Collinearity constraints for on-shell massless particle three-point functions, and implications for allowed-forbidden $n+1$-point functions

TL;DR

This paper explores collinearity constraints on massless particle three-point functions, showing how forbidden on-shell amplitudes become suppressed off-shell and affect the structure of higher-point functions, with implications for on-shell and off-shell amplitude relations.

Contribution

It introduces a collinearity-based argument for helicity conservation constraints and analyzes how off-shell effects modify forbidden amplitudes and their influence on higher-point functions.

Findings

Forbidden three-point functions vanish on-shell but can be nonzero off-shell.

Off-shell forbidden amplitudes are suppressed by $k^2$ near the mass shell.

The $k^2$ factor cancels in certain $n+1$-point functions, removing poles.

Abstract

A simple collinearity argument implies that the massless particle three-point function of helicities $h_1, h_2, h_3$ with corresponding real-valued four-momenta $k_1, k_2, k_3$ taken as all incoming or all outgoing (i.e., $k_1 +k_2 +k_3=0$), vanishes by helicity conservation unless $h_1+h_2+h_3=0$. When any one particle with four-momentum $k$ is off mass shell, this constraint no longer applies; a forbidden amplitude with $h_1+h_2+h_3\neq 0$ on-shell can be nonzero off-shell, but vanishes proportionally to $k^2$ as $k$ approaches mass shell. When an on-shell forbidden amplitude is coupled to an allowed $n$-point amplitude to form an $n+1$ point function, this $k^2$ factor in the forbidden amplitude cancels the $k^2$ in the propagator, leading to a $n+1$-point function that has no pole at $k^2=0$. We relate our results for real-valued four-momenta to the corresponding selection rules that have been derived in the on-shell literature for complexified four-momenta.