Positivity properties of five-point two-loop Wilson loops with Lagrangian insertion

TL;DR

This paper explores the geometric and algebraic structure of five-point two-loop Wilson loops with Lagrangian insertion in supersymmetric Yang-Mills theory, revealing sign properties, integrand structures, and the need for new alphabet letters at higher loops.

Contribution

It constructs the integrand for all-loop ladder geometries, analyzes two-loop integrals analytically, and introduces a bootstrap approach highlighting the necessity of new alphabet letters at three loops.

Findings

Analytic evaluation of two-loop integrals shows uniform sign properties.

Bootstrap assumptions hold at two loops but fail at three loops.

Identification of novel pentagon alphabet letters at three loops.

Abstract

In this paper we discuss the geometric integrand expansion of the five-point Wilson loop with one Lagrangian insertion in maximally supersymmetric Yang-Mills theory. We construct the integrand corresponding to an all-loop class of ladder-type geometries. We then investigate the known two-loop observable from this geometric viewpoint. To do so, we evaluate analytically the new two-loop integrals corresponding to the negative geometry contribution, using the canonical differential equations method. Inspecting the analytic result, we present numerical evidence that in this decomposition, each piece has uniform sign properties, when evaluated in the Amplituhedron region. Finally, we present an alternative bootstrap approach for the ladder-type geometries. We find that certain minimal bootstrap assumptions can be satisfied at two loops, but lead to a contradiction at three loops. This suggests to us that novel alphabet letters are required at this loop order. Indeed studying planar three-loop Feynman integrals, we do identify novel pentagon alphabet letters.