Dyonic lattices, $θ$-angles and axions in the Standard Model
We investigate the implications of the Witten effect in the Standard Model with a general global gauge group structure and determine the values of the three $θ$-parameters that lead to distinct families of allowed spectra of dyons. We construct and classify the corresponding dyonic charge lattices consistent with the Standard Model gauge structure. This approach enables us to re-derive the known global-group--dependent periodicities of the $θ$ angles and to determine all CP-invariant points in $θ$-space. The electromagnetic subgroup $U(1)_{\mathrm{em}}$ is shown to arise \emph{prior} to electroweak symmetry breaking by factoring out the effect of the anomalous $B+L$ transformations, which reduces the physical $θ$-parameter space from a three-torus to a two-torus. Our phenomenological conclusions include an observation that a discovery of a $U(1)_{\mathrm{em}}$ monopole carrying non-zero electric charge would determine the last remaining unknown parameter of the Standard Model. Lastly we study how $θ$-space shapes axion physics with emphasis on the axion-photon coupling and show that a single axion is insufficient to render the Standard Model vacuum fully CP invariant.