UCLA

We begin the dissertation in Chapter 1 with a discussion of tree-level amplitudes in Yang-

Mills theories. The DDM and BCJ decompositions of the amplitudes are described and

related to one another by the introduction of a transformation matrix. This is related to the

Kleiss-Kuijf and BCJ amplitude identities, and we conjecture a connection to the existence

of a BCJ representation via a condition on the generalized inverse of that matrix. Under

two widely-believed assumptions, this relationship is proved. Switching gears somewhat, we introduce the RSVW formulation of the amplitude, and the extension of BCJ-like features to residues of the RSVW integrand is proposed. Using the previously proven connection of BCJ representations to the generalized inverse condition, this extension is validated, including a version of gravitational double copy.

The remainder of the dissertation involves an analysis of the analytic properties of loop

amplitudes in N = 4 super-Yang-Mills theory. Chapter 2 contains a review of the planar case, including an exposition of dual variables and momentum twistors, dual conformal symmetry, and their implications for the amplitude. After defining the integrand and on-shell diagrams, we explain the crucial properties that the amplitude has no poles at infinite momentum and that its leading singularities are dual-conformally-invariant cross ratios, and can therefore be normalized to unity. We define the concept of a dlog form, and show that it is a feature of the planar integrand as well. This leads to the definition of a pure integrand basis. The proceeding setup is connected to the amplituhedron formulation, and we put forward the hypothesis that the amplitude is determined by zero conditions.

Chapter 3 contains the primary computations of the dissertation. This chapter treats

amplitudes in fully nonplanar N = 4 super-Yang-Mills, analyzing the conjecture that they

follow the pattern of having no poles at infinity, can be written in dlog form, and can

be decomposed into a pure integrand basis, with each basis element having unit leading

singularity. Through explicit calculation, we show that this is true for the two-loop fourpoint,

three-loop four-point, and two-loop five-point amplitudes, and discuss the features of

each case. We then discuss the zero condition hypothesis, showing explicitly that it holds for the two-loop four-particle amplitude, and showing the set of conditions that fix the amplitude in the three-loop four-particle and two-loop five-particle cases without explicitly performing the fixing. This concludes the body of the dissertation.

Two appendices complete the dissertation. Appendix B includes an in-depth discussion of

dlog forms, including purely mathematical examples and a discussion of their appearance in one-loop amplitudes. Finally, Appendix C redoes portions of the analysis of Chapter 3 for the two- and three-loop four-particle amplitudes, but gives representations that are not in a pure integrand basis. Instead diagram symmetry is imposed on the basis elements, and diagrams that lack maximal cuts are pushed into maximal-cut diagrams. This gives representations closer in spirit to the previously-constructed representations of these amplitudes, such as the BCJ representations. It also highlights the role of color Jacobi identities and the freedom in the amplitude representation they can generate, and contains an explicit discussion of these features that is unpublished elsewhere.