The Vacuum Polarization Function to O(alpha²) Accuracy Near Threshold and Darwin Corrections

The QED vacuum polarization function is calculated to O(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(alpha^2) contributions are of order alpha^2 in size rather than alpha^2/Pi^2. The vacuum polarization contributions to the O(alpha^6) hyperfine splitting of the positronium ground state are recalculated and differences with an older calculation are pointed out. The results are used to determine O(C_F^2 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^- 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 $b\bar b$ and $c\bar c$ systems, respectively. In the case of the $t\bar t$ 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 $\Upsilon$ system is made.