Geometric Landau Analysis and Symbol Bootstrap

TL;DR

This paper explores how positive geometry constrains the structure of loop integrals in supersymmetric gauge theory, using geometric and Landau analysis to determine the symbol alphabet and analyze ladder integrals.

Contribution

It introduces a geometric-Landau analysis method to identify physical singularities and determine the symbol alphabet for Wilson loop integrals with Lagrangian insertions in $ ext{N}=4$ super Yang-Mills.

Findings

Computed six-point two-loop and five-point three-loop ladder negative geometries at the symbol level.

Proposed a conjectural symbol alphabet for ladder negative geometries at two loops for all multiplicities.

Provided insights into the function space of Wilson loop integrals with Lagrangian insertions.

Abstract

We investigate how the positive geometry framework for loop integrands in $\mathcal{N}{=}4$ super Yang-Mills theory constrains the structure of the integrated answers. This is done in the context of a geometric expansion of Wilson loops with a Lagrangian insertion, called negative geometries, extending ideas previously used for scattering amplitudes related to the Amplituhedron. The procedure we adopt combines the knowledge of all maximal codimension boundaries of the geometry, which characterize all possible leading singularities of the integral, with a geometrically informed Landau analysis. The interplay between geometry and Landau analysis arises from associating Landau diagrams to geometric boundaries. The boundary structure of the geometry then determines which solutions to the Landau equations are spurious and which ones are physical, that is, which singularities are actually present in the integral. This method allows us to efficiently determine the symbol alphabet of the associated integral, and serves as a starting point for the symbol bootstrap. We successfully implement this procedure and compute the six-point two-loop and five-point three-loop ladder negative geometries at the symbol level. We also present the conjectural alphabet for ladder negative geometries at two loops for all multiplicities. These are finite integrals that serve as building blocks for the Wilson loop with Lagrangian insertion, and therefore provide insights into the function space of the latter.