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

We determine the full mass and q2q2 dependence of the heavy quark vacuum polarization function Π(q2)Π(q2) and its contribution to the total e+e−e+e− cross section at O(αs2) and O(αs3) in perturbative QCD. We use known results for the expansions of Π(q2)Π(q2) at high energies, in the threshold region and around q2=0q2=0, conformal mapping and the Padé approximation method. From our results for Π(q2)Π(q2) we determine numerically at O(αs3) the previously unknown non-logarithmic contributions in the high-energy expansion at order i(m2/q2)(m2/q2)i for i=0,1i=0,1 and the coefficients in the expansion around q2=0q2=0 at order q2nq2n with n⩾2n⩾2. We also determine at O(αs2) the previously unknown O(v0)O(v0) constant term in the expansion of Π(q2)Π(q2) 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 in the three kinematic regions becomes available by future multi-loop computations. For the contributions to the total e+e−e+e− cross section at O(αs2) we confirm results obtained earlier by Chetyrkin, Kühn and Steinhauser.