Numerical evaluation of multiple polylogarithms

By Jens Vollinga, Stefan Weinzierl. Computer Physics Communications, 2004

473 citations recorded by the source corpus.

Multiple polylogarithms appear in analytic calculations of higher order corrections in quantum field theory. In this article we study the numerical evaluation of multiple polylogarithms. We provide algorithms, which allow the evaluation for arbitrary complex arguments and without any restriction on the weight. We have implemented these algorithms with arbitrary precision arithmetic in C++ within the GiNaC framework.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7cea80a45551a5434a011e144a8638eb234ad321

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

The four-loop six-gluon NMHV ratio function (2015, methodology): “However, it is relatively straightforward to express any function with these symbol letters in terms of Goncharov polylogarithms, which in turn can be evaluated numerically with GiNaC [92, 93].”
Standard relations of multiple polylogarithm values at roots of unity (2007, methodology): “es of the ﬁnite and extended double shuﬄe relations. Most of the relations in this paper are discovered with the help of Maple using symbolic computations. We then veriﬁed all such relations by GiNaC [30] with an error bounded by 10−90. I would like to thank Jens Vollinga for answering some of my qu”
W±Z production at the LHC: fiducial cross sections and distributions in NNLO QCD (2016, methodology): “p helicity amplitudes we rely on a public C++ library [73] that implements the results of Ref. [48], and for the numerical evaluation of the relevant multiple polylogarithms we use the implementation [74] in the GiNaC [75] library. The contribution of the massive-quark loops is neglected in the two-”
Single-valued harmonic polylogarithms and the multi-Regge limit (2012, methodology): “Using the C++ symbolic computation framework GiNaC [63], which allows for the efficient numerical evaluation of HPLs to high precision [64], we can evaluate these functions numerically.”
Mathematical aspects of scattering amplitudes (2014, methodology): “ ai ∈ {0,1,−y,−1− y}, for y ∈ C. They appear in the computation of four-point functions with three on-shell and one oﬀ shell leg [15–17], and can be evaluated numerically using the techniques of ref. [13, 18]. (3) Generalized harmonic polylogarithms (GHPLs) [19]: GHPLs are deﬁned as iterated integra”
W+W− production at the LHC: fiducial cross sections and distributions in NNLO QCD (2015, methodology): “[48], and for the numerical evaluation of the relevant multiple polylogarithms we use the implementation [74] in the GiNaC [75] library.”
Event shapes and jet rates in electron-positron annihilation at NNLO (2009, background): “It should be mentioned that the results of this paper rely heavily on research carried out in the past years: Integration techniques for two-loop amplitudes [13–16], the calculation of the relevant tree-, one- and two-loop-amplitudes [17–29], routines for the numerical evaluation of polylogarithms [”
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, background): “es it was realized however that for certain multi-loop integrals new classes of transcendental functions arise that can no longer be expressed in terms of the classical polylogarithm functions, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,”
PolyLogTools — polylogs for the masses (2019, unknown): “[81] for the numerical evaluation of MPLs.”
Soft triple-real radiation for Higgs production at N3LO (2013, methodology): “This rational number can easily be determined by evaluating Ai1 numerically at a single point using any of the standard libraries to evaluate multiple polylogarithms [84,126–131], and running for”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, unknown): “05 46”
Feynman integrals and hyperlogarithms (2015, background): “Facilities for numeric evaluations of hyper- and polylogarithms are not included, because such are not necessary for the integration algorithms and secondly there are already established programs available for this task [14, 175].”