Abstract
The operator product expansion of massless celestial primary operators of arbitrary spin is investigated. Poincaré symmetry is found to imply a set of recursion relations on the operator product expansion coefficients of the leading singular terms at tree-level in a holomorphic limit. The symmetry constraints are solved by an Euler beta function with arguments that depend simply on the right-moving conformal weights of the operators in the product. These symmetry-derived coefficients are shown not only to match precisely those arising from momentum-space tree-level collinear limits, but also to obey an infinite number of additional symmetry transformations that respect the algebra of w1+∞. In tree-level minimally-coupled gravitational theories, celestial currents are constructed from light transforms of conformally soft gravitons and found to generate the action of w1+∞ on arbitrary massless celestial primaries. Results include operator product expansion coefficients for fermions as well as those arising from higher-derivative non-minimal couplings of gluons and gravitons.
Highlights
Poincaré symmetry is found to imply a set of recursion relations on the operator product expansion coefficients of the leading singular terms at tree-level in a holomorphic limit
A more thorough investigation of this proposal is aided by working in a language in which the underlying two dimensions are rendered manifest. 2D conformal (4D Lorentz) symmetry is the basis of this two-dimensional description, so these two dimensions are more readily apparent in the scattering of particles of definite boost weight, as opposed to the standard momentum eigenstates
The Ward identities for infinitedimensional symmetries that follow from soft theorems are powerful and encouraging examples of this behavior
Summary
We briefly review Poincaré symmetry in the context of celestial amplitudes, studied for example in [53]. It is compatible with Poincaré to regard the translation generators as the modes of a primary operator the left, right, or both conformal weights are. Such primary operators can be constructed by taking various light or shadow transforms of the ∆ = 1 conformally soft graviton. These modes do not automatically form a closed algebra with the global conformal generators and require an additional assumption of a truncated mode expansion as presented, for example, in [24, 27]. More details on this alternative mode expansion are provided in appendix B
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