Role of electromagnetic corrections in the $\pi\pi$ distributions of $\psi^\prime \to J/\psi \pi \pi$

Abstract

The cusp structure at the $\pi^+\pi^-$ threshold in the $\pi^0\pi^0$ invariant mass spectrum serves as a sensitive probe for extracting the $S$-wave $\pi\pi$ scattering lengths in processes where an $S$-wave $\pi^0\pi^0$ pair is produced in the final states. Within the framework of nonrelativistic effective field theory with coupled channels $\pi^0\pi^0$ and $\pi^+\pi^-$, we revisit the near-threshold structures in the $\pi^0\pi^0$ spectrum of $\psi^\prime \to J/\psi \pi\pi$. Our analysis incorporates the $\pi \pi$ final-state rescattering, including both strong and Coulomb interactions. It turns out that the cusp near the $\pi^+\pi^-$ threshold becomes more prominent when Coulomb interactions are included. The electromagnetic correctionsare found to alter the magnitude of the threshold cusp by about 2%-3%, underscoring the necessity of including these effects in precision determinations of the $\pi\pi$ scattering lengths. The coupled-channel amplitude constructed in this work provides a ready-to-use theoretical framework for experimental analyses of fine structures near $\pi\pi$ thresholds.