Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Coleman-Weinberg potential: resum at 2 loops?

+ 5 like - 0 dislike
2280 views

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?

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user bechira
asked Jul 19, 2014 in Theoretical Physics by bechira (80 points) [ no revision ]
@Qmechanic, thanks for the edit but the wikipedia article you linked seems to define the Coleman Weinberg potential as the potential of the QED lagrangian, however in Coleman and Weinberg's classic paper they apply their analysis also to simpler, scalar field theories. Here I'm clearly referring to such a potential for a scalar field theory.

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user bechira
HI @bechira: OK, removed the wiki link again. Consider adding more details to make post more accessible. Also consider to add author, title, etc., of links so that links can be reconstructed in case of link rot.

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user Qmechanic
Bechira, in all the cases, one simply writes down all the Feynman diagrams with the appropriate external lines containing the allowed vertices. If the insertions are zero, they're either guaranteed to vanish, or assumed to be subtracted in a different way before you write down the action for the quantum part.

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user Lubo Motl
@LuboMotl yes of course, the question is that it is not obvious to me that the diagrams you get from say, inserting a background field on a line in the setting sun diagram, vanishes.

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user bechira
@Qmechanic Thanks, I have added some details to break the problem down further.

This post imported from StackExchange Physics at 2014-07-22 07:28 (UCT), posted by SE-user bechira

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ys$\varnothing$csOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...