Conformal properties of soft operators -- 2 : Use of null-states

TL;DR

This paper explores the conformal properties of soft operators, highlighting their gauge transformation behavior and null-state decoupling, which simplifies polarization states and underpins soft-theorem derivations.

Contribution

It identifies universal properties of soft operators related to null-state decoupling and gauge transformations, independent of Lorentz invariance or asymptotic symmetries.

Findings

Null-state decoupling reduces polarization states.

Decoupling equations resemble zero field-strength conditions.

Results align with known soft-theorems for massive and massless particles.

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.