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Journal of High Energy Physics

, 2019:61 | Cite as

The cosmic Galois group and extended Steinmann relations for planar \( \mathcal{N} \) = 4 SYM amplitudes

  • Simon Caron-Huot
  • Lance J. Dixon
  • Falko Dulat
  • Matt von Hippel
  • Andrew J. McLeodEmail author
  • Georgios Papathanasiou
Open Access
Regular Article - Theoretical Physics
  • 6 Downloads

Abstract

We describe the minimal space of polylogarithmic functions that is required to express the six-particle amplitude in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory through six and seven loops, in the NMHV and MHV sectors respectively. This space respects a set of extended Steinmann relations that restrict the iterated discontinuity structure of the amplitude, as well as a cosmic Galois coaction principle that constrains the functions and the transcendental numbers that can appear in the amplitude at special kinematic points. To put the amplitude into this space, we must divide it by the BDS-like ansatz and by an additional zeta-valued constant ρ. For this normalization, we conjecture that the extended Steinmann relations and the coaction principle hold to all orders in the coupling. We describe an iterative algorithm for constructing the space of hexagon functions that respects both constraints. We highlight further simplifications that begin to occur in this space of functions at weight eight, and distill the implications of imposing the coaction principle to all orders. Finally, we explore the restricted spaces of transcendental functions and constants that appear in special kinematic configurations, which include polylogarithms involving square, cube, fourth and sixth roots of unity.

Keywords

Scattering Amplitudes Supersymmetric Gauge Theory 

Notes

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

Supplementary material

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ESM 1 (TXT 2 kb)

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Simon Caron-Huot
    • 1
  • Lance J. Dixon
    • 2
    • 3
    • 4
    • 5
  • Falko Dulat
    • 2
  • Matt von Hippel
    • 6
    • 7
  • Andrew J. McLeod
    • 2
    • 3
    • 7
    Email author
  • Georgios Papathanasiou
    • 3
    • 8
  1. 1.Department of PhysicsMcGill UniversityMontréalCanada
  2. 2.SLAC National Accelerator LaboratoryStanford UniversityStanfordU.S.A.
  3. 3.Kavli Institute for Theoretical PhysicsUC Santa BarbaraSanta BarbaraU.S.A.
  4. 4.Institut für Physik and IRIS AdlershofHumboldt-Universität zu BerlinBerlinGermany
  5. 5.Pauli CenterETH Zürich and University of ZürichZürichSwitzerland
  6. 6.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  7. 7.Niels Bohr International AcademyCopenhagenDenmark
  8. 8.DESY Theory Group, DESY HamburgHamburgGermany

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