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Nonperturbative lattice field theory simulations provide a systematicframework to investigate properties of conformal systems at strong couplings.These simulations can be performed using different lattice discretizations.Here we present numerical results for the step scaling beta function in SU(3)gauge theories with ten and twelve fundamental flavors. We calculate therenormalized $\beta$ function in the finite volume gradient flowrenormalization scheme. Using M\"obius domain wall fermions with Symanzik gaugeaction, Zeuthen gradient flow, and perturbative tree-level improvement, weimplement a fully $O(a^2)$ Symanzik improved set-up and demonstrate itsadvantages. We compare our findings to existing results in the literature. Forthe ten flavor system we observe excellent agreement with the domain wallstep-scaling function calculated by Chiu for the range in $g_c^2$ where ourdata overlap. In the case of the twelve flavor system, our $O(a^2)$ Symanzikimproved set-up predicts a conformal infrared fixed point around $g_c^2 \sim5.5$ in the $c=0.25$ scheme, which is presently in tension with staggeredfermion results in the literature. We consider possible reasons for thediscrepancy.

Nonperturbative determination of $\beta$ functions for SU(3) gauge theories with 10 and 12 fundamental flavors using domain wall fermions