VACUUM POLARIZATION FUNCTION TO O(ALPHA(2)) ACCURACY NEAR-THRESHOLD AND DARWIN CORRECTIONS

The QED vacuum polarization function is calculated to O(alpha(2)) (nex t-to-leading order) accuracy in the threshold regime by using the conc ept of effective field theories to resum diagrams with the instantaneo us Coulomb exchange of longitudinally polarized photons. It is shown t hat the O(alpha(2)) contributions are of order alpha(2) in size rather than alpha(2)/pi(2). The vacuum polarization effects in the single ph oton annihilation contributions to the O(alpha(6)) hyperfine splitting of the positronium ground state are recalculated and an error in an o lder calculation is 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 origi n and to the heavy-quark-antiquark production cross section in e(+)e(- ) collisions in the threshold region. The absolute value of the correc tions amounts to 10%-20% and 17%-34% in the modulus squared of the gro und state wave functions at the origin for the b (b) over bar and c (c ) over bar systems, respectively. In the case of the t (t) over bar pr oduction cross section in the threshold region the absolute value of t he 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.