Is it possible to write down a sensible analog to QED but without fermions? Or better yet, with only scalar particles? Would two scalar fields with an interaction term $\lambda \phi_1 \phi_2^2$ lead to a sensible theory, where $\phi_1$ and $\phi_2$ are scalar fields? At least from a dimension-counting perspective, this term looks OK.
This question came up from overthinking the answer to another question here: Scalar QED atoms - will they pass through each other?
