Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss in particular their application to the computation of Feynman integrals.Program summaryProgram title: HyperIntCatalogue identifier: AEUV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 9455No. of bytes in distributed program, including test data, etc.: 2542680Distribution format: tar.gzProgramming language: Maple [1], version 16 or higher.Computer: Any that supports Maple.Operating system: Any that supports Maple.RAM: Highly problem dependent; from a few MiB to many GiBClassification: 4.4, 5.Nature of problem:Feynman integrals and their εε-expansions in dimensional regularization can be expressed in the Schwinger parametrization as multi-dimensional integrals of rational functions and logarithms. Symbolic integration of such functions therefore serves a tool for the exact and direct evaluation of Feynman graphs.Solution method:Symbolic integration of rational linear combinations of polylogarithms of rational arguments is obtained using a representation in terms of hyperlogarithms. The algorithms exploit their iterated integral structure.Restrictions:To compute multi-dimensional integrals with this method, the integrand must be linearly reducible, a criterion we state in Section 4. As a consequence, only a small subset of all Feynman integrals can be addressed.Unusual features:The complete program works strictly symbolically and the obtained results are exact. Whenever a Feynman graph is linearly reducible, its εε-expansion can be computed to arbitrary order (subject only to time and memory restrictions) in εε, near any even dimension of space–time and for arbitrarily εε-dependent powers of propagators with integer values at ε=0ε=0. The method is not restricted to scalar integrals and applies even to (regulated) divergent integrals.Additional comments:Apart from Feynman integrals, other suitable parametric integrals may be computed (or expanded in εε) as well, like for example hypergeometric functions.An example worksheet, Manual.mw, is included. This contains an explanation of most features provided and includes plenty of examples of Feynman integral computations.Running time:Highly dependent on the particular problem through the number of integrations to be performed (edges of a graph), the number of remaining variables (kinematic invariants), the order in εε and the complexity of the geometry (topology of the graph). Simplest examples finish in seconds, but the time needed increases beyond any bound for sufficiently high orders in εε or graphs with many edges.References:[1] Maple 161. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.