  We establish a statistical two-body fractal (STF) model to study the spectrum
of $J/\psi$. $J/\psi$ serves as a reliable probe in heavy-ion collisions. The
distribution of $J/\psi$ in hadron gas is influenced by flow, quantum and
strong interaction effects. Previous models have predominantly focused on one
or two of these effects while neglecting the others, resulting in the inclusion
of unconsidered effects in the fitted parameters. Here, we study the issue from
a new point of view by analyzing the fact that all three effects induce a
self-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 of
self-similarity and strong interaction between \emph{c }and $\bar{c}$ on
quarks. By solving the probability and entropy equations, we derive the values
of $q_{TBS}$ and $q_2$ at various collision energies and centralities.
Substituting the value of $q_{TBS}$ into distribution function, we successfully
obtain the transverse momentum spectrum of low-$p_T$ $J/\psi$, which
demonstrates good agreement with experimental data. The STF model can be
employed to investigate other mesons and resonance states.
