Theta dependence of SU(N) gauge theories

TL;DR

The paper tests the large-$N$ predictions for the $ heta$-dependence of SU($N$) gauge theories by computing the expansion of the ground-state energy $F(\theta)$ around $\theta=0$ on the lattice. Using the Wilson formulation with Cabibbo-Marinari updates and a cooling-based topology measurement, the authors extract the topological susceptibility $\chi$ and the $\theta^4$ coefficient $b_2$ for $N=3,4,6$, and perform continuum extrapolations. They find a finite large-$N$ limit $\chi_\infty$ with $O(1/N^2)$ corrections and a very small $b_2$ that decreases with $N$, in line with Witten's predictions and the Gaussian approximation $F(\theta)-F(0) \approx A_2 \theta^2$ for small $\theta$. These results provide quantitative support for the Witten-Veneziano mechanism and clarify the role of higher-order $\theta$-dependence terms in non-Abelian gauge theories.

Abstract

We study the $θ$ dependence of four-dimensional SU($N$) gauge theories, for $N\geq 3$ and in the large-N limit. We use numerical simulations of the Wilson lattice formulation of gauge theories to compute the first few terms of the expansion of the ground-state energy $F(θ)$ around $θ=0$, $F(θ)-F(0) = A_2 θ^2 (1 + b_2 θ^2 + ...)$. Our results support Witten's conjecture: $F(θ)-F(0) = {\cal A} θ^2 + O(1/N)$ for sufficiently small values of $θ$, $θ< π$. Indeed we verify that the topological susceptibility has a nonzero large-N limit $χ_\infty=2 {\cal A}$ with corrections of $O(1/N^2)$, in substantial agreement with the Witten-Veneziano formula which relates $χ_\infty$ to the $η^\prime$ mass. Furthermore, higher order terms in $θ$ are suppressed; in particular, the $O(θ^4)$ term $b_2$ (related to the $η^\prime - η^\prime$ elastic scattering amplitude) turns out to be quite small: $b_2=-0.023(7)$ for N=3, and its absolute value decreases with increasing $N$, consistently with the expectation $b_2=O(1/N^2)$.