Bootstrapping a Five-Loop Amplitude Using Steinmann Relations

TL;DR

This work addresses the problem of determining the analytic form of the six-point amplitude in planar $\mathcal{N}=4$ SYM by leveraging Steinmann relations to drastically constrain the space of hexagon functions. The authors combine these relations with dual conformal symmetry, Regge exponentiation, and final-entry conditions to bootstrap the amplitude through five loops, without external input. They demonstrate explicit two-loop results, prove that Regge logarithms up to $\log^2(vw)$ suffice to fix all parameters at five loops, and verify consistency with the pentagon OPE and a MHV–NMHV relation, ultimately reconstructing $\mathcal E$, $E$, and $\tilde E$ at five loops. The findings imply that Steinmann constraints are a powerful, general tool for high-precision bootstrap in quantum field theory, with potential applications to QCD and higher-particle amplitudes.

Abstract

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude.

Figures (2)

• Figure 1: Illustration of the channels $s_{345}$ and $s_{234}$ for $3\to3$ kinematics. The discontinuity in one channel should not know about the discontinuity in the other channel.
• Figure 2: The remainder function $\mathcal{R}_6$, evaluated at ratios of successive loop orders $L$ on the line $u=v=w$. The spike is an artifact due to $\mathcal{R}_6^{(L)}(u,u,u)$ crossing zero very close to $u=1/3$ at each loop order.