Electric dipole moment (EDM) and magnetic dipole moment (MDM): CP violation in neutron

What matter are not the behavior of the dipole moments $vec{mu}$ and $vec{d}$ themselves, but the Hamiltonian terms $-vec{mu}cdotvec{B}$ and $-vec{d}cdotvec{E}$,* showing the interactions of the moments with external fields.

Under parity, the electric field (being a polar vector) changes sign, but under time reversal, it does not. (This can be understood by visualizing inverting the locations of all charges; this reverses the vector field $vec{E}$ defined explicitly by Coulomb's Law. However, inverting the direction of time does not reverse the explicit expression for $vec{E}$, in which time does not appear.) In contrast, under P, the magnetic field does not reverse sign, because the explicit formula for $vec{B}$ is the Biot-Savart Law, which contains a cross product. (Under $vec{V}rightarrow-vec{V}$ and $vec{W}rightarrow-vec{W}$, we hve $vec{V}timesvec{W}rightarrowvec{V}timesvec{W}$, with the two minus signs canceling. Thus the cross product of two vectors is a pseudovector or axial vector.) However, the time reversal T inverts the directions of all currents while leaving the spatial configuration unchanged, which naturally reverses $vec{B}$.

So the different transformation properties of $vec{E}$ and $vec{B}$ mean that the actual Hamiltonian/Lagrangian terms representing the dipole interactions have different discrete symmetries. For the magnetic dipole, $vec{B}$ changes sign under exactly the same combinations of C, P, and T as the spin vector (remembering that $vec{mu}=muvec{sigma}$ and $vec{d}=dvec{sigma}$), so the combination $-vec{mu}cdotvec{B}$ is invariant under all the discrete symmetries. However, the discrete symmetries of $vec{sigma}$ and $vec{E}$ do not match up. They are both odd under C; however, $vec{sigma}$ is P-even and T-odd, while $vec{E}$ is P-odd and T-even, making $-vec{d}cdotvec{E}$ odd under P, T, CP, and CT.

*Actually, these are the nonrelativistic limits of the general spin interactions with $mubar{psi}sigma^{munu}psi F_{munu}$ and $dfrac{1}{2}epsilon^{munurhosigma}bar{psi}sigma_{munu}psitilde{F}_{rhosigma}$. In these forms, it is relatively straightforward to see that the electric dipole moment term must be odd under P and T, because of the presence of the Levi-Civita pseudo-tensor $epsilon^{munurhosigma}$.