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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,848 answers , 20,603 comments
1,470 users with positive rep
501 active unimported users
More ...

Super-renormalizable theory and $\beta$-function

+ 2 like - 0 dislike
302 views

There is the statement that $\beta$-function vanishes for super-renormalizable theories. In $D=2$, scalar field has mass dimension zero. So any polynomial interaction is super-renormalizable. Then shouldn't all of them have vanishing $\beta$-functions? But there are many theories (e.g, sine-Gordon) in $2D$ which have nontrivial $\beta$-function. I must be missing something very basic here.

This post imported from StackExchange Physics at 2016-09-10 11:18 (UTC), posted by SE-user Physics Moron
asked Sep 10, 2016 in Theoretical Physics by Physics Moron (280 points) [ no revision ]
retagged Sep 10, 2016

There is the statement that β-function vanishes for super-renormalizable theories.

I'm rather skeptical about the statement. What's the context? Can you provide a source or an argument? 

Sine-Gordon has a non-polynomial interaction, hence is not covered by your argument as it stands, independent of whether the ingredients of the argument are valid.

@JiaYiyang First line of Page 770 of this book by Zinn-Justin (4th edition) : http://www.amazon.in/Quantum-Critical-Phenomena-International-Monographs/dp/0198509235 ;
The statement reads : "The theory is super-renormalizable and thus the β-function vanishes."

That would make all super-renormalizable theories scale (and possibly conformal) invariant theories. Is this true?

Yes. That's my confusion. What would be the statement?

As I understand it, super-renormalizable interactions are those with positive mass dimension, that also behave as relevant operators that lead away from a fixed point when following the RG flow towards lower energy scales (?). So to me super-renormalizable theories seem to be rather not scale invariant and I therefore dont see why their $\beta-$ functions should vanish ...

1 Answer

+ 1 like - 0 dislike

I still haven't got hold of Zinn-Justin's book mentioned in the comment, but here is a plausible/consistent interpretation of what that statement might mean. Generally one has to distinguish two kinds of RG flows and thus two kinds of beta functions, one is about pushing the UV cutoff to infinity and defining a QFT (i.e. UV completion), the other is about coarse graining the theory to gain IR information (Wilson-Kandanoff-Fisher).  The former slides a unphysical cutoff scale $\Lambda$ and the latter slides a physical probing/experimental scale $\mu$, and the beta functions are respectively denoted by $\beta_\text{UV}(g_\text{bare}(\Lambda))$ and $\beta_\text{Wilson}(g_\text{ren}(\mu))$.

The two are perturbatively very similar when one has dimensionless couplings ( see the "Update" section of this answer. In fact Wilson and Kogut only called the latter kind "RG trajectories", and the former "canonical curves"), but no a priori relation exists for general couplings.

For super-renomalizable theories, UV-finiteness require $g_\text{bare}(\Lambda)=g_0+o(1)$ (small o notation, with respect to $\Lambda$, and $g_0$ is independent of $\Lambda$), so the beta function in this sense must be 0 near $\Lambda=\infty$, and it might be this beta function that Zinn-Justin was talking about. On the other hand, since the coupling is dimensionful, we can almost be sure that the theory is not scale invariant hence the beta function in the sense of Wilson is not 0.

answered Aug 5 by Jia Yiyang (2,465 points) [ revision history ]
edited Aug 5 by Jia Yiyang

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$\hbar$ysicsOv$\varnothing$rflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...