In scalar QED, there is a vertex where two spin $0$ particles come in and a spin $1$ photon comes out. Naively this can't possibly be consistent with angular momentum conservation, because two spin $0$ things can't add up to spin $1$.
It is claimed here that this is okay because the photon is "off-shell", so it has spin $0$. I don't believe this argument. While it is true that the usual formulation of QED has longitudinal photons, the whole point is that they decouple. And it's perfectly possible to formulate QED with only physical states from the start. Being off-shell is weird, but not so weird that it can reach outside the Hilbert space and produce a new scalar state out of nowhere.
A more sensible possibility, in my opinion, is that the two spin zero particles must have orbital angular momentum $\ell = 1$. But that seems rather complicated, and I don't know how to show that.
What is the resolution to this puzzle?
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