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

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 ǫ-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 = 9m 2 , in an expansion in ǫ up to ǫ 1 . With the help of our code, we obtain numerical results for the threshold master integrals in an ǫ-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.

A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

“…Despite the remarkable progress that has been achieved in the analytic calculation of multiloop amplitudes and integrals in the last few years, analytical approaches are only at the beginning of a journey into largely unexplored mathematical territory if the function class of the results goes beyond multiple polylogarithms (MPLs), typically involving elliptic or hyper-elliptic functions, see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

Evaluating a family of two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence

“…For an application of series expansion methods to single scale integrals see e.g. [76,[85][86][87][88][89][90]. For integrals depending on several scales, multivariate series expansions have been used for special kinematic configurations (typically small or large energy limits, see e.g.…”

“…The problem is now one dimensional, and we can apply e.g. the methods of [89,97], which provide a formula for the solution of the differential equations near a singular point, whose coefficients are fixed by solving recurrence relations. Here we proceed in a different way, by directly integrating the differential equations in terms of (generalized) power series.…”

NNLO mixed EW-QCD corrections to single vector boson production

“…[64]. We have obtained their complete solution with a series expansion around z = 1 (see for instance [76,64,77,78,79,80,81]). In the computation of the MIs, the mass of the W boson is set equal to m Z , the mass of Z boson, to avoid the presence of an additional energy scale in the problem, which would make the analytical solution of the differential equations in terms of known functions more complicated.…”