The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes

TL;DR

<3-5 sentence high-level summary> This work constructs and analyzes the minimal function space H_hex of hexagon functions needed to express the six-particle amplitude in planar N=4 SYM through high loop orders, enforcing extended Steinmann relations and a cosmic Galois coaction principle. A coupling-dependent normalization ρ is introduced to align the coaction across all orders, and an iterative, tensor-based method is developed to build integrable symbols and promote them to full functions with consistent boundary data. The authors explore the saturation of H_hex across weights, investigate the role of zeta-values and MZVs at (1,1,1), and demonstrate the coaction principle on many special lines and root-of-unity points, including cyclotomic and alternating-sum configurations. The results provide strong all-order constraints on the transcendental content of the amplitudes and lay groundwork for deeper bottom-up and higher-point generalizations.

Abstract

We describe the minimal space of polylogarithmic functions that is required to express the six-particle amplitude in planar ${\cal N}=4$ 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.

Figures (1)

• Figure 1: Points associated with the unit cube in $(u,v,w)$ where the functions in ${\cal H}^{\rm hex}$ evaluate to interesting transcendental numbers associated with polylogarithms with indices that are square, fourth and sixth roots of unity, as indicated by the shape of the symbol. The color of the symbol indicates how many of the three cross ratios are singular (equal to zero) at that point.