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

Nonperturbative lattice field theory simulations provide a systematic framework 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 the renormalized $\beta$ function in the finite volume gradient flow renormalization scheme. Using M\"obius domain wall fermions with Symanzik gauge action, Zeuthen gradient flow, and perturbative tree-level improvement, we implement a fully $O(a^2)$ Symanzik improved set-up and demonstrate its advantages. We compare our findings to existing results in the literature. For the ten flavor system we observe excellent agreement with the domain wall step-scaling function calculated by Chiu for the range in $g_c^2$ where our data overlap. In the case of the twelve flavor system, our $O(a^2)$ Symanzik improved set-up predicts a conformal infrared fixed point around $g_c^2 \sim 5.5$ in the $c=0.25$ scheme, which is presently in tension with staggered fermion results in the literature. We consider possible reasons for the discrepancy.