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Amplitude Relations in Non-linear Sigma Model

Gang Chen, Yi-Jian Du

TL;DR

This work extends the web of tree-level amplitude relations from Yang-Mills to the $U(N)$ non-linear sigma model in Cayley parametrization by leveraging Berends-Giele recursion. It proves off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents and derives their on-shell counterparts, including the $U(1)$-decoupling identity and fundamental BCJ relation for amplitudes. It then argues that general KK and BCJ relations, the minimal-basis expansion, and total-amplitude constructions (DDM and KLT) carry over to this EFT, yielding a consistent, YM-like algebraic structure. Together, these results illuminate a color-kinematic parallel in a low-energy effective theory and enable streamlined amplitude computations via established decompositions and double-copy-like frameworks.

Abstract

In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.

Amplitude Relations in Non-linear Sigma Model

TL;DR

This work extends the web of tree-level amplitude relations from Yang-Mills to the non-linear sigma model in Cayley parametrization by leveraging Berends-Giele recursion. It proves off-shell identity and fundamental BCJ relation for even-point currents and derives their on-shell counterparts, including the -decoupling identity and fundamental BCJ relation for amplitudes. It then argues that general KK and BCJ relations, the minimal-basis expansion, and total-amplitude constructions (DDM and KLT) carry over to this EFT, yielding a consistent, YM-like algebraic structure. Together, these results illuminate a color-kinematic parallel in a low-energy effective theory and enable streamlined amplitude computations via established decompositions and double-copy-like frameworks.

Abstract

In this paper, we investigate tree-level scattering amplitude relations in non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy -decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total -point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.

Paper Structure

This paper contains 19 sections, 47 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preparation: Feynman rules and Berends-Giele recursion
  3. Feynman rules
  4. Berends-Giele recursion
  5. Off-shell and on-shell $U(1)$ identity from Berends-Giele recursion
  6. Four-point example
  7. Eight-point example
  8. General proof
  9. Off-shell and on-shell fundamental BCJ relation from Berends-Giele recursion
  10. Four-point example
  11. Eight-point example
  12. General proof
  13. General KK, BCJ relations, minimal-basis expansion and formulations of total amplitudes
  14. General KK, BCJ relations and minimal-basis expansion
  15. Formulations of total amplitudes
  16. ...and 4 more sections

Figures (8)

  • Figure 1: Sum of diagrams with $\alpha_1$ connected with the off-shell leg directly via four-point vertex in $U(1)$ identity.
  • Figure 2: A diagram with lower-point substructure of $U(1)$ identity.
  • Figure 3: Sum of diagrams with $\alpha_1$ connected with the off-shell leg directly via four-point vertex in BCJ relation.
  • Figure 4: A diagram with lower-point substructure of BCJ relation.
  • Figure 5: Convention in section 3
  • ...and 3 more figures