Correlation Functions in $\textrm{T}\bar{\textrm{T}}$-deformed Theories on the Torus

TL;DR

This paper analyzes two-point correlation functions in $ extrm{T}ar{ extrm{T}}$-deformed theories on a torus, revealing how their large momentum behavior differs from the plane case, indicating non-locality and UV-IR mixing effects.

Contribution

It provides a detailed computation of momentum-space correlation functions on a torus in $ extrm{T}ar{ extrm{T}}$-deformed theories using Jackiw-Teitelboim gravity, highlighting new large momentum behaviors.

Findings

Large momentum correlator decays then grows, indicating smearing of operators.

Different behaviors are found for small and large momenta relative to the torus scale.

The results illustrate the non-locality and UV-IR mixing in $ extrm{T}ar{ extrm{T}}$-deformed theories.

Abstract

We study the correlation functions of local operators in unitary $\textrm{T}\bar{\textrm{T}}$-deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation function in momentum space when the undeformed theory is a conformal field theory. The large momentum behavior of the correlation function is computed and compared to that of $\textrm{T}\bar{\textrm{T}}$-deformed field theories defined on a plane. For the latter, the behavior found was $\left(\frac{\sqrt{t}|q|}{\pi e}\right)^{-\frac{tq^2}{\pi}}$, where $q$ is the momentum and $t$ is the deformation parameter. For a torus, the same behavior is found for $|q|<<L/t$, where $L$ is the torus' length scale. However, for $|q|>>L/t$, a different behavior is found: $\left(\frac{2\sqrt{t}^5q^2}{\pi e L^3|T|^2}\right)^{\frac{tq^2}{\pi}}$, where $T$ is the modular parameter of the torus. Hence, at large momentum, the correlator decays and then grows. This behavior suggests that operators carrying momentum $q$ are smeared on a distance scale $t|q|$. The difference from the plane's result illustrates the non-locality of the theory and the UV-IR mixing.