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

Understand

Using finite size scaling techniques and a renormalization scheme based on the Gradient Flow, we determine non-perturbatively the $\beta$-function of the $SU(3)$ Yang-Mills theory for a range of renormalized couplings $\bar g^2\sim 1-12$.

• 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 $\bar{g}^2\sim1$.
• 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 $\beta$-function is known.
• On the other hand, we show that matching non-perturbatively the Gradient Flow to the Schr\"odinger Functional scheme allows us to make safe contact with perturbation theory with full control on truncation errors.