0000000000002493
AUTHOR
Armin Schweitzer
Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
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 $\mathr…
The kite integral to all orders in terms of elliptic polylogarithms
We show that the Laurent series of the two-loop kite integral in $D=4-2\varepsilon$ space-time dimensions can be expressed in each order of the series expansion in terms of elliptic generalisations of (multiple) polylogarithms. Using differential equations we present an iterative method to compute any desired order. As an example, we give the first three orders explicitly.
Analytic Continuation of the Kite Family
We consider results for the master integrals of the kite family, given in terms of ELi-functions which are power series in the nome q of an elliptic curve. The analytic continuation of these results beyond the Euclidean region is reduced to the analytic continuation of the two period integrals which define q. We discuss the solution to the latter problem from the perspective of the Picard–Lefschetz formula.
Differential equations for Feynman integrals beyond multiple polylogarithms
Differential equations are a powerful tool to tackle Feynman integrals. In this talk we discuss recent progress, where the method of differential equations has been applied to Feynman integrals which are not expressible in terms of multiple polylogarithms.