Strong decays of $P_{\psi }^N(4312{)}^{+}$ to $J\mathrm{/}\psi ({\eta }_c)p$ and ${\bar{D}}^{(\ast )}{\mathrm{\Lambda }}_c$ 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 $\bar{D}{\mathrm{\Sigma }}_c$ molecular state in isospin-$\frac{1}{2}$. We present the Bethe-Salpeter equation and wave function for the constituent particles to be a (pseudo)scalar meson and a $\frac{1}{2}$ baryon. By solving the Bethe-Salpeter equation, we obtain $P_{\psi }^N(4312{)}^{+}$ as the $\bar{D}{\mathrm{\Sigma }}_c$ 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 $J\mathrm{/}\psi p$, ${\eta }_{c}p$, ${\bar{D}}^{\ast 0}{\mathrm{\Lambda }}_c^{+}$ and ${\bar{D}}^0{\mathrm{\Lambda }}_c^{+}$ 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 ${\bar{D}}^{\ast 0}{\mathrm{\Lambda }}_c^{+}$ channel amounts to $\sim 90\mathrm{\%}$ 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 $\bar{D}{\mathrm{\Sigma }}_c$ molecular state with $J^P=(\frac{1}{2}{)}^{-}$ and isospin $I=\frac{1}{2}$.