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
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Anomalously broken conformal symmetry

+ 6 like - 0 dislike
1126 views

I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential contains $\mu^2\phi^2$.

Bardeen argues that

  1. Setting $\mu^2=0$ would increase the symmetry of the Standard Model by a conformal symmetry.

    But the things we can measure - which are basically the Green's functions - are no more symmetric with $\mu^2=0$ than with $\mu^2\neq0$! Because the symmetry is anomalously broken.

  2. The conformal symmetry is broken anomalously by non-zero beta-functions.

  3. Quadratic divergences do not contribute to the beta-functions.

    Whilst its true that they don't contribute to the running of the renormalised parameters, they contribute to the running of the bare mass.

  4. Quadratic divergences must therefore be a separate (explicit) source of symmetry breaking.

    Not true if you consider that quadratic divergences contribute to the running of the bare mass.

  5. Quadratic divergences must therefore be an non-physical artefact of our re-normalisation procedure, and we must remove them with, for example, counter-terms, and this is not a fine-tuning.

I have added my comments. My main question is,

If conformal symmetry is broken anomalously, why should our Lagrangian respect conformal symmetry? The Green's function are not more symmetric with $\mu^2=0$. I am interested in this for any symmetry, but especially conformal symmetry.

Also, I don't understand 3. and 4. Quadratic divergences would contribute to the running of the bare mass. Wouldn't that break conformal symmetry anomalously? or is it only the renormalised parameters that must have vanishing beta-functions? The distinction seems artificial. I can't understand why quadratic divergences are an explicit source of symmetry breaking, whereas logarithms etc are anomalous. This is key to solving the naturalness problem, and I can't follow the argument.

My feeling is that these arguments are faulty (which makes me think I must be making mistakes because Bardeen is a real expert who has surely thought a lot about it!). I'm certainly unconvinced. Have they been confirmed/refuted at length in the literature?

This post imported from StackExchange Physics at 2014-09-19 08:51 (UCT), posted by SE-user innisfree
asked Mar 27, 2014 in Theoretical Physics by innisfree (295 points) [ no revision ]
retagged Sep 19, 2014
Conformal symmetry is not a symmetry at all in a theory with a conformal anomaly. I think I pretty much agree with all of your comments.

This post imported from StackExchange Physics at 2014-09-19 08:51 (UCT), posted by SE-user Matt Reece
@matt thanks that gives me confidence :-) if you have any other corrections/insights/thoughts on this it would be great to hear them.

This post imported from StackExchange Physics at 2014-09-19 08:51 (UCT), posted by SE-user innisfree
the paper is only a write up of a talk, not in a journal or even on the arxiv. I can't find a follow up paper. did Bardeen also suspect the argument was faulty and abandon it?

This post imported from StackExchange Physics at 2014-09-19 08:51 (UCT), posted by SE-user innisfree

1 Answer

+ 1 like - 0 dislike

Unfortunately, Bardeen seems to misunderstand the naturalness problem that has nothing to do with quadratic divergences per se. In the strict SM, there is no naturalness problem because the running Higgs mass squared is proportional to itself. But this is not the setup that people care about when talking about the actual naturalness problem that emerges instead as soon as one extends the SM to include some new scale $\Lambda$ (where new particles start propagating). Any such a deformation of the UV of the SM introduce the naturalness problem, and you can not screen the Higgs mass from large corrections $\sim\Lambda^2$ by invoking conformal symmetry simply because it is not a good symmetry at $\Lambda$ whose very existence represents in fact a large explicit symmetry breaking. It is this UV sensitivity of the SM, through the Higgs potential, that is the actual naturalness problem.

This post imported from StackExchange Physics at 2014-09-19 08:51 (UCT), posted by SE-user TwoBs
answered Apr 6, 2014 by TwoBs (315 points) [ no revision ]

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$\hbar$ysics$\varnothing$verflow
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
...