The first exploration of the physical Riemann surfaces of the ratio $G_E^\Lambda/G_M^\Lambda$

Recently, the BESIII experiment renewed the interest on baryon form factors by measuring the modulus and phase of the ratio $G_E^\Lambda/G_M^\Lambda$ between the electric and the magnetic $\Lambda$ form factors with unprecedented accuracy. The BESIII measurement together with older, less precise data, can be analyzed by means of a dispersive procedure based on analyticity and a set of first-principle constraints. Such a dispersive procedure shows the unique ability to determine for the first time the complex structure of the ratio knowing its modulus and phase measured by the BESIII collaboration at only one energy point. Different classes of solutions are obtained, and in all cases, the time-like and space-like behaviors show interesting properties: space-like zeros or unexpected large determinations for the phase. More data at different energies would be crucial to enhance the predictive power of the dispersive procedure and to unravel further remarkable features of the $\Lambda$ baryon.