Celestial amplitudes from conformal correlators with bulk-point kinematics

We show that two- and three-point celestial (C)CFTd-1 amplitudes can be directly obtained from correlation functions in a unitary Lorentzian CFTd on R× Sᵈ⁻¹. The recipe involves a rescaling of the operators, followed by an expansion around a bulk point configuration and a transformation to an Sᵈ⁻¹ conformal primary basis. The first two steps project the CFTd correlators onto distributions on Sᵈ⁻¹. The final step implements a dimensional reduction yielding CCFTd-1 amplitudes that are manifestly vanishing for all in/out configurations and Poincar\'e invariant. The dimensional reduction may be implemented either by evaluating certain time integral transforms around the bulk-point limit, or by analytically continuing the CFTd operator dimensions and restricting the operators to Sᵈ⁻¹ time slices separated by π in global time. The latter prescription generates the correct normalization for both two- and three-point functions. On the other hand, the celestial three-point amplitudes obtained via the former prescription are found to only agree after evaluating a residue at an integer linear combination of the CFTd conformal dimensions. The correct normalization may also be obtained by considering a different integration path in the uplift of the complexified time plane to its universal cover.