Studying $\Lambda $ interactions in nuclear matter with the $^{\mathrm{208}}_{\mathrm{\Lambda }}$Pb(e,e'K$^{\mathrm{+}})^{\mathrm{208}}_{\mathrm{\Lambda }}$Tl reaction

In view of astrophysical implications, a proposal of an experiment aiming at measuring $\Lambda $ binding energies of several energy levels of the hypernucleus $^{\mathrm{208}}_{\mathrm{\Lambda }}$Tl was submitted to JLab Program Advisory Committee. $^{\mathrm{208}}_{\mathrm{\Lambda }}$Tl hypernuclei will be produced through the $^{\mathrm{208}}_{\mathrm{\Lambda }}$Pb(e,e'K$^{\mathrm{+}})^{\mathrm{208}}_{\mathrm{\Lambda }}$Tl reaction. Thanks to the extended region of constant density and the large neutron excess, $^{\mathrm{208}}$Pb provides the best available proxy of neutron star matter. The experiment will provide essential information, needed to constrain and improve the available models of Hyperon-Nucleon and Hyperon-Nucleon-Nucleon potentials and therefore confirm whether hyperonic three-body forces could be the solution to the so-called hyperon-puzzle, the impossibility of reconciling the existence of very massive (up to two solar masses) neutron stars with the presence of hyperons in the interior of these objects.