This dissertation is most easily understood as two distinct periods of research. The first three chapters are dedicated to phenomenological interests in physics. An anomalous measurement of the top quark forward-backward asymmetry in both detectors at the Tevatron collider is explained by particle content from beyond the Standard Model. The extra field content is assumed to have originated from a grand unified group SU(5), and so only specific content may be added. Methods for spontaneously breaking the R-symmetry of supersymmetric theories, of phenomenological interest for any realistic supersymmetric model, are studied in the context of two-loop Coleman- Weinberg potentials. For a superpotential with a certain structure, which must include two different couplings, a robust method of spontaneously breaking the R-symmetry is established. The phenomenological studies conclude with an isospin analysis of B decays to kaons and pions. When the parameters of the analysis are fit to data, it is seen that an enhancement of matrix. elements in certain representations of isospin emerge. This is highly reminiscent of the infamous and unexplained enhancements seen in the K [right arrow] [pi][pi] system. We conjecture that this enhancement may be a universal feature of the flavor group, isospin in this case, rather than of just the K [right arrow] [pi][pi] system. The final two chapters approach the problem of counting degrees of freedom in quantum field theories. We examine the form of the Weyl anomaly in six dimensions with the Weyl consistency conditions. These consistency conditions impose constraints that lead to a candidate for the [alpha]-theorem in six dimensions. This candidate has all the properties that the equivalent theorems in two and four dimensions did, and, in fact, we show that in an even number of dimensions the form of the Euler density, the generalized Einstein tensor, and the Weyl transformations guarantee such a candidate exists. We go on to show that, unlike in two and four dimensions, the [alpha]-theorem is six dimensions has the opposite sign of its counterparts in lower dimensions, at least in perturbation theory. This would imply, if the result could be extended without the use of perturbation theory, that the number of degrees of freedom accessible at a certain energy scale would increase as that energy scale is decreased. This is contrary to the intuition from two and four dimensions. We comment on what renormalization group flows, if any, we might find to exhibit this behavior

We perform an isospin analysis of B decays to two pseudoscalars. The analysis extracts appropriate Cabbibo-Kobayashi-Maskawa matrix elements and short distance loop factors to allow for comparison of nonperturbative QCD effects in the reduced matrix elements of the amplitudes. In decays where penguin diagrams compete with tree-level diagrams we find that the reduced matrix elements of the penguin diagrams, which are singlets or doublets under isospin, are significantly enhanced compared with the triplet and fourplet contributions of the weak Hamiltonian. This similarity to the ΔI=12 rule in K→ππ decays suggests that, more generally, processes mediated by Hamiltonians in lower-dimensional isospin representations see enhancement over higher-dimensional ones in QCD. © 2014 American Physical Society.