Low-energy scattering of the $J/\psi \pi$ and $J/\psi K$ system

Abstract

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 ${\psi }^{\prime }$ states. Using the resulting diagonalized Lagrangian, we construct the crossed-channel amplitudes for $J/\psi J/\psi \to \mathcal{P}\bar{\mathcal{P}}$ and incorporate the $\pi \pi$ and $K\bar{K}$ rescattering effects through dispersion relations. This framework is used consistently both in the phenomenological extraction of the transition parameters from ${\psi }^{\prime }\to J/\psi \pi \pi$ and in the continuation of the crossed amplitudes to the near-threshold $J/\psi \mathcal{P}(\mathcal{P}=\pi ,K)$ region. As a result, we determine both the scattering lengths and the effective ranges. We obtain the upper-bound estimates $a_{J/\psi \pi }\lesssim -0.0037$~fm and $a_{J/\psi K}\lesssim -0.049$~fm, where the negative sign indicates an attractive interaction without a bound state in our convention. 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 $J/\psi \pi$ and $J/\psi K$ couple to open-charm channels, reveals that both $J/\psi \pi$ and $J/\psi K$ scatterings are predominantly governed by the soft-gluon exchange mechanism.

Comments: 12 pages, 6 figures