Musings

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

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…