Cross sections for inelastic meson-meson scattering via quark-antiquark annihilation

We study inelastic meson-meson scattering that is governed by quark-antiquark annihilation and creation involving a quark and an antiquark annihilating into a gluon, and subsequently the gluon creating another quark-antiquark pair. The resultant hadronic reactions include for $I=1$: $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\rho}$, $K\overline{K}\ensuremath{\rightarrow}{K}^{*}{\overline{K}}^{*}$, $K{\overline{K}}^{*}\ensuremath{\rightarrow}{K}^{*}{\overline{K}}^{*}$, ${K}^{*}\overline{K}\ensuremath{\rightarrow}{K}^{*}{\overline{K}}^{*}$, as well as $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}K\overline{K}$, $\ensuremath{\pi}\ensuremath{\rho}\ensuremath{\rightarrow}K{\overline{K}}^{*}$, $\ensuremath{\pi}\ensuremath{\rho}\ensuremath{\rightarrow}{K}^{*}\overline{K}$, and $K\overline{K}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\rho}$. In each reaction, one or two Feynman diagrams are involved in the Born approximation. We derive formulas for the unpolarized cross section, the transition amplitude, and the transition potential for quark-antiquark annihilation and creation. The unpolarized cross sections for the reactions are calculated at six temperatures, and prominent temperature dependence is found. It is due to differences among mesonic temperature dependence in hadronic matter.