I'm trying to calculate the amplitude for an interaction between a scalar field $$\phi$$ and two identical spin 1 fields $$A_{\mu} \quad and \quad A^{\mu}$$ for the interaction $$\phi \longrightarrow A^{\mu} A_{\mu}$$(adsbygoogle = window.adsbygoogle || []).push({});

with the Lagrangian density $$L_{int} = -ik\phi A^{\mu} A_{\mu}$$ where k is a constant. My first thought is that the amplitude should evaluate to $$M = k\epsilon_{\mu}^{*}\epsilon^{\mu *}$$ where each of the epsilons has a particular spin. Thus, this would evaluate to -k if the two spins are equal and 0 if they are not equal. However, I would expect this to come out in terms of energy squared, but this is clearly not the case. Where am I erring?

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# Amplitude for scalar-proca couplings

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