Spin three gauge theory revisited

TL;DR

This paper classifies all consistent cubic interaction vertices for spin-3 gauge fields in flat spacetime, identifying unique structures and their derivative counts, and analyzing their consistency at higher orders.

Contribution

It provides a complete classification of cubic vertices for spin-3 gauge fields under Poincaré and parity invariance, revealing new consistent interactions in higher dimensions.

Findings

All vertices are cubic with three or five derivatives.

The Berends-Burgers-van Dam vertex is identified and shown to be inconsistent at second order.

A new five-derivative vertex is found that passes second-order consistency in certain dimensions.

Abstract

We study the problem of consistent interactions for spin-3 gauge fields in flat spacetime of arbitrary dimension n>3. Under the sole assumptions of Poincar\'e and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total of either three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions n>4 and for a completely antisymmetric structure constant tensor, another covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed.