Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Operator product expansions of derivative fields in the sine-Gordon model

Alex Karrila, Tuomas Virtanen, Christian Webb

TL;DR

This work initiates a rigorous analysis of operator product expansions in the two-dimensional sine-Gordon model for $β\in(0,4π)$, focusing on derivative fields and their coupling to Wick ordered vertex operators. The authors construct finite-volume sine-Gordon measures as renormalized perturbations of the Gaussian free field, develop analytic control of correlation functions with a spectator observable $O=e^{iφ(f)}$ through Onsager-type inequalities and moment bounds for Wick exponentials, and show that the derivative OPE acquires genuine logarithmic and cross term singularities not present in the pure GFF. The main technical contribution is the combination of Wick-ordered exponentials, moment bounds, and graph-based decompositions to establish the existence and precise diagonal limits of smeared correlation functions, culminating in explicit renormalized OPE formulas for $⟨∂φ(x)∂φ(y)O⟩$, $⟨∂φ(x)∂̄φ(y)O⟩$, and $⟨∂φ(x)∂φ(y)⟩$ as $x\to y$. By contrasting with the Gaussian free field analogue, the work highlights how near-critical interacting models can generate new structures such as Wick-ordered exponentials directly from the OPE. The methods provide a robust probabilistic framework for renormalized products of derivatives in a non-conformal, integrable QFT, with potential implications for bosonization, lattice-field theory, and the rigorous treatment of OPE bootstrap-like ideas in probabilistic settings.

Abstract

In this article, we initiate the study of operator product expansions (OPEs) for the sine-Gordon model. For simplicity, we focus on the model below the first threshold of collapse ($β<4π$) and on the singular terms in OPEs of derivative-type fields $\partial \varphi$ and $\bar\partial\varphi$. We prove that compared to corresponding free field OPEs, the sine-Gordon OPEs develop logarithmic singularities and generate Wick ordered exponentials. Our approach for proving the OPEs relies heavily on Onsager-type inequalities and associated moment bounds for GFF correlation functions involving Wick ordered exponentials of the free field.

Operator product expansions of derivative fields in the sine-Gordon model

TL;DR

This work initiates a rigorous analysis of operator product expansions in the two-dimensional sine-Gordon model for , focusing on derivative fields and their coupling to Wick ordered vertex operators. The authors construct finite-volume sine-Gordon measures as renormalized perturbations of the Gaussian free field, develop analytic control of correlation functions with a spectator observable through Onsager-type inequalities and moment bounds for Wick exponentials, and show that the derivative OPE acquires genuine logarithmic and cross term singularities not present in the pure GFF. The main technical contribution is the combination of Wick-ordered exponentials, moment bounds, and graph-based decompositions to establish the existence and precise diagonal limits of smeared correlation functions, culminating in explicit renormalized OPE formulas for , , and as . By contrasting with the Gaussian free field analogue, the work highlights how near-critical interacting models can generate new structures such as Wick-ordered exponentials directly from the OPE. The methods provide a robust probabilistic framework for renormalized products of derivatives in a non-conformal, integrable QFT, with potential implications for bosonization, lattice-field theory, and the rigorous treatment of OPE bootstrap-like ideas in probabilistic settings.

Abstract

In this article, we initiate the study of operator product expansions (OPEs) for the sine-Gordon model. For simplicity, we focus on the model below the first threshold of collapse () and on the singular terms in OPEs of derivative-type fields and . We prove that compared to corresponding free field OPEs, the sine-Gordon OPEs develop logarithmic singularities and generate Wick ordered exponentials. Our approach for proving the OPEs relies heavily on Onsager-type inequalities and associated moment bounds for GFF correlation functions involving Wick ordered exponentials of the free field.

Paper Structure

This paper contains 22 sections, 23 theorems, 201 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. The operator product expansion
  3. The sine-Gordon model
  4. The main result
  5. A review of the GFF and an analogue of Theorem \ref{['th:main']} for the GFF
  6. The Gaussian free field
  7. An analogue of Theorem \ref{['th:main']} for the GFF
  8. Wick ordered exponentials and analysis of their correlation functions
  9. Wick ordered exponentials
  10. Onsager inequalities
  11. Statements of the moment bounds
  12. Key notions and estimates needed for the moment bounds
  13. Nearest neighbor maps and their associated graphs
  14. Estimating an integral
  15. Proofs of the moment bounds
  16. ...and 7 more sections

Key Result

Theorem 1.1

For $\Omega\subset \mathbb{R}^2$ bounded, connected, and open with smooth boundary, $\beta\in (0,4\pi)$, $\mu\in \mathbb{R}$, $\psi\in C_c^\infty(\Omega)$, and $\mathcal{O}$ an exponential functional in the field $\varphi$, namely $\mathcal{O}=e^{i\varphi(f)}$ with $f\in C_c^\infty(\Omega)$, the fol

Figures (2)

  • Figure 3.1: An illustration of an element of $\mathcal{F}_{2,2}^{(13)}$.
  • Figure 3.2: An illustration of reorienting the graph in case ii): the original orientation is above, and the reoriented graph is below.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 37 more