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 $Q^2=30\sim 100 {\rm GeV}^2$ and $P^2=3 {\rm GeV}^2$.