  The renormalization group applied to perturbation theory is ordinarily used
to define the running coupling constant in the spacelike region. However, to
describe processes with timelike momenta transfers, it is important to have a
self-consistent determination of the running coupling constant in the timelike
region. The technique called analytic perturbation theory (APT) allows a
consistent determination of this running coupling constant. The results are
found to disagree significantly with those obtained in the standard
perturbative approach. Comparison between the standard approach and APT is
carried out to two loops, and threshold matching in APT is applied in the
timelike region.
