ar X iv : n uc l - th / 0 60 50 51 v 1 2 2 M ay 2 00 6 RANDOM PHASE APPROXIMATION AND NEUTRINO - NUCLEUS CROSS SECTIONS 1

The Random Phase Approximation theory is used to calculate the total cross sections of electron neutrinos on 12 C nucleus. The role of the excitation of the discrete spectrum is discussed. A comparison with electron scattering and muon capture data is presented. The cross section of electron neutrinos coming from muon decay at rest is calculated. The Random Phase Approximation (RPA) is an effective theory constructed to study the excitations of many-body systems. The RPA assumes that the exited states of these systems can be described as linear combinations of one-particle on-hole (1p − 1h) and one-hole one-particle (1h − 1p) excitations. The goal of the theory is to find the coefficients of the linear combinations for a given interaction between particles and holes. In nuclear physics, the RPA has been applied to study excitations on a wide energy range, from a few MeV, the discrete spectrum, up to hundreds of MeV, in a regime called quasi-elastic where the emission of a single nucleon is the dominant process. One of the great successes of the RPA is the prediction of collective surface vibrations, called giant resonances, appearing at energies between 15 and 30 MeV in all the nuclei with more than 10 nucleons. The inputs required by the RPA are the set of single particle energies and wave functions, and the effective interaction between particles and holes. In our calculations the single particle basis, which properly includes the continuum, has been obtained by solving the one-body Schrödinger equation with a spherical Woods-Saxon potential. The parameters have been taken from the literature [1], and have been fixed to reproduce the rms charge radii and the single particle energies close to the Fermi level. The theoretical uncertainty has been studied by using various ph interactions, specifically the LM1, LM2, and PP interactions of Ref. [1]. The LM1 and LM2 interactions are zero-range forces of Landau-Migdal type with slightly different values of the parameters. The PP interaction is a finite-range interaction. A common characteristic of the three interactions is that they have been rescaled to reproduce the excitation energy of the low lying 3 − state in 16 O at 6.13 MeV. Even though the three interactions produce the same excitation energy, they give different descriptions of the 3 − state.