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

146 submissions , 123 unreviewed
3,953 questions , 1,403 unanswered
4,889 answers , 20,762 comments
1,470 users with positive rep
507 active unimported users
More ...

Do fixed points of the RG flow field depend on the renormalization scheme applied?

+ 3 like - 0 dislike
215 views

In my textbook from p 438 on it is explained, that for example for the Ginzburg-Landau model defined by the effective Lagrangian

\[\mathcal{L}_{eff} = \mathcal{L}_{kin} -\frac{1}{2}(T-T_C)\phi^2 -\frac{1}{4!}\lambda\phi^4 +\cdots\]

the position of the  non-trivial (Wilson-Fisher) fixed point which is found in this case by an $\epsilon$ expansion and dimensional regularization
\[\lambda_* =\frac{16\pi^2\epsilon}{3}, \quad m^2_* = 0\]

depends on the RG scheme applied and is therefore not physical. Only the critical exponents (that describe the behavior of the RG flow near the fixed point) are universal.

Is it always the case that the position of fixed points (or even more generally the structure) of the RG flow field depends on the renormalization scheme applied?

Concerning the fixed points, I dont understand why their position (and maybe even their presence or absence?) should depend on the scheme, as I always thought that fixed points of the RG flow corresponds to a scale (or even conformal) invariant theory, their basins of attraction define universality classes etc, so they should be physical and not depend on the exact renormalization method (scheme) applied?

asked Jul 10, 2015 in Theoretical Physics by Dilaton (4,305 points) [ no revision ]

At the end of the calculation, shouldn't you set $\epsilon$ to a specific value (say 0 or -1)?

@RyanThorngren in this calculation of the fixed point $d = 4 - \epsilon$ is assumed, with $0 < \epsilon \ll 1$

The dimreg values are only physical after fixing epsilon, usually to zero, at the end of the computation.

1 Answer

+ 3 like - 0 dislike

The beta function (and others running functions) does indeed depend on the renormalization scheme (RS), as the renormalized parameters (couplings, masses) are different for distinct schemes. Thus, fixed points have different expressions in distinct schemes. 

There are however theorems that guarantee that the beta function is the same for some set of RSs up to some given order in perturbations theory. For example, the beta function is the same for all mass-independent RSs at first and second order in perturbation theory. The proof only involves two facts: in a mass-independent RS the beta function is just a power series in the coupling constant and all coupling constants are the same at tree level.

answered Jul 14, 2015 by drake (885 points) [ no revision ]

Thanks Drake!

Can you also explain what using different RSs means in the RG flow point of view, when for example looking at a two-dimensional parameter space spanned by the mass and a coupling constant? Does applying different RSs in this picture mean, that you approach a fixed point when going from higher to lower energy along different trajectories of the RG flow, such that when you apply a mass independent RS you follow a trajectory that is orthogonal to the mass axis?

@Dilaton, I don't have access to your textbook so I don't claim to understand the context of your question fully. However, a likely explanation is that, in different schemes the definitions of a coupling constant may differ. For example, in a phi^4 theory one may define $\lambda$ as the scattering amplitude of $\phi\phi \to \phi\phi$ at any external momenta(even off-shell momenta are acceptable! For details see Weinberg Vol1 page 515), hence you may say $\lambda$ can be defined differently, and all this does not matter as long as you have the same S-matrix in the end. Thus it simply doesn't make much sense to draw the RG trajectories from different schemes(if they define couplings differently) on the same graph.

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$\hbar$y$\varnothing$icsOverflow
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
...