On-shell methods and positive geometry

Abstract

The past few decades have seen the development of a number of techniques for calculating scattering amplitudes without recourse to quantum fields and Feynman diagrams. These are broadly referred to as on-shell methods. The development of these methods has birthed a new perspective on scattering amplitudes. In many cases they can be thought of as differential forms with special properties on constrained spaces - i.e. as differential forms on so called positive geometries. The constraints of positivity suffice to completely determine the singularity structure of the S-matrix in these cases. This thesis is focused on understanding the emergence of known physics from positive geometries, computations in this new formalism and developments in on shell methods for theories involving massive particles. The first two chapters study the amplituhedron which is the positive geometry relevant to $\mathcal{N}=4$ super Yang-Mills. They focus on understanding the constraints of positivity and on seeing the physics of unitarity emerge from it. The next chapter develops geometric and on-shell techniques for understanding the properties of one loop integrals from the integrand in Feynman parameter space. The final chapter takes a step closer to the real world and develops an on-shell formalism for computing amplitudes in the bosonic, electroweak sector of the Standard Model. This includes an on-shell understanding of the Higgs mechanism within the context of the Standard Model.