Six-Gluon Amplitudes in Planar ${\cal N}=4$ Super-Yang-Mills Theory at Six and Seven Loops

TL;DR

The paper tackles the computation of six-particle amplitudes in planar ${\cal N}=4$ SYM up to seven loops for MHV and six loops for NMHV by bootstrapping within a minimal space ${\cal H}^{\rm hex}$ guided by extended Steinmann relations and the cosmic Galois coaction. The authors construct a weight-$(2L)$ basis of hexagon functions, enforce symmetries and collinear constraints, and fix remaining degrees of freedom using multi-Regge kinematics and the near-collinear Pentagon OPE, notably incorporating the single-gluon bound state. A novel weight-12 function $Z(u,v,w)$ emerges at six loops that cannot be detected by MRK or strict collinear data, explaining the breakdown of a previously conjectured MHV–NMHV relation at higher loops; all other free parameters are fixed, and the self-crossing limit provides a high-precision cross-check through seven loops. The work demonstrates the power and consistency of a cosmic Galois–restricted bootstrap for highly symmetric gauge theories and lays groundwork for applying these ideas to other theories and higher-point amplitudes, with implications for infrared regularization, OPE data, and numerical convergence properties across loop orders.

Abstract

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.

Figures (9)

• Figure 1: $E^{(L)}(u,u,1)/E^{(L-1)}(u,u,1)$ evaluated at successive loop orders. As there are points where $E^{(L-1)}(u,u,1)=0$ in this interval, the plot diverges at those points.
• Figure 2: $E^{(L)}(u,1,u)/E^{(L-1)}(u,1,u)$ evaluated at successive loop orders.
• Figure 3: $\mathcal{E}^{(L)}(u,u,1)/\mathcal{E}^{(L-1)}(u,u,1)$ evaluated at successive loop orders. As there are points where $\mathcal{E}^{(L-1)}(u,u,1)=0$ in this interval, the plot diverges at those points.
• Figure 4: $E^{(L)}(u,1,1)/E^{(L-1)}(u,1,1)$ evaluated at successive loop orders.
• Figure 5: $E^{(L)}(1,v,1)/E^{(L-1)}(1,v,1)$, evaluated at successive loop orders.