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MHV diagrams in twistor space and the twistor action

MHV diagrams give an efficient Feynman diagram-like formalism for calculating gauge theory scattering amplitudes on momentum space. Although they arise as the Feynman diagrams from an action on twistor space in an axial gauge, the main ingredients were previously expressed only in momentum space and momentum twistor space. Here we show how the formalism can be elegantly derived and expressed entirely in twistor space. This brings out the underlying superconformal invariance of the framework (up to the choice of a reference twistor used to define the axial gauge) and makes the twistor support transparent. Our treatment is largely independent of signature, although we focus on Lorentz signature. Starting from the N=4 super-Yang-Mills twistor action, we obtain the propagator for the anti-holomorphic Dolbeault-operator as a delta function imposing collinear support with the reference twistor defining the axial gauge. The MHV vertices are also expressed in terms of similar delta functions. We obtain concrete formulae for tree-level N^{k}MHV diagrams as a product of MHV amplitudes with an R-invariant for each propagator; here the R-invariant manifests superconformal as opposed to dual-superconformal invariance. This gives the expected explicit support on k+1 lines linked by k further lines associated to the propagators. The R-invariants arising correspond to those obtained in the dual conformal invariant momentum twistor version of the formalism, but differences arise in the specification of the boundary terms. Surprisingly, in this framework, some finite loop integrals can be performed as simply as those for tree diagrams.