Conformal properties of soft operators -- 2 : Use of null-states
Shamik Banerjee, Pranjal Pandey
TL;DR
The paper studies conformal representations on the celestial sphere with soft operators as highest-weight states and identifies two universal properties: a gauge-like transformation generated by a primary descendant and the decoupling of gauge-invariant null-states from S-matrix elements. The decoupling equations take the form of zero-field-strength constraints, forcing soft operators into pure-gauge forms and thereby reducing polarization degrees of freedom, which underpins the derivation of soft-theorems from symmetry Ward identities. The authors explicitly derive consequences for leading and subleading soft photons and gravitons across dimensions, verify consistency with Weinberg soft-theorems for massive external legs, and discuss the construction of the soft charge alongside comparisons to Banerjee:2019aoy. They also highlight dimension-dependent subtleties (e.g., $D=4$ vs $D=6$) and discuss potential holographic interpretations via celestial CFTs. Overall, the work provides a gauge-structure-centric perspective on soft theorems independent of Lorentz invariance or asymptotic symmetry notions, while reinforcing their compatibility with known soft-factor results.
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.