Role of electromagnetic corrections in the $ππ$ distributions of $ψ^\prime \to J/ψππ$

Abstract

The cusp structure at the $π^+π^-$ threshold in the $π^0π^0$ invariant mass spectrum serves as a sensitive probe for extracting the $S$-wave $ππ$ scattering lengths in processes where an $S$-wave $π^0π^0$ pair is produced in the final states. Within the framework of nonrelativistic effective field theory with coupled channels $π^0π^0$ and $π^+π^-$, we revisit the near-threshold structures in the $π^0π^0$ spectrum of $ψ^\prime \to J/ψππ$. Our analysis incorporates the $ππ$ final-state rescattering, including both strong and Coulomb interactions. It turns out that the cusp near the $π^+π^-$ 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 $ππ$ scattering lengths. The coupled-channel amplitude constructed in this work provides a ready-to-use theoretical framework for experimental analyses of fine structures near $ππ$ thresholds.

Figures (7)

• Figure 1: Resummation in the Coulomb $T$ matrix for the $\pi^+\pi^-$ scattering. The wavy line represents a photon, and the blue blob represents resummation of the Coulomb photon exchanges.
• Figure 2: Diagrams for $\psi^\prime \to J/\psi \pi^0\pi^0$ (a-c) and $\psi^\prime \to J/\psi \pi^+\pi^-$ (d-f). The red filled circles represent the pointlike production sources $\mathrm{V_n}$ and $\mathrm{V_c}$ for $J/\psi \pi^0\pi^0$ and $J/\psi \pi^+\pi^-$ production, respectively. The black filled squares represent the $\pi\pi$ interactions described by the solution of the LSE in Eq. \ref{['Eq_TSCsl']}. The blue blob represents infinite Coulomb-photon exchanges between $\pi^+$ and $\pi^-$ as shown in Fig. \ref{['fig:Coulomb_resum']}. The Coulomb-photon resummation between the final-state $\pi^+$ and $\pi^-$ is accounted for by the $W_C$ factor in the amplitude for $\psi^\prime \to J/\psi \pi^+\pi^-$ given in Eq. \ref{['Eq_Mcc']}. The amplitude for $\psi^\prime \to J/\psi \pi^0\pi^0$ is given in Eq. \ref{['Eq_Mnn']}.
• Figure 3: The invariant mass distribution of the $\pi^0\pi^0$ pair for $\psi^\prime \to J/\psi \pi^0\pi^0$. The blue and red curves represent the distributions with and without the Coulomb effect, respectively, using the central values of $(a_0-a_2) M_{\pi^+}$ and $a_2 M_{\pi^+}$, $\alpha^{21}$ and $\kappa$. The error bands are dominated by the errors of $\alpha^{21}$ and $\kappa$. The insets enlarge the line shape with and without Coulomb effects in the very near-threshold region. The gray vertical dot-dashed line denotes the $\pi^+\pi^-$ threshold.
• Figure 4: Ratio of the difference $\Delta\mathrm{d}\Gamma/\mathrm{d} m_{\pi\pi}$ between the $\pi^0\pi^0$ invariant mass distributions with and without Coulomb interactions (denoted as "wc" and "woc", respectively) to the invariant mass distribution without Coulomb interactions.
• Figure 5: The invariant mass distribution of the $\pi^+\pi^-$ pair for $\psi^\prime \to J/\psi \pi^+\pi^-$. The blue and red line shapes represent the distributions with and without the Coulomb interactions, respectively.