Sum rules of polarized photon structure functions revisited-- NNLO corrections to the first moment of $g_1^{\gamma }(x,Q^2,P^2)$ --

Abstract

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 $Q^2\gg P^2\gg {\Lambda }^2$ in QCD. We find that the $\alpha {\alpha }_s^2$ corrections are about 3% of the sum of the leading ($\alpha$) and the next-to-leading ($\alpha {\alpha }_s$) contributions, when $Q^2=30\sim 100{\mathrm{GeV}}^2$ and $P^2=3{\mathrm{GeV}}^2$.

Comments: 5 pages, 1 eps figure, espcrc2.sty file included, Talk given at RADCOR2005, Shonan Village, Japan, October 2-7, 2005, to appear in the Proceedings