Colorful Mirror Solution to the Strong CP Problem.

We propose theories of a complete mirror world with parity (P) solving the strong CP problem. P exchanges the entire standard model with its mirror copy. We derive bounds on the two new mass scales that arise: v^{'} where parity and mirror electroweak symmetry are spontaneously broken, and v_{3} where the color groups break to the diagonal strong interactions. The strong CP problem is solved even if v_{3}≪v^{'}, when heavy colored states at the scale v_{3} may be accessible at LHC and future colliders. Furthermore, we argue that the breaking of P introduces negligible contributions to θ[over ¯]_{QCD}, starting at three-loop order. The symmetry breaking at v_{3} can be made dynamical, without introducing an additional hierarchy problem.

Shift-Invariant Orders of an Axionlike Particle.

It is generally believed that global symmetries, in particular, axion shift symmetries, can only be approximate. This motivates one to quantify the breaking of the shift invariance that characterizes the flavorful effective couplings of an axionlike particle to standard model fermions. We identify a minimal set of Jarlskog-like flavor invariants that vanish if and only if the axion is shift symmetric. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate properties of the invariants by matching to a UV model, studying the CP transformation of the invariants, calculating their renormalization group evolution, and investigating similar conditions in the low-energy effective field theory below the electroweak scale. Finally, we discuss the order parameter associated to the nonperturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.