The Virtual Photon Structure to the Next-to-next-to-leading Order in QCD

We investigate the unpolarized virtual photon structure functions F γ 2 (x,Q 2, P 2) and F γ L(x,Q 2, P 2) in perturbative QCD for the kinematical region Λ2 ≪ P 2 ≪ Q2, where −Q2(−P 2) is the mass squared of the probe (target) photon and Λ 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 γ 2 (x,Q 2, P 2) up to the next-to-next-to-leading order (NNLO) (the order ααs) and for the moments of F γ L(x,Q 2, P 2) up to the next-to-leading order (NLO) (the order ααs). The NNLO corrections to the sum rule of F γ 2 (x,Q 2, P 2) are negative and found to be 7% ∼ 10% of the sum of the LO and NLO contributions, when P 2=1GeV and Q2 = 30 ∼ 100GeV or P 2 = 3GeV and Q2 = 100GeV, and the number of active quark flavors nf is three or four. The NLO corrections to F γ L are also negative. The moments are inverted numerically to obtain the predictions for F γ 2 (x,Q 2, P 2) and F γ L(x,Q 2, P 2) as functions of x. PACS numbers: 12.38.Bx, 13.60.Hb,14.70.Bh Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected]