Flavoured neutrino mass models

The only direct experimental evidence for physics beyond the Standard Model are the oscillations of neutrino species. Explaining this surprising discovery has led to a variety of potential New Physics models. Since neutrino oscillations demonstrate that lepton flavour is not conserved in Nature, New Physics models tend to introduce additional lepton flavour and sometimes even lepton number violating physics. The validity of any New Physics setting is assessed based on the consistency of its predictions with experimental data. In the near future, lepton flavour and/or number violating conversions of bound muons are expected to undergo the most dramatic experimental advances. By improving currents limits by several orders of magnitude, these reactions will become the most sensitive probe for charged lepton flavour/number violation. Therefore, exploring new opportunities such as these is essential to unravel novel physics beyond the Standard Model. The goal of this thesis is to contribute to improving the testability of New Physics models with respect to two different aspects, focusing on neutrino models with additional lepton flavour and/or lepton number violation. First, both the lepton flavour violating $\mu^-$-- $e^-$ conversion and the lepton flavour and lepton number violating $\mu^-$-- $e^+$ conversion require solid theoretical predictions to fully exploit their potential for investigating promising New Physics models. Since both types of bound muon conversions currently lack certain elements in their theoretical treatment, we work towards closing these gaps. To that end, we present our detailed and comprehensive computations which aim at making both processes accessible to the particle physics community. Furthermore, we compare predictions from a selection of New Physics models to current experimental data and future expected sensitivities. We also show how experiments at low energies, indirectly looking for New Physics via charged lepton flavour and lepton number violating processes, and experiments at high energies, directly looking for new particles, can provide complementary constraints. Thus, our results considerably strengthen the case for low-energy lepton flavour and lepton number violation searches being vital contributions to the search for physics beyond the Standard Model. Second, when deriving model predictions, one must take into account that many New Physics models are defined at high energy scales, whereas experimental oscillation data are measured at low energies. Since the parameters of a theory change with the energy scale under consideration, it is crucial to incorporate renormalisation group effects. In this context, we present the first comprehensive renormalisation group analysis of the Littlest Seesaw model. Our analysis demonstrates that the inclusion of running effects is crucial when confronting New Physics models with oscillation data.