Axionless strong CP problem solution: the spontaneous CP violation case

We propose an alternative to the axion mechanism for addressing the charge parity (CP) problem in quantum chromodynamics (QCD). Our approach involves imposing CP as an inherent symmetry of the Lagrangian, which is then spontaneously broken. To generate the correct texture for the Yukawa matrices, we introduce a discrete ${\mathbb{Z}}_2$ symmetry that is softly broken by the scalar potential. By identifying a benchmark point for the Yukawa couplings that aligns with the measured quark masses, the CKM matrix, and low-energy flavor-changing constraints, our findings suggest that this model offers a viable solution to the CP problem.