• 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.


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


(propose a free ad)

Site Statistics

132 submissions , 111 unreviewed
3,777 questions , 1,329 unanswered
4,725 answers , 19,953 comments
1,470 users with positive rep
468 active unimported users
More ...

Super-renormalizable theory and $\beta$-function

+ 2 like - 0 dislike

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 ...

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:
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