Fuchsia and master integrals for splitting functions from differential equations in QCD

Abstract

We report on the recent progress in reducing differential equations for Feynman master integrals to canonical form with the help of a method proposed by Roman Lee. For the first time, we present Fuchsia -our open-source implementation of the Lee algorithm written in Python using mathematical routines of a free computer algebra system SageMath. We demonstrate Fuchsia by reducing differential equations for NLO contributions to splitting functions in QCD, which contain both loops and legs integrals.

Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics

“…A few years ago, Lee presented an algorithm which, starting from arbitrary basis integrals, is designed to find a change of basis so that the differential equations of the resulting integrals can be cast into canonical form [208]. By now, several implementations are publicly available [209][210][211], however they are confined to two-scale integrals, which can be expressed by a single dimensionless ratio. The same statement holds for the procedure described in Ref.…”

Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA

“…Until now, only implementations of the algorithm presented in Ref. [70] are publicly available [74][75][76]. However, this algorithm is restricted to ordinary differential equations, which are not sufficient to describe the full functional dependence of Feynman integrals depending on multiple dimensionless scales.…”

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

“…Until recently, no implementation of the Lee method was made publicly available. This fact motivated us to develop Fuchsia -the first public implementation of the Lee algorithm [Lee15] which was presented in [GM16]. Another implementation of this method, called Epsilon, was recently presented in [Pra17].…”