# $T\bar T$ deformation of correlation functions

## TL;DR

This paper investigates how correlation functions in 2D quantum field theories evolve under the $Tar T$ deformation, revealing a derivation-like action on operators, UV finiteness, and non-local but conserved currents, with implications for understanding deformed CFTs.

## Contribution

It provides an explicit framework for analyzing correlation functions under $Tar T$ deformation, including UV divergence absorption, operator algebra evolution, and conserved currents behavior.

## Key findings

- Correlation functions evolve via a derivation on the operator algebra.
- UV divergences can be absorbed into non-local field renormalization.
- Deformed two-point functions exhibit a modified power-law behavior in momentum space.

## Abstract

We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $\lambda T\bar T$ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as $k^{2\Delta+2\lambda k^2}$, where $\Delta$ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $T\bar T$ deformation to a state-dependent coordinate transformation emerges in this picture.