The stringy scaling loop expansion and stringy scaling violation

TL;DR

The paper introduces the stringy scaling loop expansion (SSLE) as a systematic, finite-term framework to compute high-energy open bosonic string amplitudes for arbitrary $n$-point configurations. It develops a vacuum-diagram representation with explicit Feynman rules and a combinatorial counting scheme based on partitions and Euler characteristics, enabling tractable analysis of leading and next-to-leading stringy scaling behavior. The authors demonstrate that leading-order results exhibit Bjorken-like, angle-independent scaling, while next-to-leading corrections produce genuine stringy scaling violations, and they extend the formalism from 4-point to generic $n$-point amplitudes. This approach provides a new tool for exploring high-energy string scatterings and potentially connecting to Regge regimes and other high-energy limits.

Abstract

We propose a systematic approximation scheme to calculate general $n$-point $HSSA$ 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.