Strong decays of $P_{\psi}^N(4312)^+$ to $J/\psi(\eta_c) p$ and $\bar D^{(*)}\Lambda_c$ within the Bethe-Salpeter framework

Based on the effective Lagrangian in the heavy quark limit, we calculate the one-boson-exchange interaction kernel of $P_\psi^N(4312)^+$ as the $\bar D\Sigma_c$ molecular state in isospin-$\frac12$. We present the Bethe-Salpeter equation and wave function for the constituent particles to be a (pseudo)scalar meson and a $\frac12$ baryon. By solving the Bethe-Salpeter equation, we 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 $J/\psi p$, $\eta_c p$, $\bar D^{*0}\Lambda_c^+$ and $\bar D^0\Lambda_c^+$ are calculated to be $0.17$, $0.085$, $8.8$, and $0.026$ MeV, respectively, which are roughly consistent with the LHCb experimental measurements and some other theoretical researches. The obtain results indicate the fraction of $\bar D^{*0}\Lambda_c^+$ channel amounts to $\sim90\%$ of $P_\psi^N(4312)^+$, and is a highly promising channel to be discovered in the near future experiments. 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 $I=\frac12$.