4D Flat-space scattering amplitude /$CFT_3$ correlator correspondence revisited

In this paper, we make connection between CFT$_3$ three point correlation function of conserved currents and 4D three point amplitude of general spin massless gauge field explicit. We do so by taking flat space limit of momentum space CFT correlation function and show how they reproduce flat space amplitudes. We then point out a mismatch between number of independent structures in 3D CFT correlator of conserved currents and 4D flat space covariant vertex of massless higher spin fields. This is in contrast with general expectation that counting of 3d CFT correlator and 4d flat space amplitude should match. This mismatch is even more pronounced in spinor helicity variables. We also point out an interesting relation between parity even and parity odd flat space amplitude in momentum space. This observation helps us to construct a new momentum space CFT strtucture which accounts for the mismatch. However we should mention that this extra CFT structure can't be constructed out of conserved currents and hence counting mismatch between CFT correlation of conserved currents and flat space amplitude of massless gauge field persists. Story in spinor helicity variable is more complicated and is discussed in detail. We further comment on the connection of CFT correlation function in spinor helicity variables to AdS amplitudes in spinor helicity variables and light cone variables.