Description
At the critical temperature $T_{\rm c}$ hadronization of quark-gluon plasma
gives hadronic matter that evolves until kinetic freeze-out. Hadronic matter is
strongly interacting matter. How to explore hadronic matter? $\psi(4040)$,
$\psi(4160)$ and $\psi(4415)$ mesons can be individually interpreted as the
$3 ^3S_1$, $2 ^3D_1$ and $4 ^3S_1$ quark-antiquark states. The three charmonia
are dissolved when the temperature is larger than $0.97T_{\rm c}$,
$0.95T_{\rm c}$ and $0.87T_{\rm c}$, respectively. In ultrarelativistic
heavy-ion collisions the three charmonia created in initial nucleus-nucleus
collisions are thus dissolved in quark-gluon plasmas, but they can be produced
by quark interchange between two colliding charmed mesons in hadronic matter.
From the three charmonia we can study hadronic matter and temperature-dependent
strong interactions of quarks or mesons. Therefore, we suggest the production
of $\psi (4040)$, $\psi (4160)$ and $\psi (4415)$ as probes of hadronic matter
that results from the quark-gluon plasma created in ultrarelativistic heavy-ion
collisions. Charmed mesons have been well measured in Pb-Pb collisions at the
Large Hadron Collider. It is shown by our theoretical studies that numbers
of $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$ produced in a central Pb-Pb
collision at the center-of-mass energy per nucleon-nucleon pair
$\sqrt{s_{NN}}=5.02$ TeV are 0.25, 0.1 and 0.18, respectively. Therefore, we
suggest measurements of $\psi (4040)$, $\psi (4160)$ and $\psi (4415)$ mesons
in Pb-Pb collisions at the Large Hadron Collider. This is a new experimental goal. Details can be found in the two papers: L.-Y. Li, X.-M. Xu and H. J. Weber,
Phys. Rev. D105 (2022) 114025; W.-X. Li, X.-M. Xu and H. J. Weber, Eur. Phys. J. C81 (2021) 225.