Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
Sven Moch, Peter Uwer, Stefan Weinzierl
TL;DR
The paper develops an algebraic framework based on nested sums ($Z$-sums and $S$-sums) to systematize the $\varepsilon$-expansion of higher transcendental functions and to tackle multi-scale, multi-loop integrals. It defines the sums, proves their algebraic structure and connections to Goncharov's multiple polylogarithms, and presents four algorithms (A–D) to reduce expansions to $Z$-sums, $S$-sums, and polylogarithms in a computer-implementable way. The authors demonstrate applications to generalized hypergeometric functions, Appell and Kampé de Fériet functions, and, as a novel result, the two-loop C-topology with arbitrary propagator powers and dimensions, enabling broader multi-scale calculations. The approach provides a concrete bridge between nested-sum algebras and the standard polylogarithm framework, with potential impact on precision calculations in high-energy physics.
Abstract
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer. Examples, such as expansions of generalized hypergeometric functions or Appell functions are discussed. As a further application, we give the general solution of a two-loop integral, the so-called C-topology, in terms of multiple nested sums. In addition, we discuss some important properties of nested sums, in particular we show that they satisfy a Hopf algebra.