Parametrization of the QCD coupling in Hard and Regge processes

We examine the parametrization of the QCD coupling in the Bethe-Salpeter equations for the hard and Regge processes and determine the argument of alpha_s of the factorized gluon. Our analysis shows that for the hard processes alpha_s = alpha_s(k^2_T/(1- beta)) where k^2_T and beta are the longitudinal and transverse moment of the soft parton. On the other hand, in the Regge processes alpha_s = alpha_s(k^2_T}/beta). We have also shown that the well-known parametrization alpha_s = alpha_s(k^2_T) in the DGLAP equations stands only if the lowest integration limit, mu^2, over k^2_T (the starting point of the Q^2 -evolution) obeys the relation mu >> Lambda_{QCD} exp {(\pi/2)}, otherwise the coupling should be replaced by the more complicated expression.