Landau-based Schubert analysis

TL;DR

This paper introduces a Landau-based Schubert analysis method that relates geometries of intersecting lines to Landau singularities, aiding in generating symbol alphabets for Feynman integrals and scattering amplitudes.

Contribution

It presents a novel geometric approach linking Landau singularities to symbol letters, with automated tools and applications to multi-loop integrals and super-Yang-Mills amplitudes.

Findings

Successfully derives two-loop alphabets for n-point MHV amplitudes

Produces alphabet for 4-point MHV form factors

Shows alphabets as unions of type-A cluster algebras

Abstract

We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to ``uplifting" Landau singularities of a Feynman integral to its symbol letters. We illustrate this {\it Landau-based Schubert analysis} using various multi-loop Feynman integrals in four dimensions and present an automated {\ttfamily Mathematica} notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar ${\cal N}=4$ super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of $n$-point MHV amplitudes and that of the $n=4$ MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type-$A$ cluster algebras.