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(θ) = A 2θ 2(1 + b 2θ 2 + ...). Our results support Witten's conjecture: F(θ) − F(0) = Aθ 2 + O(1/ N) for θ < π. We verify that the topological susceptibility has a nonzero large- N limit χ ∞ = 2A with corrections of O(1/N 2) , 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(θ 4) term b 2 (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 b 2 = O(1/ N 2).
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