Abstract
The amplituhedron determines scattering amplitudes in planar \( \mathcal{N} \) = 4 super Yang-Mills by a single “positive geometry” in the space of kinematic and loop variables. We study a closely related definition of the amplituhedron for the simplest case of four-particle scattering, given as a sum over complementary “negative geometries”, which provides a natural geometric understanding of the exponentiation of infrared (IR) divergences, as well as a new geometric definition of an IR finite observable \( \mathcal{F} \)(g, z) — dually interpreted as the expectation value of the null polygonal Wilson loop with a single Lagrangian insertion — which is directly determined by these negative geometries. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable z, from which the cusp anomalous dimension Γcusp(g) can also be straightforwardly obtained. We study an especially simple class of negative geometries at all loop orders, associated with a “tree” structure in the negativity conditions, for which the contributions to \( \mathcal{F} \)(g, z) and Γcusp can easily be determined by an interesting non-linear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these “tree” contributions to \( \mathcal{F} \)(g, z) and Γcusp for all values of the ‘t Hooft coupling. The result for Γcusp remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
M.K. Benna, S. Benvenuti, I.R. Klebanov and A. Scardicchio, A Test of the AdS/CFT correspondence using high-spin operators, Phys. Rev. Lett. 98 (2007) 131603 [hep-th/0611135] [INSPIRE].
B. Basso, G.P. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett. 100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].
M. Kruczenski, A Note on twist two operators in N = 4 SYM and Wilson loops in Minkowski signature, JHEP 12 (2002) 024 [hep-th/0210115] [INSPIRE].
R. Roiban and A.A. Tseytlin, Strong-coupling expansion of cusp anomaly from quantum superstring, JHEP 11 (2007) 016 [arXiv:0709.0681] [INSPIRE].
B. Basso, L.J. Dixon, D.A. Kosower, A. Krajenbrink and D.-l. Zhong, Fishnet four-point integrals: integrable representations and thermodynamic limits, JHEP 07 (2021) 168 [arXiv:2105.10514] [INSPIRE].
D.J. Broadhurst and A.I. Davydychev, Exponential suppression with four legs and an infinity of loops, Nucl. Phys. B Proc. Suppl. 205-206 (2010) 326 [arXiv:1007.0237] [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.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].
L.F. Alday, E.I. Buchbinder and A.A. Tseytlin, Correlation function of null polygonal Wilson loops with local operators, JHEP 09 (2011) 034 [arXiv:1107.5702] [INSPIRE].
O.T. Engelund and R. Roiban, On correlation functions of Wilson loops, local and non-local operators, JHEP 05 (2012) 158 [arXiv:1110.0758] [INSPIRE].
O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].
J.G.M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].
J. Frenkel and J.C. Taylor, Nonabelian Eikonal Exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
L.F. Alday, J.M. Henn and J. Sikorowski, Higher loop mixed correlators in N = 4 SYM, JHEP 03 (2013) 058 [arXiv:1301.0149] [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, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the Amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs, Phys. Rev. Lett. 122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The Momentum Amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE].
F. Coronado, Bootstrapping the Simplest Correlator in Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory to All Loops, Phys. Rev. Lett. 124 (2020) 171601 [arXiv:1811.03282] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Octagon at finite coupling, JHEP 07 (2020) 219 [arXiv:2003.01121] [INSPIRE].
B. Basso, L.J. Dixon and G. Papathanasiou, Origin of the Six-Gluon Amplitude in Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 124 (2020) 161603 [arXiv:2001.05460] [INSPIRE].
S. Caron-Huot and F. Coronado, Ten dimensional symmetry of N = 4 SYM correlators, arXiv:2106.03892 [INSPIRE].
J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in \( \mathcal{N} \) = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [arXiv:1911.10174] [INSPIRE].
N. Arkani-Hamed, A. Hillman and S. Mizera, Feynman Polytopes and the Tropical Geometry of UV and IR Divergences, arXiv:2202.12296 [INSPIRE].
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] [INSPIRE].
S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].
B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].
L.F. Alday, P. Heslop and J. Sikorowski, Perturbative correlation functions of null Wilson loops and local operators, JHEP 03 (2013) 074 [arXiv:1207.4316] [INSPIRE].
J.M. Henn, What Can We Learn About QCD and Collider Physics from N = 4 Super Yang-Mills?, Ann. Rev. Nucl. Part. Sci. 71 (2021) 87 [arXiv:2006.00361] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
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].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [DOI] [arXiv:1212.5605] [INSPIRE].
J.L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion, JHEP 01 (2015) 001 [arXiv:1303.4734] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].
J.L. Bourjaily and J. Trnka, Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM, JHEP 08 (2015) 119 [arXiv:1505.05886] [INSPIRE].
E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive geometry, local triangulations, and the dual of the Amplituhedron, JHEP 01 (2021) 035 [arXiv:2009.05607] [INSPIRE].
E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive Geometries for One-Loop Chiral Octagons, arXiv:2007.12191 [INSPIRE].
J. Henn, B. Mistlberger, V.A. Smirnov and P. Wasser, Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP 04 (2020) 167 [arXiv:2002.09492] [INSPIRE].
S. He, Z. Li, Y. Tang and Q. Yang, The Wilson-loop d log representation for Feynman integrals, JHEP 05 (2021) 052 [arXiv:2012.13094] [INSPIRE].
S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
L.J. Dixon, M. von Hippel, A.J. McLeod and J. Trnka, Multi-loop positivity of the planar \( \mathcal{N} \) = 4 SYM six-point amplitude, JHEP 02 (2017) 112 [arXiv:1611.08325] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the Amplituhedron Volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
O. Erdoğan and G. Sterman, Gauge Theory Webs and Surfaces, Phys. Rev. D 91 (2015) 016003 [arXiv:1112.4564] [INSPIRE].
D. Chicherin and J. Henn, Symmetry properties of Wilson loops with a Langrangian insertion, to appear.
N. Arkani-Hamed, D. Chicherin, J. Henn and J. Trnka, Finite multi-particle amplitudes from negative geometries, in progress.
D. Chicherin and J. Henn, Pentagon Wilson loop with Lagrangian insertion at two loops in \( \mathcal{N} \) = 4 super Yang-Mills theory, to appear.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2112.06956
Rights and permissions
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.
About this article
Cite this article
Arkani-Hamed, N., Henn, J. & Trnka, J. Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron. J. High Energ. Phys. 2022, 108 (2022). https://doi.org/10.1007/JHEP03(2022)108
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)108