Bridging correlation and spectroscopy measurements to access the hadron interaction behind molecular states: the case of the $Ξ$(1620) and $Ξ$(1690) in the $K^- Λ$ system
We study the compatibility between the $K^-Λ$ correlation function, recently measured by the ALICE collaboration, and the LHCb $K^-Λ$ invariant mass distribution obtained in the $Ξ^-_b \to J/ψΛK^-$ decay. The $K^-Λ$ invariant mass distribution associated with the $Ξ^-_b$ decay has been calculated within the framework of Unitary Effective Field Theories using two models, one of them constrained by the $K^-Λ$ correlation function. We consider two degenerate pentaquark $P_{cs}$ states in the $J/ψΛ$ scattering amplitude which allows us to investigate their impact on both the $K^-Λ$ and $J/ψΛ$ mass distributions assuming different spin-parity quantum numbers and multiplicity. Without any fitting procedure, the $K^-Λ$ model is able to better reproduce the experimental $K^-Λ$ 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 $K^-Λ$ distribution in the region close to the threshold, largely dominated by the presence of the still poorly known $Ξ$(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 $πΞ$ and $\bar{K}Σ$.