Transition from real to virtual polarized photon structures

We investigate the transition of the polarized photon structure function g 1 (x,Q , P ) when the target photon shifts from on-shell (P 2 = 0) to far off-shell (P 2 ≫ Λ) region. The analysis is performed to the nextto-leading order in QCD. The first moment of g 1 which vanishes for the real photon, turns to be a negative value when target photon becomes off-shell. The explicit P -dependence of the first moment sum rule as well as of the structure function g 1 (x,Q , P ) as a function of x is studied in the framework of the vector meson dominance model. YNU-HEPTh-06-104 KUNS-1993 June 2006 e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] The photon structure has been investigated both theoretically and experimentally for decades [1, 2, 3, 4]. And in recent years there has been growing interest in the spin structure of the photon. In particular, the first moment of the polarized photon structure function g 1 has attracted a lot of attention in the literature [5]-[9] due to its connection with the axial anomaly. This structure function g 1 (x,Q , P ), where q = −Q (p = −P ) is the mass squared of the probe (target) photon, can be measured in the experiments either of the polarized ep collision [10, 11] or of the polarized ee collision in the future linear collider ILC [12, 13]. A unique and interesting feature of the photon structure functions is that, in contrast with the nucleon case, the target mass squared −P 2 is not fixed but can take various values and that the structure functions show different behaviors depending on the values of P . The QCD analysis of g 1 for a real photon (P 2 = 0) target was performed in the leading order (LO) [15], and in the next-to-leading order (NLO) [11, 18]. In the case of a virtual photon target (P 2 6= 0), g 1 (x,Q , P ) was investigated up to the NLO in QCD by the present authors in [14], and in the second paper of [18]. Moreover, the polarized parton distributions inside the virtual photon were analyzed in various factorization schemes [16], and the target mass effects of g 1 (x,Q , P ) was studied in [17]. In Refs.[14, 16] the structure function g 1 (x,Q , P ) was analyzed in the kinematic region Λ ≪ P 2 ≪ Q , (1) where Λ is the QCD scale parameter. The advantage in studying the virtual photon target in the kinematic region (1) is that we can calculate structure functions by the perturbative method without any experimental data input [22], which is contrasted with the case of a real photon target where in the NLO there exist nonperturbative pieces [20]-[28]. In the present paper we analyze the transition of the polarized photon structure function g 1 when the target photon shifts from on-shell (P 2 = 0) to far off-shell in the region (1). In fact, g 1 exhibits an interesting P -dependence. At P 2 = 0, the structure function g 1 satisfies a remarkable sum rule [5]-[9]: ∫ 1 0 g 1 (x,Q )dx = 0 . (2)