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

Andre H. Hoang; Vicent Mateu; S. Mohammad Zebarjad

doi:10.1016/j.nuclphysb.2008.12.005

paper

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

arXiv:0807.4173 · doi:10.1016/j.nuclphysb.2008.12.005

Abstract

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 $O (α_{s}^{2})$ and $O (α_{s}^{3})$ 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 $O (α_{s}^{3})$ 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 $q^{2 n}$ with $n \geq 2$ . We also determine at $O (α_{s}^{2})$ the previously unknown $O (v^{0})$ 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 $O (α_{s}^{2})$ we confirm results obtained earlier by Chetyrkin, Kühn and Steinhauser.

17 pages, 5 figures; new references included, more detailed discussion

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

Abstract

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