In particle physics we deal with parity transformations and in particular when we regard weak interactions. While learning this subject I found something that needs more clarification for me. Namely, suppose that a particle has an electric dipole moment $\vec{d}$ and some magnetic moment $\vec{\mu}$. Every source I see states that a permanent dipole moment 'violates' parity symmetry, but I do not see why.
I tried the show that the energy is not invariant under parity symmetry but I am not sure whether this is a rigorous theoretical way of proving this.
$$E = -\vec{\mu}\cdot \vec{B} - \vec{d}\cdot \vec{E}$$
is the energy, suppose I apply the parity operator on this:
$$\mathcal{P}(E) = -\vec{\mu}\cdot \vec{B} + \vec{d}\cdot \vec{E}.$$
So we see that $\mathcal{P}(E)\neq E$ so I would say that parity is 'violated' in order to let the energy $E$ be invariant under parity transformation ($E$ is a scalar so it should be mapped to itself). I used that $\vec{d} \propto \vec{S}$ and that $\vec{E}$ is a vector and $\vec{\mu},\vec{B},\vec{S}$ pseudovectors. $\vec{S}$ is the spin vector. My conclusion is therefore: $\vec{d}=\vec{0}$ or that parity is violated.
Is this 'proof' of showing this theoretically rigorous or am I missing someting?