How do I find the Feynman rules for a general Lagrangian density?

For example the Lagrangian $$L = \partial_\mu \psi \partial^\mu \psi +a \psi\partial_\mu \psi \partial^\mu \psi+b \psi^2 \partial_\mu \psi \partial^\mu \psi.$$

Lagrange-Euler equation is: $$\Box \psi(1+2a \psi+2b\psi^2)=-a\partial_\mu \psi \partial^\mu \psi -2b \psi \partial_\mu \psi \partial^\mu \psi.$$ Obviously there's no interaction term like $\phi^3$ or $\phi^4$, so there's no the X vertex appearing in the rules, but how about the other rules how do these change?

And how to find them?

  • $\begingroup$ You write down the propagator and calculate every single term :p $\endgroup$ – gented May 28 at 12:38

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