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

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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 ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$ 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 Pad\'e approximation method.
• From our results for $\Pi(q^2)$ we determine numerically at ${\cal O}(\alpha_s^3)$ 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 ${\cal O}(\alpha_s^2)$ the previously unknown ${\cal O}(v^0)$ constant term in the expansion of $\Pi(q^2)$ in the threshold region, where $v$ is the quark velocity.