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On-shell constructibility of tree amplitudes in general field theories

Timothy Cohen, Henriette Elvang, Michael Kiermaier

TL;DR

This work establishes a universal, gauge-invariant criterion for the on-shell constructibility of tree amplitudes in four-dimensional local field theories using all-line momentum shifts. By deriving a simple large-z falloff bound that depends only on the sum of external helicities and the minimal mass dimension of couplings, it shows that all-line recursion suffices to construct all n-point tree amplitudes with n>4 in power-counting renormalizable theories, and extends naturally to massive particles via a massive spinor-helicity framework. The authors connect these recursion relations to the MHV vertex expansion, illustrate the framework through diverse examples (including higher-derivative operators and Higgs–gluon fusion), and provide a physical interpretation: amplitudes requiring independent n-point interactions obstruct constructibility unless those interactions are fixed by symmetries. The results unify and generalize prior recursion analyses, offering practical criteria for when on-shell recursion captures full tree-level amplitudes across broad classes of theories and particle content.

Abstract

We study "on-shell constructibility" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles, both massless and massive. Our analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. We focus on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the MHV vertex expansion. We derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depends only on the mass dimensions of the coupling constants and on the sum of helicities of the external particles. Our proof is strikingly simple since it just relies on dimensional analysis and little-group transformation properties. In particular, the results demonstrate that all tree amplitudes with n>4 external states are constructible in any power-counting renormalizable theory. Aspects of all-line shift constructibility are illustrated in numerous examples, ranging from pure scalar theory and the massless Wess-Zumino model to theories with higher-derivative interactions, gluon-Higgs fusion, and Z-boson scattering. We propose a sharp physical interpretation of our constructibility criterion: the all-line shift fails precisely for those classes of n-point amplitudes that can receive local contributions from independent gauge-invariant n-field operators.

On-shell constructibility of tree amplitudes in general field theories

TL;DR

This work establishes a universal, gauge-invariant criterion for the on-shell constructibility of tree amplitudes in four-dimensional local field theories using all-line momentum shifts. By deriving a simple large-z falloff bound that depends only on the sum of external helicities and the minimal mass dimension of couplings, it shows that all-line recursion suffices to construct all n-point tree amplitudes with n>4 in power-counting renormalizable theories, and extends naturally to massive particles via a massive spinor-helicity framework. The authors connect these recursion relations to the MHV vertex expansion, illustrate the framework through diverse examples (including higher-derivative operators and Higgs–gluon fusion), and provide a physical interpretation: amplitudes requiring independent n-point interactions obstruct constructibility unless those interactions are fixed by symmetries. The results unify and generalize prior recursion analyses, offering practical criteria for when on-shell recursion captures full tree-level amplitudes across broad classes of theories and particle content.

Abstract

We study "on-shell constructibility" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles, both massless and massive. Our analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. We focus on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the MHV vertex expansion. We derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depends only on the mass dimensions of the coupling constants and on the sum of helicities of the external particles. Our proof is strikingly simple since it just relies on dimensional analysis and little-group transformation properties. In particular, the results demonstrate that all tree amplitudes with n>4 external states are constructible in any power-counting renormalizable theory. Aspects of all-line shift constructibility are illustrated in numerous examples, ranging from pure scalar theory and the massless Wess-Zumino model to theories with higher-derivative interactions, gluon-Higgs fusion, and Z-boson scattering. We propose a sharp physical interpretation of our constructibility criterion: the all-line shift fails precisely for those classes of n-point amplitudes that can receive local contributions from independent gauge-invariant n-field operators.

Paper Structure

This paper contains 29 sections, 114 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. All-line shift recursion relations
  3. From all-line shifts to the MHV vertex expansion
  4. Examples
  5. The simplest theories
  6. The Wess-Zumino model
  7. Gluons $\to$ Higgs fusion
  8. $F^m$ and $R^m$ operators
  9. Ultra helicity violating 1-loop amplitudes
  10. Massive particles
  11. Massive all-line shifts
  12. Spinor-helicity formalism for massive particles
  13. Dirac fermions:
  14. Spinor brackets:
  15. Massive vector bosons:
  16. ...and 14 more sections

Figures (2)

  • Figure 1: The particle content (a) and the seven MHV vertices (b) of the Wess-Zumino model.
  • Figure 2: A Feynman diagram (left) and the corresponding MHV vertex diagrams (right) of the NMHV amplitude $\langle fff\bar{f}\bar{f}\bar{f}\rangle$.