# Non-zero from the Triangular Ansatz and Leptogenesis

Abstract

[3mm]

Recent experiments indicate a departure from the exact Tri-Bimaximal mixing by measuring definitive non-zero value of . Within the framework of type I seesaw mechanism, we reconstruct the triangular Dirac neutrino mass matrix from the symmetric mass matrix. The deviation from symmetry is then parametrized by adding dimensionless parameters in the triangular mass matrix. In this parametrization of the neutrino mass matrix, the non-zero value is controlled by .
We also calculate the resulting leptogenesis and show that the triangular texture can generate the observed baryon asymmetry in the universe via leptogenesis scenario.

Keywords: Neutrino Physics; Flavor Symmetry; Leptogenesis; Matter-antimatter.

PACS numbers: 14.60.Pq; 11.30.Hv; 98.80.Cq

## 1 Introduction

The discovery of the neutrino masses and the lepton mixing have played a pivotal role in probing physics beyond the Standard Model (SM). Recent experiments MINOS [1, 2], T2K [3], Double CHOOZ [4], Daya Bay [5] and RENO [6] have reported definitive non-zero result. The latest global analysis of neutrino oscillation data yields the following best-fit values with and errors for the oscillation parameters [7].

Parameter | best fit | ||

Seesaw mechanism [8]-[11] gives a natural explanation of the smallness of the masses for light neutrinos and mixing by connecting the tiny neutrino masses to a very heavy right-handed neutrinos masses.

According to the seesaw mechanism, lepton number is broken at high energies due to right-handed neutrino Majorana masses, resulting in small left-handed neutrino Majorana masses suppressed by the heavy mass scale. The seesaw mechanism also provides an attractive mechanism for generating the baryon asymmetry of the universe via leptogenesis [12]-[19].

The seesaw Lagrangian can be written as :

(1) |

where and denotes
the left-handed (light) neutrinos, the left-handed charged leptons and the right-handed (heavy) Majorana neutrinos,
respectively. We assume that the mass matrices of both the heavy and the charged lepton
are diagonal and real matrices.

After integrating out the heavy right-handed neutrinos, the symmetric Majorana mass matrix for light neutrinos is :

(2) |

The mixing matrix that diagonalizes is :

(3) |

where and

(4) |

where , is the CP-violating phase
and is diagonal matrix consisting of non-trivial Majorana phases.

Now any nonsingular matrix can be decomposed into the product of a nonsingular, upper or lower triangular matrix and a unitary matrix . In particular, the Dirac matrix can be written as :

(5) |

It has been noticed that the relevant quantities for leptogenesis are the CP-violating phases in
which means that the unitary matrix cancels out in the unflavored
leptogenesis but not in the seesaw formula. Our ansatz consists in taking the unitary matrix to be identity implying that the Dirac matrix is either lower or upper triangular matrix .

Triangular textures have been considered as the simplest form to study since all the unphysical features can be eliminated from the start [20] and [21]. Leptogenesis with triangular ansatz has been studied by [22, 23].

The paper is organized as follows. In the next section, we consider a lower or upper triangular matrix as Dirac mass matrix. We show how to reconstruct the triangular Dirac matrix from the neutrino symmetric mass matrix for the Tri-Bimaximal (TBM) mixing. In section 3, we discuss the breaking of the triangular mass by adding dimensionless parameters. We next analyze the mass texture by fitting it to the observed neutrino oscillation data. We show that the triangular mass matrix can naturally accommodate the observed neutrino oscillation parameters. In section 4, we calculate the relevant quantities for leptogenesis and estimate the baryon asymmetry that is consistent with the observed value.

## 2 Triangular Dirac Matrix for Tri-Bimaximal Mixing

In the following, we reconstruct the Dirac matrix analytically from the neutrino symmetric mass matrix elements ( i.e. light neutrinos masses, heavy Majorana masses and the Majorana phases ) for the Tri-Bimaximal mixing pattern and [24]. The general symmetric matrix that is diagonalized by the Tri-Bimaximal mixing matrix can be written as :

(6) | |||||

with the eigenvalues :

(7) |

By solving the above equation, we get the following Dirac matrix for upper or lower triangular matrix :

(8) |

where is the determinant of the neutrino mass matrix.

## 3 Deviations from Tri-Bimaximal

Now we consider deviation from TBM by breaking the triangular neutrino matrix with dimensionless parameters as :

(9) |

with and .

It is interesting to notice that by taking and for the lower triangular and and for upper triangular, the neutrino matrix will be :

(10) |

For , has symmetry leading to and .
Moreover, in the limit , the above matrix gives the TBM angles and mass eigenvalues.

In particular for , the lower or upper triangular mass matrix gives the neutrino matrix :

(11) |

which is the strong scaling Ansatz studied by [25].

To see how neutrino mass matrix given by can lead to deviations of neutrino angles from their TBM values, we consider the hermitian matrix ,

(12) |

where

(13) | |||||

(14) |

As a consequence, the hermitian matrix is diagonalized by the unitary matrix ,

(15) |

where the deviation of the from their TBM values is parametrized by three quantities :

(16) |

The angle of the atmospheric mixing is expressed in terms of the parameters as :

(17) | |||||

where and .

The Dirac CP phase is given by :

(18) | |||||

where

(19) | |||||

In this analysis, we consider only the case of . In the case, we have .

The atmospheric angle is :

(20) |

and the deviation of the atmospheric mixing will be :

(21) |

We see that the deviation vanishes for as it should be.

The Dirac CP phase becomes :

(22) |

A useful measure of CP violation is given by [26]:

(23) | |||||

Here we have :

(24) |

which shows that for , the rephasing invariant vanishes.

The reactor mixing angle is :

(25) |

The solar mixing angle is governed by :

(26) |

where

(27) | |||||

which for , we get as expected.

Finally, the mass squared differences are given by :

(28) |

In the numerical analysis, we choose the parameters and as inputs. With the help of the
allowed experimental neutrino results, we determine the allowed ranges of these inputs.

In Fig.1, we have plotted with respect to and and obtain restriction on the parameter space of and .
One can see that and are constrained in the ranges and .

In the limit and , the mass-squared differences and are :

(29) |

Now, using the best fit of the masses-squared differences, we get :

(30) |

In Fig.2 and Fig.3, we display the exact expressions for the mass-squared differences versus and the phase for and . It can be seen that for these values and are within ranges. Since depends on the , Fig. 3 shows two symmetrically regions by varying between and .

For and , we show in Fig.4 and Fig.5 the variation of the reactor and
atmospheric angles versus and . These plots indicate that and are within ranges. Moreover, since depends on the , Fig. 4 shows two symmetrically separated regions within these ranges.

## 4 Leptogenesis

The CP violating asymmetry is due to the interference of tree level contribution with the one-loop corrections for the decay of the ith heavy neutrino [12, 13] :

(31) |

The contribution of the heavier neutrinos is washed out and only the asymmetry generated by the lightest neutrino survives :

(32) |

Here is the electroweak symmetry breaking scale and the function stems from the contributions of both self-energy and vertex diagrams :

(33) |

For hierarchical heavy Majorana neutrinos i.e., , one has :

(34) |

The CP asymmetry gives rise to a lepton asymmetry in the universe :

(35) |

where denotes the entropy density, is the effective number of relativistic degrees of freedom contributing to the entropy of the early universe and is the dilution factor that is obtained by solving the Boltzmann equations. To a good approximation, is given by [27] :

(36) |

where the parameter , which is defined as the ratio of the thermal average of the decay rate and the Hubble parameter at the temperature , is :

(37) |

Here is the Planck mass.

The produced lepton asymmetry is converted into a net baryon asymmetry through the non-perturbative sphaleron processes. The baryon asymmetry is obtained :

(38) |

where in the standard model and are the number of fermion families and complex Higgs doublets respectively. Typically, one gets :

(39) |

For the lower triangular Dirac neutrino mass matrix with , the CP asymmetry is :

(40) |

The value of for the lower triangular texture is expressed as :

(41) |

which is independent of and is approximately equal to .

The baryon asymmetry generated by leptogenesis is :

(42) |

We see that for an exact symmetry, the baryon asymmetry vanishes [28]-[37].

We use as the upper and lower bound of the baryon-photon ratio from the WMPA observation [38]. Our result is shown in Fig.6 with , where the two horizontal lines represents the WMPA allowed bounds. We can see that the triangular Dirac mass matrix provides a baryon asymmetry consistent with the current observations.

## 5 Conclusion

In this paper, we have reconstructed lower and triangular textures of the neutrino Dirac mass matrix that can be obtained via type I seesaw mechanism from the neutrino symmetric mass matrix for the TBM mixing pattern.

The recent experiments have reported definitive non-zero value for the reactor mixing angle. Therefore, we have considered deviation of the triangular ansatz from TBM by adding dimensionless parameters. We have studied the phenomenological implications of these triangular textures and have obtained interesting results for the mixing angles and mass-squared
differences.

Furthermore, we have shown that the breaking of the triangular mass matrix from the exact TBM gives a non-zero baryon asymmetry via leptogenesis consistent with the current observations.

## Acknowledgments

I would like to thank S. Nasri for reading the manuscript and useful comments.

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