Heavy flavour contributions to the deep inelastic sum rules

We have calculated the first and second order corrections to several deep inelastic sum rules which are due to heavy flavour contributions. A comparison is made with the existing perturbation series which has been computed up to third order for massless quarks only. In general it turns out that the effects of heavy quarks are very small except when $Q \sim m$ or $Q \gg m$. Here $Q$ and $m$ denote the virtual mass of the vector boson and the mass of the heavy quark, respectively. For $Q \gg m$ the radiative corrections reveal large logarithms of the type $\ln Q^2/m^2$ which have to be absorbed in the running coupling constant so that the number of light flavours $n_f$ is enhanced by one unit. However this has to happen at much larger values of Q i.e. $Q \sim 6.5~ m$ than one usually assumes for the flavour thresholds which appear in the running coupling constant. An alternative description for the heavy flavour dependence of the running coupling constant in the context of the MOM-scheme is discussed.