Polarized Virtual Photon Structure Function in the Next-to-leading Order in QCD

The virtual photon structure function g 1 (x,Q , P ), which can be obtained in polarized ee colliding-beam experiments, is investigated for Λ ≪ P 2 ≪ Q, where −Q (−P ) is the mass squared of the probe (target) photon. The analysis is made to next-to-leading order in QCD, in the framework of the QCD improved parton model with the DGLAP evolution equations. The non-leading corrections significantly modify the leading log result, in particular, at large x as well as at small x. YNU-HEPTh-98-102 KUCP-124 December 1998 e-mail address: [email protected] e-mail address: [email protected] In the last several years, the nulcleon spin structure functions g p(n) 1 (x,Q ) and g p(n) 2 (x,Q ) have been extensively studied by deep-inelastic scattering of polarized leptons on polarized nucleon targets. The information on the spin structure of photon would be provided by the resolved-photon process in polarized electron and proton collision in the polarized version of HERA[1]. More directly, the spin-dependent structure function of photon can be measured by the polarized e e collision in the future linear colliders. For real photon (P 2 = 0) target, there exists only one spin struture function, g 1 (x,Q ), which is equivalent to the structure function W γ 4 (x,Q ) (g 1 ≡ 2W γ 4 ) discussed some time ago in [2]. The leading order (LO) QCD correction to g 1 was first studied in [3] and later in [4, 5]. Recently the first moment of g γ 1 attracted attention in the literatures [5, 6, 7] in connection with its relevance to the axial anomaly, which also plays an important role in the QCD analysis of the nulcleon spin structure function g p(n) 1 . Now the next-to-leading order (NLO) QCD analysis for g 1 is possible, since the required spin-dependent two-loop splitting functions in DGLAP evolution equations or, equivalently, the two-loop anomalous dimensions of the relevant operators have been calculated recently [8, 9]. Actually there has been already an analysis of g 1 for the case of the real photon target by Stratmann and Vogelsang [10]. In this paper we shall investigate the polarized virtual photon structure function g 1 (x,Q , P ) to the NLO in QCD, where −Q (−P ) is the mass squared of the probe (target) photon (Fig.1). We examine g 1 (x,Q , P ) for the kinematical region: Λ ≪ P 2 ≪ Q , where Λ is the QCD scale parameter. The advantage in studying the virtual photon target is that we can calculate the whole structure function entirely up to NLO by the perturbative method. On the other hand, for the real photon target there appear the non-perturbative pieces in NLO [11], which we may dispose of by using, say, vector-meson-dominance model. The unpolarized virtual photon structure functions F γ 2 (x,Q , P ), and F γ L(x,Q , P ) were studied in LO [12] and in NLO [13]. The analysis can be performed either in the framework of the QCD improved parton model [14] using the DGLAP parton evolution equations or in the framework of the operator product expansion (OPE) supplemented by the renormalization group