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.