Referenced internal-line decomposition of one-loop string integrand and application to gauge and gravitational beta functions

TL;DR

This paper develops a systematic decomposition method for one-loop string integrands, enabling the analysis of gauge and gravitational beta functions in heterotic string theory, revealing model-independent results through internal state decomposition.

Contribution

It introduces a generalized internal-line decomposition approach for one-loop string integrands applicable to bosonic and heterotic theories, facilitating beta function analysis.

Findings

Beta functions vanish trivially due to supersymmetry.

Decomposition yields model-independent results.

Method applicable to various string theories.

Abstract

Following the idea of Refs.[1,2], the double-copy-like decomposition of exchanged internal states in the world-line limit of one-loop string amplitudes is systematically formulated and generalized to both bosonic and heterotic string theories. As an application, the one-loop beta functions for the gauge and gravitational coupling constants are investigated by analyzing the low-energy field-theory limit of the corresponding three-point one-loop amplitudes in heterotic string theory under a naive $T^6$ compactification. Due to supersymmetry, these beta functions vanish trivially. However, by decomposing the scattering integrand according to the different internal loop-exchanged states, the most general model-independent results are obtained.