Abstract
We study the six-point NMHV ratio function in planar \( \mathcal{N} \) = 4 SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a “radial” direction. We also investigate positivity of the MHV six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].
F. Gliozzi, J. Scherk and D.I. Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys. B 122 (1977) 253 [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].
Y. Bai and S. He, The amplituhedron from momentum twistor diagrams, JHEP 02 (2015) 065 [arXiv:1408.2459] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
Y. Bai, S. He and T. Lam, The amplituhedron and the one-loop grassmannian measure, JHEP 01 (2016) 112 [arXiv:1510.03553] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the amplituhedron volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
D. Galloni, Positivity sectors and the amplituhedron, arXiv:1601.02639 [INSPIRE].
S.N. Karp and L.K. Williams, The m = 1 amplituhedron and cyclic hyperplane arrangements, arXiv:1608.08288 [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605.
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.
F.C.S. Brown, Multiple zeta values and periods of moduli spaces M 0 ,n ( R ), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
S. Müller-Stach, S. Weinzierl and R. Zayadeh, A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Num. Theor. Phys. 6 (2012) 203 [arXiv:1112.4360] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [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].
A.E. Lipstein and L. Mason, From d logs to dilogs the super Yang-Mills MHV amplitude revisited, JHEP 01 (2014) 169 [arXiv:1307.1443] [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].
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [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 and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [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].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a five-loop amplitude using Steinmann relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].
J.M. Henn and T. Huber, The four-loop cusp anomalous dimension in \( \mathcal{N} \) = 4 super Yang-Mills and analytic integration techniques for Wilson line integrals, JHEP 09 (2013) 147 [arXiv:1304.6418] [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].
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].
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].
O. Steinmann, Über den Zussamenhang Zurischen Wightmanfunktionen und Retardierten Kommutatoren I, Helv. Physica Acta 33 (1960) 257.
O. Steinmann, Über den Zussamenhang Zurischen Wightmanfunktionen und Retardierten Kommutatoren II, Helv. Physica Acta 33 (1960) 347.
K.E. Cahill and H.P. Stapp, Optical theorems and steinmann relations, Annals Phys. 90 (1975) 438 [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [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].
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].
V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
G. Georgiou, E.W.N. Glover and V.V. Khoze, Non-MHV tree amplitudes in gauge theory, JHEP 07 (2004) 048 [hep-th/0407027] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [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, Space-time S-matrix and flux-tube S-matrix IV. Gluons and fusion, JHEP 09 (2014) 149 [arXiv:1407.1736] [INSPIRE].
J.M. Drummond and G. Papathanasiou, Hexagon OPE resummation and multi-Regge kinematics, JHEP 02 (2016) 185 [arXiv:1507.08982] [INSPIRE].
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] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
J.M. Henn and T. Huber, Systematics of the cusp anomalous dimension, JHEP 11 (2012) 058 [arXiv:1207.2161] [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The multiple zeta value data mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, cs/0004015.
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [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].
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: 1611.08325
Dedicated to John Schwarz on his 75th birthday
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
Dixon, L.J., von Hippel, M., McLeod, A.J. et al. Multi-loop positivity of the planar \( \mathcal{N} \) = 4 SYM six-point amplitude. J. High Energ. Phys. 2017, 112 (2017). https://doi.org/10.1007/JHEP02(2017)112
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2017)112