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Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems. (English) Zbl 1436.81049
Summary: We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope $$\Delta_G$$ corresponding to the Lee-Pomeransky polynomial $$G$$. Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

MSC:
 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 81U05 $$2$$-body potential quantum scattering theory
Software:
TOPCOM; polymake; DLMF
Full Text:
References:
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