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We study the 4-dimensional $n$-component $|\varphi|^4$ spin model for allintegers $n \ge 1$, and the 4-dimensional continuous-time weakly self-avoidingwalk which corresponds exactly to the case $n=0$ interpreted as asupersymmetric spin model. For these models, we analyse the correlation lengthof order $p$, and prove the existence of a logarithmic correction to mean-fieldscaling, with power $\frac 12\frac{n+2}{n+8}$, for all $n \ge 0$ and $p>0$. Theproof is based on an improvement of a rigorous renormalisation group methoddeveloped previously.

Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|\varphi|^4$ spins