  We prove $|x|^{-2}$ decay of the critical two-point function for the
continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper
critical dimension $d=4$. This is a statement that the critical exponent $\eta$
exists and is equal to zero. Results of this nature have been proved previously
for dimensions $d \geq 5$ using the lace expansion, but the lace expansion does
not apply when $d=4$. The proof is based on a rigorous renormalisation group
analysis of an exact representation of the continuous-time weakly self-avoiding
walk as a supersymmetric field theory. Much of the analysis applies more widely
and has been carried out in a previous paper, where an asymptotic formula for
the susceptibility is obtained. Here, we show how observables can be
incorporated into the analysis to obtain a pointwise asymptotic formula for the
critical two-point function. This involves perturbative calculations similar to
those familiar in the physics literature, but with error terms controlled
rigorously.
