$T\bar{T}$ Deformations and Form Factor Program

TL;DR

This work addresses computing minimal form factors for 1+1D IQFTs perturbed by irrelevant $ ext{T}ar{ ext{T}}$-type operators. It extends the form factor bootstrap by introducing a deformation factor $ ext{D}_{oldsymbol{oldsymbol{\alpha}}}( heta)$ that multiplies the undeformed MFF, with a factorization $ ext{D}_{oldsymbol{oldsymbol{\alpha}}}( heta)= ext{φ}_{oldsymbol{oldsymbol{\alpha}}}( heta) ext{C}_{oldsymbol{eta}}( heta)$, where $ ext{φ}$ encodes the CDD-like perturbation and $ ext{C}$ regularizes large-$ heta$ behavior through a set of parameters $eta_n$. The authors derive an integral representation for the MFF, address convergence via a regularization scheme, and fix the $eta_n$-dependent piece to obtain a finite, UV-consistent MFF, then illustrate the construction with the Ising model perturbed by $ ext{T}ar{ ext{T}}$, showing a pole at $ heta=0$ for certain couplings and a zero otherwise. The results yield taming of the MFF asymptotics and elucidate the analytic structure of deformed MFFs, with implications for correlation functions and potential classification of UV-complete theories. Overall, the framework provides a systematic method to obtain MFFs in TTbar-perturbed IQFTs and clarifies when and why poles or zeros arise in the deformation regime.

Abstract

In this proceeding contribution, we review a recently proposed method to compute the minimal form factors (MFFs) of diagonal integrable field theories perturbed by irrelevant fields of the $T\bar{T}$ family. Our construction generalizes standard form factor techniques to deal with the deformed two-body scattering amplitudes, which are typical in this setting. The results are minimal form factors which are the product of the undeformed solution and a new function. This function can be fixed by requiring constant asymptotics for large rapidities, smoothness in the limit when the perturbation parameters go to zero, and agreement with standard MFF formulae for particular choices of the perturbation couplings. We observe that, for a certain range of parameters, the new MFF develops a pole at $θ=0$. By considering several UV-complete theories, we argue that such poles can emerge naturally from the MFF integral representation and suggest how they may be eliminated.

Figures (1)

• Figure 1: The modulo square of the MFF of the Ising field theory perturbed by $\mathrm{T}\overline{\mathrm{T}}$ for different values of the coupling $\alpha$. The black line correspond to $\alpha=0$. Other lines correspond to values $\pi, \pi/2, \pm 2\pi$ and $3\pi$. It is important to note that there is a change at $\alpha=\pi$. For $\alpha>\pi$ the MFF diverges at $\theta=0$ whilst for $\alpha\leq \pi$ it has a zero, just like for the unperturbed model.