  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.
