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

Understand

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.