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.

This dissertation explores the effects of higher-dimension operators in scattering amplitudes,and how such amplitudes can be used to gain insight into areas of physics ranging from elementary particle interactions to the tidal responses of black holes and neutron stars. Chapter 1 provides a brief introduction to higher-dimension operators and the Effective Field Theories (EFTs) which are their natural environment. In Chapter 2, we prove a theorem stating that operators which are "longer" in a specific sense cannot renormalize "shorter" operators at low loop levels. This result applies very generally, and can apply at high loop levels given the appropriate operators. We also discuss how the theorem applies to specifically to the Standard Model Effective Field Theory (SMEFT). In Chapter 3, we extend this discussion of renormalization within the SMEFT by calculating a large class of one loop amplitudes of dimension-six SMEFT operators and showing how to use these amplitudes to compute two-loop anomalous dimensions. Finally, in Chapter 4 we turn the calculation of amplitudes with higher-dimension operators to the purpose of calculating the effects of tides on the gravitational potential between, for example, orbiting black holes and neutron stars. This work, which includes in principle the leading effect of tidal operators at all loops, has direct relevance for gravitational wave detectors.

In this manuscript, we detail three recent calculations

addressing the ultraviolet behavior of supersymmetric quantum

gravity. First, we revisit the classic calculation of the two-loop

pure gravity divergence. We argue that the $\frac{1}{\epsilon}$

divergence is regulator and duality dependent. In its place, we

propose that examining the running of the scaling parameter,

$\log \mu$, is a duality and regulator independent approach to

assessing gravity divergences in four dimensions. We explicitly

calculate the $\log \mu$ coefficient at two loops for gravity with

any particle content, explicitly verifying the divergences for pure

gravity, and the finiteness for supersymmetric gravities. Second,

we analyze fully-integrated $\NeqFour$ supergravity at one loop

using the double copy. We find that there are evanescent effects at

one loop that come directly from evanescent terms in

pure-Yang--Mills. Using this observation, we lay the groundwork for

deeper analysis of the $U(1)$ anomaly with respect to the observed

evanescent behavior. Finally, we tackle the long-standing question

of the critical dimension of $\NeqEight$ supergravity at five loops.

We construct an integrand using the generalized double copy, expand

the integrand in large loop momentum, and reduce the resulting

integrals using $\mathfrak{sl}(L)$ integration-by-parts relations.

This procedure yields a critical dimension of $d_c = 24/5$.

Amplitudes methods have been successfully applied to various physical situations. In this manuscript we present their application to two quite distinct problems: classical gravitational-wave physics and constraining gravitational theories beyond general relativity. In chapter 1 we develop two methods for simplifying a computationally intense step in amplitudes calculations, specifically the sum over the physical states of photons and gravitons. These methods, even though generic, are developed with the application of gravitational-wave physics in mind. In chapters 2 and 3 we compute novel corrections to the Newtonian gravitational potential experienced by two spinning objects. These corrections originate in general-relativity effects. In chapter 4 we obtain bounds on Wilson coefficients of four-dimensional gravitational effective field theories. By calculating explicit examples of consistent gravitational theories and extracting the corresponding Wilson coefficients, we find that they occupy tiny islands in the Wilson-coefficient space derived. In chapter 5 we extend this analysis to dimensions higher than four. We again construct explicit gravitational theories and observe that the corresponding Wilson coefficients land in similar regions.

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.

In Part 1, we combine on-shell methods with the six-dimensional helicity formalism of Cheung and O'Connell to construct tree-level and multiloop scattering amplitudes. As a nontrivial multiloop example, we confirm that the recently constructed four-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions, is valid for dimensions less than or equal to six. We demonstrate that the tree-level amplitudes of maximal super-Yang-Mills theory in six dimensions, when stripped of their overall momentum and supermomentum delta functions, are covariant with respect to the six-dimensional dual conformal group. We demonstrate that this property is also present for loop amplitudes.

In Part 2, we explore consequences of the recently discovered duality between color and kinematics, which states that kinematic numerators in a diagrammatic expansion of gauge-theory amplitudes can be arranged to satisfy Jacobi-like identities in one-to-one correspondence to the associated color factors. The related squaring relations express gravity amplitudes in terms of gauge-theory ingredients. We then present a Yang-Mills Lagrangian whose diagrams through five points manifestly satisfy the duality between color and kinematics. Finally, we compute the coefficient of the potential three-loop divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments.

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.

This thesis discusses consequences of color-kinematics duality and applications towards computing quantum scattering amplitudes and classical radiation fields. Stemming from this duality, tree-level Bern-Carrasco-Johansson amplitude relations can be extended to one-loop integral coefficient relations for scattering in Yang-Mills theory. The double copy, which also follows from color-kinematics duality, allows for graviton scattering amplitudes to be found from scattering amplitudes in Yang-Mills theory. Additionally, a classical radiative double copy for obtaining gravitational waves in various theories is discussed. As a warm-up, a classical double copy of the Lienard-Wiechert potential in electrodynamics is found within a specific context, which allows for gravitational waves in linearized gravity to be found. Next, radiation in Yang-Mills and Yang-Mills-biadjoint-scalar theories is found, and the radiative double copy of these results allows for radiation in general relativity and Einstein-Yang-Mills theory, repsectively. In light of the recent detection of gravitational waves by the LIGO collaboration, this motivates the search of efficient analytic techniques for computing gravitational radiation. The double copy offers a way to apply methods from particle physics to gravitational-wave astronomy.

In Chapters 1 and 2, we introduce and review the duality between color and kinematics in Yang-Mills theory uncovered by Bern, Carrasco and Johansson (BCJ). In addition to revealing interesting structures in Yang-Mills theory, this conjectured duality immensely simplifies the computation of scattering amplitudes in theories of gravity.

In Chapter 3, we provide evidence in favor of the conjectured duality between color and kinematics for the case of nonsupersymmetric pure Yang-Mills amplitudes by constructing a form of the one-loop four-point amplitude of this theory that makes the duality manifest. Our construction is valid in any dimension. We also describe a duality-satisfying representation for the two-loop four-point amplitude with identical four-dimensional external helicities. We use these results to obtain corresponding gravity integrands for a theory containing a graviton, dilaton, and antisymmetric tensor, simply by replacing color factors with specified diagram numerators. Using this, we give explicit forms of ultraviolet divergences at one loop in four, six, and eight dimensions, and at two loops in four dimensions.

In Chapter 4, we extend the four-point one-loop nonsupersymmetric pure Yang-Mills discussion of Chapter 3 to include fermions and scalars circulating in the loop with all external gluons. This gives another nontrivial loop-level example showing that the duality between color and kinematics holds in nonsupersymmetric gauge theory. The construction is valid in any spacetime dimension and written in terms of formal polarization vectors. We also convert these expressions into a four-dimensional form with explicit external helicity states. Using this, we compare our results to one-loop duality-satisfying amplitudes that are already present in literature.

In Chapter 5, we switch from the topic of color-kinematics duality to discuss the recently renewed interest in the soft behavior of gravitons and gluons. Specifically, we discuss the subleading low-energy behavior.

Cachazo and Strominger recently proposed an extension of the soft-graviton theorem found by Weinberg. In addition, they proved the validity of their extension at tree level. This was motivated by a Virasoro symmetry of the gravity S-matrix related to BMS symmetry. As shown long ago by Weinberg, the leading soft behavior is not corrected by loops. In contrast, we show in Chapter 6 that with the standard definition of soft limits in dimensional regularization, the subleading behavior is anomalous and modified by loop effects. We argue that there are no new types of corrections to the first subleading behavior beyond one loop and to the second subleading behavior beyond two loops. To facilitate our investigation, we introduce a new momentum-conservation prescription for defining the subleading terms of the soft limit. We discuss the loop-level subleading soft behavior of gauge-theory amplitudes before turning to gravity amplitudes.

In Chapter 7, we show that at tree level, on-shell gauge invariance can be used to fully determine the first subleading soft-gluon behavior and the first two subleading soft-graviton behaviors. Our proofs of the behaviors for n-gluon and n-graviton tree amplitudes are valid in D dimensions and are similar to Low’s proof of universality of the first subleading behavior of photons. In contrast to photons coupling to massive particles, in four dimensions the soft behaviors of gluons and gravitons are corrected by loop effects. We comment on how such corrections arise from this perspective. We also show that loop corrections in graviton amplitudes arising from scalar loops appear only at the second soft subleading order. This case is particularly transparent because it is not entangled with graviton infrared singularities. Our result suggests that if we set aside the issue of infrared singularities, soft- graviton Ward identities of extended BMS symmetry are not anomalous through the first subleading order.

Finally, in Chapter 8, we conclude this dissertation with a discussion of the evanescent effects on nonsupersymmetric gravity at two loops. Evanescent operators such as the Gauss- Bonnet term have vanishing perturbative matrix elements in exactly D = 4 dimensions. Similarly, evanescent fields do not propagate in D = 4; a three-form field is in this class, since it is dual to a cosmological-constant contribution. In this chapter, we show that evanescent operators and fields modify the leading ultraviolet divergence in pure gravity. To analyze the divergence, we compute the two-loop identical-helicity four-graviton amplitude and determine the coefficient of the associated (non-evanescent) R^3 counterterm studied long ago by Goroff and Sagnotti. We compare two pairs of theories that are dual in D = 4: gravity coupled to nothing or to three-form matter, and gravity coupled to zero-form or to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the one-loop conformal anomaly---the coefficient of the Gauss-Bonnet operator---changes under p-form duality transformations. We concur, and also find that the leading R^3 divergence changes under duality transformations. Nevertheless, in both cases the physical renormalized two-loop identical-helicity four-graviton amplitude can be chosen to respect duality. In particular, its renormalization-scale dependence is unaltered.

In Chapter 1, we combine the six-dimensional helicity formalism of Cheung and O'Connell with D-dimensional generalized unitarity to obtain a formalism for computing one-loop amplitudes in dimensionally regularized Quantum Chromodynamics (QCD). We illustrate the procedure through various examples, including a next-to-leading order QCD correction to a Higgs process in the large top-quark mass limit.

In Chapter 2, we review the duality between color and kinematics and give our general procedure for extracting ultraviolet divergences in preparation for the following chapters.

In Chapter 3, we construct a form of the one-loop four-point amplitude in pure Yang-Mills, valid in any dimension, that makes the conjectured duality between color and kinematics manifest. We also describe a two-loop example. We then use these to obtain gravity integrands for a theory containing a graviton, dilaton and antisymmetric tensor, with which we study the ultraviolet structure at one and two loops.

In Chapters 4-6, we use the duality between color and kinematics to study the ultraviolet properties of half-maximal supergravity with and without matter. At one and two loops, we link cancellations to forbidden color tensors in pure Yang-Mills. Principal results for the pure theory (without matter) are the finiteness at two loops in D=5 and at three loops in D=4. These results are unexpected based on standard symmetry arguments. We also show that both cases are divergent when matter is included, which contradicts the conjectured existence of a superspace formalism used to explain the finite results in the pure theory.