We extend our recent analytic study of the strong coupling ₄₅₅ in the nonperturbative and near-perturbative regimes~deTeramond: 2024ikl by imposing rigorous renormalization-group results from asymptotically free gauge theories at Q². The asymptotic boundary conditions modify the scaling properties of ₄₅₅ at large values of the momentum transfer Q², and lead to a scale-dependent confinement strength (Q²). This requires that both (Q²) and ₄₅₅ (Q², (Q²) ) remain holomorphic in the complex Q² plane, except at the physical cuts associated with the heavy-quark thresholds and the singularity flow trajectory studied in~deTeramond: 2024ikl. For color SU (3), a precise connection is found between the scaling exponent of (Q²) in the ultraviolet, the value of the infrared fixed point of the strong coupling, and the number of flavors in agreement with observations. The nonperturbative analytic model gives an accurate description of the strong coupling at all scales, up to the highest available data.
Téramond et al. (Mon,) studied this question.