Simple Recursion Relations for General Field Theories

TL;DR

This paper develops a unified framework of recursion relations for constructing scattering amplitudes in all four-dimensional massless quantum field theories, revealing their power and limitations.

Contribution

It introduces a generalized set of recursion relations applicable to all 4D massless theories, including renormalizable, non-renormalizable, and supersymmetric models.

Findings

All amplitudes in renormalizable theories are 5-line constructible.

Amplitudes are 3-line constructible if external particles have spin or equal charge scalars.

Standard Model and supersymmetric theories are 3-line constructible.

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.