A few thoughts on $\theta$ and the electric dipole moments

I highlight a few thoughts on the contribution to the dipole moments from the so-called $\theta$ parameter. The dipole moments are known can be generated by $\theta$. In fact, the renowned strong $\cal{CP}$ problem was formulated as a result of non-observation of the dipole moments. What is less known is that there is another parameter of the theory, the $\theta_{QED}$ which becomes also a physical and observable parameter of the system when some conditions are met. This claim should be contrasted with conventional (and very naive) viewpoint that the $\theta_{\rm QED}$ is unphysical and unobservable. A specific manifestation of this phenomenon is the so-called Witten effect when the magnetic monopole becomes the dyon with induced electric charge $e'=-e \frac{\theta_{QED}}{2\pi}$. We argued that the similar arguments suggest that the electric magnetic dipole moment $\mu$ of any microscopical configuration in the background of $\theta_{QED}$ generates the electric dipole moment $\langle d_{\rm ind} \rangle$ proportional to $\theta_{QED}$, i.e. $\langle d_{\rm ind}\rangle= - \frac{\theta_{\rm QED} \cdot \alpha}{\pi} \mu$. We also argue that many $\cal{CP}$ correlations such as $\langle \vec{B}_{\rm ext} \cdot\vec{E}\rangle = -\frac{\alpha\theta_{\rm QED}}{\pi}\vec{B}^2_{\rm ext}$ will be generated in the background of an external magnetic field $\vec{B}_{\rm ext}$ as a result of the same physics.