Thesis (Ph.D.); Submitted to Massachusetts Inst. of Tech., Cambridge, MA (US); 1 Aug 2003. / Published through the Information Bridge: DOE Scientific and Technical Information. "JLAB-PHY-03-174" "DOE/ER/40150-2747" Zhengwei Chai. 08/01/2003. Report is also available in paper and microfiche from NTIS.

When calculating scattering processes in theories involving massless gauge bosons, such as gluons in Quantum Chromodynamics (QCD), one encounters infrared (IR), or soft, divergences. To obtain precise predictions, it is important to have exact expressions for these IR divergences, which are present in any on-shell scattering amplitude. Due to their long wavelength, soft gluons factorise with respect to short-distance, or hard, interactions and can be captured by correlators of semi-infinite Wilson lines. The latter obey a renormalisation group equation, which gives rise to exponentiation. The exponent can be represented diagrammatically in terms of weighted sums of Feynman diagrams, called webs. A web with L external legs, each with ni gluon attachments, is denoted (n1; n2; : : : ; nL). In this way all soft gluon interactions can be described by a soft anomalous dimension. It is currently known at three loops with lightlike kinematics, and at two loops with general kinematics. Our work is a step towards a three-loop result in general kinematics. In recent years, much progress has been made in understanding the general physical properties of scattering amplitudes and in exploiting these properties to calculate specific amplitudes. At the same time, we have discovered a lot of structure underpinning the space of multiple polylogarithms, the functions in terms of which most known amplitudes can be written. General properties include analyticity, implying that scattering amplitudes are analytic functions except on certain branch cuts, and unitarity, or conservation of probability. These two properties are both exploited by unitarity cuts. Unitarity cuts provide a diagrammatic way of calculating the discontinuities of a Feynman diagram across its branch cuts, which is often simpler than calculating the diagram itself. From this discontinuity, the original function can be reconstructed by performing a dispersive integral. In this work, we extend the formalism of unitarity cuts to incorporate diagrams involving Wilson-line propagators, where the inverse propagator is linear in the loop momenta, rather than the quadratic case which has been studied before. To exploit this for the calculation of the soft anomalous dimension, we first found a suitable momentum-space IR regulator and corresponding prescription, and then derived the appropriate largest time equation (LTE). We find that, as in the case of the scalar diagrams, most terms contributing to the LTE turn out to be zero, albeit for different reasons. This simplifies calculations considerably. This formalism is then applied to the calculation of webs with non-lightlike Wilson lines. As a test, we first looked at webs that have been previously studied using other methods. It emerges that, when using the correct variables, the dispersive integrals one encounters here are trivial, illustrating why unitarity cuts are a particularly useful tool for the calculation of webs. We observe that our technique is especially efficient when looking at diagrams involving three-gluon vertices, such as the (1; 1; 1) web and the Y diagram between two lines. We then focus on three-loop diagrams connecting three or four external non-lightlike lines and involving a three-gluon vertex. We calculate the previously unknown three-loop three-leg (1; 1; 3) web in general kinematics. We obtain a result which agrees with the recently calculated lightlike limit. We also develop a technique to test our results numerically using the computer program SecDec, and we find agreement with our analytical result. The result for the (1; 1; 3) web can then be exploited to gain insight into the more complicated three-loop four-leg (1; 1; 1; 2) web. Indeed, the (1; 1; 1; 2) web reduces to the (1; 1; 3) web in a certain collinear limit. We propose an ansatz for the (1; 1; 1; 2) web in general kinematics, based on a conjectured basis of multiple polylogarithms. The result for the (1; 1; 3) web, together with the known result for the lightlike limit of the (1; 1; 1; 2) web, imposes strong constraints on the ansatz. Using these constraints, we manage to fix all but four coefficients in the ansatz. We fit the remaining coefficients numerically, but find that the quality of the fit is not good. We find possible explanations for this poor quality. This calculation is still a work in progress. Our results provide a major step towards the full calculation of the three-loop soft anomalous dimension for non-lightlike Wilson lines. We calculated new results for three-loop webs, and also deepened the understanding of webs in general. We confirm a conjecture about the functional dependence of the soft anomalous dimension on the cusp angles. We also confirm earlier findings about the symbol alphabet of the relevant functions. This confirms the remarkable simplicity found earlier in the expressions for the soft anomalous dimension.

Pure Yang-Mills amplitudes with all external gluons carrying positive helicity, known as all-plus amplitudes, have an especially simple structure. The tree amplitudes vanish and, up to at least two loops, the loop-level amplitudes are related to those of N = 4 super-Yang-Mills (SYM) theory. This makes all-plus amplitudes a useful testing ground for new methods of simplifing more general classes of amplitudes. In this thesis we consider three new approaches, focusing on the structure before integration. We begin with the planar (leading-colour) sector. A D-dimensional local-integrand presentation, based on four-dimensional local integrands developed for N = 4 SYM, is developed. This allows us to compute the planar six-gluon, two-loop all-plus amplitude. Its soft structure is understood before integration, and we also perform checks on collinear limits. We then proceed to consider subleading-colour structures. A multi-peripheral colour decomposition is used to find colour factors based on underlying tree-level amplitudes via generalised unitarity cuts. This allows us to find the integrand of the full-colour, two-loop, five-gluon all-plus amplitude. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. Finally, we consider representations satisfying colour-kinematics duality. We discuss obstacles to finding such numerators in the context of the same five-gluon amplitude at two loops. The obstacles are overcome by adding loop momentum to our numerators to accommodate tension between the values of certain cuts and the symmetries of certain diagrams. Control over the size of our ansatz is maintained by identifying a highly constraining, but desirable, symmetry property of our master numerator.

This thesis is divided into two parts. The first part concerns the study of the ambitwistor string and the scattering equations, while the second concerns the interplay of the symmetries of the asymptotic null boundary of Minkowski space, called [scri], and scattering amplitudes. The first part begins with a review of the CHY formulas for scattering amplitudes, the scattering equations and the ambitwistor string including its pure spinor version. Next are the results of this thesis concerning these topics, they are: generalizing the ambitwistor model to higher genus surfaces; calculating the one-loop NS-NS scattering amplitudes and studying their modular and factorization properties; deriving the one-loop scattering equations and analyzing their factorization; showing that, in the case of the four graviton amplitude, the ambitwistor amplitude gives the expected kinematical prefactor; matching this amplitude to the field theory expectation in a particular kinematical regime; solving the one loop scattering equations in this kinematical regime; a conjecture for the IR behaviour of the one-loop ambitwistor integrand; computing the four graviton, two-loop amplitude using pure spinors; showing that this two-loop amplitude has the correct kinematical prefactor and factorizes as expected for a field theory amplitude; generalizing the ambitwistor string to curved backgrounds; obtaining the field equations for type II supergravity as anomaly cancellation on the worldsheet; generalizing the scattering equations for curved backgrounds. The second part begins with a review of the definition of the null asymptotic boundary of four dimensional Minkowski space, its symmetry algebra, and their relation to soft particles in the S-matrix. Next are the results of this thesis concerning these topics, they are: constructing two models consisting of maps from a worldsheet to [scri], one containing the spectrum of N=8 supergravity, and the other the spectrum of N=4 super Yang-Mills; showing how certain correlators in these theories calculate the tree-level S-matrix of N=8 sugra and N=4 sYM respectively; defining worldsheet charges which encode the action of the appropriate asymptotic symmetry algebra and showing that their Ward-identities recover the soft graviton, and soft gluon factors; defining worldsheet charges for proposed extensions of these symmetry algebras and showing that their Ward-identities give the subleading soft graviton and subleading soft gluon factors.

The density amplitude of an isolated quantum vortex line in superfluid 4He is calculated using a generalized Gross-Pitaevskii (G-P) equation. The generalized G-P equation for the order parameter extends the usual mean-field approach by replacing the interatomic potential in the ordinary G-P equation by a local, static T matrix, which takes correlations between the particles into account. The T matrix is a sum of ladder diagrams appearing in a diagrammatic expansion of the mean field term in an exact equation for the order parameter. It is an effective interaction which is much softer than the realistic interatomic Morse dipole-dipole potential from which it is calculated. A numerical solution of the generalized G-P equation is required since it is a nonlinear integro-differential equation with infinite limits. For the energy denominator in the T matrix equation, a free-particle spectrum and the observed phonon-roton spectrum are each used. For the fraction of particles in the zero-momentum state (Bose-Einstein dondensate) which enters the equation, both a theoretical value of 0.1 and an experimental value of 0.024 are used. The chemical potential is adjusted so that the density as a function of distance from the vortex core approaches the bulk density asymptotically. Solutions of the generalized G-P equation are not very dependent on the choice of energy denominator or condensate fraction. The density profile is a monotonically increasing function of the distance from the vortex core. The core radius, defined to be the distance to half the bulk density, varies from 3.7 A to 4.7 A, which is over three times the experimental value of 1.14 A at absolute zero.

Este trabalho tem como objetivo o ensino dos fundamentos da Astrofísica Estelar moderna a alunos do Ensino Médio através de uma abordagem empírica que usa como ferramentas principais da metodologia a observação, o registro, a análise e a interpretação das mudanças de brilho das estrelas variáveis ao longo do tempo. Mediante técnicas simples (fotometria visual) para a estimativa dos valores dos brilhos aparentes (magnitudes) dessas estrelas no decorrer de vários meses, o aluno coleta seus próprios dados observacionais. Em seguida, esses dados serão colocados em gráficos (curvas de luz), cuja análise permitirá a medida de seus períodos e amplitudes de variação, bem como a determinação de uma série de parâmetros que caracterizam as propriedades físicas das estrelas. O estudo é conduzido de uma forma que permite apresentar aos alunos vários dos grandes temas da Astrofísica contemporânea sem que eles percam o vínculo sensorial com a observação direta dos fenômenos celestes, seja com auxílio de instrumental modesto (binóculos ou pequenos telescópios), seja até mesmo a olho nu. O produto final de nosso trabalho é um Guia de Estudos intitulado Astrofísica Estelar para o Ensino Médio, em doze capítulos e mais de trezentas páginas, que está incluído em sua íntegra nesta monografia. O Guia foi desenvolvido tendo como alvos principais os alunos e professores do Ensino Médio formal. Não obstante, ele pode ser também utilizado por qualquer pessoa que tenha interesse no assunto e que possua os conceitos necessários de Matemática e Física do Ensino Médio; ou ainda em cursos do ensino não-formal que sejam ministrados, por exemplo, por planetários, museus de ciência e astrônomos amadores (situação essa testada com ótimos resultados pelo autor). Por fim, vale mencionar que, neste tipo de aprendizado (conhecido como research-based education, que poderíamos traduzir livremente como aprendizado vinculado à pesquisa), utilizamos intensivamente diversos recursos tecnológicos disponibilizados gratuitamente na internet pelas grandes entidades internacionais que se dedicam à pesquisa e ao estudo das estrelas variáveis tais como softwares específicos, bancos de dados, simulações e cartas para a estimativa das magnitudes. O uso dessa metodologia vinculada à pesquisa, em nossa experiência, se constitui em um fator valioso para desenvolver nos alunos a criatividade, o raciocínio crítico, a familiarização com o método científico e as habilidades para a pesquisa. / This work aims at teaching the fundamental aspects of Stellar Astrophysics to high school students through an empirical approach which uses variable stars observations, data collection, and the analysis and interpretation of brightness changes as key methodological tools. By using some simple techniques (visual photometry) to estimate the values of apparent brightness (magnitudes) of these stars over several months, the student collects his/her own observational data. Subsequently, that data will be plotted in graphs (light curves), whose analysis will in turn allow the determination of its periods and amplitudes of variation, as well as of a series of physical parameters that characterize the physical properties of the stars. The whole study is conducted in a way that shows students several of the major themes of contemporary Astrophysics without losing their sensory link with the direct observation of celestial phenomena - whether with the aid of modest instruments (binoculars or small telescopes) or even naked eye. The final product of our work is a Study Guide called \"Stellar Astrophysics for High School\" in twelve chapters and more than three hundred pages, which is included in their entirety in this monograph. The Guide was developed with the main targets being students and teachers of the formal high school level. Nevertheless, it can also be used by anyone who has an interest in the subject and who possesses the required concepts of high school Mathematics and Physics; or even in non-formal teaching courses that are taught, for example, by planetariums, science museums and amateur astronomers (a situation that has been tested with great results by the author). Last but not least, it is worth mentioning that in this type of learning (known as \"research-based education\") we intensively use various technological resources freely available on the internet from the great international entities that are dedicated to the research and study of variable stars such as specific softwares, databases, simulations and charts for estimating magnitudes. The use of research-based education, in our experience, is a valuable tool to foster and develop students creativity, critical thinking, and the familiarization with the scientific method and research skills.

Este trabalho tem como objetivo o ensino dos fundamentos da Astrofísica Estelar moderna a alunos do Ensino Médio através de uma abordagem empírica que usa como ferramentas principais da metodologia a observação, o registro, a análise e a interpretação das mudanças de brilho das estrelas variáveis ao longo do tempo. Mediante técnicas simples (fotometria visual) para a estimativa dos valores dos brilhos aparentes (magnitudes) dessas estrelas no decorrer de vários meses, o aluno coleta seus próprios dados observacionais. Em seguida, esses dados serão colocados em gráficos (curvas de luz), cuja análise permitirá a medida de seus períodos e amplitudes de variação, bem como a determinação de uma série de parâmetros que caracterizam as propriedades físicas das estrelas. O estudo é conduzido de uma forma que permite apresentar aos alunos vários dos grandes temas da Astrofísica contemporânea sem que eles percam o vínculo sensorial com a observação direta dos fenômenos celestes, seja com auxílio de instrumental modesto (binóculos ou pequenos telescópios), seja até mesmo a olho nu. O produto final de nosso trabalho é um Guia de Estudos intitulado Astrofísica Estelar para o Ensino Médio, em doze capítulos e mais de trezentas páginas, que está incluído em sua íntegra nesta monografia. O Guia foi desenvolvido tendo como alvos principais os alunos e professores do Ensino Médio formal. Não obstante, ele pode ser também utilizado por qualquer pessoa que tenha interesse no assunto e que possua os conceitos necessários de Matemática e Física do Ensino Médio; ou ainda em cursos do ensino não-formal que sejam ministrados, por exemplo, por planetários, museus de ciência e astrônomos amadores (situação essa testada com ótimos resultados pelo autor). Por fim, vale mencionar que, neste tipo de aprendizado (conhecido como research-based education, que poderíamos traduzir livremente como aprendizado vinculado à pesquisa), utilizamos intensivamente diversos recursos tecnológicos disponibilizados gratuitamente na internet pelas grandes entidades internacionais que se dedicam à pesquisa e ao estudo das estrelas variáveis tais como softwares específicos, bancos de dados, simulações e cartas para a estimativa das magnitudes. O uso dessa metodologia vinculada à pesquisa, em nossa experiência, se constitui em um fator valioso para desenvolver nos alunos a criatividade, o raciocínio crítico, a familiarização com o método científico e as habilidades para a pesquisa. / This work aims at teaching the fundamental aspects of Stellar Astrophysics to high school students through an empirical approach which uses variable stars observations, data collection, and the analysis and interpretation of brightness changes as key methodological tools. By using some simple techniques (visual photometry) to estimate the values of apparent brightness (magnitudes) of these stars over several months, the student collects his/her own observational data. Subsequently, that data will be plotted in graphs (light curves), whose analysis will in turn allow the determination of its periods and amplitudes of variation, as well as of a series of physical parameters that characterize the physical properties of the stars. The whole study is conducted in a way that shows students several of the major themes of contemporary Astrophysics without losing their sensory link with the direct observation of celestial phenomena - whether with the aid of modest instruments (binoculars or small telescopes) or even naked eye. The final product of our work is a Study Guide called \"Stellar Astrophysics for High School\" in twelve chapters and more than three hundred pages, which is included in their entirety in this monograph. The Guide was developed with the main targets being students and teachers of the formal high school level. Nevertheless, it can also be used by anyone who has an interest in the subject and who possesses the required concepts of high school Mathematics and Physics; or even in non-formal teaching courses that are taught, for example, by planetariums, science museums and amateur astronomers (a situation that has been tested with great results by the author). Last but not least, it is worth mentioning that in this type of learning (known as \"research-based education\") we intensively use various technological resources freely available on the internet from the great international entities that are dedicated to the research and study of variable stars such as specific softwares, databases, simulations and charts for estimating magnitudes. The use of research-based education, in our experience, is a valuable tool to foster and develop students creativity, critical thinking, and the familiarization with the scientific method and research skills.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

[pt] O presente trabalho de tese é destinado ao estudo do decaimento do méson D+ KS(pi)−(pi)+(pi)+ com dados coletados pelo experimento FOCUS do Fermilab. O estado final, composto por 4 mésons pseudo- escalares, pode ser alcançado através de uma considerável variedade de sub- estruturas ressonantes. O proposito da análise é encontrar quais são estes possíveis canais intermediários através de suas contribuições e fases relativas. Para tanto, é utilizado o formalismo de Análise de Amplitudes usando o chamado Modelo Isobárico. A dinâmica do decaimento é definida através de uma função que contém as características (formas funcionais das ressonâncias, distribuição angular, etc) dos canais que intervêm no decaimento e cujo domínio é um espaço o de fase determinado por 5 invariantes (devido às restrições de decaimento em quatro corpos sem spin). A função é então ajustada ao conjunto de dados coletados. Nos resultados, verificamos uma grande contribuição do méson vetor-axial a1(1260) (52%) seguido do vetor-axial K1(1400) (34%). Além disso, o modelo apresenta contribuição da ressonância (sigma) (cerca de 8%, vinda de a1(sigmapi)KS e sigmaKSpi) e uma razoável contribuição da ressonância escalar K− (14%). O estado K já foi visto em seu modo neutro em outros decaimentos de charme, porém ainda não em seu modo carregado. Não encontramos contribui ção significativa do decaimento direto em 4 corpos (não- ressonante). Esta tese vem se somar ao esforço no entendimento da dinâmica das intera ções fortes a baixas energias, que nos últimos anos tem ganhado da física de mésons charmosos uma importante contribuição. / [en] This thesis is devoted to the study of the decay D+ KS(pi)− (pi)+(pi)+ with data collected from the FOCUS experiment, at Fermilab. The final state composed of 4 pseudo-scalars can be produced through a number of resonant sub-structures. The purpose of this analysis is to find the contributing intermediate states by measuring their relative strenghts and phases. For that, the Amplitude Analysis formalism is used, with the so-called Isobar Model. The decay dynamics is described through a function which has the features of the contributing channels (functional forms of the resonances, angular distribution, etc) and which domain is a phase space determined by 5 invariants (due to the kinematical constraints of a 4- body spinless decay). The data sample is thus fitted to this function. Our results show a dominant contribution of the axial- vector meson a1(1260) (52%), followed by the K1(1400) axial-vector (34%). Moreover, the model presents a contribution from the sigma meson (about 8% as a1(sigmapi)KS and sigmaKSpi) and a significant contribution from the scalar k−. The k state has been reported in its neutral mode in other charm decays but not is its charged mode. We find no significant contribution from the direct 4-body decay (non-resonant). This work adds to the effort in the understanding of the strong-interaction dynamics at low energies, which in recent years have been receiving an important contribution from charm meson physics.

In dieser Dissertation stellen wir moderne analytische Methoden zur Untersuchung von konformen Feldtheorien (CFTs) in mehr als zwei Dimensionen vor. Mit Hilfe dieser Methoden können das Spektrum der Theorie und die Operatorprodukt-Koeffizienten (OPE-Koeffizienten) ermittelt werden. Zunächst untersuchen wir das Spektrum lokaler Operatoren in CFTs auf einem Defekt mit Kodimension größer eins. Wir zeigen, dass für großen transversalen Spin s das Spektrum jeder Theorie abzählbar unendlich viele Häufungspunkte aufweist. Der Spin s ist die Quantenzahl, die zu der Untergruppe der Lorentzgruppe gehört, welche den Defekt invariant lässt. Des Weiteren finden wir die OPE-Koeffizienten und die anomalen Dimensionen der zu den Häufungspunkten gehörenden Operatoren in einer Entwicklung in 1/s mit Hilfe von Lichtkegel-Bootstrap-Techniken. Außerdem leiten wir aus der Diskontinuität der kausalen Zweipunktfunktion die Operatordimensionen und OPE-Koeffizienten als analytische Funktionen von s her. Im zweiten Teil dieser Arbeit führen wir die Mellindarstellung von konformen Korrelationsfunktionen ein. In dieser Darstellung sind das Spektrum und die OPE-Koeffizienten manifest enthalten. Wir legen den Fokus auf die Beschreibung von Vierpunktfunktionen in drei Dimensionen von entweder ausschließlich Spin 1/2 Operatoren oder einer Mischung aus Spin 1/2 und skalaren Operatoren. Nachdem wir für diese Vierpunktfunktionen die Mellinamplituden definieren, untersuchen wir die Polstruktur dieser genauer. Im Anschluss illustrieren wir die Analyse an konkreten Mellinamplituden von fermionischen Wittendiagrammen und konformen fermionischen Feynmandiagrammen. Im letzten Teil untersuchen wir die OPE im Kontext der Holographie. Hierbei leiten wir theorieunabhängige Beziehungen zwischen den OPE-Koeffizienten der Weltflächen-CFT einer Stringtheorie in Anti-de-Sitter-Raumzeit und der dualen CFT her. / In this thesis, we discuss some modern analytical approaches to studying conformal field theories (CFTs) in dimensions greater than two. The results thus derived pertain to the dynamical data that define a generic CFT, namely the spectrum of operators and the coefficients in the operator product expansion (OPE). We begin with an investigation of the spectrum of local operators supported on conformal defects of codimension greater than one and establish the existence therein of a countably infinite number of universal accumulation points at large transverse spin s. Here, s is a quantum number associated with the symmetry under the Lorentz transformations that preserve the defect. Using lightcone bootstrap techniques, we calculate the anomalous dimensions and OPE coefficients of the operators that populate these accumulation points in a large s expansion. Furthermore, we derive an integral formula to obtain the CFT data associated with the defect theory from the discontinuity in the causal two-point function of scalar operators in the ambient theory, thereby inverting the expansion of this correlator in the defect channel. This formula extracts the operator dimensions and OPE coefficients in an analytic function in s and also enables us to resum the large s expansion obtained using lightcone bootstrap. Thereafter we move on to a discussion of the Mellin representation of fermionic conformal correlators. The dynamical data in CFTs is manifest in the analytic properties of Mellin amplitudes. We define, concretely for three spacetime dimensions, the Mellin amplitudes associated with the four-point function of spin-half operators and the mixed four-point function of spin-half and scalar operators. We analyze the pole structure of these Mellin amplitudes and illustrate the general features thus unraveled with some explicit computations of Mellin amplitudes associated with Witten diagrams and conformal Feynman integrals with fermionic legs. Finally we look at the OPE in the context of holography and derive a set of theory independent relations between OPE coefficients in the worldsheet CFT of a string theory in anti-de Sitter spacetime and those in the dual CFT.