Currents and the quantum effective action

[muppet]

Currents and the "quantum" effective action

Hi all,
I've been reading Burgess' Primer on effective field theory: arXiv:hep-th/0701053v2. I can't follow the reasoning here:

W[J] [is] the sum of all connected graphs that are constructed using vertices and propagators built from the classical lagrangian, L, and having the currents, J, as external lines. But $\Gamma$[ϕ] just differs from W[J] by subtracting $\int$d4x Jϕ, and evaluating the result at the specific configuration J[ϕ] = −(δ$\Gamma$/δϕ). This merely lops off all of the 1-particle reducible graphs, ensuring that $\Gamma$[ϕ] is given by summing 1-particle irreducible graphs.

I could maybe begin to make sense of this if I were allowed to assume that −(δ$\Gamma$/δϕ) is the sum of 1PI irreducible graphs with one external line, so that using bubble graphs as "sources" leads to a cancellation of 1 particle reducible diagrams (although in this case it would still be far from obvious to me that this cancellation still works when you have more than two external sources). But as this is what we're puporting to show here, I can't follow the logic at all. Any comments would be greatly appreciated.

 

[Bill_K]

I don't follow it either. And I guess it's a too late to ask Schwinger. Anyway for what it's worth, the same argument is made here in Sect 3.8. This one's a little bit more detailed.

 

[muppet]

Thanks for your reply, Bill_K. If I now understand the problem correctly (and this is the third time I've started penning a reply only to realize that I actually didn't...), the essential points are that
1) The path integral expression for $\Gamma[\phi]$ should be thought of as integrating over fluctations about the mean value $\phi$;
2) 1PI subgraphs are essentially factors that multiply any diagrams that contain them.

The first point means that, as we've fixed our current to give a particular value of the VEV, tadpoles that look like they should contribute to the VEV must sum to zero. The second then implies that any diagram containing a tadpole subgraph must itself vanish. I was originally getting hung up on how vanishing tadpoles lead to the "lopping off" of all reducible graphs such as s-channel 2->2 scattering processes in a theory with a 3 point vertex; the point, I think, is that the external legs of such diagrams now vanish, dragging the rest of the diagram along with them.