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

Abstract

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 Λ 2 ≪ P 2 ≪ Q 2 , where −Q 2 (−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 αα s ) and for the moments of F γ L (x, Q 2 , P 2 ) up to the next-to-leading order (NLO) (the order αα s ). 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 2 and Q 2 = 30 ∼ 100GeV 2 or P 2 = 3GeV 2 and Q 2 = 100GeV 2 , and the number of active quark flavors n f 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.

Virtual Photon Structure Functions to NNLO in QCD

“…(6.6)-(6.8) of Ref. [6]. Inverting the moments (5), we plot in Fig.3 the longitudinal virtual photon structure function F γ L (x, Q 2 , P 2 ) predicted by pQCD for the case of n f = 4, Q 2 = 100 GeV 2 and P 2 = 1 GeV 2 .…”

“…(2.34)-(2.37) of Ref. [6] and they are written in terms of the 1-, 2-and 3-loop anomalous dimensions, the 1-and 2-loop coefficient functions and the 1-and 2-loop photon matrix elements of hadronic operators.…”

Parton distributions in the virtual photon up to NNLO in QCD and factorization scheme dependence

“…The A i (1) n and A i (2) n terms represent the operator mixing between the hadronic operators and photon operators in the NLO and NNLO, respectively, and the operator mixing implies that there exist parton distributions in the photon. Thus we have, at μ 2 = P 2 (or at…”

“…In the previous talk (see this Proceedings), Uematsu presented our analysis on the structure functions of the virtual photon in perturbative QCD (pQCD) up to the next-to-next-to-leading order (NNLO) [1]. The structure functions F γ 2 and F γ L were studied in the kinematical region, Λ 2 P 2 Q 2 , where ;Q 2 (;P 2 ) is the mass squared of the probe (target) photon and Λ is the QCD scale parameter.…”

Real photon structure functions in the massive parton model in next-to-leading order

“…For more information on the study of the virtual photon structure functions F γ 2 (x, Q 2 , P 2 ), F γ L (x, Q 2 , P 2 ) and g γ 1 (x, Q 2 , P 2 ), see, for example, Ref. [12]. So far in most of the QCD analyses of the photon structure functions, all the active quarks have been treated as massless.…”