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For sub-GeV dark matter coupled to any of the minimal anomaly-free U(1) vector mediators, thermal freeze-out fixes the electron-recoil cross section in one-to-one correspondence with the annihilation cross section, giving direct-detection m

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load-bearing objection A useful systematic catalog of sub-GeV thermal-relic targets for U(1) mediators, but the 'one-to-one' predictivity claim is overstated because the relic condition leaves m_Z'/m_chi free. the 3 major comments →

arxiv 2603.03444 v2 pith:E665XBNR submitted 2026-03-03 hep-ph astro-ph.CO

New Thermal-Relic Targets for sub-GeV Dark Matter Direct Detection

Xu Han , Gordan Krnjaic This is my paper
classification hep-ph astro-ph.CO
keywords thermal relic dark mattersub-GeV dark mattervector mediatoranomaly-free U(1)electron-recoil direct detectionkinetic mixingL_mu-L_tau modelLee-Weinberg bound
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.

The paper argues that sub-GeV dark matter coupled to the new vector bosons of minimal anomaly-free U(1) extensions of the Standard Model has fully predictive direct-detection targets. Because the same couplings govern both the early-universe annihilation that sets the relic density and the present-day electron-scattering rate, the thermal-relic condition converts dark matter mass into a specific predicted electron-recoil cross section for each mediator. For mediators with tree-level electron couplings (dark photon, L_mu-Le, Le-L_tau, B-L, B-3L_e), current electron-recoil searches have almost fully excluded the predictions. For electrophobic mediators that couple to electrons only through loop-induced kinetic mixing (L_mu-L_tau, B-3L_mu, B-3L_tau), large viable parameter space remains, and the predicted cross sections sit within reach of near-future experiments. The paper presents these curves as milestones: seeing a signal would discover the model, and seeing nothing would falsify it.

Core claim

The central claim is that the minimal anomaly-free U(1) extensions of the Standard Model — the dark photon, gauged L_i-L_j, B-L, and B-3L_i — provide a complete set of predictive thermal-relic milestones for sub-GeV dark matter. When the mediator is heavier than the dark matter, both annihilation and electron scattering proceed through the same s-channel mediator, so the freeze-out condition ⟨σv⟩ ≈ 10^-26 cm^3/s fixes the effective electron cross section σ̄_e as a function of m_χ with no free parameters beyond the dark-sector coupling and mass ratio. Under this assumption, models with tree-level electron couplings are strongly constrained: the dark-photon scalar model is already excluded, an

What carries the argument

The load-bearing object is a heavier-than-dark-matter vector mediator Z′ (m_Z′ > m_χ) whose s-channel exchange controls both annihilation and scattering. The thermally-averaged annihilation cross section (Eqs. 12–15) and the effective electron cross section σ̄_e (Eqs. 22–23) depend on the same combination of the dark coupling α_D, the mediator–Standard Model coupling (g_Z′ or kinetic mixing ε), and m_Z′; thermal freeze-out fixes ⟨σv⟩, so σ̄_e is determined once m_χ is chosen. For electrophobic mediators, the one-loop kinetic mixing ε of Eq. (6) or Eq. (10) is the only pathway to electrons, and for B-3L_μ,τ this ε depends on an unknown ultraviolet scale Λ, which turns the milestone into a ban

Load-bearing premise

The predictivity of the entire milestone list depends on the assumption that the new U(1) is the minimal anomaly-free extension with only Standard Model field content (plus decoupled right-handed neutrinos where needed), so that no extra charged states alter the loop-induced kinetic mixing or add other constraints.

What would settle it

A measurement of the electron-recoil rate in a silicon detector — for example the upcoming Oscura exposure — that places an upper limit below the predicted L_μ-L_τ thermal-relic curve at any m_χ in the 1–100 MeV range would falsify that minimal model; the same logic applies to the B-3L_μ,τ band across its 100 GeV-to-M_Pl range of Λ.

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

If this is right

  • The dark-photon complex-scalar thermal-relic model is already excluded by DAMIC-M data, and the other tree-level electrophilic mediators leave at most small corners that existing exposures will soon cover.
  • For the L_μ-L_τ mediator, the thermal-relic target sits at lower σ̄_e due to loop suppression but remains within reach of SENSEI, Oscura, and upgraded DAMIC-M exposures.
  • For the B-3L_μ and B-3L_τ mediators, the thermal-relic target is a band whose thickness reflects the uncertainty in the kinetic-mixing scale; Λ=100 GeV gives the bottom edge and Λ=M_Pl the top edge.
  • Dirac fermion dark matter is viable only for the electrophobic mediators, and only for m_χ below the lightest charged lepton the mediator couples to, because annihilation then proceeds to neutrinos and avoids CMB energy-injection limits.
  • A null result at the predicted curves would falsify the minimal anomaly-free thermal-relic vector-mediator framework for sub-GeV dark matter in those channels.

Where Pith is reading between the lines

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

  • Because the paper's mediator list exhausts the minimal anomaly-free U(1) extensions with only Standard Model field content, a null search across all the listed milestones would eliminate the entire class of sub-GeV thermal dark matter with s-channel vector mediators, not just individual models.
  • The L_μ-L_τ and B-3L_μ,τ mediators are the same kinds of states often invoked for muon-related anomalies; a direct-detection signal or exclusion would cross-constrain those interpretations, a connection the paper leaves implicit.
  • The B-3L band's dependence on Λ suggests that measuring the thermal-relic curve would indirectly constrain the ultraviolet scale at which loop-induced kinetic mixing is generated, effectively turning direct detection into a probe of high-scale physics.
  • A natural extension would be to check whether non-minimal 'anomalon' scenarios can mimic these milestones at lower σ̄_e, since the paper notes that adding SM-charged states would introduce new constraints and modify the loop mixing.

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

3 major / 3 minor

Summary. The paper studies sub-GeV dark matter coupled to massive vector mediators from minimal anomaly-free U(1) extensions to the Standard Model. For complex scalar and Dirac fermion dark matter with m_Z' > m_chi, the thermal relic abundance fixes the annihilation cross section, and the same couplings determine electron-recoil direct detection. The authors present thermal-relic targets for the dark photon, L_i - L_j, B-L, and B-3L_i models, and overlay current and future constraints. They conclude that electrophilic mediators are nearly excluded, while electrophobic mediators L_mu - L_tau and B-3L_mu,tau retain large viable parameter space.

Significance. If the targets are correct, the paper provides a useful systematic map for sub-GeV direct-detection searches. Its analytic formulas are standard and internally consistent, and the enumeration of minimal anomaly-free U(1) mediators is a valuable contribution. The claimed 'robustly discovered or falsified' status, however, is stronger than what the paper demonstrates: the r = m_Z'/m_chi degeneracy near the s-channel resonance and the Lambda-dependence for B-3L_i mean that the targets are not always unique curves. These issues are fixable but need explicit treatment before the central claim can be accepted.

major comments (3)
  1. [Sec. VI, Eqs. (13)-(15), (22)-(23)] The one-to-one correspondence between relic density and sigma_bar_e is stated for all m_Z' > m_chi. For fixed m_chi, the relic condition leaves r = m_Z'/m_chi free. Off the s-channel pole the target is indeed r-independent since both annihilation and scattering scale as 1/m_Z'^4, but for r near 2 the Breit-Wigner denominator in Eqs. (13)/(15) reduces the required alpha_D and suppresses sigma_bar_e relative to the plotted line. The dark-photon resonance is acknowledged in footnote 1, but the same caveat applies to all mediators and is not quantified. The figures should state the r assumed for the target curves (or show the resonant band); otherwise a null result excludes only the benchmark, not the full m_Z' > m_chi parameter space.
  2. [Sec. IV (CMB), Table I, Eq. (9)] The statement that Dirac DM is viable below the lightest charged particle to which the mediator couples is incorrect for B-3L_tau. The B-3L_tau current in Eq. (9) includes quarks, so for m_chi > m_pi hadronic annihilation opens; the CMB bound Eq. (24) then excludes thermal-relic Dirac DM for m_chi > m_pi, not only for m_chi > m_tau. Table I's '(nu nu only for m_chi < m_tau)' for B-3L_tau should read m_chi < m_pi (or the hadronic threshold). This affects the right panel of Fig. 6 and the summary of viable windows.
  3. [Sec. IV, Eq. (10), Fig. 6] For B-3L_mu and B-3L_tau, sigma_bar_e depends on epsilon^2, and epsilon depends on the unknown scale Lambda through Eq. (10). The paper acknowledges this and shows a band, but the abstract and Sec. VI still call these 'firmly predictive milestones.' Since the target is a band spanning the Lambda range, the falsifiability claim should be qualified: a null result only excludes the assumed Lambda range, and a positive signal would not determine m_chi without knowing Lambda. Please quantify the band thickness and soften the wording accordingly.
minor comments (3)
  1. [Sec. II, Eqs. (3)-(4)] The width formulas apply only for m_Z' > 2m_chi, while the analysis assumes only m_Z' > m_chi. Clarify how the region m_chi < m_Z' < 2m_chi is treated (the DM decay channel is closed, so the width should be SM-dominated rather than given by Eqs. (3)/(4)).
  2. [Sec. II, Eq. (6)] The sign of epsilon in Eq. (6) depends on the sign convention for the Z' charges. For L_mu-L_tau the log is finite, but the absolute value is what enters the cross section; consider stating |epsilon| or defining the convention.
  3. [Fig. 2 caption] The caption's statement that for B-3L_e,tau 'xi = epsilon Q_mu' is confusing because the hadronic R-ratio for these models uses the tree-level quark coupling, not the kinetic-mixing-induced muon coupling. Please clarify the normalization.

Circularity Check

0 steps flagged

No significant circularity; the target curve is an equation-solving benchmark, though the 'one-to-one' phrasing overstates predictivity due to the free m_Z'/m_chi ratio.

full rationale

The derivation chain is self-contained and non-circular: the paper defines anomaly-free U(1) mediators, computes s-channel annihilation cross sections (Eqs. 13-21), imposes the thermal-relic condition by solving the Boltzmann equation (Eqs. 11-12), and then translates the same couplings into the electron-recoil cross section (Eqs. 22-23). No electron-scattering data are used as inputs to set the model parameters, and experimental constraints are applied after the relic curves are constructed. The dark-photon panel is adapted from the same-author preprint [15], but it is included for completeness and is anchored by the external DAMIC-M measurement [14]; it is not load-bearing for the new L_i-L_j, B-L, or B-3L_i targets. The hadronic R-ratio prescription is taken from the independent Ref. [21], not from the present authors. The limitations the paper itself flags in Sec. VI—the anomalon loophole and the B-3L_i Lambda-dependence of the kinetic mixing—are honest caveats and do not reduce the derivation to its inputs. The main issue is the overstatement that the relic condition puts the direct-detection cross section in 'one-to-one correspondence' with the annihilation cross section: for fixed m_chi, the ratio m_Z'/m_chi is a free parameter, so a plotted target line is a benchmark member of a one-parameter family unless that ratio is specified. This is a predictive-ambiguity/correctness caveat, not a circular fit or a self-referential derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The paper introduces no new particles or forces; it reuses well-known mediator models. The free parameters are choices for plotting accelerator constraints and the UV-scale Lambda that sets the kinetic-mixing band for B-3L models. The main load-bearing assumptions are the m_Z'>m_chi regime, the completeness of the minimal U(1) list without anomalons, and the standard freeze-out cosmology.

free parameters (3)
  • alpha_D = 0.5 (chosen for constraint overlays) = 0.5
    Used to translate B-factory and CCFR bounds onto the sigma_e-m_chi plane (Figs. 4-5 captions); the paper asserts smaller values make constraints more severe but does not quantify the shift.
  • m_Z'/m_chi = 3 (chosen for constraint overlays) = 3
    Same overlay conversions; not a fit, but the plotted limits move if this ratio is changed.
  • Lambda (UV scale in loop-induced kinetic mixing for B-L and B-3L_i) = 100 GeV - M_Pl band
    Appears in Eqs. (8) and (10); for B-3L_mu,tau it controls epsilon and therefore the entire direct-detection cross section, producing the target band in Fig. 6.
axioms (7)
  • domain assumption m_Z' > m_chi so annihilation is s-channel and direct-detection rates depend on the same coupling combination as the relic abundance
    Stated in Sec. II and Table I; defines the predictive regime but excludes the resonant and on-shell decay regimes.
  • standard math Standard thermal freeze-out with standard cosmology and the Lee-Weinberg bound
    Sec. III; all targets rely on the standard Boltzmann relic condition.
  • domain assumption The anomaly-free U(1) list is complete assuming only SM field content plus right-handed neutrinos and no anomalons
    Sec. VI explicitly excludes anomalon-extended models and B_i-B_j baryonic extensions; the 'complete list' claim in the abstract depends on this assumption.
  • domain assumption g_D >> g_Z' so the mediator width is dominated by decays to dark matter
    Sec. II after Eq. (4); affects the Z' propagator width in the annihilation cross section.
  • standard math R-ratio modeling for hadronic annihilation in B-L and B-3L_i models follows Ref. [21]
    Sec. III and Fig. 2; used for the hadronic contribution to the relic abundance.
  • standard math Loop-induced kinetic mixing formulas, Eqs. (6), (8), and (10), give the dominant electron coupling for electrophobic mediators
    Fig. 1; for L_mu-L_tau, B-3L_mu and B-3L_tau this is the entire direct-detection interaction.
  • domain assumption Existing CMB and BBN constraints from Refs. [34-40] are correctly applied
    Used to exclude Dirac fermion DM in most models and to set the low-mass BBN bound.

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read the original abstract

Dark matter direct detection experiments involving electron recoils are beginning to test highly-predictive, thermal-relic milestones for sub-GeV dark matter models. Due to the Lee-Weinberg bound, thermal dark matter candidates in this mass range necessarily require comparably-light mediator particles to achieve a suitably large annihilation cross section. Here we present new thermal-relic milestones for sub-GeV dark matter candidates that couple to vector mediators. In these models, the mediators are massive gauge bosons of anomaly-free abelian extensions to the Standard Model, including the dark photon, gauged $L_i - L_j, B-L$, and $B-3L_i$ models, where $B$ is the baryon number, $L$ is the lepton number, and $i,j$ index the lepton families. Since the same interactions that govern cosmological production also govern electron scattering, the targets we present are firmly predictive and allow for these models to be robustly discovered or falsified. Furthermore, since the mediators we study exhaust the minimal anomaly-free U(1) extensions to the Standard Model, our results offer a complete list of predictive milestones for sub-GeV dark matter coupled to vector mediators.

Figures

Figures reproduced from arXiv: 2603.03444 by Gordan Krnjaic, Xu Han.

Figure 1
Figure 1. Figure 1: FIG. 1: In each of the models we consider, there is an [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Normalized R-ratios for mediators with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Limits on the effective DM-electron scattering [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Thermal relic milestones for complex scalar DM coupled to the following mediators: [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Limits on the effective DM-electron scattering cross section ¯σ [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Top row: limits on the effective DM-electron scattering cross section ¯σ [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

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