The Calculation of the Two-Loop Spin Splitting Functions $P_{ij}^{(1)}(x)$

TL;DR

This paper computes the two-loop spin-dependent splitting functions $P_{ij}^{(1)}(x)$ in the $\overline{MS}$ scheme by evaluating the two-loop operator matrix elements and extracting the anomalous dimensions $\gamma_{ij}^{m}$ via the OPE for polarized deep inelastic scattering. It provides explicit expressions for the singlet and non-singlet spin splitting functions, confirms the supersymmetric relation among anomalous dimensions up to $O(\alpha_s^2)$, and discusses finite renormalizations arising from the $\gamma_5$ prescription. The results enable consistent NLO analyses of the spin structure function $g_1(x,Q^2)$ when combined with known coefficient functions. Cross-checks using alternative renormalization approaches (e.g., BPHZ) and scheme transformations are discussed to ensure robustness of the spin-dependent anomalous dimensions.

Abstract

We present the calculation of the two-loop spin splitting functions $P_{ij}^{(1)}(x)\; (i,j = q,g)$ contributing to the next-to-leading order corrected spin structure function $g_1(x,Q^2)$. These splitting functions, which are presented in the \MSbs, are derived from the order $α_s^2$ contribution to the anomalous dimensions $γ_{ij}^{m} \; (i,j = q,g)$. The latter correspond to the local operators which appear in the operator product expansion of two electromagnetic currents. Some of the properties of the anomalous dimensions will be discussed. In particular our findings are in agreement with the supersymmetric relation $γ_{qq}^{m}+γ_{gq}^{m}-γ_{qg}^{m}-γ_{gg}^{m}=0$ up to order $α_s^2$.

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