Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral. (English) Zbl 1373.81293
Summary: 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 \(| q | \leq 1\), where \(| q | = 1\) is attained only at the singular points \(t \in \{m^2, 9 m^2, \infty \}\). This ensures the convergence of the \( q\)-series expansion of the ELi-functions and provides a fast and efficient evaluation of these Feynman integrals.
MSC:
| 81T18 | Feynman diagrams |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 11G55 | Polylogarithms and relations with \(K\)-theory |
| 14H52 | Elliptic curves |
Software:
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