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Topics in Nonsupersymmetric Scattering Amplitudes in Gauge and Gravity Theories
 Nohle, Joshua David
 Advisor(s): Bern, Zvi
Abstract
In Chapters 1 and 2, we introduce and review the duality between color and kinematics in YangMills theory uncovered by Bern, Carrasco and Johansson (BCJ). In addition to revealing interesting structures in YangMills 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 YangMills amplitudes by constructing a form of the oneloop fourpoint amplitude of this theory that makes the duality manifest. Our construction is valid in any dimension. We also describe a dualitysatisfying representation for the twoloop fourpoint amplitude with identical fourdimensional 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 fourpoint oneloop nonsupersymmetric pure YangMills discussion of Chapter 3 to include fermions and scalars circulating in the loop with all external gluons. This gives another nontrivial looplevel 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 fourdimensional form with explicit external helicity states. Using this, we compare our results to oneloop dualitysatisfying amplitudes that are already present in literature.
In Chapter 5, we switch from the topic of colorkinematics duality to discuss the recently renewed interest in the soft behavior of gravitons and gluons. Specifically, we discuss the subleading lowenergy behavior.
Cachazo and Strominger recently proposed an extension of the softgraviton 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 Smatrix 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 momentumconservation prescription for defining the subleading terms of the soft limit. We discuss the looplevel subleading soft behavior of gaugetheory amplitudes before turning to gravity amplitudes.
In Chapter 7, we show that at tree level, onshell gauge invariance can be used to fully determine the first subleading softgluon behavior and the first two subleading softgraviton behaviors. Our proofs of the behaviors for ngluon and ngraviton 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 threeform field is in this class, since it is dual to a cosmologicalconstant 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 twoloop identicalhelicity fourgraviton amplitude and determine the coefficient of the associated (nonevanescent) 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 threeform matter, and gravity coupled to zeroform or to twoform matter. Duff and van Nieuwenhuizen showed that, curiously, the oneloop conformal anomalythe coefficient of the GaussBonnet operatorchanges under pform duality transformations. We concur, and also find that the leading R^3 divergence changes under duality transformations. Nevertheless, in both cases the physical renormalized twoloop identicalhelicity fourgraviton amplitude can be chosen to respect duality. In particular, its renormalizationscale dependence is unaltered.
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