The gradient flow coupling at high-energy and the scale of SU(3) Yang–Mills theory

Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the β-function of the SU(3) Yang–Mills theory for a range of renormalized couplings g¯ 2∼ 1–12. We perform a detailed study of the matching with the asymptotic NNLO perturbative behavior at high-energy, with our non-perturbative data showing a significant deviation from the perturbative prediction down to g¯ 2∼ 1. We conclude that schemes based on the Gradient Flow are not competitive to match with the asymptotic perturbative behavior, even when the NNLO expansion of the β-function is known. On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schrödinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors. This strategy allows us to obtain a precise determination of the Λ -parameter of the SU(3) Yang–Mills theory in units of a reference hadronic scale (8t0ΛMS¯=0.6227(98)), showing that a precision on the QCD coupling below 0.5% per-cent can be achieved using these techniques.