On the structure of interactions of mass dimension one fermions: a functional renormalization group perspective

In this paper, we provide the first systematic investigation of renormalization group properties of mass dimension one fermions described by ELKO spinors. By construction, ELKOs must be neutral under any Standard Model charge, therefore, providing a natural candidate for dark matter. We consider two versions of scalar-ELKO systems: the first characterized by a derivative Yukawa-like interaction, while the second involves ELKO four-fermion interactions as well as a scalar-ELKO portal. We also considered a system composed of ELKOs interacting with an Abelian gauge field via Pauli-like term. In all cases, we identified the minimal set of interactions that are required by a consistent renormalization group flow, and we discussed the possibility of constructing UV-complete trajectories based on asymptotic freedom. We used the functional renormalization group as a method of investigation.

💡 Research Summary

This paper presents the first systematic study of the renormalization‑group (RG) properties of mass‑dimension‑one fermions described by ELKO spinors, using the functional renormalization group (FRG) as the main tool. ELKOs are eigenstates of the charge‑conjugation operator, neutral under all Standard Model (SM) gauge charges, and therefore natural dark‑matter candidates. The authors consider three classes of perturbatively renormalizable interactions: (i) a derivative Yukawa‑type coupling between ELKO and a real scalar, (ii) a combination of ELKO four‑fermion self‑interaction and a scalar‑ELKO portal, and (iii) a Pauli‑like coupling of ELKO to an Abelian gauge field.

The FRG framework is introduced with an infrared cutoff k and the Litim regulator, leading to the Wetterich flow equation for the effective average action Γ_k. A truncation ansatz is adopted that includes kinetic terms for ELKO, the scalar φ, and the gauge field A_μ, together with wave‑function renormalizations Z_i(k) and scale‑dependent masses. In the one‑loop approximation the flow of the wave‑function factors is neglected, focusing on the dimensionless couplings g_{φξ}, λ_φ, λ_ξ and κ.

For the scalar sector, the derivative Yukawa interaction g_{φξ} φ \bar ξ γ^μ∂μ ξ is shown to generate a φ^4 term λ_φ even if λ_φ is set to zero at the UV scale. The β‑functions reveal that g{φξ} runs with a negative canonical dimension (−1) and receives contributions proportional to λ_φ and g_{φξ}^2, while λ_φ is driven by g_{φξ}^2. Consequently, a pure derivative Yukawa theory is not closed under RG flow; the scalar self‑interaction is unavoidable.

When the ELKO four‑fermion operator (λ_ξ (\bar ξ ξ)^2) is added together with the scalar‑ELKO portal, the system acquires an additional dimension‑negative coupling λ_ξ (canonical dimension −2). The FRG analysis shows that λ_ξ receives positive contributions from g_{φξ}^2 and mixed λ_φ g_{φξ} terms. In certain regions of parameter space λ_ξ can flow to a non‑trivial fixed point, suggesting the possibility of an asymptotically safe or asymptotically free UV completion despite the non‑renormalizable power counting.

For the gauge sector, the only allowed interaction is a Pauli‑like term κ \bar ξ σ^{μν}F_{μν} ξ, since minimal coupling is forbidden by ELKO neutrality. The β‑function for κ is proportional to κ itself (through the anomalous dimension η_A) and, in the absence of scalar portals, κ exhibits asymptotic freedom. When the scalar portal is present, mixed κ g_{φξ} contributions appear, allowing κ to approach a finite interacting fixed point.

Across all three models the authors identify a minimal set of interactions that must be present for a self‑consistent RG flow: the derivative Yukawa term forces a scalar quartic coupling; the four‑fermion term forces the scalar portal; the Pauli term is stable under RG but can be linked to the portal. They discuss the conditions under which UV‑complete trajectories exist, emphasizing that asymptotic freedom can be achieved in the gauge‑ELKO system and that non‑trivial fixed points may arise in the scalar‑ELKO system with both four‑fermion and portal couplings.

Methodologically, the paper demonstrates that the FRG provides a non‑perturbative yet tractable framework to explore theories with unconventional mass dimensions and non‑Hermitian interactions (the latter being recast as pseudo‑Hermitian). The automated Mathematica pipeline (xAct, DoFun, FormTracer, FORM) enables the handling of the intricate tensor algebra required for ELKO spinors.

In conclusion, the work establishes that ELKO fermions can be embedded in a renormalizable quantum field theory with a well‑defined RG flow, provided the minimal interaction set identified is included. The results open the door to constructing UV‑complete dark‑matter models based on mass‑dimension‑one fermions and illustrate the power of functional RG techniques in probing beyond‑Standard‑Model physics.