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

We investigate the unpolarized virtual photon structure functions F 2 x;Q2; P2 and F L x;Q2; P2 in perturbative QCD for the kinematical region 2 P2 Q2, where Q2 ( P2) is the mass squared of the probe (target) photon and 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 2 x;Q2; P2 up to the next-to-next-to-leading order (NNLO) (the order s) and for the moments of F L x;Q2; P2 up to the next-to-leading order (NLO) (the order s). The NNLO corrections to the sum rule of F 2 x;Q2; P2 are negative and found to be 7%–10% of the sum of the LO and NLO contributions, when P2 1 GeV2 and Q2 30 100 GeV2 or P2 3 GeV2 and Q2 100 GeV2, and the number of active quark flavors nf is three or four. The NLO corrections to F L are also negative. The moments are inverted numerically to obtain the predictions for F 2 x;Q2; P2 and F L x;Q2; P2 as functions of x.