The large-$N$ limit of the topological susceptibility of $\mathrm{SU}(N)$ Yang-Mills theories via Parallel Tempering on Boundary Conditions

TL;DR

This paper determines the large-$N$ limit of the topological susceptibility $\chi$ in SU($N$) Yang–Mills theories by performing nonperturbative lattice simulations for $N=3,4,5,6$ using Parallel Tempering on Boundary Conditions (PTBC) to mitigate topological freezing. It demonstrates that, at fixed physical smoothing radius, the continuum limit of $\chi$ is independent of the smoothing scale and obtains a precise large-$N$ extrapolation through a global continuum–large-$N fit, yielding $\frac{1}{\sigma^2}\chi_{\infty}=0.02088(39)$, with $\chi_{\infty}^{1/4} = 180.9(0.8)$ MeV in the chosen scale setting. The results are consistent with the Witten–Veneziano mechanism and the Di Vecchia–Veneziano framework, and they show good agreement with prior continuum-extrapolated determinations across $N$, while clarifying a few tensions due to extrapolation ranges in earlier work. The work also demonstrates the efficacy of PTBC in reducing autocorrelations and enabling finer lattice spacings, paving the way for future QCD applications and studies of topological observables at large $N$. Overall, the study provides a robust, high-precision large-$N$ portrait of $\chi$ that informs nonperturbative aspects of the axial anomaly and hadron phenomenology.

Abstract

I present a large-$N$ determination of the topological susceptibility $χ$ of $\mathrm{SU}(N)$ Yang--Mills theories using non-perturbative numerical Monte Carlo simulations of the lattice-discretized theory for $3\le N \le 6$, and adopting the Parallel Tempering on Boundary Conditions (PTBC) algorithm to bypass topological freezing for $N>3$. Thanks to this algorithm I am able to explore a uniform range of lattice spacing across all values of $N$, and to precisely determine $χ$ for finer lattice spacings compared to previous studies with periodic or open boundary conditions. By taking the continuum limit at fixed smoothing radius in physical units, I am also able to show the independence of the continuum limit of $χ$ from this choice. I conclude providing a comprehensive comparison of my new PTBC results with previous determinations of the topological susceptibility in the literature, both at finite $N$ and in the large-$N$ limit.

Figures (16)

• Figure 1: Example of the Monte Carlo evolution of the cooled topological charge $Q_{{\mathrm{L}}}$ for $R_{{\mathrm{s}}}\sqrt{\sigma}\simeq 1.18$ for the finest lattice spacing and largest $N$ explored in this study.
• Figure 2: Top panel shows the autocorrelation times $\tau_0$ of $Q^2$ obtained with PTBC algorithm on the periodic system as a function of the lattice spacing $a$ and of the rank of the gauge group $N$. The corresponding number of replicas required is shown in the bottom panel. The empirical rescalings reported in these plots have been determined by looking for a collapse of the data around an approximate constant value, see the text for more details.
• Figure 3: Computation of the continuum limit of $r_0\sqrt{\sigma}$ for $N=3$ (left panel), and of $\sqrt{8t_0\sigma}$ for $N=3,4,5,6$ (center panel) and $N=\infty$ (right panel).
• Figure 4: Continuum limits of $\chi_{{{\mathrm{L}}}}$ for various values of $N$ for 3 values of the smoothing radius $R_{{\mathrm{s}}}$. This quantity is kept fixed as a function of both $a$ and $N$. Each continuum limit is taken according to the fit function in Eq. \ref{['eq:fit_function_contlim_N_by_N']}.
• Figure 5: Same data as in Fig. \ref{['fig:contlim_chi_N_by_N_1']}, but plotted for various values of $N$ at fixed $R_{{\mathrm{s}}}\sqrt{\sigma}$. Lattice artifact slopes $C(R_{{\mathrm{s}}},N)$ are practically independent of $N$, and become very small for $R_{{\mathrm{s}}}\sqrt{\sigma}\gtrsim 1.2$. In the bottom right plot I also show the full $R_{{\mathrm{s}}}$-dependence of $\chi(N)/\sigma^2$ and $C(R_{{\mathrm{s}}},N)$ in the whole explored range of smoothing radii.