Variable number schemes for heavy quark electroproduction

Heavy quark production has been a major topic of investigation at hadron-hadron, electron-proton and electron-positron colliders. Here a review is given of some topics which are of interest primarily for electronproton colliders. We concentrate on this reaction because a theoretical treatment can be based on the operator product expansion, and also because data are available for deep-inelastic charm production at HERA. How all this relates to Fermilab experiments will be discussed at the end. In QCD perturbation theory one needs to introduce a renormalization scale and a mass factorization scale to perform calculations. We choose both equal to μ, which will be a function of Q and the square of the mass of the charm quark m. At small μ, where kinematic effects due to quark masses are important, the best way to describe charm quark production is via heavy quark pair production where parton densities are used for the initial state light mass u,d,s quarks and the gluon and m only appears in the heavy quark coefficient functions (or partonic cross sections) like H S,(2) i,g (z,Q ,m, μ), etc., [ 1]. Here the superscripts refer to their flavor decomposition and the order in perturbation theory, while the subscripts refer to the projection i = 2, L and the partonic initial state. The arguments refer to the partonic Bjorken variable z = Q/(s + Q) and to the fact that these functions depend upon invariants and scales. The renormalization necessary to calculate these NLO expressions follows the CWZ method [ 2]. The symbol H refers to those coefficient functions which are derived from Feynman diagrams where the virtual photon couples to a heavy quark line. Analytic expressions for these functions are not known. Numerical fits are available in [ 3]. Asymptotic expressions for Q ≫ m are available in [ 4] which contain terms like ln(Q2/m2) and ln(Q/m) ln(Q/μ) multipled by functions of z and demonstrate that these functions are singular in the limit that m → 0. There are other heavy quark coeffi-