Various of the single scale quantities in massless and massive QCD up to 3-loop order can be expressed by iterative integrals over certain classes of alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples are the anomalous dimensions to 3-loop order, the massless Wilson coefficients and also different massive operator matrix elements. Starting at 3-loop order, however, also other letters appear in the case of massive operator matrix elements, the so called iterative non-iterative integrals, which are related to solutions based on complete elliptic integrals or any other special function with an integral representation that is definite but not a Volterra-type integral. After outlining the formalism leading to iterative non-iterative integrals,we present examples for both of these cases with the 3-loop anomalous dimension γqg(2) and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the ρ-parameter.
@article{arxiv.1711.09742,
title = {Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations},
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and E. Imamoglu and M. van Hoeij A. von Manteuffel and C. G. Raab and C. -S. Radu and C. Schneider},
journal= {arXiv preprint arXiv:1711.09742},
year = {2017}
}