Virtual photon structure functions to the next-to-next-to-leading order in QCD

We investigate the unpolarized virtual photon structure functions F{sub 2}{sup {gamma}}(x,Q{sup 2},P{sup 2}) and F{sub L}{sup {gamma}}(x,Q{sup 2},P{sup 2}) in perturbative QCD for the kinematical region {lambda}{sup 2}<<P{sup 2}<<Q{sup 2}, where -Q{sup 2} (-P{sup 2}) is the mass squared of the probe (target) photon and {lambda} is the QCD scale parameter. Using the framework of the operator product expansion supplemented by the renormalization group method, we derive the definite predictions for the moments of F{sub 2}{sup {gamma}}(x,Q{sup 2},P{sup 2}) up to the next-to-next-to-leading order (NNLO) (the order {alpha}{alpha}{sub s}) and for the moments of F{sub L}{sup {gamma}}(x,Q{sup 2},P{sup 2}) up to the next-to-leading order (NLO) (the order {alpha}{alpha}{sub s}). The NNLO corrections to the sum rule of F{sub 2}{sup {gamma}}(x,Q{sup 2},P{sup 2}) are negative and found to be 7%-10% of the sum of the LO and NLO contributions, when P{sup 2}=1 GeV{sup 2} and Q{sup 2}=30{approx}100 GeV{sup 2} or P{sup 2}=3 GeV{sup 2} and Q{sup 2}=100 GeV{sup 2}, and the number of active quark flavors n{sub f} is three or four. The NLO corrections to F{sub L}{sup {gamma}} are also negative. The moments are inverted numerically to obtain the predictions for F{sub 2}{sup {gamma}}(x,Q{sup 2},P{sup 2}) and F{sub L}{sup {gamma}}(x,Q{supmore » 2},P{sup 2}) as functions of x.« less