In the leptogenesis scenario, decays of heavy Majorana neutrinos generate lepton family asymmetries, Y e , Y µ and Y τ . These asymmetries are sensitive to CP violating phases in seesaw models. The time evolution of the lepton family asymmetries is derived by solving Boltzmann equations. Considering a minimal seesaw model, we show how each family asymmetry varies with one particular CP violating phase. For instance, we find the case in which the lepton asymmetry is dominated by Y µ or Y τ , depending on the choice of the CP violating phase. We also find the case in which the signs of the lepton family asymmetries Y µ and Y τ are opposite. Their absolute values can be larger than the total lepton asymmetry, and baryon asymmetry may result from the cancellation of the lepton family asymmetries. * ) by guest on April 9, 2015 http://ptp.oxfordjournals.org/ Downloaded from where i = 1, 2, 3 and k = 1, 2, · · · , N. Here, L i are SU (2) lepton doublet fields, N R k are the heavy Majorana right-handed neutrinos, and l R i are the right-handed charged leptons. M N R is the N ×N Majorana mass matrix of the right-handed neutrinos, and it is diagonal, i.e., M N R = diag.(M 1 , M 2 , · · · , M N ). y i l are the Yukawa terms for charged leptons. We can choose the basis in which both M N R and y i l are real and diagonal, without loss of generality. In this basis, flavor violating processes occur through off-diagonal elements of the 3×N Yukawa matrix y ν . In the broken phase, the Higgs field has the vacuum expectation value v = 246 GeV, and a Dirac mass term is generated as m D = v √ 2 y ν . The minimal seesaw model, which is compatible with the present neutrino oscil-by guest on April 9, 2015 http://ptp.oxfordjournals.org/ Downloaded from by guest on April 9, 2015 http://ptp.oxfordjournals.org/ Downloaded from

We study the correlation between CP violation in neutrino oscillations and leptogenesis in the framework with two heavy Majorana neutrinos and three light neutrinos. Among three unremovable CP phases, a heavy Majorana phase contributes to leptogenesis. We show how the heavy Majorana phase contributes to Jarlskog determinant J as well as neutrinoless double β decay by identifying a low energy CP violating phase which signals the CP violating phase for leptogenesis. For some specific cases of the Dirac mass term of neutrinos, a direct relation between lepton number asymmetry and J is obtained. For the most general case of the framework, we study the effect on J coming from the phases which are not related to leptogenesis, and also show how the correlation can be lost in the presence of those phases.PACS numbers: 11.30.e, 11.30.f, 14.60 p Finding any relation between baryogenesis via leptogenesis [1] and low energy CP violation observed in the laboratory is a very interesting issue [2]. The CP violation required for leptogenesis stems from the CP phases in the heavy Majorana sector, whereas CP violation measurable from the neutrino oscillations [3] can be described by the neutrino mixing matrix. One interesting question concerned with the low energy leptonic CP violation is whether it can be affected by the CP violating phases responsible for leptogenesis. Several people [4] have already discussed some potential connections between low energy CP violation and leptogenesis by using some ansatz, but it is still unclear how large the former can affect the latter in general. The major difficulty to quantify such a connection occurs due to lack of the available low energy data to fix parameters of the seesaw model.The purpose of this paper is to examine in a rather general framework how leptogenesis can be related to the low energy CP violation by determining the parameters as many as possible from available low energy experimental results and cosmological observations. In order to make a quantitative analysis of the connection between low energy leptonic CP violation and leptogenesis, we consider * E-mail:endoh@theo.phys.hiroshima-u.ac.jp † E-mail:kaneko@muse.sc.niigata-u.ac.jp ‡ E-mail:kang@theo.phys.hiroshima-u.ac.jp § E-mail:morozumi@theo.phys.sci.hiroshima-u.ac.jp * * E-mail:tanimoto@muse.sc.niigata-u.ac.jp the minimal CP violating seesaw model which has two heavy Majorana neutrinos and three light left-handed neutrinos; (3,2) seesaw model. As will be shown later, to break CP symmetry, the required minimal number of singlet heavy Majorana neutrino is two in the seesaw model with three light lepton doublets. This (3,2) seesaw model is consistent with recent data of neutrino oscillations and contains 8 real parameters and 3 CP violating phases in the neutrino sectors which make this model more constrained and predictive compared with the general (3,3) seesaw model [5] with 18 parameters. We will show that while all three CP violating phases contribute to low energy leptonic CP violation, only a single CP violating...

We study the structure of CP violating phases in the seesaw model. We find that the 3ϫ6 MNS matrix contains six independent phases, three of which are identified as a Dirac phase and two Majorana phases in the light neutrino sector while the remaining three arise from the mixing of the light neutrinos and heavy neutrinos. We show how to determine these phases from physical observables.

We have overlooked a Feyman diagram which contributes to the lepton number asymmetry. In Fig. 1, in the self-energy diagram, there is a loop diagram coming from neutrino and neutral Higgs boson.In order to evaluate the diagram, we have to include the interaction term for neutral Higgs boson in Eq. ͑15͒:where 0 is a neutral complex scalar which forms SU͑2͒ doublet with ϩ , T ϭ( ϩ ,v/ͱ2ϩ 0 ). In the exact calculation, we have to keep all the six neutrinos. However, the mixings among heavy neutrinos are suppressed as v 2 /M 2 , where M denotes the heavy neutrino mass. Therefore, to a good approximation, we can keep only light three neutrinos inside the loop. By evaluating the mixing matrix and using the approximate 3ϫ3 unitarity, i.e., ͚ jϭ1 3 V lЈ j* V l j Ӎ␦ lЈl , the contribution from ( 0 , L ) loop is the same as ( ϩ ,l L Ϫ ) loop in the limit of vanishing masses for Higgs bosons, light neutrinos, and charged leptons. As a result, we need to multiply a factor of 2 to the self-energy contribution. The changes are summarized in Table I below. We also correct some notational errors.

How is CP violation of low energy related to CP violation required from baryon number asymmetry ? We give an example which shows a direct link between CP violation of neutrino oscillation and baryogenesis through leptogenesis.