Abstract
We construct an effective Quantum Field Theory for the wrapping effects in 1+1 dimensional models of factorised scattering. The recently developed graph-theoretical approach to TBA gives the perturbative desctiption of this QFT. For the sake of simplicity we limit ourselves to scattering matrices for a single neutral particle and no bound state poles, such as the sinh-Gordon one. On the other hand, in view of applications to AdS/CFT, we do not assume that the scattering matrix is of difference type. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation.
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References
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. Scaling three state Potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
A.B. Zamolodchikov, TBA equations for integrable perturbed SU(2)k × SU(2)l/SU(2)k+l coset models, Nucl. Phys. B 366 (1991) 122 [INSPIRE].
C.-N. Yang and C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
G. Kato and M. Wadati, Bethe Ansatz Cluster Expansion Method for Quantum Integrable Particle Systems, J. Phys. Soc. Jap. 73 (2004) 1171.
G. Kato and M. Wadati, Direct calculation of thermodynamic quantities for the Heisenberg model, J. Math. Phys. 43 (2002) 5060 [cond-mat/0212325].
G. Kato and M. Wadati, Graphical representation of the partition function of a one-dimensi onal δ-function Bose gas, J. Math. Phys. 42 (2001) 4883 [cond-mat/0212323].
G. Kato and M. Wadati, Partition function for a one-dimensional δ-function Bose gas, Phys. Rev. E 63 (2001) 036106 [cond-mat/0212321].
I. Kostov, D. Serban and D.-L. Vu, TBA and tree expansion, Springer Proc. Math. Stat. 255 (2017) 77 [arXiv:1805.02591] [INSPIRE].
I. Kostov, D. Serban and D.-L. Vu, Boundary TBA, trees and loops, Nucl. Phys. B 949 (2019) 114817 [arXiv:1809.05705] [INSPIRE].
D.-L. Vu and T. Yoshimura, Equations of state in generalized hydrodynamics, SciPost Phys. 6 (2019) 023 [arXiv:1809.03197] [INSPIRE].
D.-L. Vu, Cumulants of conserved charges in GGE and cumulants of total transport in GHD: exact summation of matrix elements?, arXiv:1909.08852 [INSPIRE].
F. Woynarovich, O(1) contribution of saddle point fluctuations to the free energy of Bethe Ansatz systems, Nucl. Phys. B 700 (2004) 331 [cond-mat/0402129] [INSPIRE].
B. Pozsgay, On O(1) contributions to the free energy in Bethe Ansatz systems: The Exact g-function, JHEP 08 (2010) 090 [arXiv:1003.5542] [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Structure Constants in \( \mathcal{N} \) = 4 SYM at Finite Coupling as Worldsheet g-Function, arXiv:1906.07733 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09): Prague, Czech Republic, August 3–8, 2009, pp. 265–289, 2009, DOI [arXiv:0908.4052] [INSPIRE].
J. Balog, Field theoretical derivation of the TBA integral equation, Nucl. Phys. B 419 (1994) 480 [INSPIRE].
A.M. Polyakov and P.B. Wiegmann, Theory of nonabelian Goldstone bosons in two dimensions, Phys. Lett. 131B (1983) 121 [INSPIRE].
E. Ogievetsky, N. Reshetikhin and P. Wiegmann, The principal chiral field in two-dimension and classical Lie algebra, NORDITA-84/38.
L.D. Faddeev and N.Yu. Reshetikhin, Integrability of the Principal Chiral Field Model in (1+1)-dimension, Annals Phys. 167 (1986) 227 [INSPIRE].
C. Destri and H.J. De Vega, Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories, Nucl. Phys. B 438 (1995) 413 [hep-th/9407117] [INSPIRE].
C. Destri and H.J. de Vega, Non-linear integral equation and excited-states scaling functions in the sine-Gordon model, Nucl. Phys. B 504 (1997) 621 [hep-th/9701107] [INSPIRE].
P. Zinn-Justin, Quelques applications de l’ansatz de Bethe, Ph.D. Thesis, Paris University, France (1998).
D. Volin, Quantum integrability and functional equations: Applications to the spectral problem of AdS/CFT and two-dimensional σ-models, J. Phys. A 44 (2011) 124003 [arXiv:1003.4725] [INSPIRE].
J. Teschner, On the spectrum of the Sinh-Gordon model in finite volume, Nucl. Phys. B 799 (2008) 403 [hep-th/0702214] [INSPIRE].
S.L. Lukyanov, Free field representation for massive integrable models, Commun. Math. Phys. 167 (1995) 183 [hep-th/9307196] [INSPIRE].
F. Woynarovich, On the normalization of the partition function of Bethe Ansatz systems, Nucl. Phys. B 852 (2011) 269 [arXiv:1007.1148] [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Exact Three-Point Functions of Determinant Operators in Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 123 (2019) 191601 [arXiv:1907.11242] [INSPIRE].
T.R. Klassen and E. Melzer, Purely Elastic Scattering Theories and their Ultraviolet Limits, Nucl. Phys. B 338 (1990) 485 [INSPIRE].
V. Pestun, Review of localization in geometry, J. Phys. A 50 (2017) 443002 [arXiv:1608.02954] [INSPIRE].
D.-L. Vu, I. Kostov and D. Serban, Boundary entropy of integrable perturbed SU(2)k WZNW, JHEP 08 (2019) 154 [arXiv:1906.01909] [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys. A 39 (2006) 12863 [hep-th/0005181] [INSPIRE].
S.L. Lukyanov, Finite temperature expectation values of local fields in the sinh-Gordon model, Nucl. Phys. B 612 (2001) 391 [hep-th/0005027] [INSPIRE].
A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: Modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [INSPIRE].
S. Negro and F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B 875 (2013) 166 [arXiv:1306.1476] [INSPIRE].
Z. Bajnok and F. Smirnov, Diagonal finite volume matrix elements in the sinh-Gordon model, Nucl. Phys. B 945 (2019) 114664 [arXiv:1903.06990] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N} \) = 4 Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS5/CFT4 , JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
P.P. Kulish and N.Yu. Reshetikhin, Diagonalization of GL(N) invariant transfer matrices and quantum N-wave system (Lee model), J. Phys. A 16 (1983) L591 [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].
F. Coronado, Bootstrapping the simplest correlator in planar \( \mathcal{N} \) = 4 SYM at all loops, arXiv:1811.03282 [INSPIRE].
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Exact null octagon, arXiv:1907.13131 [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons I: Combinatorics and Non-Planar Resummations, JHEP 08 (2019) 162 [arXiv:1904.00965] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons II: Strong Coupling, arXiv:1909.04077 [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, The Octagon as a Determinant, JHEP 11 (2019) 178 [arXiv:1905.11467] [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, Determinant formula for the octagon form factor in \( \mathcal{N} \) = 4 SYM, Phys. Rev. Lett. 122 (2019) 231601 [arXiv:1903.05038] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
B. Basso, V. Goncalves and S. Komatsu, Structure constants at wrapping order, JHEP 05 (2017) 124 [arXiv:1702.02154] [INSPIRE].
S. Komatsu, Lectures on Three-point Functions in N = 4 Supersymmetric Yang-Mills Theory, in Proceedings, Les Houches Summer School: Integrability: From Statistical Systems to Gauge Theory: Les Houches, France, vol. 106, 2019, DOI [arXiv:1710.03853] [INSPIRE].
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ArXiv ePrint: 1911.07343
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Kostov, I. Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz. J. High Energ. Phys. 2020, 43 (2020). https://doi.org/10.1007/JHEP02(2020)043
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DOI: https://doi.org/10.1007/JHEP02(2020)043