GKZ-system of the 2-loop self energy with 4 propagators

Abstract

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

Restrictions of Pfaffian systems for Feynman integrals

“…But this is a contradiction, because f (t) is the minimal polynomial. 35 Suppose we are given a pair of irreducible polynomials…”

“…We take our inspiration from a particularly well-studied holonomic D-module: the GKZ hypergeometric system [24] -though, as we show, the algorithms presented here also apply beyond this case. In the GKZ framework [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], one generalizes parametric representations of a Feynman integral to include extra variables, such that now z = (z 1 , . .…”