  We explore the phenomenology of an $SU(2)$-singlet doubly charged scalar at
the high and low energy frontier. Such a particle is predicted in different new
physics models, like left-right symmetric models or the Zee-Babu model.
Nonetheless, since its interactions with Standard Model (SM) leptons are gauge
invariant, it can be consistently studied as a UV complete SM extension. Its
signatures range from same-sign di-lepton pairs to flavour changing decays of
charged leptons to muonium-antimuonium oscillations. In this article, we use a
systematic effective-field-theory approach for studying the low-energy
observables and comparing them consistently to collider bounds. For this
purpose, experimental searches for doubly charged scalars at the Large Hadron
Collider are reinterpreted, including large width effects, and projections for
exclusion and discovery reaches in the high-luminosity phase are provided. The
sensitivities of the future International Linear Collider and Compact Linear
Collider for the doubly charged scalar are presented with focus on di-lepton
final states and resonant production. Theoretically and phenomenologically
motivated benchmark scenarios are considered showing the different impact of
low- and high-energy observables. We find that future low- and high-energy
experiments display strong complementarity in studying the parameter space of
the model.
