On the Tree-Level Structure of Scattering Amplitudes of Massless Particles

TL;DR

<3-5 sentence high-level summary> The paper extends the on-shell recursion program to all theories of massless particles by proving the existence of generalized BCFW-like recursion relations that include a boundary term expressed through a subset of zeros of the amplitude. It shows that the boundary term can be written as a sum over lower-point amplitudes weighted by factors determined by the locations of zeros and poles, making the full tree-level amplitude reconstructible from three-particle amplitudes. The authors analyze the UV (large-z) behavior and the collinear/multi-particle limits to fix these zeros and demonstrate consistency via explicit examples in gauge and gravity–coupled theories, including gluons and Einstein–Maxwell theory. The framework provides a unifying perspective on tree-level constructibility and offers practical formulas for computing amplitudes from minimal input data, with implications for understanding the analytic structure of scattering in massless theories.

Abstract

We provide a new set of on-shell recursion relations for tree-level scattering amplitudes, which are valid for any non-trivial theory of massless particles. In particular, we reconstruct the scattering amplitudes from (a subset of) their poles and zeroes. The latter determine the boundary term arising in the BCFW-representation when the amplitudes do not vanish as some momenta are taken to infinity along some complex direction. Specifically, such a boundary term can be expressed as a sum of products of two on-shell amplitudes with fewer external states and a factor dependent on the location of the relevant zeroes and poles. This allows us to recast the amplitudes to have the standard BCFW-structure, weighted by a simple factor dependent on a subset of zeroes and poles of the amplitudes. We further comment on the physical interpretation of the zeroes as a particular kinematic limit in the complexified momentum space. The main implication of the existence of such recursion relations is that the tree-level approximation of any consistent theory of massless particles can be fully determined just by the knowledge of the corresponding three-particle amplitudes.

Figures (5)

• Figure 1: Generalised on-shell recursion relation. This new recursion relation shows the same structure of the usual BCFW one, with a further factor which depends on a sub-set of the zeroes of the amplitude.
• Figure 2: Multi-particle limit $P_{\mathcal{K}}^2\,\rightarrow\,0$. In this figure, the behaviour of the terms of the generalised recursion relation \ref{['Mfin']} under the multi-particle limits is depicted. In particular, in the terms contributing, all the particles in the set $\mathcal{K}$ belong to one of the two sub-amplitudes.
• Figure 3: Collinear limit $P_{k_1 k_2}^2\,\rightarrow\,0$. For theories with $\delta$-derivative interactions ($\delta\,>\,0$), the analysis of this collinear limit returns a non-trivial equality which, for theories with internal quantum numbers, gives the algebra of the internal group \ref{['Pk1k2lim3']} and \ref{['Pk1k2lim4']}
• Figure 4: Collinear limit $[i,j]\,\rightarrow\,0$. There are just one class of terms contributing to this limit, which is characterised by a three-particle amplitude of type $M_{3}\left(\hat{i},k,-\hat{P}_{ij}\right)$.
• Figure 5: Collinear limit $\langle i,j\rangle\,\rightarrow\,0$. There are just one class of terms contributing to this limit, which is characterised by a three-particle amplitude of type $M_{3}\left(\hat{P}_{ij},k,\hat{j}\right)$.