Azurite: An algebraic geometry based package for finding bases of loop integrals

TL;DR

The work addresses the challenge of finding a compact basis for the space of loop integrals associated with a given Feynman diagram and its subdiagrams. It introduces Azurite, a Mathematica/Singular-based tool that constructs IBP relations on generalized-unitarity cuts and employs syzygy computations within the Baikov representation to produce a minimal set of master integrals without squared propagators. Through adaptive parametrization, graph-theory symmetries, finite-field arithmetic, and parallelization, Azurite achieves rapid determinations of master integrals for two- and three-loop topologies and can analytically generate IBP identities on maximal cuts. This approach enhances the efficiency of IBP reductions and differential-equation methods in multi-loop amplitude calculations, with future directions including improved syzygy generation and public full IBP-reduction capabilities.

Abstract

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package {\sc Azurite} ({\bf A ZUR}ich-bred method for finding master {\bf I}n{\bf TE}grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems {\sc Singular} and {\sc Mathematica}. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts.

Figures (4)

• Figure 1: Pentagon-box diagram and one of its subdiagrams.
• Figure 2: Some diagrams which can be simplified in the adaptive parametrization.
• Figure 3: Irreducible topologies for the massless triple-box diagram.
• Figure 4: Computation time and number of master integrals for different topologies and mass configurations.