The Structure of Amplitudes in N=4 SYM

Abstract

The study of amplitudes and related quantities in the N = 4 Super Yang-Mills theory is a subject undergoing rapid evolution at the moment. In this work we present a review of some of the key ideas and concepts which we use to calculate `l-loop, n-point amplitudes of varying helicity. We show that performing a restriction on the external data of being in 1 + 1-dimensions allows remarkably compact expressions to be obtained at both MHV and NMHV levels. We use this data to motivate in 1 + 1-dimensions remarkably simple formulae for all collinear-limits and ultimately a universal uplifting formula which generates all n-point amplitudes of a particular loop-order and helicity configuration from a small set of lower-loop amplitudes. We also use the mechanism of the correlation function ↔ amplitude duality to construct the integrand for the five-point amplitude in full four-dimensional kinematics to
six-loops in the parity-even sector and five-loops in the parity-odd sector. Finally we consider a rewriting of certain known momentum-twistor amplitudes in terms of bi-twistor, six-dimensional X-variables and dimensionally regularise these equations to match known O(ε) results. From this we make some observations about the requirements for this process to be successful in the limited number of cases where the full O(ε) solution is known and provide an ansatz for constructing the terms for more complicated amplitudes.