Chiral effective field theory description of neutrino nucleon-nucleon Bremsstrahlung in supernova matter

We revisit the rates of neutrino pair emission and absorption from nucleon-nucleon bremsstrahlung in supernova matter using the $T$-matrix formalism in the long-wavelength limit. Based on two-body potentials of chiral effective field theory ($\chi$EFT), we solve the Lippmann-Schwinger equation for the $T$-matrix including non-diagonal contributions. We consider final-state Pauli blocking and hence our calculations are valid for nucleons with an arbitrary degree of degeneracy. We also explore the in-medium effects on the $T$-matrix and find that they are relatively small for supernova matter. We compare our results with one-pion exchange rates, commonly used in supernova simulations, and calculations using an effective on-shell diagonal $T$-matrix from measured phase shifts. We estimate that multiple-scattering effects and correlations due to the random phase approximation introduce small corrections on top of the $T$-matrix results at subsaturation densities. A numerical table of the structure function is provided that can be used in supernova simulations.