The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients

We report on the three-loop unpolarized and polarized massive operator matrix elements, with single- and two-mass corrections, and the associated deep-inelastic massive Wilson coefficients in the region $Q^2 \gg m_Q^2$, the calculation of which has been completed recently. We also provide fast and precise numerical representations of the massless Wilson coefficients, splitting functions to tree-loop order, and target-mass corrections in $x$-space well suited for QCD-fitting codes.

💡 Research Summary

The paper presents a comprehensive calculation of three‑loop massive operator matrix elements (OMEs) and the associated asymptotic Wilson coefficients for deep‑inelastic scattering (DIS) in the kinematic regime where the virtuality $Q^2$ is much larger than the heavy‑quark mass squared, $Q^2\gg m_Q^2$. Both unpolarized and polarized cases are treated, and the authors include single‑mass as well as genuine two‑mass contributions (charm–bottom). The work completes a long‑standing program to provide the full set of NNLO (next‑to‑next‑to‑leading order) heavy‑flavor corrections needed for precision QCD analyses of DIS data, such as the extraction of the strong coupling constant $\alpha_s(M_Z)$ and the charm quark mass $m_c$ at the 1 % level or better.

Methodology
The calculation starts from the generation of all three‑loop Feynman diagrams with QGRAF. Algebraic manipulation, colour algebra and integration‑by‑parts (IBP) reduction are performed with FORM, Color and Reduze 2, yielding a set of master integrals. The master integrals are evaluated in Mellin space (variable $N$) and in $x$‑space. For Mellin‑space treatment, high‑precision moments are computed and used as input for guessing algorithms that reconstruct the underlying recurrence relations. The Sigma package is employed to analyse whether the recurrences factorise; when they do, sum‑product solutions are obtained. In many cases, especially for the gluonic OMEs $A_{gg}^{Q}$ and $A_{Qg}$, the recurrences do not factorise, and the authors resort to a generating‑function approach: local operator insertions are resummed into a continuous variable $t\in