  We consider a model of half-filled bilayer graphene, in which the three
dominant Slonczewski-Weiss-McClure hopping parameters are retained, in the
presence of short range interactions. Under a smallness assumption on the
interaction strength $U$ as well as on the inter-layer hopping $\epsilon$, we
construct the ground state in the thermodynamic limit, and prove its
analyticity in $U$, uniformly in $\epsilon$. The interacting Fermi surface is
degenerate, and consists of eight Fermi points, two of which are protected by
symmetries, while the locations of the other six are renormalized by the
interaction, and the effective dispersion relation at the Fermi points is
conical. The construction reveals the presence of different energy regimes,
where the effective behavior of correlation functions changes qualitatively.
The analysis of the crossover between regimes plays an important role in the
proof of analyticity and in the uniform control of the radius of convergence.
The proof is based on a rigorous implementation of fermionic renormalization
group methods, including determinant estimates for the renormalized expansion.
