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 ...

  Some questions about the large-N Gross-Neveu-Yukawa model

+ 5 like - 0 dislike
412 views

Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$,

$S = \int d^dx \{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu \sigma )^2 + m^2\sigma^2 }{2g^2 } + \frac{\lambda \sigma^4 }{4!g^4 } ] \} - (N'-1)Trln(\gamma^\mu \partial_\mu + \sigma )$

Assuming that this has a large-N saddle with uniform $\sigma$ one gets the large-N free energy density as,

$E(\sigma) = \Lambda^{d-4}(\frac{m^2\sigma^2}{2g^2} + \frac{\lambda \sigma^4 }{4!g^4 } ) - \frac{N}{2}\int^\Lambda \frac{d^dq}{(2\pi)^d}ln [\frac{q^2 + \sigma^2 }{q^2 } ]$

And the large-N saddle value of $\sigma$ is determined by the large-N gap equation, $E'(\sigma)=0$

Now from here how do the following conclusions come?

  • Firstly that a non-trivial solution to the gap equation exists only when, $\frac{m^2}{g^2} < N\Lambda^{4-d}(\frac{1}{(2\pi)^d} \int^\Lambda \frac{d^dk }{ k^2 } ) $

How does this one come?

  • Secondly from this apparently follows that the inverse $\sigma$ propagator in the massive phase is,

$\Delta_\sigma^{-1}(p) = \Lambda^{d-4}(\frac{p^2}{g^2} + \frac{\lambda \sigma^2}{3g^4} ) + \frac{N(p^2+4\sigma^2) } {2(2\pi)^d}\int^\Lambda \frac{d^dq }{(q^2+\sigma^2)((p+q)^2 + \sigma^2)} $

How does this equation come?

  • Now from this one can show that $\Delta_\sigma \sim \frac{2}{N b(d) p^{d-2} }$

    From the above it follows that the canonical dimension of $\sigma$ is 1. How does one understand that the mass dimension of the field $\sigma$ does not depend on the space-time dimension?

  • Now I don't understand this argument which says that now since $[\sigma] =1$, both the terms $(\partial_\mu \sigma)^2$ and $\sigma^4$ are of dimension $4$ and hence for $2\leq d \leq 4$ these terms vanish in the IR critical theory? For this argument to work was it necessary that the IR theory was critical?

This post imported from StackExchange Physics at 2014-08-11 14:51 (UCT), posted by SE-user user6818
asked Feb 9, 2014 in Theoretical Physics by user6818 (960 points) [ no revision ]
retagged Aug 11, 2014

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
...