Zero-norm states and High-energy Symmetries of String Theory

TL;DR

This paper derives stringy Ward identities from zero-norm states in open bosonic string theory, enabling algebraic determination of high-energy scattering amplitude relations across all energy levels and loop orders, revealing new amplitude components.

Contribution

It introduces a method to derive all-energy Ward identities from zero-norm states, avoiding saddle point calculations, and uncovers new high-energy amplitude components crucial for gauge invariance.

Findings

Derived Ward identities valid at all energies and loops.

Determined proportionality constants between string scattering amplitudes.

Discovered new high-energy amplitude components essential for gauge invariance.

Abstract

We derive stringy Ward identities from the decoupling of two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. These Ward identities are valid to all energy and all loop orders in string perturbation theory. The high-energy limit of these stringy Ward identities can then be used to fix the proportionality constants between scattering amplitudes of different string states algebraically without referring to Gross and Mende's saddle point calculation of high-energy string-loop amplitudes. As examples, all Ward identities for the mass level 4 and 6 are derived, their high-energy limits are calculated and the proportionality constants between scattering amplitudes of different string states are determined. In addition to those identified before, we discover some new nonzero components of high-energy amplitudes not found previously by Gross and Manes. These components are essential to preserve massive gauge invariances or decouple massive zero-norm states of string theory. A set of massive scattering amplitudes and their high energy limits are calculated explicitly for each mass level to justify our results.