CP violation is an important condition to explain the preponderance of baryons in our universe, yet the available CP violation in the Standard Model (SM) via the so-called Cabibbo-Kobayashi-Maskawa mechanism seems to not provide enough CP violation. Thus searching for new sources of CP violation is one of the central tasks of modern physics. In this thesis, we focus on a new possible source of CP violation which generates triple-product correlations in momenta which can appear in neutron and nuclear radiative β decay. We show that at low energies such a CP violating correlation may arise from the exotic coupling of nucleon, photon and neutrino that was proposed by Harvey, Hill, and Hill (HHH). One specialty of such an exotic HHH coupling is that it does not generate the well-known CP-violating terms such as ``D-term'', ``R-term'', and neutron electric dipole moment, in which particle's spins play critical role. We show that such a new HHH-induced CP violating effect is proportional to the imaginary part of c5gv, where gv is the vector coupling constant in neutron and nuclear β decay, and c5 is the phenomenological coupling constant that appears in chiral perturbation theory at O(M-2) with M referring to the nucleon or nuclear mass. We consider a possible non-Abelian hidden sector model, which is beyond the SM and may yield a nontrivial Im(c5). The available bounds on both Im(c5) and Im(gv) are considered, and a better limit on Im(c5) can come from a direct measurement in radiative beta decay. We calculate the competitive effect that arises from the general parameterization of the weak interaction that was proposed by Lee and Yang in 1956. We also show that in the proposed measurements, the CP-violating effect can be mimicked by the SM via final-state interactions (FSI). For a better determination of the bound of Im(c5), we consider the FSI-induced mimicking effect in full detail in O(α) as well as in leading recoil order. To face ongoing precision measurements of neutron radiative β decay of up to 1% relative error, we sharpen our calculations of the CP conserving pieces of neutron radiative β decay by considering the largest contributions in O(α2): the final-state Coulomb corrections as well as the contributions from two-photon radiation.

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δg are all simple for a residual set of Cr metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of Cr metrics such that the nonzero eigenvalues of the Hodge Laplacian Δg(k) on k-forms are all simple for 0 ≤ k ≤ 3. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of Cr metrics, the nonzero eigenvalues of the Hodge Laplacian Δg(2) acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *gd, which is related to the Hodge Laplacian by Δg(2) = -(*gd)2 when the operators are restricted to coexact 2-forms on a 5-manifold. We use techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics. We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n = 4 ℓ + 1 and k = 2 ℓ for ℓ ϵ N.

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At the present time it is unknown how the U(1)A anomaly of Quantum Chromodynamics behaves at high temperatures. We therefore want to look for thermal changes of the effects of the anomaly. For example, by studying the properties of the η' meson at high temperatures it would be possible to deduce important information on the axial anomaly, thanks to the deep connection between them. In this thesis the width of the η' as a function of the temperature is studied in the framework of large-Nc Chiral Perturbation Theory, at next-to-leading order, and in the corresponding Resonance Chiral Theory. We calculate the width increase due to scattering with particles from the heat bath, which we assume to consist of a pion gas. We compare the results obtained in both frameworks and as expected we find a smaller, but still consistent width increase when the more realistic resonance exchange is taken into account. The results suggest that the in-medium width of the η' may increase up to ΔΓ<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Capprox" /> 10 MeV at a temperature of T<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Capprox" /> 120 MeV. We find therefore a width increase of considerable size, comparable to the inverse lifetime of the fireball created in relativistic heavy-ion collisions. In other words, our results suggest that it may be possible to study experimentally how the properties of the η' change at high temperatures.

Semidefinite programming is a powerful modeling tool for a wide range of optimization and feasibility problems. Its prevalent use in practice relies on the fact that a (nearly) optimal solution of a semidefinite program can be obtained efficiently in both theory and practice, provided that the semidefinite program and its dual satisfy the Slater condition. This thesis focuses on the situation where the Slater condition (i.e., the existence of positive definite feasible solutions) does not hold for a given semidefinite program; the failure of the Slater condition often occurs in structured semidefinite programs derived from various applications. In this thesis, we study the use of the facial reduction technique, originally proposed as a theoretical procedure by Borwein and Wolkowicz, as a preprocessing technique for semidefinite programs. Facial reduction can be used either in an algorithmic or a theoretical sense, depending on whether the structure of the semidefinite program is known a priori. The main contribution of this thesis is threefold. First, we study the numerical issues in the implementation of the facial reduction as an algorithm on semidefinite programs, and argue that each step of the facial reduction algorithm is backward stable. Second, we illustrate the theoretical importance of the facial reduction procedure in the topic of sensitivity analysis for semidefinite programs. Finally, we illustrate the use of facial reduction technique on several classes of structured semidefinite programs, in particular the side chain positioning problem in protein folding.

Chiral perturbation theory is a useful tool in the study of low energy reactions involving light particles. However the inclusion of heavy particles in chiral perturbation theory results in large contributions from loop diagrams which violate the standard power counting scheme. We review two methods, referred to as heavy baryon chiral perturbation theory and infrared regularization, which remove the high energy effects of the heavy particles and which therefore do not violate the power counting scheme. We then use these two methods to calculate the amplitude for pion photoproduction to fourth order and prove that the two amplitudes are equivalent.

Three different aspects of the sinh-Gordon model are explored in this thesis. We begin, in chapter one, with a summary of the model and the necessary background. Chapter two studies the model with two boundary conditions. Two approaches are presented to investigate the reflection factors off the boundaries and the energy of the theory. In chapter three, perturbation theory is developed to study the theory with one general boundary condition. A contribution to the quantum reflection factor is obtained and compared with the result obtained for the special boundary condition. Chapters four and five investigate the supersymmetric extension of the model in the presence of a single boundary. Firstly, the classical limits of the supersymmetric reflection matrices are checked. The exact reflection factors are studied perturbatively up to the second order of the coupling constant. Secondly, the perturbation theory and the path integral formalism are employed in the supersymmetric model to study the quantum reflection factors. We conclude with a brief sixth chapter describing the outlook for further investigations.

19 Dec 2004. / Published through the Information Bridge: DOE Scientific and Technical Information. "IS-T 1948" Jamie Marie Rintelman. 12/19/2004. Report is also available in paper and microfiche from NTIS.

Chiral perturbation theory is a useful tool in the study of low energy reactions involving light particles. However the inclusion of heavy particles in chiral perturbation theory results in large contributions from loop diagrams which violate the standard power counting scheme. We review two methods, referred to as heavy baryon chiral perturbation theory and infrared regularization, which remove the high energy effects of the heavy particles and which therefore do not violate the power counting scheme. We then use these two methods to calculate the amplitude for pion photoproduction to fourth order and prove that the two amplitudes are equivalent.