GKZ-hypergeometric systems for Feynman integrals

Abstract

Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller's transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.

32 pages, 1 figure

References in corpus (18)

• 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
• Analytical Result for Dimensionally Regularized Massless On-shell Double Box
• Non-planar massless two-loop Feynman diagrams with four on-shell legs
• Explicit results for all orders of the epsilon-expansion of certain massive and massless diagrams
• A New Hypergeometric Representation of One-Loop Scalar Integrals in $d$ Dimensions
• HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
• Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
• Application and explicit solution of recurrence relations with respect to space-time dimension
• Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case
• Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions
• Counting master integrals: integration by parts vs. differential reduction
• Geometrical methods in loop calculations and the three-point function
• HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalizedhypergeometric functions: Fc hypergeometric function of three variables
• Evaluating Feynman integrals by the hypergeometry
• The system of partial differential equations for the $C_{{}_0}$ function
• Periods of Feynman Diagrams and GKZ D-Modules

Cited by in corpus (19)

• Collider Physics at the Precision Frontier
• Analytic Structure of all Loop Banana Amplitudes
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Landau Discriminants
• Tropical Monte Carlo quadrature for Feynman integrals
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• FeynGKZ: a Mathematica package for solving Feynman integrals using GKZ hypergeometric systems
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• Algebraic aspects of hypergeometric differential equations
• Cohen-Macaulay Property of Feynman Integrals
• Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
• Feynman Integrals of Grassmannians
• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• From equations in coordinate space to Picard-Fuchs and back
• Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems
• Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions
• Vanishing cycles and analysis of singularities of Feynman diagrams
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals