Electric dipole moment from QCD $ \theta$ and how it vanishes for mixed states

In a previous paper (A.P. Balachandran et al., JHEP 05, 012 (2012)), we studied the \( \eta{^\prime}\) mass and formulated its chirally symmetric coupling to fermions which induces electric dipole moment (EDM) maintaining chiral symmetry throughout in contrast to earlier works. 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 exactly as for \(\ensuremath \theta=0\) . The coupling does not lead to new divergences (not present for \(\ensuremath \cos\theta=1\) in soft photon processes either. Furthermore, as was argued previously (A.P. Balachandran et al., JHEP 05, 012 (2012)), 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 \(\ensuremath \vert\sin\theta\vert\lesssim10^{-11}\) . This fact eliminates the need to fine tune \( \theta\) or for the axion field.