Inter- and intra-valley exciton transition due to electron-hole exchange interaction in monolayer MoS$_{2}$
We investigate the exciton transition due to the electron-hole exchange interaction in monolayer MoS$_{2}$. Both the long- and short-range parts of the intra- and inter-valley electron-hole exchange interaction are considered. We find that the short-range (long-range) exchange interaction can cause both the inter- and intra-valley (only the intra-valley) bright exciton transitions. With the intra-valley bright exciton transition channel nearly forbidden due to the large splitting of the valence bands, the inter-valley channel due to the short-range exchange interaction can cause a {\em fast} bright exciton transition. From experiment data, the short-range exchange interaction energy is estimated to be 1.4 meV.