On the classification of UV completions of integrable TT¯ deformations of CFT

It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant TT¯ perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [TT¯]_s>1 irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge c_UV> c_IR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [TT¯]_s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with c_IR = 12 we find 3 UV completions based on a single massless Majorana fermion description with c_{UV =}710 and 32, which both have N = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E₈ appropriate to a magnetic perturbation with c_UV = 2122,1512, and 312. We argue that it is likely there are more cases for this E₈ spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with c_IR = 45 and we find 3 completions with 45< c_UV≤165. For the su(4) case, which has 3 particles and c_IR = 1, and we find 11 UV completions with 1 < c_UV ≤ 5, most of which were previously unknown.