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String amplitudes from field-theory amplitudes and vice versa

Song He, Fei Teng, Yong Zhang

TL;DR

The paper develops a general IBP-based algorithm to reduce any massless string correlator to a logarithmic CHY half-integrand, enabling a CHY-field-theory amplitude that, via a disk/sphere double copy, reconstructs the original string amplitude. This two-step process—IBP reduction to ${ m I}'_n$ followed by SE simplification to ${ m I}''_n$—provides a practical and generic framework for open bosonic and heterotic strings, and it yields closed-form CHY integrands for the tree-level $(DF)^2+ ext{YM}+ ext{φ}^3$ theory across single- and multi-trace sectors. The approach unifies string and field-theory amplitudes under a double-copy perspective, clarifies the role of ${ m Z}$-type disk integrals, and generates all-multiplicity CHY representations that respect BCJ relations and SE. Potential extensions include massive string states, loop-level amplitudes, and connections to ambitwistor-string models, offering a path to deeper understanding of string/field-theory dualities and computational techniques in scattering amplitudes.

Abstract

We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk/sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL(2) weight. By applying the reduction to open bosonic/heterotic strings, we get a closed-form CHY integrand for the $(DF)^2+\text{YM}+φ^3$ theory.

String amplitudes from field-theory amplitudes and vice versa

TL;DR

The paper develops a general IBP-based algorithm to reduce any massless string correlator to a logarithmic CHY half-integrand, enabling a CHY-field-theory amplitude that, via a disk/sphere double copy, reconstructs the original string amplitude. This two-step process—IBP reduction to followed by SE simplification to —provides a practical and generic framework for open bosonic and heterotic strings, and it yields closed-form CHY integrands for the tree-level theory across single- and multi-trace sectors. The approach unifies string and field-theory amplitudes under a double-copy perspective, clarifies the role of -type disk integrals, and generates all-multiplicity CHY representations that respect BCJ relations and SE. Potential extensions include massive string states, loop-level amplitudes, and connections to ambitwistor-string models, offering a path to deeper understanding of string/field-theory dualities and computational techniques in scattering amplitudes.

Abstract

We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk/sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL(2) weight. By applying the reduction to open bosonic/heterotic strings, we get a closed-form CHY integrand for the theory.

Paper Structure

This paper contains 18 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. CHY integrand from string correlator
  3. General claim
  4. Derivation: IBP reduction and double copy
  5. IBP reduction
  6. String double copy
  7. New formulas for bosonic and heterotic strings
  8. The single-trace integrand
  9. Conclusion and discussion
  10. More on IBP
  11. Single trace integrand with three gravitons
  12. Single-trace examples
  13. Consistency checks
  14. Towards multi-trace integrands
  15. Pure gluon integrand
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Double copy for string amplitudes, and CHY half-integrands from string correlator via IBP and SE.
  • Figure 2: Examples of rational functions of $z_{ij}$ with no numerators or connected subcycles after gauge fixing. For this case, gauge fixing always results in a single labeled tree.