Structural Relations of Harmonic Sums and Mellin Transforms at Weight w = 6 1 Johannes Blümlein

We derive the structural relations between nested harmonic sums and the corresponding Mellin transforms of Nielsen integrals and harmonic polylogarithms at weight w = 6. They emerge in the calculations of massless single–scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, to 3– and 4–loop order. We consider the set of the multiple harmonic sums at weight six without index {−1}. This restriction is sufficient for all known physical cases. The structural relations supplement the algebraic relations, due to the shuffle product between harmonic sums, studied earlier. The original amount of 486 possible harmonic sums contributing at weight w = 6 reduces to 99 sums with no index {−1}. Algebraic and structural relations lead to a further reduction to 20 basic functions. These functions supplement the set of 15 basic functions up to weight w = 5 derived formerly. We line out an algorithm to obtain the analytic representation of the basic sums in the complex plane. Proceedings of the “Motives, Quantum Field Theory, and Pseudodifferential Operators”, held at the Clay Mathematics Institute, Boston University, June 2–14, 2008