Completeness and Complementarity for $\mu \to e \gamma$, $\mu \to 3e$ and $\mu \to e$ conversion

Lepton Flavour Violation(LFV) is New Physics that must occur, but is stringently constrained by experiments searching for mu to e flavour change, such as $\mu \to e \gamma$, $\mu \to 3e$ and $\mu \to e$ conversion. However, in an Effective Field Theory(EFT) parametrisation, there are many more $\mu \leftrightarrow e$ operators than restrictive constraints, so determining operator coefficients from data is a remote dream. It is nonetheless interesting to learn about New Physics from data, so this manuscript introduces "observable-vectors" in the space of operator coefficients, which identify at any scale the combination of coefficients probed by the observable. These vectors have at least partpermil overlap with most of the coefficients, and are used to study whether $\mu \to e \gamma$, $\mu \to 3e$ and $\mu \to e$ conversion give complementary information about New Physics. The appendix gives updated sensitivities of these processes, (and a subset of LFV tau decays), to operator coefficients at the weak scale in the SMEFT and in the EFT below mW.