  Using many-body diagrammatic perturbation theory we consider carrier density-
and substrate-dependent many-body renormalization of doped or gated graphene
induced by Coulombic electron-electron interaction effects. We quantitatively
calculate the many-body spectral function, the renormalized quasiparticle
energy dispersion, and the renormalized graphene velocity using the
leading-order self-energy in the dynamically screened Coulomb interaction
within the ring diagram approximation. We predict experimentally detectable
many-body signatures, which are enhanced as the carrier density and the
substrate dielectric constant are reduced, finding an intriguing instability in
the graphene excitation spectrum at low wave vectors where interaction
completely destroys all particle-like features of the noninteracting linear
dispersion. We also make experimentally relevant quantitative predictions about
the carrier density and wave-vector dependence of graphene velocity
renormalization induced by electron-electron interaction. We compare on-shell
and off-shell self-energy approximations within the ring diagram approximation,
finding a substantial quantitative difference between their predicted velocity
renormalization corrections in spite of the generally weak-coupling nature of
interaction in graphene.
