Abstract

We discuss the approach to the critical point of the $phi^4$ model in 4 dimensions. One of the major successes of the renormalization group technique has been to explain why this model features logarithmic corrections to the scaling predictions for the blow up of thermodynamic quantities. We review the strategy of the proof in the “symmetric regime” with zero external magnetic field, in which case this is a classic result. We then present the proof of logarithmic corrections to the magnetization as the magnetic field tends to zero. Despite being a central aspect of the model, these have been an open problem until now, probably because technical complications where expected due to the broken symmetry. We have found these concerns to be unfounded, and our proof only needs a single cluster expansion on top of the classic RG construction for the critical point.