  We introduce a finite dimensional anharmonic soft spin glass in a field and
show how it allows the construction a field theory at zero temperature and the
corresponding loop expansion. The mean field level of the model coincides with
a recently introduced fully connected model, the KHGPS model, and it has a spin
glass transition in a field at zero temperature driven by the appearance of
pseudogapped non-linear excitations. We analyze the zero temperature limit of
the theory and the behavior of the bare masses and couplings on approaching the
mean field zero temperature critical point. Focusing on the so called replicon
sector of the field theory, we show that the bare mass corresponding to
fluctuations in this sector is strictly positive at the transition in a certain
region of control parameter space. At the same time the two relevant cubic
coupling constants $g_1$ and $g_2$ show a non-analytic behavior in their bare
values: approaching the critical point at zero temperature, $g_1\to \infty$
while $g_2\propto T$ with a prefactor diverging at the transition. Along the
same lines we also develop the field theory to study the density of states of
the model in finite dimension. We show that in the mean field limit the density
of states converges to the one of the KHGPS model. However the construction
allows a treatment of finite dimensional effects in perturbation theory.
