Supernova neutrino opacity from nucleon-nucleon bremsstrahlung and related processes

Elastic scattering on nucleons, nu N --> N nu, is the dominant supernova (SN) opacity source for mu and tau neutrinos. The dominant energy- and number-changing processes were thought to be nu e(-) --> e(-)nu and nu<(nu)over bar> <-> e(+)e(-) until Suzuki showed that the bremsstrahlung process nu<(nu)over bar>NN <-> NN was actually more important. We find that for energy exchange, the related "inelastic scattering process" nu NN <-> NN nu is even more effective by about a factor of 10. A simple estimate implies that the nu(mu) and nu(tau) spectra emitted during the Kelvin-Helmholtz cooling phase are much closer to that of <(nu)over bar>(e) than had been thought previously. To facilitate a numerical study of the spectra formation we derive a scattering kernel that governs both bremsstrahlung and inelastic scattering and give an analytic approximation formula. We consider only neutron-neutron interactions; we use a one-pion exchange potential in Born approximation, nonrelativistic neutrons, and the long-wavelength limit, simplifications that appear justified for the surface layers of an SN core. We include the pion mass in the potential, and we allow for an arbitrary degree of neutron degeneracy. Our treatment does not include the neutron-proton process and does not include nucleon-nucleon correlations. Our perturbative approach applies only to the SN surface layers, i.e., to densities below about 10(14) g cm(-3).