$Δ(1232)$ isobars and neutrino opacities in hot dense matter
We calculated the diffusion mean free paths of electron neutrinos and antineutrinos in hot, neutrino-trapped matter containing $Δ(1232)$ isobars, as found in proto-neutron stars and neutron star merger remnants. We included charged-current absorption with the $NΔ$ weak vertex of weak pion production, neutral-current scattering on all baryons and three-body antineutrino absorption on electron--baryon pairs, with relativistic mean-field kinematics. The reaction $\barν_e+e^-+n\toΔ^-$ dominates the antineutrino opacity whenever the neutron and electron chemical potentials exceed the in-medium $Δ^-$ rest energy, even where the $Δ$ population is negligible. At two to four times saturation density, where the reaction is open, it shortens the antineutrino mean free path from \nubarLamNucMin--\nubarLamNucMax~m to \nubarLamDMin--\nubarLamDMax~cm. Because antineutrinos are scarce at high lepton fraction, this changes lepton-number diffusion only at low lepton fraction, by up to \DtwoRedHi\% at these densities and by a factor of \DtwoRedLow\ at 1.5 times saturation density. For electron neutrinos, absorption on the $Δ^-$ and, at low lepton fraction, the opening of nucleonic absorption shorten the mean free path by factors of up to \nuRatioYLthreeMax\ at lepton fraction 0.3 and up to about \nuRatioYLoneMax\ at lepton fraction 0.1. We also showed that, for a given axial coupling, the spin of the $Δ$ reduces its Gamow--Teller strength per particle from 3 to $5/3$.