High energy photon colliders (γγ,γe) are based on e-e-linear colliders where high energy photons are produced using Compton scattering of laser light on high energy electrons just before the interaction point. This paper is a part of the Technical Design Report of the linear collider TESLA.1Physics program, possible parameters and some technical aspects of the photon collider at TESLA are discussed.

We present a detailed investigation of the renormalization scheme dependence of the next-to-next-to-leading order QCD predictions for the processes e+e-~hadrons and x-* v,+hadrons. Based on it the comparison of the results obtained in three frequently used approaches to resolving the renormalization scheme ambiguities with experimental data is carried out.

QCD analysis of F γ 2 (x, Q 2 ) is revisited. It is emphasized that the presence of the inhomogeneous term in the evolution equations for quark distribution functions of the photon implies important difference in the way factorization mechanism works in photon-hadron and photon-photon collisions as compared to the hadronic ones. Moreover, a careful definitions of the very concepts of the "leading order" and "next-to-leading order" QCD analysis of F γ 2 are needed in order to separate genuine QCD effects from those of pure QED origin. After presenting such definitions, I show that all existing allegedly LO, as well as NLO analyses of F γ 2 (x, Q 2 ) are incomplete. The source of this incompleteness of the conventional approach is traced back to the lack of clear identification of QCD effects and to the misinterpretation of the behaviour of q γ (x, M ) as a function of α s (M ). Complete LO and NLO QCD analyses of F γ 2 (x, Q 2 ) are shown to differ substantially from the conventional ones. Whereas complete NLO analysis requires the knowledge of two so far uncalculated quantities, a complete LO one is currently possible, but compared to the conventional formulation requires the inclusion of four known, but in the existing LO analyses unused quantities. The arguments recently advanced in favour of the conventional approach are analyzed and shown to contain a serious flaw. If corrected, they actually lend support to my claim.

We present the next-to-next-to-leading order QCD analysis of the Gross-Llewellyn Smith (GLS) sum rule in deep inelastic lepton-nucleon scattering, taking into account dimensiontwo, twist-four power correction. We discuss in detail the renormalization scheme dependence of the perturbative QCD approximations, propose a procedure for an approximate treatment of the quark mass threshold effects and compare the results of our analysis to the recent experimental data of the CCFR collaboration. From this comparison we extract the value of the strong coupling constant α nnl s (M Z , MS) = 0.115 ± 0.001(stat) ± 0.005(syst) ± 0.003(twist) ± 0.0005(scheme). We stress the importance of an accurate measurement of the GLS sum rule and in particular of its Q 2 dependence.

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