High quality Nelson-Barr solution to the strong CP problem with θ = π

We discuss composite UV completions of the Nelson-Barr(NB) solution to the strong CP problem. In our construction, the CP symmetry is broken spontaneously by the dynamics of a hidden QCD at $\theta=\pi$. We focus on the minimal implementation of the NB construction where the visible sector contains one extra pair of vector-like up/down quarks. We show that the minimal NB theory suffers from a quality problem, and discuss how composite UV completions may resolve it. We present a simple calculable scheme, free of a quality problem, where dynamical CP violation in the hidden sector is mediated through a scalar portal to the visible sector, which successfully realizes the NB construction.
Comment: 28 pages, 6 Figures