  We consider the weakly coupled $\phi^4 $ theory on $\mathbb Z^4 $, in a weak
magnetic field $h$, and at the chemical potential $\nu_c $ for which the theory
is critical if $h=0$. We prove that, as $h\to 0$, the magnetization of the
model behaves as $(h\log h^{-1})^{\frac 13} $, and so exhibits a logarithmic
correction to mean field scaling behavior. This result is well known to
physicists, but had never been proven rigorously. Our proof uses the classic
construction of the critical theory by Gawedzki and Kupiainen, and a cluster
expansion with large blocks.
