  We consider a model with a complex scalar field that couples to $(e,\mu)$ or
$(\mu,\tau)$ within the "longevity" window: $[|m_{l_1} - m_{l_2}|, m_{l_1} +
m_{l_2}]$ in which $l_1$ and $l_2$ are the two different charged leptons.
Within such a mass window, even a relatively large coupling (e.g. of the size
commensurate with the current accuracy/discrepancy in the muon $g-2$
experiment) leads to long lifetimes and macroscopic propagation distance
between production and decay points. We propose to exploit several existing
neutrino experiments and one future experiment to probe the parameter space of
this model. For the $\mu-e$ sector, we exploit the muonium decay branching
ratio and the production and decay sequence at the LSND experiment, excluding
the parametric region suggested by $g_\mu -2$ anomaly. For the $\tau-\mu$
sector, we analyze three main production mechanisms of scalars at beam dump
experiments: the Drell-Yan process, the heavy meson decay, and the muon
scattering. We explore the constraints from the past CHARM and NuTeV
experiments, and evaluate sensitivity for the proposed beam dump experiment,
SHiP. The latter can thoroughly probe the parameter space relevant for the
$g_\mu - 2$ anomaly.
