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We construct a twistor space action for N = 4 super Yang-Mills theory and show that it is equivalent to its four dimensional spacetime counterpart at the level of perturbation theory. We compare our partition function to the original twistor-string proposal, showing that although our theory is closely related to string theory, it is free from conformal supergravity. We also provide twistor actions for gauge theories with N < 4 supersymmetry, and show how matter multiplets may be coupled to the gauge sector.

Recently a powerful duality between color and kinematics has been proposed for integrands of scattering amplitudes in quite general gauge theories. In this paper the duality proposal is extended to the more general class of gauge theory observables formed by form factors. After a discussion of the general setup the existence of the duality is verified in twoand three-loop examples in four dimensional maximally supersymmetric Yang-Mills theory which involve the stress energy tensor multiplet. In these cases the duality reproduces known results in a particularly transparent and uniform way. As a non-trivial application we obtain a very simple form of the integrand of the four-loop two-point (Sudakov) form factor which passes a large set of unitarity cut checks.

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

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In this article a first step is made towards the extension of Britto-CachazoFeng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their 'single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choice employed earlier at tree level by Arkani-Hamed and Kaplan. The relation between shifts of integrands and shifts of its integrals is investigated explicitly at one loop. Two particular sources of discrepancy between the integral and integrand are identified related to UV and IR divergences. This is cross-checked with known results for helicity equal amplitudes at one loop. The nature of the on-shell residue at each of the single-cut singularities of the integrand is commented upon. Several natural conjectures and opportunities for further research present themselves.