Theta dependence of SU(N) 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 $b_2=O(1/N^2)$.