Light neutrino mass spectrum with one or two right-handed singlet fermions added

We analyse two cases of the minimal extension of the Standard Model when one or two right-handed fields are added to the three left-handed fields. A second Higgs doublet (two Higgs doublet model - 2HDM) is included in our model. We calculate one-loop radiative corrections to the mass parameters which produce mass terms for the neutral leptons. In both cases we numerically analyse light neutrino masses as functions of the heavy neutrino masses. Parameters of the model are varied to find light neutrino masses that are compatible with experimental data of solar and atmospheric neutrino mass differences for normal hierarchy. We choose values for the parameters of the tree-level by numerical scans, where we look for the best agreement between computed and experimental neutrino oscillation angles.


Description of the model
The mass terms for the neutrinos can be written in a compact form as a mass term with a (n L + n R ) × (n L + n R ) symmetric mass matrix where M D is a n L ×n R Dirac neutrino mass matrix, while the hat indicates thatM R is a diagonal matrix. M ν can be diagonalized as where the m i are real and non-negative. In order to implement the seesaw mechanism [1] we assume that the elements of M D are of order m D and those of M R are of order m R , with m D m R . Then, the neutrino masses m i with i = 1, 2, . . . , n L are of order m 2 D /m R , while those with i = n L + 1, . . . , n L + n R are of order m R .
In the standard seesaw, one-loop corrections to the mass matrix, i.e. the self energies, are determined by the neutrino interactions with the Z boson, the neutral Goldstone boson G 0 , and the Higgs boson h 0 . Each diagram contains a divergent piece but the sum of these three contributions turns out to be finite.
Once the one-loop corrections are taken into account the neutral fermion mass matrix is given by where the 0 3×3 matrix appearing at tree level (1) is replaced by the contribution δM L . This correction is a symmetric matrix, it has the largest influence as compared to other corrections. The expression for one-loop corrections is given by [2] with ∆ b = k b k ∆ k , where b k are two-dimensional complex unit vectors, ∆ k are the Yukawa matrices, and the sum b runs over all neutral physical Higgses H 0 b . Neutral Higgses are characterized by the unit-length vectors b which hold special orthogonality conditions. We use the set of values It coincides with a special case of the invariant mixing angles that define the diagonalization matrix of the neutral Higgs squared-mass matrix [3]. The Higgs potential of the 2HDM becomes CP-conserving in this case.

Case n R = 1
First we consider the minimal extension of the standard model by adding only one right-handed field ν R to the three left-handed fields contained in ν L . We The It is possible to estimate masses of the light neutrinos from experimental data of solar and atmospheric neutrino oscillations [4] assuming that the lightest neutrino has initial mass value m in . Numerically varying the value of m in we choose parameters for the tree level that agree with the experimental oscillation data most accurately.
For the calculation of radiative corrections we use the orthogonal complex vectors b defined in eq. (5). Diagonalization of the mass matrix corrected by oneloop contributions is performed with a unitary matrix ν M (1) † ν , and U ϕ is a phase matrix. The second light neutrino obtains its mass from radiative corrections. The third light neutrino remains massless.
The numerical analysis shows (see Fig. 1) that we can reach the allowed neutrino mass ranges for a heavy singlet with the mass close to 10 4 GeV when the angle of oscillations θ atm is fixed to the experimental 3σ range [4].

Case n R = 2
When we add two singlet fields ν R to the three lefthanded fields ν L , the radiative corrections give masses to all three light neutrinos.
1, and | b| = 1. Diagonalizing the symmetric mass matrix M ν (1) in block form we write: The non zero masses inM l andM h are determined by the seesaw mechanism: m 2 D i ≈ m h i m l i and m 2 R i ≈ m 2 h i , i = 1, 2. We use m 1 > m 2 > m 3 ordering of masses. At tree level the third light neutrino is massless.
In numerical calculations the model parameters as well as the derived masses of the light neutrinos are obtained in several steps. First, the diagonal mass matrix for the tree level is constructed. Having chosen the mass for the third neutrino m in , the masses of the other two light neutrinos are estimated from experimental data on solar and atmospheric neutrino oscillations [4]. The masses of the heavy right-handed neutrinos are input parameters. This diagonal tree level matrix is used to constrain the parameters that enter the tree-level mass matrix M ν and its diagonalization matrix. Then the diagonalization matrix is used to evaluate one-loop corrections to the mass matrix. Diagonalization of the corrected mass matrix yields masses for the three light neutrinos. If the calculated mass difference is compatible with the experimental oscillations data, the parameter set is kept. Otherwise, another set of parameters is generated. Figure 3 illustrates the obtained results for normal neutrino mass ordering.   In the case of n R = 2 the inverted hierarchy is also allowed [5].

Conclusions
1. For the case n R = 1 we can match the differences of the calculated light neutrino masses to ∆m 2 and ∆m 2 atm with the mass of a heavy singlet larger than 10 4 GeV. The parametrization used for this case and restrictions from the neutrino oscillation data limit the values of free parameters. One light neutrino remains massless. 2. In the case n R = 2 we obtain three non vanishing masses. The numerical analysis shows that the values of parameters and Higgs masses depend on the choice of the set of b vectors and the heavy neutrino masses. The radiative corrections generate the lightest neutrino mass and have a big impact on the second lightest neutrino mass.