  The competition between the radiative and nonradiative lifetimes determines
the optical quantum yield and plays a crucial role in the potential
optoelectronic applications of transition metal dichalcogenides (TMDC). Here,
we show that, in the presence of free carriers, an additional nonradiative
decay channel opens for excitons in TMDC monolayers. Although the usual Auger
decay channel is suppressed at low doping levels by the simultaneous momentum
and energy conservation laws, exciton-phonon coupling relaxes this suppression.
By solving a Bethe-Salpeter Equation, we calculate the phonon-assisted Auger
decay rates in four typical TMDCs as a function of doping, temperature, and
dielectric environment. We find that even for a relatively low doping of
10$^{12}$ cm$^{-2}$, the nonradiative lifetime ranges from 16 ps to 165 ps in
different TMDCs, offering competition to the radiative decay channel.
