Finite-order correlation length of the $$|\varphi|^4$$ spin model in four dimensions

The correlation length of order $$p$$ for the $$|\varphi|^4$$ spin model (a continuous-spin version of the O(n) model) is a normalization of the $$p$$-th moment of its two-point function. We will outline the proof (based on a renormalisation group method of Bauerschmidt, Brydges, and Slade) that, in the upper-critical dimension 4, this quantity undergoes mean-field scaling with a logarithmic correction as the critical point for this model is approached from above (for sufficiently weak coupling). Via a supersymmetric integral representation, this result also extends to the weakly self-avoiding walk with a contact attraction, for which the correlation length of order $$p$$ is closely related to the mean $$p$$-th displacement of the walk. This is joint work with Roland Bauerschmidt, Gordon Slade, and Alexandre Tomberg.