  Utilizing the Baym-Kadanoff formalism with the polarization function
calculated in the random phase approximation, the dynamics of the $\nu=0$
quantum Hall state in bilayer graphene is analyzed. Two phases with nonzero
energy gap, the ferromagnetic and layer asymmetric ones, are found. The phase
diagram in the plane $(\tilde{\Delta}_0,B)$, where $\tilde{\Delta}_0$ is a
top-bottom gates voltage imbalance, is described. It is shown that the energy
gap scales linearly, $\Delta E\sim 14 B[T]K, with magnetic field.
