Abstract
It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant \( T\overline{T} \) 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 [\( T\overline{T} \)]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 cUV > cIR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [\( T\overline{T} \)]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 cIR = \( \frac{1}{2} \) we find 3 UV completions based on a single massless Majorana fermion description with cUV = \( \frac{7}{10} \) and \( \frac{3}{2} \), which both have \( \mathcal{N} \) = 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 cUV = \( \frac{21}{22},\frac{15}{12} \), and \( \frac{31}{2} \). 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 cIR = \( \frac{4}{5} \) and we find 3 completions with \( \frac{4}{5} \) < cUV ≤ \( \frac{16}{5} \). For the su(4) case, which has 3 particles and cIR = 1, and we find 11 UV completions with 1 < cUV ≤ 5, most of which were previously unknown.
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References
S. Weinberg, Critical Phenomena for Field Theorists, in The Subnuclear Series. Vol. 14: Understanding the Fundamental Constituents of Matter, A. Zichichi eds., Springer, New York, U.S.A., pg. 1.
S. Weinberg, Asymptotically Safe Inflation, Phys. Rev. D 81 (2010) 083535 [arXiv:0911.3165] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS2 holography and \( T\overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].
R. Conti, L. Iannella, S. Negro and R. Tateo, Generalised Born-Infeld models, Lax operators and the \( \mathrm{T}\overline{\mathrm{T}} \) perturbation, JHEP 11 (2018) 007 [arXiv:1806.11515] [INSPIRE].
R. Conti, S. Negro and R. Tateo, The \( \mathrm{T}\overline{\mathrm{T}} \) perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [INSPIRE].
A.B. Zamolodchikov, Integrable Field Theory from Conformal Field Theory, Adv. Studies Pure Math. 19 (1989) 641.
G. Mussardo, Statistical Field Theory. An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press, Oxford, U.K. (2010).
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Massless factorized scattering and sigma models with topological terms, Nucl. Phys. B 379 (1992) 602 [INSPIRE].
A. LeClair, deformation of the Ising model and its ultraviolet completion, J. Stat. Mech. 2111 (2021) 113104 [arXiv:2107.02230] [INSPIRE].
C. Ahn, C. Kim, C. Rim and A.B. Zamolodchikov, RG flows from superLiouville theory to critical Ising model, Phys. Lett. B 541 (2002) 194 [hep-th/0206210] [INSPIRE].
L. Castillejo, R.H. Dalitz and F.J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101 (1956) 453 [INSPIRE].
A. LeClair, Thermodynamics of \( T\overline{T} \) perturbations of some single particle field theories, J. Phys. A 55 (2022) 185401 [arXiv:2105.08184] [INSPIRE].
G. Hernández-Chifflet, S. Negro and A. Sfondrini, Flow Equations for Generalized \( T\overline{T} \) Deformations, Phys. Rev. Lett. 124 (2020) 200601 [arXiv:1911.12233] [INSPIRE].
B. Doyon, J. Durnin and T. Yoshimura, The Space of Integrable Systems from Generalised \( T\overline{T} \)-Deformations, arXiv:2105.03326 [INSPIRE].
G. Camilo, T. Fleury, M. Lencsés, S. Negro and A. Zamolodchikov, On factorizable S-matrices, generalized TTbar, and the Hagedorn transition, JHEP 10 (2021) 062 [arXiv:2106.11999] [INSPIRE].
L. Córdova, S. Negro and F.I. Schaposnik Massolo, Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model, JHEP 01 (2022) 035 [arXiv:2110.14666] [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
F. Ravanini, Thermodynamic Bethe ansatz for Gk ⨂ Gℓ/Gk+ℓ coset models perturbed by their phi(1,1,Adj) operator, Phys. Lett. B 282 (1992) 73 [hep-th/9202020] [INSPIRE].
C. Ahn, D. Bernard and A. LeClair, Fractional Supersymmetries in Perturbed Coset CFTs and Integrable Soliton Theory, Nucl. Phys. B 346 (1990) 409 [INSPIRE].
H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Affine Toda Field Theory and Exact S Matrices, Nucl. Phys. B 338 (1990) 689 [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [INSPIRE].
V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].
D. Gepner and E. Witten, String Theory on Group Manifolds, Nucl. Phys. B 278 (1986) 493 [INSPIRE].
P. Goddard, A. Kent and D.I. Olive, Virasoro Algebras and Coset Space Models, Phys. Lett. B 152 (1985) 88 [INSPIRE].
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Ahn, C., LeClair, A. On the classification of UV completions of integrable \( T\overline{T} \) deformations of CFT. J. High Energ. Phys. 2022, 179 (2022). https://doi.org/10.1007/JHEP08(2022)179
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DOI: https://doi.org/10.1007/JHEP08(2022)179