Strong decays of $P_ψ^N(4312)^+\to J/ψ(η_c) p$ within the Bethe-Salpeter framework

Qiang Li, Chao-Hsi Chang, Tianhong Wang, Guo-Li Wang

Published 2023-01-05Version 1

Based on the effective Lagrangian in the heavy quark limit, we calculate the one-boson-exchange interaction kernel of $\bar D\Sigma_c$ in the isospin-$\frac12$ state. We present the Bethe-Salpeter equation and wave function for the constituent particles to be a (pseudo)scalar meson and a $\frac12$ baryon. By using the Bethe-Salpeter equation, we can obtain $P_\psi^N(4312)^+$ as the $\bar D\Sigma_c$ molecular state with $J^P=(\frac{1}{2})^-$. Combining the effective Lagrangian and the obtained BS wave function, the partial decay widths of $P_\psi^N(4312)^+$ to $J/\psi p$ and $\eta_c p$ are calculated to be $0.11^{-0.04}_{+0.06}$MeV and $5.6^{-1.9}_{+2.6}\times10^{-2}$MeV, respectively, which are consistent with the LHCb experimental measurements. Our results indicate the fraction of $J/\psi p$ channel amounts to $\sim1\%$ of $P_\psi^N(4312)^+$, and can reach to $\sim30\%$ when the interaction kernel is reduced by half. Our results favor the interpretation of $P_\psi^N(4312)^+$ as the $\bar D\Sigma_c$ molecular state with $J^P=(\frac{1}{2})^-$ and isospin $I=\frac12$.