Manifesting Hidden Structure in Scattering Amplitudes

This dissertation explores hidden structure within scattering amplitudes in quantum field theory, both at tree-level and at loop-level, and introduces some novel methods for making this hidden structure manifest. In Chapter 1 we give a brief introduction to the field of scattering amplitudes. In Chapter 2, we examine two-loop amplitudes in half-maximal supergravity (SUGRA) and discuss different ways of manifesting the underlying and surprising UV-finiteness. In particular, we apply state of the art techniques for performing integration by parts (IBP) reduction on families of multiloop integrals and introduce a new way of exposing UV-cancellations by performing IBP reduction directly on vacuum diagrams in a way that does not mix up UV and IR divergences. In Chapter 3 we continue to explore IBP reduction, now from a different perspective. Here we focus on planar Feynman diagrams and use the properties of dual conformal transformations to identify a family of IBP vectors that do not double propagators, and are therefore compatible with the highly successful unitarity method of computing loop amplitudes. A natural extension of these ideas is to try to apply similar methods to nonplanar diagrams, and doing so leads to some very preliminary steps towards uncovering a nonplanar analog of dual conformal symmetry, which is believed to exist from other considerations. Initial steps along this direction are discussed at the end of this chapter. In Chapter 4 we continue on from Chapter 3 in uncovering a nonplanar analog of dual conformal symmetry. Here we show that through three loops in N = 4 super-Yang–Mills (sYM) theory at four points and through two loops at five points, a representation of the amplitudes exist such that every relevant nonplanar diagram enjoys this hidden nonplanar symmetry. In Chapter 5 we leave the world of loop amplitudes and consider one of the simplest classes of scattering amplitudes—tree amplitudes in planar N = 4 sYM. A novel geometric perspective on this class of amplitudes is afforded by the amplituhedron picture. We introduce a new formalism for computing these amplitudes in the NMHV helicity sector, and a new and purely combinatorial description of the underlying polytopes. Our formalism makes the equivalence of different triangulations of the underlying space manifest by introducing a new set of objects that can be used to express the amplitude uniquely. In Chapter 6 we discuss ongoing work. In the first section, we discuss some progress that has been made in extending the formalism introduced in Chapter 5 to different MHV sectors. This would provide new insight on the geometric underpinning of these amplitudes, which is currently not understood. Finally, in the last section of this chapter we describe an algorithm for finding complete sets of numerical solutions to the scattering equations, and doing so faster than other implementations currently available