Electric Dipole Moment from QCD $\theta$ and How It Vanishes for Mixed   States
  In a previous paper [1], we studied the $\eta'$ mass and formulated its chirally symmetric coupling to fermions which induces electric dipole moment (EDM). Here we calculate the EDM to one-loop. It is finite, having no ultraviolet divergence while its infrared divergence is canceled by soft photon emission processes \emph{exactly} as for $\theta=0$. The coupling does not lead to new divergences (not present for $\sin\theta=0$) in soft photon processes either. Furthermore, as it was argued previously [1], the EDM vanishes if suitable mixed quantum states are used. This means that in a quantum theory based on such mixed states, a strong bound on EDM will not necessarily lead to a strong bound such as $|\sin \theta|\lesssim 10^{-11}$ . This fact eliminates the need to fine-tune $\theta$ or for the axion field. 
A. P. Balachandran, T. R. Govindarajan and Amilcar R. de Queiroz
hep-th hep-ph math-ph math.MP