  We reconsider Ising spins in a Gaussian random field within the replica
formalism. The corresponding continuum model involves several coupling
constants beyond the single one which was considered in the standard $\phi^4$
theory approach. These terms involve more than one replica, and therefore in a
mean field theory they do not contribute to the zero-replica limit. However the
fluctuations involving those extra terms are singular on the Curie line below
eight dimensions, and by the time one reaches the dimension six, it is
necessary to keep them in the renormalization group analysis. As a result it is
found that there is no stable fixed point of order $(6-d)$. Whether this means
that there is no expansion in powers of $(6-d)$, or that the transition is
driven to first order by these fluctuations, is difficult to decide at this
level, but it explains the failure of the $(d, d-2)$ correspondence. In this
work we have not considered the possibility of a glassy phase in the would-be
paramagnetic region, but in a subsequent work we shall present evidence for
replica symmetry breaking in dimension lower than six.

Condensed Matter