Magnus and Dyson Series for Master Integrals
Mario Argeri, Stefano Di Vita, Pierpaolo Mastrolia, Edoardo Mirabella, Johannes Schlenk, Ulrich Schubert, Lorenzo Tancredi
TL;DR
This work develops a Magnus-Dyson based framework to cast differential equations for master integrals in dimensional regularization into a canonical form where the $\epsilon$-dependence is factorized from kinematics. By constructing a transformation via the Magnus expansion, the authors obtain a basis in which the differential system is linear in $\epsilon$ and integrable using Dyson or Magnus series, yielding solutions in terms of Harmonic Polylogarithms with uniform transcendentality. The method is demonstrated on representative two-loop QED processes, including one-loop Bhabha scattering, two-loop electron form factors, and the two-loop non-planar box, with boundary conditions fixed by regularity and known results. The framework is further extended to systems with polynomial $\epsilon$-dependence, indicating wide applicability to master integrals across dimensional regularization.
Abstract
We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 -> 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.