Transcendental weight in supersymmetric field theories

To accommodate the ever-growing need for accurate theoretical predictions of scattering events, recent years have seen a tremendous amount of effort towards understanding the mathematical structure of scattering amplitudes. As a testing ground for developing new computational techniques, theorists have turned to idealised setups such as N = 4 SYM, the maximally supersymmetric Yang- Mills theory in four dimensions. In this theory, scattering amplitudes display an enormous amount of interesting structure. One of the most remarkable traits is the property of maximal transcendental weight, which implies that all polylogarithmic functions appearing in an L-loop amplitude have weight 2L. As of yet for generic amplitudes this property remains an observation. Furthermore, N = 4 SYM is the only gauge theory known to exhibit this feature. In this thesis, we aim to investigate this property. First, we identify a special kinematic regime called the multi-Regge or high-energy limit. Turning to planar N = 4 SYM, we develop a computational technique in this regime which is recursive in the perturbative order, allowing the maximal transcendentality of the high-energy limit of planar N = 4 SYM to be shown to all orders. Building on this, we study part of the parton-parton cross section in the high- energy limit for generic gauge theories, aiming to find other theories which may display maximal transcendentality. We find simple constraints in terms of the matter content which imply maximal transcendental weight for this object. For theories with matter in the adjoint and fundamental representation of the gauge group, these constraints are satisfied by supersymmetric theories with a vanishing one-loop beta function, hinting towards a connection between maximal transcendentality and superconformal symmetry. Finally, we study a two-loop amplitude in general kinematics in N = 2 superconformal quantum chromodynamics, one of the theories found to display maximal transcendentality in our computation in the high energy limit. Our aim is to see whether this maximal transcendentality continues to hold beyond the high-energy limit. We find that maximal transcendentality is broken, but the lower-weight terms at leading-colour are given by a single multiple zeta value, confirming published results. Beyond leading-colour, we observe an interplay between the lower-weight terms and Catani infrared subtraction.(SC - Sciences) -- UCL, 201