Correlation functions, scattering amplitudes and the superconformal partial wave

Abstract

In this thesis we explore aspects of correlation functions and scattering amplitudes in supersymmetric field theories.

Firstly, we study correlation functions and scattering amplitudes in the perturbative regime of N=4 supersymmetric Yang-Mills theory. Here we begin by giving a new method for computing the supercorrelation functions of the chiral part of the stress-tensor supermultiplet by making use of twistor theory. We derive Feynman rules and graphical rules which involve a new set of building blocks which we can identify as a new class of N=4 off-shell superconformal invariants. This class of off-shell superconformal invariant is related to the known N=4 on-shell superconformal invariant pertinent to planar scattering amplitudes.

We then move onto the six-point tree-level NMHV scattering amplitude. Previous results are given in terms of a manifestly dual superconformal invariant basis called the R-invariant. We define and analyse a generalisation of this invariant which contains half of the dual superconformal invariance. We apply it to the six-point tree-level NMHV scattering amplitude and find a new representation which manifestly contains half of the dual superconformal invariance and physical pole structure. This is in contrast to the R-invariant basis which manifests symmetry properties but does not manifest physical pole structures.

Finally, we find the superconformal partial wave for four-point correlation functions of scalar operators on a super Grassmannian space (the space of m|n-planes in 2m|2n-dimensions) for theories with space-time symmetry SU(m,m|2n). This contains N=0,2,4 four-dimensional superconformal field theories in analytic superspace as well as a certain class of representations for the compact SU(2n) coset spaces. As an application we then specialise to N=4 supersymmetric Yang-Mills theory and use these results to perform a detailed superconformal partial wave analysis of the four-point functions of arbitrary weight half-BPS operators. We discuss the non-trivial separation of protected and unprotected sectors for the <2222>, <2233> and <3333> cases in an SU(N) gauge theory at finite N (where <ijkl> = <tr(W^i)tr(W^j)tr(W^k)tr(W^l)>). The <2233> correlator predicts a non-trivial protected twist four sector for <3333> which we can completely determine using the knowledge that there is one such protected twist-4 operator for each spin.