Scaling Limits and Critical Behaviour of the 4-Dimensional n-Component |\varphi |^4 Spin Model

We consider the \(n\) -component \(|\varphi |^4\) spin model on \({\mathbb {Z}}^4\) , for all \(n \ge 1\) , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\) ; double logarithmic scaling for \(n=4\) ; and is bounded when \(n>4\) . In addition, for the model defined on the \(4\) -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi |^4\) model.