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.

Comments: 13 pages, 7 figures, comments are welcome