Two-loop operator matrix elements calculated up to finite terms for polarized deep inelastic lepton-hadron scattering

TL;DR

This work advances the NNLO program for the polarized deep inelastic scattering structure function $g_1(x,Q^2)$ by computing two-loop operator matrix elements with finite terms in $ε$ to enable renormalization up to ${\cal O}(α_s^3)$. It tackles the technical challenge of a consistent $γ_5$ extension in $N$ dimensions using the HVBM prescription and introduces renormalization constants $Z_{qq}^{5,r}$ to restore Ward identities. The authors develop a complete framework for NS and singlet OME decomposition, derive unrenormalized two-loop OMEs, and provide renormalization formulas that yield the renormalized OMEs $A_{ij}$ in the ${\overline{MS}}$ scheme, enabling partial three-loop contributions via one-loop insertions. These results lay the groundwork for a full NNLO anomalous-dimension analysis of $g_1$, with extensive explicit expressions provided in Appendix A and cross-checked against existing literature.

Abstract

We present the two-loop corrected operator matrix elements contributing to the scale evolution of the longitudinal spin structure function $g_1(x,Q^2)$ calculated up to finite terms which survive in the limit $ε= N - 4 \to 0$. These terms are needed to renormalize the local operators up to third order in the strong coupling constant $α_s$. Further the expressions for the two-loop corrected operator matrix elements can be inserted into 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 third order anomalous dimensions so that a complete next-to-next-to-leading order (NNLO) analysis of the above mentioned structure function can be carried out. In our calculation particular attention is paid to the renormalization constant which is needed to restore the Ward-identities violated by the HVBM prescription for the $γ_5$-matrix in $N$-dimensional regularization.