Remarks on Strong CP-Violating Lagrangians

TL;DR

This paper clarifies the relationship between two formulations of the strong CP-violating Lagrangian, showing their consistency and highlighting the residual CP violation in the topological sector, with implications for understanding CP violation in QCD.

Contribution

It demonstrates the equivalence of the DVW and Baluni formulations and emphasizes the importance of the SU(3)-singlet eta_0 tadpole contribution for accurate CP violation analysis.

Findings

The DVW and Baluni Lagrangians are consistent despite different treatments.

Residual CP violation exists in the topological sector after vacuum rotation.

Including eta_0 tadpole contributions is essential for correct CP violation limits.

Abstract

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 $\theta G\tilde G$ 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 $\eta_0$ 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.