Dissecting polytopes: Landau singularities and asymptotic expansions in $2\to 2$ scattering

TL;DR

This paper explores the geometric structure of Landau singularities in Feynman integrals, extending existing algorithms to handle complex pinched singularities in $2\to 2$ scattering, and identifies new asymptotic regions crucial for accurate expansions.

Contribution

It introduces a method to dissect Newton polytopes at singular loci, enabling evaluation of integrals with pinched singularities and systematic identification of new regions for asymptotic analysis.

Findings

Pinched singularities occur starting from three loops in certain nonplanar graphs.

Standard sector decomposition fails for these integrals, but the new method succeeds.

New regions characterized by multiple connected subgraphs or Glauber modes are essential for correct asymptotics.

Abstract

Parametric representations of Feynman integrals have a key property: many, frequently all, of the Landau singularities appear as endpoint divergences. This leads to a geometric interpretation of the singularities as faces of Newton polytopes, which facilitates algorithmic evaluation by sector decomposition and asymptotic expansion by the method of regions. Here we identify cases where some singularities appear instead as pinches in parametric space for general kinematics, and we then extend the applicability of sector decomposition and the method of regions algorithms to such integrals, by dissecting the Newton polytope on the singular locus. We focus on $2\to 2$ massless scattering, where we show that pinches in parameter space occur starting from three loops in particular nonplanar graphs due to cancellation between terms of opposite sign in the second Symanzik polynomial. While the affected integrals cannot be evaluated by standard sector decomposition, we show how they can be computed by first linearising the graph polynomial and then splitting the integration domain at the singularity, so as to turn it into an endpoint divergence. Furthermore, we demonstrate that obtaining the correct asymptotic expansion of such integrals by the method of regions requires the introduction of new regions, which can be systematically identified as facets of the dissected polytope. In certain instances, these hidden regions exclusively govern the leading power behaviour of the integral. In momentum space, we find that in the on-shell expansion for wide-angle scattering the new regions are characterised by having two or more connected hard subgraphs, while in the Regge limit they are characterised by Glauber modes.