Strong CP solution with soft PQ breaking

A recent work combined the popular left-right parity (LR) and Peccei-Quinn (PQ) symmetries to explain the alignment in quark masses. Since axions may not exist, we break PQ softly and discover a new solution to the strong CP problem. Remarkably, since the soft PQ breaking terms respect parity, the strong CP phase is absent at tree-level, while $\bar{\theta} \gtrsim 7 \times 10^{-11}$ is generated in two loops. Thus a neutron electric dipole moment close to the current experimental bound is predicted, making the theory falsifiable, even if the scale of new physics is well beyond the reach of colliders. In LR models leptonic CP phases (such as Dirac phase $\delta_{CP}$) can generate the strong CP phase in one loop, which is a challenge for their existence. PQ symmetry sets one of the leptonic Yukawa matrices to zero at the tree level and automatically suppresses this contribution, and makes a way for leptonic CP violation to be present.