Abstract
The study of \( \textrm{T}\overline{\textrm{T}} \)-perturbed quantum field theories is an active area of research with deep connections to fundamental aspects of the scattering theory of integrable quantum field theories, generalised Gibbs ensembles, and string theory. Many features of these theories, such as the peculiar behaviour of their ground state energy and the form of their scattering matrices, have been studied in the literature. However, so far, very few studies have approached these theories from the viewpoint of the form factor program. From the perspective of scattering theory, the effects of a \( \textrm{T}\overline{\textrm{T}} \) perturbation (and higher spin versions thereof) is encoded in a universal deformation of the two-body scattering matrix by a CDD factor. It is then natural to ask how these perturbations influence the form factor equations and, more generally, the form factor program. In this paper, we address this question for free theories, although some of our results extend more generally. We show that the form factor equations admit general solutions and how these can help us study the distinct behaviour of correlation functions at short distances in theories perturbed by irrelevant operators.
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References
A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19 (1989) 641 [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
G. Delfino and G. Niccoli, Matrix elements of the operator \( T\overline{T} \) in integrable quantum field theory, Nucl. Phys. B 707 (2005) 381 [hep-th/0407142] [INSPIRE].
G. Delfino and G. Niccoli, The composite operator \( T\overline{T} \) in sinh-Gordon and a series of massive minimal models, JHEP 05 (2006) 035 [hep-th/0602223] [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].
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].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].
S. Datta and Y. Jiang, \( T\overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
R. Conti, S. Negro and R. Tateo, Conserved currents and \( T{\overline{T}}_s \) irrelevant deformations of 2D integrable field theories, JHEP 11 (2019) 120 [arXiv:1904.09141] [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].
G. Camilo et al., On factorizable S-matrices, generalized \( T\overline{T} \), 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 and A.B. Zamolodchikov, Factorized S-matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [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].
S. Negro, Integrable structures in quantum field theory, J. Phys. A 49 (2016) 323006 [arXiv:1606.02952] [INSPIRE].
P. Christe and G. Mussardo, Elastic S-matrices in (1 + 1)-dimensions and Toda field theories, Int. J. Mod. Phys. A 5 (1990) 4581 [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. Fring, C. Korff and B.J. Schulz, On the universal representation of the scattering matrix of affine Toda field theory, Nucl. Phys. B 567 (2000) 409 [hep-th/9907125] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Natural tuning: towards a proof of concept, JHEP 09 (2013) 045 [arXiv:1305.6939] [INSPIRE].
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, \( T\overline{T} \) in AdS2 and quantum mechanics, Phys. Rev. D 101 (2020) 026011 [arXiv:1907.04873] [INSPIRE].
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, Hamiltonian deformations in quantum mechanics, \( T\overline{T} \), and the SYK model, Phys. Rev. D 102 (2020) 046019 [arXiv:1912.06132] [INSPIRE].
F. Giordano, S. Negro and R. Tateo, The generalised Born oscillator and the Berry-Keating Hamiltonian, arXiv:2307.15025 [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
R. Conti, J. Romano and R. Tateo, Metric approach to a \( T\overline{T} \)-like deformation in arbitrary dimensions, JHEP 09 (2022) 085 [arXiv:2206.03415] [INSPIRE].
R. Conti, S. Negro and R. Tateo, The \( T\overline{T} \) perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [INSPIRE].
R. Conti, L. Iannella, S. Negro and R. Tateo, Generalised Born-Infeld models, Lax operators and the \( T\overline{T} \) perturbation, JHEP 11 (2018) 007 [arXiv:1806.11515] [INSPIRE].
S. Dubovsky, S. Negro and M. Porrati, Topological gauging and double current deformations, JHEP 05 (2023) 240 [arXiv:2302.01654] [INSPIRE].
F. Aramini, N. Brizio, S. Negro and R. Tateo, Deforming the ODE/IM correspondence with \( T\overline{T} \), JHEP 03 (2023) 084 [arXiv:2212.13957] [INSPIRE].
P. Dorey, C. Dunning, S. Negro and R. Tateo, Geometric aspects of the ODE/IM correspondence, J. Phys. A 53 (2020) 223001 [arXiv:1911.13290] [INSPIRE].
M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, Quantisation of the effective string with TBA, JHEP 07 (2013) 071 [arXiv:1305.1278] [INSPIRE].
A. LeClair, Thermodynamics of perturbations of some single particle field theories, J. Phys. A 55 (2022) 185401 [arXiv:2105.08184] [INSPIRE].
A. LeClair, Deformation of the Ising model and its ultraviolet completion, J. Stat. Mech. 2111 (2021) 113104 [arXiv:2107.02230] [INSPIRE].
C. Ahn and A. LeClair, On the classification of UV completions of integrable \( T\overline{T} \) deformations of CFT, JHEP 08 (2022) 179 [arXiv:2205.10905] [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].
J. Cardy, \( T\overline{T} \) deformation of correlation functions, JHEP 12 (2019) 160 [arXiv:1907.03394] [INSPIRE].
O. Aharony and T. Vaknin, The \( T\overline{T} \) deformation at large central charge, JHEP 05 (2018) 166 [arXiv:1803.00100] [INSPIRE].
O. Aharony et al., Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
M. Guica and R. Monten, Infinite pseudo-conformal symmetries of classical \( T\overline{T} \), \( J\overline{T} \) and JTa-deformed CFTs, SciPost Phys. 11 (2021) 078 [arXiv:2011.05445] [INSPIRE].
M. Guica, \( J\overline{T} \)-deformed CFTs as non-local CFTs, arXiv:2110.07614 [INSPIRE].
M. Guica, R. Monten and I. Tsiares, Classical and quantum symmetries of \( T\overline{T} \)-deformed CFTs, arXiv:2212.14014 [INSPIRE].
M. Baggio and A. Sfondrini, Strings on NS-NS backgrounds as integrable deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].
A. Dei and A. Sfondrini, Integrable S matrix, mirror TBA and spectrum for the stringy AdS3 × S3 × S3 × S1 WZW model, JHEP 02 (2019) 072 [arXiv:1812.08195] [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, \( T\overline{T} \), \( J\overline{T} \), \( T\overline{J} \) and string theory, J. Phys. A 52 (2019) 384003 [arXiv:1905.00051] [INSPIRE].
N. Callebaut, J. Kruthoff and H. Verlinde, \( T\overline{T} \) deformed CFT as a non-critical string, JHEP 04 (2020) 084 [arXiv:1910.13578] [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].
A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and \( T\overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
P. Kraus, J. Liu and D. Marolf, Cutoff AdS3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
M. Guica and R. Monten, \( T\overline{T} \) and the mirage of a bulk cutoff, SciPost Phys. 10 (2021) 024 [arXiv:1906.11251] [INSPIRE].
Y. Jiang, Expectation value of \( T\overline{T} \) operator in curved spacetimes, JHEP 02 (2020) 094 [arXiv:1903.07561] [INSPIRE].
G. Jafari, A. Naseh and H. Zolfi, Path integral optimization for \( T\overline{T} \) deformation, Phys. Rev. D 101 (2020) 026007 [arXiv:1909.02357] [INSPIRE].
A.J. Tolley, \( T\overline{T} \) deformations, massive gravity and non-critical strings, JHEP 06 (2020) 050 [arXiv:1911.06142] [INSPIRE].
L.V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde, JT gravity at finite cutoff, SciPost Phys. 9 (2020) 023 [arXiv:2004.07242] [INSPIRE].
S. Okumura and K. Yoshida, \( T\overline{T} \)-deformation and Liouville gravity, Nucl. Phys. B 957 (2020) 115083 [arXiv:2003.14148] [INSPIRE].
S. Ebert, C. Ferko, H.-Y. Sun and Z. Sun, \( T\overline{T} \) in JT gravity and BF gauge theory, SciPost Phys. 13 (2022) 096 [arXiv:2205.07817] [INSPIRE].
M. Medenjak, G. Policastro and T. Yoshimura, \( T\overline{T} \)-deformed conformal field theories out of equilibrium, Phys. Rev. Lett. 126 (2021) 121601 [arXiv:2011.05827] [INSPIRE].
M. Medenjak, G. Policastro and T. Yoshimura, Thermal transport in \( T\overline{T} \)-deformed conformal field theories: from integrability to holography, Phys. Rev. D 103 (2021) 066012 [arXiv:2010.15813] [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Boosting nearest-neighbour to long-range integrable spin chains, J. Stat. Mech. 0811 (2008) L11001 [arXiv:0807.5081] [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Long-range deformations for integrable spin chains, J. Phys. A 42 (2009) 285205 [arXiv:0902.0956] [INSPIRE].
B. Pozsgay, Y. Jiang and G. Takács, \( T\overline{T} \)-deformation and long range spin chains, JHEP 03 (2020) 092 [arXiv:1911.11118] [INSPIRE].
E. Marchetto, A. Sfondrini and Z. Yang, \( T\overline{T} \) deformations and integrable spin chains, Phys. Rev. Lett. 124 (2020) 100601 [arXiv:1911.12315] [INSPIRE].
J. Cardy and B. Doyon, \( T\overline{T} \) deformations and the width of fundamental particles, JHEP 04 (2022) 136 [arXiv:2010.15733] [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].
R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cim. Suppl. 3 (1965) 147 [INSPIRE].
J. Mossel and J.-S. Caux, Generalized TBA and generalized Gibbs, J. Phys. A 45 (2012) 255001 [arXiv:1203.1305] [INSPIRE].
K.G. Wilson and J.B. Kogut, The renormalization group and the epsilon expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
O.A. Castro-Alvaredo, S. Negro and F. Sailis, Completing the bootstrap program for \( T\overline{T} \)-deformed massive integrable quantum field theories, arXiv:2305.17068 [INSPIRE].
M. Karowski and P. Weisz, Exact form-factors in (1 + 1)-dimensional field theoretic models with soliton behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].
F.A. Smirnov, Form-factors in completely integrable models of quantum field theory, World Scientific, Singapore (1992) [INSPIRE].
G. Mussardo and P. Simon, Bosonic type S-matrix, vacuum instability and CDD ambiguities, Nucl. Phys. B 578 (2000) 527 [hep-th/9903072] [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Correlation functions of integrable 2D models of relativistic field theory. Ising model, Int. J. Mod. Phys. A 6 (1991) 3419 [INSPIRE].
A. Fring, G. Mussardo and P. Simonetti, Form-factors for integrable Lagrangian field theories, the sinh-Gordon theory, Nucl. Phys. B 393 (1993) 413 [hep-th/9211053] [INSPIRE].
A. Koubek and G. Mussardo, On the operator content of the sinh-Gordon model, Phys. Lett. B 311 (1993) 193 [hep-th/9306044] [INSPIRE].
S.L. Lukyanov, Free field representation for massive integrable models, Commun. Math. Phys. 167 (1995) 183 [hep-th/9307196] [INSPIRE].
J.B. Zuber and C. Itzykson, Quantum field theory and the two-dimensional Ising model, Phys. Rev. D 15 (1977) 2875 [INSPIRE].
B. Schroer and T.T. Truong, The order/disorder quantum field operators associated to the two-dimensional Ising model in the continuum limit, Nucl. Phys. B 144 (1978) 80 [INSPIRE].
O. Babelon and D. Bernard, From form-factors to correlation functions: the Ising model, Phys. Lett. B 288 (1992) 113 [hep-th/9206003] [INSPIRE].
J.L. Cardy and G. Mussardo, Form-factors of descendent operators in perturbed conformal field theories, Nucl. Phys. B 340 (1990) 387 [INSPIRE].
J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive QFT with a boundary: the Ising model, J. Statist. Phys. 134 (2009) 105 [arXiv:0810.0219] [INSPIRE].
D.X. Horváth and P. Calabrese, Symmetry resolved entanglement in integrable field theories via form factor bootstrap, JHEP 11 (2020) 131 [arXiv:2008.08553] [INSPIRE].
H. Babujian and M. Karowski, Towards the construction of Wightman functions of integrable quantum field theories, Int. J. Mod. Phys. A 19S2 (2004) 34 [hep-th/0301088] [INSPIRE].
O.A. Castro-Alvaredo and M. Mazzoni, Two-point functions of composite twist fields in the Ising field theory, J. Phys. A 56 (2023) 124001 [arXiv:2301.01745] [INSPIRE].
D. A Marca, S. Beltraminelli and D. Merlini, Mean staircase of the Riemann zeros: a comment on the Lambert W function and an algebraic aspect, arXiv:0901.3377 [INSPIRE].
R.B. Mann and T. Ohta, Exact solution for the metric and the motion of two bodies in (1 + 1)-dimensional gravity, Phys. Rev. D 55 (1997) 4723 [gr-qc/9611008] [INSPIRE].
G. Delfino, P. Simonetti and J.L. Cardy, Asymptotic factorization of form-factors in two-dimensional quantum field theory, Phys. Lett. B 387 (1996) 327 [hep-th/9607046] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [INSPIRE].
G. Delfino and G. Mussardo, The spin spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc, Nucl. Phys. B 455 (1995) 724 [hep-th/9507010] [INSPIRE].
G. Delfino, Integrable field theory and critical phenomena: the Ising model in a magnetic field, J. Phys. A 37 (2004) R45 [hep-th/0312119] [INSPIRE].
Acknowledgments
The authors thank John Donahue, Benjamin Doyon, Fedor Smirnov, Roberto Tateo and Alexander Zamolodchikov for useful discussions. Olalla A. Castro-Alvaredo thanks EPSRC for financial support under Small Grant EP/W007045/1. Fabio Sailis is grateful for his PhD Studentship which is funded by City, University of London. The work of Stefano Negro is partially supported by the NSF grant PHY-2210349 and by the Simons Collaboration on Confinement and QCD Strings. This project was partly inspired by a meeting at the Kavli Institute for Theoretical Physics (Santa Barbara) in September 2022. Olalla A. Castro-Alvaredo and Stefano Negro thank the Institute for financial support from the National Science Foundation under Grant No. NSF PHY-1748958, and hospitality during the conference “Talking Integrability: Spins, Fields and Strings”, August 29-September 1 (2022) and the related extended program on “Integrability in String, Field and Condensed Matter Theory”, August 22-October 14 (2022).
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Castro-Alvaredo, O.A., Negro, S. & Sailis, F. Form factors and correlation functions of \( \textrm{T}\overline{\textrm{T}} \)-deformed integrable quantum field theories. J. High Energ. Phys. 2023, 48 (2023). https://doi.org/10.1007/JHEP09(2023)048
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DOI: https://doi.org/10.1007/JHEP09(2023)048