One-loop Massive Scattering Amplitudes and Ward Identities in String Theory

TL;DR

This paper computes one-loop scattering amplitudes for open bosonic strings, proving infinite Ward identities and analyzing their high-energy limits to determine proportionality constants among amplitudes of different string states.

Contribution

It explicitly proves one-loop Ward identities from zero-norm states and explores high-energy limits to fix amplitude proportionality constants, advancing understanding of string scattering at one-loop.

Findings

Proved infinite one-loop Ward identities in open bosonic string theory.

Derived high-energy proportionality constants between different string states.

Compared one-loop and tree-level Ward identities to address subtleties.

Abstract

We calculate bosonic open string one-loop massive scattering amplitudes for some low-lying string states. By using the periodicity relations of Jacobi theta functions, we explicitly prove an infinite number of one-loop type I stringy Ward identities derived from type I zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. The subtlety in the proofs of one-loop type II stringy Ward identities is discussed by comparing with those of string-tree cases. High-energy limit of these stringy Ward identities can be used to fix the proportionality constants between one-loop massive high-energy scattering amplitudes of different string states with the same momenta. These proportionality constants can not be calculated directly from sample calculations as we did previously in the cases of string-tree scattering amplitudes.