On the Interface between Perturbative and Nonperturbative QCD

The QCD running coupling $α_s(Q^2)$ sets the strength of the interactions of quarks and gluons as a function of the momentum transfer $Q$. The $Q^2$ dependence of the coupling is required to describe hadronic interactions at both large and short distances. In this article we adopt the light-front holographic approach to strongly-coupled QCD, a formalism which incorporates confinement, predicts the spectroscopy of hadrons composed of light quarks, and describes the low-$Q^2$ analytic behavior of the strong coupling $α_s(Q^2)$. The high-$Q^2$ dependence of the coupling $α_s(Q^2)$ is specified by perturbative QCD and its renormalization group equation. The matching of the high and low $Q^2$ regimes of $α_s(Q^2)$ then determines the scale $Q_0$ which sets the interface between perturbative and nonperturbative hadron dynamics. The value of $Q_0$ can be used to set the factorization scale for DGLAP evolution of hadronic structure functions and the ERBL evolution of distribution amplitudes. We discuss the scheme-dependence of the value of $Q_0$ and the infrared fixed-point of the QCD coupling. Our analysis is carried out for the $\bar{MS}$, $g_1$, $MOM$ and $V$ renormalization schemes. Our results show that the discrepancies on the value of $α_s$ at large distance seen in the literature can be explained by different choices of renormalization schemes. We also provide the formulae to compute $α_s(Q^2)$ over the entire range of space-like momentum transfer for the different renormalization schemes discussed in this article.