I have the following system of two coupled real scalar fields $\sigma$ and $\phi$:
What would the Feynman rules be for this system? I realise that something will be different about the propagator for $\sigma$ as it has no kinetic term, but I'm not sure how that translates into the rules.
Furthermore, how would you draw the Feynman diagrams which determine the amplitude for a 2 -> 2 scattering process associated to the field $\phi$ (at tree level)?
The whole idea is perplexing me a bit and I was wondering if there was any insight here!