We’ve known since the 1960s that CP symmetry—the simultaneous interchage of all particles with their antiparticles and inversion of the spatial coordinates—is violated in the Standard Model. This is now well understood to be a result of a physical complex phase in the quark-mixing matrix, and particle physicists are engaged in an intense theoretical and experimental program to scrutinize the nature of CP violation in the quark sector to test the consistency of the Standard Model and potentially uncover hints of new physics.
However, there is a second source of CP violation that appears in the Standard Model, the so-called “theta-term”: $$ \mathcal{L} \supset \frac{\bar{\theta}}{16\pi^2} \int \mathrm{d}^4 x ~ \mathrm{tr} G_{\mu\nu} \widetilde{G}^{\mu\nu} $$ Here, $\theta$ is an angular parameter which can in principle take any value between $0$ and $2\pi$. Despite appearing like a topological term, which would naively have no physical consequences, this operator leads to a non-zero electric dipole moment for the neutron which can be searched for experimentally. The current constraints, however, tell us that $\bar{\theta} \lesssim 10^{-10}$! This apparent fine-tuning is known as the Strong CP Problem: it is a strong motivation for physics beyond the Standard Model which may explain the smallness of $\bar{\theta}$ through some dynamical or symmetry-based mechanism.
There are two well-studied solutions to the Strong CP problem: the first is the axion, or Peccei–Quinn, solution, which essentially promotes $\bar{\theta}$ to a dynamical field that has a potential minimized at $\bar{\theta} = 0$. The second is to suppose that CP (or parity) is a fundamental symmetry that is spontaneously broken by dynamics at very short distances. Models invoking this solution with a fundamental CP symmetry are usually referred to as Nelson–Barr models. My research examines both of these solutions, with the broad aim of trying to better understand their viability from a theoretical point of view, their implications for Cosmology, and their potential connections to other puzzles of the Standard Model, such as the flavor puzzle.