A geometric method of sector decomposition

TL;DR

The paper addresses IR divergences in multi-loop integrals by reformulating sector decomposition as a deterministic geometric problem. By mapping to convex geometry, it treats factorization as constructing intersections of dual convex cones and triangulating these intersections, thereby avoiding iterative loops. The approach yields potentially smaller sector counts and provides a clear, algorithmic path via cone construction, dualization, and triangulation, implemented in a Python prototype. Though the current proof-of-concept is slow, the method promises more reliable pole extraction and numerical integration in perturbative QCD calculations.

Abstract

We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method provides a deterministic algorithm and never falls into an infinite loop. The number of resulting sectors depends on the algorithm of triangulation. Our test implementation shows smaller number of sectors comparing with other existing methods with iterations.

Figures (2)

• Figure 1: An example of cone and dual cone
• Figure 2: An example of different triangulations