On the analytically-improved running coupling in QCD

TL;DR

This paper derives an exact integral representation for the analytic running coupling in QCD within the 't Hooft scheme, highlighting its infrared limit and expressing it via the Lambert function's imaginary part.

Contribution

It introduces a novel integral form of the analytic QCD coupling using the Lambert function, enhancing understanding of its infrared behavior.

Findings

Exact integral representation of the coupling derived

Infrared limit of the coupling confirmed as 4π/β₀

Coupling expressed in terms of Lambert function's imaginary part

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 $\bar\alpha_t(Q^2=0)= 4 \pi /\beta_0$.