We present the next-to-next-to-leading order ($\alpha {\alpha }_s^2$) corrections to the first moment of the polarized virtual photon structure function $g_1^{\gamma }(x,Q^2,P^2)$ in the kinematical region ${\Lambda }^2\ll P^2\ll Q^2$, where $-Q^2(-P^2)$ is the mass squared of the probe (target) photon and $\Lambda$ is the QCD scale parameter. In order to evaluate the three-loop-level photon matrix element of the flavor singlet axial current, we resort to the Adler-Bardeen theorem for the axial anomaly and we calculate in effect the two-loop diagrams for the photon matrix element of the gluon operator. The $\alpha {\alpha }_s^2$ corrections are found to be about 3% of the sum of the leading order ($\alpha$) and the next-to-leading order ($\alpha {\alpha }_s$) contributions, when $Q^2=30\sim 100{\mathrm{GeV}}^2$ and $P^2=3{\mathrm{GeV}}^2$, and the number of active quark flavors $n_f$ is three to five.

• Received 17 April 2006

DOI:https://doi.org/10.1103/PhysRevD.73.094024