Geometric approach to asymptotic expansion of Feynman integrals

TL;DR

The paper introduces a covariant, geometrically grounded method to identify the regions that contribute to non-threshold asymptotic expansions of Feynman integrals. By mapping the monomials of the alpha-representation polynomials $\mathcal{U}$ and $\mathcal{F}$ to points in a high-dimensional space, expansion regions correspond to bottom facets of the convex hull of these points, with region scalings given by normals to these facets. The approach is implemented in Mathematica using QHull, enabling automatic region discovery for complex multi-loop integrals and including non-trivial Minkowski-space examples (e.g., ultrasoft-collinear regions). The work provides a practical algorithm and a usable tool for systematic, automated asymptotic analysis of parametric integrals inHigh-energy physics.

Abstract

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.

Figures (2)

• Figure 1: Graphical representation of sets $\{F\}$ (crossed points) and $\{U\}$ (diamonds) corresponding to the integral of Eq. \ref{['eqn:1e1l2s']}.
• Figure 2: Double-scale two-loop vertex integral.