The sensitivity of $\mu\to e$ processes to $\tau$ flavour change

Transforming a $\mu$ to a $\tau$, then the $\tau$ to to an $e$, results in $\mu \to e$. In an EFT framework, we explore the sensitivity of $\mu\to e$ observables to products of $(\mu\to \tau)\times (\tau \to e)$ interactions, and show that the exceptional sensitivity of upcoming $\mu \to e$ experiments could allow to probe parameter space beyond the reach of upcoming $\tau \to l$ searches in Higgs, $\tau$ and $B$ decays. We describe the $\tau \to l$ interactions as dimension six operators in the SM EFT, identify pairs of them giving interesting contributions to $\mu \to e$ processes, and obtain the anomalous dimensions mixing those pairs into dimension eight $\mu\to e$ operators. We find that $\mu \to e$ processes are sensitive to $\tau$ flavour-changing $B$ decays at rates comparable to current $B$ anomalies, but cannot reduce rates -- as appropriate in many current $B$ anomalies -- because they do not interfere with the SM.