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Production of $ψ(4040)$, $ψ(4160)$, and $ψ(4415)$ mesons in hadronic matter

We present the first study of production of $ψ(4040)$, $ψ(4160)$, and $ψ(4415)$ mesons in hadronic matter. Quark interchange between two colliding charmed mesons leads to the production of the three mesons. We calculate unpolarized cross sections for the reactions, $D\bar{D} \to ρR$, $D\bar{D}^* \to πR$, $D\bar{D}^* \to ρR$, $D^*\bar{D} \to πR$, $D^*\bar{D} \to ρR$, $D^*\bar{D}^* \to πR$, and $D^*\bar{D}^* \to ρR$, where $R$ stands for $ψ(4040)$, $ψ(4160)$, and $ψ(4415)$. In the temperature region covering hadronic matter the peak cross sections of producing $ψ(4040)$ are similar to or larger than the ones of producing $ψ(4415)$, and the latter are generally larger than those of producing $ψ(4160)$. With the cross sections we establish new master rate equations for $ψ(4040)$, $ψ(4160)$, and $ψ(4415)$. The equations are solved for central Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV at the Large Hadron Collider. Solutions of the equations show that the $ψ(4040)$ number density at kinetic freeze-out of hadronic matter is larger than the $ψ(4415)$ number density which is larger than the $ψ(4160)$ number density.