alpha alpha(2)(s) corrections to the first moment of the polarized virtual photon structure function g(1)(gamma)(x, Q(2), P-2)

We present the next-to-next-to-leading order (alpha alpha(2)(s)) 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 P-2 <<(2)<< 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(2)(s) 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 similar to 100 GeV2 and P-2=3 GeV2, and the number of active quark flavors n(f) is three to five