Generalisations of the MHV lagrangian theory to D-dimensions and to super-space

Abstract

It has been known that the CSW rules correctly reproduce all tree-level scattering amplitudes for perturbative non-abelian gauge theory but fail to explain some of the loop order results. In this thesis we generalise the lagrangian derivation of the rules and account for the missing amplitudes in dimensional regularisation scheme. We analyse generically when the equivalence between MHV rules lagrangian and Yang-Mills lagrangian theory is violated and hence the CSW rules do not apply. We find a type of generalised measure-preserving transformations which when applied to the Yang-Mills lagrangian also produce vertices that have same the helicity structure as the CSW rules. Among these transformations we find in 4-dimensions the canonical transformation generates the MHV vertices that are described by the Parke-Taylor formula. Finally we generalise the canonical transformation on supersymmetric theories. In light-cone gauge the physical components of the N=1 SYM lagrangian are closed under a subgroup of the SUSY transformations. We find the N=1 super Yang-Mills lagrangian can be rewritten in terms of chiral and anti-chiral superfields. In both N=1 and N=4 theories we perform a fermionic integral transformation on superfields analogous to Fourier transform which takes functions from coordinate space into momentum space. The on-shell SUSY generators we derive from the integral transformation agree with the prescription commonly used in the supersymmetry BCFW recursion formula. We apply the canonical transformation on both supersymmetric theories and compute the generic n-point MHV super-vertex. The N=4 MHV super-vertices are shown to agree with Nair's formula which was originally derived from WZW model.