  In monolayer Transition Metal Dichalcogenides (TMDs) the valence and
conduction bands are spin split because of the strong spin-orbit interaction.
In tungsten-based TMDs the spin-ordering of the conduction band is such that
the so-called dark exciton, consisting of an electron and a hole with opposite
spin orientation, has lower energy than the A exciton. A possible mechanism
leading to the transition from bright to dark excitons involves the scattering
of the electrons from the upper to the lower conduction band state in K. Here
we exploit the valley selective optical selection rules and use two-color
helicity-resolved pump-probe spectroscopy to directly measure the intravalley
spin-flip relaxation dynamics of electrons in the conduction band of
single-layer WS$_2$. This process occurs on a sub-ps time scale and it is
significantly dependent on the temperature, indicative of a phonon-assisted
relaxation. These experimental results are supported by time-dependent
ab-initio calculations which show that the intra-valley spin-flip scattering
occurs on significantly longer time scales only exactly at the K point. In a
realistic situation the occupation of states away from the minimum of the
conduction band leads to a dramatic reduction of the scattering time.
