Scattering lengths of the $J/ψπ$ and $J/ψK$ systems

TL;DR

We address the problem of near-threshold interactions between $J/\psi$ and light pseudoscalars $\pi$ and $K$ using dispersion relations, separating two OZI-suppressed mechanisms: multi-gluon exchange and open-charm coupled channels. An explicit chiral-symmetry-breaking mixing term $c_{m}^{(\alpha\beta)}$ is diagonalized by a rotation with angle $\theta$, yielding physical $J/\psi$ and $\psi'$ and redefined couplings $\tilde{c}_{i}^{(\alpha\beta)}$; the diagonalized Lagrangian preserves contact interactions while keeping $\tilde{c}_{m}^{(12)}$ nonzero, which can be constrained by $\psi'\to J/\psi\pi\pi$. From the diagonalized theory, we compute the $S$-wave scattering amplitudes at threshold via a dispersion relation with Muskhelishvili–Omnès final-state interactions, and extract the scattering lengths through crossing, obtaining $a_{J/\psi\pi}\lesssim -0.0021$ fm and $a_{J/\psi K}\lesssim -0.028$ fm. The results show that soft-gluon exchange dominates near threshold while coupled-channel effects are subleading, with tiny open-charm contributions, a pattern that aligns with analogous findings in $J/\psi$–nucleon scattering and may reflect a universal gluon-driven mechanism in charmonium–hadron interactions; these predictions are amenable to lattice QCD validation.

Abstract

We investigate the low-energy interactions between the charmonium state $J/ψ$ and the light pseudoscalar mesons ($π$ 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 $J/ψ$ and $ψ'$ states. Following this diagonalization, we relate the threshold scattering amplitudes of $J/ψ{\cal P}~({\cal P}=π,K)$, which determine the scattering lengths, to the $J/ψJ/ψ\to {\cal P}\bar{\cal P}$ amplitudes via crossing symmetry. The ${\cal P}\bar{\cal P}$ rescattering effects are incorporated using dispersion relations. We obtain the $S$-wave scattering lengths $a_{J/ψπ} \lesssim -0.0021$~fm and $a_{J/ψK} \lesssim -0.028$~fm, where the negative sign indicates an attractive interaction. Our results show that the $J/ψK$ 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 $J/ψπ$ couples to $D\bar{D}^*$ and $J/ψK$ couples to $D^*\bar{D}_s/D\bar{D}_s^*$, reveals that both $J/ψπ$ and $J/ψK$ scatterings are predominantly governed by the soft-gluon exchange mechanism.

Figures (3)

• Figure 1: Unitary relation for the scattering amplitudes of processes $J/\psi J/\psi \to {\cal P}\bar{\cal P}$. The black dots and squares denote the full $J/\psi J/\psi \to {\cal P}\bar{\cal P}$ amplitudes and the $\pi \pi$-$K\bar{K}$ coupled-channel rescattering, respectively. The red dashed line signifies the cut.
• Figure 2: Real (blue solid) and imaginary (red dashed) parts of the $S$-wave $\pi\pi$-$K\bar{K}$ coupled-channel Omnès matrix elements from Refs. Hoferichter:2012wfCao:2024zlf.
• Figure 3: Real (blue solid) and imaginary (red dashed) parts of the $S$-wave scattering amplitudes for the processes $J/\psi J/\psi \to {\cal P}\bar{\cal P}\ ({\cal P} = \pi,K)$, where $\alpha_{11} = \alpha_{12}$ is taken for definiteness.