Two-component dark matter from a flavor-dependent $U(1)$ gauge extension

Abstract: We revisit the dark matter phenomenology of a flavor-dependent $U(1)_X$ gauge extension of the Standard Model, where anomaly cancellation predicts the existence of exactly three fermion generations and requires the presence of three right-handed neutrinos. In Ref.~\cite{VanLoi:2023utt}, a strong hierarchy between the vacuum expectation values of two singlet scalars, $\La_2 \gg \La_1$, renders all $\mathbb{Z}_2$-odd scalar states heavy, resulting in a two-component dark matter scenario composed exclusively of fermions. In the present work, we relax this simplifying assumption and consider a more general mass spectrum. In particular, scalar mixing can naturally lead to a situation in which the lightest $\mathbb{Z}_2$-odd particle is a scalar rather than a fermion. As a consequence, the model admits a qualitatively new realization of two-component dark matter consisting of one fermionic and one scalar component, in addition to the purely fermionic scenario studied previously. We perform a dedicated phenomenological analysis of these two-component dark matter realizations, focusing on the coupled thermal freeze-out dynamics and the resulting relic abundance. Constraints from the observed relic density and current direct-detection limits are taken into account, and viable regions of parameter space are identified.

• The paper shows that relaxing the singlet-VEV hierarchy enables two viable dark-matter realizations: two stable fermions or a fermion–scalar pair, while retaining scotogenic neutrino masses.
• Coupled Boltzmann-equation calculations find viable two-fermion dark matter above roughly 2–2.5 TeV, whereas the fermion–scalar case permits the stable neutrino near 1.65 TeV through additional annihilation and conversion channels.
• The fermion-only scenario predicts extremely weak spin-independent scattering, while scalar dark matter typically yields cross sections near 10⁻⁴⁷–10⁻⁴⁸ cm², making it a promising target for next-generation detectors.

Overview and motivation

This paper revisits the dark matter (DM) phenomenology of a flavor-dependent $U(1)_X$ gauge extension of the Standard Model (SM), originally proposed in Ref. [VanLoi:2023utt]. The defining feature of the original construction is that gauge anomaly cancellation uniquely fixes the number of fermion generations to three and mandates the presence of three right-handed neutrinos. After spontaneous symmetry breaking, a residual $\mathbb{Z}_2$ symmetry stabilizes the lightest $\mathbb{Z}_2$-odd particle, while one $\mathbb{Z}_2$-even right-handed neutrino ($\nu_{3R}$) is accidentally stabilized by gauge invariance, yielding two DM components together with scotogenic neutrino masses.

The prior analysis imposed a strong hierarchy between the vacuum expectation values (VEVs) of the two singlet scalars, $\Lambda_2 \gg \Lambda_1$, which rendered all $\mathbb{Z}_2$-odd scalars heavy and restricted the DM sector to two fermions. The present work relaxes this assumption to the more general case $\Lambda_1 \sim \Lambda_2$. Under these conditions, scalar mixing allows the lightest $\mathbb{Z}_2$-odd state to be a scalar rather than a fermion, opening a qualitatively new fermion–scalar two-component DM realization alongside the previously studied fermion–fermion scenario.

Model structure and particle spectrum

The $U(1)_X$ charge is defined as $\mathbb{Z}_2$0 with family-dependent coefficients. The $\mathbb{Z}_2$1 anomaly requires $\mathbb{Z}_2$2, so combined with QCD asymptotic freedom ($\mathbb{Z}_2$3) only $\mathbb{Z}_2$4 or $\mathbb{Z}_2$5 are allowed; the authors select $\mathbb{Z}_2$6 ($\mathbb{Z}_2$7, $\mathbb{Z}_2$8). Cancellation of the mixed gravitational and cubic anomalies then forces three right-handed neutrinos with charges $\mathbb{Z}_2$9 and $\mathbb{Z}_2$0. The scalar sector comprises the SM Higgs doublet $\mathbb{Z}_2$1, two singlets $\mathbb{Z}_2$2 and $\mathbb{Z}_2$3 with VEVs $\mathbb{Z}_2$4 GeV, and two inert fields—a doublet $\mathbb{Z}_2$5 and a singlet $\mathbb{Z}_2$6—required for scotogenic neutrino mass generation.

In the general spectrum without the $\mathbb{Z}_2$7 hierarchy, the CP-even singlet states mix into $\mathbb{Z}_2$8, while the inert doublet and singlet mix into $\mathbb{Z}_2$9 and $\mathbb{Z}_2$0. The mixing angles $\mathbb{Z}_2$1 and $\mathbb{Z}_2$2 are strongly suppressed by the condition $\mathbb{Z}_2$3, so an approximate alignment holds in which $\mathbb{Z}_2$4 maps predominantly to $\mathbb{Z}_2$5 and $\mathbb{Z}_2$6 to $\mathbb{Z}_2$7. A key structural result is that $\mathbb{Z}_2$8 is always the lightest $\mathbb{Z}_2$9-odd scalar: its squared mass differs from the other dark scalar masses by a term proportional to $\nu_{3R}$0, which is sizable at the TeV scale. This guarantees a stable scalar DM candidate whenever the scalar sector is light enough. The model also predicts a neutral gauge boson $\nu_{3R}$1 with mass $\nu_{3R}$2; no tree-level $\nu_{3R}$3–$\nu_{3R}$4 mass mixing arises since $\nu_{3R}$5 is uncharged under $\nu_{3R}$6. Collider and flavor constraints push $\nu_{3R}$7 to the multi-TeV regime, which the authors adopt as input.

Residual symmetry and DM scenarios

The breaking of $\nu_{3R}$8 by the $\nu_{3R}$9 VEVs leaves a residual discrete symmetry. Requiring invariance of both singlet VEVs yields transformations $\Lambda_2 \gg \Lambda_1$0, and combining with spin-parity gives the physically relevant conserved subgroup $\Lambda_2 \gg \Lambda_1$1 with $\Lambda_2 \gg \Lambda_1$2. All SM fields, $\Lambda_2 \gg \Lambda_1$3, $\Lambda_2 \gg \Lambda_1$4 are even; $\Lambda_2 \gg \Lambda_1$5, $\Lambda_2 \gg \Lambda_1$6, and $\Lambda_2 \gg \Lambda_1$7 are odd. Consequently, the model admits exactly two multicomponent DM realizations: a two-fermion scenario ($\Lambda_2 \gg \Lambda_1$8 plus $\Lambda_2 \gg \Lambda_1$9) and a fermion–scalar scenario ($\mathbb{Z}_2$0 plus $\mathbb{Z}_2$1).

Coupled freeze-out dynamics

The relic abundances are obtained by solving coupled Boltzmann equations including pair annihilation into $\mathbb{Z}_2$2-even states, coannihilation among odd particles, and DM conversion processes between components, with Heaviside functions enforcing kinematic thresholds. The system is solved numerically with micrOMEGAs 6.2.4. The phenomenologically relevant parameters are $\mathbb{Z}_2$3, the trilinear coupling $\mathbb{Z}_2$4, Yukawa couplings $\mathbb{Z}_2$5 and $\mathbb{Z}_2$6, and the masses $\mathbb{Z}_2$7 and $\mathbb{Z}_2$8. The Yukawa couplings are restricted to narrow ranges ($\mathbb{Z}_2$9) because they enter the one-loop scotogenic neutrino mass matrix directly.

Two-fermion dark matter

Taking $\Lambda_1 \sim \Lambda_2$0 as the lightest $\Lambda_1 \sim \Lambda_2$1-odd fermion (with no $\Lambda_1 \sim \Lambda_2$2–$\Lambda_1 \sim \Lambda_2$3 mixing assumed), points reproducing $\Lambda_1 \sim \Lambda_2$4 cluster in the upper-right region of the $\Lambda_1 \sim \Lambda_2$5 plane, giving lower bounds of approximately $\Lambda_1 \sim \Lambda_2$6 TeV and $\Lambda_1 \sim \Lambda_2$7 TeV. Over most of the viable space the two components contribute comparably to the total relic density, a direct consequence of $\Lambda_1 \sim \Lambda_2$8 producing similar annihilation cross sections. This contrasts sharply with the hierarchical limit of the earlier study, where $\Lambda_1 \sim \Lambda_2$9 was adopted and the relic fractions were strongly asymmetric.

The predicted spin-independent (SI) cross sections lie in the range $\mathbb{Z}_2$0, several orders of magnitude below XENONnT, LZ, and PandaX-4T limits. The suppression follows because elastic scattering proceeds via $\mathbb{Z}_2$1-channel exchange of the heavy scalars $\mathbb{Z}_2$2 with couplings proportional to light quark masses. Notably, this analysis improves on Ref. [VanLoi:2023utt] by including the heavy-scalar mediators, whereas the earlier work considered only the $\mathbb{Z}_2$3 portal, which yields predominantly suppressed spin-dependent interactions.

Fermion–scalar dark matter

When $\mathbb{Z}_2$4 is the lightest $\mathbb{Z}_2$5-odd particle, the DM system consists of $\mathbb{Z}_2$6 and $\mathbb{Z}_2$7. The scalar component annihilates efficiently through Higgs-portal channels ($\mathbb{Z}_2$8) and participates in conversion processes such as $\mathbb{Z}_2$9, producing a freeze-out pattern qualitatively distinct from the purely fermionic case. The fermionic component now requires only $U(1)_X$0 TeV—a weaker bound than in the two-fermion scenario—because the additional scalar-mediated annihilation and conversion channels deplete the total abundance more efficiently. The scalar mass $U(1)_X$1 spans a broad range within the scan.

The central tension of the paper concerns the scalar component's direct-detection prospects. The SI scattering of $U(1)_X$2 off nucleons is dominated by $U(1)_X$3-channel exchange of $U(1)_X$4, $U(1)_X$5, and $U(1)_X$6, controlled by the same couplings $U(1)_X$7 and $U(1)_X$8 that govern its annihilation. Large $U(1)_X$9 are needed to avoid overclosing the Universe but simultaneously raise the SI rate. As a result, the fermion–scalar scenario is considerably more constrained than the purely fermionic one: viable solutions exist, but the predicted cross section for $\mathbb{Z}_2$00 typically lies near $\mathbb{Z}_2$01, close to the projected sensitivity of next-generation direct-detection experiments. This makes the new realization decisively testable—and potentially falsifiable—in the near future.

Limitations and open questions

Several assumptions bound the scope of the results. The quartic couplings $\mathbb{Z}_2$02 are omitted from the numerical analysis on the grounds that they play no significant role in the DM phenomenology, though their full impact on the scalar spectrum and vacuum stability is not quantified here. Kinetic mixing between $\mathbb{Z}_2$03 and $\mathbb{Z}_2$04 is neglected, and the Yukawa matrices are taken flavor diagonal for simplicity, so flavor-violating coannihilation and conversion effects remain unexplored. The claim that $\mathbb{Z}_2$05 is always the lightest dark scalar relies on $\mathbb{Z}_2$06 being TeV-scale; the behavior outside this regime is not mapped. Finally, indirect-detection signals and collider signatures of the scalar DM component are not addressed, leaving open how the fermion–scalar scenario could be discriminated from the fermion–fermion case at colliders given the multi-TeV $\mathbb{Z}_2$07 mass.

Conclusion

By relaxing the $\mathbb{Z}_2$08 hierarchy of the original flavor-dependent $\mathbb{Z}_2$09 model, this work enlarges the DM sector to include a stable scalar candidate $\mathbb{Z}_2$10 alongside the accidentally stable $\mathbb{Z}_2$11. The two-fermion scenario remains viable with TeV-scale masses and SI cross sections far below current limits, while the newly identified fermion–scalar scenario features an intrinsic tension between relic-density and direct-detection constraints through the shared couplings $\mathbb{Z}_2$12 and $\mathbb{Z}_2$13. The latter scenario, though consistent with present data, sits within reach of next-generation direct-detection experiments, making it a concrete target for near-future tests of this class of flavor-dependent gauge extensions.