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Dirac right-handed neutrinos turn ΔN_eff into a decisive filter on Z' portal dark matter, excluding resonance WIMPs below ~400 GeV and leaving secluded and FIMP regions that future CMB can still test.

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2026-07-10 13:23 UTC pith:OQM2ULGB

load-bearing objection Solid, incremental B-L portal scan that correctly folds the ν_R–χ channel and latest ΔN_eff bounds into concrete (m_χ, Q_χ) windows for resonance, secluded, and FIMP cases. the 2 major comments →

arxiv 2607.08082 v1 pith:OQM2ULGB submitted 2026-07-09 hep-ph

Z^prime Portal Dark Matter with Observable Delta N_(rm eff)

Ang Liu , Zhi-Long Han , Fei Huang This is my paper
classification hep-ph PACS 95.35.+d14.70.Pw98.80.Cq12.60.Cn
keywords Z' portalDirac neutrinosΔ N_effWIMPFIMPU(1)_{B-L}dark matter relic densityCMB-S4
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard Z' portal dark-matter models usually assume Majorana right-handed neutrinos, so the only experimental handles are collider searches for the Z' and direct detection of the dark-matter particle. This paper replaces those neutrinos with Dirac fermions protected by an unbroken U(1)_{B-L} symmetry. The same Z' that couples the dark matter to the Standard Model also keeps the three Dirac neutrinos in (or out of) thermal equilibrium, generating a measurable extra radiation density ΔN_eff. Because dark-matter freeze-out or freeze-in and neutrino decoupling are controlled by the same gauge coupling and charge, every viable dark-matter abundance automatically predicts a definite ΔN_eff. Current bounds already force resonant WIMP dark matter above roughly 400 GeV and charge Q_χ ≳ 0.7; secluded and freeze-in candidates can sit below the thermal floor ΔN_eff = 0.14. Future CMB experiments that reach ΔN_eff ~ 0.03 will either discover the signal or rule out the entire resonant branch, while still leaving open windows for the other two production mechanisms.

Core claim

When right-handed neutrinos are Dirac particles charged under the same U(1)_{B-L} that mediates dark-matter interactions, the requirement that the observed relic density be produced through the Z' portal forces a lower bound ΔN_eff ≥ 0.14 in the resonant WIMP case and permits smaller, still-testable values only for secluded WIMPs and FIMPs; the combined cosmological, direct-detection, indirect-detection and collider constraints therefore carve out sharply defined, experimentally accessible regions in the (m_χ, Q_χ) plane.

What carries the argument

The shared Z' portal: both the dark-matter annihilation (or freeze-in) rate and the ν_R ν_R o f f̄ / χ χ̄ scattering rates are controlled by the same product g' Q, so the temperature at which the Dirac neutrinos decouple—and hence the value of ΔN_eff—is locked to the dark-matter abundance.

Load-bearing premise

The B-L symmetry that keeps the right-handed neutrinos Dirac and generates a light Z' via the Stueckelberg mechanism must remain unbroken; any spontaneous breaking that gives the neutrinos Majorana masses or removes the light Z' collapses the whole ΔN_eff prediction.

What would settle it

A future CMB measurement that sets ΔN_eff < 0.14 with no residual excess would completely exclude the resonant WIMP branch of the model while leaving only the secluded and FIMP windows still open.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Resonant WIMP dark matter is forced into a narrow high-mass window (roughly 400 GeV to 10^5 GeV) with charges between ~1 and a few thousand, all of which will be covered by CMB-S4/HD.
  • Secluded and freeze-in candidates can produce ΔN_eff well below 0.14, so a null CMB result does not kill the entire Z' portal scenario.
  • The same parameter space that survives cosmology is already within reach of next-generation direct-detection experiments for TeV-scale masses and of MeV-scale gamma-ray telescopes for the secluded annihilation channel.
  • Collider searches for a light Z' become secondary once ΔN_eff is measured; the cosmological bound is stronger than present LHC and B-factory limits over most of the interesting range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If Dirac neutrinos are the only light species charged under B-L, a precision ΔN_eff measurement effectively becomes a dark-matter mass and charge spectrometer for any Z' portal model.
  • The same logic can be ported to other anomaly-free U(1) extensions (e.g., L_μ-L_τ) that keep right-handed neutrinos Dirac, potentially turning every future CMB stage into a simultaneous probe of neutrino nature and dark-matter production.
  • A confirmed ΔN_eff excess near 0.14 would favour thermal production of both ν_R and χ, while a value significantly below that floor would point toward freeze-in or secluded dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies a minimal U(1)_{B-L} extension with Dirac right-handed neutrinos u_R (Q_ u R = −1) and a vector-like Dirac dark-matter fermion u_R-protected by unbroken B−L and a Z_2 parity, with the Z′ mass generated by the Stueckelberg mechanism. Free parameters are {m_ u, m_Z′, g′, Q_ u}. Both WIMP (resonant u ū o f f̄ and secluded u ū o Z′Z′) and FIMP production via the Z′ portal are analyzed. The central claim is that thermal and non-thermal u_R contributions to u N_eff, together with perturbativity, direct/indirect detection, and Z′ collider bounds, carve out concrete viable windows: resonant WIMP survives only for roughly 400 GeV ≲ m_ u ≲ 1.54 imes10^5 GeV and 0.7 ≲ Q_ u ≲ 3400 (with u N_eff less 0.14), while secluded and FIMP regions can yield u N_eff < 0.14 and remain testable by CMB-S4/HD. A future null result u N_eff < 0.14 would exclude the resonant case.

Significance. If the results hold, the work supplies a clean, falsifiable link between Z′ portal dark matter and upcoming precision u N_eff measurements. The multi-constraint scans (Figs. 4, 5, 7) and the explicit statement that a null u N_eff < 0.14 kills the resonant WIMP window are concrete predictions that can be tested by CMB-S4/HD, future direct/indirect detection, and colliders. The treatment of the additional u_R ū_R o u ū channel, the piecewise reaction rates, and the consistent use of micrOMEGAs for thermal averages are standard and reproducible within the stated model assumptions. The paper therefore adds a useful, observationally sharp corner to the Z′ portal literature.

major comments (2)
  1. Section II (after Eq. (1) and the Stueckelberg paragraph): the unbroken U(1)_{B-L} that both forbids Majorana masses for u_R and generates the light Z′ is a model-building premise, not an inconsistency. The paper should, however, state more explicitly that any spontaneous breaking that generates Majorana masses or removes the light Z′ collapses the entire u N_eff calculation and the portal structure; a short paragraph quantifying the scale at which this occurs would strengthen the claim’s robustness.
  2. Section III.A, Eqs. (10)–(12) and the discussion of T_ u R^dec vs T_ u^dec: the claim that u_R ū_R o u ū dominates the resonant decoupling for large Q_ u is load-bearing for the floor u N_eff less 0.14. The paper should quantify more carefully the extreme-resonance regime (where phase-space suppression of u Z′ o u ū makes the channel negligible) and show that the quoted mass window 400 GeV ≲ m_ u remains intact when that regime is included.
minor comments (5)
  1. Figure 1 caption and panels: the Unicode/encoding artifacts (e.g. “/uni0000000b”) make the legends hard to read; clean PDF fonts would improve clarity.
  2. Eq. (13) and the surrounding text: the nucleon mass is written m_n less 0.939 GeV; a brief note that the reduced-mass factor is already included would avoid ambiguity.
  3. Section IV.A, freeze-in Boltzmann equations (15)–(16): the neglect of t-channel Z′Z′ and of u ū o u_R ū_R is stated but not quantified; a short numerical check (or reference to the verification mentioned in the text) would be helpful.
  4. Throughout: the notation r_Z′ = m_Z′/m_ u is introduced early but occasionally written inconsistently (r_Z vs r_Z′); unify for readability.
  5. References: a few recent related works on Dirac u_R and u N_eff (e.g. those already cited in the introduction) could be cross-linked more explicitly when the non-thermal results of Ref. [18] are adopted.

Circularity Check

0 steps flagged

No significant circularity: relic-density contours and ΔN_eff floors are computed from free parameters against external experimental bounds.

full rationale

The paper constructs a minimal U(1)_{B-L} model with Dirac ν_R and vector-like DM χ, then solves the Boltzmann equations for freeze-out (WIMP resonance/secluded) and freeze-in (FIMP) production of both χ and ν_R. Relic-density contours (black lines in Figs. 4, 5, 7) are obtained by requiring Ω_χ h^{2} = 0.12 for free parameters {m_χ, m_Z', g', Q_χ}; they are not forced by a normalization chosen to match the target. The thermal ΔN_eff floor of 0.14 follows directly from Eq. (10) once g_*(T_dec) o 106.75, an external SM input. All exclusion regions (DESI, P-ACT, CMB-S4/HD, LZ/XENONnT, AMS/Fermi/HESS, BaBar/LHCb/LEP/CMS/ATLAS) are taken from independent experimental literature. Self-citations (e.g., [53], [54]) supply only prior related calculations and do not close a logical loop that defines the quoted mass/charge windows. The unbroken B-L + Stueckelberg premise is a model-building assumption, not an internal circular construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

The central claim rests on four free parameters of the minimal U(1)_{B-L} extension, standard cosmological Boltzmann evolution, the assumption that B-L remains unbroken (protecting Dirac u_R and generating Z' via Stueckelberg), and the usual experimental upper limits treated as hard cuts. No additional ad-hoc scales are introduced beyond the free parameters themselves.

free parameters (4)
  • m_χ
    Dark-matter mass; scanned freely and fixed only by the requirement of correct relic density plus experimental bounds.
  • m_Z'
    Z' mass (or equivalently the ratio r_Z' = m_Z'/m_χ); free input that sets the resonance or secluded kinematics.
  • g'
    U(1)_{B-L} gauge coupling; free and constrained by perturbativity, collider searches and relic density.
  • Q_χ
    U(1)_{B-L} charge of the dark-matter fermion; arbitrary free parameter that controls both annihilation and direct-detection rates.
axioms (4)
  • domain assumption Standard cosmological Boltzmann equations for freeze-out and freeze-in with entropy and Hubble rates that include three Dirac u_R degrees of freedom when thermally populated.
    Used throughout Sections III and IV to compute Y_χ and Y_ u_R; taken as standard without re-derivation.
  • ad hoc to paper Unbroken U(1)_{B-L} that both forbids Majorana masses for u_R and generates the Z' mass via the Stueckelberg mechanism.
    Stated in Section II; if violated the Dirac nature and the light Z' portal disappear.
  • domain assumption Yukawa coupling y ≲ 10^{-11} that generates sub-eV Dirac neutrino masses contributes negligibly to ΔN_eff (≈ 7.5 imes10^{-12}).
    Cited from Luo et al.; used to drop the Yukawa contribution entirely.
  • domain assumption Current experimental upper limits on ΔN_eff (DESI ≲ 0.4, P-ACT ≲ 0.17), σ_SI, ⟨σv⟩ and Z' production can be treated as hard exclusion boundaries.
    Applied throughout the figures; standard practice in the field.
invented entities (2)
  • Vector-like Dirac fermion χ with arbitrary U(1)_{B-L} charge Q_χ and Z_2-odd parity no independent evidence
    purpose: Serves as the dark-matter candidate that couples to the Z' portal.
    Introduced in Section II; stability is imposed by hand via an extra Z_2; no independent evidence outside the model.
  • Massive Z' boson generated by the Stueckelberg mechanism under unbroken B-L no independent evidence
    purpose: Mediates the portal interactions between SM fermions, u_R and χ.
    Stated after Eq. (1); the Stueckelberg choice keeps B-L unbroken so that u_R remain Dirac.

pith-pipeline@v1.1.0-grok45 · 31527 in / 3348 out tokens · 32494 ms · 2026-07-10T13:23:42.409199+00:00 · methodology

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read the original abstract

In the conventional $Z^\prime$ portal dark matter scenario, the prediction of detectable dark matter $\chi$ typically relies on the collider sensitivities of $Z^\prime$ and direct detection, where the Majorana type right-handed neutrinos are usually assumed. However, if the right-handed neutrinos $\nu_R$ are Dirac type, they will contribute to the additional effective number of relativistic species $\Delta N_{\rm eff}$, which brings different detectable predictions for $Z^\prime$ portal dark matter. In light of the great improvement of $\Delta N_{\rm eff}$ for the upcoming experiments, we investigate the $Z^\prime$ portal dark matter with Dirac type $\nu_R$. Under the $U(1)_{B-L}$ symmetry, this model includes $\nu_R$ with $U(1)_{B-L}$ charge $Q_{\nu_R}=-1$ and $\chi$ with arbitrary $Q_\chi$ beyond the SM. Based on the relation in the production of $\chi$ and $\nu_R$, both the WIMP and FIMP dark matter through the $Z^\prime$ portal scenario are considered. We perform a comprehensive exploration of the viable parameter space under the constraints from $\Delta N_{\rm eff}$ induced by thermal and non-thermal $\nu_R$, perturbative limit, dark matter direct and indirect detection, and collider searches of $Z^\prime$.

Figures

Figures reproduced from arXiv: 2607.08082 by Ang Liu, Fei Huang, Zhi-Long Han.

Figure 1
Figure 1. Figure 1: FIG. 1. Panel (a): The dependence of DM relic density in the WIMP scenario. Panel (b): Influence of different [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Constraints from DM direct detection experiments and cosmological experiments related to [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The DM indirect detection and [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comprehensive constraints in the resonance scenario. Panel (a), (b), and (c) correspond to the comprehensive [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Figure 4 but for the secluded scenario. Panels (a), (b), and (c) correspond to cases [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Panel (a): The dependence of DM relic density in the FIMP scenario. Panel (b): The evolution of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Figure 4, but for the FIMP scenario. Panel (a), (b), and (c) correspond to cases of [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

discussion (0)

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