Discrete symmetries on the light front and a general relation connecting the nucleon electric dipole and anomalous magnetic moments

We consider the electric dipole form factor, ${F}_{3}({q}^{2})$, as well as the Dirac and Pauli form factors, ${F}_{1}({q}^{2})$ and ${F}_{2}({q}^{2})$, of the nucleon in the light-front formalism. We derive an exact formula for ${F}_{3}({q}^{2})$ to complement those known for ${F}_{1}({q}^{2})$ and ${F}_{2}({q}^{2})$. We derive the light-front representation of the discrete symmetry transformations and show that time-reversal- and parity-odd effects are captured by phases in the light-front wave functions. We thus determine that the contributions to ${F}_{2}({q}^{2})$ and ${F}_{3}({q}^{2})$, Fock state by Fock state, are related, independent of the fundamental mechanism through which $CP$ violation is generated. Our relation is not specific to the nucleon, but, rather, is true of spin-$1/2$ systems in general, be they lepton or baryon. The empirical values of the anomalous magnetic moments, in concert with empirical bounds on the associated electric dipole moments, can better constrain theories of $CP$ violation. In particular, we find that the neutron and proton electric dipole moments echo the isospin structure of the anomalous magnetic moments, ${\ensuremath{\kappa}}^{n}\ensuremath{\sim}\ensuremath{-}{\ensuremath{\kappa}}^{p}$.