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.
Highlights
As usual, these recursion relations rely on a complex deformation of the external momenta parameterized by a complex number z
We show that all amplitudes in a renormalizable theory are 5-line constructible
Our results apply to a general quantum field theory of massless particles in four dimensions, which we summarize as follows: Renormalizable theories
Summary
Let us define a broad covering space of recursion relations subject to a loose set of criteria. We demand that the external momenta remain on-shell and conserve momenta for all values of z. In four dimensions, these conditions are automatically satisfied if the momentum deformation is a complex shift of the holomorphic and anti-holomorphic spinors of external legs, λi → λi(z) = λi + zηi, λi → λi(z) = λi + zηi, i∈I i ∈ I,. When the specific elements of I and I are not very important, we will sometimes refer to this as an [|I|, |I| -line shift, where the labels are the orders of I and I. An on-shell recursion relation is obtained by applying Cauchy’s theorem to deform the contour out to z = ∞, in the process picking up all the residues of M(z) in the complex plane. Which should be considered as four constraints on ηi and ηi which are satisfied provided the number of reference spinors is sufficient
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