Free energy and θ dependence of SU (N) gauge theories

Abstract

We study the dependence of the free energy on the CP violating angle θ, in four-dimensional SU(N) gauge theories with N ≥ 3, and in the large-N limit. Using the Wilson lattice formulation for numerical simulations, we 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) = A θ2 + O(1/N) for θ < π. We verify that the topological susceptibility has a nonzero 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′ - ηpm 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).