On the analytically-improved running coupling in QCD

Abstract

We examine, in the `t Hooft renormalization scheme, the analytic running coupling ${\bar{\alpha }}_t(Q^2)$ in QCD, using the two-loop $\beta$-function with positive expansion parameters ${\beta }_0$ and ${\beta }_1$. An exact integral representation is derived for this causal coupling, which is fully expressed in terms of the imaginary part of the Lambert function $W$. This integral form manifestly accounts for the universal value of the infrared limit ${\bar{\alpha }}_t(Q^2=0)=4\pi /{\beta }_0$.

Comments: 9 pages, one figure; accepted for publication in Modern Physics Letters A