From twistor actions to MHV diagrams

Abstract

We show that MHV diagrams are the Feynman diagrams of certain twistor actions for gauge theories in an axial gauge. The gauge symmetry of the twistor action is larger than that on space-time and this allows us to fix a gauge that makes the MHV formalism manifest but which is inaccessible from space-time. The framework is extended to describe matter fields: as an illustration we explicitly construct twistor actions for an adjoint scalar with arbitrary polynomial potential and a fermion in the fundamental representation and show how this leads to additional towers of MHV vertices in the MHV diagram formalism.An important outcome of Witten's twistor-string theory [1] is the "MHV formalism" [2], in which scattering amplitudes in four dimensional Yang-Mills theory are described in terms of diagrams whose vertices are the MHV vertices with two positive and arbitrarily many negative helicity gluons. Much work (see e.g. [3,4,5,6]) has since gone into developing unitarity methods to compute loop amplitudes, or on extending the MHV formalism to include matter coupled to the Yang-Mills field [7,8].A key question here is the connection between these twistor-inspired developments and the usual, Lagrangian-based approach to gauge theory. Lagrangians have the obvious advantages of leading to a systematic perturbation theory including loops, and of making manifest the symmetry properties of a theory. Progress towards a full derivation of the MHV formalism from the standard space-time Lagrangian in lightcone gauge has been made in [9,10,11,12,13]. The key idea in these approaches is to find new field variables in which the anti-self-dual sector is linearized. These new field variables are related to the old ones by non-linear and non-local field transformations, expressed in the form of an infinite series. The possibility of such a transformation of the anti-self-dual sector into a free theory relies, in effect, on the

Wilson Loop Diagrams and Positroids

“…Z n } are chosen such that any ordered subset of 4 + k twistors defines a positive volume. [(2, 6), [8]…”

“…Scattering amplitudes for N = 4 SYM can be obtained from a formulation of the theory via an action in twistor space that yields MHV diagrams as the Feynman diagrams [8,9]. In a separate development, it emerges that amplitudes can be obtained in a completely different way as certain correlation functions of Wilson loops in planar N = 4 SYM [3].…”

MHV, CSW and BCFW: field theory structures in string theory amplitudes

“…However, there may very well be some sense to be made of them. It should be noted though that in field theory it is not straightforward to derive the vanishing of the appropriate residues directly [41], despite the fact that this is known to be true from an independent action-based derivation of the N = 4 CSW rules [22,42] for all types of external particles.…”

CSW rules for massive matter legs and glue loops

“…Coupling the fermion action (18) to Yang-Mills gauge fields follows the lines of the appendix in [5] which leads to the results of [16]. Basically, the difference with the scalar action is the difference in weight on the twistor space of the fields.…”

“…Similar twistor techniques as worked out for scalars can also be applied to fermions 3 as in [16] where the action was obtained. However, there the field transformations for the fermions were not written explicitly.…”