How to extract the $\Lambda N$ scattering length from production reactions

A dispersion integral is derived that allows one to relate directly (spin dependent) $\Lambda N$ invariant mass spectra, measured in a large-momentum transfer reaction such as $pp\to K^+p\Lambda$ or $\gamma d\to K^+n\Lambda$, to the scattering length for elastic $\Lambda N$ scattering. The involved systematic uncertainties are estimated to be smaller than 0.3 fm. This estimate is confirmed by comparing results of the proposed formalism with those of microscopic model calculations. We also show, for the specific reaction $pp\to K^+\Lambda p$, how polarization observables can be used to separate the two spin states of the $\Lambda N$ system.