Abstract
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 \( \overline{D}{\Sigma}_c \) 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 \( \overline{D}{\Sigma}_c \) molecular state with JP = \( {\left(\frac{1}{2}\right)}^{-} \). Combining the effective Lagrangian and the obtained BS wave function, the partial decay widths of \( {P}_{\psi}^N{(4312)}^{+} \) to J/ψp, ηcp, \( {\overline{D}}^{\ast 0}{\Lambda}_c^{+} \) and \( {\overline{D}}^0{\Lambda}_c^{+} \) 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 \( {\overline{D}}^{\ast 0}{\Lambda}_c^{+} \) channel amounts to ~ 90% 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 \( \overline{D}{\Sigma}_c \) molecular state with JP = \( {\left(\frac{1}{2}\right)}^{-} \) and isospin I = \( \frac{1}{2} \).
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Acknowledgments
The author Q. Li thanks Prof. Fen-Kun Guo of ITP-CAS, and Dr. Xu-Chang Zheng of Chongqing Univ., and Dr. Hao Xu of Northwest Normal Univ. for helpful suggestions and discussions. This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 12005169, 12075301, 11821505, 12047503, and 12075073. It is also supported by the National Key R&D Program of China (2022YFA1604803), the Natural Science Basic Research Program of Shaanxi (Program No. 2021JQ-074), and the Fundamental Research Funds for the Central Universities.
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Li, Q., Chang, CH., Wang, T. et al. Strong decays of \( {P}_{\psi}^N{(4312)}^{+} \) to J/ψ(ηc)p and \( {\overline{D}}^{\left(\ast \right)}{\Lambda}_c \) within the Bethe-Salpeter framework. J. High Energ. Phys. 2023, 189 (2023). https://doi.org/10.1007/JHEP06(2023)189
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DOI: https://doi.org/10.1007/JHEP06(2023)189