# CP Violation in neutrino oscillation and leptogenesis

###### Abstract

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 as well as neutrinoless double 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 is obtained. For the most general case of the framework, we study the effect on 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:

PACS numbers:11.30.e, 11.30.f, 14.60 pFinding 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 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 phase contributes to leptogenesis. We will also investigate how large the CP phase responsible for leptogenesis contributes to low energy CP violation by determining the independent parameters from available experimental results and cosmological observations. Finally, we will discuss the potential implication of CP violation measurable from neutrino oscillations on leptogenesis.

Let us begin our study by considering the leptonic sector of the (3,2) seesaw model. In a basis where both heavy Majorana and charged lepton mass matrices are real diagonal, the Lagrangian is given by:

(1) |

where and the Dirac mass term is matrix. Here, we remark that the Dirac mass matrix contains unremovable CP phases if we take singlet heavy Majorana neutrinos in this basis. Thus, one can easily see that at least two singlet heavy Majorana neutrinos are required to break CP symmetry in the seesaw model with three lepton doublets. The matrix can be generally parameterized as:

(2) |

with,

(3) |

where and denote the rotations of plane, , and and are real and positive. Without loss of generality, we can choose . The allowed range for the angles and the phases is . There are three CP violating phases, which appears in , and in . In a different basis with complex , can be interpreted as a heavy Majorana phase. The lepton number asymmetry for the lightest heavy Majorana neutrino () decays into [7] is given by;

(4) |

where with GeV, and

(5) |

We see that CP violation concerned with leptogenesis can be possible only if the mixing angle and CP violating phases for the heavy Majorana neutrinos are non-zero. For our purpose, let us now study how the phase contributes to CP violation in the neutrino oscillations which is usually described in terms of the MNS neutrino mixing matrix [6]. The effective mass matrix for light neutrinos is given by . Then, the MNS mixing matrix is decomposed into two mixing matrices as follows; , and is diagonalized by the MNS mixing matrix as

(6) |

where is a unitary matrix diagonalizing the matrix and parameterized by,

(7) |

Then, . Note that the (3,2) seesaw model predicts one massless neutrino. In addition, , and are determined as:

(8) | |||

We remark that the mixing angle and CP violating phase have been transferred to and . As one can see from the above formulae, leptogenesis occurs only if the mixing angle and CP violating phase are non-zero, which in turn implies non-vanishing , , via Eq.(8). As we will see below, the CP phase contributes to CP violation in neutrino oscillations, so that it is anticipated that there is correlation between CP violation generated from neutrino mixings and leptogenesis. To see this concretely, let us compute Jarlskog determinant [8] which is proportional to the CP asymmetry in neutrino oscillation,

(9) | |||||

From the expression of , it is obvious that all three CP violating phases and contribute to CP violation in the neutrino oscillations, and that the CP phase always hangs around . Since only is closely related to leptogenesis, in order to investigate the interplay between CP violation for leptogenesis and low energy leptonic CP violation, we should determine the contributions of the phases ) and separately as well as to fix the parameters .

Before discussing the correlation between both CP violations, let us study how we can get some information on the mixing angles and CP phases from the available experimental and cosmological results. The mixing angles and CP phases can be classified into two categories, one contains and which are related to phenomena at high energy and the other contains parameters in . First of all, we show how we can estimate the allowed values of CP violating phase and mixing angle . The information on and may come from the constraints on light neutrino mass spectra as well as cosmological condition for leptogenesis. To see this, we first present the parameters and lepton number asymmetry in terms of some physical quantities which will be taken as inputs in numerical calculation. Here, we choose the heavy Majorana neutrino masses (), their decay widths (), and light neutrino masses as the physical input parameters. As will be clear later, it is convenient to define two parameters ;

(10) |

Then, by considering the light neutrino mass eigenvalue equation, , the lepton number asymmetry and the phase can be written in terms of and ,

(11) |

(12) |

where and . There are two solutions of Eq.(12) leading to negative ; and for , which in turn gives positive baryon number via sphaleron process. Next, let us present the parameters, , and in terms of the above 6 physical quantities. From the eigenvalue equations for we can express , , and as follows;

(13) | |||

(14) |

where , and . We also determine and with a given set of parameters by using the same procedure given in Eqs.(7,8). We take . In order to determine the values of and , it is necessary to determine those of . Let us now show how the variables can be constrained. From the neutrino mass eigenvalue equation, it follows that

(15) |

From the experimental results for the neutrino oscillation, let us take eV and eV (LMA)[9]. From Eq.(15), the lower bound on is eV. By solving the Boltzmann equation [11], we can obtain a value of , i.e., lepton number density () normalized by entropy density (). When solving the Boltzmann equation, we need the value of . For fixed , one can get the maximum value of which gives the maximum via Boltzmann equation. In Fig.1, we plot the maximum lepton number density predicted from Eq.(11) as a function of for several fixed .We set the initial conditions for Boltzmann equation at GeV and we take the distribution of the heavy majorana particle in thermal equilibrium and at the temperature.

The allowed values of consistent with baryogenesis are presented by shaded band in Fig.1. Thus, we can obtain the allowed region of for a fixed . However, there is no allowed value of for a rather lower value of GeV, which in turn leads to the lower bound on . By using the allowed region for as given in the above, we can estimate the allowed region of via Eq.(11) again. Figure 2 shows how we can get the allowed region of . For example, for a given set GeV, and eV, we obtain the allowed range for as eV.

Let us move to the other category of the parameters, in , which are not determined from high energy phenomena, but must be related to the low energy MNS mixing matrix, Thus two of them can be determined from the neutrino oscillation experimental results. For simplicity, we focus on the case with the small mixing angles and , which is consistent with Chooz experiment [10]. In this case, the MNS mixing matrix is given, in the leading order, by

(16) |

where , and . Note that we do not present the subleading contributions in , which are comparable to . Taking which lead to bi-large mixing pattern, in this approximation, , and are given by:

(17) |

In principle, we are able to fix three unknown parameters; and once the left-hand sides of Eq.(17) are measured. It is then possible to quantitatively see whether the low energy CP violation denoted by is dominated by leptogenesis phase or by the CP violating phases and which are not related to leptogenesis. First of all, let us study the interesting case of , which makes the analysis more predictive because a CP violating phase is simultaneously suppressed. This can be understood as the extreme case of . Interestingly enough, this case dictates that the origin of may come from the mixing angle which is related to heavy Majorana neutrino sector. Jarlskog determinant is then simply given by,

(18) |

Only is a completely arbitrary parameter in this case and thus we can easily investigate how can be affected by . In other word, can be estimated through in this case. If is turned out to be much smaller than , the measurement of CP violation in low energy experiment may directly indicate leptogenesis.

In Fig.3, we show the correlation between lepton number asymmetry and in Eq.(18) with . In each contour, and are fixed and is varied. For eV, we obtain and eV, while for eV, is in the range and is in the range eV. On the other hand, the case with , we see from Eq.(17) that mainly depends on , which has nothing to do with leptogenesis.

We have studied how CP violation responsible for baryogenesis manifests itself in MNS matrix and Jarlskog determinant which signals low energy CP violation in neutrino oscillation. We have obtained cosmological constraints on CP violation and mixing which originate from high energy phenomena. Then using the low energy constraints we have showed it is possible to estimate the size and sign of baryon number in the most general case once , neutrinoless double beta decay and CP violation of neutrino oscillation are measured. In a specific case of the framework, a correlation between CP violation in neutrino oscillation and leptogenesis has been studied and the size of has been estimated.

The works of T. M. and M. T. are supported by the Grand-in-Aid for Scientific Research of the MEXT, Japan, No.13640290 and No.12047220 respectively. S.K.K is supported by JSPS Invitation Fellowship (No.L02515) and by BK21 program of the Ministry of Education in Korea. The authors thank H. So, K. Funakubo, T. Kobayashi, M. Plumacher, A. Purwanto, KEK, YITP, and SI2002. A part of this work was completed during the YITP-W-02-05 on ”Flavor mixing, CP violation and Origin of matter”.

## References

- [1] M. Fukugita and T. Yanagida, Phys. Lett. B175,45(1986).
- [2] G. Branco, T. Morozumi, B. Nobre and M.N. Rebelo, Nucl. Phys. B 617, 475 (2001).
- [3] For recent proposal of the search for CP violation in neutrino oscillation, see Y. Itow et.al., hep-ex/0106019.
- [4] A. Joshipura, E. Pascos, and W. Rodejohann, hep-ph/0105175; W. Buchmuller and D. Wyler, Phys. Lett. B 521, 291 (2001); T. Endoh, T. Morozumi and A. Purwanto, hep-ph/0201309; J. Ellis and M. Raidal, hep-ph/0206174; S. Davidson and A. Ibarra, hep-ph/0206304; G. Branco, R. Gonzalez Felipe, F. Joaquim and M. Rebelo, hep-ph/0202030; M.N. Rebelo, hep-ph/0207236; P. Frampton, S. Glashow and T. Yanagida, hep-ph/0208157.
- [5] T.Yanagida, in Proceeding of the Workshop on the Unified Theory and the Baryon Numberin the Universe, edited by O.Sawada and A. Sugamoto (KEK Report No.79-18, Tsukuba, Japan, 1979), p.95; M.Gell-Mann, P.Ramond, and R.Slansky, in Supergravity, edited by P.van Nieuwenhuizen and D.Z.Freedman (North-Holland, Amsterdam,1979),p.315. R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
- [6] Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. 28 870(1962).
- [7] L. Covi, E. Roulet and F. Vissani, Phys. Lett. B 384, 169 (1996).
- [8] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
- [9] SNO Collaboration, Q. R. Ahmadet al., nucl-ex/0106015; Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 86, 5651 (2001);Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998); K2K Collaboration, S. H. Ahn et al., Phys.Lett. B 511, 178 (2001).
- [10] CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420, 397 (1998);Palo Verde Collaboration, F. Boehm et al., hep-ex/0003022.
- [11] M. Plumacher, Z. Phys. C 74, 549 (1997); M. Luty, Phys. Rev. D 45, 455 (1992).