# Soldering spin-3 opposite helicities in $D=2+1$

## TL;DR

This paper develops a method to combine opposite helicity states of a spin-3 particle in 2+1 dimensions into a single parity doublet, resulting in a new, ghost-free sixth-order Lagrangian analogous to linearized 'New Massive Gravity' and confirming its particle content through multiple analyses.

## Contribution

It introduces a novel soldering procedure for high-derivative spin-3 models, producing a parity doublet Lagrangian without auxiliary fields, advancing higher-spin field theory.

## Key findings

- Successfully soldered opposite helicity states into a parity doublet.
- Derived a new sixth-order spin-3 Lagrangian free of ghosts.
- Confirmed particle content via covariant and non-covariant analyses.

## Abstract

Here we present the `soldering' of opposite helicity states of a spin-3 particle, in $D=2+1$, into one parity doublet. The starting points may be either the sixth- or the fifth-order (in derivatives) spin-3 self-dual models of opposite helicities. The high number of derivatives avoids the use of auxiliary fields which has been so far an obstacle for a successful soldering procedure. The resulting doublet model is a new Lagrangian with six orders in derivatives and no auxiliary field. It may be regarded as a spin-3 analogue of the linearized `New Massive Gravity'. We check its particle content via a gauge invariant and Lorentz covariant analysis of the analytic structure of the two-point amplitude with the help of spin-3 analogues of the Barnes and Rivers projection operators. The particle content is alternatively confirmed in a specific non-covariant gauge by a decomposition in helicity variables. The soldered model is ghost free and contains two physical states as expected for a parity doublet.