Two-loop operator matrix elements calculated up to finite terms

TL;DR

To enable NNLO analyses of deep inelastic scattering, third-order anomalous dimensions are required but not yet complete. This work computes the two-loop operator matrix elements in $N$-dimensional regularization up to finite terms in the Feynman gauge, including singlet/non-singlet sectors and mixing with non-gauge-invariant operators, and extracts renormalization-group coefficients from pole structures while verifying Ward identities. The authors provide full unrenormalized two-loop OME expressions (with finite terms) and discuss their use in inserting into one-loop graphs to generate parts of three-loop contributions, with explicit results and NGI sectors detailed in the appendices. This constitutes a foundational step toward a complete NNLO analysis of DIS, enabling renormalization to $O(α_s^3)$ and improving predictions for $F_2$ and $F_3$ at NNLO.

Abstract

We present the two-loop corrected operator matrix elements calculated in N-dimensional regularization up to the finite terms which survive in the limit $ε= N - 4 \to 0 $. The anomalous dimensions of the local operators have been previously extracted from the pole terms and determine the scale evolution of the deep inelastic structure functions measured in unpolarized lepton hadron scattering. The finite $ε$-independent terms in the two-loop expressions are needed to renormalize the local operators up to third order in the strong coupling constant $α_s$. Further the unrenormalized expressions for the two-loop corrected operator matrix elements can be inserted into specific one loop graphs to obtain a part of the third order contributions to these matrix elements. This work is a first step in obtaining the anomalous dimensions up to third order so that a complete next-to-next-to-leading order (NNLO) analysis can be carried out for deep inelastic electroproduction.