Spin-Dependent Structure Functions of the Photon

The implications of the positivity constraint on the presently unknown polarized structure function of the photon, g 1 (x,Q 2), are studied in detail. In particular the non–trivial consequences of this constraint, |g 1 (x,Q 2)| ≤ F γ 1 (x,Q 2), in the next– to–leading order analysis of g 1 are pointed out by employing appropriate (DIS) factorization schemes related to g 1 and F γ 1 (rather than to F γ 2 ). The spin dependent structure function g 1 (x,Q ) of a longitudinally polarized photon was studied [1, 2] within the framework of the radiative parton model, developed [3] for the presently well measured and known structure function F γ 2 (x,Q ) of an unpolarized photon. In particular the next–to–leading order (NLO) analysis [2] of g 1 adopted a perturbatively stable DISγ factorization scheme, as advocated in [3], and implemented some boundary conditions [1] at the low input scale Q = μ ≃ 0.3 GeV of the radiative parton model. These boundary conditions led, however, to a violation of the positivity constraint |Aγ1(x,Q )| ≡ |g 1 (x,Q )/F γ 1 (x,Q )| ≤ 1 . (1) It was therefore suggested [2] to repeat the analysis [3] of F γ 2 in a DISγ factorization scheme related to F γ 1 rather than to F γ 2 which was the source of the above mentioned violation. Such a reanalysis is obviously rather time consuming and leads, moreover, to a diminished perturbative stability of the resulting parton distributions. In the present letter we propose an alternative solution to the positivity constraint which avoids the need for the above mentioned reanalysis of the data on F γ 2 . For this purpose we recall that the DISγ factorization scheme, suggested and adopted in [3] for unpolarized photon structure functions, is related to F γ 2 as given in NLO(MS) by 1 x F γ 2 (x,Q ) = ∑