  We analyze the many-body properties of bilayer graphene (BLG) at charge
neutrality, governed by long range interactions between electrons. Perturbation
theory in a large number of flavors is used in which the interactions are
described within a random phase approximation, taking account of dynamical
screening effect. Crucially, the dynamically screened interaction retains some
long range character, resulting in $\log^2$ renormalization of key quantities.
We carry out the perturbative renormalization group calculations to one loop
order, and find that BLG behaves to leading order as a marginal Fermi liquid.
Interactions produce a log squared renormalization of the quasiparticle residue
and the interaction vertex function, while all other quantities renormalize
only logarithmically. We solve the RG flow equation for the Green function with
logarithmic accuracy, and find that the quasiparticle residue flows to zero
under RG. At the same time, the gauge invariant quantities, such as the
compressibility, remain finite to $\log^2$ order, with subleading logarithmic
corrections. The key experimental signature of this marginal Fermi liquid
behavior is a strong suppression of the tunneling density of states, which
manifests itself as a zero bias anomaly in tunneling experiments in a regime
where the compressibility is essentially unchanged from the non-interacting
value.
