Neutrino mass matrix solutions and neutrinoless double beta decay

We present a determination of the neutrino mass matrix which holds for values of the neutrinoless double beta decay effective mass mee larger than the neutrino mass differences. We find eight possible solutions and discuss for each one the corresponding neutrino mass eigenvalues and zero texture. A minimal structure of the perturbations to add to these zero textures to recover the full mass matrix is also determined. Implications for neutrino hot dark matter are discussed for each solution. 1) Introduction: The neutrino mass matrix in the three neutrino framework has been severely constrained over the last few years by atmospheric [1] and solar [2] neutrino experiments, and by recent positive indications from laboratory experiments [3, 4]. In the basis in which the charged lepton mass matrix is diagonal, the neutrino mass matrix contains all the information about the mixing schemes in the leptonic sector. The mixing angle required for the atmospheric neutrinos appears to be maximal or very close to maximal. This is one of the main constraints on the neutrino mass matrix, when combined with the mass squared difference required for atmospheric neutrinos δmatm, which is in the range (1.5 − 4.8) · 10[5]. There are a few solutions to the solar neutrinos, which give mass squared differences δmsol and mixing angle θ with νe which can lie in various ranges (see e.g. Ref. [6]). The reactor experiment CHOOZ gives in addition a strong upper limit on the Ue3 element of the neutrino mixing matrix [7]. Besides these five constraints on the neutrino mass matrix, a sixth one would be given by the measurement of the effective mass mee in neutrinoless double beta decay. This parameter would provide a scale to the neutrino mass matrix. Very recently a positive indication of neutrinoless double has been reported with mee in the following range [8]: mee = (0.05− 0.86) eV (at 95% c.l.) (1) and with a best fit for mee = 0.39 eV. In the following we study what could be the consequences for the neutrino mass matrix of a value of mee within the range of Eq. (1). By considering this range we study more generally what would be the effect on the mass matrix of an effective mass mee determined with a value greater than the neutrino mass differences and smaller than the dark matter constraints (see below). This range covers also the values of mee which could be further tested experimentally in a relatively near future [9]. Since the scale mee is then higher than the mass squared differences required by the neutrino oscillation results, this implies a degenerate or partially degenerate solution [10]. The hierarchical solution is not allowed by a value of mee within the range of Eq. (1). An inverse hierarchy solution is still possible but only for mee close to the lower edge of this range. Considering the fact that the atmospheric mixing is close to maximal, several texture mass matrices have been constructed [11], including in particular the ones where the mass squared difference δmatm is considered to be the dominant entry in the mass matrix and where the solar neutrino solutions could be added as perturbations. These textures are very useful to construct models, where the zeroes could have their origin in some symmetry of the model, so that the perturbations come from the symmetry breaking. For a value of mee within the range of Eq. (1), δm 2 atm would likely becomes a perturbation in the neutrino 2 mass matrix with respect to the scale mee. This would be an important information for the determination of the texture of the mass matrix. Moreover, given a value of mee in the range of Eq. (1) the neutrino contribution to the hot dark matter would be significant. With these assumptions we now attempt to look into this problem. Instead of starting the analysis from the possible textures, we start with the experimental inputs and then try to determine the neutrino mass matrices and in the process get the possible textures. In the following we shall work in the basis in which the charged lepton mass matrix is diagonal. The physical states |να >, (α = e, μ, τ), which enter in the charged current, are related to the mass eigenstates |νi >, (with masses mi, i = 1, 2, 3), by the mixing matrix |να >= Uαi|νi > (2) and the effective mass entering in the neutrinoless double beta decay is mee = (Mν)ee = (Mν)11 = ∑ i U eimi. (3) (Mν)11 is the (11) element of the neutrino mass matrix in the flavour basis. In this article we do not consider the imaginary part of the mass matrix assuming that there is no CP violation, or very small CP violation. In this limit, we can parametrize the mass matrix in the flavor basis as a function of the three real mass eigenvalues and the three angles coming from the mixing matrix Uαi. For simplicity and also as a first step of a more involved analysis we consider in addition only the case with exact maximal mixing between the νμ and ντ (motivated by the best fit value from SuperKamiokande) and we consider Ue3 = 0 (motivated by the result of the CHOOZ experiment). In this framework we determine what are the effects on these six mass matrix parameters of the five experimental constraints mentioned above taking a given value of mee. We find that there are eight different solutions for the mass matrix, each one being determined by these six constraints. For each solution we display the corresponding zero texture and show how the perturbations arise around these zero textures to give the full mass matrix. We also discuss the consequences for hot dark matter for the different solutions. 2) The neutrino mass matrix solutions: In a general way the diagonal neutrino mass matrix can be written as: M ν = 