Abstract
The three-loop four-point function of stress-tensor multiplets in \( \mathcal{N}=4 \) super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus obtaining the three-loop correlation function analytically. The integrals have the generic structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of leading singularities to obtain the rational coefficients, the symbol — with an appropriate ansatz for its structure — as a means of characterising multiple polylogarithms, and the technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then lift to find the corresponding polylogarithmic functions. The final formulae are numerically confirmed. The techniques we develop can be applied more generally, and we illustrate this by analytically evaluating one of the integrals contributing to the same four-point function at four loops. This example shows a connection between the leading singularities and the entries of the symbol.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.T. Grisaru, M. Roček and W. Siegel, Zero three loop β-function in N = 4 super Yang-Mills theory, Phys. Rev. Lett. 45 (1980) 1063 [INSPIRE].
W.E. Caswell and D. Zanon, Vanishing three loop β-function in N = 4 supersymmetric Yang-Mills theory, Phys. Lett. B 100 (1981) 152 [INSPIRE].
P.S. Howe, K. Stelle and P. Townsend, The relaxed hypermultiplet: an unconstrained N = 2 superfield theory, Nucl. Phys. B 214 (1983) 519 [INSPIRE].
S. Mandelstam, Light cone superspace and the vanishing of the β-function for the N = 4 model, in Proc. 21st Int. Conf. on High Energy Physics, P. Petiau and M. Proneuf eds., J. Phys. (France) 43 (1982) C-3 [INSPIRE].
S. Mandelstam, Light cone superspace and the ultraviolet finiteness of the N = 4 model, Nucl. Phys. B 213 (1983) 149 [INSPIRE].
L. Brink, O. Lindgren and B.E. Nilsson, N = 4 Yang-Mills theory on the light cone, Nucl. Phys. B 212 (1983) 401 [INSPIRE].
L. Brink, O. Lindgren and B.E. Nilsson, The ultraviolet finiteness of the N = 4 Yang-Mills theory, Phys. Lett. B 123 (1983) 323 [INSPIRE].
P.S. Howe, K. Stelle and P. Townsend, Miraculous ultraviolet cancellations in supersymmetry made manifest, Nucl. Phys. B 236 (1984) 125 [INSPIRE].
J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [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].
N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe ansätze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [INSPIRE].
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].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
G. Korchemsky, J. Drummond 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. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [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].
G. Korchemsky and A. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [INSPIRE].
S. Ivanov, G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD: contour gauges, Yad. Fiz. 44 (1986) 230 [Sov. J. Nucl. Phys. 44 (1986) 145] [INSPIRE].
G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Quark and gluon propagators, Yad. Fiz. 45 (1987) 198 [Sov. J. Nucl. Phys. 45 (1987) 127] [INSPIRE].
G. Korchemsky and A. Radyushkin, Infrared asymptotics of perturbative QCD. Vertex functions, Yad. Fiz. 45 (1987) 1466 [Sov. J. Nucl. Phys. 45 (1987) 910] [INSPIRE].
G. Korchemsky and A. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
G. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys. B 406 (1993) 225 [hep-ph/9210281] [INSPIRE].
L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].
B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].
B. Eden, G.P. Korchemsky and E. Sokatchev, More on the duality correlators/amplitudes, Phys. Lett. B 709 (2012) 247 [arXiv:1009.2488] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].
E. D’Hoker, S.D. Mathur, A. Matusis and L. Rastelli, The operator product expansion of N = 4 SYM and the 4 point functions of supergravity, Nucl. Phys. B 589 (2000) 38 [hep-th/9911222] [INSPIRE].
G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM 4 in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].
G. Arutyunov, F. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
F. Gonzalez-Rey, I. 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] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469 [INSPIRE].
G. Hartwell and P.S. Howe, (N,p,q) harmonic superspace, Int. J. Mod. Phys. A 10 (1995) 3901 [hep-th/9412147] [INSPIRE].
B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys. B 607 (2001) 191 [hep-th/0009106] [INSPIRE].
B. Eden, P.S. Howe, A. Pickering, E. Sokatchev and P.C. West, Four point functions in N = 2 superconformal field theories, Nucl. Phys. B 581 (2000) 523 [hep-th/0001138] [INSPIRE].
F. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [INSPIRE].
P. Heslop and P. Howe, Four point functions in N = 4 SYM, JHEP 01 (2003) 043 [hep-th/0211252] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Hidden symmetry of four-point correlation functions and amplitudes in N = 4 SYM, Nucl. Phys. B 862 (2012) 193 [arXiv:1108.3557] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [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].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky, V.A. Smirnov and E. Sokatchev, Five-loop Konishi in N = 4 SYM, Nucl. Phys. B 862 (2012) 123 [arXiv:1202.5733] [INSPIRE].
B. Eden, Three-loop universal structure constants in N = 4 SUSY Yang-Mills theory, arXiv:1207.3112 [INSPIRE].
V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].
J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
N. Usyukina and A.I. Davydychev, An approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [INSPIRE].
N. 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 [INSPIRE].
F.C.S. Brown, Single-valued multiple polylogarithms in one variable Comptes Rendus Math. 338 (2004) 527.
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
S. Caron-Huot, Loops in spacetime, in ECT*, Trento workshop — Scattering amplitudes: from QCD to maximally supersymmetric Yang-Mills theory and back, http://sites.google.com/site/trentoworkshop/program/, Trento Italy July 16-20 2012.
F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [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].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J. Fleischer, A. Kotikov and O. Veretin, Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass, Nucl. Phys. B 547 (1999) 343 [hep-ph/9808242] [INSPIRE].
J. Drummond, Generalised ladders and single-valued polylogarithms, JHEP 02 (2013) 092 [arXiv:1207.3824] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1 [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].
A. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
S. Gorishnii, S. Larin, L. Surguladze and F. Tkachov, MINCER: program for multiloop calculations in quantum field theory for the Schoonschip system, Comput. Phys. Commun. 55 (1989) 381 [INSPIRE].
J. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083] [INSPIRE].
E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].
C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, cs/0004015 [INSPIRE].
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].
A. Smirnov and M. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [INSPIRE].
A. Smirnov, V. Smirnov and M. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].
O. Schnetz, Graphical functions and single-valued multiple polylogarithms, arXiv:1302.6445 [INSPIRE].
D.E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Alg. 58 (1979) 432.
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059.
F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].
P. Baikov and K. Chetyrkin, Four loop massless propagators: an algebraic evaluation of all master integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].
R. Lee, A. Smirnov and V. Smirnov, Master integrals for four-loop massless propagators up to transcendentality weight twelve, Nucl. Phys. B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].
R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R. Lee, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, JHEP 07 (2008) 031 [arXiv:0804.3008] [INSPIRE].
A. Smirnov and V. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, arXiv:1302.5885 [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘d log’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1303.6909
In memory of Francis Dolan.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Drummond, J., Duhr, C., Eden, B. et al. Leading singularities and off-shell conformal integrals. J. High Energ. Phys. 2013, 133 (2013). https://doi.org/10.1007/JHEP08(2013)133
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2013)133