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

TL;DR

Azurite is a new algebraic geometry-based software package that efficiently finds bases of Feynman integral vector spaces using syzygy computations and symmetry analysis, aiding in simplifying complex quantum field theory calculations.

Contribution

It introduces Azurite, a novel package that constructs IBP identities on unitarity cuts using algebraic geometry, improving the efficiency of finding master integrals.

Findings

Successfully computes bases of Feynman integrals.

Utilizes syzygy computations and diagram symmetries.

Supports analytical calculation of IBP identities on cuts.

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.