QFT. Effective action and the skeleton expansion, how the legendre transform works!!

yaboidjaf

1. The problem statement, all variables and given/known data

I've written a presentation on the effective action and have been posed a few questions to look out for. I think I know the answer to the first but am stumped by the second.

"you've said we can get the correct scattering amplitudes by computing tree diagrams with exact propagators and vertices. Do you mean here just the exact propagators and exact 3-vertices? Or exact propagators and exact vertices of all valencies? More specifically, if you consider a 2-to-2 scattering amplitude (4 external lines), do you need an exact 4-vertex for this amplitude?

Another question: effective action is a sum of 1PI diagrams. I.e. by the Legendre transform trick we have managed to get rid of 1-particle-reducible diagrams. How then does it manage to generate exact n-vertices? Exact propagators would be understandable, for this is what the Legendre transform has done. But why exact vertices of any valency? Can you see how this works on the example of a 3-valent vertex?"

2. Relevant equations



3. The attempt at a solution

For the first part I'm pretty sure that we evaluate them using the exact vertices. so we use the exact 4-vertex for the 2 to 2 scattering.

For the second part i'm not too sure i understand. the idea of the legendre transform is to remove the 1pr diagrams from the sum of all connected diagrams, does that not mean it generates both exact propagators as well as exact vertices?