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Simple Recursion Relations for General Field Theories

Clifford Cheung, Chia-Hsien Shen, Jaroslav Trnka

TL;DR

This work develops a unified, minimal set of on-shell recursion relations for massless four-dimensional field theories by constructing a covering space that extends BCFW and Risager into general m-line shifts. By analyzing large-z behavior with a background-field skeleton, the authors derive universal bounds that determine when a given shift yields a valid recursion, separating Q^2=0 and Q^2≠0 cases and identifying how helicity, vertex valency, and derivatives affect constructibility. They classify theories by the minimal shift required to reconstruct all amplitudes, showing renormalizable theories are 5-line constructible, while many gauge theories, supersymmetric theories, and the Standard Model are 3-line constructible under reasonable charge assumptions. Non-renormalizable theories can be fully constructible only in restricted cases (no derivatives or limited derivative order), while some amplitudes in higher-derivative theories remain accessible. The paper demonstrates both the power and the limitations of an entirely on-shell, recursion-based formulation of quantum field theory and outlines future directions for soft shifts, higher dimensions, and loops.

Abstract

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.

Simple Recursion Relations for General Field Theories

TL;DR

This work develops a unified, minimal set of on-shell recursion relations for massless four-dimensional field theories by constructing a covering space that extends BCFW and Risager into general m-line shifts. By analyzing large-z behavior with a background-field skeleton, the authors derive universal bounds that determine when a given shift yields a valid recursion, separating Q^2=0 and Q^2≠0 cases and identifying how helicity, vertex valency, and derivatives affect constructibility. They classify theories by the minimal shift required to reconstruct all amplitudes, showing renormalizable theories are 5-line constructible, while many gauge theories, supersymmetric theories, and the Standard Model are 3-line constructible under reasonable charge assumptions. Non-renormalizable theories can be fully constructible only in restricted cases (no derivatives or limited derivative order), while some amplitudes in higher-derivative theories remain accessible. The paper demonstrates both the power and the limitations of an entirely on-shell, recursion-based formulation of quantum field theory and outlines future directions for soft shifts, higher dimensions, and loops.

Abstract

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.

Paper Structure

This paper contains 29 sections, 49 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Covering Space of Recursion Relations
  3. Definition
  4. Factorization
  5. Recursion Relations
  6. Large $z$ Behavior of Amplitudes
  7. Ansatz
  8. Large $z$ Behavior
  9. External Wavefunctions.
  10. Numerator and Denominator.
  11. Full Amplitude.
  12. ($Q^2 \neq0$)
  13. ($Q^2 =0$)
  14. $[0,m\rangle$-Line and $[m,0\rangle$-Line Shifts.
  15. $[1,m-1\rangle$-Line and $[m-1,1\rangle$-Line Shifts.
  16. ...and 14 more sections

Figures (2)

  • Figure 1: A skeleton diagram for an $Q^2 \neq 0$ shift. Here straight lines are hard particles and curved lines are soft backgrounds. Color segments are propagators, and red and green denotes those that scale as $1/z$ and $1/z^2$ at large $z$, respectively.
  • Figure 2: Factorization channels in the 6-point scalar amplitude in $\phi^4$ theory. The left and right diagrams show the factorization channels for the general case and the case of a $U(1)$ charged scalar, respectively.