Heavy Quark Vacuum Polarization Function at O(alpha_s^2) and O(alpha_s^3)

We determine the full mass and $q^2$ dependence of the heavy quark vacuum polarization function $Π(q^2)$ and its contribution to the total $e^+e^-$ cross section at ${\cal O}(α_s^2)$ and ${\cal O}(α_s^3)$ in perturbative QCD. We use known results for the expansions of $Π(q^2)$ at high energies, in the threshold region and around $q^2=0$, conformal mapping and the Padé approximation method. From our results for $Π(q^2)$ we determine numerically at ${\cal O}(α_s^3)$ the previously unknown non-logarithmic contributions in the high-energy expansion at order $(m^2/q^2)^i$ for $i=0,1$ and the coefficients in the expansion around $q^2=0$ at order $q^{2n}$ with $n\ge 2$. We also determine at ${\cal O}(α_s^2)$ the previously unknown ${\cal O}(v^0)$ constant term in the expansion of $Π(q^2)$ 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 in the three kinematic regions becomes available by future multi-loop computations. For the contributions to the total $e^+e^-$ cross section at ${\cal O}(α_s^2)$ we confirm results obtained earlier by Chetyrkin, Kühn and Steinhauser.

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