Rationalizing Loop Integration

TL;DR

This paper demonstrates that direct Feynman-parametric loop integration is feasible for many planar multi-loop integrals by leveraging dual-conformal representations and momentum-twistor space to rationalize algebraic roots, simplifying calculations.

Contribution

It introduces a method to directly integrate planar multi-loop Feynman integrals using dual-conformal and momentum-twistor techniques, enabling rationalization of algebraic roots and strategic coordinate choices.

Findings

Successful direct integration of multi-loop integrals up to four loops.

Use of momentum-twistor space to rationalize algebraic roots.

Examples covering integrals with up to eight particles.

Abstract

We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals. Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop integrals, and the fact that many of the algebraic roots associated with (e.g. Landau) leading singularities are automatically rationalized in momentum-twistor space---facilitating direct integration via partial fractioning. We describe how momentum twistors may be chosen non-redundantly to parameterize particular integrals, and how strategic choices of coordinates can be used to expose kinematic limits of interest. We illustrate the power of these ideas with many concrete cases studied through four loops and involving as many as eight particles. Detailed examples are included as ancillary files to this work's submission to the arXiv.