Abstract
We start with an n−point correlation function in a conformal gauge theory. We show that a special limit produces a polygonal Wilson loop with n sides. The limit takes the n points towards the vertices of a null polygonal Wilson loop such that successive distances \( x_{i,i + 1}^2 \to 0 \). This produces a fast moving particle that generates a “frame” for the Wilson loop. We explainin detail how the limit is approached, including some subtle effects from the propagation of a fast moving particle in the full interacting theory. We perform perturbative checks by doing explicit computations in \( \mathcal{N} = 4 \) super-Yang-Mills theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171.
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [SPIRES].
I.A. Korchemskaya and G.P. Korchemsky, On light-like Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].
A. Bassetto, I.A. Korchemskaya, G.P. Korchemsky and G. Nardelli, Gauge invariance and anomalous dimensions of a light cone Wilson loop in lightlike axial gauge, Nucl. Phys. B 408 (1993) 62 [hep-ph/9303314] [SPIRES].
A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Loop Space Formalism And Renormalization Group For The Infrared Asymptotics Of QCD, Phys. Lett. B 171 (1986) 459 [SPIRES].
S.V. Ivanov, G.P. Korchemsky and A.V. Radyushkin, Infrared Asymptotics Of Perturbative QCD: Contour Gauges, Yad. Fiz. 44 (1986) 230 [Sov. J. Nucl. Phys. 44 (1986) 145] [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Infrared asymptotics of perturbative QCD. Quark and gluon propagators, Sov. J. Nucl. Phys. 45 (1987) 127 [Yad. Fiz. 45 (1987) 198] [SPIRES].
G.P. Korchemsky and A.V. Radyushkin, Infrared asymptotics of perturbative QCD. Vertex functions, Sov. J. Nucl. Phys. 45 (1987) 910 [Yad. Fiz. 45 (1987) 1466] [SPIRES].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
G.P. Korchemsky, J.M. Drummond and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES].
A. Brandhuber, P. Heslop and G. Travaglini, MHV Amplitudes in N = 4 Super Yang-Mills and Wilson Loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [SPIRES].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [SPIRES].
Z. Bern et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [SPIRES].
N. Berkovits and J. Maldacena, Fermionic T-duality, Dual Superconformal Symmetry and the Amplitude/Wilson Loop Connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [SPIRES].
N. Beisert, R. Ricci, A.A. Tseytlin and M. Wolf, Dual Superconformal Symmetry from AdS5 x S5 Superstring Integrability, Phys. Rev. D 78 (2008) 126004 [arXiv:0807.3228] [SPIRES].
D.J. Gross and S.B. Treiman, Light cone structure of current commutators in the gluon quark model, Phys. Rev. D 4 (1971) 1059 [SPIRES].
L. Andrianopoli and S. Ferrara, K-K excitations on AdS 5 × S 5 as N = 4 *primary* superfields, Phys. Lett. B 430 (1998) 248 [hep-th/9803171] [SPIRES].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].
S. Penati, A. Santambrogio and D. Zanon, T wo-point functions of chiral operators in N = 4 SYM at order g 4, JHEP 12 (1999) 006 [hep-th/9910197] [SPIRES].
S. Penati, A. Santambrogio and D. Zanon, More on correlators and contact terms in N = 4 SYM at order g 4, Nucl. Phys. B 593 (2001) 651 [hep-th/0005223] [SPIRES].
E. D’Hoker, D.Z. Freedman and W. Skiba, Field theory tests for correlators in the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 045008 [hep-th/9807098] [SPIRES].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three-point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [SPIRES].
P.S. Howe, E. Sokatchev and P.C. West, 3-point functions in N = 4 Yang-Mills, Phys. Lett. B 444 (1998) 341 [hep-th/9808162] [SPIRES].
B. Eden, C. Schubert and E. Sokatchev, Three-loop four-point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [SPIRES].
B. Eden, C. Schubert and E. Sokatchev, unpublished (2000).
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g 4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [SPIRES].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Four-point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B 557 (1999) 355 [hep-th/9811172] [SPIRES].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Simplifications of four-point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Phys. Lett. B 466 (1999) 20 [hep-th/9906051] [SPIRES].
A. Gorsky and A. Zhiboedov, One-loop derivation of the Wilson polygon — MHV amplitude duality, J. Phys. A 42 (2009) 355214 [arXiv:0904.0381] [SPIRES].
V.M. Braun, G.P. Korchemsky and D. Mueller, The uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys. 51 (2003) 311 [hep-ph/0306057] [SPIRES].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [SPIRES].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [SPIRES].
F. Gonzalez-Rey, I.Y. Park and K. Schalm, A note on four-point functions of conformal operators in N = 4 super-Yang-Mills, Phys. Lett. B 448 (1999) 37 [hep-th/9811155] [SPIRES].
N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N = 4 super Yang-Mills, JHEP 04 (2009) 052 [arXiv:0901.3653] [SPIRES].
N. Drukker and J. Plefka, The structure of n-point functions of chiral primary operators in N = 4 super Yang-Mills at one-loop, JHEP 04 (2009) 001 [arXiv:0812.3341] [SPIRES].
N.I. Usyukina and A.I. Davydychev, An Approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [SPIRES].
N.I. Usyukina and A.I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [SPIRES].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [SPIRES].
F.A. Dolan and H. Osborn, Conformal partial wave expansions for N = 4 chiral four point functions, Annals Phys. 321 (2006) 581 [hep-th/0412335] [SPIRES].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett. A 4 (1989) 1257 [SPIRES].
G.P. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [hep-ph/9210281] [SPIRES].
L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An Operator Product Expansion for Polygonal null Wilson Loops, JHEP 04 (2011) 088 [arXiv:1006.2788] [SPIRES].
B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, arXiv:1007.3246 [SPIRES].
I.I. Balitsky and V.M. Braun, Evolution Equations for QCD String Operators, Nucl. Phys. B 311 (1989) 541 [SPIRES].
D. Mueller, D. Robaschik, B. Geyer, F.M. Dittes and J. Horejsi, W ave functions, evolution equations and evolution kernels from light-ray operators of QCD, Fortschr. Phys. 42 (1994) 101 [hep-ph/9812448] [SPIRES].
G.P. Korchemsky, Sudakov Form-Factor In QCD, Phys. Lett. B 220 (1989) 629 [SPIRES].
G.P. Korchemsky, Double Logarithmic Asymptotics in QCD, Phys. Lett. B 217 (1989) 330 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1007.3243
1Unité de Recherche Associée au CNRS URA 2306
2Laboratoire d’Annecy-le-Vieux de Physique Théorique, UMR 5108.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Alday, L.F., Eden, B., Korchemsky, G.P. et al. From correlation functions to Wilson loops. J. High Energ. Phys. 2011, 123 (2011). https://doi.org/10.1007/JHEP09(2011)123
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2011)123