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

Abstract

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 various applications. In particular, many Feynman integrals can be computed by this method.Keywords: Feynman integrals, hyperlogarithms, polylogarithms, computer algebra, symbolic integration, ε-expansions 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 ε-expansion can be computed to arbitrary order (subject only to time and memory restrictions) in ε, near any even dimension of space-time and for arbitrarily ε-dependent powers of propagators with integer values at ε = 0. Also the method is not restricted to scalar integrals only, but arbitrary tensor integrals can be computed directly. Additional comments: Further applications to parametric integrals, outside the application to Feynman integrals in the Schwinger parametrization, are very likely. 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 ε 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 ε or graphs with many edges. Program summary References[1] Maple 16 1 . Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

“…. ; z) for any rational functions in z (at least in principle), but it is not always possible for non-rational expressions (see [43]). Actually both these problems are related to the fact that not all hypergeometric functions can be expanded as combinations of GPLs, but require more general functions.…”

“…In [28] a minimal basis of functions at higher weights were proposed, and specifically it was proposed that at weight four the additional [28,34,37,40,43].…”

Simplified Differential Equations Approach for Master Integrals

“…At the present stage we are not setting a fully-fledged numerical implementation, which will be done when all families will be computed. Our experience with double-box computations show that using for instance HyperInt [50] to bring all GPs in their range of convergence, before evaluating them numerically by GiNaC, increases efficiency by two orders of magnitude. Moreover expressing GPs in terms of classical polylogarithms and Li 2,2 , could also reduce substantially the CPU time [51].…”

Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

“…If it turns out that there is an order in which the dependence of the denominator of the integrand on the Feynman parameters is linear then the whole integral can be solved in terms of multiple polylogarithms. This strategy was successfully applied for example in [20][21][22][23][24] and implemented as the computer code HyperInt in [25].…”

“…The goal of this paper is thus to evaluate the coordinate-space Feynman integral associated with the graph of figure 1. Although this integral is linearly reducible, in the sense of [19], and an analytical result can be obtained with HyperInt in [25], we are going to evaluate it with differential equations, keeping in mind that many cases in our set of 20 remaining four-loop conformal integrals will be linearly irreducible, although knowledge on polynomial reduction is constantly increasing. Indeed, it is helpful to be able to test our result against a different method.…”