Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals

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.

28 pages, minor changes

References in corpus (10)

• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• Two-loop QCD corrections to gluon-gluon scattering
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
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• Two-loop QCD corrections to massless quark-gluon scattering
• Calculation of master integrals by difference equations
• Application and explicit solution of recurrence relations with respect to space-time dimension
• Analytical Result for Dimensionally Regularized Massless Master Non-planar Double Box with One Leg off Shell
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