Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral

Abstract

We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of $t\in \mathbb{R}$. Furthermore, the nome $q$ of the elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le 1$, where $|q|=1$ is attained only at the singular points $t\in \{m^2,9m^2,\infty \}$. This ensures the convergence of the $q$-series expansion of the $\mathrm{ELi}$-functions and provides a fast and efficient evaluation of these Feynman integrals.

30 pages, version to be published

References in corpus (13)

• Lectures on differential equations for Feynman integrals
• Feynman Diagrams and Differential Equations
• Resolution of singularities for multi-loop integrals
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Elliptic integral evaluations of Bessel moments
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• Numerical integration of one-loop Feynman diagrams for N-photon amplitudes
• The kite integral to all orders in terms of elliptic polylogarithms
• Numerical NLO QCD calculations
• Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
• A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques
• Evaluation of multi-loop multi-scale integrals and phenomenological two-loop applications

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• Singularities of Feynman Integrals
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