  We speculate about the possible interpretations of the recently observed
excess in the $h \to \tau \mu$ decay. We derive a robust lower bound on the
Higgs boson coupling strength to a tau and a muon, even in presence of the most
general new physics affecting other Higgs properties. Then we reevaluate
complementary indirect constraints coming from low energy observables as well
as from theoretical considerations. In particular, the tentative signal should
lead to $\tau \to \mu \gamma$ at rates which could be observed at Belle II. In
turn we show that, barring fine-tuned cancellations, the effect can be
accommodated within models with an extended scalar sector. These general
conclusions are demonstrated in explicit new physics models. Finally we show
how, given the $h \to \tau \mu$ signal, the current and future searches for
$\mu \to e \gamma$ and $\mu \to e$ nuclear conversions unambiguously constrain
the allowed rates for $h \to \tau e$.

High Energy Physics - Phenomenology