Texture Dirac Mass Matrices and Lepton Asymmetry in the Minimal Seesaw Model with Tri-Bimaximal Mixing

We examined the minimal seesaw mechanism of 3 × 2 Dirac matrix by starting our analysis with the masses of light neutrinos with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We found all possible Dirac mass textures which contain one zero entry or two in the matrix.

The seesaw mechanism is arguably the most attractive way to explain the smallnes of neutrino masses. In its conventional form, the seesaw invokes three heavy singlet right-handed neutrinos ν R [1]. However, the number three is not sacret. The two non-zero light neutrino mass difference required by experiment could be explained with just two heavy right-handed neutrinos [2]. The seesaw mechanism with two right-handed neutrinos predicts one of the physical light neutrino mass to be exactly zero; which is permissible within the current knowledge of neutrino masses and mixings.
The results of the LEP experiments on the measurement of the invisible width of the Z boson imply that only three flavor neutrinos exist in nature (see Ref. [4]). The simplest one to give 3 × 3 mass matrix m ν is both Dirac mass matrix m D and right-handed massive Majorana mass matrix M R are 3 × 3 matrices.

II. THE MINIMAL SEESAW MODEL
The most economical seesaw model which compatible with solar and atmospheric neutrinos is satisfied by two righthanded neutrinos. The leptonic part of the Yukawa interactions in presence of three left-handed and two right-handed neutrinos can be written as where N R denote the right-handed neutrino fields which are singlet under the standard model gauge group, φ is SU(2) higgs doublet withφ = iσ 2 φ * , ψ Li is the lepton doublet of flavor i, and E Ri are the right-handed charged leton singlet. the Yukawa coupling constants Y ν and Y are comlex-valued matrices. After the electroweak symmetry breaking one gets the charged mass matrix m = vY and the Dirac mass matrix for the neutrino as m D = vY ν where v is the vacuum exectation value of the neutral component of the higgs doublet φ. The Majorana mass matrix M R is 2 × 2 comlex symmetric matrix. The mass matrix for the neutral fermions can be written as c Jurusan Fisika FMIPA ITS The light neutrino mass matrix m ν after the seesaw diagonalization is given by the seesaw formula This master formula (3) is valid when the eigenvalues of M R are much larger than the elements of m D and in such a case the eigenvalues of m ν come out very small with respect to those of m D .
In general the Majorana mass matrix M R is non-diagonal form in the basis where the charged current is flavor diagonal. In this form one can make a basis rotation so that the righthanded Majorana mass matrix becomes diagonal by the unitary matrix. All possible heavy Majorana mass matrices M R , their inverse and their diagonal form are given in the Table I, with the values and In this form the Dirac mass matrix must be 3 × 2 form. It implies the the determinant of light massive neutrino matrix is zero det m ν = 0 (6) Since it gives at least one of the eigenvalues of m ν is exactly zero; i.e, either m 1 = 0 for normal neutrino mass hierarchy ( m 1 < m 2 < m 3 ) or m 3 = 0 for inverted neutrino mass hierarchy (m 1 > m 2 > m 3 ).

III. TRI-BIMAXIMAL MIXING AND DIRAC MASS TEXTURE
It is an experimental fact that within measurement errors the observed neutrino mixing matrix is compatible with the so called tri-bimaximal form, introduced by Harrison, Perkins and Scott (HPS). The matrix is given by One purpose of this paper is just to find the Dirac neutrino mass matrix for either m1 = 0 or m3 = 0. Looking back to Eqs. (1), we have 3 × 2 matrix m D = vY ν , we can write it in the form Using the diagonal inverse matrix M −1 R we obtain Then applying the HPS matrix to diagonalize the neutrino mass matrix m ν we have m 11 = 4y 2 1 + y 2 2 + y 2 3 − 4y 1 (y 2 + y 3 ) + 2y 2 y 3 M 1 + 4x 2 The lightest neutrino is allowed to be massless; i.e., either m 1 = 0 (normal neutrino mass hierarchy, NH) or m 3 = 0 (inverted neutrino mass hierarchy, IH) has no conflict with the present neutrino oscillation measurements. In both cases, the non-vanishing neutrino masses can be determined in terms of m 2 sol anad m 2 atm : for normal hierarchy, and M 1 = 9.12 × 10 12 GeV ≈ 10 13 GeV M 2 = 9.07 × 10 13 GeV ≈ 10 14 GeV for inverse hierarchy.

IV. LEPTON ASYMMETRY
When the Majorana right-handed neutrinos decay into leptons and Higgs scalars, they violate the lepton number since right-handed neutrino fermionic lines do not have any preferred arrow The interference between the tree-level decay amplitude and the absorptive part of the one-loop vertex leads to a lepton asymmetry We assume a hierarchical mass pattern of the heavy neutrinos M 1 << M 2 . in this case, the interactions of N 1 can be in thermal equilibrium when N 2 decay is washed-out by the lepton number violating processes with 1 . Thus only the decays of N 1 are relevant for generation of the final lepton asymmetry ≈ 1 . in this case, the CP asymmetry paarameter in the