Higher order self-dual models for spin-3 particles in $D=2+1$

TL;DR

This paper constructs a 5th order self-dual model for spin-3 particles in 2+1 dimensions, filling a gap in the literature, and demonstrates its ghost-free nature using a master action approach.

Contribution

It introduces the missing 5th order spin-3 self-dual model and shows its ghost-free property, extending the understanding of higher order self-dual models for spin-3 particles.

Findings

Constructed the 5th order spin-3 self-dual model.

Proved the model is ghost free via a master action.

Connected the model with the 6th order top model.

Abstract

In $D=2+1$ dimensions, elementary particles of a given helicity can be described by local Lagrangians (parity singlets). By means of a "soldering" procedure two opposite helicities can be joined together and give rise to massive spin-$s$ particles carrying both helicities $\pm s$ (parity doublets), such Lagrangians can also be used in $D=3+1$ to describe massive spin-$s$ particles. From this point of view the parity singlets (self-dual models) in $D=2+1$ are the building blocks of real massive elementary particles in $D=3+1$. In the three cases $s=1,\, 3/2,\, 2$ there are $2s$ self-dual models of order $1,2, \cdots, 2s$ in derivatives. In the spin-3 case the 5th order model is missing in the literature. Here we deduce a 5th order spin-3 self-dual model and fill up this gap. It is shown to be ghost free by means of a master action which relates it with the top model of 6th order. We believe that our approach can be generalized to arbitrary integer spin-$s$ in order to obtain the models of order $2s$ and $2s-1$. We also comment on the difficulties in relating the 5th order model with their lower order duals.