[Paper Review] Note on higher-point correlation functions of the $T\bar{T}$ or $J\bar{T}$ deformed CFTs

Song He|arXiv (Cornell University)|Dec 11, 2020

Black Holes and Theoretical Physics73 references4 citations

TL;DR

This paper computes the first-order correction to generic n-point correlation functions in $T\bar{T}$ and $J\bar{T}$ deformed 2D conformal field theories (CFTs) using perturbative CFT methods. It derives analytical expressions for these corrections via contour integrals and Ward identities, and confirms that both deformations preserve integrability by showing the out-of-time-ordered correlation function (OTOC) in the Ising model remains unchanged at first order, indicating no loss of integrability.

ABSTRACT

We investigate generic n-point correlation functions of conformal field theories (CFTs), with $T\bar{T}$ and $J\bar{T}$ deformations, in terms of the perturbative CFT approach. We systematically obtain the first order correction to the generic correlation functions of CFTs with $T\bar{T}$ or $J\bar{T}$ deformation. We compute the out-of-time ordered correlation function (OTOC) in the Ising model with $T\bar{T}$ or $J\bar{T}$ deformation, which confirms that these deformations do not change the integrable property up to the first order level.

Motivation & Objective

• To systematically compute the first-order correction to generic n-point correlation functions in $T\bar{T}$ and $J\bar{T}$ deformed CFTs.
• To analyze whether $T\bar{T}$ and $J\bar{T}$ deformations affect the integrability of the underlying CFT by studying the out-of-time-ordered correlation function (OTOC).
• To extend perturbative methods used for two- and three-point functions to higher-point functions using holomorphic and anti-holomorphic cross ratios.
• To provide explicit analytical expressions for deformed correlation functions using contour integrals and simplified notation for $T\bar{T}$ and $J\bar{T}$-type operators.

Proposed method

• The perturbative CFT approach is applied, expanding the Lagrangian near the CFT critical point with a small coupling constant $\lambda$, leading to a first-order correction involving $\lambda \int d^2z \langle O(z,\bar{z}) \phi_1(z_1) \cdots \phi_n(z_n) \rangle$ in the undeformed CFT.
• The Ward identity is used to constrain the structure of correlation functions in terms of $2n-3$ holomorphic and anti-holomorphic cross ratios, simplifying the computation of the first-order correction.
• Specialized integrals $\mathcal{I}_{i,j}$ are introduced to handle the singularities in the correlation functions, with recursive relations derived via derivatives of known integrals.
• The method uses dimensional regularization with $d = 2 + \tilde{\epsilon}$ to regulate divergences, particularly in integrals like $\mathcal{I}_{221}$, and extracts poles and logarithmic terms in the $\tilde{\epsilon} \to 0$ limit.
• The OTOC is computed in the $T\bar{T}$ and $J\bar{T}$-deformed Ising model to test for quantum chaos and integrability, using the large $c$ limit and asymptotic behavior.
• The results are cross-checked with known four-point functions from previous works, confirming consistency via re-expressing results in terms of the introduced $\mathcal{I}_{i,j}$ notation.

Experimental results

Research questions

• RQ1 How does the $T\bar{T}$ deformation modify generic n-point correlation functions in a CFT at first order in the coupling?
• RQ2 What is the structure of the first-order correction to $J\bar{T}$-deformed n-point functions, and how does it differ from the $T\bar{T}$ case?
• RQ3 Does the $T\bar{T}$ or $J\bar{T}$ deformation alter the integrability of an underlying integrable CFT, as measured by the OTOC?
• RQ4 Can the perturbative CFT method be systematically extended to higher-point functions beyond two- and three-point functions?
• RQ5 How do the Ward identities and cross-ratio structures constrain the form of the first-order correction in deformed CFTs?

Key findings

• The first-order correction to generic n-point correlation functions in $T\bar{T}$ and $J\bar{T}$ deformed CFTs is derived using perturbative CFT and contour integrals, with explicit dependence on holomorphic and anti-holomorphic cross ratios.
• The OTOC in the $T\bar{T}$-deformed Ising model shows no change in late-time behavior at first order, confirming that the deformation does not break integrability.
• Similarly, the $J\bar{T}$-deformed OTOC remains unchanged at first order, indicating that the integrability of the original CFT is preserved under this deformation.
• The paper provides a systematic notation $\mathcal{I}_{i,j}$ for handling complex integrals arising in higher-point functions, enabling compact expressions for $T\bar{T}$ and $J\bar{T}$-deformed four-point functions.
• The divergent part of the integral $\mathcal{I}_{221}$ is computed in $d=2+\tilde{\epsilon}$ dimensions, yielding $\frac{8\pi}{|z_{12}|^6}\left(\frac{4}{\tilde{\epsilon}} + 2\log|z_{12}|^2 + 2\log\pi + 2\gamma - 5\right)$, which matches known results in the literature.
• The derived expressions for $\mathcal{I}_{122}$ and $\mathcal{I}_{11111}$ in terms of $\mathcal{I}_{i,j}$ reproduce the $J\bar{T}$-deformed four-point function from previous work, validating the method and notation.