  Recently, the QCD renormalization group (RG) equation at higher orders in
MS-like renormalization schemes has been solved for the running coupling as a
series expansion in powers of the exact 2-loop order coupling. In this work, we
prove that the power series converges to all orders in perturbation theory.
Solving the RG equation at higher orders, we determine the running coupling as
an implicit function of the 2-loop order running coupling. Then we analyze the
singularity structure of the higher order coupling in the complex 2-loop
coupling plane. This enables us to calculate the radii of convergence of the
series solutions at the 3- and 4-loop orders as a function of the number of
quark flavours $n_{\rm f}$. In parallel, we discuss in some detail the
singularity structure of the ${\bar{\rm MS}}$ coupling at the 3- and 4-loops in
the complex momentum squared plane for $ 0\leq n_{\rm f} \leq 16 $. The
correspondence between the singularity structure of the running coupling in the
complex momentum squared plane and the convergence radius of the series
solution is established. For sufficiently large $n_{\rm f}$ values, we find
that the series converges for all values of the momentum squared variable
$Q^2=-q^2>0$. For lower values of $n_{\rm f}$, in the ${\bar{\rm MS}}$ scheme,
we determine the minimal value of the momentum squared $Q_{\rm min}^2$ above
which the series converges. We study properties of the non-power series
corresponding to the presented power series solution in the QCD Analytic
Perturbation Theory approach of Shirkov and Solovtsov. The Euclidean and
Minkowskian versions of the non-power series are found to be uniformly
convergent over whole ranges of the corresponding momentum squared variables.

High Energy Physics - Phenomenology