Scattering lengths of the $J/\psi \pi$ and $J/\psi K$ systems

We investigate the low-energy interactions between the charmonium state $J/\psi$ and the light pseudoscalar mesons ($\pi$ and $K$) within the framework of dispersion relations.We demonstrate that the symmetry-breaking terms in the chiral Lagrangian induce mixing between the bare charmonium fields, necessitating a diagonalization procedure to correctly identify the physical $J/\psi$ and $\psi'$ states. Following this diagonalization, we relate the threshold scattering amplitudes of $J/\psi{\cal P}~({\cal P}=\pi,K)$, which determine the scattering lengths, to the $J/\psi J/\psi \to {\cal P}\bar{\cal P}$ amplitudes via crossing symmetry. The ${\cal P}\bar{\cal P}$ rescattering effects are incorporated using dispersion relations. We obtain the $S$-wave scattering lengths $a_{J/\psi\pi} \lesssim -0.0021$~fm and $a_{J/\psi K} \lesssim -0.028$~fm, where the negative sign indicates an attractive interaction. Our results show that the $J/\psi K$ interaction is moderately enhanced relative to the pion channel, driven by explicit chiral symmetry breaking. Furthermore, a quantitative comparison of the coupled-channel mechanism, where $J/\psi\pi$ couples to $D\bar{D}^*$ and $J/\psi K$ couples to $D^*\bar{D}_s/D\bar{D}_s^*$, reveals that both $J/\psi\pi$ and $J/\psi K$ scatterings are predominantly governed by the soft-gluon exchange mechanism.