Virtual photon structure functions and positivity constraints

Abstract

We study the three positivity constraints on the eight virtual photon structure functions, derived from the Cauchy-Schwarz inequality and which are, hence, model independent. The photon structure functions obtained from the simple parton model show quite different behaviors in a massive or a massless quark case, but they satisfy, in both cases, the three positivity constraints. We then discuss an inequality which holds among the unpolarized and polarized photon structure functions ${F}_{1}^{\ensuremath{\gamma}},$ ${g}_{1}^{\ensuremath{\gamma}},$ and ${W}_{\mathrm{TT}}^{\ensuremath{\tau}},$ in the kinematic region ${\ensuremath{\Lambda}}^{2}\ensuremath{\ll}{P}^{2}\ensuremath{\ll}{Q}^{2},$ where $\ensuremath{-}{Q}^{2}(\ensuremath{-}{P}^{2})$ is the mass squared of the probe (target) photon, and we examine whether this inequality is satisfied by the perturbative QCD results.