  The competing ground states of bilayer graphene are studied by applying
renormalization group techniques to a bilayer honeycomb lattice with nearest
neighbor hopping. In the absence of interactions, the Fermi surface of this
model at half-filling consists of two nodal points with momenta $\mathbf{K}$,
$\mathbf{K}'$, where the conduction band and valence band touch each other,
yielding a semi-metal. Since near these two points the energy dispersion is
quadratic with perfect particle-hole symmetry, excitonic instabilities are
inevitable if inter-band interactions are present. Using a perturbative
renormalization group analysis up to the one-loop level, we find different
competing ordered ground states, including ferromagnetism, superconductivity,
spin and charge density wave states with ordering vector
$\mathbf{Q}=\mathbf{K}-\mathbf{K}'$, and excitonic insulator states. In
addition, two states with valley symmetry breaking are found in the excitonic
insulating and ferromagnetic phases. This analysis strongly suggests that the
ground state of bilayer graphene should be gapped, and with the exception of
superconductivity, all other possible ground states are insulating.
