The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements

TL;DR

This paper computes polarized three-loop anomalous dimensions in QCD using massive operator matrix elements, introducing new methods to handle complex integrals and providing comprehensive results that extend existing literature.

Contribution

It presents the first complete calculation of polarized three-loop anomalous dimensions including the $ ext{T}_F$ contributions, using advanced techniques to handle elliptic integrals.

Findings

Calculated all $ ext{T}_F$ contributions to three-loop anomalous dimensions.

Recalculated $ ext{T}_F$ contributions to the three-loop QCD $eta$-function.

Developed methods to handle elliptic contributions in master integrals.

Abstract

We calculate all contributions $\propto T_F$ to the polarized three-loop anomalous dimensions in the M-scheme using massive operator matrix elements and compare to results in the literature. This includes the complete anomalous dimensions $\gamma_{qq}^{(2),\rm PS}$ and $\gamma_{qg}^{(2)}$. We also obtain the complete two-loop polarized anomalous dimensions in an independent calculation. While for most of the anomalous dimensions the usual direct computation methods in Mellin $N$-space can be applied since all recurrences factorize at first order, this is not the case for $\gamma_{qg}^{(2)}$. Due to the necessity of deeper expansions of the master integrals in the dimensional parameter $\varepsilon = D-4$, we had to use the method of arbitrary high moments to eliminate elliptic contributions in intermediate steps. 4000 moments were generated to determine this anomalous dimension and 2640 moments turned out to be sufficient. As an aside, we also recalculate the contributions $\propto T_F$ to the three-loop QCD $\beta$-function.