Vacuum polarization in pQCD: first complete result

We report the results of the first complete QCD five-loop calculation, viz. analytical contribution of order m q 2 α s 4 , to the (absorptive part of) the vacuum polarization function. The contribution directly leads to the knowledge of the the full O ( α s 4 m q 2 / s ) correction to the total cross-section of e+e− annihilation into hadrons. We compare our results with predictions following from various optimization schemes.