Cross sections for inelastic meson-meson scattering

We study two kinds of inelastic meson-meson scattering. The first kind is inelastic 2-to-2 meson-meson scattering that is governed by quark interchange as well as quark-antiquark annihilation and creation. Cross-section formulas are provided to get unpolarized cross sections for $\pi K \to \rho K^\ast$ for $I=1/2$, $\pi K^\ast \to \rho K$ for $I=1/2$, $\pi K^\ast \to \rho K^\ast$ for $I=1/2$, and $\rho K \to \rho K^\ast$ for $I=1/2$. Near threshold, quark interchange dominates the reactions near the critical temperature. The second kind is 2-to-1 meson-meson scattering with the process that a quark in an initial meson and an antiquark in another initial meson annihilate into a gluon and subsequently the gluon is absorbed by the other antiquark or quark. The transition potential for the process is derived. Four Feynman diagrams at tree level contribute to the 2-to-1 meson-meson scattering. Starting from the $S$-matrix element, the isospin-averaged unpolarized cross section with transition amplitudes is derived. The cross sections for $\pi \pi \to \rho$ and $\pi K \to K^*$ decrease with increasing temperature.