Abstract
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of special numbers. Starting with harmonic sums and polylogarithms we discuss recent extensions of these quantities as cyclotomic, generalized (cyclotomic), and binomially weighted sums, associated iterated integrals and special constants and their relations.
Highlights
During the late 1990ies several massless and massive two-loop calculations in Quantum Chromodynamics reached a complexity, see e.g. [1,2,3,4,5], which made it necessary to introduce new functions a systematic manner to represent the analytic results in an adequate form
At the side of the nested sums they belong to the cyclotomic harmonic sums [40]
Elliptic integrals emerge in the calculation of massive Feynman diagrams [42,43,44,45,46,47]
Summary
During the late 1990ies several massless and massive two-loop calculations in Quantum Chromodynamics reached a complexity, see e.g. [1,2,3,4,5], which made it necessary to introduce new functions a systematic manner to represent the analytic results in an adequate form. Root-valued letters occur in the alphabets of iterated integrals [41] They correspond to binomially-weighted generalized cyclotomic sums. The stuffle and shuffle relations imply relations between the harmonic sums and harmonic polylogarithms, respectively, which do not depend on their arguments N and x and are called algebraic relations [65] Both these algebras can be applied to the multiple zeta values. By iteration of this structure the general cyclotomic sums are obtained They occur in the calculation of massive Feynman diagrams. The special constants being associated to the cyclotomic sums and polylogarithms extend the multiple zeta values. An even wider class of special numbers is associated to the generalized (cyclotomic) harmonic sums and polylogarithms. A convenient way to work with these and the more special functions being listed above is provided by the Mathematica-package HarmonicSums [39, 75]
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