Recently, a canonical change of field variables was proposed that converts the Yang-Mills Lagrangian into an MHV-rules Lagrangian, i.e. one whose tree level Feynman diagram expansion generates CSW rules. We solve the relations defining the canonical transformation, to all orders of expansion in the new fields, yielding simple explicit holomorphic expressions for the expansion coefficients. We use these to confirm explicitly that the three, four and five point vertices are proportional to MHV amplitudes with the correct coefficient, as expected. We point out several consequences of this framework, and initiate a study of its implications for MHV rules at the quantum level. In particular, we investigate the wavefunction matching factors implied by the Equivalence Theorem at one loop, and show that they may be taken to vanish in dimensional regularisation.

We demonstrate that the canonical change of variables that yields the MHV lagrangian, also provides contributions to scattering amplitudes that evade the equivalence theorem. This 'ET evasion' in particular provides the tree-level (−++) amplitude, which is non-vanishing off shell, or on shell with complex momenta or in (2, 2) signature, and is missing from the MHV (a.k.a. CSW) rules. At one loop there are ET-evading diagrammatic contributions to the amplitudes with all positive helicities. We supply the necessary regularisation in order to define these contributions (and quantum MHV methods in general) by starting from the light-cone Yang-Mills lagrangian in D dimensions and making a canonical change of variables for all D − 2 transverse degrees of freedom of the gauge field. In this way, we obtain dimensionally regularised three-and four-point MHV amplitudes. Returning to the one-loop (++++) amplitude, we demonstrate that its quadruple cut coincides with the known result, and show how the original light-cone Yang-Mills contributions can in fact be algebraically recovered from the ET-evading contributions. We conclude that the canonical MHV lagrangian, supplemented with the extra terms brought to correlation functions by the non-linear field transformation, provide contributions which are just a rearrangement of those from light-cone Yang-Mills and thus coincide with them both on and off shell.

We characterise the one-loop amplitudes for N = 6 and N = 4 supergravity in four dimensions. For N = 6 we find that the one-loop n-point amplitudes can be expanded in terms of scalar box and triangle functions only. This simplification is consistent with a loop momentum power count of n − 3, which we would interpret as being n + 4 for gravity with a further −7 from the N = 6 superalgebra. For N = 4 we find that the amplitude is consistent with a loop momentum power count of n, which we would interpret as being n + 4 for gravity with a further −4 from the N = 4 superalgebra. Specifically the N = 4 amplitudes contain non-cut-constructible rational terms.

We perform a canonical change of the field variables of light-cone gauge massless QCD to obtain a lagrangian whose terms are proportional up to polarisation factors to MHV amplitudes and continued off shell by the CSW prescription. We solve for this transformation as a series expansion to all orders in the new fields, and use this to prove that the resulting vertices are indeed MHV vertices as claimed. We also demonstrate how this works explicitly for the vertices with: two quarks and two gluons, four quarks, and a particular helicity configuration of two quarks and three gluons. Finally, we generalise the construction to massive QCD.

We present a semi-recursive method for calculating the rational parts of one-loop gravity amplitudes which utilises axial gauge diagrams to determine the non-factorising pieces of the amplitude. This method is used to compute the amplitudes M 1-loop