Integrability and Renormalization under $T\bar{T}$

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.

arXiv comment38 pages, 6 figures; v2, typos fixed

References in corpus (40)

• On space of integrable quantum field theories
• $T\bar{T}$-deformed 2D Quantum Field Theories
• Analyticity in Spin in Conformal Theories
• Asymptotic Fragility, Near $Ad{S}_2$ Holography and $T\bar{T}$
• Solving the Simplest Theory of Quantum Gravity
• Cutoff AdS${}_3$ versus the $T\bar{T}$ deformation
• The $T\overline{T}$ deformation of quantum field theory as random geometry
• $T\bar{T}$ Partition Function from Topological Gravity
• An integrable Lorentz-breaking deformation of two-dimensional CFTs
• A spacetime derivation of the Lorentzian OPE inversion formula
• Holography at finite cutoff with a $T^2$ deformation
• The $\text{T}\bar{\text{T}}$ perturbation and its geometric interpretation
• Modular invariance and uniqueness of $T\bar{T}$ deformed CFT
• $T\bar{T}$-deformations in closed form
• $T\bar{T}$ deformed partition functions
• All point correlation functions in SYK
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Generalised Born-Infeld models, Lax operators and the $\text{T}\bar{\text{T}}$ perturbation
• $T\overline{T}$ in AdS${}_2$ and Quantum Mechanics
• Natural Tuning: Towards A Proof of Concept
• Multipoint Conformal Blocks in the Comb Channel
• Conformal Blocks for Arbitrary Spins in Two Dimensions
• On $T\bar{T}$ deformations and supersymmetry
• Supersymmetry and $T\overline{T}$ Deformations
• $T\bar{T}$-deformations, AdS/CFT and correlation functions
• The TT* deformation at large central charge
• Expectation value of composite field $T\bar{T}$ in two-dimensional quantum field theory
• Moving the CFT into the bulk with $T\bar{T}$
• TT deformations in general dimensions
• On correlation functions in $J\bar{T}$-deformed CFTs
• $T\bar{T}$ and LST
• Anyons from Strings
• Conformal bootstrap for the BFKL Pomeron
• $T\bar{T}$ deformations as TsT transformations
• Expectation value of $\mathrm{T}\overline{\mathrm{T}}$ operator in curved spacetimes
• KdV charges in $T\bar{T}$ theories and new models with super-Hagedorn behavior
• Solving a family of $T\bar{T}$-like theories
• TTbar deformation and the light-cone gauge
• No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit
• Four Loop Scattering in the Nambu-Goto Theory

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• Infinite pseudo-conformal symmetries of classical $T\bar{T}$, $J\bar{T}$ and $J{T}_a$ - deformed CFTs
• Field Theory of Interacting Boundary Gravitons
• Correlation Functions in $\text{T}\bar{\text{T}}$-deformed Conformal Field Theories
• $T\bar{T}$-flow effects on torus partition functions
• $T\overline{T}$ deformation in SCFTs and integrable supersymmetric theories
• The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations
• Quantization of the ModMax Oscillator
• $T\bar{T}$ deformation of the Ising model and its ultraviolet completion
• A definition of primary operators in $J\bar{T}$-deformed CFTs
• Flows in the Space of Interacting Chiral Boson Theories
• Large N Analysis of $T\bar{T}$-deformation and Unavoidable Negative-norm States
• Celestial amplitude for 2d theory
• Irrelevant Deformations of Chiral Bosons
• Perturbative renormalization of the $T\bar{T}$-deformed free massive Dirac fermion
• Renormalisation in $\text{T}\bar{\text{T}}$-Deformed (Non-)Integrable Theories
• Thermodynamics of $T\bar{T}$ perturbations of some single particle field theories
• Correlation Functions in $\text{T}\bar{\text{T}}$-deformed Theories on the Torus
• Composite operators in $T\bar{T}$-deformed free QFTs
• Systematic approach to correlators in $T\bar{T}$ deformed CFTs
• Note on higher-point correlation functions of the $T\bar{T}$ or $J\bar{T}$ deformed CFTs
• Soliton, breather and shockwave solutions of the Heisenberg and the $T\bar{T}$ deformations of scalar field theories in 1+1 dimensions
• Defining the $T\overline{T}$ Deformation on ${\mathrm{AdS}}_2$
• Entanglement Renormalization of a $T\bar{T}$-deformed CFT
• Nonperturbative aspects of two-dimensional $T\bar{T}$-deformed scalar theory from functional renormalization group
• Locality and Conserved Charges in $T\overline{T}$-Deformed CFTs
• A $T\bar{T}$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering
• Flat limit of massless scalar scattering in ${\mathrm{AdS}}_2$