Bridging correlation and spectroscopy measurements to access the hadron interaction behind molecular states: the case of the $\Xi$(1620) and $\Xi$(1690) in the $K^- \Lambda$ system

We study the compatibility between the $K^-\Lambda$ correlation function, recently measured by the ALICE collaboration, and the LHCb $K^-\Lambda$ invariant mass distribution obtained in the $\Xi^-_b \to J/\psi \Lambda K^-$ decay. The $K^-\Lambda$ invariant mass distribution associated with the $\Xi^-_b$ decay has been calculated within the framework of Unitary Effective Field Theories using two models, one of them constrained by the $K^-\Lambda$ correlation function. We consider two degenerate pentaquark $P_{cs}$ states in the $J/\psi \Lambda$ scattering amplitude which allows us to investigate their impact on both the $K^-\Lambda$ and $J/\psi \Lambda$ mass distributions assuming different spin-parity quantum numbers and multiplicity. Without any fitting procedure, the $K^-\Lambda$ model is able to better reproduce the experimental $K^-\Lambda$ mass spectrum in the energy region above $1680$ MeV as compared to previous unitarized scattering amplitudes constrained to a large amount of experimental data in the neutral $S=-1$ meson-baryon sector. We observe a tension between our model and the LHCb $K^-\Lambda$ distribution in the region close to the threshold, largely dominated by the presence of the still poorly known $\Xi$(1620) state. We discuss in detail the different production mechanisms probed via femtoscopy and spectroscopy that could provide valid explanations for such disagreement, indicating the necessity to employ future correlation data in other $S=-2$ channels such as $\pi\Xi$ and $\bar{K}\Sigma$.