The symbology of Feynman integrals from twistor geometries

TL;DR

This paper explores the geometric structure of Feynman integrals using momentum twistor space, revealing how their symbol letters can be derived from intersection points and cross-ratios, and connecting to cluster algebras.

Contribution

It introduces a geometric approach to determine symbol letters of planar Feynman integrals via momentum twistor intersections, extending to two-loop cases and suggesting non-planar generalizations.

Findings

All one-loop symbol letters obtained from quadruple cuts match known results.

Two-loop symbol letters derived from intersection points agree with differential equation methods.

The approach hints at a connection between symbol letters and cluster algebras.

Abstract

We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities (LS). Cutting propagators in momentum twistor space amounts to intersecting lines associated with loop and external dual momenta, including the special line associated with the point at infinity, which breaks dual conformal symmetry. We show that cross-ratios of intersection points on these lines, especially those on the infinity line, naturally produce symbol letters for Feynman integrals in $D=4-2\epsilon$, which include and generalize their LS. At one loop, we obtain all symbol letters using intersection points from quadruple cuts for integrals up to pentagon kinematics with two massive corners, which agree perfectly with canonical differential equation (CDE) results. We then obtain all two-loop letters, for up to four-mass box and one-mass pentagon kinematics, by considering more intersections arising from two-loop cuts. Finally we comment on how cluster algebras appear from this construction, and importantly how we may extend the method to non-planar integrals.