θ dependence of SU(N) gauge theories

We study the θ dependence of four-dimensional SU(N) gauge theories, for N ⩾ 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) = A₂θ²(1+b₂θ² + ···). Our results support Witten's conjecture: F(θ)−F(0) =  θ²+O(1/N) for sufficiently small values of θ, θ < π. Indeed we verify that the topological susceptibility has a non-zero large-N limit χ_∞ = 2 with corrections of O(1/N²), in substantial agreement with the Witten-Veneziano formula which relates χ_∞ to the η' mass. Furthermore, higher order terms in θ are suppressed; in particular, the O(θ⁴) term b₂ (related to the η'−η' elastic scattering amplitude) turns out to be quite small: b₂ = −0.023(7) for N = 3, and its absolute value decreases with increasing N, consistently with the expectation b₂ = O(1/N²).