Investigating the invisible Light and weakly-interacting new physics

Several shortcomings of the Standard Model of particle physics can be addressed by the introduction of new particles. This thesis explores the phenomenology of such particles that potentially exist at relatively low energy scales, eluding detection owing to their feeble interactions with ordinary matter. The first part of the thesis focuses on sterile neutrinos and the lepton-number-violating process of neutrinoless double beta decay, which can shed light on the nature of neutrinos and the origin of their observed masses. The theory behind the prediction of the rate of this process is developed further. The phenomenology of a minimal extension to the Standard Model with two sterile neutrinos is studied, combining experimental results with cosmological observations, and it is shown that this minimal model will be completely probed in the inverted hierarchy by future experimental programmes. The second part primarily deals with the QCD axion, a hypothetical pseudoscalar particle that can solve the strong CP problem while also being a dark matter candidate. The extended hadronic axion model space that can serve as a target region for future experiments is discussed, along with the effects of CP-violating axion interactions on the values of fundamental constants. Finally, neutrinoless double beta decay processes accompanied by scalar emission are studied, and it is shown how features of these new processes can be disentangled from Standard Model backgrounds.