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Remarks on Strong CP-Violating Lagrangians

Owing to a different treatment of the vacuum alignment, the strong CP-violating Lagrangian obtained by Di Vecchia, Veneziano and Witten (DVW) 3 decades ago do not look quite the same as the one originally derived by Baluni at the quark or hadron level. We show that they are consistent with each other and emphasize that, within the DVW approach, the $θG\tilde G$ term is not entirely removed away after the vacuum is rotated from the CP-odd state to the CP-even one; strong CP violation resides not only in the quark mass terms but also in the residual topological sector. While the latter relies on the Dashen's theorem, the latter is based on the minimization of a potential energy with respect to the phase of the quark condensate. Contrary to some claims, it is necessary to include the SU(3)-singlet $η_0$ tadpole contribution for strong CP-odd effects induced by the Baluni-type Lagrangian to ensure that strong CP violation vanishes in the zero axial anomaly limit.