An Early Route to MHV Tree Amplitudes

In the perturbative expansion of a gauge theory, large numbers of Feynman amplitudes combine to produce mathematically simple (and elegant) expressions. So many people had long suspected that deeper symmetry structures were involved, but those structures remained tantalizingly beyond reach until recently. The Search and Discovery story in the July 2004 issue of Physics Today (page 19) gives a lucid and detailed explanation of the evolving understanding of symmetry structures that underlie Feynman amplitudes. Identification of the connection between the maximally helicity-violating (MHV) amplitudes for the N = 4 super Yang–Mills theory and supertwistor space was a crucial ingredient in the recent developments.

I would like to share the reasoning that led me, some years ago, to make an early proposal, ¹ which unfortunately was not mentioned in the Physics Today story. I noticed that the Parke–Taylor formula for the MHV amplitudes involved the inverse of scalar products of spinor momenta that could be related to free fermion propagators (or current correlations of a Wess-Zumino-Witten theory) on the complex projective space CP ¹. The space arises naturally (as the fiber) in twistor space. Earlier, Edward Witten had observed that the constraints defining the N = 4 super Yang–Mills theory could be nicely interpreted in supertwistor space. ² Putting these observations together, I wrote a formula for MHV amplitudes in N = 4 Yang–Mills theory in twistor language. Conformal symmetry plays an important role. Witten’s recent work is more comprehensive, generalizing this complicated series of relationships to string theory and non-MHV amplitudes, and leading to many beautiful results that, for the special case of MHV amplitudes, are identical to mine.

The work I’ve just described may be useful for practical calculations; but beyond that use, I hope there will emerge a new organizational principle for perturbation theory other than expansion in terms of Feynman amplitudes.