Parametrization of the QCD coupling in the Evolution Equations

TL;DR

This paper investigates how the parametrization of the QCD coupling in evolution equations depends on the choice of the starting scale, showing that standard forms are valid only under certain conditions on the initial scale.

Contribution

It clarifies the conditions under which the common parametrization of the QCD coupling in evolution equations is valid, proposing a more complex form when initial scales are lower.

Findings

Standard parametrization valid if initial scale mu >> Lambda_QCD * exp(pi/2)

For lower initial scales, a more complex coupling expression is required

Provides criteria for choosing the correct coupling parametrization in QCD evolution

Abstract

We examine the parametrization of the QCD coupling in the Evolution Equations, including DGLAP. Our conclusion is that the well-known parametrization, where the argument of the coupling is k^2_{\perp}/\beta or just k^2_{\perp}, stands only if the lowest integration limit in the transverse momentum space (the starting point mu^2 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 presented in Eq. (37).