  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.
