Abstract
The spectrum of the light-cone AdS5×S5 superstring contains states composed of particles with complex momenta including in particular those which turn into bound states in the decompactification limit. We propose the mirror TBA description for these states. We focus on a three-particle state which is a finite-size representative of a scattering state of a fundamental particle and a two-particle bound state and dual to an operator from the \( \mathfrak{s}\mathfrak{u}(2) \) sector of \( \mathcal{N} = 4\;{\text{SYM}} \). We find that the analytic behavior of Y-functions differs drastically from the case of states with real momenta. Most importantly, Y Q -functions exhibit poles in the analyticity strip which leads to the appearance of new terms in the formula for the energy of this state. In addition, the TBA equations are supplied by quantization conditions which involve Y 2. Considering yet another example of a three- particle state, we find that the corresponding quantization conditions do not even involve Y 1. Our treatment can be generalized to a wide class of states with complex momenta.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].
A. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011) 103001 [arXiv:1010.1344] [INSPIRE].
Z. Bajnok, Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz, arXiv:1012.3995 [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5×S 5 superstring. Part I, J. Phys. A A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, arXiv:1012.3982 [INSPIRE].
M. Takahashi, One-dimensional Hubbard model at finite temperature, Prog. Theor. Phys. 47 (1972) 69.
G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
G. Arutyunov and S. Frolov, String hypothesis for the AdS 5×S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5×S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].
N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [INSPIRE].
J. Balog and A. Hegedus, The Bajnok-Janik formula and wrapping corrections, JHEP 09 (2010) 107 [arXiv:1003.4303] [INSPIRE].
A. Sfondrini and S.J. van Tongeren, Lifting asymptotic degeneracies with the mirror TBA, JHEP 09 (2011) 050 [arXiv:1106.3909] [INSPIRE].
G. Arutyunov and S. Frolov, Simplified TBA equations of the AdS 5×S 5 mirror model, JHEP 11 (2009) 019 [arXiv:0907.2647] [INSPIRE].
J. Balog and A. Hegedus, Quasi-local formulation of the mirror TBA, arXiv:1106.2100 [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [INSPIRE].
S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].
N. Gromov, D. Serban, I. Shenderovich and D. Volin, Quantum folded string and integrability: from finite size effects to Konishi dimension, JHEP 08 (2011) 046 [arXiv:1102.1040] [INSPIRE].
R. Roiban and A. Tseytlin, Semiclassical string computation of strong-coupling corrections to dimensions of operators in Konishi multiplet, Nucl. Phys. B 848 (2011) 251 [arXiv:1102.1209] [INSPIRE].
B.C. Vallilo and L. Mazzucato, The Konishi multiplet at strong coupling, JHEP 12 (2011) 029 [arXiv:1102.1219] [INSPIRE].
M. Beccaria and G. Macorini, Quantum folded string in S 5 and the Konishi multiplet at strong coupling, JHEP 10 (2011) 040 [arXiv:1108.3480] [INSPIRE].
N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Five-loop Konishi from the mirror TBA, JHEP 04 (2010) 069 [arXiv:1002.1711] [INSPIRE].
J. Balog and A. Hegedus, 5-loop Konishi from linearized TBA and the XXX magnet, JHEP 06 (2010) 080 [arXiv:1002.4142] [INSPIRE].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].
Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [INSPIRE].
T. Lukowski, A. Rej and V. Velizhanin, Five-loop anomalous dimension of twist-two operators, Nucl. Phys. B 831 (2010) 105 [arXiv:0912.1624] [INSPIRE].
R.A. Janik, Review of AdS/CFT integrability, Chapter III.5: Lúscher corrections, arXiv:1012.3994 [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Wrapping at four loops in N = 4 SYM, Phys. Lett. B 666 (2008) 100 [arXiv:0712.3522] [INSPIRE].
V. Velizhanin, The four-loop anomalous dimension of the Konishi operator in N = 4 supersymmetric Yang-Mills theory, JETP Lett. 89 (2009) 6 [arXiv:0808.3832] [INSPIRE].
G. Arutyunov and S. Frolov, Comments on the mirror TBA, JHEP 05 (2011) 082 [arXiv:1103.2708] [INSPIRE].
A. Klümper and P.A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchy, Physica A 183 (1992) 304.
A. Klümper and P.A. Pearce, New results for exactly solvable critical RSOS models and vertex models in two dimensions, Physica A 194 (1993) 397.
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [INSPIRE].
N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe Ansatze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].
N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of conformal N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [INSPIRE].
N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].
V. Kazakov and S. Leurent, Finite size spectrum of SU(N ) principal chiral field from discrete Hirota dynamics, arXiv:1007.1770 [INSPIRE].
J. Balog, SU(k) principal model TBA and energy formula, unpublished.
A. Cavaglia, D. Fioravanti, M. Mattelliano and R. Tateo, On the AdS 5 /CFT 4 TBA and its analytic properties, arXiv:1103.0499 [INSPIRE].
G. Arutyunov, S. Frolov and M. Staudacher, Bethe ansatz for quantum strings, JHEP 10 (2004) 016 [hep-th/0406256] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
A. Kuniba, T. Nakanishi and J. Suzuki, Functional relations in solvable lattice models. 1: Functional relations and representation theory, Int. J. Mod. Phys. A 9 (1994) 5215 [hep-th/9309137] [INSPIRE].
R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981) 3785.
R. Suzuki, Hybrid NLIE for the mirror AdS 5 ×S 5 , J. Phys. A 44 (2011) 235401 [arXiv:1101.5165] [INSPIRE].
A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5 /CFT 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].
R.A. Janik, The AdS 5 ×S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
G. Arutyunov and S. Frolov, On AdS 5 ×S 5 string S-matrix, Phys. Lett. B 639 (2006) 378 [hep-th/0604043] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N=4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
J. Balog and A. Hegedus, AdS 5 ×S 5 mirror TBA equations from Y-system and discontinuity relations, JHEP 08 (2011) 095 [arXiv:1104.4054] [INSPIRE].
P. Pearce and A. Klüemper, Finite size corrections and scaling dimensions of solvable lattice models: an analytic method, Phys. Rev. Lett. 66 (1991) 974 [INSPIRE].
A. Klümper and P.A. Pearce, Analytic calculation of scaling dimensions: tricritical hard squares and critical hard hexagons, J. Stat. Phys. 64 (1991) 13.
N. Gromov and V. Kazakov, talk at Conference on Integrability in Gauge and String Theory 2010, Nordita, Stockholm Sweden, June 2010, http://agenda.albanova.se/contributionDisplay.py?contribId=258&confId=1561.
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, arXiv:1110.0562 [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Finite-size effect for four-loop Konishi of the β-deformed N = 4 SYM, Phys. Lett. B 693 (2010) 380 [arXiv:1006.2209] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, Orbifolded Konishi from the mirror TBA, J. Phys. A A 44 (2011) 325404 [arXiv:1103.5853] [INSPIRE].
M. Beccaria and G. Macorini, Y-system for Z S orbifolds of N = 4 SYM, JHEP 06 (2011) 004 [arXiv:1104.0883] [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].
G. Arutyunov and S. Frolov, The dressing factor and crossing equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [INSPIRE].
G. Arutyunov and S. Frolov, The S-matrix of string bound states, Nucl. Phys. B 804 (2008) 90 [arXiv:0803.4323] [INSPIRE].
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ArXiv ePrint: 1111.0564
Correspondent fellow at Steklov Mathematical Institute, Moscow. (Gleb Arutyunov, Sergey Frolov)
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Arutyunov, G., Frolov, S. & van Tongeren, S.J. Bound states in the mirror TBA. J. High Energ. Phys. 2012, 14 (2012). https://doi.org/10.1007/JHEP02(2012)014
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DOI: https://doi.org/10.1007/JHEP02(2012)014