Constraining bulk-to-boundary correlators under Poincar\'e symmetry

It is well known that a general two-point function cannot be uniquely determined by Poincar\'e symmetry. In this paper, we show that bulk-to-boundary correlators are highly constrained after imposing suitable fall-off conditions near future/past null infinity. More precisely, scalar bulk-to-boundary correlators are fixed to a unique form up to a normalization constant, whereas fermionic bulk-to-boundary correlators are fixed to a linear superposition of scalar and fermionic branches. This is established by asymptotically expanding the Ward identities, where upon the leading terms decouple from the subleading ones. In the fermionic branch, the power-law exponent of the bulk-to-boundary correlator is greater by one than the fall-off index. Consequently, we revisit the relation between Carrollian correlators and momentum space scattering amplitudes for fermionic operators. In this context, we find that the Fourier transform bridging the two acquires an extra factor of $\sqrt{\omega}$ for each fermionic operator. Furthermore, we reduce the bulk-to-boundary correlator to the boundary-to-boundary correlator and identify a critical fall-off index $\Delta=1$. For $0 < \Delta < 1$, only a magnetic branch exists for scalars. For $\Delta > 1$, the electric branch is always divergent for both scalar and fermionic branches and thus requires regularization.