A New Treatment of Neutrino Oscillations in Medium

A new more rigorous and accurate method for treating neutrino oscillations in the context of the MSW effect in a medium is proposed. This leads to a new type of resonance condition which for small mixing angles puts rather stringent conditions on Eν/δm . The implications on the solar neutrino problem are discussed. IOA-304/94 April 1994 One of the most important questions of the physics beyond the standard model is the problem of neutrino masses. Furthermore if the neutrinos are massive, the neutral leptons produced in weak interactions are not stationary. They are linear combinations of the neutrino mass eigenstates (neutrino mixing). Even though at present there is no theory which can predict the mass and mixing of the neutrinos, most Grand Unified Models predict small masses and mixing. If the neutrinos are almost degenerate, neutrino oscillation experiments [1, 2] are the best candidates to measure small δm (from 1eV 2 down to 10eV ). Furthermore, neutrino oscillations may explain the solar neutrino problem [3, 4, 5, 6], i.e. the apparent reduction of the νe flux at earth compared to that predicted by the standard solar model [7, 8] (SSM). The mechanism of neutrino oscillations however, is not effective if the neutrino mixing is small. It has been observed, though, that under the conditions of high density encountered in the sun’s interior the oscillation can be enhanced due to the MSW effect [9, 10]. In other words small mixing angles can be converted into large effective mixing angles due to the resonant scattering of νe neutrinos by electrons. Calculations of neutrino oscillations involving the MSW effect have hitherto involved the following steps: 1) Convert the evolution equation from flavor space into neutrino mass eigenstate basis by locally diagonalizing the space dependent Hamiltonian[1, 11]. 2) Ignore the transitions between the mass eigenstates (adiabatic approximation) or treat such transitions in perturbation treatment. In the last case one assumes that [12]. δEν/2 dθm/dx ≪ 1 (1) where δEν is the difference in neutrino energies and dθm dx the variation of the mixing angle in the medium with distance. In particular the above equation must be true at the resonance point. In the present paper we will provide an exact solution which does not go through the local neutrino mass eigenstates. Our method is quite simple and offers itself to a simple interpretation. For illustration purposes we will exhibit our method in the case of two generations but it can easily be extended to any number of generations. Following standard procedure [11] we can write the neutrino state at any time as |ν(t) >= ae(t)|νe > +aα(t)|να > where |νe > is the electron neutrino and να any other flavor (e.g. νμ). The