21 small values of η are. And of course, in order to perform a correct analysis of such effects, one should only compare results for either spectral functions or linear conductance obtained numerically with the same value of η . real part of the interacting self-energy ℜ e Σ r e -vib = 0 at ω = 0 by definition because it is an odd function of ω . Our electron-vibron interacting system conserves its Fermi-liquid properties at zero temperature and equilib- rium, as also obtained for interacting electron systems (see for example Refs.[88–91]). We expect that such a behaviour also hold for other electron-vibron diagrams than the Fock diagram. Out of equilibrium, the system may lose its conventional Fermi-liquid properties, as we have already shown in the main part of the paper. In conclusion, we want to stress that by introducing a finite, but tiny, value for the imaginary part η , one might not get the exact linear conductance value. 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