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

TL;DR

This study investigates how the free energy of SU(N) gauge theories depends on the CP-violating angle theta, confirming theoretical predictions and providing numerical estimates for the topological susceptibility and higher-order terms.

Contribution

The paper provides numerical evidence supporting Witten's conjecture on the theta dependence of SU(N) gauge theories and quantifies the suppression of higher-order theta terms.

Findings

Topological susceptibility approaches a nonzero large-N limit consistent with the Witten-Veneziano formula.

Higher-order theta terms are small and decrease with increasing N.

Results support the theta dependence form F(theta) - F(0) = A theta^2 + O(1/N).

Abstract

We study the dependence of the free energy on the CP violating angle theta, 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(theta) around theta = 0, F(theta) - F(0) = A_2 theta^2 (1 + b_2 theta^2 + ...). Our results support Witten's conjecture: F(theta) - F(0) = A theta^2 + O(1/N) for theta < pi. We verify that the topological susceptibility has a nonzero large-N limit chi_infinity = 2A with corrections of O(1/N^2), in substantial agreement with the Witten-Veneziano formula which relates chi_infinity to the eta' mass. Furthermore, higher order terms in theta are suppressed; in particular, the O(theta^4) term b_2 (related to the eta' - eta' 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).