In this paper we present the analytic form of the heavy quark coefficient functions for deep inelastic lepton-hadron scattering in the kinematical regime Q2 >> m 2. Here Q2 and m 2 stand for the masses squared of the virtual photon and heavy quark, respectively. The calculations have been performed up to next-to-leading order in the strong couphng constant a, using operator product expansion techniques. Apart from a check on earlier calculations, which however are only accessible via large computer programs, the asymptotic forms of the coefficient functions are useful for charm production at HERA when the condition Q2 >> mc2 is satisfied. Furthermore, the analytical expressions can also be used when one applies the variable tlavour number scheme up to next-to-leading order in a,.

Starting from fixed-order perturbation theory (FOPT) we derive expressions for the heavy-flavour components of the deep-inelastic structure functions (F i,H (x, Q 2 , m 2 H ), i = 2, L; H = c, b, t) in the variable-flavour number scheme (VFNS). These expressions are valid in all orders of perturbation theory. This derivation establishes a relation between the parton densities 1 supported by the Foundation for Fundamental Research on Matter (FOM) 2 on leave from ITP, SUNY at Stony Brook, New York 11794-3840, USA 1 parametrized at n f and n f + 1 light flavours. The consequences for the existing parametrizations of the parton densities are discussed. Further we show that in charm electroproduction the exact and asymptotic expressions for the heavy-quark coefficient functions yield identical results for F 2,c (x, Q 2 , m 2 c ) when Q 2 ≥ 20 (GeV/c) 2 . We also study the differences between the FOPT and the VFNS descriptions for F 2,c (x, Q 2 , m 2 c ). It turns out that the charm structure function in the VFNS is larger than the one obtained in FOPT over the whole Q 2 -range. Furthermore inspection of the perturbation series reveals that the higher order corrections in the VFNS are smaller than those present in FOPT for Q 2 ≥ 10 (GeV/c) 2 . Therefore the VFNS gives a better prediction for the charm structure function at large Q 2 -values than FOPT.2 c [10]. Although it was shown in NLO [11] that these logarithms lead to rather stable charm structure functions F i,c (x, Q 2 , m 2 c ) with respect to variations in the factorization and renormalization scales µ, their size still warrants some special treatment (for a discussion of the scale dependence of the charm con-4

In this paper we present the analytic form of the heavy flavour coefficient functions for polarized deep inelastic lepton-hadron scattering. The expressions are valid in the kinematical regime Q 2 ≫ m 2 where Q 2 and m 2 stand for the masses squared of the virtual photon and heavy quark respectively. Using these coefficient functions we have computed the next-to-leading order α s corrections to polarized charm production at HERA collider energies, where both the electron and proton beams are polarized. We also give an estimate of these corrections at fixed target experiments where the typical Q 2 values are much smaller than at HERA.

The most important part of the order α 2 s corrections to the charm component of the charged-current structure functions F 2 (x, Q 2 ) and F 3 (x, Q 2 ) have been calculated. This calculation is based on the asymptotic form of the heavy-quark coefficient functions corresponding to the higher order corrections to the W-boson-gluon fusion process. These coefficient functions which are in principle only valid for Q 2 ≫ m 2 can be also used to estimate the order α 2 s contributions at lower Q 2 values provided x < 0.1. It turns out that the above corrections are appreciable in the large Q 2 -region and they explain the discrepancy found for the structure functions between the fixed-flavour scheme (FFS) and the variable-flavour-number scheme (VFNS). These corrections also hamper the extraction of the strange-quark density from the data obtained for the charged-current and the electromagnetic-current processes.