Heavy quark vacuum polarization function at O(αs2) and O(αs3)

We determine the full mass and q(2) dependence of the heavy quark vacuum polarization function Pi(q(2)) and its contribution to the total e(+)e(-) cross section at O(alpha(2)(s)) and O(alpha(3)(s)) in perturbative QCD. We use known results for the expansions of Pi(q(2)) at high energies, in the threshold region and around q(2) = 0, conformal mapping and the Pade approximation method. From our results for Pi(q(2)) we determine numerically at O(alpha(3)(s)) the previously unknown non-logarithmic contributions ill the high-energy expansion at older (m(2)/q(2))(i) for i = 0, 1 and the coefficients in the expansion around q(2) = 0 at order q(2n) with n >= 2. We also determine at O(alpha(2)(s)) the previously unknown O(nu(0)) constant term in the expansion of Pi(q(2)) in the threshold region, where v is the quark velocity. Our method allows for a quantitative estimate of uncertainties; and can be systematically improved once more information ill the three kinematic regions becomes available by future multi-loop computations. For the contributions to the total e(+)e(-) cross section at O(alpha(2)(s)) we confirm results obtained earlier by Chetyrkin, Kuhn and Steinhauser, (C) 2008 Elsevier B.V. All rights reserved.