In-flight cLFV conversion: $e-\mu$, $e-\tau$ and $\mu-\tau$ in minimal extensions of the Standard Model with sterile fermions

We revisit charged lepton flavour in-flight conversions, in which a beam of electrons or muons is directed onto a fixed target, $e + N \to \mu +N$, $e + N \to \tau +N$ and $\mu + N \to \tau +N$, focusing on elastic interactions with a nucleus $N$. After a general discussion of this observable, we carry a full phenomenological analysis in the framework of minimal Standard Model extensions via sterile neutrinos, with a strong emphasis on the r\^ole of the increasingly more stringent constraints arising from other (low-energy) charged lepton flavour violation observables. Despite the potential interest of this observable, in particular in the light of certain upcoming facilities with the capability of very intense lepton beams, our study suggests that due to current bounds on three-body decays ($\ell_i \to 3 \ell_j$) and $\mu-e$ conversion in Nuclei, the expected number of conversions in such a minimal framework is dramatically reduced. An experimental observation of such a conversion would thus signal the presence of another source of flavour violation, possibly at tree-level.