quantum field theory - Feynman rules for coupled systems

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!