Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums

TL;DR

This paper computes the six-gluon disk amplitude in superstring theory, expressing it through hypergeometric functions and deriving algebraic identities that relate these functions, advancing understanding of string interactions with six external legs.

Contribution

It introduces a systematic method to calculate tree-level string amplitudes using world-sheet supersymmetry and hypergeometric function identities, providing new tools for analyzing six-point scattering processes.

Findings

Expressed the six-gluon amplitude via generalized hypergeometric functions.

Derived algebraic identities linking different hypergeometric functions.

Connected hypergeometric function expansions to Euler-Zagier sums and zeta-functions.

Abstract

The six gluon disk amplitude is calculated in superstring theory. This amplitude probes the gauge interactions with six external legs on Dp-branes, in particular including e.g. F^6-terms. The full string S-matrix can be expressed by six generalized multiple hypergeometric functions (triple hypergeometric functions), which in the effective action play an important role in arranging the higher order alpha' gauge interaction terms with six external legs (like F^6, D^4 F^4, D^2 F^5, D^6 F^4, D^2 F^6, ...). A systematic and efficient method is found to calculate tree-level string amplitudes by equating seemingly different expressions for one and the same string S-matrix: Comparable to Riemann identities appearing in string-loop calculations, we find an intriguing way of using world-sheet supersymmetry to generate a system of non-trivial equations for string tree-level amplitudes. These equations result in algebraic identities between different multiple hypergeometric functions. Their (six-dimensional) solution gives the ingredients of the string S-matrix. We derive material relevant for any open string six-point scattering process: relations between triple hypergeometric functions, their integral representations and their alpha'-(momentum)-expansions given by (generalized) Euler-Zagier sums or (related) Witten zeta-functions.