Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

Abstract

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 ∆ 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.

From infinity to four dimensions: higher residue pairings and Feynman integrals

“…where the middle-dimensional integration cycle is Γ := R P + and the hat denotes the fact that we expect (3.14) to agree with (3.1) only after taking the limit δ a → 0. Closely related ways of rewriting Feynman integrals as twisted periods were introduced in [30], see also [31,68,70,72,[74][75][76][77].…”

The l-loop banana amplitude from GKZ systems and relative Calabi-Yau periods

“…The periods are completely determined by modular functions as follows from [20]: We can bring the constraint P 2 = 0 (3.12) defining the elliptic curve into Weierstrass form y 2 = 4x 3 − xg 2 (u, m) − g 3 (u, m). This defines the modular parameter τ (u, m) from the definition of the Hauptmodul j of PSL(2, Z) as 17) where q = exp(2πiτ ). Then the period a Ω/u which yields the maximal cut integral is given in terms of the Eisenstein series as…”

“…Their form can be readily generalized to arbitrary dimensions as in equation (2.18) and will be called Barth-Nieto Calabi-Yau (l − 1)-folds and describe the geometry of the l-loop graph (2.1) for differential equaions. 7 See also [16,17] for a different discussion of the GKZ system in the context of Feynman integrals with generic mass dependencies. 8 In the two-loop case the diagram is also called sunset diagram.…”

GKZ-hypergeometric systems for Feynman integrals

“…Using the triangulations of the Newton polytope of Lee-Pomeransky polynomial, the author of Ref. [38] presents GKZ-hypergeometric system of the sunset diagram of codimension= 6. He also constructs canonical series solutions which contain three redundant variables besides three independent dimensionless ratios among the external momentum squared p 2 and three virtual mass squared m 2 i (i = 1, 2, 3) under the assumption…”