Abstract
We demonstrate that the large-spin expansion of the energy of Gubser-Klebanov-Polyakov (GKP) strings that rotate in ℝ × S2 and AdS3 can be expressed in terms of Lambert’s W-function. We compute the leading, subleading and next-to-subleading series of exponential corrections to the infinite-volume dispersion relation of GKP strings that rotate in ℝ × S2. These strings are dual to the long \( \mathcal{N} \) = 4 SYM operators \( \mathrm{Tr}\left[ {\varPhi {{\mathcal{Z}}^m}\varPhi {{\mathcal{Z}}^{J-m }}} \right] \)+… and provide their scaling dimensions at strong coupling. We also show that the strings obey a short-long (strings) duality. For the folded GKP strings that spin inside AdS3 and are dual to twist-2 operators, we confirm the known formulas for the leading and next-to-leading coefficients of their anomalous dimensions and derive the corresponding expressions for the next-to-next-to-leading coefficients.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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) 1113] [hep-th/9711200] [INSPIRE].
S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS /CFT correspondence, hep-th/0201253 [INSPIRE].
C. Kristjansen, M. Staudacher and A. Tseytlin, Gauge-string duality and integrability: Progress and outlook, J. Phys. A 42 (2009) 250301 [INSPIRE].
N. Beisert, C. Ahn, L.F. Alday, Z. Bajnok, J.M. Drummond et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
A.A. Tseytlin, Review of AdS/CFT Integrability, Chapter II.1: Classical AdS5xS5 string solutions, Lett. Math. Phys. 99 (2012) 103 [arXiv:1012.3986] [INSPIRE].
M. Axenides, E. Floratos and A. Kehagias, Scaling violations in Yang-Mills theories and strings in AdS 5, Nucl. Phys. B 662 (2003) 170 [hep-th/0210091] [INSPIRE].
H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [INSPIRE].
D.J. Gross and F. Wilczek, Asymptotically free gauge theories. 2., Phys. Rev. D 9 (1974) 980 [INSPIRE].
E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories: The Anomalous Dimensions of Wilson Operators, Nucl. Phys. B 129 (1977) 66 [Erratum ibid. B 139 (1978) 545-546] [INSPIRE].
E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher Order Effects in Asymptotically Free Gauge Theories. 2. Flavor Singlet Wilson Operators and Coefficient Functions, Nucl. Phys. B 152 (1979) 493 [INSPIRE].
G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].
E.G. Floratos, C. Kounnas and R. Lacaze, Higher Order QCD Effects in Inclusive Annihilation and Deep Inelastic Scattering, Nucl. Phys. B 192 (1981) 417 [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
A. Vogt, S. Moch and J.A.M. Vermaseren, The Three-loop splitting functions in QCD: The Singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].
A.V. Kotikov and L.N. Lipatov, DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory, hep-ph/0112346 [INSPIRE].
A.V. Kotikov, L.N. Lipatov and V.N. Velizhanin, Anomalous dimensions of Wilson operators in N = 4 SYM theory, Phys. Lett. B 557 (2003) 114 [hep-ph/0301021] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754-756] [hep-th/0404092] [INSPIRE].
B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. 0611 (2006) P11014 [hep-th/0603157] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
B. Basso, G.P. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett. 100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].
I. Kostov, D. Serban and D. Volin, Functional BES equation, JHEP 08 (2008) 101 [arXiv:0801.2542] [INSPIRE].
S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [INSPIRE].
R. Roiban, A. Tirziu and A.A. Tseytlin, Two-loop world-sheet corrections in AdS 5 × S 5 superstring, JHEP 07 (2007) 056 [arXiv:0704.3638] [INSPIRE].
R. Roiban and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly from quantum superstring, JHEP 11 (2007) 016 [arXiv:0709.0681] [INSPIRE].
A.V. Kotikov, A. Rej and S. Zieme, Analytic three-loop Solutions for \( \mathcal{N} \) = 4 SYM Twist Operators, Nucl. Phys. B 813 (2009) 460 [arXiv:0810.0691] [INSPIRE].
M. Beccaria, A.V. Belitsky, A.V. Kotikov and S. Zieme, Analytic solution of the multiloop Baxter equation, Nucl. Phys. B 827 (2010) 565 [arXiv:0908.0520] [INSPIRE].
Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].
T. Lukowski, A. Rej and V.N. Velizhanin, Five-Loop Anomalous Dimension of Twist-Two Operators, Nucl. Phys. B 831 (2010) 105 [arXiv:0912.1624] [INSPIRE].
G. Georgiou, Two and three-point correlators of operators dual to folded string solutions at strong coupling, JHEP 02 (2011) 046 [arXiv:1011.5181] [INSPIRE].
G. Georgiou, SL(2) sector: weak/strong coupling agreement of three-point correlators, JHEP 09 (2011) 132 [arXiv:1107.1850] [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].
M. Beccaria, V. Forini, A. Tirziu and A.A. Tseytlin, Structure of large spin expansion of anomalous dimensions at strong coupling, Nucl. Phys. B 812 (2009) 144 [arXiv:0809.5234] [INSPIRE].
M. Beccaria, G.V. Dunne, V. Forini, M. Pawellek and A.A. Tseytlin, Exact computation of one-loop correction to energy of spinning folded string in AdS 5 × S 5, J. Phys. A 43 (2010) 165402 [arXiv:1001.4018] [INSPIRE].
G. Georgiou and G. Savvidy, Large spin behavior of anomalous dimensions and short-long strings duality, J. Phys. A 44 (2011) 305402 [arXiv:1012.5580] [INSPIRE].
G. Savvidy, Non-Abelian Tensor Gauge Fields, Proc. Steklov Inst. Math. 272 (2011) 201 [arXiv:1004.4456] [INSPIRE].
Z. Chong, H. Lü and C. Pope, Rotating strings in massive type IIA supergravity, hep-th/0402202 [INSPIRE].
D.M. Hofman and J.M. Maldacena, Giant Magnons, J. Phys. A 39 (2006) 13095 [hep-th/0604135] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, Dressing and wrapping, J. Stat. Mech. 0710 (2007) P10003 [arXiv:0704.3586] [INSPIRE].
S. Schäfer-Nameki, M. Zamaklar and K. Zarembo, Quantum corrections to spinning strings in AdS 5 × S 5 and Bethe ansatz: A Comparative study, JHEP 09 (2005) 051 [hep-th/0507189] [INSPIRE].
S. Schäfer-Nameki and M. Zamaklar, Stringy sums and corrections to the quantum string Bethe ansatz, JHEP 10 (2005) 044 [hep-th/0509096] [INSPIRE].
S. Schäfer-Nameki, Exact expressions for quantum corrections to spinning strings, Phys. Lett. B 639 (2006) 571 [hep-th/0602214] [INSPIRE].
S. Schäfer-Nameki, M. Zamaklar and K. Zarembo, How Accurate is the Quantum String Bethe Ansatz?, JHEP 12 (2006) 020 [hep-th/0610250] [INSPIRE].
J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [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].
Z. Bajnok, Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz, Lett. Math. Phys. 99 (2012) 299 [arXiv:1012.3995] [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].
G. Arutyunov, S. Frolov and M. Zamaklar, Finite-size Effects from Giant Magnons, Nucl. Phys. B 778 (2007) 1 [hep-th/0606126] [INSPIRE].
D. Astolfi, V. Forini, G. Grignani and G.W. Semenoff, Gauge invariant finite size spectrum of the giant magnon, Phys. Lett. B 651 (2007) 329 [hep-th/0702043] [INSPIRE].
J.A. Minahan and O. Ohlsson Sax, Finite size effects for giant magnons on physical strings, Nucl. Phys. B 801 (2008) 97 [arXiv:0801.2064] [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].
T.R. Klassen and E. Melzer, On the relation between scattering amplitudes and finite size mass corrections in QFT, Nucl. Phys. B 362 (1991) 329 [INSPIRE].
R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling: The Giant magnon, Phys. Rev. D 76 (2007) 126008 [arXiv:0708.2208] [INSPIRE].
M.P. Heller, R.A. Janik and T. Lukowski, A New derivation of Lüscher F-term and fluctuations around the giant magnon, JHEP 06 (2008) 036 [arXiv:0801.4463] [INSPIRE].
N. Gromov, S. Schäfer-Nameki and P. Vieira, Quantum Wrapped Giant Magnon, Phys. Rev. D 78 (2008) 026006 [arXiv:0801.3671] [INSPIRE].
K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].
T. Klose and T. McLoughlin, Interacting finite-size magnons, J. Phys. A 41 (2008) 285401 [arXiv:0803.2324] [INSPIRE].
C.-S. Chu, G. Georgiou and V.V. Khoze, Magnons, classical strings and β-deformations, JHEP 11 (2006) 093 [hep-th/0606220] [INSPIRE].
N. Bobev and R. Rashkov, Multispin Giant Magnons, Phys. Rev. D 74 (2006) 046011 [hep-th/0607018] [INSPIRE].
D.V. Bykov and S. Frolov, Giant magnons in TsT-transformed AdS 5 × S 5, JHEP 07 (2008) 071 [arXiv:0805.1070] [INSPIRE].
C. Ahn and P. Bozhilov, Finite-Size Dyonic Giant Magnons in TsT-transformed AdS 5 × S 5, JHEP 07 (2010) 048 [arXiv:1005.2508] [INSPIRE].
D. Gaiotto, S. Giombi and X. Yin, Spin Chains in \( \mathcal{N} \) = 6 Superconformal Chern-Simons-Matter Theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [INSPIRE].
G. Grignani, T. Harmark and M. Orselli, The SU(2) × SU(2) sector in the string dual of \( \mathcal{N} \) =6 superconformal Chern-Simons theory, Nucl. Phys. B 810 (2009) 115 [arXiv:0806.4959] [INSPIRE].
M. Abramowitz and I. Stegun eds., Handbook of Mathematical Functions, Dover, New York (1972).
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge (1958).
I.N. Galidakis, On an Application of Lambert’s W Function to Infinite Exponentials, Complex Var. Theory App. 49 (2006) 759.
M. Beccaria, V. Forini and G. Macorini, Generalized Gribov-Lipatov Reciprocity and AdS/CFT, Adv. High Energy Phys. 2010 (2010) 753248 [arXiv:1002.2363] [INSPIRE].
T. Fukushima, Numerical Computation of Inverse Complete Elliptic Integrals of First and Second Kinds, J. Comput. Appl. Math. 249 (2013) 37.
N. Khuri and H. Ren, Explicit Solutions for the Running Coupling Constant and the Separatrix of Quantum Field Theories, Annals Phys. 189 (1989) 142 [INSPIRE].
T. Appelquist, A. Ratnaweera, J. Terning and L.C.R. Wijewardhana, The Phase structure of an SU(N) gauge theory with N f flavors, Phys. Rev. D 58 (1998) 105017 [hep-ph/9806472] [INSPIRE].
E. Gardi, G. Grunberg and M. Karliner, Can the QCD running coupling have a causal analyticity structure?, JHEP 07 (1998) 007 [hep-ph/9806462] [INSPIRE].
B.A. Magradze, Analytic approach to perturbative QCD, Int. J. Mod. Phys. A 15 (2000) 2715 [hep-ph/9911456] [INSPIRE].
C. Csáki and M. Reece, Toward a systematic holographic QCD: A Braneless approach, JHEP 05 (2007) 062 [hep-ph/0608266] [INSPIRE].
H. Sonoda, Solving renormalization group equations with the Lambert W function, Phys. Rev. D 87 (2013) 085023 [arXiv:1302.6069] [INSPIRE].
T.L. Curtright and C.K. Zachos, Renormalization Group Functional Equations, Phys. Rev. D 83 (2011) 065019 [arXiv:1010.5174] [INSPIRE].
M. Kruczenski, Spiky strings and single trace operators in gauge theories, JHEP 08 (2005) 014 [hep-th/0410226] [INSPIRE].
K. Zoubos, Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
C. Ahn and P. Bozhilov, Finite-size effects of Membranes on AdS 4 × S 7, JHEP 08 (2008) 054 [arXiv:0807.0566] [INSPIRE].
S.A. Hartnoll and C. Nuñez, Rotating membranes on G 2 manifolds, logarithmic anomalous dimensions and N = 1 duality, JHEP 02 (2003) 049 [hep-th/0210218] [INSPIRE].
M. Axenides, E. Floratos and G. Linardopoulos, Stringy Membranes in AdS/CFT, JHEP 08 (2013) 089 [arXiv:1306.0220] [INSPIRE].
R. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329 [INSPIRE].
L. Comtet, Advanced Combinatorics, Reidel (1974).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1311.5800
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Floratos, E., Georgiou, G. & Linardopoulos, G. Large-spin expansions of GKP strings. J. High Energ. Phys. 2014, 18 (2014). https://doi.org/10.1007/JHEP03(2014)018
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2014)018