Scale setting of SU($N$) Yang--Mills theory, topology and large-$N$ volume independence

Abstract

We set the scale of SU($N$) Yang--Mills theories for $N=3,5,8$ and in the large-$N$ limit via gradient flow, as a first step towards the computation of the large-$N$ $Λ$-parameter using step scaling. We adopt twisted boundary conditions to achieve large-$N$ volume reduction and the Parallel Tempering on Boundary Conditions algorithm to tame topological freezing. This setup allows accurate determinations of the gradient-flow scales even on extremely fine lattices for all the explored values of $N$. Moreover, we are able to precisely estimate the finite-size systematics related to topological freezing, and to show the suppression of finite-volume effects expected by virtue of large-$N$ twisted volume reduction.

Figures (15)

• Figure 1: Example of the determination of $t_0$ on an $\mathrm{SU}(3)$ lattice with $L=38$, $L_s=24$, $b=0.37583$ and TBCs, with and without projection to $Q=0$.
• Figure 2: Dependence on the short size $\hat{l}_s$ of the non-projected scale $t_0$ with improvement of lattice artifacts, see \ref{['eq:improvement-artifacts']}. The scales $T_0(N,b,L_s, L_{\mathrm{max}})$ are determined at the largest long size $l=l_{\mathrm{max}}$ available, and satisfy $l_{\max}\gtrsim 3\sqrt{8t_0}$. All finite-volume scales are normalized with their corresponding infinite-volume result $T_0(N,b)$ obtained from the $N$-by-$N$ fits.
• Figure 3: Dependence on $N$ of the best-fit parameters $\hat{A}(N)$ and $M(N)$ in \ref{['eq:fit-volume-NbyN-anyQ']}, obtained from the $N$-by-$N$ fits of the improved $t_0$ shown in \ref{['fig:t0+_NbyN_twst_imp2_anyQ_vs_Ls']}. As expected, $\hat{A}(N) = \mathcal{O}(N^{-2})$ and $M(N) = \mathcal{O}(N^{0})$.
• Figure 4: Left panels: dependence of the non-projected scale $t_0$ on the long size $l$ after subtraction of the exponentially-suppressed corrections in $l_s$. Right panels: infinite-volume results for the projected scale $t_0^{(0)}$. Colored bands represent the infinite-volume result for the non-projected scale.
• Figure 5: Dependence on $N$ of the best-fit parameters $\hat{A}(N)$, $M(N)$ and $\hat{C}(N)$ in \ref{['eq:fit-volume-NbyN-Q=0']}, obtained from the $N$-by-$N$ fits of the improved $t_0^{(0)}$. The values resulting from the non-projected fits are also shown. Also in this case, $\hat{A}(N) = \mathcal{O}(N^{-2})$ and $M(N) = \mathcal{O}(N^{0})$, and values obtained with and without projection are compatible at each $N$. Moreover, also $\hat{C}(N) = \mathcal{O}(N^{-2})$ holds as expected.