We consider chiral perturbation theory with a nonzero $\theta$ term. Because of the $CP$ violating term, the vacuum of chiral fields is shifted to a nontrivial element on the $SU(N_f)$ group manifold. The $CP$ violation also provides mixing of different $CP$ eigenstates, between scalar and pseudoscalar, or vector and axialvector, operators. We investigate up to $\mathcal{O}({\theta }^2)$ effects on the mesonic two-point correlators of chiral perturbation theory to the one-loop order. We also address the effects of fixing topology, by using saddle-point integration in the Fourier transform with respect to $\theta$.

• Received 12 July 2009

DOI:https://doi.org/10.1103/PhysRevD.81.034022