TWISTOR DIAGRAMS

Recent papers by Andrew Hodges

Twistor diagram for the interaction of three positive-helicity and three negative-helicity gluons in tree-level QCD

My area of research has been greatly advanced since the end of 2003, when Ed Witten explained how a twistor string model could describe gauge-theoretic interactions.

The possibility of a twistor-string connection had been raised long before. In fact in a talk of September 1988, I showed results on gauge field scattering which anticipate the work I am doing now. This was published in The interface of mathematics and particle physics, eds. D. G. Quillen, G. B. Segal and Tsou S. T. (Clarendon Press, 1990). I have prepared a .pdf file of this article (108Kb). Download here.

These ideas were pursued throughout the 1990s, with the collaboration of several graduate students. The results of this work will eventually all be useful, but what we missed in that period was that there are amazing simplifications to be found in the theory of pure gauge fields, as had in fact been shown in the 1986 work of Parke and Taylor. Another factor is that it was not until about 2000 that the gauge theory of the SU(3) strong interaction could be taken as a completely valid arena for scattering theory. In early 2004, when I caught up with what Witten explained, I could see that the twistor diagram structures I had studied in the 1990s were highly relevant both to the general theory and to practical calculations in gauge-theoretic scattering.

I gave a presentation at the Oxford conference on strings and twistors, January 2005. It is in pdf form (220Kb). Click here to download. (For the other presentations at this conference go here.) My presentation ended with a conjecture about the representation of more advanced scattering processes which is now completely superseded by a much more general and powerful observation.

This is because during February 2005 I saw how I could use the recursion relation recently discovered by Ruth Britto, Freddy Cachazo and Bo Feng. By translating it into twistor diagrams, which turned out to be completely natural, I could represent all the tree-level amplitudes of gauge theory as twistor diagrams. For a first paper on this development, Twistor diagram recursion for all gauge-theoretic tree amplitudes, 7 March 2005, download a .pdf (336Kb) from arXiv at http://arXiv.org/abs/hep-th/0503060.

Results of the 'quadruple cut' analysis of one-loop diagrams (which was also first worked out by the BCF group) are also naturally expressed in twistor diagrams. Some of this material appeared in the talk I gave at the Queen Mary workshop, London University, 3 November 2005. You can download a pdf file of the presentation I gave by going to this site.

At the end of 2005, however, I saw an important new line of development, which extends the BCF recursion relation by making it helicity-independent. (It uses super-symmetric twistors, making the formalism manifestly super-convariant.) From a computational point of view this helps by bringing all the various helicity patterns into one formula. For a first paper on this development, Twistor diagrams for all tree amplitudes in gauge theory: a helicity-independent formalism, 29 December 2005, download a .pdf (204Kb) from arXiv at http://arxiv.org/abs/hep-th/0512336.

A second paper appeared on 13 March 2006, Scattering amplitudes for eight gauge fields, applying this new formalism to a particular problem. As far as I know, no-one else had worked out all the 8-field helicity-conserved amplitudes before, because it looked too complicated. The new formalism simplifies the problem a lot, by bringing all the different helicity patterns into a common form. But it also shows that there are further simplifications to be found. Download from arXiv at http://arxiv.org/abs/hep-th/0603101.

I spoke about this work at the workshop on "Twistors, Strings, Gauge Theory and Gravity" organized by Freddy Cachazo at the Perimeter Institute, Waterloo, Ontario, 10-13 September 2006.

I gave a review talk at the London Mathematical Society Durham Symposium on Twistors, Strings and Scattering Amplitudes, 19-26 August 2007. My presentation is available on that site, along with all the other talks, or you can download it here.

Very few people seem to have taken any notice of this, presumably because the diagram formalism is so unfamiliar, but in December 2008 I was invited to another Workshop organized by Freddy Cachazo at the Perimeter Institute. One particular reason for holding this workshop was that Freddy Cachazo had been studying my diagram calculus with Nima Arkani-Hamed and his colleagues, and my work had assisted them in arriving at some new and very interesting results. Their first paper, The S-matrix in Twistor Space, by Nima Arkani-Hamed, Freddy Cachazo, Clifford Cheung, and Jared Kaplan, appeared in March 2009 and is available on arXiv here.

After the International Workshop on gauge and string amplitudes, at Durham University, UK, 30 March - 2 April 2009, I saw that the new geometrical framework of 'dual conformal invariance' could also be best represented with twistor coordinates, and that this would make it possible to eliminate the 'spurious poles' which have plagued the description of gauge-theoretic amplitudes. A new paper, Eliminating spurious poles from gauge theoretic amplitudes, appeared on 11 May 2009. Download from arXiv at http://arxiv.org/abs/0905.1473.

I participated in a workshop on Hidden structures in field theory amplitudes at the Niels Bohr Institute, Copenhagen, 12-14 August 2009. A pdf file of my presentation is available on that site, along with all the others.

Forthcoming: I expect to make an extended visit to the Institute for Advanced Study, Princeton, N.J. in spring 2010.

Twistor diagrams home page

Contact email: andrew[AT]synth.co.uk