Charged-current reactions in the supernova neutrino-sphere

We calculate neutrino absorption rates due to charged-current reactions $\nu_e+n \rightarrow e^- + p $ and $\bar{\nu}_e+p \rightarrow e^+ + n $ in the outer regions of a newly born neutron star called the neutrino-sphere. To improve on recent work which has shown that nuclear mean fields enhance the $\nu_e$ cross-section and suppress the $\bar{\nu}_e$ cross-section, we employ realistic nucleon-nucleon interactions that fit measured scattering phase shifts. Using these interactions we calculate the momentum-, density-, and temperature-dependent nucleon self-energies in the Hartree-Fock approximation. A potential derived from chiral effective field theory and a pseudo-potential constructed to reproduce nucleon-nucleon phase shifts at the mean-field level are used to study the equilibrium proton fraction and the charged-current rates are studied in detail. We compare our results to earlier calculations obtained using phenomenological mean-field models and to those obtained in the virial expansion valid at low density. We find that for typical ambient conditions in the neutrino-sphere, $T=5-10$ MeV and $\rho =10^{11}-10^{13}$ g/cm$^3$, the difference between the $\nu_{e}$ and $\bar{\nu}_e$ absorption rates is much larger than predicted earlier. Our results have important implications for heavy-element nucleosynthesis in supernovae and for supernova neutrino detection.