Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5

Understand

We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale.

• The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3--loop anomalous dimensions.
• The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier.
• Unlike the shuffle relations, they depend on the value of the quantities considered.