A Novel Series Solution to the Renormalization-Group Equation in QCD

Abstract

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 two-loop-order coupling. In this work, we prove that the power series converge to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the two-loop-order running coupling. Then we analyze the singularity structure of the higher-order coupling in the complex two-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the three- and four-loop orders as a function of the number of quark flavours n f . In parallel, we discuss in some detail the singularity structure of the $\overline{\rm MS}$ coupling at the three- and four-loops in the complex-momentum squared plane for 0 ≤ n f ≤ 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 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 f , in the $\overline{\rm MS}$ scheme, we determine the minimal value of the momentum-squared Q 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 the whole ranges of the corresponding momentum-squared variables.