A new conformal gauge theory of vector-spinors and spin-3/2 particles

TL;DR

This work presents a consistent conformal gauge theory for vector-spinors by introducing a unique off-shell fermionic gauge invariance. The resulting gauge-invariant kinetic operator remains Weyl invariant in the massless limit, and gamma-trace configurations can be globally gauged away, yielding a gamma-traceless field $\psi_\mu$ with $\gamma^\mu\psi_\mu=0$. The classical theory propagates eight degrees of freedom for a massive spin-$\tfrac{3}{2}$ plus four spin-$\tfrac{1}{2}$ degrees of freedom with mass $2m$, with no Velo–Zwanziger instabilities; the quantum theory provides a causal mode decomposition and a propagator consistent with the gauge structure. The one-loop conformal anomaly from this vector-spinor sector has an opposite sign for the $a$-coefficient compared to lower-spin fields, offering new insights into higher-spin conformal dynamics and potential phenomenological implications. This approach preserves a Lagrangian description for higher-spin particles and motivates extending the framework to more general fermionic tensor fields.

Abstract

The unique off-shell fermionic gauge invariance of a vector-spinor field theory is found, and the invariant action is derived. The latter is Weyl invariant in any dimension in the massless limit, and it coincides with the singular point of the one-parameter family of Rarita-Schwinger Lagrangians. Pure gauge configurations are represented by gamma-trace vector-spinors, which can be gauged away in a global way. Previous claims that this theory is inconsistent are shown to be flawed, and the Velo-Zwanziger instability is proved to be absent. The theory propagates a massive spin-3/2 particle together with a spin-1/2 state whose mass is twice that of the j=3/2 mode. This physical prediction is derived in a consistent fashion both from the classical field equations in d=4 and from the causal construction of a gamma-traceless quantum vector-spinor field. The conformal anomaly is derived using known results for the heat kernel of non-minimal second-order operators, and it is shown that the a-anomaly has an opposite sign w.r.t. known results for lower spin fields.