Graphical functions in parametric space

Abstract

Graphical functions are positive functions on the punctured complex plane $\mathbb{C}\setminus \{0,1\}$ which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover we prove a formula that relates graphical functions of planar dual graphs.

v2: extended introduction, minor changes in notation and correction of misprints

References in corpus (8)

• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• The massless higher-loop two-point function
• On Motives Associated to Graph Polynomials
• Feynman graph polynomials
• Leading singularities and off-shell conformal integrals
• On hyperlogarithms and Feynman integrals with divergences and many scales
• The Galois coaction on ${\varphi }^4$ periods
• Spanning forest polynomials and the transcendental weight of Feynman graphs

Cited by in corpus (11)

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• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Energy-energy correlations at next-to-next-to-leading order
• Multi-loop techniques for massless Feynman diagram calculations
• All five-loop planar four-point functions of half-BPS operators in $\mathcal{N}=4$ SYM
• Recursive computation of Feynman periods
• Statistics of Feynman amplitudes in ${\varphi }^4$-theory
• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values