Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

TL;DR

The paper addresses evaluating four-loop master integrals when a canonical $ε$-form of the differential equations is not available. It employs a numerical algorithm based on generalized power-series expansions near regular singular points, solving coefficient-difference equations, and matching expansions to connect distant points. Focusing on a four-loop generalized sunset diagram with three massive and two massless propagators, it computes high-precision $ε$-expansions at the threshold $p^2=9m^2$ and uses PSLQ against bases of multiple polylogarithms at sixth roots of unity (with $\sqrt{3}$) to extract analytic results up to $ε^1$. The results demonstrate that, even without a canonical basis, the threshold values can be expressed in terms of conventional polylogarithmic constants, and outline the practical considerations for extending to higher weights and the role of basis choice in PSLQ analyses.

Abstract

This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $ε$-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, $p^2=9 m^2$, in an expansion in $ε$ up to $ε^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $ε$-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.

Figures (1)

• Figure 1: The generalized sunset graph with two massless and three massive lines with the same mass.