# String Correlators: Recursive Expansion, Integration-by-Parts and Scattering Equations

## TL;DR

This paper develops recursive methods to expand and simplify string correlators, deriving new CHY formulas for complex multitrace amplitudes in heterotic and bosonic string theories, advancing the understanding of string and gauge theory relations.

## Contribution

It introduces a recursive expansion for multitrace string correlators and derives a novel CHY formula for $(DF)^2 + {
m YM} + ^3$ amplitudes, extending previous single-trace results.

## Key findings

- Recursive expansion reduces complex correlators to simpler Parke-Taylor factors.
- Derived a closed-form CHY formula for multitrace amplitudes in heterotic and bosonic strings.
- Validated the formula through factorization and consistency checks.

## Abstract

We further elaborate on the general construction proposed in arXiv:1812.03369, which connects, via tree-level double copy, massless string amplitudes with color-ordered QFT amplitudes that are given by Cachazo-He-Yuan formulas. The current paper serves as a detailed study of the integration-by-parts procedure for any tree-level massless string correlator outlined in the previous letter. We present two new results in the context of heterotic and (compactified) bosonic string theories. First, we find a new recursive expansion of any multitrace mixed correlator in these theories into a logarithmic part corresponding to the CHY integrand for Yang-Mills-scalar amplitudes, plus correlators with the total number of traces and gluons decreased. By iterating the expansion, we systematically reduce string correlators with any number of subcycles to linear combinations of Parke-Taylor factors and similarly for the case with gluons. Based on this, we then derive a CHY formula for the corresponding $(DF)^2 + {\rm YM} + \phi^3$ amplitudes. It is the first closed-form result for such multitrace amplitudes and thus greatly extends our result for the single-trace case. As a byproduct, it gives a new CHY formula for all Yang-Mills-scalar amplitudes. We also study consistency checks of the formula such as factorizations on massless poles.