How does one find the Feynman diagrams?

[leo.]

I'm studying Quantum Field Theory and the main books I'm reading (Peskin and Schwartz) present Feynman diagrams something like this: one first derive how to express with perturbation theory the $n$-point correlation functions, and then represent each term by a diagram. It is then derived the Feynman rules that allows one to do the backwards process: given a diagram, find out the terms in the expansion.

The point that all books make, at least in my opinion, is: computing the expansion is hard and using Wick's theorem is quite hard, so one develops these Feynman rules so that the process becomes: (1st) find the Feynman diagrams and (2nd) associate the number to the diagrams according to the rules.

My whole question is: considering a particular interaction lagrangian, how does one finds the Feynman diagrams up to some specific order?

It obviously isn't by expanding the $n$-point function, because if it was there would be no point in drawing the diagrams anyway, since the expansion would already be known.

The only thing I can figure out is that in the expansion of the $n$-point each diagram has $n$ external points and expansion to order $\lambda^k$ will have $k$ internal points.

So considering for example the $\mathcal{L}_{\mathrm{int}} = \lambda \phi^4/4!$ theory, how can I find the Feynman diagrams up to order $\lambda^2$?

 

[mfb]

Make a list of all couplings.
Then find all diagrams up to a given order that only include these couplings and have the right external legs.

While you can do this manually for low orders, there are tools that can do this automatically. Just try everything and discard what doesn't work.