Abstract
We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in \( \mathcal{N} = 4 \) super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar \( \mathcal{N} = 4 \) super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
A. Brandhuber, P. Heslop and G. Travaglini, A note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [SPIRES].
J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [SPIRES].
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] [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].
C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES].
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] [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].
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].
L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [SPIRES]
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation and Regge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [SPIRES].
R.H. Boels, No triangles on the moduli space of maximally supersymmetric gauge theory, JHEP 05 (2010) 046 [arXiv:1003.2989] [SPIRES].
Z. Bern, J.J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized unitarity and six-dimensional helicity, arXiv:1010.0494 [SPIRES].
A. Sever and P. Vieira, Symmetries of the N = 4 SYM S-matrix, arXiv:0908.2437 [SPIRES].
T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [SPIRES].
G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [SPIRES].
N. Beisert, J. Henn, T. McLoughlin and J. Plefka, One-loop superconformal and yangian symmetries of scattering amplitudes in N = 4 super Yang-Mills, JHEP 04 (2010) 085 [arXiv:1002.1733] [SPIRES].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [SPIRES].
R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [SPIRES].
N. Arkani-Hamed, F. Cachazo and C. Cheung, The grassmannian origin of dual superconformal invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [SPIRES].
L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [SPIRES].
J.M. Drummond and L. Ferro, The yangian origin of the grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [SPIRES].
G.P. Korchemsky and E. Sokatchev, Superconformal invariants for scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 839 (2010) 377 [arXiv:1002.4625] [SPIRES].
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] [SPIRES].
L.F. Alday and R. Roiban, Scatteringamplitudes, Wilson loops andthestring/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [SPIRES].
J.M. Henn, Duality between Wilson loops and gluon amplitudes, Fortsch. Phys. 57 (2009) 729 [arXiv:0903.0522] [SPIRES].
L.J. Mason and D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [SPIRES].
S. Caron-Huot, Notes on the scattering amplitude/Wilson loopduality, arXiv:1010.1167 [SPIRES].
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] [SPIRES].
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] [SPIRES].
V. Del Duca, C. Duhr and V.A. Smirnov, A two-loop octagon Wilson loop in N = 4 SYM, JHEP 09 (2010) 015 [arXiv:1006.4127] [SPIRES].
P. Heslop and V.V. Khoze, Analytic results for MHV Wilson loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [SPIRES].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, More loops and legs in Higgs-regulated N = 4 SYM amplitudes, JHEP 08 (2010) 002 [arXiv:1004.5381] [SPIRES].
D.A. Kosower, R. Roiban and C. Vergu, The six-point NMHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 83 (2011) 065018 [arXiv:1009.1376] [SPIRES].
L.F. Alday, Some analytic results for two-loop scattering amplitudes, arXiv:1009.1110 [SPIRES].
L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, arXiv:1007.3243 [SPIRES].
B. Eden, G.P. Korchemsky and E. Sokatchev, More on the duality correlators/amplitudes, arXiv:1009.2488 [SPIRES].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158.
A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123.
T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [SPIRES].
V. Smirnov, Feynman integral calculus, Springer, U.S.A. (2006).
J.M. Drummond and J.M. Henn, Simple loop integrals and amplitudes in N = 4 SYM, arXiv:1008.2965 [SPIRES].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [SPIRES].
A. Hodges, The box integrals in momentum-twistor geometry, arXiv:1004.3323 [SPIRES].
L. Mason and D. Skinner, Amplitudes at weak coupling as polytopes in AdS 5, J. Phys. A 44 (2011) 135401 [arXiv:1004.3498] [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].
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].
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.
A.P. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics, Nucl. Phys. B 662 (2003) 461 [hep-th/0303056] [SPIRES].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo, S. Caron-Hout and J. Trnka, Local integrals for planar scattering amplitudes, arXiv:1012.6032 [SPIRES].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, in Strings, branes and extra dimensions, S.S. Gubser and J.D. Lykken eds., World Scientific, Singapore(2004), hep-th/0201253 [SPIRES].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [SPIRES].
T. Gehrmann and E. Remiddi, Two-loop master integrals for γ * → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [SPIRES].
D.J. Broadhurst and A.I. Davydychev, Exponential suppression with four legs and an infinity of loops, Nucl. Phys. Proc. Suppl. 205-206 (2010) 326 [arXiv:1007.0237] [SPIRES].
C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP 01 (2007) 082 [hep-ph/0611236] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
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
Drummond, J.M., Henn, J.M. & Trnka, J. New differential equations for on-shell loop integrals. J. High Energ. Phys. 2011, 83 (2011). https://doi.org/10.1007/JHEP04(2011)083
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2011)083