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Abstract

By combining the effective Lagrangian and Bethe-Salpeter framework, we studied the mass spectra, wave functions, and strong decay widths of the two pentaquark states $P_\psi^N(4440)^+$ and $P_\psi^N(4457)^+$ reported by LHCb in 2019. Taking into account both the mass ordering and the decay widths, our results favor the interpretation of $P_\psi^N(4440)^+$ and $P_\psi^N(4457)^+$ as the isospin-$\frac12$ $[\bar D^*\Sigma_c]$ molecular states with $J^P$ configuration $(\frac{3}{2})^-$ and $(\frac12)^-$, respectively. We first calculate the one-boson-exchange interaction kernel of $[\bar D^*\Sigma_c]$ in the isospin-$\frac12$ configuration. Then we present the Bethe-Salpeter equation\,(BSE) and wave functions for the bound states of a vector meson and a $\frac12$ baryon with $J^P={\frac12}^-$ and ${\frac32}^-$. The obtained mass results for the $(\frac32)^-$ and $(\frac12)^-$ are $4.442$ and $4.457$ GeV, respectively. Combining the effective Lagrangians and the BS wave functions, we further calculate the strong decay channels $\bar D^{(*)0}\Lambda_c^+$, $J/\psi(\eta_c) p$, and $\bar D\Sigma_c^{(*)}$ for the two $P_\psi^N$ states. In the favored $\frac32^-$ and $\frac12^-$ configuration, the obtained total widths are $21.8$ MeV and $13.0$ MeV, respectively, which are substantially consistent with the LHCb data. Our results suggest that $\bar D^{0}\Lambda_c^+$ and $\bar D^{(*)0}\Lambda_c^+$ are the dominant decay channels to detect $P_\psi^N(4440)^+$ and $P_\psi^N(4457)^+$, respectively.