  Plasmons in ordinary electron liquids are collective excitations whose
long-wavelength limit is rigid center-of-mass motion with a dispersion relation
that is, as a consequence of Galileian invariance, unrenormalized by many-body
effects. The long-wavelength plasmon frequency is related by the f-sum rule to
the integral of the conductivity over the electron-liquid's Drude peak,
implying that transport properties also tend not to have important
electron-electron interaction renormalizations. In this article we demonstrate
that the plasmon frequency and Drude weight of the electron liquid in a doped
graphene sheet, which is described by a massless Dirac Hamiltonian and not
invariant under ordinary Galileian boosts, are strongly renormalized even in
the long-wavelength limit. This effect is not captured by the Random Phase
Approximation (RPA), commonly used to describe electron fluids. It is due
primarily to non-local inter-band exchange interactions, which, as we show,
reduce both the plasmon frequency and the Drude weight relative to the RPA
value. Our predictions can be checked using inelastic light scattering or
infrared spectroscopy.
