Theta dependence of SU(N) gauge theories

Abstract

We study the $\theta$ dependence of four-dimensional SU($N$) gauge theories, for $N\ge 3$ 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 $F(\theta )$ around $\theta =0$, $F(\theta )-F(0)=A_2{\theta }^2(1+b_2{\theta }^2+...)$. Our results support Witten's conjecture: $F(\theta )-F(0)=\mathcal{A}{\theta }^2+O(1/N)$ for sufficiently small values of $\theta$, $\theta <\pi$. Indeed we verify that the topological susceptibility has a nonzero large-N limit ${\chi }_{\infty }=2\mathcal{A}$ with corrections of $O(1/N^2)$, in substantial agreement with the Witten-Veneziano formula which relates ${\chi }_{\infty }$ to the ${\eta }^{\prime }$ mass. Furthermore, higher order terms in $\theta$ are suppressed; in particular, the $O({\theta }^4)$ term $b_2$ (related to the ${\eta }^{\prime }-{\eta }^{\prime }$ 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 $b_2=O(1/N^2)$.

Comments: 12 pages. Longer version, with additional clarifications. Results and conclusions are unchanged