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On Primary Relations at Tree-level in String Theory and Field Theory

Qian Ma, Yi-Jian Du, Yi-Xin Chen

TL;DR

The paper shows that all tree-level KK and BCJ relations in both string theory and field theory can be generated from two primaries, dramatically reducing the independent color-ordered amplitudes from $N!$ to $(N-3)!$. It introduces a new momentum kernel for open-string amplitudes and derives a $(N-2)!$ color-decomposition via nested kinematic-phase commutators, with explicit field-theory limits. The authors establish a comprehensive framework linking cyclic symmetry, KK/BCJ relations, generalized U(1)-like decoupling identities, and general monodromy relations, including a constructive minimal-basis expansion for open-string disk amplitudes. These results yield practical, explicit formulas for amplitude decomposition and provide a rigorous bridge between string-theoretic monodromy and field-theoretic BCJ structures, with potential computational simplifications for gauge and gravity amplitudes.

Abstract

By the use of cyclic symmetry, KK relations and BCJ relations, one can reduce the number of independent $N$-point color-ordered tree amplitudes in gauge theory and string theory from $N!$ to $(N-3)!$. In this paper, we investigate these relations at tree-level in both string theory and field theory. We will show that there are two primary relations. All other relations can be generated by the primary relations. In string theory, the primary relations can be chosen as cyclic symmetry as well as either the fundamental KK relation or the fundamental BCJ relation. In field theory, the primary relations can only be chosen as cyclic symmetry and the fundamental BCJ relation. We will further show a kind of more general relation which can also be generated by the primary relations. The general formula of the explicit minimal-basis expansions for color-ordered open string tree amplitudes will be given and proven in this paper.

On Primary Relations at Tree-level in String Theory and Field Theory

TL;DR

The paper shows that all tree-level KK and BCJ relations in both string theory and field theory can be generated from two primaries, dramatically reducing the independent color-ordered amplitudes from to . It introduces a new momentum kernel for open-string amplitudes and derives a color-decomposition via nested kinematic-phase commutators, with explicit field-theory limits. The authors establish a comprehensive framework linking cyclic symmetry, KK/BCJ relations, generalized U(1)-like decoupling identities, and general monodromy relations, including a constructive minimal-basis expansion for open-string disk amplitudes. These results yield practical, explicit formulas for amplitude decomposition and provide a rigorous bridge between string-theoretic monodromy and field-theoretic BCJ structures, with potential computational simplifications for gauge and gravity amplitudes.

Abstract

By the use of cyclic symmetry, KK relations and BCJ relations, one can reduce the number of independent -point color-ordered tree amplitudes in gauge theory and string theory from to . In this paper, we investigate these relations at tree-level in both string theory and field theory. We will show that there are two primary relations. All other relations can be generated by the primary relations. In string theory, the primary relations can be chosen as cyclic symmetry as well as either the fundamental KK relation or the fundamental BCJ relation. In field theory, the primary relations can only be chosen as cyclic symmetry and the fundamental BCJ relation. We will further show a kind of more general relation which can also be generated by the primary relations. The general formula of the explicit minimal-basis expansions for color-ordered open string tree amplitudes will be given and proven in this paper.

Paper Structure

This paper contains 27 sections, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A new defined momentum kernel and the $(N-2)!$-formula of color decomposition in string theory
  3. A new defined momentum kernel in string theory
  4. The $(N-2)!$-formula of color decomposition in string theory
  5. Example
  6. General discussion
  7. Generating KK and BCJ relations by primary relations
  8. Generating $U(1)$-like decoupling identity by primary relations
  9. Fundamental BCJ relation and cyclic symmetry as the primary relations
  10. Fundamental KK relation and cyclic symmetry as the primary relations
  11. Generating generalized $U(1)$-like decoupling identity by $U(1)$-like decoupling identity
  12. Examples
  13. General discussion
  14. Generating KK-BCJ relation by generalized $U(1)$-like decoupling identity
  15. Examples
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Contour for KK-BCJ relation
  • Figure 2: Contour for general monodromy relation