Abstract
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.
Highlights
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
We can see that some of the primary operators have interpretation as soft-operators arising from physical creation-annihilation operators of massless gauge particles, but, for our purpose, we have to assume the existence of more general primary operators which do not have any such interpretation
Summary
We study the consequences of decoupling for various soft operators. We will assume throughout the rest of the paper that the space-time dimension D = n + 2 is even. The decoupling of null-states Fab(x) effectively reduces the number of polarization states of the soft-photon from n = D − 2 to 1. This is consistent with the fact that the transformation parameter φ(x) is a scalar. Sa = ∂aφ encodes this fact in the sense that the soft photons of every helicity is essentially a derivative of a scalar function φ which is the (large) gauge transformation parameter. Another point of view will be the solvability of the Ward-identity for asymptotic symmetries.
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