High-precision theoretical predictions for particle physics at the intensity frontier

The intensity frontier of particle physics explores the existence of a new theory of fundamental interactions through precision measurements at relatively low energies. The design of more precise experiments has to be accompanied by the development of more accurate Monte Carlo event generators for the simulation of the relevant processes. In order to match the experimental precision, such essential tools must include appropriate higher-order radiative corrections. Specifically, the new generation of precision experiments demands the computation of next-to-leading order or next-to-next-to-leading order corrections with the resummation of logarithmically enhanced terms. This thesis presents new cutting-edge calculations of various processes that are relevant at the intensity frontier. The positron energy spectrum of muon decay is computed with unprecedented precision to support the search for charged lepton flavour violation. The other processes are motivated by the data-driven determination of the hadronic contribution to the anomalous magnetic moment of the muon, both in the time-like and space-like approaches. The latter is the basis of the MUonE experiment, which aims to evaluate the hadronic vacuum polarisation contribution by measuring the running of the electromagnetic coupling in the space-like region through the scattering of high-energy muons on atomic electrons. In this regard, two different processes are considered, the elastic muon-electron scattering and the lepton pair production in muon-nucleus scattering, which constitute the signal and the main background of the experiment, respectively. The pion pair production in electron-positron annihilation is instead computed in the context of the traditional time-like dispersive approach, assuming inclusive radiation. This process is the dominant contribution to the R-ratio and it is particularly important for experiments based on the energy scan method, such as CMD-3. Special attention is given to the treatment of the pion form factor in the computation of loop diagrams, with the aim of improving the theoretical description of the forward-backward asymmetry. The implementation of these calculations into fully differential Monte Carlo tools is essential to provide theoretical support to present and future experiments at the intensity frontier.