Different OPE channels in bootstrap

Question

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

Comments

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.

Answer 1

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.

Comments

@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..

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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).

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