Magnus and Dyson series for Master Integrals

Abstract

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.

Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering

“…Of those 49 MIs, 8 contain only massless internal lines, 24 involve one massive line and 17 involve two massive lines. The system of differential equations obeyed by the MIs is cast in a canonical form [90], following the algorithm based on the use of the Magnus exponential, introduced in [91,92]. 1 Boundary conditions are retrieved either from the knowledge of simpler integrals emerging in specific kinematic limits, or by requiring the regularity of the solution at pseudothresholds.…”

“…By means of a suitable basis transformation, built with the help of the Magnus exponential [91,116] following the procedure outlined in section 2 of [92], we obtain a canonical set of MIs [90]. Such a basis obeys a system of differential equation where the dependence on is factorized from the kinematics.…”

The two-loop master integrals for $ q\overline{q} $ → VV

“…Improving upon our earlier results [18], we are now transforming the differential equations to a canonical form [31] which renders their integration trivial after an expansion in . The algorithm applied for this transformation is described in detail in section 3, similar procedures have been put forward most recently in [32,33]. With this, the remaining non-trivial step in the calculation of the master integrals is the determination of the boundary terms, which we describe in section 4.…”

“…Moreover, even in those cases where it is known that the final result will contain only GHPLs, no algorithm for finding such basis is known, while only some general criteria have been pointed out recently [32,33,57,58].…”

On the calculation of soft phase space integral

“…The advantage of the optimal basis is that the resulting differential equation can be solved almost trivially, up to some integration constants, as demonstrated in many non-trivial examples [13,14,[49][50][51][52][56][57][58][59][60][61][62][63][64][65][66]. Specifically, in d = 4 − 2ǫ dimension, the differential equation for the optimal basis can be written as…”