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This thesis uses heavy-mass effective field theory for gravitons and scalars to compute the Schwarzschild-Tangherlini metric. By deducing the Schwarzschild metric order by order, we develop a method to derive a general form for the metric around Schwarzschild black holes at arbitrary order. After calculating the bending angle along time-like and null world lines and dropping all the terms that do not contribute to the bending angle, we find some gauge conditions in momentum space that can be transformed into the harmonic gauge in position space. Finally, we present our work on off-shell currents within our gauge conditions, these currents allow a new perspective on the computation of the metric.

This thesis uses heavy-mass effective field theory for gravitons and scalars to compute the Schwarzschild-Tangherlini metric. By deducing the Schwarzschild metric order by order, we develop a method to derive a general form for the metric around Schwarzschild black holes at arbitrary order. After calculating the bending angle along time-like and null world lines and dropping all the terms that do not contribute to the bending angle, we find some gauge conditions in momentum space that can be transformed into the harmonic gauge in position space. Finally, we present our work on off-shell currents within our gauge conditions, these currents allow a new perspective on the computation of the metric.

The post-Minkowskian expansion of Einstein's general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein-Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck's constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck's constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years.

Розраховані і досліджені c-функції, що визначають ступінь впливу зовнішнього збуджуючого електромагнітного поля на прямокутний діелектричний резонатор (ДР) у відкритому просторі. Показано наявність напрямків з ''нульовою'' проекцією поля збудження на поле ДР. За допомогою теорії збурень, вивчено просторовий розподіл полів розсіювання, що виникає при падінні плоскої електромагнітної хвилі p- або s-типу, на квадратну решітку прямокутних діелектричних резонаторів. Побудована електродинамічна модель розсіювання на решітці ДР прямокутної форми. Продемонстровано появу відбитої і тіньової пелюстки при розсіянні на решітці прямокутних ДР з основними магнітними видами коливань. Досліджуються особливості розсіювання на решітці кубічної форми з виродженими магнітними коливаннями основного типу. Показано, що виродження власних коливань резонаторів призводить до більш складної картині розсіювання: появи додаткових пелюсток, а також зміни їх форми. Відзначено, що форма просторового розподілу розсіяного поля решітки може помітно змінюватися з варіацією частоти в межах смуги частот зв'язаних коливань резонаторів решітки. Отримані практичні результати моделювання дозволяють значно скоротити час обчислень і оптимізувати складні багаторезонаторні структури мікрохвильових та оптичних систем зв'язку, які одночасно виконують функції поділу або об'єднання каналів.

The c-functions that determine the degree of influence of an external exciting electromagnetic field on a rectangular dielectric resonator (DR) in open space are calculated and investigated. The presence of directions with ''zero'' projection of the exciting electromagnetic field onto the DR field is shown. Using the perturbation theory, the spatial distribution of the scattered electromagnetic fields, which arises when a plane electromagnetic wave of p-, or s-type, is incident on a square lattice of rectangular dielectric resonators is studied. An electromagnetic model of scattering on a rectangular DR lattice is constructed. The appearance of a reflected and shadow lobe during scattering by a lattice of rectangular DRs with basic magnetic modes is demonstrated. The features of scattering by a cubic lattice with degenerate magnetic oscillations of the main type are investigated. It is shown that the degeneracy of the eigenoscillations of the resonators leads to a more complex scattering pattern: the appearance of additional lobes, as well as a change in their shape. It is noted that the shape of the spatial distribution of the scattered electromagnetic field of the grating can change noticeably with frequency variation within the frequency band of coupled oscillations of the resonators of the lattice. The obtained practical simulation results make it possible to significantly reduce the computation time and optimize complex multi-cavity structures of microwave and optical communication systems that simultaneously perform the functions of separation or combining of channels., Рассчитаны и исследованы c-функции, определяющие степень воздействия внешнего возбуждающего электромагнитного поля на прямоугольный диэлектрический резонатор (ДР) в открытом пространстве. Показано наличие направлений с ''нулевой'' проекцией возбуждающего поля на поле ДР. С помощью теории возмущений, изучено пространственное распределение полей рассеяния, возникающее при падении плоской электромагнитной волны p- или s-типа, на квадратную решетку прямоугольных диэлектрических резонаторов. Построена электродинамическая модель рассеяния на решетке ДР прямоугольной формы. Продемонстрировано появление отраженного и теневого лепестка при рассеянии на решетке прямоугольных ДР с основными магнитными типами колебаний. Исследуются особенности рассеяния на решетке кубической формы с вырожденными магнитными колебаниями основного типа. Показано, что вырождение собственных колебаний резонаторов приводит к более сложной картине рассеяния: появлению дополнительных лепестков, а также изменению их формы. Отмечено, что форма пространственного распределения рассеянного поля решетки может заметно меняться с вариацией частоты в пределах полосы частот связанных колебаний резонаторов решетки. Полученные практические результаты моделирования позволяют значительно сократить время вычислений и оптимизировать сложные многорезонаторные структуры микроволновых и оптических систем связи, одновременно выполняющие функции разделения или объединения каналов., Розраховані і досліджені c-функції, що визначають ступінь впливу зовнішнього збуджуючого електромагнітного поля на прямокутний діелектричний резонатор (ДР) у відкритому просторі. Показано наявність напрямків з ''нульовою'' проекцією поля збудження на поле ДР. За допомогою теорії збурень, вивчено просторовий розподіл полів розсіювання, що виникає при падінні плоскої електромагнітної хвилі p- або s-типу, на квадратну решітку прямокутних діелектричних резонаторів. Побудована електродинамічна модель розсіювання на решітці ДР прямокутної форми. Продемонстровано появу відбитої і тіньової пелюстки при розсіянні на решітці прямокутних ДР з основними магнітними видами коливань. Досліджуються особливості розсіювання на решітці кубічної форми з виродженими магнітними коливаннями основного типу. Показано, що виродження власних коливань резонаторів призводить до більш складної картині розсіювання: появи додаткових пелюсток, а також зміни їх форми. Відзначено, що форма просторового розподілу розсіяного поля решітки може помітно змінюватися з варіацією частоти в межах смуги частот зв'язаних коливань резонаторів решітки. Отримані практичні результати моделювання дозволяють значно скоротити час обчислень і оптимізувати складні багаторезонаторні структури мікрохвильових та оптичних систем зв'язку, які одночасно виконують функції поділу або об'єднання каналів.

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 andrelated to one another by the introduction of a transformation matrix. This is related to theKleiss-Kuijf and BCJ amplitude identities, and we conjecture a connection to the existenceof a BCJ representation via a condition on the generalized inverse of that matrix. Undertwo 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 loopamplitudes 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 treatsamplitudes in fully nonplanar N = 4 super-Yang-Mills, analyzing the conjecture that theyfollow the pattern of having no poles at infinity, can be written in dlog form, and canbe decom

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 andrelated to one another by the introduction of a transformation matrix. This is related to theKleiss-Kuijf and BCJ amplitude identities, and we conjecture a connection to the existenceof a BCJ representation via a condition on the generalized inverse of that matrix. Undertwo 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 loopamplitudes 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 treatsamplitudes in fully nonplanar N = 4 super-Yang-Mills, analyzing the conjecture that theyfollow the pattern of having no poles at infinity, can be written in dlog form, and canbe decom

We investigate the regularization-scheme dependence of scattering amplitudes in massless QCD and find that the four-dimensional helicity scheme (FDH) and dimensional reduction (DRED) are consistent at least up to NNLO in the perturbative expansion if renormalization is done appropriately. Scheme dependence is shown to be deeply linked to the structure of UV and IR singularities. We use jet and soft functions defined in soft-collinear effective theory (SCET) to efficiently extract the relevant anomalous dimensions in the different schemes. This result allows us to construct transition rules for scattering amplitudes between different schemes (CDR, HV, FDH, DRED) up to NNLO in massless QCD. We also show by explicit calculation that the hard, soft and jet functions in SCET are regularization-scheme independent.

Integrands for colour ordered scattering amplitudes in planar N=4 SYM are dual to those of correlation functions of the energy-momentum multiplet of the theory. The construction can relate amplitudes with different numbers of legs.By graph theory methods the integrand of the four-point function of energy-momentum multiplets has been constructed up to six loops in previous work. In this article we extend this analysis to seven loops and use it to construct the full integrand of the five-point amplitude up to five loops, and in the parity even sector to six loops.All results, both parity even and parity odd, are obtained in a concise local form in dual momentum space and can be displayed efficiently through graphs. We have verified agreement with other local formulae both in terms of supertwistors and scalar momentum integrals as well as BCJ forms where those exist in the literature, i.e. up to three loops.Finally we note that the four-point correlation function can be extracted directly from the four-point amplitude and so this uncovers a direct link from four- to five-point amplitudes.

We compute four-point scattering amplitudes in N = 2 SCQCD with general external matter configurations using N = 1 superspace Feynman diagrams, at one loop in the general case and up to two loops in the fundamental sector. In the pure adjoint sector at one loop we confirm exact agreement with the corresponding amplitudes in N = 4 SYM theory, supporting the idea that a closed subsector of the SCQCD might be exactly integrable. External matter in the fundamental representation breaks dual conformal invariance already at one-loop order while the principle of maximal transcendentality is respected up to two loops.

Monostatic synthetic aperture sonar (SAS) data sets for the scattering from a target in a waveguide can be simulated via a fast model that combines an acoustic ray approximation for propagation in the waveguide with far-field scattering from a target in free-space. With the assumption that acoustic rays arise from the source and receiver and their images, the problem of wave propagation and scattering from a target within the waveguide is replaced by a superposition of a set of free-field scattering problems. Under normal operating conditions, the separation distance between a SAS platform and a target is large compared to the carrier wavelength of the transmitted signal, and hence, the free-field scattered signal can be reduced to a far-field approximation where a spherically diverging wave is weighted by a scattering amplitude. The scattering amplitude contains all of the information about the scatterer (e.g., its material properties) and the directionality of the scattered field. A scattering amplitude can be obtained from a direct measurement or a finite-element analysis of a target. The fast ray model allows one, who is interested in SAS processing algorithms, to generate data sets with variations in the environment or source-receiver-target geometry without incurring the expense associated with the collection of actual SAS data. Results from the fast ray model will be compared to data and finite-element results for several targets in shallow water. The horizontal range to the targets and time gating permit the air-water interface to be ignored., The original document contains color images.

We compute the four-point amplitude in 3d N=8 SYM at two loops, by solving the three dimensional scalar doublebox in dimensional regularization. We compare it to the same result in the ABJM theory, to which maximal SYM should flow in the infrared at strong coupling. After proper rescalings, we find that the two amplitudes coincide in the Regge limit

The inverse problem considered herewith consists of determining the shape of a two dimensional, perfectly conducting obstacle from knowledge of the incident monochromatic plane electromagnetic wave and of a set of laboratory measurements related to the scattered wave: the Ipswich data. The solution algorithm and its numerical implementation rely on approximate back propagation (ABP). The latter is an algebraic transformation, which is deduced from the properties of complete families of base functions and relates the far field scattering coefficients to those on the obstacle boundary. The features presented herewith for the first time are: a) a consistency result for a forward propagator, which transforms least squares boundary coefficients into far field scattering coefficients; b) the development and testing of a scalar ABP method for two dimensional electromagnetic problems, where polarization is vertical; c) the outline and the implementation of a vector ABP method for horizontal polarization; d) the application of both methods to the inversion of experimental data. Since general conditions for the well posedness of ABP methods are still unknown, the main related open problems are also stated.

The paper is theoretical in nature and the objective is to investigate several problems in high energy and elementary particle physics of current interest and to attempt to gain a better understanding of these problems. The investigations in this field are normally divided into theoretical and experimental studies. For some of the proposed problems of investigations, there is no lack of experimental data, and constant comparisons with the experimental results guide the direction of pursuit. An outline of these studies is presented in this report.

The paper presents a study of the analytic properties in the Khuri plane of the invariant amplitudes, that define a scattering process of a 1-spin 'current' on a 1/2-spin target. Some current algebra and superconvergence identities are obtained. The latter are derived in two different ways. The formalism developed through the paper is used to rederive the well known low energy theorem for the Compton scattering. (Author), See also AD-670 269.

The real part of the virtual Compton amplitude can be directly determined from measurements of electron (or muon) bremsstrahlung or pair photoproduction. In general, the interference of the Compton amplitude with the Bethe-Heitler amplitude for pair production or bremsstrahlung yields a contribution to the cross section which is antisymmetric when the leptons are interchanged. This interference contribution thus produces different cross sections for electron and positron bremsstrahlung at a given scattering energy and angle. Also, the counting rate for pair production will depend on which lepton has the greater momentum. The determination of the real part of the Compton amplitude would supply information on the isobar resonance shape, test the dispersion relation for the forward amplitude, and resolve uncertainties in the determination of the nucleon resonances. A simple estimate for the lepton asymmetry of the pair production cross sections due to the forward Compton amplitude is given in addition to a complete calculation of the effect of the first nucleon resonance using the isobar model. The results are also discussed for nuclear targets. For the latter case, a broadening of the isobar decay width due to absorption in the nuclear medium must be taken into account. (Author), Prepared in cooperation with Stanford Univ., Calif., Inst. of Theoretical Physics, Rept. no. ITP-336 and Texas Univ., Austin, Center for Particle Theory, Rept. no. CPT-10.

A general unitary integral is analytically continued, the purpose being to obtain the domain of holomorphy of the integral in terms of the domains of holomorphy of the amplitudes occurring in the integrand. The amplitudes are assumed to be analytic on Riemann domains whose structure is restricted only by certain assumptions found to be prerequisites for an analytic continuation of the unitary integral. The Riemann domains allow for the possibility of an infinite number of signularities of an arbitrary type, and natural boundaries may occur. The analytic continuation is carried out explicitly in terms of scalar variables. All difficulties associated with the use of scalar variables are resolved to give an analytic continuation paralleling that in momentum space. (Author)

The variational method of Schwinger is extended, and leads to several new stationary expressions for the phase shifts, and for the scattering amplitude. Numerical examples are given. (Author)

Sum rules satisfied by the real part of a crossing-symmetric forward-scattering amplitude are presented. These sum rules involve integration of the real part up to finite energies only and hold for any energy., In cooperation with The Rockefeller Inst., New York. Pub. in Physical Review Letters v14 n3 p84-6 Jan 18 1965 (Copies not available to DDC or Clearinghouse customers). Revision of manuscript submitted 4 Nov 64.