On-Shell Recursion Relations for Generic Theories

TL;DR

The authors prove that tree-level on-shell recursion relations (BCFW-type) hold for a broad class of two-derivative theories with multiple particle species, provided amplitudes vanish as complex momentum deformations become large. They demonstrate this vanishing behavior for amplitudes with at least one gluon (spin ≤1) and at least one graviton (spin ≤2) by employing a background-field method, carefully chosen gauges, and a dilaton-assisted field redefinition to reveal spin-Lorentz symmetry. The work extends known recursion results to generic gauge and gravity theories with scalars and fermions, clarifying the conditions under which recursion relations exist and providing explicit tensor structures X and Y that control high-z behavior. It also notes connections to KLT relations and existing YM/gravity correspondences, underscoring the broad applicability of on-shell recursive techniques beyond conventional Lagrangian-dependent computations.

Abstract

We show that on-shell recursion relations hold for tree amplitudes in generic two derivative theories of multiple particle species and diverse spins. For example, in a gauge theory coupled to scalars and fermions, any amplitude with at least one gluon obeys a recursion relation. In (super)gravity coupled to scalars and fermions, the same holds for any amplitude with at least one graviton. This result pertains to a broad class of theories, including QCD, N=4 SYM, and N=8 supergravity.

Figures (3)

• Figure 1: The unique diagram. Since gluons 1 and 2 meet directly at a vertex, the momentum flowing into the soft gluon is $p_1 + p_2$. For this reason, background field light-cone gauge cannot be chosen for this diagram.
• Figure 2: All diagrams with no hard propagators. The blobs represent insertions of a classical background that parameterizes all of the soft physics. In diagrams $a)-c)$ particles 1 and 2 are both gluons, while $d)$ and $e)$ are mixed diagrams. Diagrams $a)$ (the unique diagram) and $b)$ occur in pure gauge theory; after dotting into the appropriate polarizations, they vanish at large $z$ArkaniHamed:2008yf. Diagram $c)$ is proportional to $\eta^{ab}$, so it preserves the spin Lorentz symmetry and is $\mathcal{O}(1/z)$ after dotting into polarizations. Diagrams $d)$ and $e)$ are $\mathcal{O}(1/z)$ after dotting into polarizations.
• Figure 3: An example of a diagram with only hard fermion propagators. Naively, the leading $z$ contribution goes as $\mathcal{O}(1)$ and comes from taking a $q$ from every propagator numerator. However, after dotting into the external polarization for particle 1, we find that every such diagram vanishes.