Integrability and Renormalization under $T \bar T$

TL;DR

This paper derives the renormalized Lagrangian for the $T\bar T$ deformed free scalar field theory, connecting the deformation, integrability, and $S$-matrix, and discusses implications for computing observables.

Contribution

It explicitly constructs the renormalized Lagrangian for the $T\bar T$ deformation of a free scalar up to second order, linking $S$-matrix factorization and integrability.

Findings

Renormalized Lagrangian obtained to second order in deformation parameter.

Deformed theories remain integrable with factorized $S$-matrix.

Discussion on computing observables from the renormalized Lagrangian.

Abstract

Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this $T\bar T$ flow equation one can find a simple expression for both the energy spectrum and the $S$-matrix of the $T\bar T$ deformed theories. Our goal is to find the renormalized Lagrangian of the $T\bar T$ deformed theories. In the context of the $T\bar T$ deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the $S$-matrix factorizes into two-to-two $S$-matrices. One may thus hope to be able to extract the renormalized Lagrangian from the $S$-matrix. We do this explicitly for the $T\bar T$ deformation of a free massive scalar, to second order in the deformation parameter. Once one has the renormalized Lagrangian one can, in principle, compute all other observables, such as correlation functions. We briefly discuss this, as well as the relation between the renormalized Lagrangian, the $T\bar T$ flow equation, and the $S$-matrix. We also mention a more general class of integrability-preserving deformations of a free scalar field theory.