Abstract
We review the computation of the anomalous dimension of twist operators in the planar limit of \({\mathcal{N}=4}\) SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations; this aspect of the story is emphasized.
Similar content being viewed by others
References
Janik, R.: Review of AdS/CFT Integrability, Chapter III.5: Lüscher corrections, arxiv:1012.3994
Kazakov, V., Gromov, N.: Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability, arxiv:1012.3996
Bajnok, Z.: Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz, arxiv:1012.3995
Korchemsky, G.: Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and \({\fancyscript{N}}\) < 4 SYM, arxiv:1012.4000
Staudacher, M.: Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism, arxiv:1012.3990
Ahn, C., Nepomechie, R.I.: Review of AdS/CFT Integrability, Chapter III.2: Exact world-sheet S-matrix, arxiv:1012.3991
Vieira, P., Volin, D.: Review of AdS/CFT Integrability, Chapter III.3: The dressing factor, arxiv:1012.3992
Staudacher M.: The factorized S-matrix of CFT/AdS. JHEP 0505, 054 (2005) hep-th/0412188
Beisert N., Staudacher M.: Long-range PSU(2,2/4) Bethe ansaetze for gauge theory and strings. Nucl. Phys. B 727, 1 (2005) hep-th/0504190
Janik R.A.: The AdS5 × S5 superstring worldsheet S-matrix and crossing symmetry. Phys. Rev. D 73, 086006 (2006) hep-th/0603038
Gomez C., Hernandez R.: The magnon kinematics of the AdS/CFT correspondence. JHEP 0611, 021 (2006) hep-th/0608029
Plefka J., Spill F., Torrielli A.: On the Hopf algebra structure of the AdS/CFT S-matrix. Phys. Rev. D 74, 066008 (2006) hep-th/0608038
Eden B., Staudacher M.: Integrability and transcendentality. J. Stat. Mech. 0611, P014 (2006) hep-th/0603157
Beisert N., Eden B., Staudacher M.: Transcendentality and crossing. J. Stat. Mech. 0701, P021 (2007) hep-th/0610251
Anastasiou C., Bern Z., Dixon L.J., Kosower D.A.: Planar amplitudes in maximally supersymmetric Yang-Mills theory. Phys. Rev. Lett. 91, 251602 (2003) hep-th/0309040
Bern Z., Dixon L.J., Smirnov V.A.: Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond. Phys. Rev. D 72, 085001 (2005) hep-th/0505205
Bern Z., Czakon M., Dixon L.J., Kosower D.A., Smirnov V.A.: The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. Phys. Rev. D 75, 085010 (2007) hep-th/0610248
Volin, D.: Minimal solution of the AdS/CFT crossing equation, arxiv:0904.4929
Collins J.C.: Sudakov form factors. Adv. Ser. Direct. High Energy Phys. 5, 573 (1989) hep-ph/0312336
Belitsky A.V., Gorsky A.S., Korchemsky G.P.: Logarithmic scaling in gauge/string correspondence. Nucl. Phys. B 748, 24 (2006) hep-th/0601112
Korchemsky G.P., Radyushkin A.V.: Loop space formalism and renormalization group for the infrared asymptotics of QCD. Phys. Lett. B 171, 459 (1986)
Korchemsky G.P.: Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of Parton distributions. Mod. Phys. Lett. A 4, 1257 (1989)
Korchemsky G.P., Marchesini G.: Structure function for large x and renormalization of Wilson loop. Nucl. Phys. B 406, 225 (1993) hep-ph/9210281
Kruczenski M.: A note on twist two operators in N = 4 SYM and Wilson loops in Minkowski signature. JHEP 0212, 024 (2002) hep-th/0210115
Alday L.F., Maldacena J.M.: Gluon scattering amplitudes at strong couplin. JHEP 0706, 064 (2007) arxiv:0705.0303
Sieg, C.: Review of AdS/CFT Integrability, Chapter I.2: The spectrum from perturbative gauge theory, arxiv:1012.3984
Tseytlin, A.: Review of AdS/CFT Integrability, Chapter II.1: Classical AdS5 × S5 string solutions, arxiv:1012.3986
McLoughlin, T.: Review of AdS/CFT Integrability, Chapter II.2: Quantum Strings in AdS5 × S5, arxiv:1012.3987
Gubser S.S., Klebanov I.R., Polyakov A.M.: A semi-classical limit of the gauge/string correspondence. Nucl. Phys. B 636, 99 (2002) hep-th/0204051
Frolov S., Tseytlin A.A.: Semiclassical quantization of rotating superstring in AdS5 × S5. JHEP 0206, 007 (2002) hep-th/0204226
Roiban R., Tirziu A., Tseytlin A.A.: Two-loop world-sheet corrections in AdS5 × S5 superstring. JHEP 0707, 056 (2007) arxiv:0704.3638
Roiban R., Tseytlin A.A.: Strong-coupling expansion of cusp anomaly from quantum superstring. JHEP 0711, 016 (2007) arxiv:0709.0681
Beccaria, M., Dunne, G.V., Forini, V., Pawellek, M., Tseytlin, A.A.: Exact computation of one-loop correction to energy of spinning folded string in AdS5 × S5, arxiv:1001.4018
Kotikov A.V., Lipatov L.N.: On the highest transcendentality in \({\fancyscript{N} =}\) 4 SUSY. Nucl. Phys. B 769, 217 (2007) hep-th/0611204
Alday L.F., Arutyunov G., Benna M.K., Eden B., Klebanov I.R.: On the strong coupling scaling dimension of high spin operators. JHEP 0704, 082 (2007) hep-th/0702028
Kostov I., Serban D., Volin D.: Strong coupling limit of Bethe ansatz equations. Nucl. Phys. B 789, 413 (2008) hep-th/0703031
Beccaria M., De Angelis G.F., Forini V.: The scaling function at strong coupling from the quantum string Bethe equations. JHEP 0704, 066 (2007) hep-th/0703131
Basso B., Korchemsky G.P., Kotanski J.: Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling. Phys. Rev. Lett. 100, 091601 (2008) arxiv:0708.3933
Casteill P.Y., Kristjansen C.: The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz. Nucl. Phys. B 785, 1 (2007) arxiv:0705.0890
Kostov I., Serban D., Volin D.: Functional BES equation. JHEP 0808, 101 (2008) arxiv:0801.2542
Basso B., Korchemsky G.P.: Nonperturbative scales in AdS/CFT. J. Phys. A 42, 254005 (2009) arxiv:0901.4945
Belitsky A.V., Gorsky A.S., Korchemsky G.P.: Gauge/string duality for QCD conformal operators. Nucl. Phys. B 667, 3 (2003) hep-th/0304028
Belitsky A.V., Korchemsky G.P., Pasechnik R.S.: Fine structure of anomalous dimensions in \({\fancyscript{N} =}\) 4 super Yang-Mills theory. Nucl. Phys. B 809, 244 (2009) arxiv:0806.3657
Korchemsky G.P.: Quasiclassical QCD pomeron. Nucl. Phys. B 462, 333 (1996) hep-th/9508025
Freyhult L., Rej A., Staudacher M.: A generalized scaling function for AdS/CFT. J. Stat. Mech. 0807, P07015 (2008) arxiv:0712.2743
Feverati G., Fioravanti D., Grinza P., Rossi M.: On the finite size corrections of anti-ferromagnetic anomalous dimensions in \({\fancyscript{N} =}\) 4 SYM. JHEP 0605, 068 (2006) hep-th/0602189
Feverati G., Fioravanti D., Grinza P., Rossi M.: Hubbard’s adventures in \({\fancyscript{N} =}\) 4 SYM-land? Some non-perturbative considerations on finite length operators. J. Stat. Mech. 0702, P001 (2007) hep-th/0611186
Fioravanti D., Rossi M.: On the commuting charges for the highest dimension SU(2) operator in planar \({\fancyscript{N} =}\) 4 SYM. JHEP 0708, 089 (2007) arxiv:0706.3936
Bombardelli, D., Fioravanti, D., Rossi, M.: Non-linear integral equations in \({\fancyscript{N}=}\) 4 SYM, arxiv:0711.2934
Bombardelli D., Fioravanti D., Rossi M.: Large spin corrections in \({\fancyscript{N} =}\) 4 SYM sl(2): still a linear integral equation. Nucl. Phys. B 810, 460 (2009) arxiv:0802.0027
Freyhult L., Zieme S.: The virtual scaling function of AdS/CFT. Phys. Rev. D 79, 105009 (2009) arxiv:0901.2749
Fioravanti, D., Grinza, P., Rossi, M.: On the logarithmic powers of sl(2) SYM4, arxiv:0911.2425
Kotikov A.V., Lipatov L.N.: DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory. Nucl. Phys. B 661, 19 (2003) hep-ph/0208220
Georgi H., Politzer H.D.: Electroproduction scaling in an asymptotically free theory of strong interactions. Phys. Rev. D 9, 416 (1974)
Gross D.J., Wilczek F.: Asymptotically free gauge theories. 1. Phys. Rev. D 8, 3633 (1973)
Dolan F.A., Osborn H.: Conformal four point functions and the operator product expansion. Nucl. Phys. B 599, 459 (2001) hep-th/0011040
Kotikov A.V., Lipatov L.N., Velizhanin V.N.: Anomalous dimensions of Wilson operators in N = 4 SYM theory. Phys. Lett. B 557, 114 (2003) hep-ph/0301021
Kotikov A.V., Lipatov L.N., Onishchenko A.I., Velizhanin V.N.: Three-loop universal anomalous dimension of the Wilson operators in \({\fancyscript{N} =}\) 4 SUSY Yang-Mills model. Phys. Lett. B 595, 521 (2004) hep-th/0404092
Moch S., Vermaseren J.A.M., Vogt A.: The three-loop splitting functions in QCD: the non-singlet case. Nucl. Phys. B 688, 101 (2004) hep-ph/0403192
Vermaseren J.: Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A 14, 2037 (1999) hep-ph/9806280
Beccaria M.: Anomalous dimensions at twist-3 in the sl(2) sector of \({\fancyscript{N} =}\) 4 SYM. JHEP 0706, 044 (2007) arxiv:0704.3570
Kotikov A.V., Lipatov L.N., Rej A., Staudacher M., Velizhanin V.N.: Dressing and Wrapping. J. Stat. Mech. 0710, P10003 (2007) arxiv:0704.3586
Beccaria M., Dokshitzer Y.L., Marchesini G.: Twist 3 of the sl(2) sector of \({\fancyscript{N} =}\) 4 SYM and reciprocity respecting evolution. Phys. Lett. B 652, 194 (2007) arxiv:0705.2639
Beccaria M., Forini V., Lukowski T., Zieme S.: Twist-three at five loops, Bethe Ansatz and wrapping. JHEP 0903, 129 (2009) arxiv:0901.4864
Lukowski, T., Rej, A., Velizhanin, V.N.: Five-Loop Anomalous Dimension of Twist-Two Operators, arxiv:0912.1624
Beccaria M., Catino F.: Sum rules for higher twist sl(2) operators in \({\fancyscript{N} =}\) 4 SYM. JHEP 0806, 103 (2008) arxiv:0804.3711
Derkachov S.E., Korchemsky G.P., Manashov A.N.: Evolution equations for quark gluon distributions in multi-color QCD and open spin chains. Nucl. Phys. B 566, 203 (2000) hep-ph/9909539
Belitsky A.V., Korchemsky G.P., Mueller D.: Towards Baxter equation in supersymmetric Yang-Mills theories. Nucl. Phys. B 768, 116 (2007) hep-th/0605291
Belitsky A.V.: Long-range SL(2) Baxter equation in \({\fancyscript{N} =}\) 4 super-Yang-Mills theory. Phys. Lett. B 643, 354 (2006) hep-th/0609068
Belitsky A.V.: Baxter equation beyond wrapping. Phys. Lett. B 677, 93 (2009) arxiv:0902.3198
Kotikov A.V., Rej A., Zieme S.: Analytic three-loop Solutions for \({\fancyscript{N} =}\) 4 SYM Twist Operators. Nucl. Phys. B 813, 460 (2009) arxiv:0810.0691
Beccaria, M., Belitsky, A.V., Kotikov, A.V., Zieme, S.: Analytic solution of the multiloop Baxter equation, arxiv:0908.0520
Kuraev E., Lipatov L., Fadin V.S.: The Pomeranchuk singularity in nonabelian gauge theories. Sov. Phys. JETP 45, 199 (1977)
Balitsky I., Lipatov L.: The Pomeranchuk singularity in quantum chromodynamics. Sov. J. Nucl. Phys. 28, 822 (1978)
Bajnok Z., Janik R.A., Lukowski T.: Four loop twist two, BFKL, wrapping and strings. Nucl. Phys. B 816, 376 (2009) arxiv:0811.4448
Beccaria M., Forini V.: Four loop reciprocity of twist two operators in \({\fancyscript{N} =}\) 4 SYM. JHEP 0903, 111 (2009) arxiv:0901.1256
Dokshitzer Y.L., Marchesini G., Salam G.P.: Revisiting parton evolution and the large-x limit. Phys. Lett. B 634, 504 (2006) hep-ph/0511302
Basso B., Korchemsky G.P.: Anomalous dimensions of high-spin operators beyond the leading order. Nucl. Phys. B 775, 1 (2007) hep-th/0612247
Dokshitzer Y.L., Marchesini G.: \({\fancyscript{N} =}\) 4 SUSY Yang-Mills: three loops made simple(r). Phys. Lett. B 646, 189 (2007) hep-th/0612248
Beccaria, M., Forini, V., Macorini, G.: Generalized Gribov-Lipatov reciprocity and AdS/CFT, arxiv:1002.2363
Braun V.M., Korchemsky G.P., Mueller D.: The uses of conformal symmetry in QCD. Prog. Part. Nucl. Phys. 51, 311 (2003) hep-ph/0306057
Beccaria M., Forini V.: Reciprocity of gauge operators in \({\fancyscript{N} =}\) 4 SYM. JHEP 0806, 077 (2008) arxiv:0803.3768
Gribov V.N., Lipatov L.N.: e+ e- pair annihilation and deep inelastic e p scattering in perturbation theory. Sov. J. Nucl. Phys. 15, 675 (1972)
Vogt A., Moch S., Vermaseren J.A.M.: The three-loop splitting functions in QCD: The singlet case. Nucl. Phys. B 691, 129 (2004) hep-ph/0404111
Beccaria M.: Universality of three gaugino anomalous dimensions in \({\fancyscript{N} =}\) 4 SYM. JHEP 0706, 054 (2007) arxiv:0705.0663
Beccaria M.: Three loop anomalous dimensions of twist-3 gauge operators in \({\fancyscript{N}}\) = 4 SYM. JHEP 0709, 023 (2007) arxiv:0707.1574
Giombi, S., Ricci, R., Roiban, R., Tseytlin, A.A., Vergu, C.: Quantum AdS 5 × S 5 superstring in the AdS light-cone gauge, arxiv:0912.5105
Klose, T.: Review of AdS/CFT Integrability, Chapter IV.3: \({\fancyscript{N} =\, }\) 6 Chern-Simons and Strings on AdS4 × CP3, arxiv:1012.3999
Benna M.K., Benvenuti S., Klebanov I.R., Scardicchio A.: A test of the AdS/CFT correspondence using high-spin operators. Phys. Rev. Lett. 98, 131603 (2007) hep-th/0611135
Belitsky A.V.: Strong coupling expansion of Baxter equation in \({\fancyscript{N} =\, }\) 4 SYM. Phys. Lett. B 659, 732 (2008) arxiv:0710.2294
Kotikov A.V., Lipatov L.N.: Eden-Staudacher and Beisert-Eden-Staudacher equations in the \({\fancyscript{N} = }\) 4 supersymmetric gauge theory. Theor. Math. Phys. 155, 606 (2008)
Eden, B.: The BES equation and its strong coupling limit. In: Talk at 12th Itzykson meeting, Saclay, Paris
Volin, D.: Quantum integrability and functional equations, arxiv:1003.4725
Beccaria M., Forini V., Tirziu A., Tseytlin A.A.: Structure of large spin expansion of anomalous dimensions at strong coupling. Nucl. Phys. B 812, 144 (2009) arxiv:0809.5234
Basso, B.: Exciting the GKP string at any coupling, arxiv:1010.5237
Freyhult, L., Rej, A., Zieme, S.: From weak coupling to spinning strings, arxiv:0911. 2458
Rej A., Staudacher M., Zieme S.: Nesting and dressing. J. Stat. Mech. 0708, P08006 (2007) hep-th/0702151
Kotanski J.: Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. Acta Phys. Polon. B 39, 3127 (2008) arxiv:0811.2667
Alday L.F., Maldacena J.M.: Comments on operators with large spin. JHEP 0711, 019 (2007) arxiv:0708.0672
Basso B., Korchemsky G.P.: Embedding nonlinear O(6) sigma model into \({\fancyscript{N} = }\) 4 super-Yang-Mills theory. Nucl. Phys. B 807, 397 (2009) arxiv:0805.4194
Bajnok Z., Balog J., Basso B., Korchemsky G.P., Palla L.: Scaling function in AdS/CFT from the O(6) sigma model. Nucl. Phys. B 811, 438 (2009) arxiv:0809.4952
Giombi, S., Ricci, R., Roiban, R., Tseytlin, A.A., Vergu, C.: Generalized scaling function from light-cone gauge AdS5 × S5 superstring, arxiv:1002.0018
Fioravanti, D., Grinza, P., Rossi, M.: The generalised scaling function: a note, arxiv:0805.4407
Buccheri, F., Fioravanti, D.: The integrable O(6) model and the correspondence: checks and predictions, arxiv:0805.4410
Gromov, N.: Generalized Scaling Function at Strong Coupling. JHEP 0811, 085 (2008). arxiv:0805.4615
Beccaria M.: The generalized scaling function of AdS/CFT and semiclassical string theory. JHEP 0807, 082 (2008) arxiv:0806.3704
Fioravanti, D., Grinza, P., Rossi, M.: The generalised scaling function: a systematic study, arxiv:0808.1886
Volin, D.: The 2-loop generalized scaling function from the BES/FRS equation, arxiv:0812.4407
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Freyhult, L. Review of AdS/CFT Integrability, Chapter III.4: Twist States and the Cusp Anomalous Dimension. Lett Math Phys 99, 255–276 (2012). https://doi.org/10.1007/s11005-011-0483-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-011-0483-z