Wing Sheung Chan

Introduction “You see things; and you say ‘Why?’ But I dream things that never were; and I say ‘Why not?’ ” — George B. Shaw, “the Serpent”, Back to Methuselah Physics is the study of Nature. Its main goal is to understand the universe and the world around us. As such, the ultimate destination of physics (whether or not reachable) is to obtain a self-consistent theory that accurately describes everything that exists/happens, existed/happened, or will exist/will happen in the universe. Such a hypothetical theory is commonly referred to as a theory of everything (TOE). It is without a doubt still a distant goal. Nonetheless, physicists have made remarkable progress in the past centuries. In particular, they brought us the two most important theoretical frameworks in modern physics, namely the general theory of relativity (or simply general relativity, GR) and the Standard Model of particle physics (or simply the Standard Model, SM). Together they are the closest thing we have to a TOE. Over the past century, both theories have been put under the scrutiny of countless experiments, none of which was able to refute the theories definitely. Yet, despite the enormous success, the facts that GR and the SM are incompatible with each other and that there are observations that could not be explained by the theories imply that neither of them is a complete theory in explaining Nature. Thus, understanding what is wrong or missing in these theories has become a great challenge for physicists today. Luckily, we are not searching in complete darkness. Driven and constrained by high-energy experiments, precision measurements and cosmological observations, theorists are able to conjecture beyond-the-Standard-Model (BSM) theories that better explain the universe. There are many of these theories and they are all waiting to be tested by all the ever improving, limit-pushing physics experiments. Among others, (charged) lepton flavour violation is one of the most promising and plausible BSM phenomena. Leptons are a class of elementary particles in the SM that come in three generations, or so-called flavours. According to the SM in its current formulation, the number of charged leptons in each flavour does not change in any physical proces s † . Violation of this rule is known as lepton flavour violation. However, such a rule † Unless neutrino mixing is considered. Nonetheless, violations of charged lepton flavour in point interactions due to neutrino mixing have only vanishingly small probabilities, and are negligible when actual observations are concerned. There are also ongoing debates on whether and how neutrino mixing should be considered as part of the SM. More will be discussed in the upcoming chapter. 1