The FBSDE approach to sine-Gordon up to $6\pi$

TL;DR

This paper introduces a stochastic analysis framework using FBSDEs to study the sine-Gordon quantum field theory up to the second threshold, enabling analysis without cut-offs and providing insights into its properties.

Contribution

It develops a novel FBSDE-based approach to analyze the sine-Gordon quantum field theory up to $6\pi$, extending previous methods and removing the need for cut-offs.

Findings

Describes the interacting field without cut-offs.

Provides results on large deviations and decay of correlations.

Shows singularity with respect to the free field.

Abstract

We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (\beta \varphi))_2$ on the full space up to the second threshold, i.e. for $\beta^2 < 6 \pi$. The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition $(X_t)_{t \geqslant 0}$ of the interacting Euclidean field $X_{\infty}$ along a scale parameter $t \geqslant 0$. This FBSDE describes the optimiser of the stochastic control representation of the Euclidean QFT introduced by Barashkov and one of the authors. We show that the FBSDE provides a description of the interacting field without cut-offs and that it can be used effectively to study the sine-Gordon measure to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties.