Interactions of virtual photons are analyzed in terms of photon structure. It is argued that the concept of parton distribution functions is phenomenologically very useful even for highly virtual photons involved in hard collisions. The role of longitudinal photons for proper interpretation of jet cross-sections in the region of moderate virtualities accessible at HERA is investigated.

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The accuracy of the conventional treatment of quark mass thresholds in the QCD running coupling constant α s , based on the step approximation to the β-function, is investigated. The errors of extrapolating α s (µ) from low energies to µ = M Z , implied by this approximation, are shown to be of the same magnitude as the typical next-to-next-to-leading order contributions to α s and tend to increase the resulting α s . The importance of the proper choice of matching points is emphasized.In this note I shall discuss the importance of the proper treatment of quark mass effects on the QCD running coupling constant α s . Although the quark mass effects are not large, steady improvement in the precision of experimental data, coming in particular from a new generation of experiments at CERN and Fermilab, combined with significant progress in higher order QCD calculations, have led to a renewed interest in quantitative aspects of these effects [1,2]. There are essentially two reasons for it.The first concerns the exploitation of the next-to-next-to-leading-order (NNLO) QCD calculation that have recently become available for a number of physical quantities [3, 4, 5, 6] and which exist basically for massless quarks only. The NNLO corrections are tiny effects and to include them makes sense only if they are large compared to errors resulting from the approximate treatment of the quark mass thresholds in massless QCD. To quantify the importance of the NNLO correction to the couplant α s , consider the difference between the values of α s (MS, M Z ) in the NLO and NNLO approximations, taking, as an example, Λ (5) MS = 0.2 GeV. At the NLO the couplant a ≡ α s /π is given as a solution to the equationwhere for n f massless quarksare the first two, universal, coefficients of the QCD β-function appearing on the r.h.s. of the definition equation for the couplant a(µ)For µ = M Z , n f = 5 and Λ 1