Calculations of Zero-norm States and Reduction of Stringy Scattering Amplitudes

TL;DR

This paper introduces a simplified method to generate zero-norm states in open bosonic string theory, demonstrating their role in fixing scattering amplitudes across mass levels and revealing a decoupling phenomenon valid at all energies.

Contribution

It provides a new simplified approach to generate zero-norm states and shows their universal decoupling effect on string scattering amplitudes at all energy levels.

Findings

Zero-norm states up to the fourth massive level are calculated.

String scattering amplitudes are fixed by higher spin states at each mass level.

Decoupling phenomenon is valid at all energies, not just high-energy regimes.

Abstract

We give a simplified method to generate two types of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. Zero-norm states up to the fourth massive level and general formulas of some zero-norm tensor states at an arbitrary mass level are calculated. On-shell Ward identities generated by zero-norm states and the factorization property of stringy vertex operators can then be used to argue that the string-tree scattering amplitudes of the degenerate lower spin propagating states are fixed by those of higher spin propagating states at each fixed mass level. This decoupling phenomenon is, in contrast to Gross's high-energy symmetries, valid to all energy. As examples, we explicitly demonstrate this stringy phenomenon up to the fourth massive level (spin-five), which justifies the calculation of two other previous approaches based on the massive worldsheet sigma-model and Witten's string field theory (WSFT).