Cross sections for 2-to-1 meson-meson scattering

We study the processes $K\bar{K} \to \phi$, $\pi D \to D^\ast$, $\pi \bar{D} \to \bar{D}^\ast$, and the production of $\psi (4160)$ and $\psi (4415)$ mesons in collisions of charmed mesons or charmed strange mesons. The 2-to-1 meson-meson scattering involves a process where a quark and an antiquark from the two initial mesons annihilate into a gluon and subsequently the gluon is absorbed by the spectator quark or antiquark. Transition amplitudes for the scattering process derive from the transition potential in conjunction with mesonic quark-antiquark wave functions and the relative-motion wave function of the two initial mesons. We derive these transition amplitudes in the partial wave expansion of the relative-motion wave function of the two initial mesons so that parity and total-angular-momentum conservation are maintained. We calculate flavor and spin matrix elements in accordance with the transition potential and unpolarized cross sections for the reactions using the transition amplitudes. Cross sections for the production of $\psi (4160)$ and $\psi (4415)$ generally increase as the colliding mesons go through the cases of $D\bar{D}$, $D^*\bar{D}$, and $D^*\bar{D}^*$ or the cases of $D_s^+D_s^-$, $D_s^{*+}D_s^-$, and $D_s^{*+}D_s^{*-}$. We suggest the production of $\psi (4160)$ and $\psi (4415)$ as a probe of hadronic matter that results from the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.