The stringy scaling loop expansion and stringy scaling violation

We propose a systematic approximation scheme to calculate general string-tree level n-point hard string scattering amplitudes (HSSA) of open bosonic string theory. This stringy scaling loop expansion contains finite number of vacuum diagram terms at each loop order of scattering energy due to a vacuum diagram contraint and a topological graph constraint. In addition, we calculate coefficient and give the vacuum diagram representation and its Feynman rules for each term in the expansion of the HSSA. As an application to extending our previous calculation of n-point leading order stringy scaling behavior of HSSA, we explicitly calculate some examples of 4-point next to leading order stringy scaling violation terms.