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A simple collinearity argument implies that the massless particle three-pointfunction of helicities $h_1, h_2, h_3$ with corresponding real-valuedfour-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 $h_1+h_2+h_3=0$. Whenany one particle with four-momentum $k$ is off mass shell, this constraint nolonger applies; a forbidden amplitude with $h_1+h_2+h_3\neq 0$ on-shell can benonzero off-shell, but vanishes proportionally to $k^2$ as $k$ approaches massshell. When an on-shell forbidden amplitude is coupled to an allowed $n$-pointamplitude to form an $n+1$ point function, this $k^2$ factor in the forbiddenamplitude cancels the $k^2$ in the propagator, leading to a $n+1$-pointfunction that has no pole at $k^2=0$. We relate our results for real-valuedfour-momenta to the corresponding selection rules that have been derived in theon-shell literature for complexified four-momenta.

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