Non-minimal coupling between a neutral atom and the EM field

Let us say that I have neutral bosonic atoms interacting with an EM field. I can write down the Lagrangian as

\begin{align} \mathcal{L}=\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi^{\dagger}-m^2|\phi|^2-\lambda|\phi|^4-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+A_{\mu}J^{\mu}+\mathcal{L}_{int} \end{align}

where $$\mathcal{L}_{int}$$ is the interaction term. However, I am not sure what form this term should take. As the atom is neutral, $$\partial_{\mu}\to\partial_{\mu}+iq A_{\mu}$$ no longer works. This makes me think that some form of non-minimal coupling is required. According to to Maggiore (pg. 72), the interaction occurs through higher electric and magnetic multipoles of the scalar field, where he gives the example $$\mathcal{L}_{int}=a\phi F_{\mu\nu}F^{\mu\nu}$$, where $$a$$ is the coupling constant and $$\phi$$ is a real scalar field. I have two sets of questions associated with this:

$$1$$. Will $$\mathcal{L}_{int}=a|\phi| F_{\mu\nu}F^{\mu\nu}$$ work for a complex scalar field, or do I instead need to have $$\mathcal{L}_{int}=a|\phi|^2 F_{\mu\nu}F^{\mu\nu}$$?

$$2$$. What is the form of interaction for the $$n^{\rm{th}}$$ order multipole moment? I am assume that the dominant term is dipole in nature, then quadrupole, octupole, etc?

• Are you specifically asking about a relativistic model of a neutral atom, as the notation seems to suggest? – Chiral Anomaly Feb 27 at 1:03
• Yes; I am interested in the interaction between a neutral atom and the EM field from a quantum field theory point of view. – user85503 Feb 27 at 15:38