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

TL;DR

This work extends integrand-reduction techniques from one to two loops in pure four-dimensional space, using Gröbner-basis methods to classify the full two-loop integrand basis across all topologies and external-momentum configurations. It distinguishes integrand vs integral bases and analyzes the on-shell variety and its branch structure to determine coefficients efficiently, providing explicit basis element sets for each topology and kinematic case. The study demonstrates the utility of algebraic-geometry approaches for systematic coefficient extraction and highlights the complexity of branch structures, especially under degeneracies, underscoring the need for further IBP-based reduction for practical amplitude computations. Overall, the paper lays a comprehensive groundwork for automated, topology- and kinematics-aware two-loop integrand basis classification in four dimensions, with implications for higher-loop extensions.

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 Gröbner 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.