We study the decay processes of mediator particles, which are predicted
in the light dark matter scenario, especially for the case in which their masses lie
near the threshold region. We consider a vector mediator particle as an example,
whose mass is around twice the muon or the bottom quark mass. We develop a
method to quantitatively deal with the threshold singularity of the decay processes.
We found that for the mass of the mediator being above the threshold, the decay
width is enhanced by the so-called Sommerfeld effect. On the other hand, for the
mass below the threshold, the decay width is also enhanced due to the mixing
between the mediator and the bound states of the final state particles.