P and CP solution of the strong CP puzzle

Abstract

We use parity (P) to set $\theta_{QCD}$ to zero in the minimal left-right symmetric model with a bi-doublet Higgs, add a heavy vectorlike quark family, and obtain in a novel manner the Nelson Barr (NB) form associated so far only with spontaneous CP solution to the strong CP Puzzle. Our solution does not have the `coincidence of scales problem', that typically plagues NB models. P protects $\bar{\theta}$, if it breaks at a scale $v_R$ below the mass $M$ of the heavy quarks, and $\bar{\theta} \sim 10^{-9} (v_R/M)^2$ is radiatively generated, which can be acceptably small. On the other hand, if $M < v_R$, the $\bar{\theta} \sim 10^{-9}$ generated by the NB mechanism is too large, but if $\delta_{CKM}$ is obtained without the NB form, surprisingly a lower irreducible $\bar{\theta} \sim (10^{-13}~to~10^{-10}) ln( {v_R/M)}$, testable by neutron EDM experiments is generated. No leptonic CP violation is generated (Dirac phase $\delta_{CP} = 0~or~\pi$ in PMNS matrix) which makes the minimal model testable by neutrino experiments. We also discuss some challenges in a non-minimal model that generates leptonic CP violation. Lastly but importantly, we find with doublet rather than bi-doublet Higgses, that there is an automatic NB solution on imposing CP (the NB form is accidental due to $SU(2)_R$), which does not require generalized parity and needs just one mirror generation.