Virtual Photon Structure Functions to NNLO in QCD

The unpolarized virtual photon structure functions $F_2^{\gamma }(x,Q^2,P^2)$ and $F_L^{\gamma }(x,Q^2,P^2)$ are investigated in perturbative QCD for the kinematical region ${\mathrm{\Lambda }}^2\ll P^2\ll Q^2$, where $-Q^2(-P^2)$ is the mass squared of the probe (target) photon and $\mathrm{\Lambda }$ is the QCD scale parameter. In the framework of operator product expansion supplemented by the renormalization group method, the definite predictions are derived for the moments of $F_2^{\gamma }(x,Q^2,P^2)$ up to the next-to-next-to-leading order (the order $\alpha {\alpha }_s$) and for the moments of $F_L^{\gamma }(x,Q^2,P^2)$ up to the next-to-leading order (the order $\alpha {\alpha }_s$)