  Carrier doping by the electric field effect has emerged recently as an ideal
route for monitoring many-body physics in two-dimensional (2D) materials where
the Fermi level is tuned in a way that -- indirectly -- the strength of the
interactions can also be scanned. The possibility of systematic doping in
combination with high resolution photoemission has allowed to uncover a
genuinely many-body electron spectrum in single-layer MoS2 transition metal
dichalcogenide, resolving three clear quasi-particle states, where only one
state should be expected if the electron-phonon interaction vanished. Our
analysis combines first-principles and consistent complex plane analytic
approaches and brings into light the presence and the physical origin of two
gaps and the three quasi-particle bands which are unambiguously present in the
photoemission spectrum. One of these states, though being strongly interacting
with the accompanying virtual phonon cloud, presents a notably long lifetime
which is an appealing property when trying to understand and take advantage of
many-body interactions to modulate the transport properties.
