Rationalizing Loop Integration

TL;DR

The paper demonstrates that a broad class of planar multi-loop Feynman integrals can be integrated directly in Feynman parameters by leveraging manifest dual-conformal structures and momentum-twistor coordinates to rationalize algebraic roots. It introduces non-redundant momentum-twistor parameterizations and canonical charts on the positive Grassmannian to expose the essential kinematic degrees of freedom while preserving single-valuedness in the Euclidean domain. Through explicit hexagon, heptagon, and octagon examples up to four loops and eight particles, the authors show how Gramian determinants and other algebraic obstructions can be tamed, enabling direct integration in many nontrivial cases. The ancillary files provide concrete results and illustrate the potential of this approach to improve analytic control over planar amplitudes in N=4 SYM and to inform future extensions to higher-point and higher-loop computations.

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.