Heavy flavor mass corrections to the longitudinal and transverse cross-sections in $e^+ e^-$-collisions

We present the heavy flavour mass corrections to the order α s corrected longitudinal ( σ L ) and transverse ( σ T ) cross sections in e + e − -collisions. Its effect on the value of the running coupling constant extracted from the longitudinal cross section is investigated. Furthermore we make a comparison between the size of these mass corrections and the magnitude of the order α s 2 contribution to σ T and σ L which has been recently calculated for massless quarks. Also studied will be the changes in the above quantities when the fixed pole mass scheme is replaced by the running mass approach