Supernova Neutrino Scattering Rates Reduced by Nucleon Spin Fluctuations: Perturbative Limit

In a nuclear medium, spin-dependent forces cause the nucleon spins to fluctuate with a rate $\Gamma_\sigma$. We have previously shown that as a consequence the effective axial-current neutrino-nucleon scattering cross section is reduced. Here, we calculate this reduction explicitly in the perturbative limit $\Gamma_\sigma\ll T$. By virtue of an exact sum rule of the spin-density structure function, we express the modified cross section in terms of the matrix element for neutrino-nucleon scattering in the presence of a spin-dependent nuclear potential. This representation allows for a direct comparison with and confirmation of Sawyer's related perturbative result. In a supernova core with a typical temperature $T=10\,\rm MeV$, the perturbative limit is relevant for densities $\rho\alt10^{13}\,\rm g\,cm^{-3}$ and thus applies around the neutrino sphere. There, the cross-section reduction is of order a few percent and thus not large; however, a new mode of energy transfer between neutrinos and nucleons is enabled which may be important for neutrino spectra formation. We derive an analytic perturbative expression for the rate of energy transfer.