On factorizable S-matrices, generalized TTbar, and the Hagedorn transition

Giancarlo Camilo,Máté Lencsés,Stefano Negro,Thiago Fleury,Alexander Zamolodchikov

doi:10.1007/jhep10(2021)062

Abstract

We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable S-matrix of an integrable QFT deformed by CDD factors. Such S-matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators. The TBA equations, of course, determine the ground state energy E(R) of the finite-size system, with the spatial coordinate compactified on a circle of circumference R. We limit attention to theories involving just one kind of stable particles, and consider deformations of the trivial (free fermion or boson) S-matrix by CDD factors with two elementary poles and regular high energy asymptotics — the “2CDD model”. We find that for all values of the parameters (positions of the CDD poles) the TBA equations exhibit two real solutions at R greater than a certain parameter-dependent value R*, which we refer to as the primary and secondary branches. The primary branch is identified with the standard iterative solution, while the secondary one is unstable against iterations and needs to be accessed through an alternative numerical method known as pseudo-arc-length continuation. The two branches merge at the “turning point” R* (a square-root branching point). The singularity signals a Hagedorn behavior of the density of high energy states of the deformed theories, a feature incompatible with the Wilsonian notion of a local QFT originating from a UV fixed point, but typical for string theories. This behavior of E(R) is qualitatively the same as the one for standard TTbar deformations of local QFT.

Highlights

Rendering the theory UV incomplete, and possibly violating causality at short scales
We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable S-matrix of an integrable QFT deformed by CDD factors
Such S-matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators

Summary

Introduction

Rendering the theory UV incomplete, and possibly violating causality at short scales. We believe it is important to understand the physical origin of the above abnormal short-distance properties Another exact result about the theory Aα concerns the deformation of its S-matrix, whose elements differ from the corresponding undeformed ones by a universal phase factor, available in closed form [6]. We will study certain generalizations of the TTbar deformation which can be defined whenever the original QFT is integrable [1] In most of such deformations, the scattering phases present a less exotic high-energy behavior — i.e., they have finite limit at θ → ∞ — while, at the same time, the overall density of states grows exponentially with the energy, leading to the Hagedorn singularity. The deformations of an Integrable QFT (IQFT) by the operators T T(s) generate an infinite-dimensional family of flows generalizing (1.1),

Objectives

Results

Conclusion