Feynman Integrals of Grassmannians

Abstract

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain GKZ-systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the GKZ-system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the 2-loop self-energy diagrams which are convenient to embed in the subvarieties of Grassmannians.

latex, 132 pages, including 11 figures + 1 appendix

References in corpus (10)

• Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
• Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC
• Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm
• Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• GKZ-hypergeometric systems for Feynman integrals
• Counting master integrals: integration by parts vs. differential reduction
• Geometrical methods in loop calculations and the three-point function
• Evaluating Feynman integrals by the hypergeometry

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• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• From equations in coordinate space to Picard-Fuchs and back
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals