Finite-Order Correlation Length for Four-Dimensional Weakly Self-Avoiding Walk and $${|\varphi|^4}$$ Spins

We study the four-dimensional n-component ({|varphi|^4}) spin model for all integers ({n ge 1}) and the four-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case ({n=0}) interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power ({frac 12frac{n+2}{n+8}}), for all ({n ge 0}) and ({p u003e 0}). The proof is based on an improvement of a rigorous renormalisation group method developed previously.