Referenced internal-line double copy and application to gauge and gravitational beta functions

TL;DR

This work develops a systematic internal-line double-copy framework for one-loop string amplitudes in bosonic and heterotic theories by leveraging chiral splitting to expose left–right factorization. It analyzes the low-energy three-point amplitudes under $T^6$ compactification to extract universal, model-independent beta-function results, showing that the gauge beta function reproduces the standard field-theory coefficient while the gravitational beta function vanishes in this perturbative setting. Gravitational corrections to gauge running are identified as higher-dimensional-operator effects rather than genuine renormalizations of the gauge coupling. The results provide a universal string-based perspective on running couplings, clarifying how gravity can influence gauge dynamics only through higher-dimensional operators, and demonstrate a consistent, model-agnostic framework across bosonic and heterotic strings.

Abstract

Following the approach 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.

Figures (2)

• Figure 1: One-loop contributions to the three-point amplitude of closed strings. For diagram-(a) the distance between $z_1^{}$ and $z_2^{}$ are infinitesimally small $\sim \frac{1}{\tau_2}$ as $\tau_2\!\rightarrow\! \infty$ and form a propagator $\frac{1}{\,s_{12}^{}\,}$, whereas $z_3^{}$ is far away from $z_1^{}$ and $z_2^{}$. There are two similar diagrams with $z_{23}^{}\!\rightarrow\! 0$ and $z_{31}^{}\!\rightarrow\! 0$ (forming the pole structures $\frac{1}{\,s_{23}^{}\,}$ and $\frac{1}{\,s_{31}^{}\,}$) respectively. For the diagram-(b), all $\{z_1^{},z_2^{},z_3^{}\}$ are far away from each other.
• Figure 2: The fundamental region is shown as the regions where $|\tau|\space\!>\space\!1$ and $-\frac{1}{2}\!\!<\!\tau_1^{}\!\!<\!\frac{1}{2}\space$. A cutoff $L\!\!\gg\!\! 1$ on the $\tau_2^{}$ axis is introduced for the later calculation. The $\tau_2^{}\!>\!L$ region corresponds to the region of field-theory limit and $\tau_2^{} \space\!<\!L$ is the stringy region.