Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex

TL;DR

This paper proves that the nonabelian cubic vertex for spin-3 fields discovered in the 1980s cannot be extended to higher orders, confirming an obstruction to consistent interactions in flat spacetime.

Contribution

It provides a rigorous proof that the Berends-Burgers-van Dam obstruction for spin-3 interactions cannot be resolved, even with additional fields or higher spins.

Findings

The BBvD cubic vertex is inconsistent at higher order.

No modifications can cure the obstruction.

The obstruction applies to all multiplets of spin-three fields.

Abstract

In the eighties, Berends, Burgers and van Dam (BBvD) found a nonabelian cubic vertex for self-interacting massless fields of spin three in flat spacetime. However, they also found that this deformation is inconsistent at higher order for any multiplet of spin-three fields. For arbitrary symmetric gauge fields, we severely constrain the possible nonabelian deformations of the gauge algebra and, using these results, prove that the BBvD obstruction cannot be cured by any means, even by introducing fields of spin higher (or lower) than three.