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

TL;DR

This paper analyzes the large N limit of the 2D O(N) Linear Sigma Model, showing exponential decay of correlations and convergence to a massive Gaussian Free Field without restrictions on coupling constants.

Contribution

It demonstrates the emergence of a massive Gaussian Free Field in the large N limit of the 2D O(N) Linear Sigma Model, extending previous results to all coupling constants.

Findings

Correlations decay exponentially fast in the large N limit.

Marginals converge to a massive Gaussian Free Field.

Results hold without restrictions on coupling constants.

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/\"Ust\"unel extension of Talagrand's inequality with some classical tools in Euclidean Quantum Field Theory.