The soft limit also implies that massless spin-2 particles (gravitons) must have self-interactions.
((a)) To warm up, consider the soft limit of massless spin-1 particles coupled to scalars (as in scalar QED). Just assuming generic interactions (not the scalar QED Lagrangian), show that there must be an AAϕ∗ϕ interaction for Compton scattering to be Lorentz invariant.
((b)) Now consider Compton scattering of gravitons h off scalars. Show that there must be an hhϕϕ interaction. Then show that unlike the massless spin-1 case the new interaction must have a pole at (q1+q2)2=0. This pole should be resolved into a graviton exchange graph. Derive a relationship between the form of the graviton self-coupling and the hϕϕ coupling.