  We calculate the density-density response function and electron self-energy
for undoped bilayer graphene, within the Random Phase Approximation (RPA). We
show that the quasiparticle decay rate scales linearly with the quasiparticle
energy, and quasiparticle weight vanishes logarithmically in the low-energy
limit, indicating non-Fermi liquid behavior. This is a consequence of the
absence of a Fermi surface for neutral bilayer graphene and corresponding
larger phase space available for scattering processes. Experimental
consequences of our results as well as their differences from those of
single-layer graphene are discussed.
