A note on higher-derivative actions for free higher-spin fields

TL;DR

The work develops a systematic hierarchy of free higher-spin actions with increasing derivatives, introducing Einstein-like, Maxwell-like, and Weyl-like formulations and organizing them via deWit-Freedman generalized Christoffel symbols. It shows that higher-derivative actions can retain gauge and constrained Weyl symmetries, with factorized constructions using Ricci- and Maxwell-like tensors and Francia's higher-spin mass operator as a unifying trace modifier. A key result is that the 4n and 4n+2 derivative Weyl-like actions are controlled by specific constraint structures on gauge and Weyl parameters and, in special cases, reproduce the Fradkin-Tseytlin conformal higher-spin action in four dimensions. The discussion addresses deformations to (A)dS backgrounds and implications for interacting theories, suggesting connections to conformal higher-spin theories and potential nonlinear completions, albeit with non-unitarity caveats. The framework provides a compact, symmetry-driven perspective on the landscape between Fronsdal theory and conformal higher-spin dynamics, with explicit factorization and mass-term relations.

Abstract

Higher-derivative theories of free higher-spin fields are investigated focusing on their symmetries. Generalizing familiar two-derivative constrained formulations, we first construct less-constrained Einstein-like and Maxwell-like higher-derivative actions. Then, we construct Weyl-like actions - the actions admitting constrained Weyl symmetries - with different numbers of derivatives. They are presented in a factorized form making use of Einstein-like and Maxwell-like tensors. The last (highest-derivative) member of the hierarchy of the Weyl-like actions coincides with the Fradkin-Tseytlin conformal higher-spin action in four dimensions.