Strong Diffusion Effect of Charm Quarks on $J/\psi$ Production in Pb-Pb collisions at the LHC

We study the $J/\psi$ production based on coalescence model at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV Pb-Pb collisions. With the colliding energy increasing from 2.76 TeV to 5.02 TeV, the number of charm pairs is enhanced by more than 50%. However, the ratio of $J/\psi$ inclusive nuclear modification factors $R^{5.02TeV}_{AA}/R^{2.76TeV}_{AA}$ is only about 1.1 $\sim$ 1.2. We find that the regeneration of $J/\psi$ is proportional to the densities of charm and anti-charm quarks, instead of their total numbers. The charm quark density is diluted by the strong expansion of quark gluon plasma, which suppresses the combination probability of heavy quarks and $J/\psi$ regeneration. This effect is more important in higher colliding energies where QGP expansion is strong. We also propose the ratio $N_{J/\psi}/(N_D)^2$ as a measurement of $c$ and $\bar c$ coalescence probability, which is only affected by the heavy quark diffusions in QGP, and does not depend on the inputs such as cold nuclear matter effects and cross sections of charm quark production. Further more, we give the predictions at the energy of Future Circular Collider ($\sqrt{s_{NN}}$ = 39 TeV).