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 ...

  Different OPE channels in bootstrap

+ 2 like - 0 dislike
953 views

Can someone quickly explain what exactly are those different channels (namely s,t,u) in OPE expansions frequently used in conformal bootstrap. Explanation with a simple example will be really helpful.

This post imported from StackExchange Physics at 2015-11-18 15:17 (UTC), posted by SE-user Physics Moron
asked Nov 17, 2015 in Theoretical Physics by Physics Moron (285 points) [ no revision ]

Could these be the Mandelstem variables (or channels) for two particle to two particle scattering,  that are denoted u,s, and t too?

If not, it would probably help to answer the question, if you could provide a specific example of how they appear in an OPE.

1 Answer

+ 4 like - 0 dislike

The different channels in bootstrap always refer to a four-point function; for simplicity we can take a scalar primary $\phi$, so the object of interest is

$$\langle \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \rangle.$$

Up to a scaling factor, this equals a function of two cross ratios $u$ and $v$, say $g(u,v)$. To analyze $g(u,v)$, we can pick any conformal frame we like. For now, we'll map $x_1,x_3,x_4$ to

$$x_1 = 0, \quad x_3 = (1,0,\ldots,0), \quad x_4 = \infty.$$

By a suitable rotation, we can always assume that $x_2$ lies on the plane spanned by the first two unit vectors. Let's parametrize this plane by a complex coordinate $z$ with conjugate $\bar{z}$, so $x_2 \equiv z$.

We now use the fact that the OPE $$\phi(y_1) \phi(y_2) \sim  \sum_i c_{\phi \phi i} \mathcal{O}_k(y_2) $$ converges inside a correlator $\langle \phi(y_1) \phi(y_2) \ldots \rangle$ if there's a sphere separating $y_1$ and $y_2$ from all other operator insertions. This is a consequence of radial quantisation. Let's apply this to the four-point function

$$  \langle \phi(0) \phi(z) \phi(1) \phi(\infty) \rangle \sim g(z,\bar{z}).$$

In passing, we've changed coordinates from $u,v$ to $z,\bar{z}$.

If $|z| < 1$ both the OPEs $\phi(0) \phi(z)$ and $\phi(1) \phi(\infty)$ converge at the same time. This means that that the four-point function admits a conformal block expansion

$$ g(z,\bar{z}) \sim \sum_i c_{\phi \phi i}^2 G_i(z,\bar{z}) $$

where $G_i(z,\bar{z})$ is the conformal block corresponding to the primary operator $\mathcal{O}_i$. We can call this the s-channel.

Now consider a different case: the disk $|z-1| < 1$. In this case the OPEs $\phi(z)\phi(1)$ and $\phi(0)\phi(\infty)$ converge. This is a different double OPE expansion. However, it should give back the same four-point function. In a formula:

$$ g(z,\bar{z}) \sim  \sum_i c_{\phi \phi i}^2 G_i(1-z,1-\bar{z}).$$

This is a different channel, let's call it the t-channel.

Finally you can consider a third channel, when $z$ is far away from 0 and 1.

The magic of the (numerical) conformal bootstrap relies on the fact that if $z$ lies in the intersection of the disks $|z|<1$ and $|z-1| < 1$, the s and t channels converge at the same time. This gives you constraints on the spectrum without knowing $g(z,\bar{z})$ explicitly.

I want to stress that there is no consistent naming scheme for the different channels. So don't focus too much on the name "s-channel" or "t-channel" since different authors may use different names; it should be clear from the context what they mean. Also the $z$ coordinate above is nothing special: there are infinitely many other ones (other conformal frames) you can use.

I want to end with the message that by now there are many sets of lecture notes, MSc and PhD theses discussing the bootstrap program. With a quick Google search you will find a lot of pedagogical material. There are many too many tiny details that cannot be explained in a single forum post.

answered Dec 1, 2015 by Bootstrapper [ no revision ]

@Bootstrapper  Thanks a lot for this detailed answer! Can you please refer to some of those pedagogical materials which you find very basic and well written..

There are three sets of notes you can find online: at Slava Rychkov's website, Jared Kaplan's "AdS/CFT from the Bottom Up" (strong focus on AdS, but you can skip that part) and David Simmons-Duffin's notes for TASI 2015. Finally there's Alessandro Vichi's PhD thesis from 2011 ( background-position: 0px 14px; background-repeat: repeat no-repeat;">doi:10.5075/epfl-thesis-5116). There may be more material out there (I only learned of Simmons-Duffin's notes just now).

@Bootstrapper Thanks a lot! References are really useful.. 

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$\hbar$ysic$\varnothing$Overflow
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
...