# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1906.01650