Axionless strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:math> problem solution: The spontaneous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"

We propose an alternative to the axion mechanism for addressing the charge parity (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>C</a:mi><a:mi>P</a:mi></a:math>) problem in QCD. Our approach involves imposing <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>C</c:mi><c:mi>P</c:mi></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi mathvariant="double-struck">Z</e:mi><e:mn>2</e:mn></e:msub></e:math> 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>C</h:mi><h:mi>P</h:mi></h:math> problem.