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I have the following system of two coupled real scalar fields $\sigma$ and $\phi$:

$S[\phi,\sigma]=\int{d^4x[-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2-\frac{1}{2}M^2\sigma^2-\frac{\lambda}{2}\sigma\phi^2]}$.

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!

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    $\begingroup$ The propagator for $\phi$ is just the regular scalar propagator with mass $m$; you'll also have an interaction vertex with two $\phi$ lines, and a single $\sigma$ line, the vertex factor would be $\lambda$. In addition, $[\lambda] =1$, so I'd expect the interaction to be super-renormalizable, i.e. only a specific finite set of Feynman diagrams will be divergent. $\endgroup$ – JamalS Jun 1 '14 at 11:52
  • $\begingroup$ Ok that makes sense. So each graph (at 2nd order) would have 2 three point vertices, and each line would be labeled with what the specific propagator it is connecting? $\endgroup$ – Josh Cork Jun 1 '14 at 16:11
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    $\begingroup$ Also, as the $\sigma$ field doesn't have a kinetic term, I don't think you can have internal $\sigma$ lines. $\endgroup$ – JamalS Jun 1 '14 at 16:17
  • $\begingroup$ Integrate σ out in the functional integral, exactly, and trivially. $\endgroup$ – Cosmas Zachos Sep 19 '16 at 20:49

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