Neutrino - nucleon reaction rates in the supernova core in the relativistic random phase approximation

In view of the application to supernova simulations, we calculate neutrino reaction rates with nucleons via the neutral and charged currents in the supernova core in the relativistic random phase approximation (RPA) and study their effects on the opacity of the supernova core. The formulation is based on the Lagrangian employed in the calculation of nuclear equation of state (EOS) in the relativistic mean field theory (RMF). The nonlinear meson terms are treated appropriately so that the consistency of the density correlation derived in RPA with the thermodynamic derivative obtained from EOS by RMF is satisfied in the static and long wave length limit. We employ pion and rho meson exchange interactions together with the phenomenological Landau-Migdal parameters for the isospin-dependent nuclear interactions. We find that both the charged and neutral current reaction rates are suppressed from the standard Bruenn's approximate formula considerably in the high density regime. In the low density regime, on the other hand, the vector current contribution to the neutrino-nucleon scattering rate is enhanced in the vicinity of the boundary of the liquid-gas phase transition, while the other contributions are moderately suppressed there also. In the high temperature regime or in the regime where electrons have a large chemical potential, the latter of which is important only for the electron capture process and its inverse process, the recoil of nucleons cannot be neglected and further reduces the reaction rates with respect to the standard approximate formula which discards any energy transfer in the processes. These issues could have a great impact on the neutrino heating mechanism of collapse-driven supernovae.