Why is tree-level interaction between neutral scalar and photons non-renormalizable? I've read that the decay of a neutral scalar particle into two photons, i.e.,
$$
S(p+q) \to \gamma(p) + \gamma(q)
$$
can't happen via tree diagrams and instead is caused by loop diagrams (such as a triangle diagram with an internal fermion); otherwise the theory won't be renormalizable. But why is this the case? This decay caused by a tree diagram means I need to add a term such as $gA_\mu A^\mu \phi$ to my Lagrangian. I understand why this is stupid: I'm coupling a neutral field to photons (how is $g$ even defined if $\phi$ has no charge?), but I don't see how it violates renormalizability in particular.
 A: Regardless of renormalizability, the term that you wrote down $(gA_\mu A^\mu \phi)$ does not describe photons because it is not gauge invariant. This would be a theory of a massless vector boson with three dynamical propagating degrees of freedom (two transverse and one longitudinal), which is inconsistent with Lorentz invariance and irrelevant to considerations of renormalizability because it simply does not describe photons.
Now for photons, the lowest order term that you can write down is 
$$
\frac{g}{\Lambda} F^{\mu\nu}F_{\mu\nu}\phi
$$
This would be gauge invariant, and could physically describe the electromagnetic interactions of the scalar field (note that a neutral "composite" field can effectively interact with the electromagnetic field). Dimensional analysis then tells us that this term is not renormalizable. Which is by the way totally okay, if we consider this interaction as an effective interaction valid up the energy scale $\Lambda$, usually resulting from integrating out very massive dof that would have lead to this interaction in a renormalizable way, or as a term in the effective potential resulting from loop corrections.
