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From ten-flavor tests of the $β$-function to $α_s$ at the Z-pole

New tests are applied to two $β$-functions of the much-discussed BSM model with ten massless fermion flavors in the fundamental representation of the SU(3) color gauge group. The renormalization scheme of the two $β$-functions is defined on the gauge field gradient flow in respective finite or infinite physical volumes at zero lattice spacing. Recently published results in the ten-flavor theory led to indicators of an infrared fixed point (IRFP) in the finite-volume step $β$-function in the strong coupling regime of the theory arXiv:2004.00754. We analyze our substantially extended set of ten-flavor lattice ensembles at strong renormalized gauge couplings and find no evidence or hint for IRFP in the finite-volume step $β$-function within controlled lattice reach. We also discuss new ten-flavor tests of the recently introduced lattice definition and algorithmic implementation of the $β$-function defined on the gradient flow of the gauge field over infinite Euclidean space-time in the continuum. Originally we introduced this new algorithm to match finite-volume step $β$-functions in massless near-conformal gauge theories with the infinite-volume $β$-function reached in the chiral limit from small fermion mass deformations of spontaneous chiral symmetry breaking. Results from the lattice analysis of the ten-flavor infinite-volume $β$-function are consistent with the absence of IRFP from our step $β$-function based analysis. We make important contact at weak coupling in infinite volume with gradient flow based three-loop perturbation theory, serving as a first pilot study toward the long-term goal of developing alternate approach to the determination of the strong coupling $α_s$ at the Z-boson pole in QCD.