Dark matter from the quadratic spinor Lagrangian II: A spin-3/2 no-go and the uniqueness of the spin-1/2 candidate

Abstract

The composite Quadratic Spinor Lagrangian (QSL) propagates a spin-1/2 Dirac fermion whose mass is generated geometrically by cosmological trace torsion. It is natural to ask whether promoting the spinor 1-form $\Psi_\mu$ to an independent Dirac-vector field yields a genuine spin-3/2 dark-matter candidate. We prove that it does not. Three results combine into a no-go theorem. First, the torsional term, computed exactly by Clifford reduction, is a frame-aligned mass confined to the time-component sector -- not a uniform spin-3/2 mass. Second, the second-order form $2 D\Psi \gamma_5 D\Psi$ has identically vanishing kinetic and cross terms for the independent field: as a component expression it is the boundary part of the spinor-curvature identity and carries no bulk dynamics. Third, the genuine dynamics therefore reside in the curvature side of that identity, $S=-\int\bar\psi\psi R\sqrt{-g}$, in which the metric $g=\Psi\otimes_S\Psi$ and the scalar $\bar\psi\psi$ are both composites of $\Psi$; the second variation consequently factors, $\delta^2S=\mathcal Q(h_{\mu\nu}[\delta\Psi],\delta\Phi[\delta\Psi])$, through the linearized metric and a scalar, both massless. Every propagating pole therefore lies on the metric light cone $k^2=0$ -- the graviton and a scalar -- and no massive spin-3/2 mode exists, on any background. This is the dynamical completion of the kinematic fact that the composite spinor 1-form has no spin-3/2 part, and it establishes the composite spin-1/2 Dirac fermion as the unique propagating matter excitation, and the unique dark-matter candidate, of the QSL. Through the super-SL(2,C) structure this surviving mode is naturally read as the Goldstino of the local supersymmetry broken by the metric condensate -- a composite, gravitational descendant of the light-gravitino dark matter of Fayet and of Pagels and Primack.