Toward a novel determination of the strong QCD coupling at the Z-pole

We test here our recently introduced new lattice method for the $\beta$-function defined over infinite Euclidean space-time in the continuum from scale changes generated by infinitesimal or finite steps of the renormalized gauge coupling on the gradient flow. Harlander and Neumann calculated in this scheme the three-loop approximation to the continuum $\beta$-function. Our goal is the nonperturbative lattice implementation of the scheme which we tested originally in the chiral limit of the sextet model and in multi-flavor QCD with ten and twelve flavors of massless fermions. Results are reported here in the SU(3) Yang-Mills gauge sector without dynamical fermions and in ten-flavor QCD with massless femions. The three-loop gradient flow based $\beta$-function of Harlander and Neumann is used to connect the $\Lambda_{\overline{\rm MS}}$ scale of the SU(3) Yang-Mills gauge theory with the nonperturbative flow time scale $t_0$, or the equivalent Sommer scale $r_0$. Similarly, the $\Lambda_{\overline{\rm MS}}$ scale is connected with a selected nonperturbative scale in the ten-flavor theory, a pilot study of our new lattice based nonperturbative $\beta$-function for high precision determination of the strong coupling $\alpha_s$ at the Z-boson pole in QCD with three massless fermion flavors. This goal is an important alternative to results from the finite volume based step $\beta$-function of the Alpha collaboration. Work is ongoing on direct application of the method to QCD with three massless fermion flavors.