The four-loop six-gluon NMHV ratio function

TL;DR

This work extends the hexagon function bootstrap to the four-loop NMHV six-point amplitude in planar $\mathcal{N}=4$ SYM, achieving a unique ratio function constrained by dual superconformal symmetry via the $\bar{Q}$ equation, collinear limits, and multi-Regge kinematics. The authors build a weight-eight hexagon-function basis and express the NMHV data in terms of six primary coproduct components, ultimately fixing all parameters through a combination of MRK and OPE checks. They provide extensive analytic and numeric characterizations across multiple kinematic regimes, including near-collinear and multi-particle factorization limits, and offer a comprehensive coproduct-based description of the hexagon basis through weight eight. The results not only corroborate integrability-based predictions but also pave the way for higher-loop explorations and a deeper understanding of the analytic structure of scattering amplitudes in this highly symmetric theory.

Abstract

We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar $\mathcal{N} = 4$ super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a $\bar{Q}$ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N$^3$LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. We also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.

Figures (12)

• Figure 1: $V^{(1)}(u,u,1)$, $V^{(2)}(u,u,1)$, $V^{(3)}(u,u,1)$, and $V^{(4)}(u,u,1)$ normalized to one at $(1,1,1)$. One loop is in red, two loops is in green, three loops is in yellow, and four loops is in blue.
• Figure 2: $V^{(1)}(u,1,u)$, $V^{(2)}(u,1,u)$, $V^{(3)}(u,1,u)$, and $V^{(4)}(u,1,u)$ normalized to one at $(1,1,1)$. One loop is in red, two loops is in green, three loops is in yellow, and four loops is in blue.
• Figure 3: $V^{(1)}(u,1,1)$, $V^{(2)}(u,1,1)$, $V^{(3)}(u,1,1)$, and $V^{(4)}(u,1,1)$ normalized to one at $(1,1,1)$. One loop is in red, two loops is in green, three loops is in yellow, and four loops is in blue.
• Figure 4: $V^{(1)}(1,v,1)$, $V^{(2)}(1,v,1)$, $V^{(3)}(1,v,1)$, and $V^{(4)}(1,v,1)$ normalized to one at $(1,1,1)$. One loop is in red, two loops is in green, three loops is in yellow, and four loops is in blue.
• Figure 5: $\tilde{V}^{(2)}(u,1,1)$, $\tilde{V}^{(3)}(u,1,1)$ and $\tilde{V}^{(4)}(u,1,1)$ normalized so that the coefficient of the $\ln^2 u$ term in the $u\rightarrow 0$ limit is unity. Two loops is in green, three loops is in yellow, and four loops is in blue.