Mass generation for the two dimensional O(N) Linear Sigma Model in the large N limit

Abstract

This work studies the $O(N)$ Linear Sigma Model on ${\mathbb{R}}^2$ under a scaling dictated by the formal $1/N$ expansion. We show that in the large $N$ limit, correlations decay exponentially fast, where the acquired mass decays exponentially in the inverse temperature. In fact, each marginal converges to a massive Gaussian Free Field (GFF) on ${\mathbb{R}}^2$, quantified in the $2$-Wasserstein distance with a weighted $H^1({\mathbb{R}}^2)$ cost function. In contrast to prior work on the torus via parabolic stochastic quantization, our results hold without restrictions on the coupling constants, allowing us to also obtain a massive GFF in a suitable double scaling limit. Our proof combines the Feyel/Üstünel extension of Talagrand's inequality with some classical tools in Euclidean Quantum Field Theory.