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Different OPE channels in bootstrap

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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 (280 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

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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.. 

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