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

Matías G. Delgadino, Scott A. Smith

TL;DR

The paper addresses mass generation in the two-dimensional $O(N)$ Linear Sigma Model under a $1/N$ expansion, proving exponential decay of correlations and that each component converges to a massive Gaussian Free Field in the large-$N$ limit. The authors combine the Feyel/Üstünel extension of Talagrand's inequality with classical Euclidean QFT tools, including Wick renormalization and Moshe–Kupiainen-type gap equations, to obtain uniform-in-$N$ ultraviolet stability and quantitative Wasserstein-distance convergence to a massive GFF. A key result is that the plane model admits a mass $m_*$ given by a gap equation, and the marginals converge to $\mu_{m_*}^{\otimes N}$ with rate $O(N^{-1/2})$ in an appropriate Sobolev-based Wasserstein metric; in a double scaling limit, the mass scales as $m_* = e^{-2\pi\beta}$, matching Polyakov-type predictions. The work thus provides a rigorous large-$N$ mass-generation mechanism in 2D sigma models, supports continuum-limit behavior, and offers quantitative transport-entropy tools that may influence rigorous QFT analyses and 1/N expansions.

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.

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

TL;DR

The paper addresses mass generation in the two-dimensional Linear Sigma Model under a expansion, proving exponential decay of correlations and that each component converges to a massive Gaussian Free Field in the large- limit. The authors combine the Feyel/Üstünel extension of Talagrand's inequality with classical Euclidean QFT tools, including Wick renormalization and Moshe–Kupiainen-type gap equations, to obtain uniform-in- ultraviolet stability and quantitative Wasserstein-distance convergence to a massive GFF. A key result is that the plane model admits a mass given by a gap equation, and the marginals converge to with rate in an appropriate Sobolev-based Wasserstein metric; in a double scaling limit, the mass scales as , matching Polyakov-type predictions. The work thus provides a rigorous large- mass-generation mechanism in 2D sigma models, supports continuum-limit behavior, and offers quantitative transport-entropy tools that may influence rigorous QFT analyses and 1/N expansions.

Abstract

This work studies the Linear Sigma Model on under a scaling dictated by the formal expansion. We show that in the large 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 , quantified in the -Wasserstein distance with a weighted 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.
Paper Structure (15 sections, 27 theorems, 231 equations)

This paper contains 15 sections, 27 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Large $N$ Limit of The Linear Sigma Model on the Torus
  4. The Massive Gaussian Free Field
  5. Uniform in $N$ Ultraviolet Stability
  6. Large N Limit
  7. Large N Limit of the Linear Sigma Model on the Plane
  8. The Gap Equation
  9. Statement of the Main Result
  10. Scaling Optimal Bounds on the Wasserstein Distance
  11. Proof of the Main Theorem
  12. Some elementary integral estimates
  13. Talagrand's inequality for infinite dimensional Gaussian Measures
  14. Stroock's formula
  15. Construction and Properties of the Linear Sigma Model on the Torus

Key Result

Theorem 1.1

For all $\lambda>0$ and $\beta \geq 0$, there exists a mass $m_{*}=m_{*}(\lambda,\beta)$ and a constant $C=C(\lambda,\beta)$ independent of $N$ with the following property. Any infinite volume limit $\nu^{N}=\nu^{N,\lambda,\beta}$ of the Linear Sigma Model on $\mathbb{R}^{2}$ obtained from periodic for all $i \in [N]$ and suitable cylindrical functionals $F: \mathcal{D}'(\mathbb{R}^{2}) \mapsto \

Theorems & Definitions (68)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • ...and 58 more