ar X iv : h ep - p h / 06 04 13 0 v 1 1 4 A pr 2 00 6 The αα 2 s corrections to the first moment of the polarized virtual photon structure function

We present the next-to-next-to-leading order (αα s) corrections to the first moment of the polarized virtual photon structure function g 1 (x,Q , P ) in the kinematical region Λ ≪ P 2 ≪ Q, where −Q(−P ) is the mass squared of the probe (target) photon and Λ 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 αα s corrections are found to be about 3% of the sum of the leading order (α) and the next-to-leading order (ααs) contributions, when Q 2 = 30 ∼ 100GeVand P 2 = 3GeV, and the number of active quark flavors nf is three to five. YNU-HEPTh-06-102 KUNS-1992 April 2006 e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] e ⇒ (⇐) e γ q ⇒ (⇐ ) ‘probe’ Q = −q > 0