Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

TL;DR

This paper explores advanced integral solutions in 3-loop massive QCD calculations, highlighting the emergence of iterative non-iterative integrals involving elliptic functions and their application to key QCD quantities.

Contribution

It introduces the formalism for iterative non-iterative integrals in 3-loop massive QCD and provides explicit examples involving anomalous dimensions and the rho-parameter.

Findings

Identification of iterative non-iterative integrals involving elliptic functions.

Application of the formalism to 3-loop anomalous dimensions.

Analysis of the structure of solutions in complex QCD calculations.

Abstract

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 $\gamma_{qg}^{(2)}$ and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the $\rho$-parameter.