Few thoughts on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math> and the electric dipole moments

I highlight a few thoughts on the contribution to the dipole moments from the so-called $\ensuremath{\theta}$ parameter. The dipole moments are known can be generated by $\ensuremath{\theta}$. In fact, the renowned strong $\mathcal{CP}$ problem was formulated as a result of nonobservation of the dipole moments. What is less known is that there is another parameter of the theory, the ${\ensuremath{\theta}}_{\mathrm{QED}}$ 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 ${\ensuremath{\theta}}_{\mathrm{QED}}$ 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 ${e}^{\ensuremath{'}}=\ensuremath{-}e\frac{{\ensuremath{\theta}}_{\mathrm{QED}}}{2\ensuremath{\pi}}$. We argued that the similar arguments suggest that the electric magnetic dipole moment $\ensuremath{\mu}$ of any microscopical configuration in the background of ${\ensuremath{\theta}}_{\mathrm{QED}}$ generates the electric dipole moment $⟨{d}_{\mathrm{ind}}⟩$ proportional to ${\ensuremath{\theta}}_{\mathrm{QED}}$, i.e., $⟨{d}_{\mathrm{ind}}⟩=\ensuremath{-}\frac{{\ensuremath{\theta}}_{\mathrm{QED}}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\alpha}}{\ensuremath{\pi}}\ensuremath{\mu}$. We also argue that many $\mathcal{CP}$ odd correlations such as $⟨{\stackrel{\ensuremath{\rightarrow}}{B}}_{\mathrm{ext}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{E}⟩=\ensuremath{-}\frac{\ensuremath{\alpha}{\ensuremath{\theta}}_{\mathrm{QED}}}{\ensuremath{\pi}}{\stackrel{\ensuremath{\rightarrow}}{B}}_{\mathrm{ext}}^{2}$ will be generated in the background of an external magnetic field ${\stackrel{\ensuremath{\rightarrow}}{B}}_{\mathrm{ext}}$ as a result of the same physics.

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