Hexagonal Wilson loop with Lagrangian insertion at two loops in $\mathcal{N}=4$ super Yang-Mills theory

TL;DR

This paper computes the two-loop hexagonal Wilson loop with Lagrangian insertion in planar $ ext{N}=4$ SYM, revealing its relation to three-loop amplitudes and pure Yang-Mills theory, using a bootstrap approach with physical constraints.

Contribution

It provides the first two-loop bootstrap computation of the hexagonal Wilson loop with Lagrangian insertion in $ ext{N}=4$ SYM, connecting it to higher-loop amplitudes and special function spaces.

Findings

Two-loop result matches the maximal transcendental part of six-point three-loop all-plus amplitude.

Physical constraints fix all indeterminates in the bootstrap ansatz.

Verification of Steinmann relations supports the conjectured correspondence.

Abstract

In this work, we compute the two-loop result of the null hexagonal Wilson loop with a Lagrangian insertion in planar, maximally supersymmetric Yang-Mills theory via a bootstrap approach. Normalized by the null polygonal Wilson loop itself, the integrand-level result of this observable corresponds to the logarithm of the six-point three-loop amplitude in this theory, while its integrated result is conjectured to match the maximal transcendental part of the six-point three-loop all-plus amplitude in pure Yang-Mills theory. Our work builds on two recent advances. On the one hand, the set of leading singularities relevant to this observable was recently classified. On the other hand, the relevant space of special functions that may in principle accompany these leading singularities was determined at two loops and for six particles by a dedicated Feynman integral calculation. These two ingredients serve as the foundation of our bootstrap ansatz. We fix all indeterminates in this ansatz by imposing physical constraints, such as symmetries, absence of spurious divergences, and correct behavior in soft and collinear limits. Finally, we discuss and verify certain physical properties of our symbol result, including physical singularities, behavior under multi-Regge limit, as well as Steinmann relations between symbol entries. The latter relations are motivated by the correspondence to all-plus amplitudes in pure Yang-Mills theory, and successfully checking them constitutes a consistency check of this conjectured correspondence.