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.

中文翻译：

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软算子的共形特性。第二部分。空状态的使用

以软算子作为最高权重向量的（洛伦兹）保形群的表示具有两个通用属性，我们在本文中清楚地说明了这一点。给定一个具有一定维度和自旋的软算子，第一个性质是关于作用于软算子的“（大）规范变换”的存在。第二个属性是软算子的（大）规范不变空状态与 S 矩阵元素的解耦。在每种情况下，去耦方程都具有零场强条件的形式，软算子作为（规范）势。零态去耦有效地减少了软粒子的偏振态数量，并且对于从渐近对称性的 Ward 恒等式推导出软定理至关重要。据我们所知，这些性质与 S 矩阵的洛伦兹不变性或渐近对称性的存在没有直接关系。我们还验证了从零状态解耦获得的结果与外部腿中具有有限能量的有质量和无质量粒子的领先和次领先软定理一致。