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

TL;DR

This paper presents an advanced algorithm for solving differential equations of master integrals using series expansions near singular points, enabling high-precision numerical and analytical evaluations at special kinematic values.

Contribution

The authors extend their previous algorithm to evaluate four-loop master integrals at threshold with high precision, deriving analytical results from numerical data using PSLQ and multiple polylogarithms.

Findings

Achieved 6000-digit numerical precision for master integrals at threshold.

Derived analytical expressions for master integrals using PSLQ and polylogarithm bases.

Successfully evaluated master integrals at $p^2=9 m^2$ up to $ ext{O}(\e)$.

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 $\epsilon$-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 $\epsilon$ up to $\epsilon^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $\epsilon$-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.