Detailed QCD analysis of the photon structure function

Abstract

The effects of nonasymptotic terms in the photon structure function $F_2^{\gamma }(x,Q^2)$ predicted by QCD are studied directly in $x$ space in leading and subleading order. The full (regular) solution including the hadronic nonasymptotic (${Q_0}^2$-dependent input) terms is shown to be free from the unphysical singularities of the asymptotic solution and is furthermore positive definite for suitable chosen boundary conditions (at $Q^2={Q_0}^2$) and in the physically relevant $x$ region ($W\gtrsim 2$ GeV). The implications of the nonasymptotic solution and of their perturbatively uncalculable boundary conditions for the determination of $\Lambda$ and for the predictive power of purely perturbative QCD in the determination of $F_2^{\gamma }(x,Q^2)$ are critically analyzed. Furthermore, taking carefully into account charm production, detailed predictions are given for present and future (unfolded) data. Similarly to the case of deepinelastic lepton-nucleon scattering, the (nonperturbative) photonic input parton distributions at $Q^2={Q_0}^2$ are different for leading- and higher-order calculations. This implies that the differences between leading- and higher-order predictions for $F_2^{\gamma }(x,Q^2)$ are too small to be distinguished by present experiments. The only clean test of QCD, independent of the hadronic input, can be achieved by observing an increase of $F_2^{\gamma }(x,Q^2)$ with $\ln Q^2$ for fixed values of $x$.