Thermodynamics of $T \bar{T}$ perturbations of some single particle field theories

We study the Thermodynamic Bethe Ansatz (TBA) equations for pure $T\bar{T}$ perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for $T\bar{T}$ deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories (CFT's) are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference $R$ which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c-theorem in this context which is discussed in some detail. For positive infra-red (IR) central charge $c_{IR}$, for flows consistent with the c-theorem the ground state energy develops a (previously known) square-root singularity towards the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with $c_{IR} < 0$ do not have this singularity and are interpreted as being UV complete with $c_{UV} = 0$. We extend our results to a continuously variable $c_{IR}$ by introducing a chemical potential and suggest this as a possible toy model for the $T\bar{T}$ perturbed Liouville theory.Comment: 17 pages, version 2: Includes some short remarks suggesting the singularity indicates a tachyonic vacuum instability. References added; version 3: More pedagogical; care is taken to match conventions of previous work;, references added; no change in previous results nor additional results. Version 4: published version in J. Phys. A. Mathematical and Theoretica