Weyl double copy and massless free fields in curved spacetimes

In spinor formalism, since any massless free-field spinor with spin higher than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We determine the corresponding Dirac-Weyl spinors in a given empty spacetime. Regarding them as basic units, other higher spin massless free-field spinors are then identified. Along this way, we find some hidden fundamental features related to these fields. In particular, for non-twisting vacuum Petrov type N solutions, we show that all higher spin massless free-field spinors can be constructed with one type of Dirac-Weyl spinor and the zeroth copy. Furthermore, we systematically rebuild the Weyl double copy for non-twisting vacuum type N and vacuum type D solutions. Moreover, we show that the zeroth copy not only connects the gravity fields with a single copy but also connects the degenerate Maxwell fields with the Dirac-Weyl fields in the curved spacetime, both for type N and type D cases. Besides, we extend the study to non-twisting vacuum type III solutions. We find a particular Dirac-Weyl scalar independent of the proposed map and whose square is proportional to the Weyl scalar. A degenerate Maxwell field and an auxiliary scalar field are then identified. Both of them play similar roles as the Weyl double copy. The result further inspires us that there is a deep connection between gravity theory and gauge theory.