The large-$N$ Yang--Mills $Λ$-parameter from step scaling

We use the step-scaling method and results obtained at $N = 3, 5$ and $8$ to determine the $N$-dependence of the dynamically generated scale $\Lambda$ of $\mathrm{SU}(N)$ Yang--Mills theories. We implement the step-scaling method in a suitable finite-volume renormalization scheme based on twisted boundary conditions, introduced to effectively achieve large-$N$ volume independence, and on a coupling defined through the gradient flow. In the $\overline{\mathrm{MS}}$ scheme, we obtain the following values in terms of the gradient flow scale $t_0$: $\sqrt{8t_0}\Lambda_{\scriptscriptstyle{\overline{\mathrm{MS}}}} = 0.577(23)$, $0.632(32)$, and $0.611(43)$ for $N=3,5$ and $8$, respectively. They extrapolate to a large-$N$ value of: $\sqrt{8t_0}\Lambda_{\scriptscriptstyle{\overline{\mathrm{MS}}}} (N=\infty) = 0.639(36)$, and the $N$-dependence is given by $\sqrt{8t_0}\Lambda_{\scriptscriptstyle{\overline{\mathrm{MS}}}}(N)=0.639(36)[1-0.85(62)/N^2+\mathcal{O}(1/N^4)]$. This work represents the first calculation of the Yang--Mills $\Lambda$-parameter in the large-$N$ limit that does not rely on asymptotic scaling strategies.