CP parity of the Yang-Mills theta density (${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$)

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.