Representations of the (Lorentz) conformal group with the soft operators as highest weight vectors have two universal properties, which we clearly state in this paper. Given a soft operator with a certain dimension and spin, the first property is about the existence of "(large) gauge transformation" that acts on the soft operator. The second property is the decoupling of (large) gauge-invariant null-states of the soft operators from the S-matrix elements. In each case, the decoupling equation has the form of zero field-strength condition with the soft operator as the (gauge) potential. Null-state decoupling effectively reduces the number of polarisation states of the soft particle and is crucial in deriving soft-theorems from the Ward identities of asymptotic symmetries. To the best of our understanding, these properties are not directly related to the Lorentz invariance of the S-matrix or the existence of asymptotic symmetries. We also verify that the results obtained from the decoupling of null-states are consistent with the leading and subleading soft-theorems with finite energy massive and massless particles in the external legs.

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In this paper we evaluate the modified celestial amplitude for gravitons and gluons, as defined in arXiv:1801.10171 [hep-th]. We find that the modified (tree) amplitude is finite for gravitons in Einstein gravity. The modified amplitude behaves like correlation function of operators inserted at various points of null-infinity in the Minkowski space-time. Therefore, unlike the standard celestial amplitudes, these are three dimensional objects. We also show that this amplitude admits conformal soft factorization recently studied in the literature.