QCD corrections up to order α<SUB>s</SUB><SUP>2</SUP> to polarized quark production in e<SUP>+</SUP>e<SUP>-</SUP> - annihilation

We present the calculation of the order α<SUB>s</SUB><SUP>2</SUP> contributions to the cross section e<SUP>+</SUP>+e<SUP>-</SUP> → q<SUP>-</SUP>+q where the incoming leptons as well as one of the outgoing (anti) quarks are longitudinally polarized. The computation is carried out for massless quarks so that it can be applied to light flavour production ( u,d,s ). Unfortunately the massless quark approach is not valid for heavy flavour production like c,b,t even in the case when the centre of mass energy Q is much larger than the quark mass m . This is in contrast to unpolarized scattering where this approach works rather well for Q»m . The reason for this can be attributed to the anomalous terms which are characteristic of polarized coefficient functions. Furthermore we also computed the order α<SUB>s</SUB> corrections to the longitudinal, transverse and normal polarizations of heavy flavours with m≠0 . The latter have been presented earlier in the literature except for some contributions which are shown here for the first time. It turns out that the corrections to the longitudinal and transverse (in the plane) polarization are rather small. However the order α<SUB>s</SUB> corrections to the normal (out of the plane) polarization are large so that second order contributions (for m≠0 ) are needed to get a better determination of this quantity