  The photoluminescence (PL) spectrum of transition metal dichalcogenides
(TMDs) shows a multitude of emission peaks below the bright exciton line and
not all of them have been explained yet. Here, we study the emission traces of
phonon-assisted recombinations of momentum-dark excitons. To this end, we
develop a microscopic theory describing simultaneous exciton, phonon and photon
interaction and including consistent many-particle dephasing. We explain the
drastically different PL below the bright exciton in tungsten- and
molybdenum-based materials as result of different configurations of bright and
dark states. In good agreement with experiments, we show that WSe$_2$ exhibits
clearly visible low-temperature PL signals stemming from the phonon-assisted
recombination of momentum-dark excitons.
