  The spin-glass transition in a field in finite dimension is analyzed directly
at zero temperature using a perturbative loop expansion around the Bethe
lattice solution. The loop expansion is generated by the $M$-layer construction
whose first diagrams are evaluated numerically and analytically. The
generalized Ginzburg criterion reveals that the upper critical dimension below
which mean-field theory fails is $D_U \le 8$, at variance with the classical
result $D_U = 6$ yielded by finite-temperature replica field theory. Our
expansion around the Bethe lattice has two crucial differences with respect to
the classical one. The finite connectivity $z$ of the lattice is directly
included from the beginning in the Bethe lattice, while in the classical
computation the finite connectivity is obtained through an expansion in $1/z$.
Moreover, if one is interested in the zero temperature ($T = 0$) transition,
one can directly expand around the $T = 0$ Bethe transition. The expansion
directly at $T = 0$ is not possible in the classical framework because the
fully connected spin glass does not have a transition at $T = 0$, being in the
broken phase for any value of the external field.
