Disk scattering of open and closed strings (I)

At the tree level, the scattering processes involving open and closed strings are described by a disk world-sheet with vertex operator insertions at the boundary and in the bulk. Such amplitudes can be decomposed as certain linear combinations of pure open string amplitudes. While previous relations have been established on the double cover (complex sphere) in this letter we derive them on the disk (upper complex half plane) allowing for different momenta of the left- and right-movers of the closed string. Formally, the computation of disk amplitudes involving both open and closed strings is reduced to considering the monodromies on the underlying string world-sheet.