The Stringy Scaling Loop Expansion and Stringy Scaling Violation

Abstract We propose a systematic approximation scheme to calculate general string-tree level n-point hard string scattering amplitudes ($\text{HSSA}$) of open bosonic string theory. This stringy scaling loop expansion contains a 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 the coefficient and give the vacuum diagram representation and its Feynman rules for each term in the expansion of the $\text{HSSA}$. As an application to extending our previous calculation of n-point leading-order stringy scaling behavior of $\text{HSSA}$, we explicitly calculate some examples of 4-point next-to-leading-order stringy scaling violation terms.