Multiple polylogarithms and elliptic generalisations

Multiple polylogarithms

Multiple polylogarithms are generalisations of the usual logarithm towards several variables and higher transcendental weight. They occur in particle physics in higher loop calculations. Multiple polylogarithms have a rich algebraic structure: They obey two distinct Hopf algebras, and in addition there are convolution and conjugation operations. Based on these algebraic properties we derived algorithms, which allowed us to solve certain classes of integrals in terms of multiple polylogarithms. We further developed methods for the numerical evaluation of multiple polylogarithms and implemented these in the open-source computer algebra system GiNaC.

Elliptic generalisations

Multiple polylogarithms are an important class of functions, but not sufficient to express all Feynman integrals. The simplest counter-example is given by the two-loop sunrise integral with three non-zero internal masses. Recently, we made a first step beyond the class of multiple polylogarithms. Using ideas from the theory of mixed Hodge structures we were able to derive a differential equation for the sunrise integral. A generalisation lead us then to a procedure to find the Picard-Fuchs equation for any Feynman integral. In the analytic solution for the sunrise integral one encounters elliptic generalisations of multiple polylogarithms.

Talks on this subject:

• Talk given at Radcor 2015, Los Angeles, June 2015, arXiv:1601.03646 [hep-ph].

Selected publications on this subject:

• Luise Adams, Christian Bogner, Stefan Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J.Math.Phys. 57 (2016), 032304, arXiv:1512.05630 [hep-ph].
• Stefan Müller-Stach, Stefan Weinzierl, Raphael Zayadeh, Picard-Fuchs equations for Feynman integrals, Commun.Math.Phys. 326 (2014), 237, arXiv:1212.4389 [hep-ph].
• Jens Vollinga, Stefan Weinzierl, Numerical evaluation of multiple polylogarithms, Comput.Phys.Commun. 167 (2005) 177, hep-ph/0410259.
• Sven Moch, Peter Uwer, Stefan Weinzierl, Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals, J.Math.Phys. 43 (2002), 3363, hep-ph/0110083.