  When mu-e-gamma (or tau-mu-gamma) loop involves a vector boson, the amplitude
is suppressed by more than two powers of heavy particle masses. However we show
that the scalar boson loop diagrams are much less damped. Particularly, the
loop amplitude in which the intermediate fermion and scalar boson have
comparable masses is as large as possible, as allowed by the decoupling
theorem. Such a situation is realized in the "universal extra dimension
theory", and can yield a large enough rate to be detectable in current
experiments. Our investigation involves precise calculation of the scalar boson
loop's dependence on the masses of the intermediate states.
