Axionless strong CP problem solution: The spontaneous CP-violation case

We propose an alternative to the axion mechanism for addressing the charge parity (𝐶⁢𝑃) problem in QCD. Our approach involves imposing 𝐶⁢𝑃 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 ℤ2 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 Cabibbo-Kobayashi-Maskawa matrix, and low-energy flavor-changing constraints, our findings suggest that this model offers a viable solution to the 𝐶⁢𝑃 problem.