Say we want to compute the Coleman-Weinberg potential at 2 loops.
The general strategy as we know is to expand the field ϕ around some background classical field ϕ→ϕb+ϕ, and do a path integral over the quantum part of the field, ϕ.
We can retrieve the effective action by doing a path integral, something like eq.42 in this reference.
There are 2 ways to do this at 1 loop, we can either evaluate a functional determinant or do the classic Coleman-Weinberg thing where we sum up all diagrams we get by inserting any number of background fields ϕ2b into the loop integral. This is eq. (56) of that same reference again.
My question is, why do we not need to do this resummation over background field insertions at 2 loops? For example, in this (quite standard) reference, as well as in chapter 11 in Peskin and Schroeder, the authors seem to claim that the 2 loop contribution to the path integral are simply the "rising sun" and "figure 8" vacuum diagrams, and no summing over classical field insertions is even mentioned.
What am I missing?
EDIT:
To give some more details, in perturbation theory, each diagram contributing to the path integral is spacial integral of some functional derivative acting on the free field path integral with a source:
the loop diagram with n insertions of external field ϕb is the term:
(ϕ2b∫dx(δδJ(x))2)nZ0[J]
The 2 loop figure 8 is
∫dx(δδJ(x))4Z0[J]
The 2 loop diagrams that it seems like the papers cited above are excluding are contributions like
(ϕ2b∫dx(δδJ(x))2)n∫dx(δδJ(x))4Z0[J]
It seems to me that these terms will indeed arise in the exponential expansion of the interacting lagrangian, so it seems that a resummation over n, as in the 1 loop case, is still necessary. Where is my error?
This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user bechira