Coleman-Weinberg potential: resum at 2 loops?

Question

Say we want to compute the Coleman-Weinberg potential at 2 loops.

The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + \phi$, and do a path integral over the quantum part of the field, $\phi$.

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 $\phi_b^2$ 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 $\phi_b$ is the term: $$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n Z_0[J]$$

The 2 loop figure 8 is

$$\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$

The 2 loop diagrams that it seems like the papers cited above are excluding are contributions like

$$\left( \phi_b^2 \int dx \left( \frac {\delta}{\delta J(x)}\right)^2 \right)^n\int dx \left( \frac {\delta}{\delta J(x)}\right)^4 Z_0[J]$$

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?

Answer 1

When computing the effective potential $V(\phi)$ in the ordered phase ($\phi_b>0$), one has to use the classical propagator $G_c[\phi_b]$ given by the inverse of $$\frac{\delta^2 S[\phi]}{\delta\phi^2},$$ which is a functional of $\phi$. The vacuum energy is given by $V(\phi_b)$, where one should remember that $\phi_b$ is also computed consistently in perturbation by $$V'(\phi_b)=0.$$ Using the classical propagator is equivalent to a consistent resummation of the $\phi_b^2$ to all order. In particular, the effective action at two-loops is given by $$\Gamma[\phi]=S[\phi]+\frac{1}{2}Tr \log G_c[\phi]+{\rm 2-loops\; diagrams},$$ where the 2-loops diagrams are the 8 and the rising sun, that have to be computed using the classical propagator.

Answer 2

Indeed you also resum at n>1 loop order.

The thing is that at 1-loop this resummation corresponds to having logarithmic form, while at n>1, it corresponds to using "dressed", massive propagators.

It all comes down to some exciting combinatorics, for instance, see Appendix A in the original paper of Coleman, Weinberg "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking"