Heavy Quark Effects in the Virtual Photon Structure Functions

We investigate the heavy quark mass effects in the virtual photon structure functions $F_{2}^{\gamma}(x, Q^2, P^2)$ and $F_{L}^{\gamma}(x, Q^2, P^2)$ in the framework of the mass-independent renormalization group equation (RGE). We study a formalism in which the heavy quark mass effects are treated based on parton picture as well as on the operator product expansion (OPE), and perform the numerical evaluation of $F_{\rm eff}^{\gamma}(x, Q^2, P^2)$ to the next-leading order (NLO) in QCD.