Vacuum polarization function toO(α2)accuracy near threshold and Darwin corrections

Abstract

The QED vacuum polarization function is calculated to $O({\ensuremath{\alpha}}^{2})$ (next-to-leading order) accuracy in the threshold regime by using the concept of effective field theories to resum diagrams with the instantaneous Coulomb exchange of longitudinally polarized photons. It is shown that the $O({\ensuremath{\alpha}}^{2})$ contributions are of order ${\ensuremath{\alpha}}^{2}$ in size rather than ${\ensuremath{\alpha}}^{2}/{\ensuremath{\pi}}^{2}.$ The vacuum polarization effects in the single photon annihilation contributions to the $O({\ensuremath{\alpha}}^{6})$ hyperfine splitting of the positronium ground state are recalculated and an error in an older calculation is pointed out. The results are used to determine ${O(C}_{F}^{2}{\ensuremath{\alpha}}_{s}^{2})$ (next-to-next-to-leading order) Darwin corrections to heavy-quark--antiquark bound state $l=0$ wave functions at the origin and to the heavy-quark--antiquark production cross section in ${e}^{+}{e}^{\ensuremath{-}}$ collisions in the threshold region. The absolute value of the corrections amounts to 10%--20% and 17%--34% in the modulus squared of the ground state wave functions at the origin for the $b\overline{b}$ and $c\overline{c}$ systems, respectively. In the case of the $t\overline{t}$ production cross section in the threshold region the absolute value of the corrections is between 2% and 6% around the $1S$ peak and between 1% and 2% for higher energies. A critical comment on recent QCD sum rule calculations for the \ensuremath{\Upsilon} system is made.