The Lauricella Functions and Exact String Scattering Amplitudes

TL;DR

This paper expresses 26D open bosonic string scattering amplitudes using Lauricella functions, revealing underlying symmetries and deriving new recurrence relations applicable across all energy scales.

Contribution

It introduces a novel representation of string scattering amplitudes with Lauricella functions, unifying previous results and deriving a new universal recurrence relation.

Findings

Reexpression of SSA in terms of Lauricella functions with SL(K+3,C) symmetry.

Derivation of known relations in various scattering limits from the Lauricella function framework.

Establishment of a new recurrence relation valid for all energies.

Abstract

We discover that the 26D open bosonic string scattering amplitudes (SSA) of three tachyons and one arbitrary string state can be expressed in terms of the D-type Lauricella functions with associated SL(K+3,C) symmetry. As a result, SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These include the linear relations first conjectured by Gross [1-3] and later corrected and proved in [4-9] in the hard scattering limit, the recurrence relations in the Regge scattering limit with associated SL(5,C) symmetry [19-21] and the extended recurrence relations in the nonrelativistic scattering limit with associated SL(4,C) symmetry [24] discovered recently. Finally, as an application, we calculate a new recurrence relation of SSA which is valid for all energies.