The Smatrix in twistor space
Abstract
The marvelous simplicity and remarkable hidden symmetries recently uncovered in (Super) YangMills and (Super)Gravity scattering amplitudes strongly suggests the existence of a “weakweak” dual formulation of these theories where these structures are made more manifest at the expense of manifest locality. In this note we suggest that in four dimensions, this dual description lives in (2, 2) signature and is naturally formulated in twistor space. We begin at treelevel, by recasting the momentumspace BCFW recursion relation in a completely onshell form that begs to be transformed into twistor space. Our transformation is strongly inspired by Witten’s twistor string theory, but differs in treating twistor and dual twistor variables on a more equal footing; a related transcription of the BCFW formula using only twistor space variables has been carried out independently by Mason and Skinner. Using both twistor and dual twistor variables, the three and fourpoint amplitudes are strikingly simplefor YangMills theories they are “1” or “1”. The BCFW computation of higherorder amplitudes can be represented by a simple set of diagrammatic rules, concretely realizing Penrose’s program of relating “twistor diagrams” to scattering amplitudes. More specifically, we give a precise definition of the twistor diagram formalism developed over the past few years by Andrew Hodges. The “Hodges diagram” representation of the BCFW rules allows us to compute amplitudes and study their remarkable properties in twistor space. For instance the diagrams for YangMills theory are topologically disks and not trees, and reveal striking connections between amplitudes that are not manifest in momentum space. Twistor space also suggests a new representation of the amplitudes directly in momentum space, that is naturally determined by the Hodges diagrams. The BCFW rules and Hodges diagrams also enable a systematic twistorial formulation of gravity. All tree amplitudes can be combined into an “SMatrix” scattering functional which is the natural holographic observable in asymptotically flat space; the BCFW formula turns into a simple quadratic equation for this “SMatrix” in twistor space, providing a holographic description of mathcal{N} = 4 SYM and mathcal{N} = 8 Supergravity at tree level. We move on to initiate the exploration of loop amplitudes in (2, 2) signature and twistor space, beginning with a discussion of their IR behavior. We find that the natural pole prescriptions needed for transformation to twistor space make the amplitudes perfectly welldefined objects, free of IR divergences. Indeed in momentum space, the loop amplitudes so regulated vanish for generic momenta, and transformed to twistor space, are even simpler than their treelevel counterparts: the full 4pt oneloop amplitudes in mathcal{N} = 4 SYM are simply equal to “1” or “0”! This further supports the idea that there exists a sharply defined object corresponding to the SMatrix in (2, 2) signature, computed by a dual theory naturally living in twistor space.
 Publication:

Journal of High Energy Physics
 Pub Date:
 March 2010
 DOI:
 10.1007/JHEP03(2010)110
 arXiv:
 arXiv:0903.2110
 Bibcode:
 2010JHEP...03..110A
 Keywords:

 Supersymmetric gauge theory;
 Duality in Gauge Field Theories;
 Classical Theories of Gravity;
 High Energy Physics  Theory
 EPrint:
 V1: 46 pages + 23 figures. Less telegraphic abstract in the body of the paper. V2: 49 pages + 24 figures. Largely expanded set of references included. Some diagrammatic clarifications added, minor typo fixed