September 27, 2004

Collinear

A lot of people (myself included) got very excited by the fact that perturbative $N=4$ super Yang-Mills amplitudes seemed to take a very simple form when written in (super)-Twistor space and that, moreover, the tree-level amplitudes can be recovered very elegantly from a topological string theory with target space the aforementioned super-Twistor space. But my ardour cooled considerably when it became apparent that, when one went to the one-loop level in the Yang-Mills, the aforementioned topological string theory would produce not just super Yang-Mills, but super Yang-Mills coupled to conformal supergravity.

Moreover, it appeared that the known one-loop amplitudes were not easily interpretable in terms of a twistor string theory. One could easily identify contributions in which the external gluons are supported on

1. a pair of lines in twistor space (connected by two twistor-space “propagators”)
2. a degree-two genus-zero curve (with a single twistor-space “propagator”)
3. $(n-1)$ of the gluons inserted as above, but with the $n^{th}$ gluon inserted somewhere in the same plane as the rest.

This last type of contribution is hard to reconcile with some sort of twistor string theory.

It now appears that this pessimistic conclusion was a bit too hasty. Cachazo Svrček and Witten have traced the problem in their earlier analysis to a sort of “holomorphic anomaly.” Their criterion for collinearity in twistor space was that the amplitude should obey a certain differential equation. However, the differential operator in question, rather than annihilating the amplitude, give a $\delta$-function whenever the momentum on an internal line is parallel to one of the external gluon momenta. It’s just a glorified version of

(1)$\overline{\partial} \frac{1}{z} = 2\pi \delta^{(2)} (z)$

The amplitude “really” receives contributions only of types (1) and (2). The apparent contributions of type (3) come from exceptional points in the integration over loop momenta, where an internal momentum is collinear with one of the external gluons.

I wish I’d thought of that…

Posted by distler at September 27, 2004 2:31 AM

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String theory

The way I saw things, Wittens string theory on $\mathbb{C}P^{3|4}$ and Berkovits alternative string theory are two different theories. There are certainly many technical differences.

But you make it sound like they are the same. Are you really saying that, and on what grounds?

Posted by: Volker Braun on September 27, 2004 8:32 AM | Permalink | Reply to this

Berkovits and Witten

They’re not manifestly the same. But B&W certainly imply that a similar conclusion holds in Witten’s theory. I thought the point of Cachazo et al was to examine the known 1-loop results and try to divine from them a set of Feynman rules for a new twistor string theory.

The hunt for that would seem to be on again…

Posted by: Jacques Distler on September 27, 2004 9:01 AM | Permalink | PGP Sig | Reply to this

Re: Berkovits and Witten

Hi Jacques,

Mysteriously, the same CSW rules seem to work for loop amplitudes, at least the simplest ones (MHV in N=4), as shown by Brandhuber et. al. Their paper is very nice, and seems to show a connection between the off-shell continuation in CSW and the original ways of deriving these amplitudes, using cuts and collinear limits. The relation to some twistor string theory is less clear, I guess.

Posted by: Moshe Rozali on September 27, 2004 4:17 PM | Permalink | Reply to this

Re: Berkovits and Witten

That’ll teach me to get behind in my reading!

Brandhuber, Spence & Travaglini do, indeed, show that — contra what appears to follow from the earlier CSW paper — the one-loop MHV amplitudes are reproduced by sewing together tree-level MHV amplitudes (ie, contributions of type “1” above).

The current CSW paper reconciles this result with their previous analysis.

Posted by: Jacques Distler on September 27, 2004 4:38 PM | Permalink | PGP Sig | Reply to this

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