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We establish a statistical two-body fractal (STF) model to study the spectrumof $J/\psi$. $J/\psi$ serves as a reliable probe in heavy-ion collisions. Thedistribution of $J/\psi$ in hadron gas is influenced by flow, quantum andstrong interaction effects. Previous models have predominantly focused on oneor two of these effects while neglecting the others, resulting in the inclusionof unconsidered effects in the fitted parameters. Here, we study the issue froma new point of view by analyzing the fact that all three effects induce aself-similarity structure, involving a $J/\psi$-$\pi$ two-meson state and a$J/\psi$, $\pi$ two-quark state, respectively. We introduce modification factor$q_{TBS}$ and $q_2$ into the probability and entropy of charmonium. $q_{TBS}$denotes the modification of self-similarity on $J/\psi$, $q_2$ denotes that ofself-similarity and strong interaction between \emph{c }and $\bar{c}$ onquarks. By solving the probability and entropy equations, we derive the valuesof $q_{TBS}$ and $q_2$ at various collision energies and centralities.Substituting the value of $q_{TBS}$ into distribution function, we successfullyobtain the transverse momentum spectrum of low-$p_T$ $J/\psi$, whichdemonstrates good agreement with experimental data. The STF model can beemployed to investigate other mesons and resonance states.

The Spectrum of Low-$p_T$ $J/\psi$ in Heavy-Ion Collisions in a Statistical Two-Body Fractal Model