Abstract
We explore a direct connection between the collinear limit and the multi-Regge limit for scattering amplitudes in the \( \mathcal{N} \) = 4 super Yang-Mills theory. Starting with the collinear expansion for the six-gluon amplitude in the Euclidean kinematic region, we perform an analytic continuation term by term to the so-called Mandelstam region. We find that the result coincides with the collinear expansion of the analytically continued amplitude. We then take the multi-Regge limit, and conjecture that the final result precisely reproduces the one from the BFKL approach. Combining this procedure with the OPE for null polygonal Wilson loops, we explicitly compute the leading contribution in the “collinear-Regge” limit up to five loops. Our results agree with all the known results up to four loops. At five-loop, our results up to the next-to-next-to-leading logarithmic approximation (NNLLA) also reproduce the known results, and for the N3LLA and the N4LLA give non-trivial predictions. We further present an all-loop prediction for the imaginary part of the next-to-double-leading logarithmic approximation. Our procedure has a possibility of an interpolation from weak to strong coupling in the multi-Regge limit with the help of the OPE.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
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] [INSPIRE].
Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
R.M. Schabinger, The imaginary part of the N = 4 super-Yang-Mills two-loop six-point MHV amplitude in multi-Regge kinematics, JHEP 11 (2009) 108 [arXiv:0910.3933] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05(2009) 046[arXiv:0902.2987] [INSPIRE].
T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Thermodynamic Bethe ansatz equations for minimal surfaces in AdS 3, JHEP 04 (2010) 108 [arXiv:1002.2941] [INSPIRE].
Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Six-point gluon scattering amplitudes from Z 4 -symmetric integrable model, JHEP 09 (2010) 064 [arXiv:1005.4487] [INSPIRE].
Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, g-functions and gluon scattering amplitudes at strong coupling, JHEP 04 (2011) 100 [arXiv:1102.2477] [INSPIRE].
Y. Hatsuda, K. Ito and Y. Satoh, T-functions and multi-gluon scattering amplitudes, JHEP 02 (2012) 003 [arXiv:1109.5564] [INSPIRE].
Y. Hatsuda, K. Ito and Y. Satoh, Null-polygonal minimal surfaces in AdS 4 from perturbed W minimal models, JHEP 02 (2013) 067 [arXiv:1211.6225] [INSPIRE].
Y. Hatsuda, K. Ito, Y. Satoh and J. Suzuki, Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling, JHEP 08 (2014) 162 [arXiv:1406.5904] [INSPIRE].
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] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping null polygon Wilson loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Spacetime and flux tube S-matrices at finite coupling for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111(2013) 091602 [arXiv:1303.1396] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux tube S-matrix II. Extracting and matching data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions, JHEP 08 (2014) 085 [arXiv:1402.3307] [INSPIRE].
J. Bartels, L.N. Lipatov and A. Sabio Vera, N = 4 supersymmetric Yang-Mills scattering amplitudes at high energies: the Regge cut contribution, Eur. Phys. J. C 65 (2010) 587 [arXiv:0807.0894] [INSPIRE].
V.S. Fadin, E.A. Kuraev and L.N. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60 (1975) 50 [INSPIRE].
L.N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642] [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-reggeon processes in the Yang-Mills theory, Sov. Phys. JETP 44 (1976) 443 [Zh. Eksp. Teor. Fiz. 71 (1976) 840] [Erratum ibid. 45 (1977) 199] [INSPIRE].
I.I. Balitsky and L.N. Lipatov, The pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597] [INSPIRE].
J. Bartels, J. Kotanski and V. Schomerus, Excited hexagon Wilson loops for strongly coupled N = 4 SYM, JHEP 01 (2011) 096 [arXiv:1009.3938] [INSPIRE].
J. Bartels, V. Schomerus and M. Sprenger, Multi-Regge limit of the n-gluon bubble ansatz, JHEP 11 (2012) 145 [arXiv:1207.4204] [INSPIRE].
J. Bartels, J. Kotanski, V. Schomerus and M. Sprenger, The excited hexagon reloaded, arXiv:1311.1512 [INSPIRE].
J. Bartels, L.N. Lipatov and A. Prygarin, Collinear and Regge behavior of 2 → 4 MHV amplitude in N = 4 super Yang-Mills theory, arXiv:1104.4709 [INSPIRE].
L.N. Lipatov and A. Prygarin, BFKL approach and six-particle MHV amplitude in N = 4 super Yang-Mills, Phys. Rev. D 83 (2011) 125001 [arXiv:1011.2673] [INSPIRE].
V.S. Fadin and L.N. Lipatov, BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N = 4 SUSY, Phys. Lett. B 706 (2012) 470 [arXiv:1111.0782] [INSPIRE].
S. Caron-Huot, When does the gluon reggeize?, arXiv:1309.6521 [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].
L.N. Lipatov and A. Prygarin, Mandelstam cuts and light-like Wilson loops in N = 4 SUSY, Phys. Rev. D 83 (2011) 045020 [arXiv:1008.1016] [INSPIRE].
J. Pennington, The six-point remainder function to all loop orders in the multi-Regge limit, JHEP 01 (2013) 059 [arXiv:1209.5357] [INSPIRE].
F.C.S. Brown, Single-valued multiple polylogarithms in one variable, C. R. Acad. Sci. Paris Ser. I 338 (2004) 527.
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [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].
B. Basso, Exciting the GKP string at any coupling, Nucl. Phys. B 857 (2012) 254 [arXiv:1010.5237] [INSPIRE].
N. Dorey and M. Losi, Giant holes, J. Phys. A 43 (2010) 285402 [arXiv:1001.4750] [INSPIRE].
N. Dorey and P. Zhao, Scattering of giant holes, JHEP 08 (2011) 134 [arXiv:1105.4596] [INSPIRE].
B. Basso and A. Rej, Bethe ansätze for GKP strings, Nucl. Phys. B 879 (2014) 162 [arXiv:1306.1741] [INSPIRE].
D. Fioravanti, S. Piscaglia and M. Rossi, On the scattering over the GKP vacuum, Phys. Lett. B 728 (2014) 288 [arXiv:1306.2292] [INSPIRE].
G. Papathanasiou, Hexagon Wilson loop OPE and harmonic polylogarithms, JHEP 11 (2013) 150 [arXiv:1310.5735] [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].
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].
A. Prygarin, M. Spradlin, C. Vergu and A. Volovich, All two-loop MHV amplitudes in multi-Regge kinematics from applied symbology, Phys. Rev. D 85 (2012) 085019 [arXiv:1112.6365] [INSPIRE].
J. Bartels, A. Kormilitzin, L.N. Lipatov and A. Prygarin, BFKL approach and 2 → 5 maximally helicity violating amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. D 86 (2012) 065026 [arXiv:1112.6366] [INSPIRE].
J. Bartels, A. Kormilitzin and L. Lipatov, Analytic structure of the n = 7 scattering amplitude in \( \mathcal{N} \) = 4 SYM theory at multi-Regge kinematics: conformal Regge pole contribution, Phys. Rev. D 89 (2014) 065002 [arXiv:1311.2061] [INSPIRE].
J. Bartels, V. Schomerus and M. Sprenger, Heptagon amplitude in the multi-Regge regime, JHEP (2014) [arXiv:1405.3658] [INSPIRE].
R.A. Janik, Twist-two operators and the BFKL regime — nonstandard solutions of the Baxter equation, JHEP 11 (2013) 153 [arXiv:1309.2844] [INSPIRE].
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: 1404.6506
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
Hatsuda, Y. Wilson loop OPE, analytic continuation and multi-Regge limit. J. High Energ. Phys. 2014, 38 (2014). https://doi.org/10.1007/JHEP10(2014)038
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)038