Strong decays of $P_{\psi }^N(4440{)}^{+}$ and $P_{\psi }^N(4457{)}^{+}$ within the Bethe-Salpeter framework

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-$\frac{1}{2}$ $[{\bar{D}}^{\ast }{\Sigma }_c]$ molecular states with $J^P$ configuration $(\frac{3}{2}{)}^{-}$ and $(\frac{1}{2}{)}^{-}$, respectively. We first calculate the one-boson-exchange interaction kernel of $[{\bar{D}}^{\ast }{\Sigma }_c]$ in the isospin-$\frac{1}{2}$ configuration. Then we present the Bethe-Salpeter equation (BSE) and wave functions for the bound states of a vector meson and a $\frac{1}{2}$ baryon with $J^P={\frac{1}{2}}^{-}$ and ${\frac{3}{2}}^{-}$. The obtained mass results for the $(\frac{3}{2}{)}^{-}$ and $(\frac{1}{2}{)}^{-}$ are $4.442$ and $4.457$ GeV, respectively. Combining the effective Lagrangians and the BS wave functions, we further calculate the strong decay channels ${\bar{D}}^{(\ast )0}{\Lambda }_c^{+}$, $J/\psi ({\eta }_c)p$, and $\bar{D}{\Sigma }_c^{(\ast )}$ for the two $P_{\psi }^N$ states. In the favored ${\frac{3}{2}}^{-}$ and ${\frac{1}{2}}^{-}$ 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 ${\bar{D}}^{(\ast )0}{\Lambda }_c^{+}$ are the dominant decay channels to detect $P_{\psi }^N(4440{)}^{+}$ and $P_{\psi }^N(4457{)}^{+}$, respectively.

54 pages, 6 figures, 8 tables. Accepted by JHEP

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