Double thresholds distort the line shapes of the PψsΛ(4338)0 resonance

Abstract

Very recently, the LHCb Collaboration reported the first observation of the hidden-charm pentaquark with strangeness, ${P}_{\ensuremath{\psi}s}^{\mathrm{\ensuremath{\Lambda}}}(4338{)}^{0}$. Considering this state is very close to the ${\mathrm{\ensuremath{\Xi}}}_{c}^{0}{\overline{D}}^{0}$ and ${\mathrm{\ensuremath{\Xi}}}_{c}^{+}{D}^{\ensuremath{-}}$ thresholds, we explore the possible bias of the Breit-Wigner parametrization, with emphasis on the effect of its coupling to the double thresholds ${\mathrm{\ensuremath{\Xi}}}_{c}^{0}{\overline{D}}^{0}$ and ${\mathrm{\ensuremath{\Xi}}}_{c}^{+}{D}^{\ensuremath{-}}$. We first use a qualitative picture based on the ``uniformization'' of the Riemann surface of the two-channel system to understand the positions of the enhancement. Then we use the Lippmann-Schwinger equation formalism (equivalent to the $K$-matrix parametrization) with two models, the zero-range model and the Flatt\'e model to investigate the $J/\ensuremath{\psi}\mathrm{\ensuremath{\Lambda}}$ line shapes. Our results show that the nominal peak of the ${P}_{\ensuremath{\psi}s}^{\mathrm{\ensuremath{\Lambda}}}(4338{)}^{0}$ could arise either from the pole well above the ${\mathrm{\ensuremath{\Xi}}}_{c}^{+}{D}^{\ensuremath{-}}$ threshold on the $(\ensuremath{-},+)$ sheet or from the pole well below the ${\mathrm{\ensuremath{\Xi}}}_{c}^{0}{\overline{D}}^{0}$ threshold on the $(\ensuremath{-},\ensuremath{-})$ sheet in the two-channel system. Using the Breit-Wigner distribution to depict the above two line shapes could be misleading. We also find a novel type of line shapes with the enhancement constrained by the threshold difference. We urge the LHCb Collaboration to perform the refined experimental analysis considering the unitarity and analyticity, e.g., using the $K$-matrix parametrization. As a by-product, we obtain that the ratio of the isospin violating decay ${\mathrm{\ensuremath{\Gamma}}}_{{P}_{\ensuremath{\psi}s}^{\mathrm{\ensuremath{\Lambda}}}\ensuremath{\rightarrow}J/\ensuremath{\psi}\mathrm{\ensuremath{\Sigma}}}/{\mathrm{\ensuremath{\Gamma}}}_{{P}_{\ensuremath{\psi}s}^{\mathrm{\ensuremath{\Lambda}}}\ensuremath{\rightarrow}J/\ensuremath{\psi}\mathrm{\ensuremath{\Lambda}}}$ could be up to 10%.