Cross sections for 2-to-3 meson-meson scattering

We study 2-to-3 meson-meson scattering based on the process that a gluon is created from a constituent quark or antiquark and subsequently the gluon creates a quark-antiquark pair. The transition potential for the process is derived in QCD. Eight Feynman diagrams at tree level are involved in the 2-to-3 meson-meson scattering. Starting from the $S$-matrix element, we derive the unpolarized cross section from the eight transition amplitudes corresponding to the eight Feynman diagrams. The transition amplitudes contain color, spin, and flavor matrix elements. The 2-to-3 meson-meson scattering includes $\pi \pi \to \pi K\bar{K}$, $\pi K \to \pi \pi K$, $\pi K \to KK\bar{K}$, $KK \to \pi KK$, and $K\bar{K} \to \pi K\bar{K}$. Cross sections for the reactions are calculated. The cross sections depend on temperature obviously, and the cross section for $\pi K \to \pi\pi K$ for total isospin $I=3/2$ at zero temperature is compared to experimental data. By comparison with inelastic 2-to-2 meson-meson scattering, we find that 2-to-3 meson-meson scattering may be as important as inelastic 2-to-2 meson-meson scattering.