Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

TL;DR

This paper introduces an algorithm for analytically solving linear systems of first order factorizable differential equations, with applications in quantum field theory calculations, using iterated integrals and without requiring special basis representations.

Contribution

The paper presents a novel algorithm for solving first order factorizable differential systems analytically, applicable to complex quantum field theory integrals, utilizing iterated integrals over structured alphabets.

Findings

Successfully applied to three-loop massive form factors

Results expressed in cyclotomic harmonic polylogarithms

No need for special basis of master integrals

Abstract

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three--loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.