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Hermitian formulation for mass dimension one fermions: Flat and curved space-times

Gabriel Brandão de Gracia, Rodolfo José Bueno Rogerio

TL;DR

This work develops a Hermitian, renormalizable field-theoretic framework for mass-dimension-one Elko fermions as dark matter candidates, emphasizing rotational covariance and a consistent adjoint via the parity operator $\mathcal{P} = m^{-1} \gamma_\mu p^\mu$. It introduces a derivative Higgs portal $\mathcal{L}_I = g' \overset{\boldsymbol{\neg}}{[t]{\uplambda}} i\slashed{\partial} \uplambda \phi$, demonstrates one-loop renormalizability with divergences matching bare terms, and provides explicit scattering and annihilation amplitudes in both flat and curved space-time contexts. The paper further presents a first-order path-integral formulation that generalizes to curved space-time by promoting $\partial$ to $\nabla$ and discusses the implications for coupling to gravity, including the necessity of a spin-connection framework. Collectively, these results offer a coherent, Hermitian DM model with a calculable phenomenology and a practical route to gravity-compatible extensions. The approach yields a physically consistent dark sector with a viable Higgs-portal interaction, enabling potential connections to cosmological relic abundance and Higgs-scale collider signatures while avoiding non-Hermitian minimal couplings to gauge fields.

Abstract

Throughout this paper, we conduct our discussion by a partial review of Elko's Hermiticity, introducing the Hermitian formulation for interacting mass dimension one fermions based on Elko spinor. It includes pivotal observations about renormalizability and the study of some allowed interactions. Beyond these points, since dark-matter phenomenology is mainly connected to gravitation, we introduce original remarks on how the Hermitian prescription can be readily generalized to include curved space-time, considering the very definition of the Elko spinor structure. We establish the Elko dual as arising from the path-integral formulation of a more fundamental structure. It enables one to include a curved background space-time and also quantum gravity into our investigations.

Hermitian formulation for mass dimension one fermions: Flat and curved space-times

TL;DR

This work develops a Hermitian, renormalizable field-theoretic framework for mass-dimension-one Elko fermions as dark matter candidates, emphasizing rotational covariance and a consistent adjoint via the parity operator . It introduces a derivative Higgs portal , demonstrates one-loop renormalizability with divergences matching bare terms, and provides explicit scattering and annihilation amplitudes in both flat and curved space-time contexts. The paper further presents a first-order path-integral formulation that generalizes to curved space-time by promoting to and discusses the implications for coupling to gravity, including the necessity of a spin-connection framework. Collectively, these results offer a coherent, Hermitian DM model with a calculable phenomenology and a practical route to gravity-compatible extensions. The approach yields a physically consistent dark sector with a viable Higgs-portal interaction, enabling potential connections to cosmological relic abundance and Higgs-scale collider signatures while avoiding non-Hermitian minimal couplings to gauge fields.

Abstract

Throughout this paper, we conduct our discussion by a partial review of Elko's Hermiticity, introducing the Hermitian formulation for interacting mass dimension one fermions based on Elko spinor. It includes pivotal observations about renormalizability and the study of some allowed interactions. Beyond these points, since dark-matter phenomenology is mainly connected to gravitation, we introduce original remarks on how the Hermitian prescription can be readily generalized to include curved space-time, considering the very definition of the Elko spinor structure. We establish the Elko dual as arising from the path-integral formulation of a more fundamental structure. It enables one to include a curved background space-time and also quantum gravity into our investigations.

Paper Structure

This paper contains 10 sections, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. On a rotationaly invariant basis and a prescription for the adjoint field
  3. On action formulation, interactions, and the Hermitian nature
  4. Hermitian interactions and a criterion for a dark coupling
  5. Elko-Higgs derivative coupling
  6. Renormalizability at one loop
  7. Mø ller-like scattering
  8. On the Elko annihilation processes
  9. Elko action from a more fundamental structure and a possible curved space-time completion
  10. Concluding Remarks

Figures (2)

  • Figure 1: Elko-Higgs coupling. The arrows denote mass dimension one fermions and the dashed line refers to the Higgs particle.
  • Figure 2: Feynman rules for external Elko fields.