings and , where are the three neutrino masses, and correspond to normal ordering (NO) and inverted ordering (IO) mass spectrum respectively. All these mixing parameters except have been measured with good accuracy [Capozzi:2013csa, Forero:2014bxa, Gonzalez-Garcia:2014bfa, Capozzi:2016rtj, Esteban:2016qun], the experimentally allowed regions at confidence level (taken from Ref. [Capozzi:2013csa]) are:
for NO (IO) neutrino mass spectrum. At present, both T2K [Abe:2015awa, T2K_delta_CP, Abe:2017uxa] and NOA [NovA_delta_CP, Adamson:2017gxd] report a weak evidence for a nearly maximal CP violating phase , and hits of also show up in the global fit of neutrino oscillation data [Capozzi:2013csa, Forero:2014bxa, Gonzalez-Garcia:2014bfa, Capozzi:2016rtj, Esteban:2016qun]. Moreover, several experiments are being planned to look for CP violation in neutrino oscillation, including long-baseline facilities, superbeams, and neutrino factories. The above structure of lepton mixing, so different from the the small mixing in the quark sector, provides a great theoretical challenge. The idea of flavor symmetry has been extensively exploited to provide a realistic description of the lepton masses and mixing angles. The finite discrete non-abelian flavor symmetries have been found to be particularly interesting as they can naturally lead to certain mixing patterns [Lam:2007qc], please see Refs. [Altarelli:2010gt, Ishimori:2010au, King:2013eh] for review.
Although the available data are not yet able to determine the individual neutrino mass , the neutrino masses are known to be of order eV from tritium endpoint, neutrinoless double beta decay and cosmological data. The smallness of neutrino masses can be well explained within the see-saw mechanism [Minkowski:1977sc], in which the Standard Model (SM) is extended by adding new heavy states. The light neutrino masses are generically suppressed by the large masses of the new states. In type I seesaw model [Minkowski:1977sc] the extra states are right-handed (RH) neutrinos which have Majorana masses much larger than the electroweak scale, unlike the standard model fermions which acquire mass proportional to electroweak symmetry breaking. Apart from elegantly explaining the tiny neutrino masses, the seesaw mechanism provides a simple and attractive explanation for the observed baryon asymmetry of the Universe, one of the most longstanding cosmological puzzles. The CP violating decays of heavy RH neutrinos can produce a lepton asymmetry in the early universe, which is then converted into a baryon asymmetry through violating anomalous sphaleron processes at the electroweak scale. This is the so-called leptogenesis mechanism [Fukugita:1986hr].
It is well-known that in the paradigm of the unflavored thermal leptogenesis the CP phases in the neutrino Yukawa couplings in general are not related to the the low energy leptonic CP violating parameters (i.e. Dirac and Majorana phases) in the mixing matrix. However, the low energy CP phases could play a crucial role in the flavored thermal leptogenesis [flavored_leptogenesis] in which the flavors of the charged leptons produced in the heavy RH neutrino decays are relevant. In models with flavor symmetry, the total number of free parameters is greatly reduced, therefore the observed baryon asymmetry could possibly be related to other observable quantities [Jenkins:2008rb_leptogenesis]. In general, the leptogenesis CP asymmetries would vanish if a Klein subgroup of the flavor symmetry group is preserved in the neutrino sector [Chen:2016ptr].
Recent studies show that the extension of discrete flavor symmetry to include CP symmetry is a very predictive framework [Feruglio:2012cw, Ding:2013bpa, Li:2015jxa, Branco:2015hea, Ding:2013nsa, King:2014rwa, Ding:2014ssa, Hagedorn:2014wha, Ding:2014ora, Ding:2015rwa, Yao:2016zev, Lu:2016jit, Chen:2014wxa]. If the given flavor and CP symmetries are broken to an abelian subgroup and in the charged lepton and neutrino sectors respectively, the resulting lepton mixing matrix would be determined in terms of a free parameter whose value can be fixed by the reactor angle . Hence all the lepton mixing angles, Dirac CP violating phase and Majorana CP phases can be predicted [Chen:2014wxa]. Moreover, other phenomena involving CP phases such as neutrinoless double beta decay and leptogenesis are also strongly constrained in this approach [Chen:2016ptr, Yao:2016zev, Hagedorn:2016lva]. In fact, we find that the leptogenesis CP asymmetries are exclusively due to the Dirac and Majorana CP phases in the lepton mixing matrix, and the -matrix depends on only a single real parameter in this scenario [Chen:2016ptr].
In this paper we shall extend upon the work of [Chen:2016ptr] in which the SM is extended to introduce three RH neutrinos. Here we shall study the interplay between residual symmetry and leptogenesis in seesaw model with two RH neutrinos. We find that all the leptogenesis CP asymmetries would be exactly vanishing if two residual CP transformations or a cyclic residual flavor symmetry are preserved by the seesaw Lagrangian. On the other hand, if only one remnant CP transformation is preserved in the neutrino sector, all mixing angles and CP phases are then fixed in terms of three real parameters which can take values between 0 and , and the -matrix would be constrained to depend on only one free parameter. The total CP asymmetry in leptogenesis is predicted to be zero. Hence our discussion will be entirely devoted to the flavored thermal leptogenesis scenario in which the lightest RH neutrino mass is typically in the interval of GeV GeV. Our approach is quite general and it is independent of the explicit form of the residual symmetries and how the vacuum alignment achieving the residual symmetries is dynamically realized. In order to show concrete examples, we apply this general formalism to the flavor group combined with CP symmetry which is broken down to an abelian subgroup in the charged lepton sector and a remnant CP transformation in the neutrino sector. The expressions for lepton mixing matrix as well as mixing parameters in each possible cases are presented. We find that for small values of the flavor group index , the experimental data on lepton mixing angles can be accommodated for certain values of the parameters . The corresponding predictions for the cosmological matter-antimatter asymmetry are discussed.
The rest of the paper is organized as follows. In section 2 we briefly review some generic aspects of leptogenesis in two RH model and present some analytic approximations which will be used later. In section LABEL:sec:LepG_one_CP we study the scenario that one residual CP transformation is preserved in the neutrino sector. The lepton mixing matrix is determined up to an arbitrary real orthogonal matrix multiplied from the right hand side. The -matrix contains only one free parameter, and each element is either real or purely imaginary. The total CP asymmetry is vanishing, consequently the unflavored leptogenesis is not feasible unless subleading corrections are taken into account. The scenario of two remnant CP transformations or a cyclic residual flavor symmetry is discussed in section LABEL:sec:Lep_CP_flavor. All leptogenesis CP asymmetries are found to vanish in both cases. Leptogenesis could become potentially viable only when higher order contributions lift the postulated residual symmetry. In section LABEL:sec:example we apply our general formalism to the case that the single residual CP transformation of the neutrino sector arises from the breaking of the most general CP symmetry compatible with flavor group which is broken down to an abelian subgroup in the charged lepton sector. The predictions for lepton flavor mixing and baryon asymmetry are studied analytically and numerically. Finally, in section LABEL:sec:Conclusions we summarize our main results and draw the conclusions.
2 General set-up of leptogenesis in two right-handed neutrino model
The seesaw mechanism is a popular extension of the Standard Model (SM) to explain the smallness of neutrino masses. In the famous type I seesaw mechanism [Minkowski:1977sc], one generally introduces additional three right-handed neutrinos which are singlets under the SM gauge group. Although the seesaw mechanism describes qualitatively well the observations in neutrino oscillation experiments, it is quite difficult to make quantitative predictions for neutrino mass and mixing without further hypothesis for underlying dynamics. The reason is that the seesaw mechanism involves a large number of undetermined parameters at high energies whereas much less parameters could be measured experimentally.
A intriguing way out of this problem is to simply reduce the number of right-handed neutrinos from three to two [King:1999mb, Frampton:2002qc, Raidal:2002xf]. The two right-handed neutrino (2RHN) model can be regarded as a limiting case of three right-handed neutrinos where one of the RH neutrinos decouples from the seesaw mechanism either because it is very heavy or because its Yukawa couplings are very weak. Since the number of free parameters is greatly reduced, the 2RHN model is more predictive than the standard scenario involving three RH neutrinos. Namely, the lightest left-handed neutrino mass automatically vanishes, while the masses of the other two neutrinos are fixed by and . Hence only two possible mass spectrums can be obtained
Moreover there is only one Majorana CP violating phase corresponding to the phase difference between these two nonzero mass eigenvalues. The Lagrangian responsible for lepton masses in the 2RHN model takes the following form
where and indicate the lepton doublet and singlet fields with flavor respectively, is the RH neutrino with mass (), and is the Higgs doublet with . The Yukawa couplings form an arbitrary complex matrix, here we have worked in the basis in which both the Yukawa couplings for the charged leptons and the Majorana mass matrix for the RH neutrinos are diagonal and real. After electroweak symmetry breaking, the light neutrino mass matrix is given by the famous seesaw formula
where GeV refers to the vacuum expectation value of the Higgs field , and with for NO and for IO, and is the lepton mixing matrix. It is convenient to express the Yukawa coupling in terms of the neutrino mass eigenvalues, mixing angles and CP violation phases as333For other parameterizations of the neutrino Yukawa coupling, see Ref. [Pascoli:2003uh].
where is a complex orthogonal matrix having the following structure [Casas:2001sr, Ibarra:2003xp]