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?

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

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