Charged-current reactions in the supernova neutrino-sphere

We compute 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. Using realistic nucleon-nucleon potentials that fit measured scattering phase shifts, we calculate the momentum-, density- and temperature-dependent nucleon self-energies in the Hartree-Fock approximation, which leads to an enhancement of the $\nu_e$ cross-section and a suppression of the $\bar{\nu}_e$ cross section. A potential based on chiral effective field theory and a pseudo-potential constructed to reproduce nucleon-nucleon phase shifts are employed. The effect of Hartree-Fock corrections to the nucleon self-energies on the equilibrium proton/electron fraction, and on the charged current rates is studied in detail. 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 are not as large as in previous calculations. Our results have implications for heavy element nucleosynthesis in supernovae and supernova neutrino detection.