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The Standard Model of particle physics is a phenomenologically successful description of the strong, weak, and electromagnetic interactions at all currently accessible energy scales with few exceptions \cite{Agashe:2014kda}. The notable deficiencies of the Standard Model are its inability to explain the matter anti-matter asymmetry, the existence of neutrino oscillations \cite{Fukuda:1998mi,Ahmad:2002jz}, the anomalous magnetic moment of the muon \cite{Bennett:2006fi,Hagiwara:2011af}, and its failure to provide a suitable candidate for the gravitationally observed dark matter \citep{Dolgov:1995np}. We explore an extension of the Standard Model that introduces a new gauge symmetry $L_\mu-L_\tau$ along with three right-handed neutrinos, and a symmetry breaking scalar field. The inclusion of right-handed neutrinos are motivated by the aforementioned neutrino oscillation data while the scalar field is motivated by cosmological bounds on a new $Z'$. We attempt to fit our model to the observed neutrino mass textures in the see-saw limit. Despite having a Lagrangian density with three Yukawa couplings, and four right-handed mass parameters we found the left handed neutrino mass matrix was controlled by only four independent quantities. We were attempting to fit to a set of five measured parameters $\{ \Delta m_{12}^2,\Delta m_{13}^2,\theta_{12},\theta_{23},\theta_{13} \}$. This was found to be impossible with our proposed model. Higher dimensional operators were introduced to allow the model to generate neutrino textures that agree with experiment. Our first minimal model was able to reproduce the correct neutrino textures with the exception of one of either $\theta_{13}$ or $\theta_{12}$ the disagreements was at the level of $25\%$. We found that our model was able to fit to the central value of neutrino data after the introduction of various combinations of dimension-five operators. The parametric dependence of these solutions were found to be incompatible with the $Z'$ as a progenitor of dark matter scenario proposed by Shuve and Yavin \cite{Shuve:2014doa}. The $Z'$ progenitor scenario and the see-saw mechanism seem to be distinct entities in the sense that for the former to be viable the dark matter candidate cannot play a significant role in the generation of neutrino textures. / Thesis / Master of Science (MSc)

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