μ and τ neutrino thermalization and production in supernovae: Processes and time scales

We investigate the rates of production and thermalization of ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ and ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ neutrinos at temperatures and densities relevant to core-collapse supernovae and protoneutron stars. Included are contributions from electron scattering, electron-positron annihilation, nucleon-nucleon bremsstrahlung, and nucleon scattering. For the scattering processes, in order to incorporate the full scattering kinematics at arbitrary degeneracy, the structure function formalism developed by Reddy, Prakash, and Lattimer [Phys. Rev. D $58,$ 013009 (1998)] and Burrows and Sawyer [Phys. Rev. C $58,$ 554 (1998)] is employed. Furthermore, we derive formulas for the total and differential rates of nucleon-nucleon bremsstrahlung for arbitrary nucleon degeneracy in asymmetric matter. We find that electron scattering dominates nucleon scattering as a thermalization process at low neutrino energies $({\ensuremath{\varepsilon}}_{\ensuremath{\nu}}\ensuremath{\lesssim}10 \mathrm{MeV}),$ but that nucleon scattering is always faster than or comparable to electron scattering above ${\ensuremath{\varepsilon}}_{\ensuremath{\nu}}\ensuremath{\simeq}10 \mathrm{MeV}.$ In addition, for $\ensuremath{\rho}\ensuremath{\gtrsim}{10}^{13} {\mathrm{g}\mathrm{}\mathrm{}\mathrm{cm}}^{\ensuremath{-}3},$ $T\ensuremath{\lesssim}14 \mathrm{MeV},$ and neutrino energies $\ensuremath{\lesssim}60 \mathrm{MeV},$ nucleon-nucleon bremsstrahlung always dominates electron-positron annihilation as a production mechanism for ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ and ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ neutrinos.