On the running electromagnetic coupling constant at M-Z

We present a discussion on how to define the running electromagnetic coupli ng constant at M-z or some other intermediate scale! e.g., m(r). We argue t hat a natural definition consistent with general requirements of the renorm alization group should use Euclidean values of the momentum of the photon p ropagator as the appropriate scale. We demonstrate explicitly, through the evaluation of the running coupling constant at the scale of the Y-resonance mass, that the usual definition of the hadronic contribution with a princi pal-value prescription is inconsistent. In the determination of the value o f alpha at M-z, we use a Euclidean definition rather than the principal-val ue one, and as a result, the numerical difference is comparable in size to the errors caused by existing experimental and QCD inputs to the evaluation of alpha(M-z).