Improved study of the $\beta$-function of $SU(3)$ gauge theory with $N_f=10$ massless domain-wall fermions

I perform an improved study of the $\beta$-function of $SU(3)$ lattice gauge theory with $N_f=10$ massless optimal domain-wall fermions in the fundamental representation, which serves as a check to what extent the scenario in the previous work [arXiv:1603.08854; Proc. Sci. LATTICE2016 (2017) 228] is valid. In the finite-volume gradient flow scheme with $c=\sqrt{8t}\mathrm{/}L=0.3$, the renormalized couplings $g^2(L,a)$ of four primary lattices ($ L/a = { 8, 10, 12, 16 }$)aretuned(in$ 6/g_0^2 $)tothesame$ g_c^2 $ with statistical error less than $0.5\mathrm{\%}$, in contrast to the previous work where $g^2(L,a)$ were obtained by the cubic-spline interpolation. Then the renormalized couplings $g^2(sL,a)$ of the scaled lattices ($ sL/a = {16, 20, 24, 32} $with$s=2$)arecomputedatthesame$ 6/g_0^2 $ of the corresponding primary lattices. Using the renormalized couplings of four lattice pairs $(sL,L)\mathrm{/}a=\{(16,8),(20,10),(24,12),(32,16)\}$, the step-scaling $\beta$-function $[g^2(sL,a)-g^2(L,a)]\mathrm{/}\ln (s^2)$ is computed and extrapolated to the continuum limit $\beta (s,g_c^2)$, as summarized in Table III. Based on the four data points of $\beta (s,g_c^2)$ at $ g_c^2 = { 6.86(2), \ 6.92(3), \ 7.03(2), \ 7.16(2) } $, I infer that the theory is infrared near-conformal, or conformal with the fixed-point $g_{\ast }^2=7.55(36)$. This corrects the scenario in the previous work with $g_{\ast }^2\sim 7.0$, and also suggests that the interpolation method cannot give a reliable determination of the $\beta$-function, especially in the regime close to the infrared fixed-point.