Neutrino-Nucleus Reactions and Muon Capture in 12C

The neutrino-nucleus cross section and the muon capture rate are discussed within a simple formalism which facilitates the nuclear structure calculations. The corresponding formulae only depend on four types of nuclear matrix elements, which are currently used in the nuclear beta decay. We have also considered the non-locality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in 12C the higher order relativistic corrections are of the order of ~5 only, and therefore do not play a significant role. As nuclear model framework we use the projected QRPA (PQRPA) and show that the number projection plays a crucial role in removing the degeneracy between the proton-neutron two quasiparticle states at the level of the mean field. Comparison is done with both the experimental data and the previous shell model calculations. Possible consequences of the present study on the determination of the $\nu_\mu ->\nu_e$ neutrino oscillation probability are briefly addressed.