  Broken symmetry states in bilayer graphene in perpendicular electric
$E_\perp$ and in-plane magnetic $B_\parallel$ fields are studied in the
presence of the dynamically screened long-range Coulomb interaction and the
symmetry-breaking contact four-fermion interactions. The integral gap equations
are solved numerically, and it is shown that the momentum dependence of gaps is
essential: It diminishes by an order of magnitude the gaps compared to the case
of momentum-independent approximation, and the obtained gap magnitudes are
found to agree well with existing experimental values. We derived a phase
diagram of bilayer graphene at the neutrality point in the plane
$(B_\parallel,E_\perp)$ showing that the (canted) layer antiferromagnetic (LAF)
state remains a stable ground state of the system at large $B_\parallel$. On
the other hand, while the LAF phase is realized at small values of $E_{\perp}$,
the quantum valley Hall (QVH) phase is the ground state of the system at values
$E_{\perp}>E_{cr}(B_\parallel)$, where a critical value $E_{cr}(B_\parallel)$
increases with in-plane magnetic field $B_{||}$.
