  Understanding quasiparticle band structures of transition metal
dichalcogenides (TMDs) is critical for technological advances of these
materials for atomic layer electronics and photonics. Although theoretical
calculations to date have shown qualitatively similar features, there exist
subtle differences which can lead to important consequences in the device
characteristics. For example, most calculations have shown that all single
layer (SL) TMDs have direct band gaps, while some have shown that $SL-WSe_2$
have an indirect gap. Moreover, there are large variations in the reported
quasiparticle gaps, corresponding to large variations in exciton binding
energies. By using a comprehensive form of scanning tunneling spectroscopy, we
have revealed detailed quasiparticle electronic structures in TMDs, including
the quasi-particle gaps, critical point energy locations and their origins in
the Brillouin Zones (BZs). We show that $SL-WSe_2$ actually has an indirect
quasi-particle gap with the conduction band minimum located at the Q point
(instead of K), albeit the two states are nearly degenerate. Its implications
on optical properties are discussed. We have further observed rich
quasi-particle electronic structures of TMDs as a function of atomic structures
and spin-orbital couplings.
