Five-point partial waves, splitting constraints and hidden zeros

TL;DR

This work develops a concrete partial-wave framework for planar five-point tree amplitudes of identical scalars by constructing a $d$-dimensional basis of double-residue residues, validated in four dimensions via massive spinor-helicity gluing. It shows how five-point splitting and hidden-zero constraints translate into linear relations among the five-point partial-wave coefficients $a^{(k_1,k_2)}_{jln}$ and induce compatibility conditions on the four-point data, with explicit demonstrations using the Veneziano amplitude. At low mass levels, these constraints propagate to fix much of the five-point data in terms of four-point coefficients, while allowing a genuine kernel when both channels admit spin-2 exchanges, suggesting that higher-point input (e.g., six-point splitting or multipositivity) may be required for full rigidity. Remarkably, in the residue-space, the splitting identities imply hidden zeros at intersections of the loci, revealing a nontrivial cross-talk between higher-point consistency and lower-point EFT constraints. The framework provides a transparent analytic handle on higher-point constraints and offers a bridge to broader bootstrap approaches, including potential loop generalizations and dispersion-relations analyses, with implications for rigidity and string-like behavior in higher-point amplitudes.

Abstract

We study the partial-wave expansion of residues of five-point tree-amplitude involving identical scalar particles in the external legs. We check the construction using massive spinor-helicity building blocks and by matching to the tree-level five-point Veneziano amplitude at fixed mass levels. As an application, we express five-point splitting constraints - the reduction of the five-point amplitude to products of four-point amplitudes on special kinematic loci - as linear relations among the five-point partial-wave coefficients. At low mass levels these constraints, together with spin truncation, fix the full five-point partial-wave data in terms of the four-point coefficients and imply simple compatibility conditions; remarkably, imposing two independent splitting loci also forces the residue to vanish on their intersection, making the associated hidden zero manifest in partial-wave space. We also show that once both channels allow spin-2 exchange a genuine kernel can remain, indicating the need for additional higher-point input to achieve complete rigidity.

Figures (4)

• Figure 1: Toller (dihedral) angle for the five-point double-residue kinematics. We define $Q_{12}=p_1+p_2$ and $Q_{34}=p_3+p_4$ and choose a frame where $p_5$ sets the $\hat{z}$ axis. The polar angles $\theta_{12}$ and $\theta_{34}$ are the angles between $Q_{12}$ and $p_5$, and between $Q_{34}$ and $p_5$, respectively. The Toller angle $\omega$ is the dihedral angle between the planes $\Pi_{12}=\mathrm{span}(p_5,Q_{12})$ and $\Pi_{34}=\mathrm{span}(p_5,Q_{34})$, equivalently the angle between the transverse projections of $Q_{12}$ and $Q_{34}$ onto the plane orthogonal to $p_5$.
• Figure 2: Allowed region in the $(\hat{s}_{12},\hat{s}_{34})$ plane for equal-mass $2\to3$ kinematics. The lower boundary is $\hat{s}_{34}=4$ and the upper boundary is $\hat{s}_{34}=(\sqrt{\hat{s}_{12}}-1)^2$, corresponding to $\hat{\lambda}\ge 0$ on the physical branch.
• Figure 3: At fixed $\hat{s}_{12}=25$, the constraint $|\cos\theta_{12}|\le 1$ implies that $\hat{s}_{51}$ must lie in the interval \ref{['eq:s51-interval']} as $\hat{s}_{34}$ varies over its allowed range.
• Figure 4: Projection of the allowed physical region at fixed $(\hat{s}_{12},\hat{s}_{34})=(25,9)$ into the $(\hat{s}_{51},\hat{s}_{23})$ plane, obtained by sampling $\theta_{12},\theta_{34},\omega$ over their full ranges. The thickness in $\hat{s}_{23}$ reflects the residual $\omega$ dependence encoded by the band \ref{['eq:omega-band']}.