  We extend and apply a rigorous renormalisation group method to study critical
correlation functions, on the 4-dimensional lattice $\mathbb{Z}^4$, for the
weakly coupled $n$-component $|\varphi|^4$ spin model for all $n \geq 1$, and
for the continuous-time weakly self-avoiding walk. For the $|\varphi|^4$ model,
we prove that the critical two-point function has $|x|^{-2}$ (Gaussian) decay
asymptotically, for $n \ge 1$. We also determine the asymptotic decay of the
critical correlations of the squares of components of $\varphi$, including the
logarithmic corrections to Gaussian scaling, for $n \geq 1$. The above extends
previously known results for $n = 1$ to all $n \ge 1$, and also observes new
phenomena for $n > 1$, all with a new method of proof. For the continuous-time
weakly self-avoiding walk, we determine the decay of the critical generating
function for the "watermelon" network consisting of p weakly mutually- and
self-avoiding walks, for all $p \ge 1$, including the logarithmic corrections.
This extends a previously known result for $p = 1$, for which there is no
logarithmic correction, to a much more general setting. In addition, for both
models, we study the approach to the critical point and prove existence of
logarithmic corrections to scaling for certain correlation functions. Our
method gives a rigorous analysis of the weakly self-avoiding walk as the $n =
0$ case of the $|\varphi|^4$ model, and provides a unified treatment of both
models, and of all the above results.
