Revealing the temperature effect on the nucleon-nucleon inelastic cross section in isospin-asymmetric nuclear medium

The nucleon-nucleon ($NN$) inelastic cross section plays an important role in constraining the nuclear equation of state at high baryon density and in describing the formation and evolution of compact astrophysical objects. In this study, the temperature $T$ dependence of the $\Delta^{++}$ and $\Delta^{-}$ production cross sections in the isospin-symmetric and -asymmetric nuclear medium is investigated within the self-consistent and relativistic Boltzmann--Uehling--Uhlenbeck (RBUU) framework. Two relativistic mean-field parameterizations are employed: the density-dependent parameterization (called DD-ME$\delta$) and the nonlinear-dependent parameterization (called OMEG). Both parameterizations yield similar $T$-dependent baryon effective masses and mass splittings, although the OMEG set exhibits a stronger density dependence, particularly at higher densities ($> 1.5\rho_{0}$). Consequently, at lower densities, the energy, density, temperature, and isospin dependence of both $\Delta^{++}$ and $\Delta^{-}$ production cross sections are comparable for both sets, whereas at higher densities, the OMEG set predicts a stronger temperature and density sensitivity. Moreover, the $T$ dependence of the $NN$ inelastic cross section is enhanced with increasing density, but is suppressed in isospin-asymmetric nuclear matter compared to that in isospin-symmetric nuclear matter. The isospin dependence of the cross section remains nearly $T$-independent at small asymmetries, yet becomes more intricate in highly asymmetric systems. These findings provide valuable testing inputs for improving the thermal treatment of $\Delta$-related dynamical processes in transport models and offer insights into the behavior of $\Delta$ in astrophysical environments, such as core-collapse supernovae and binary neutron star mergers.