Charged current neutrino interactions in hot and dense matter

We derive the charged current absorption rate of electron and antielectron neutrinos in dense matter using a fully relativistic approach valid at arbitrary matter degeneracy. We include mean field energy shifts due to nuclear interactions and the corrections due to weak magnetism. The rates are derived both from the familiar Fermi's golden rule and from the techniques of finite temperature field theory, and are shown to be equivalent. In various limits, these results can also be used to calculate neutral current opacities. We find that some pieces of the response have been left out in previous derivations and their contribution at high density can be significant. Useful formulas and detailed derivations are presented and we provide a new open-source implementation of these opacities for use in radiation hydrodynamic simulations of core-collapse supernovae and neutron star mergers.

Figure 1

Contours of the ${\nu }_e+n\to e^{-}+p$ double-differential absorption rate as a function of electron scattering angle ${\mu }_{13}$ and energy transfer $q_0$, normalized to the integrated absorption rate.

Figure 2

The angle integrated cross section as a function of energy transfer $q_0$ for ${\nu }_e+n\to e^{-}+p$. The left panel shows the differential cross section at a density of $0.001{\mathrm{fm}}^{-3}$ and the right panel shows it at a density of $0.1{\mathrm{fm}}^{-3}$. The dotted black lines shows the position of the energy transfer peak when final state blocking is not included, $U_4-U_2+M_4^{*}-M_2^{*}$. The solid lines show our full expressions for the cross section, the dot dashed lines show the full expressions neglecting tensor corrections, and the dashed lines show the [10] prescription for the cross section.

Figure 3

Comparison of various approximations to the charged current cross section per nucleon as a function of density in beta-equilibrated matter. The reference cross section, ${\sigma }_{\mathrm{Ref}}$ is taken to be the full cross section without weak magnetism corrections (i.e., $F_2=0.0$). The neutrino energy is assumed to be $\pi T$. The left panel shows the reaction ${\nu }_e+n\to e^{-}+p$ while the right panel shows ${\overline{\nu }}_e+p\to e^{+}+n$. The electron antineutrino cross sections are only shown up to threshold. Full denotes the fully relativistic cross section including weak magnetism corrections, $\vec{q}=0$ is the cross section given by Eq. (76), RPL98 denotes the cross section given by the expressions in [10], and Constant ${\mathrm{\Lambda }}^{\mu \nu }$ refers to the cross section given by Eq. (74).

Figure 4

The ${\nu }_e+n\to e^{-}+p$ mean free path as a function of total density for neutrino energies $\{3,10,33,100\}\mathrm{MeV}$. The full cross section (solid lines), the full cross section neglecting weak magnetism (dashed lines), the expression from [10] for the cross section (dot-dashed lines), the nonrelativistic elastic limit (dotted lines), and the constant ${\mathrm{\Lambda }}_{\mu \nu }$ matrix element limit (dot-dot-dashed lines) of the cross section are all shown. The left panel shows the results for a free gas of protons and neutrons, while the right panel shows the results for a gas with a mean field potential as described in the text. At low density, all of the approximations agree reasonably well (although there are significant deviations for higher energy neutrinos), but at higher density and for finite $\mathrm{\Delta }U$ they diverge significantly.

Figure 5

Variation of the charged current neutrino mean free path with the nucleon effective masses at saturation density. The nucleon effective mass is assumed to be isospin independent and matter is assumed to be in beta equilibrium. The neutrino energy is taken to be $E_{\nu }=\pi T$. Full denotes the fully relativistic cross section including weak magnetism corrections, RPL98 denotes the cross section given by the expressions in [10], and Constant ${\mathrm{\Lambda }}^{\mu \nu }$ refers to the cross section given by Eq. (74).