Heavy Flavor Contributions to QCD Sum Rules and the Running Coupling Constant

We have calculated first and second order corrections to several sum rules measured in deep inelastic lepton-hadron scattering. These corrections, which are due to heavy flavors only, are compared with the existing perturbation series which is computed for massless quarks up to third order in the strong coupling constant \alpha_s. A study of the perturbation series reveals that the large logarithms of the type ln^i Q^2/m^2 dominate the perturbation series at much larger values than those given by the usual matching conditions imposed on the \alpha_s(\mu). Therefore these matching conditions cannot be used to extrapolate the running coupling constant from small \mu to very large scales like \mu=M_Z. An alternative description of the running coupling constant in the MOM-scheme is proposed.