TTP04-29 Vacuum polarization in pQCD: first complete O(\alpha_s^4) result

We report the results of the first complete QCD five-loop calculation, viz. analytical contribution of order m_q^2 alpha_s^4, to the (absorptive part of) the vacuum polarization function. The contribution directly leads to the knowledge of the the full O( \als^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.