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 $Ψμ$ to an \emph{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 $2 DΨγ_5 DΨ$ 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, $S=-\int\barψψR\sqrt{-g}$, in which the metric $g=Ψ\otimes_SΨ$ and the scalar $\barψψ$ are \emph{both} composites of $Ψ$; the second variation consequently factors, $δ^2S=\mathcal Q(h{μν}[δΨ],δΦ[δΨ])$, 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 Pagels and Primack.

• The paper establishes that the QSL framework uniquely propagates a composite spin-1/2 Dirac fermion as the dark matter candidate, effectively excluding a massive spin-3/2 field.
• It employs rigorous kinematic and dynamic analyses, showing that torsional mass terms and vanishing kinetic contributions prevent the activation of spin-3/2 propagation.
• The study links the surviving Goldstino-like spin-1/2 mode to supersymmetry breaking and cosmological mass generation tied to the Hubble parameter.

Spin-3/2 No-go and Uniqueness of Spin-1/2 Dark Matter in the Quadratic Spinor Lagrangian

Introduction and Context

This paper rigorously analyzes the dark-matter phenomenology stemming from the Quadratic Spinor Lagrangian (QSL) formalism for general relativity. The QSL action $$\mathcal{L}_{\mathrm{QSL}} = 2\,D\Psi\,\gamma_5\,D\Psi$$ employs a Clifford-algebra-valued 1-form $\Psi = \vartheta^a\gamma_a\psi$, where $\psi$ is a Dirac spinor and $\vartheta^a$ is a tetrad. Previous work ("Dark matter from the quadratic spinor Lagrangian. I" [TungCQG2026]) established that the composite QSL propagates a spin-$\frac{1}{2}$ Dirac fermion whose mass is geometrically induced by cosmological trace torsion. This paper confronts the natural question: can the QSL, when its spinor 1-form $\Psi_\mu$ is promoted to an independent Dirac-vector field, also propagate a genuine massive spin-$\frac{3}{2}$ field and thus provide a higher-spin dark-matter candidate?

Lorentz Structure of Vector-Spinor Fields

The vector-spinor $\Psi_\mu$ carries reducible Lorentz content: $$[(\frac12,0)\oplus(0,\frac12)]\otimes(\frac12,\frac12) = (\frac12,0)\oplus(0,\frac12)\oplus(1,\frac12)\oplus(\frac12,1)$$, decomposing cleanly into spin-$\frac{1}{2}$ and spin-$\Psi = \vartheta^a\gamma_a\psi$0 sectors. The composite QSL restricts to $\Psi = \vartheta^a\gamma_a\psi$1, for which the gamma-traceless spin-$\Psi = \vartheta^a\gamma_a\psi$2 part vanishes identically. This kinematic restriction evokes the question of whether an independent $\Psi = \vartheta^a\gamma_a\psi$3 could activate the spin-$\Psi = \vartheta^a\gamma_a\psi$4 sector.

Pillar I: Non-uniform Torsional Mass Structure

The torsional mass term in the QSL arises from the Einstein--Cartan trace torsion $\Psi = \vartheta^a\gamma_a\psi$5, with contorsion $\Psi = \vartheta^a\gamma_a\psi$6 (Appendix A). The mass-like term is $\Psi = \vartheta^a\gamma_a\psi$7. For the composite field, this yields a pure scalar Dirac mass:

$\Psi = \vartheta^a\gamma_a\psi$8

locked to the Hubble rate $\Psi = \vartheta^a\gamma_a\psi$9 at production. For the independent vector-spinor, the mass term is frame-aligned, coupling only the time-component (i.e., $\psi$0), not the spatial or diagonal components, and the spin-$\psi$1 sector $\psi$2 receives zero mass (Appendix A).

Figure 1: Curvature-induced transverse mass $\psi$3 and non-adiabaticity parameter $\psi$4 versus scale factor; the mass drops below $\psi$5 during matter domination, but auxiliary nature prevents relic production.

Pillar II: Absence of a Spin-3/2 Kinetic Term

The component expression of the QSL action,

$\psi$6

does not provide a bulk kinetic term for the independent field. The kinetic part and the cross term vanish identically; only the pure torsional mass survives (Appendix B). This follows from the underlying spinor-curvature identity, which renders the kinetic term a boundary operator rather than a bulk dynamic term.

Pillar III: Second Variation Factorization and No Propagation

Dynamical content resides only in the curvature side, via the action:

$\psi$7

with both metric and scalar constructed as composites of $\psi$8. The second variation factors through linearized metric $\psi$9 and scalar $\vartheta^a$0, both massless:

$\vartheta^a$1

No massive spin-$\vartheta^a$2 mode is found, the kinetic structure supporting only graviton and scalar propagations. This dynamical completion corresponds to a constrained system with only auxiliary (non-propagating) gamma-traceless vector-spinor components.

Constraint Sector and Degree-of-Freedom Counting

Hamiltonian analysis via Dirac–Bergmann confirms the absence of spin-$\vartheta^a$3 propagation: the time component acts as a Lagrange multiplier, enforcing primary and secondary constraints that remove all would-be lower-spin excitations. The remaining degrees of freedom match those of the graviton and a scalar.

Spin-1/2 Uniqueness and Dark Matter Implications

Combining kinematic and dynamic arguments, the QSL framework is shown to propagate uniquely the composite spin-$\vartheta^a$4 Dirac fermion. This mode inherits geometric mass tied to the Hubble parameter, justifying its identification as the sole dark-matter candidate. Contrast is drawn with Rarita–Schwinger and supergravity paradigms wherein independent spin-$\vartheta^a$5 fields propagate (with potential pathologies); the QSL evades such pathologies by structural exclusion rather than by constraint engineering.

Curved Backgrounds and Mass-Induced Activation

The analysis is robust against background curvature. Cosmological corrections induce only algebraic mass-like couplings (proportional to pressure $\vartheta^a$6) for the transverse spin-$\vartheta^a$7 components:

$\vartheta^a$8

No kinetic term emerges, so even at points where $\vartheta^a$9 (e.g., matter domination), the auxiliary nature of the mode persists and no healthy propagation nor relic production occurs. The mode acts as a redundant parametrization of graviton data.

Supergravity Connection and Goldstino Interpretation

Via the super-$\frac{1}{2}$0 structure, the surviving spin-$\frac{1}{2}$1 mode is interpreted as the Goldstino of spontaneously broken local supersymmetry, as the metric condensate breaks the super-Lorentz symmetry. The supercurrent is sourced by the gamma-trace spin-$\frac{1}{2}$2 mode, not the transverse spin-$\frac{1}{2}$3.

Numerical Results and Null Space Structure

The null space of the inverse propagator $\frac{1}{2}$4 remains invariant across spacelike and timelike $\frac{1}{2}$5, jumping only at $\frac{1}{2}$6, corroborating analytically derived results and confirming the absence of timelike (massive) poles. Detailed computations sample several backgrounds and directions; the characteristic cone is strictly luminal.

Implications and Outlook

The QSL framework structurally precludes propagating massive spin-$\frac{1}{2}$7 excitations, uniquely selecting a spin-$\frac{1}{2}$8 composite fermion as the dark-matter candidate. It represents a maximally economic model, containing only graviton and Goldstino-like excitations derived from the same field. Higher-spin generalizations would require operators external to the QSL family.

The theoretical implications include a sharpened perspective on spinor and higher-spin dynamics in gravitational backgrounds, providing a powerful no-go theorem. Practically, the model offers a parameter-less prediction for dark-matter mass and abundance tied to early-universe Hubble scale, potentially constraining observational explorations and model-building in cosmological and gravitational dark matter.

Conclusion

This paper provides an authoritative no-go theorem for propagating spin-$\frac{1}{2}$9 dark matter in the Quadratic Spinor Lagrangian context, rooted in both kinematic and dynamical analyses. The unique propagating matter is the composite spin-$\Psi_\mu$0 Dirac fermion, whose geometric mass and cosmological relic abundance are protected against potential higher-spin generalizations. No modification within the QSL structure itself admits massive spin-$\Psi_\mu$1 modes, reinforcing the distinction between QSL and supergravity frameworks. The Goldstino interpretation links the findings to established supersymmetry breaking mechanisms, while the luminosity of the characteristic cone ensures robust absence of pathologies typically afflicting higher-spin fields.

Figure 1: Curvature-induced transverse mass $\Psi_\mu$2 and non-adiabaticity parameter $\Psi_\mu$3 versus scale factor, indicating persistent auxiliary nature and the absence of relic production across cosmological evolution.