  We prove that the susceptibility of the continuous-time weakly self-avoiding
walk on $\mathbb{Z}^d$, in the critical dimension $d=4$, has a logarithmic
correction to mean-field scaling behaviour as the critical point is approached,
with exponent 1/4 for the logarithm. The susceptibility has been well
understood previously for dimensions $d \geq 5$ using the lace expansion, but
the lace expansion does not apply when $d=4$. The proof begins by rewriting the
walk two-point function as the two-point function of a supersymmetric field
theory. The field theory is then analysed via a rigorous renormalisation group
method developed in a companion series of papers. By providing a setting where
the methods of the companion papers are applied together, the proof also serves
as an example of how to assemble the various ingredients of the general
renormalisation group method in a coordinated manner.
