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

Understand

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.