Feynman integrals as A-hypergeometric functions

Abstract

We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel'fand-Kapranov-Zelevinsky (GKZ) system. In order to define such GKZ system, we consider the polynomial obtained from the Symanzik polynomials $g=\mathcal{U}+\mathcal{F}$ as having indeterminate coefficients. Noncompact integration cycles can be determined from the coamoeba---the argument mapping---of the algebraic variety associated with $g$. In general, we add a deformation to $g$ in order to define integrals of generic graphs as linear combinations of their canonical series. We evaluate several Feynman integrals with arbitrary non-integer powers in the propagators using the canonical series algorithm.

arXiv comment45 pages, 11 figures, codimension 2 example added, references added. Version to be published

References in corpus (33)

• Harmonic Polylogarithms
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Numerical evaluation of multiple polylogarithms
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• Resolution of singularities for multi-loop integrals
• Explicit solutions of the multi-loop integral recurrence relations and its application
• Feynman Integrals and Intersection Theory
• Feynman graph polynomials
• Critical points and number of master integrals
• Cuts of Feynman Integrals in Baikov representation
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Scalar One-Loop Integrals using the Negative-Dimension Approach
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• Feynman integral relations from parametric annihilators
• Application of the negative-dimension approach to massless scalar box integrals
• Maximal Cuts in Arbitrary Dimension
• Baikov-Lee Representations Of Cut Feynman Integrals
• 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
• All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters
• Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions
• Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case
• A new approach to the momentum expansion of multiloop Feynman diagrams
• Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation
• Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
• Counting master integrals: integration by parts vs. differential reduction
• HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalizedhypergeometric functions: Fc hypergeometric function of three variables

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• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• Principal Landau Determinants
• From positive geometries to a coaction on hypergeometric functions
• Kinematic singularities of Feynman integrals and principal A-determinants
• Symbol Alphabets from the Landau Singular Locus
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Towards Systematic Evaluation of de Sitter Correlators via Generalized Integration-By-Parts Relations
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• 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
• On Feynman graphs, matroids, and GKZ-systems
• Analytic continuations of the Horn $H_1$ and $H_5$ functions
• GKZ-system of the 2-loop self energy with 4 propagators
• Feynman Integrals of Grassmannians
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• $\text{AlgRel.wl}$: Algebraic Relations for the Product of Propagators in Feynman integrals
• From equations in coordinate space to Picard-Fuchs and back
• Singularities of Feynman Integrals
• Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems
• Vanishing cycles and analysis of singularities of Feynman diagrams
• The diagrammatic coaction beyond one loop
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions