Lifting Heptagon Symbols to Functions

TL;DR

This paper advances the understanding of seven-point amplitudes in planar ${ m N}=4$ super-Yang-Mills theory by elevating symbol-level results to explicit functions, revealing their structure and properties through boundary conditions and coaction analysis.

Contribution

It introduces a method to lift heptagon symbols to explicit functions, specifying derivatives and boundary conditions, and explores their mathematical properties at weight six.

Findings

Constructed explicit heptagon functions from symbols at weight six.

Analyzed the coaction structure of these functions.

Provided plots of amplitudes in the Euclidean region.

Abstract

Seven-point amplitudes in planar ${\cal N}=4$ super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.