Spin-Dependent Scattering of Sub-GeV Dark Matter: Models and Constraints

Bethany Suter; Pankaj Munbodh; Simon Knapen; Stefania Gori; Tongyan Lin

REVIEW 3 major objections 5 minor 1 cited by

Spin-Dependent Scattering of Sub-GeV Dark Matter: Models and Constraints

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper calculates spin-dependent sub-GeV dark matter scattering in crystals and finds that only the scalar-mediator model with $m_\chi \gtrsim 100$ MeV has open parameter space, through the SN1987A trapping window, with a maximal…

desk verdict A genuinely new multiphonon rate calculation for spin-dependent sub-GeV DM, wrapped in an honest but heavily caveat-dependent constraint survey whose headline open window rests on the SN1987A trapping gap. read the letter →

arxiv 2506.11191 v1 pith:DYDXZLQB submitted 2025-06-12 hep-ph

Stefania Gori , Simon Knapen , Tongyan Lin , Pankaj Munbodh , Bethany Suter This is my paper

classification hep-ph

keywords sub-GeVdarkmatterspin-dependentscatteringphononproductiondirectdetectionlightmediatorssupernovacoolingSN1987Aself-interactions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper calculates how often sub-GeV dark matter scatters off nuclei in crystals through spin-dependent interactions, using phonon excitations for light dark matter and nuclear recoils for heavier dark matter. It then asks which of three simple mediator models — a scalar $\phi$, a pseudoscalar $a$, and an axial vector $A'$ — could still produce a signal in near-future detectors after all current bounds on the mediators are applied. The answer is that only the scalar model, with dark matter heavier than about 100 MeV, has open parameter space, and that space exists only because a narrow 'trapping window' in the SN1987A supernova cooling bound is currently unconstrained. In that window the maximal signal in a sapphire target is roughly a hundred events per gram-year of exposure. The pseudoscalar and axial-vector models are effectively closed off by meson-decay, beam-dump, stellar-cooling, self-interaction, and LHC bounds.

What carries the argument

The machinery has two halves. One is the crystal response: the rate is built from the phonon structure factor $S(q,\omega)$, obtained from per-site correlation functions $C_{\ell,d}(q,\omega)$ and summed incoherently over lattice sites as $\lambda_d^2 J_d(J_d+1)$, where $\lambda_d$ encodes the nuclear spin matrix elements through the odd-group or shell model. This turns the spin-dependent DM-nucleon Hamiltonians (momentum-suppressed for $\phi$ and $a$, spin-only for $A'$) into an integral over DM velocity, momentum transfer, and phonon energy, with an explicit switch to nuclear recoil above about 100 MeV. The other half is the constraint map: measured bounds on the mediator's coupling to protons and neutrons — from rare meson decays, CHARM and E137 beam dumps, horizontal branch stars, and SN1987A — plus self-interaction bounds on the dark matter coupling, are each converted into caps on the reference cross sections.

What would settle it

Recompute the SN1987A cooling bound for $\phi$ and $a$ mediators including diffusive energy transport inside the proto-neutron star, over the mass range 0.1 to 100 MeV and couplings $2\times10^{-6} \lesssim g_p \lesssim 7\times10^{-6}$; if a cooling bound reappears in this window, the open parameter space claimed for the $\phi$ model disappears.

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Extended reading notes

Core claim

The paper's central claim is that spin-dependent scattering of sub-GeV dark matter can be computed in crystals by an incoherent multi-phonon rate for $m_\chi \lesssim 100$ MeV, matched to a nuclear-recoil rate for heavier masses, and that after combining this with all current bounds on light mediators, the only viable direct-detection target is a scalar mediator $\phi$ with mixed scalar and pseudoscalar couplings ($\phi\bar\chi\chi$ and $g_p\,\bar p\gamma_5 p$). For $m_\chi \gtrsim 100$ MeV, the allowed cross section can produce a rate of order a hundred events per gram-year in Al$_2$O$_3$, provided the SN1987A 'trapping window' between the meson and supernova bounds is real. The pseudoscalar $a$ and axial-vector $A'$ models have no viable parameter space: the $a$ is momentum-suppressed and strongly bounded by rare meson decays, while the $A'$ requires $g_\chi \sim g_p$ for a consistent UV completion, which suppresses the rate below any planned experiment. If the trapping window closes, even the $\phi$ model's prospects vanish.

Load-bearing premise

The positive result depends on the SN1987A 'trapping window' being a real gap in constraints: for mediator masses around 0.1 to 100 MeV and couplings $2\times10^{-6} \lesssim g_p \lesssim 7\times10^{-6}$, the mediator is trapped in the proto-neutron star so the cooling bound disappears, and if future diffusive-transport calculations close that window, no direct detection prospects remain.

Editorial extensions

If this is right

If the scalar-$\phi$ model is right, near-future phonon detectors with Al$_2$O$_3$ targets could see of order a hundred events per gram-year for $m_\chi \gtrsim 100$ MeV, meaning a few gram-day exposure probes new parameter space.
The pseudoscalar-axion model is excluded as a direct detection target regardless of background assumptions; no event rate from this model can exceed current mediator bounds.
For axial-vector mediators, any hint of spin-dependent sub-GeV dark matter would point not to a simple $A'$ but to a more elaborate model with $g_\chi \gg g_p$.
The 'trapping window' is the deciding factor: if it is closed by better supernova calculations, the $\phi$ model also falls below detection reach, so the paper's positive conclusion stands or falls with that window.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A dedicated calculation of diffusive energy transport by scalar and pseudoscalar mediators in the SN1987A proto-neutron star, analogous to recent studies for photon-coupled axions, could close the trapping window; if so the paper's only open model would be excluded — the paper flags this as a possibility but does not evaluate it.
The incoherent-phonon rate method applies to any target with an odd-group nucleus, so targets rich in $^{19}$F or $^{27}$Al should give the largest spin-dependent rates, and a broader target survey beyond Al$_2$O$_3$ and GaAs follows naturally from the paper's tables.
The same rate formalism could be adapted to magnetic-dipole dark matter, but that model is dominated by spin-independent dipole-charge scattering, so the spin-dependent channel alone would not be observable — a point the paper notes in its conclusions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real advance here is Section V: the first full incoherent multiphonon scattering rate for spin-dependent DM-nucleon interactions in crystals, with explicit structure functions, Debye-Waller factors, and a clean factorization of spin and phonon degrees of freedom that is justified for unpolarized, non-magnetic targets. It is implemented in DarkELF and the formulas check out as a straightforward extension of the SI phonon formalism. That part is solid and worth building on.

The constraint survey is competent and unusually honest about its own inputs. The authors flag the CHARM reinterpretation caveats, the crudeness of the A' supernova rescaling, the O(1) uncertainties in the SN1987A bound, and the model-dependence of the LHC anomalon limits. The UV completion for the axial vector model is a real piece of work, and the appendix argument that g_chi >> g_p is very hard to achieve in a heavy-mediator axial model is persuasive and a useful caution for the field.

The soft spot is exactly where the reader puts it: the entire positive conclusion for the phi mediator lives in the SN1987A trapping window. The paper says this plainly, and even shows what happens if the window closes—the blue lines in Figs. 6 and 7 place the bounds below any 3-events/kg-yr sensitivity. The authors acknowledge that diffusive transport calculations could close the window, and they impose no cosmology bounds inside it, with the BBN/Neff discussion explicitly deferred. So the headline claim of "open parameter space" and "~100 events/g-yr" is conditional on an astrophysical assumption that is plausible but not established. That is not a fatal flaw; it is a clearly labeled dependence, and the rate calculation does not depend on it. But anyone citing the bottom line should carry the caveat with it.

I also think the axial-vector exclusion relies on a fairly crude supernova rescaling, though the authors admit it and note the conclusion would survive an order-of-magnitude weakening. The omission of the Belle II excess reinterpretation is conservative, which is fine.

Who is this for? Anyone working on sub-GeV direct detection, phonon-based experiments, or light mediator constraints. The rate calculation and DarkELF implementation will be used; the constraint maps will be cited. It deserves a serious referee and publication after the usual minor revisions. My own verdict would be conditional, not reject: the physics is sound, the new calculation is real, and the caveats are stated rather than hidden.

Referee Report

3 major / 5 minor

Summary. The paper develops the multiphonon scattering formalism for spin-dependent sub-GeV dark matter in crystals, applying it to three benchmark mediators: a scalar phi with mixed scalar/pseudoscalar couplings, a pseudoscalar a, and an axial-vector A'. The authors combine the rate calculation with a survey of mediator constraints (meson decays, SN1987A, beam dumps, SIDM, LHC), implement the rates in DarkELF, and compare the resulting upper bounds on reference cross sections with the sensitivity of idealized 3-events/kg-yr experiments. Their main findings are that the a- and A'-models have no viable direct-detection parameter space, while the phi model retains open parameter space for m_chi ≳ 100 MeV, with maximal Al2O3 rates of order a hundred events per g-yr, provided the SN1987A trapping window survives.

Significance. If the rate calculation and constraint mapping are correct, the paper supplies a useful public tool and a clear benchmark map for spin-dependent sub-GeV direct detection: it explicitly separates the phonon-rate formalism from the mediator-bound analysis, provides UV completions for the axial-vector case, and is transparent about the astrophysical assumptions behind its positive conclusion. The DarkELF implementation and the explicit treatment of nuclear-spin randomness are concrete strengths. The headline result, however, is conditional on the SN1987A trapping window and on the absence of BBN/Neff bounds, which the paper itself flags; this conditionality is not merely cosmetic, as closing the window eliminates the quoted event rate.

major comments (3)

[Sec. V, Eqs. (80)-(82); Appendix E, Eqs. (E1)-(E4)] The overall normalization of the rate formulas appears internally inconsistent. From Eq. (74), the rate contains the factor N/V, and after Eq. (79) the conversion to rate per unit target mass gives (N/V)/rho_T = 1/(sum_d m_d). Equations (80)-(82) instead multiply by sum_d m_d, and for the phi and a cases also contain extra factors of 1/m_p^2 and 1/m_p^4 in the integrand. Taking the large-omega limit of Eq. (E1) does not reproduce the nuclear-recoil formula in Eq. (E10); the two differ by a factor of order (sum_d m_d)^2/m_p^4. Since the 3-events/kg-yr curves in Figs. 6-8 are derived from these expressions, please clarify whether the printed equations are the ones implemented in DarkELF. If they are, the numerical results need to be recomputed; if they are typographical errors, the manuscript formulas must be corrected to match the code.
[Sec. III A and Sec. VI A] The open parameter space for the phi model depends entirely on the SN1987A trapping window, as the authors state. The paper itself cautions in Sec. III A that future diffusive-transport calculations could close the window, and Figs. 6-7 show that with the window closed no direct-detection parameter space remains. Given that the abstract and introduction claim accessible parameter space for m_chi ≳ 100 MeV without this qualification, the abstract should explicitly state that the open parameter space and the quoted maximal rate are conditional on the trapping window and on the absence of additional cosmological bounds.
[Sec. III A and Eq. (83)] The paper does not impose its own BBN/Neff estimate inside the trapping window. The text estimates that ma ≳ 10 MeV is needed to avoid 2-sigma tension with BBN and CMB constraints, but for the heavy-mediator benchmark m_phi = 3 m_chi v_0, the open region m_chi = 100 MeV to 1 GeV corresponds to m_phi approximately 0.22 to 2.2 MeV, which lies below that estimate. If the estimate is applied, the open parameter space in Fig. 6 disappears independently of the supernova trapping window. Please either impose this constraint, present the BBN-consistent subregion separately, or explain in Sec. VI why the estimate does not apply (for example, low reheating) and quantify the model-building cost of that assumption.

minor comments (5)

[Abstract] The sentence 'for m_chi ≳ 100 MeV there is parameter space' should be qualified to indicate that this holds only for the phi model and only if the SN1987A trapping window remains open and the BBN/Neff caveats are satisfied.
[Sec. II and Sec. IV, Eqs. (40)-(41)] The relation g_p,n ≈ 0.22 g' and g_chi = -g'/2 gives |g_chi/g_p| ≈ 2.3, so the statement 'g_p ∼ g_n ∼ g_chi' should be phrased as order-one rather than as an equality, to avoid reader confusion about the numerical hierarchy.
[Appendix B] The CHARM reinterpretation caveats are welcome, but their potential impact on Fig. 1 and on the mediator-mass exclusion in the 0.1-1 GeV range should be summarized either in Sec. III A or in the caption of Fig. 1, since the main text treats the CHARM bound as definitive.
[Appendix D, Eq. (D29)] For the light-mediator double-A' model, the statement that kinetic mixing 'vanishes in the q -> 0 limit' is correct for the mixing operator itself, but the resulting scattering Hamiltonian in Eq. (D29) contains a 1/|q|^4 term; the distinction should be spelled out to avoid the impression that the effect is identically zero.
[Appendix E, Eq. (E5)] The integration limits q_± in Eq. (E5) use v_max but the surrounding text defines v_max only implicitly; please state v_max = v_esc + v_earth explicitly before the equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: mediator constraints are independent inputs, and the rate calculation is self-contained; the trapping-window dependence is an explicit, non-circular astrophysical assumption.

full rationale

The paper's two main deliverables are (i) a rate calculation for spin-dependent sub-GeV dark matter scattering into phonons and (ii) an upper-bound map obtained by combining independent mediator constraints (meson decays, SN1987A, horizontal-branch stars, beam dumps, LHC searches, and dark-matter self-interactions) with those rates. The reference cross sections in Eqs. (14)-(16) are definitions, not fits, and the benchmark mediator masses in Eqs. (18)-(19) are explicitly declared choices that approximately maximize the allowed parameter space, not quantities fitted to the predicted rates. The quoted phi-mediator open parameter space and the roughly one-hundred-events-per-g-yr rate come from maximizing the independently allowed gp and gchi within the SN1987A trapping window; the paper discloses this dependence, stating that the open parameter space 'critically relies on the trapping window,' and it explicitly shows the blue 'window closed' curves that eliminate all detection prospects. Conditional dependence on the trapping window is a stated astrophysical uncertainty, not a circular reduction, because the window is an external constraint map rather than an output of the rate calculation. Self-citations, including DarkELF [38], the phonon correlation function of [25], and the earlier multiphonon formalism [13,22,23], are to published, reproducible calculations and public code; they are not used to forbid alternatives or to smuggle in the central result, so under the review rules they do not raise the circularity score. No equation was found in which a predicted quantity is defined in terms of itself or in which a fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The calculation relies on the standard multiphonon formalism and a series of domain assumptions about spin disorder, spin-phonon decoupling, the degree to which benchmark mediators reflect the model space, and the status of the supernova trapping window. The benchmarks ma,phi = 0.3 q0 and 3 q0 are hand-chosen to optimize the allowed parameter space; Appendix A shows the qualitative conclusions are robust to marginalizing over mediator mass. The invented entities are the anomaly-cancelling fermions in the axial vector UV completion, which have no independent evidence.

free parameters (2)

Benchmark mediator mass ratios m_med/q0 = 0.3 and 3 (light and heavy benchmarks)
Eqs. (18)-(19): chosen to represent optimal light and heavy mediator regimes for maximizing direct detection parameter space; not fitted to data. Appendix A shows marginalizing over mediator mass does not change the qualitative conclusions.
Reference momentum q0 = m_chi * v0 = q0 = m_chi * 220 km/s
Sec. II: convention for defining reference cross sections and the form factor scale. Standard choice, not fitted to data.

assumptions (5)

domain assumption Nuclear spins in the target crystals are randomly oriented, so the spin-dependent scattering is fully incoherent.
Sec. V (text before Eq. 67): the rate is built on the replacement J^α J^β → (1/3) δαβ δℓℓ' δdd' J^2 (Eq. 68). If spins were coherently aligned across cells, the rate would be enhanced; the paper notes this but excludes polarized targets.
domain assumption Phonon degrees of freedom do not affect nuclear spin orientations during the scattering process.
Sec. V B (after Eq. 67): the crystal matrix element is factorized into spin and phonon parts under this assumption, stated to be a good approximation for non-magnetic materials.
domain assumption The supernova 'trapping window' is a real gap in astrophysical bounds, and no diffusive-energy-transport or cosmology bound closes it.
Sec. III A and Sec. VI A: the open parameter space for the phi and a mediators relies on the allowed coupling region between the SN1987A and meson-decay bounds; the paper states future work on diffusive transport could close the window.
domain assumption The UV completion of the axial vector model implies gχ ~ gp (Eqs. 41, C11); no consistent UV completion allows gχ >> gp while preserving the heavy-mediator Hamiltonian (8).
Sec. IV and Appendices C-D: this relation is derived for the benchmark model and argued to be generic in Appendix D, but the paper does not provide a no-go theorem; if such a model existed, the A' direct detection prospects would reopen.
domain assumption The DM is a Dirac fermion, and the mediator-nucleon couplings are generated by the specific operators in Eqs. (1)-(3) with the stated UV origins (gluon G G~ for spin-0; anomaly-free U(1)' for the A').
Sec. II: these benchmark models define the scope; alternative operators would have different q dependence and are argued to be more suppressed.

invented entities (2)

Anomalon fermions (Q', u', d', N', and conjugates)
purpose: Cancel the U(1)' gauge anomalies and generate the A' couplings in the axial vector benchmark model (Appendix C).
These heavy colored fermions are introduced to make the model UV-complete; their only observable handle is LHC searches for vector-like quarks, and there is no independent evidence for them.
Second axial U(1)' gauge field A'_2 and the scalar φ in the generalized model
purpose: Attempt to build a UV completion with gχ >> gp for the heavy-mediator limit (Appendix D).
This is an exploratory model to test whether the gχ ~ gp constraint can be relaxed; the authors find the construction does not circumvent the bound, so it is not advanced as a viable model.

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Cite this review

Pith. "Pith review of Spin-Dependent Scattering of Sub-GeV Dark Matter: Models and Constraints." pith.science (2026). https://pith.science/paper/DYDXZLQB

@misc{pith2026250611191,
  author       = {Pith},
  title        = {Pith review of: Spin-Dependent Scattering of Sub-GeV Dark Matter: Models and Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYDXZLQB}},
  note         = {Machine review of arXiv:2506.11191}
}

abstract

We calculate the scattering rate of sub-GeV dark matter in solid-state targets for spin-dependent dark matter -- nucleon interactions. For dark matter particles with mass below 100 MeV, the scattering occurs predominantly through incoherent phonon production. For dark matter heavier than 100 MeV, we match onto the nuclear recoil calculation. To compare the sensitivity of future direct detection experiments with existing constraints, we consider three models with interactions which are mediated by spin-0 or spin-1 particles. This allows us to derive bounds on the cross section from searches for the mediating particle, including bounds from stellar cooling, beam dump experiments, meson factories and dark matter self-interactions. The existing bounds are very stringent, though for $m_\chi\gtrsim 100$ MeV there is parameter space which may be accessible with direct detection, depending on the exposure and background rates.

Figures

Figures reproduced from arXiv: 2506.11191 by the authors.

**Figure 2.** FIG. 2. Contours for the self-interacting DM bound on [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. SIDM constraints on [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Bounds on [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. The phonon partial density of states for each individual atom in both GaAs and Al [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. Reference cross sections for a signal rate of 3 events/kg-yr, through an interaction mediated by the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. Reference cross sections for a signal rate of 3 events/kg-yr, through an interaction mediated by the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Reference cross sections for a signal rate of 3 events [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: FIG. 9. Reference cross sections for a signal rate of 3 events/kg-yr, through an interaction mediated by the [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10. Reference cross sections for a signal rate of 3 events/kg-yr, through an interaction mediated by the [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Coherence from interference: a solvable model of sub-GeV dark matter-nucleus scattering
hep-ph 2026-07 conditional novelty 6.0 of 10

In an exactly solvable 1D lattice, coherent and incoherent DM-nucleus structure factors differ only by a crystal-momentum delta function that becomes unimportant for n≥2 phonons, validating hybrid Inc+LW rate calculations.

Reference graph

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a mediator Repeating the same procedure, we calculated the vis- cosity cross section with the a mediator at tree-level in quantum field theory and took the non-relativistic limit σa V ≃ g4 χ 64πm2χR6(R2 + 2) " 4(2R2 + 3)2 log 1 + R2 + (R6 − 4R4 − 30R2 − 36)R2 # R≪1 ≈ g4 χm2 χv4 240πm4a . (37) 8 For the a mediator, it was shown that terms in the non-relati...
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The mass eigenstates couple to the visible and dark sector currents through L ⊃Aµ ℓ g1J1,µ − g2 δm2 m2 2 − m2 1 J2,µ (D9) + Aµ h g2J2,µ + g1 δm2 m2 2 − m2 1 J1,µ , (D10) where J1,µ and J2,µ are the SM and dark matter axial currents J µ 1 ≡ ¯qγ µγ5q (D11) J µ 2 ≡ −1 2 ¯χγµγ5χ. (D12) The leading contribution to the direct detection scatter- ing rate will be...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

and beyond the reach of experiments that are cur- rently being considered

and do not consider sub-component DM since, in this case, SIDM bounds do not have constraining power. and beyond the reach of experiments that are cur- rently being considered. • For the a (axion-like particle) mediator, direct de- tection is both spin-dependent and momentum sup- pressed. If light enough, the a can be produced in rare meson decays, which ...

[2] [2]

a mediator Repeating the same procedure, we calculated the vis- cosity cross section with the a mediator at tree-level in quantum field theory and took the non-relativistic limit σa V ≃ g4 χ 64πm2χR6(R2 + 2) " 4(2R2 + 3)2 log 1 + R2 + (R6 − 4R4 − 30R2 − 36)R2 # R≪1 ≈ g4 χm2 χv4 240πm4a . (37) 8 For the a mediator, it was shown that terms in the non-relati...

[3] [3]

heavy” or “light

SIDM summary We summarize constraints on all mediators in Fig. 3, again for the two direct detection benchmarks in (18) and (19). As mχ grows, the SIDM bound becomes weaker (i.e. gχ increases) until it ceases to be a meaningful bound since we demand that the model remains perturbative. Here we require that gχ ≤ 1. We conclude that SIDM constraints are par...

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We further assume that ¯yχ > yχ, such that the Dirac fermion associated with ¯χ1 and ¯χ2 is always heavier than χ

with the dark mat- ter. We further assume that ¯yχ > yχ, such that the Dirac fermion associated with ¯χ1 and ¯χ2 is always heavier than χ. Going forward, we will assume that the annihilation rate of this heavier fermion to either χ or A′ is efficient enough that we can neglect its residual relic density. The mass parameters of the dark sector particles ar...

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In Appendix C, we assigned charges Qχ = −1/2 so that a single scalar field ϕ could generate both the Yukawa terms for SM fields as well as the dark sector masses

A second gauge field is needed First, one may simply attempt to assign a large charge |Qχ| to χ1,2, such that gχ ≡ |Qχ|g′ ≫ g′ ∼ gp,n. In Appendix C, we assigned charges Qχ = −1/2 so that a single scalar field ϕ could generate both the Yukawa terms for SM fields as well as the dark sector masses. There, the charge of the scalar field was fixed by the need...

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We therefore consider the heavy mediator limit, where mA1 , mA2 ≫ q

Heavy mediator limit Our primary interest is to investigate whether we can construct a model that is not subject to the gp ∼ gχ con- straint and reduces to the Hamiltonian (8). We therefore consider the heavy mediator limit, where mA1 , mA2 ≫ q. First, one may consider mixing A′ 1 and A′ 2 through a kinetic mixing operator L ⊃ϵ 2 F ′µν 1 F ′ 2,µν. (D4) Be...

[8] [8]

(D12) The leading contribution to the direct detection scatter- ing rate will be through the exchange of the Aℓ

The mass eigenstates couple to the visible and dark sector currents through L ⊃Aµ ℓ g1J1,µ − g2 δm2 m2 2 − m2 1 J2,µ (D9) + Aµ h g2J2,µ + g1 δm2 m2 2 − m2 1 J1,µ , (D10) where J1,µ and J2,µ are the SM and dark matter axial currents J µ 1 ≡ ¯qγ µγ5q (D11) J µ 2 ≡ −1 2 ¯χγµγ5χ. (D12) The leading contribution to the direct detection scatter- ing rate will be...