Pole determination of $P_{\psi s}^\Lambda(4338)$ and possible $P_{\psi s}^\Lambda(4255)$ in $B^-\to J/\psi\Lambda\bar{p}$

First hidden-charm pentaquark candidate with strangeness, $P_{\psi s}^\Lambda(4338)$, was recently discovered in $B^-\to J/\psi\Lambda\bar{p}$ by the LHCb Collaboration. $P_{\psi s}^\Lambda(4338)$ shows up as a bump at the $\Xi_c\bar{D}$ threshold in the $J/\psi\Lambda$ invariant mass ($M_{J/\psi\Lambda}$) distribution. The $M_{J/\psi\Lambda}$ distribution also shows a large fluctuation at the $\Lambda_c\bar{D}_s$ threshold, hinting the existence of a possible $P_{\psi s}^\Lambda(4255)$. In this work, we determine the $P_{\psi s}^\Lambda(4338)$ and $P_{\psi s}^\Lambda(4255)$ pole positions for the first time. For this purpose, we fit a $B^-\to J/\psi\Lambda\bar{p}$ model to the $M_{J/\psi\Lambda}$, $M_{J/\psi\bar{p}}$, $M_{\Lambda\bar{p}}$, and $\cos\theta_{K^*}$ distributions from the LHCb simultaneously; $\chi^2/{\rm ndf}\sim 1.21$. Then we extract $P_{\psi s}^\Lambda$ poles from a unitary $\Xi_c\bar{D}$-$\Lambda_c\bar{D}_s$ coupled-channel scattering amplitude built in the model. In our default fit, the $P_{\psi s}^\Lambda(4338)$ pole is found at $( 4338.2\pm 1.4)-( 1.9\pm 0.5 )\,i$ MeV while the $P_{\psi s}^\Lambda(4255)$ pole at $4254.7\pm 0.4$ MeV. The $P_{\psi s}^\Lambda(4338)$ and $P_{\psi s}^\Lambda(4255)$ are mostly $\Xi_c \bar{D}$ bound and $\Lambda_c\bar{D}_s$ virtual states, respectively. Through our analysis, the data disfavors a hypothesis of $P_{\psi s}^\Lambda(4338)$ as merely a kinematical effect. This pole determination, which is important in its own right, sets a primary basis to study the nature of the $P_{\psi s}^\Lambda$ states.