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

TL;DR

This work shows that the kite-family Feynman integrals, including the equal-mass sunrise, can be expressed to all orders in the dimensional regulator via ELi-functions tied to an elliptic curve, and that analytical continuation reduces to the continuation of two periods $\psi_1$ and $\psi_2$. The authors derive explicit formulas for these periods for all real $t$ and prove that the nome $q=\exp(i\pi \psi_2/\psi_1)$ satisfies $|q|\le 1$, with equality only at singular points $t\in\{m^2,9m^2,\infty\}$, ensuring rapid convergence of the $q$-series. They also establish the master-integral differential system in a way that the $q$-space equations hold for all real $t$, and compute the monodromy via Picard-Lefschetz theory. Numerical results demonstrate perfect agreement with sector-decomposition methods across the whole real axis and threshold regions, highlighting the practical efficiency of ELi-functions for precise, fast evaluations in high-energy calculations. Overall, the paper provides a complete, efficient framework for analytic continuation and numerical evaluation of elliptic Feynman integrals over the full kinematic range.

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.

Figures (12)

• Figure 1: The variation of the roots $e_1$, $e_2$ and $e_3$ of the elliptic curve $y^2 = 4 (x-e_1)(x-e_2)(x-e_3)$ with $t$. The roots $e_1$ and $e_2$ acquire an imaginary part for $m^2 < t < 9m^2$. At $t=0$ the roots $e_2$ and $e_3$ coincide. At $t=m^2$ and at $t=9m^2$ the roots $e_1$ and $e_2$ coincide. At $t=\infty$ the roots $e_1$ and $e_3$ coincide. In the interval $m^2 \le t \le 9 m^2$ the real parts of the roots $e_1$ and $e_2$ coincide.
• Figure 2: The path for the analytic continuation in the variable $t$. Feynman's $i0$-prescription avoids the singular points at $0$, $m^2$ and $9m^2$ as shown in the figure.
• Figure 3: The path in the complex $k^2$-space, as $t$ varies along the path of fig. (\ref{['fig_t_path']}). In the regions IV, I and II the path in $k^2$-space is at an infinitesimal distance below the real axis. The path crosses the branch cut $[1,\infty]$ in the region $C_1$. In region III we have $\mathrm{Re}(k^2)=1/2$.
• Figure 4: The path in the complex $k'{}^2$-space, as $t$ varies along the path of fig. (\ref{['fig_t_path']}). In the regions IV, I and II the path in $k'{}^2$-space is at an infinitesimal distance above the real axis. In region III we have $\mathrm{Re}(k^2)=1/2$.
• Figure 5: The path in $\tau$-space and in $q$-space, as $t$ varies along the path of fig. (\ref{['fig_t_path']}). The value $t=-\infty$ corresponds to $\tau=0$ and $q=1$, the value $t=0$ corresponds to $\tau=i\infty$ and $q=0$, the value $t=m^2$ corresponds to $\tau=-1$ and $q=-1$, the value $t=9m^2$ corresponds to $\tau=-\frac{1}{3}$ and $q=\frac{1}{2} - \frac{i \sqrt{3}}{2}$.