θ dependence of SU (N) gauge theories

Abstract

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) = A2 θ2(1 + b 2θ2 + ⋯). Our results support Witten's conjecture: F(θ) - F(0) = Aθ2 + 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 X∞ = 2A with corrections of O(1/N2), in substantial agreement with the Witten-Veneziano formula which relates X∞ 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: b 2 = -0.023(7) for N = 3, and its absolute value decreases with increasing N, consistently with the expectation b2 = O(1/N 2). © SISSA/ISAS 2002.