KEK-TH-1117

VEC/PHYSICS/P/1/2006-2007

A texture for neutrino oscillations and leptogenesis

Biswajoy Brahmachari
^{1}^{1}1electronic address:
and
Nobuchika Okada
^{2}^{2}2electronic address:

Department of Physics, Vidyasagar Evening College

39, Sankar Ghosh Lane, Kolkata 700006, India

Theory Division, KEK

Tsukuba, Ibaraki 305-0801, Japan

Abstract

In an economical system with only two heavy right handed neutrinos, we postulate a new texture for Dirac mass matrix . This model implies one massless light neutrino and thus displays only two patterns of mass spectrum for light neutrinos, namely hierarchical or inverse-hierarchical. Both the cases can correctly reproduce all the current neutrino oscillation data with a unique prediction and for the hierarchical and the inverse-hierarchical cases, respectively, which can be tested in next generation neutrino-less double beta decay experiments. Introducing a single physical CP phase in , we examine baryon asymmetry through leptogenesis. Interestingly, through the CP phase there are correlations between the amount of baryon asymmetry and neutrino oscillation parameters. We find that for a fixed CP phase, the hierarchical case also succeeds in generating the observed baryon asymmetry in our universe, plus a non-vanishing which is accessible in future baseline neutrino oscillation experiments.

## 1 Introduction

The origin of the observed baryon asymmetry in our universe, ratio of number of baryons to photons[1]

(1) |

is one of the major problems in cosmology. This number has been deduced from two independent observations. (1) From the existing abundance of light elements formed after big bang[2]. (2) Precision measurements of cosmic microwave background[1].

Leptogenesis[3, 4] may explain this observed asymmetry between matter and antimatter content of the universe. In explaining this asymmetry one first creates a tiny lepton asymmetry in the early universe. This lepton asymmetry is recycled into observed baryon asymmetry above the electroweak scale via sphaleron interactions[5]. This is possible since sphaleron interactions remain in thermal equilibrium above the electroweak scale, they violate , and since they conserve .

It is widely believed that the lepton asymmetry is formed by out-of-equilibrium, lepton number violating, CP violating decay of heavy right handed neutrinos. Existence of heavy right handed neutrinos also give a natural framework for explaining smallness of neutrino mass via see-saw[6] mechanism. If there are no symmetry structures in the theory to make the right handed neutrinos stable, they must decay. They have non-vanishing Yukawa couplings with Higgs scalars and left handed doublets, complex in general, for them to do so. Therefore we study lepton asymmetry generated by the CP violating decays of heavy right handed neutrinos (with Majorana mass) at the early stage of our universe. Since leptogenesis involves no new interactions apart from those required for see-saw mechanism to succeed, we may expect that the Physics of neutrino oscillations would clarify some deep mystery of cosmology such as the observed asymmetry between matter and antimatter with which it is linked.

We know from pioneering works of Sakharov[7] that CP violation is an essential ingredient for theories of matter-antimatter asymmetry. If there is just one right handed neutrino, the Dirac mass matrix is dimensional. Lepton fields can absorb all complex phases and there is no source of CP violation. Therefore one fails to have leptogenesis with just one right handed neutrino. If there are two right handed neutrinos , the Dirac mass matrix is dimensional. Let us choose a basis where the charged lepton mass matrix as well as the heavy right handed neutrino mass matrix are diagonal. In this case we cannot absorb all six complex phases in the Dirac matrix plus two complex phases in the Majorana matrix by redefining five lepton fields . After re-phasing, we have three physical CP phases in the Dirac mass matrix. Therefore at the minimum, two right handed neutrinos[8] are enough to bring in a CP violating decay and successful leptogenesis.

Neutrino oscillations show that neutrinos have non-zero mass and that there are off-diagonal entries in the mass matrix written in flavor basis. Solar and atmospheric neutrino oscillation experiments have explored neutrino masses and mixing patterns. Current best fit values are[9, 10, 11],

(2) |

Any model of leptogenesis is required to reproduce these masses and mixing angles. It is indeed interesting to see that, via see-saw mechanism, existing neutrino data can give desired mass spectrum of heavy right handed neutrinos plus right magnitudes of primordial lepton asymmetry. There are many studies of this kind where there are three heavy right handed neutrinos and generated lepton asymmetry depends on the form of the Yukawa texture[16]. In this paper, we examine the system with only two right handed neutrinos. As discussed above, this system is the minimum one to bring physical CP phases in the lepton sector. Number of free parameters in the neutrino sector is much reduced compared to the usual three right handed neutrino case. However the system still contains an enough number of free parameters to reproduce the current neutrino oscillation data. We introduce a texture for 3 2 Dirac neutrino mass matrix by which the number of free parameters is further reduced. With a small number of free parameters, we investigate neutrino oscillation parameters and the amount of baryon asymmetry through leptogenesis. We will see correlations between them through a CP phase.

This paper is organized as follows. In the next section, we introduce Dirac mass matrix and a texture for it. We begin with the CP invariant case and apply a simple ansatz to the light neutrino mass matrix so as to reproduce the current best fit values for the neutrino oscillation parameters. In Sec. 3, we introduce a single CP phase and examine baryon asymmetry generated through leptogenesis and neutrino oscillations. With four input parameters (two light and two heavy neutrino mass eigenvalues), all neutrino oscillation parameters as well as the baryon asymmetry through leptogenesis are shown as a function of only the CP phase. Also, the averaged neutrino mass relevant to neutrino-less double beta decay experiments and the Jarlskog invariant characterizing CP violation in the lepton sector are presented as a function of the CP phase. We see correlations among these outputs, and presents some predictions for a fixed CP phase. The last section is devoted to conclusions.

## 2 Texture and a simple ansatz in CP invariant case

Without loss generality, we begin with a reference basis in which the charged lepton mass matrix , Dirac mass matrix and the right handed Majorana mass matrix are written as

(3) |

where all parameters are real and . We have three physical CP phases in by re-phasing. There is no triplet Higgs in the model, so left handed neutrinos do not have a Majorana mass at the beginning.

After see-saw mechanism, the light neutrino mass matrix becomes

(4) |

Note that the system with Dirac mass matrix leads to

(5) |

Therefore, at least, one mass eigenvalue of light neutrinos is zero. Concerning the current best fit values of neutrino oscillation data, we can conclude that only two patterns of diagonalized mass matrix for light neutrinos are possible. One is the so-called hierarchical case,

(6) |

with and . The other is the so-called inverse-hierarchical case,

(7) |

with and .

Now we introduce a texture for the Dirac mass matrix as and , and becomes a more simple form,

(8) |

with a single CP phase . The texture reduces the number of free parameters into six and allows us to analyze the correlations between the amount of baryon asymmetry and neutrino oscillation parameters with the single CP-phase. Similar textures have been discussed in Ref. [8]. The explicit form of the light neutrino mass matrix is given by

(9) |

Six parameters in the Dirac mass matrix and two heavy neutrino masses, eight parameters in total, correspond to physics of neutrinos.

We first tackle only neutrino oscillations in the CP invariant case, . CP violation will be introduced in the next section. As our stating point, we impose a simple ansatz that is diagonalized by the so-called tri-bimaximal mixing matrix[12],

(10) |

In fact, the tri-bimaximal mixing matrix is in excellent agreement with the current best fit values in Eq. (2). This ansatz strongly constrains the parameters in Eq. (8). For the hierarchical case, solving Eq. (10) with Eq. (9), we can describe each component in the Dirac mass matrix in terms of , , and ,

(11) |

For the inverse-hierarchical case, we find

(12) |

in terms of , , and . Thanks to our ansatz, only four parameters, (or ), , and are left free. These parameters will be used as inputs for our analysis in the next section.

Now we can discuss experimental tests of the model. Neutrino-less double beta decay experiments give upper bounds on the averaged neutrino mass, which can be extracted from the element of the Majorana mass matrix in the flavor basis. Here we see that for the hierarchical case,

(13) |

while for the inverse-hierarchical case,

(14) |

Here we have used an approximation for , in the inverse-hierarchical case. Therefore we have a chance of testing this relations in future neutrino-less double beta decay experiments with the sensitivity eV.

## 3 Numerical analysis in CP violating case

In this section we will introduce non-zero CP phase in the texture of Eq. (8). Except the CP phase, parameters in the Dirac mass matrix are described in terms of four free parameters as in Eq. (11) or in Eq. (12). With non-zero CP phase, we can obtain baryon asymmetry through leptogenesis. In addition, the Dirac mass matrix becomes complex and resultant neutrino oscillation parameters are deviating from those in the CP invariant case. We will see correlations among resultant neutrino oscillation parameters and the amount of baryon asymmetry created via leptogenesis.

Let us first consider leptogenesis. Primordial lepton asymmetry in the universe is generated through CP violating out-of-equilibrium decay of the lightest heavy neutrinos, which is characterized by the CP violating parameter [13],

(15) |

This formula for asymmetry is valid in the basis where the right handed neutrino is diagonal. Here, GeV is the vacuum expectation value of Higgs field, and

(16) |

Here is the efficiency factor, that parameterizes dilution effects for generated lepton asymmetry through washing-out processes. To evaluate the baryon asymmetry precisely, numerical calculations [14] are necessary. We use a fitting formula of the efficiency factor given in terms of effective light neutrino mass such that[15]

(17) |

Using the above formulas, we estimate the baryon asymmetry as a function of only the single CP phase with inputs (or ), , and . Numerical results are shown in Fig. 1 for the hierarchical and the inverse-hierarchical cases, respectively. Here we have taken eV, eV, GeV and GeV for the hierarchical case, while eV, eV, GeV and GeV for the inverse-hierarchical case. We can see that in the hierarchical case, or (rad) provides the baryon asymmetry consistent with the current observations. On the other hand, the inverse-hierarchical case cannot provide sufficient baryon asymmetry.

To understand these results, it is useful to give explicit formulas for leptogenesis in terms of parameters in the texture of Eq. (8). The CP violating parameter and the effective mass are, respectively, written as

(18) |

In the hierarchical case, the parameters fixed in Eq. (11) gives

(19) |

Here we have used an approximation formula , assuming . Our result is independent of as long as . To obtain a formula for the inverse-hierarchical case, we use an approximation, . Thus the parameters in Eq. (12) lead to

(20) |

Here we have again used , assuming , and the result is independent of . The baryon asymmetry is suppressed by the factor, . If is very large, for example GeV, we can give sufficient baryon asymmetry even with the suppression. However, in thermal leptogenesis re-heating temperature after inflation would be larger than the lightest heavy neutrino mass. It would be difficult to achieve such a quite high reheating temperature in usual reheating scenarios. We need some other mechanism such as a resonant leptogenesis[17] to enhance the primordial lepton asymmetry.

Now we analyze the neutrino oscillation parameters in the case of non-zero CP phase. Parameters in the texture are fixed as discussed in the previous section, and lead to the tri-bimaximal mixing matrix in the CP invariant case. When we switch CP phase on, the Dirac mass matrix becomes complex and, as a result, output oscillation parameters are deviating from the CP invariant case. In particular, we will find non-vanishing .

Substituting parameters given in Eq. (11) or Eq. (12) into the light neutrino mass matrix of Eq. (9), we find that is independent of and even for non-zero CP phase. Therefore, with input parameters (or ) and , output oscillation parameters are functions of only the CP phase .

In the hierarchical case with inputs eV and eV, resultant oscillation parameters are depicted in Fig. 2-4. For CP phase (rad), outputs corresponding to the solar neutrino oscillation are consistent with the best fit values, while other outputs are within the best fit region for any values of CP phase. For (rad), which provides the observed baryon asymmetry , we find the following neutrino oscillation parameters:

(21) |

They are all consistent with observations. Non-vanishing is our prediction, whose value would be covered in future baseline neutrino oscillation experiments. As can be seem from Eq. (9), the element of is independent of the CP phase and we obtain the same result as Eq. (13), numerically,

(22) |

In the inverse-hierarchical case with input parameters eV and eV, resultant oscillation parameters are depicted in Fig. 5 and 6. In this case, we find eV and , (almost) independent of the CP-phase. Although the inverse-hierarchical case cannot provide the observed baryon asymmetry, output oscillation parameters are consistent with the current data for a small CP phase (rad). Again, the element of is independent of the CP phase, and we obtain the same result as Eq. (14), numerically,

(23) |

This is an order of magnitude larger than the value in the hierarchical case.

It is also interesting to see a correlation between the baryon asymmetry through leptogenesis and the leptonic CP violating phase (Dirac phase)[18]. CP violation in the lepton sector is characterized by the Jarlskog invariant[19],

(24) | |||||

where is the Dirac phase. The Jarlskog invariant as a function of the CP phase is depicted in Fig. 7 for (a) the hierarchical and (b) the inverse-hierarchical cases, respectively. We obtain a small but non-vanishing correlating with other outputs. In the hierarchical case, we find

(25) |

for (rad).

## 4 Conclusions

Neutrino oscillation experiments have explored neutrino masses and mixing patterns. Tiny neutrino masses compared to the ordinary quark masses are naturally explained by the see-saw mechanism with heavy right handed neutrinos. Right handed neutrinos play the important role to generate the baryon asymmetry in our universe through leptogenesis. Leptogenesis requires CP violation in the lepton sector. For CP to violate we must have at least two right handed neutrinos. Keeping this minimal possibility in mind we have introduced only two heavy right handed neutrinos and studied a Dirac type mass matrix . Without loss of generality one can choose a reference basis where both charged lepton mass matrix as well as the heavy right handed Majorana mass matrices are real and diagonal. In this basis, three physical CP phases appear in .

After the see-saw mechanism we obtain an effective Majorana mass matrix for light neutrinos. As a result from the Dirac mass matrix , light neutrino mass spectrum should contain (at least) one zero mass eigenvalue. This fact allows only two patterns for neutrino mass spectrum, normal hierarchical or inverse-hierarchical.

We have chosen a simple texture for in our reference basis. To start with we have set all CP phases in to be zero. Although there is no CP violation in this case, one can study this real texture in the context of ongoing neutrino experiments. We have imposed an ansatz that the light neutrino mass matrix is diagonalized by the tri-bimaximal mixing matrix. This ansatz strongly constrains model parameters, and only four parameters (two light neutrino mass eigenvalues and two heavy neutrino mass eigenvalues) have been left free. Appropriate choice of two light neutrino mass eigenvalues reproduces the current neutrino oscillation data.

Next, we have introduced a single CP phase in . With four input parameters, we have examined baryon asymmetry generated through leptogenesis as well as neutrino oscillations, as a function of only the CP phase. We can see interesting correlations between resultant baryon asymmetry and neutrino oscillation parameters. For a special choice of the CP phase, the hierarchical case can reproduce both the observed baryon asymmetry and neutrino oscillation data. For a fixed CP phase reproducing the observed baryon asymmetry, we have a prediction for a non-vanishing which is accessible in future baseline neutrino oscillation experiments. In the inverse-hierarchical case, we have not obtained sufficient baryon asymmetry while resultant neutrino oscillation parameters can be consistent with the current data.

Independently of the CP phase, our texture leads to a unique relation for the averaged neutrino mass relevant to neutrino-less double beta decay experiments: and in the hierarchical and the inverse-hierarchical cases, respectively. These results can be tested in next generation experiments of neutrino-less double beta decay. We have worked in the context of a specific texture. However, as an extension to our approach one can introduce small and check whether the solutions reported in this article get drastically modified. This is so because, often in real world models, one may be able to restrict such that it is very small yet not exactly zero. If a real is introduced in the complex case (), and its magnitude is of order of the rest of the s, we see a variation in and . However, remain almost the same.

## 5 Acknowledgments

B.B is supported by UGC, New Delhi, India, under the grant number F.PSU-075/05-06. He would also like to thank theory division KEK, Japan, for hospitality and financial support where this work was performed. The work of N.O is supported in part by Scientific Grants from the Ministry of Education and Science of Japan.

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