The density- and isospin- dependent $\Delta$-formation cross section and its decay width

The energy-, density-, and isospin-dependent $\Delta$-formation cross section $\sigma_{N\pi \rightarrow \Delta}^*$ and $\Delta$-decay width are calculated based on the relativistic BUU approach in which the effective mass splitting of nucleon and $\Delta$ baryons in isospin-asymmetric matter is considered by the inclusion of the $\delta$ meson exchange in the effective Lagrangian density. With the density-dependent couplings for baryons of F. Hofmann et al., the $\sigma_{N\pi \rightarrow \Delta}^*$ is decreased (increased) moderately with increasing density with (without) the consideration of the density dependent pion effective mass. Meanwhile, if the invariant mass of the system is not far from the $\Delta$ pole mass, the $\Delta$-decay width is also weakly dependent on density. The mass splitting effect of differently charged nucleon and $\Delta$ baryons on $\sigma_{N\pi \rightarrow \Delta}^*$ are found to be more obvious than that of pion mesons but much weaker than the mass splitting in the hard $\Delta$ production channel $NN\rightarrow N\Delta$. Further, the largest mass-splitting influence is seen in the $\pi^-p\rightarrow \Delta^0$ and $\pi^+n\rightarrow \Delta^+$ channels but not in the production of $\Delta^-$ and $\Delta^{++}$ isobars.