Strong decays of $P_ψ^N(4440)^+$ and $P_ψ^N(4457)^+$ within the Bethe-Salpeter framework

TL;DR

This work investigates the nature of the LHCb-observed pentaquarks $P_ψ^N(4440)^+$ and $P_ψ^N(4457)^+$ by modeling them as isospin-$\frac{1}{2}$ $[\bar{D}^*\Sigma_c]$ molecular states. It develops a relativistic Bethe-Salpeter framework with an instantaneous, one-boson-exchange kernel derived from heavy-quark and chiral Lagrangians, solving for the $J^P=\frac{3}{2}^-$ and $\frac{1}{2}^-$ bound states and their wave functions. By combining the BS wave functions with effective Lagrangians for hadronic decays, the paper computes strong decay widths into eight channels, finding total widths of $Γ(P_ψ^N(4440)^+) \approx 21.8$ MeV and $Γ(P_ψ^N(4457)^+) \approx 13.0$ MeV in the favored $(\tfrac{3}{2}^-,\tfrac{1}{2}^-)$ assignment, in good agreement with LHCb values. The dominant decays are $\bar{D}^0\Lambda_c^+$ for $P_ψ^N(4440)^+$ and $\bar{D}^{*0}\Lambda_c^+$ or $\bar{D}\Sigma_c$ for $P_ψ^N(4457)^+$, supporting the molecular interpretation and providing testable predictions for experimental searches.

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 $P_ψ^N(4440)^+$ and $P_ψ^N(4457)^+$ reported by LHCb in 2019. Taking into account both the mass ordering and the decay widths, our results favor the interpretation of $P_ψ^N(4440)^+$ and $P_ψ^N(4457)^+$ as the isospin-$\frac12$ $[\bar D^*Σ_c]$ molecular states with $J^P$ configuration $(\frac{3}{2})^-$ and $(\frac12)^-$, respectively. We first calculate the one-boson-exchange interaction kernel of $[\bar D^*Σ_c]$ 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 $\bar D^{(*)0}Λ_c^+$, $J/ψ(η_c) p$, and $\bar DΣ_c^{(*)}$ for the two $P_ψ^N$ 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 $\bar D^{0}Λ_c^+$ and $\bar D^{(*)0}Λ_c^+$ are the dominant decay channels to detect $P_ψ^N(4440)^+$ and $P_ψ^N(4457)^+$, respectively.

Figures (4)

• Figure 1: Bethe-Salpeter equation of the molecular states consisting of a vector meson and a spin-$\frac{1}{2}$ baryon. The red letters denote the Lorentz indices; the blue letters $P,~p_1(k_1),~p_2(k_2)$ denote the momenta of the pentaquark, constituent meson and baryon, respectively.
• Figure 2: Strong decay of $P_\psi^N[\bar{D}^*\Sigma_c]^+$ to the $J/\psi p$ by exchanging a virtual mediator $D$ (left panel) and $D^*$ (right panel). $P,~k_1,~k_2, ~P_1,~P_2$ denote the momenta of $P_\psi^N$, constituent meson, constituent baryon, the final $J/\psi$, and the final $p$ respectively.
• Figure 3: The interaction kernel $K_i~(i=1,\cdots,3)$ in the isospin-$\frac{1}{2}$ with $m_\Lambda=0.64\,GeV$.
• Figure 4: The BS wave functions of $P_{\psi1/2}^N$ with mass $4.457\,GeV$ (left) and $P_{\psi3/2}^N$ with mass $4.442\,GeV$ (right), where two bound states masses are solved with $m_\Lambda=0.64\,GeV$.