This dissertation reviews work in computing N = 4 super-Yang–Mills (sYM) and N = 8

maximally supersymmetric gravity (mSUGRA) scattering amplitudes in D = 4 spacetime

dimensions in novel ways.

After a brief introduction and overview in Ch. 1, the various techniques used to construct amplitudes in the remainder of the dissertation are discussed in Ch. 2. This includes several new concepts such as d log and pure integrand bases, as well as how to construct the amplitude using exactly one kinematic point where it vanishes. Also included in this chapter is an outline of the Mathematica package on shell diagrams and numerics.m (osdn) that was developed for the computations herein. The rest of the dissertation is devoted to explicit examples.

In Ch. 3, the starting point is tree-level sYM amplitudes that have integral representations with residues that obey amplitude relations. These residues are shown to have corresponding residue numerators that allow a double copy prescription that results in mSUGRA residues.

In Ch. 4, the two-loop four-point sYM amplitude is constructed in several ways, showcasing many of the techniques of Ch. 2; this includes an example of how to use osdn. The two-loop five-point amplitude is also presented in a pure integrand representation with comments on how it was constructed from one homogeneous

cut of the amplitude. On-going work on the two-loop n-point amplitude is presented at the end of Ch. 4.

In Ch. 5, the three-loop four-point amplitude is presented in the d log representation and

in the pure integrand representation.

In Ch. 6, there are several examples of four- through seven-loop planar diagrams that illustrate how considerations of the singularity structure of the amplitude underpin dual-conformal invariance. Taken with the previous examples, this is additional evidence that the structure known to exist in the planar sector extends to the full theory. At the end of this chapter is a proof that all mSUGRA amplitudes have a pole at infinity for (L ≥ 4)-loops.

Finally in Ch. 7, the current status of ultraviolet divergences in the five-loop four-point mSUGRA amplitude is addressed. This includes a discussion of ongoing work aimed at resolving the mSUGRA finiteness question.

The following Mathematica scripts are submitted with this dissertation:

• on shell diagrams and numerics.m with dependencies:

– all_trees *.m

– external_kinematics_*_point.m

– rational_external_*_point.m

where “*” is a wild-card string of any set of characters of any length – either an integer

or a number spelled out.