Lorentz covariant on-shell cubic vertices for continuous-spin fields and integer-spin fields

TL;DR

This work develops a Lorentz covariant on-shell framework for cubic interactions between continuous-spin fields and integer-spin fields in flat space, using a Lorentz vector superspace for CSFs and an oscillator approach for ISFs. It provides a complete classification of parity-even cubic vertices, including self-interactions of CSFs and cross-interactions with ISFs, and demonstrates a precise correspondence with light-cone gauge vertices. The authors show that the manifestly Lorentz invariant formal cubic action is divergent and introduce a finite, Lorentz-invariant modification via delta-function constraints, which matches the light-cone results upon projection. This covariantization of the light-cone CSF/ISF vertices offers a promising path toward off-shell Lagrangian formulations and BRST treatments, with potential extensions to AdS and worldline approaches. Overall, the paper delivers a thorough, on-shell Lorentz covariant construction of CSF/ISF cubic vertices and clarifies their local/non-local structure and covariant-LC correspondence.

Abstract

In the framework of Lorentz covariant on-shell approach, interacting continuous-spin fields and integer-spin fields in flat space are investigated. Continuous-spin fields are considered by using a Lorentz vector superspace formulation, while integer-spin fields are considered by using oscillator formulation. All parity-even cubic vertices for self-interacting continuous-spin fields realized as functions on the Lorentz vector superspace are obtained. Cross-interactions of continuous-spin fields and integer-spin fields are also derived. Several representatives of cubic vertices realized as distributions are obtained. We show that manifestly Lorentz invariant formal cubic action involving at least one continuous-spin field turns out to be divergent. We find the modification of such action which maintains Lorentz invariance and leads to finite cubic action. One-to-one correspondence of Lorentz covariant cubic vertices and light-cone gauge cubic vertices is demonstrated explicitly.