The classification of two-loop integrand basis in pure four-dimension

TL;DR

This paper classifies the integrand basis of all two-loop diagrams in four-dimensional space-time, analyzing their topology and numerators using Gröbner basis methods, which aids in amplitude reduction.

Contribution

It provides a comprehensive classification of two-loop integrand bases in four dimensions, including topology and numerator sets under various kinematic conditions, using algebraic geometry techniques.

Findings

Classification of all two-loop diagram topologies.

Analysis of the variety structure and its irreducible branches.

Implications for numerical and analytical amplitude reduction.

Abstract

In this paper, we have made the attempt to classify the integrand basis of all two-loop diagrams in pure four-dimension space-time. Our classification includes the topology of two-loop diagrams which determines the structure of denominators, and the set of numerators under different kinematic configurations of external momenta by using Gr\"{o}bner basis method. In our study, the variety defined by setting all propagators to on-shell has played an important role. We discuss the structure of variety and how it splits to various irreducible branches when external momenta at each corner of diagrams satisfy some special kinematic conditions. This information is crucial to the numerical or analytical fitting of coefficients for integrand basis in reduction process.