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The theory would be CP invariant if one could eliminate both sources of CP violation through such a field redefinition. But this cannot be done unless <math>\theta = -\theta'</math>. This is because even under such field redefinitions, the combination <math>\theta'+ \theta \rightarrow (\theta'-\alpha) + (\theta + \alpha) = \theta'+\theta</math> remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking <math>\alpha = \theta'</math>, but then all the CP violation goes to the θ-term which is now proportional to <math>\bar \theta</math>. If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase <math>\bar \theta</math>. Practically, it is usually useful to put all the CP violation into the θ-term and thus only deal with real masses.
In the Standard Model where one deals with six quarks whose masses are described by the [[Yukawa interaction|Yukawa matrices]] <math>Y_u</math> and <math>Y_d</math>, the physical CP violating angle is <math>\bar \theta = \theta - \arg \det(Y_u Y_d)</math>. Since the θ-term has no contributions to perturbation theory, all effects from strong CP violation is entirely non-perturbative. Notably, it gives rise to
:<math>
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