The double thresholds distort the lineshapes of the P_(psi s)^Λ(4338)⁰ resonance

Very recently, the LHCb Collaboration reported the first observation of the hidden charm pentaquark with strangeness, $P_{\psi s}^\Lambda(4338)^0$. Considering this state is very close to the $\Xi_c^0\bar{D}^0$ and $\Xi_c^+D^-$ thresholds, we explore the possible bias of the Breit-Wigner parameterization, with emphasis on the effect of its coupling to the double thresholds $\Xi_c^0\bar{D}^0$ and $\Xi_c^+D^-$. 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 (LSE) formalism (equivalent to the $K$-matrix parameterization) with two models, the zero-range model and Flatt\'e model to investigate the $J/\psi \Lambda$ lineshapes. Our results show that the nominal peak of the $P_{\psi s}^\Lambda(4338)^0$ could arise either from the pole well above the $\Xi_c^+ D^-$ threshold on the $(-,+)$ sheet or from the pole well below the $\Xi_c^0 \bar{D}^0$ threshold on the $(-,-)$ sheet in the two-channel system. Using the Breit-Wigner distribution to depict the above two lineshapes could be misleading. We also find a novel type of lineshapes 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 parameterization. As a by-product, we obtain that the ratio of the isospin violating decay ${\Gamma_{P_{\psi s}^{\Lambda}\to J/\psi\Sigma}}/{\Gamma_{P_{\psi s}^{\Lambda}\to J/\psi\Lambda}}$ could be up to $10\%$.