  We calculate the single-particle spectral function for doped bilayer graphene
in the low energy limit, described by two parabolic bands with zero band gap
and long range Coulomb interaction. Calculations are done using thermal Green's
functions in both the random phase approximation (RPA) and the fully
self-consistent GW approximation. RPA (in line with previous studies) yields a
spectral function which apart from the Landau quasiparticle peaks shows
additional coherent features interpreted as plasmarons, i.e. composite
electron-plasmon excitations. In GW the plasmaron becomes incoherent and peaks
are replaced by much broader features. The deviation of the quasiparticle
weight and mass renormalization from their non-interacting values is small
which indicates that bilayer graphene is a weakly interacting system. The
electron energy loss function, $Im[-\epsilon^{-1}_q(\omega)]$ shows a sharp
plasmon mode in RPA which in GW approximation becomes less coherent and thus
consistent with the weaker plasmaron features in the corresponding
single-particle spectral function.
