  We consider the tree-level scattering of massless particles in
$(d+2)$-dimensional asymptotically flat spacetimes. The $\mathcal{S}$-matrix
elements are recast as correlation functions of local operators living on a
space-like cut $\mathcal{M}_d$ of the null momentum cone. The Lorentz group
$SO(d+1,1)$ is nonlinearly realized as the Euclidean conformal group on
$\mathcal{M}_d$. Operators of non-trivial spin arise from massless particles
transforming in non-trivial representations of the little group $SO(d)$, and
distinguished operators arise from the soft-insertions of gauge bosons and
gravitons. The leading soft-photon operator is the shadow transform of a
conserved spin-one primary operator $J_a$, and the subleading soft-graviton
operator is the shadow transform of a conserved spin-two symmetric traceless
primary operator $T_{ab}$. The universal form of the soft-limits ensures that
$J_a$ and $T_{ab}$ obey the Ward identities expected of a conserved current and
energy momentum tensor in a Euclidean CFT$_d$, respectively.
