Axionless Solution to the Strong CP Problem—Two-Zeros Textures of the Quark and Lepton Mass Matrices and Neutrino CP Violation

Abstract CP invariance is a very attractive solution to the strong CP problem in quantum chromodynamics (QCD). This solution requires the vanishing ${\rm arg}\, [{\rm det}\, M_d\, {\rm det} M_u]$, where $M_d$ and $M_u$ are the mass matrices for the down- and up-type quarks, respectively. It happens if we have several zeros in the quark mass matrices. We proceed with a systematic construction, in this paper, of two-zeros textures for the down-type quark mass matrix while the mass matrix for the up-type quarks is always diagonal. We find only three types of the mass matrices can explain the observed Cabibbo–Kobayashi–Maskawa (CKM) matrix, the masses of the quarks and the charged leptons, and the small enough vacuum angle $\theta < 10^{-10}$. We extend the mass construction to the neutrino sector and derive predictions on the CP-violating parameter $\delta _{\mathrm{ CP}}$ in the neutrino oscillation and the mass parameter $m_{\beta \beta }$. It is extremely remarkable that the normal (NH) and inverted (IH) hierarchies in the neutrino masses are equally possible in the case where we introduce only two right-handed neutrinos Ns. Furthermore, we have a strict prediction that the $\delta _{\mathrm{ CP}} \simeq 200^\circ$ or $250^\circ$ in the NH case. If it is the case we can naturally explain the positive sign of the baryon asymmetry in the present universe.