Exploring Fermionic Dark Matter in the Presence of Scalar Leptoquarks

Santosh Kumar Rai; Shyamashish Dey

REVIEW 4 major objections 6 minor 59 references

Exploring Fermionic Dark Matter in the Presence of Scalar Leptoquarks

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Adding two inert scalar leptoquarks splits the neutral vector-like lepton into two pseudo-Dirac states, suppressing its Z-mediated scattering with nuclei and opening a viable dark-matter window near 1–2.5 TeV.

desk verdict A plausible and new way to rescue VLL dark matter via scalar-leptoquark-induced pseudo-Dirac splitting, but the printed loop formula for that splitting is inconsistent and must be pinned down before the direct-detection claim can be trusted. read the letter →

arxiv 2509.02744 v1 pith:7CEYKN2Z submitted 2025-09-02 hep-ph

Shyamashish Dey , Santosh Kumar Rai This is my paper · ORCID

classification hep-ph PACS 95.35.+d12.60.-i14.80.Sv

keywords vector-likeleptondarkmatterscalarleptoquarkpseudo-Diracfermioninelasticdirectdetectionrelicdensityradiativemasssplitting

topics Dark Matter

open problems Dark Matter

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 claims that adding two inert scalar leptoquarks to the Standard Model can rescue the otherwise-excluded vector-like lepton doublet as a dark matter candidate. In the minimal model, the neutral lepton is a Dirac fermion that scatters elastically off nuclei through Z exchange, overshooting the LUX-ZEPLIN bound; the leptoquarks generate a one-loop Majorana mass that splits it into two pseudo-Dirac states, and the splitting kinematically forbids that scattering. The same extension opens new co-annihilation channels that bring the relic density into the Planck-observed range for dark matter masses roughly between 1 and 2.5 TeV. The paper further finds that the viable parameter space satisfies indirect-detection limits from H.E.S.S. and lies partly within reach of next-generation direct-detection experiments.

What carries the argument

The load-bearing object is the one-loop Majorana mass $m_L$ of the neutral vector-like lepton, obtained from a bottom-quark/leptoquark loop and proportional to $m_b y_a y_b^\dagger \sum_{i,j} U_{RS}(i,i)U_{RS}^\dagger(i,j) B_0(m_b^2,0,m_{\Sigma^d_i}^2)$, together with the physical masses $m_{f^0_1}$ and $m_{f^0_2}$ obtained from Eq. (37). Here $B_0$ is the Passarino-Veltman scalar loop integral, and $U_{RS}$ rotates the down-type leptoquark gauge states into mass eigenstates, so the mixing angle $\theta$ controls the size of the splitting. A pseudo-Dirac fermion is a Dirac fermion that has received a small Majorana mass, splitting it into two nearly degenerate Majorana eigenstates; the paper's key observation is that the splitting is typically larger than about 250 keV, the inelastic-scattering threshold that closes the $Z$-mediated direct-detection channel. The companion Dirac-mass correction $\delta m_D$ from the same leptoquark loops also enlarges the charged-neutral splitting in the doublet.

What would settle it

Re-evaluate the one-loop Majorana mass with all flavor and mixing indices carried explicitly at a benchmark point such as $m_{f^0_1}\simeq 1.5$ TeV, $y_a=y_b=1$, $\sin2\theta=1$; if the resulting splitting falls below 250 keV, the direct-detection evasion fails. Experimentally, a search for inelastic up-scattering of $f_1^0$ into $f_2^0$ -- the recoil spectrum and possible de-excitation signal of the heavier state -- would test the claimed splitting independently of the loop calculation.

Watch

Extended reading notes

Core claim

The central claim is that two inert scalar leptoquarks -- an $SU(2)_L$ doublet $R$ and an $SU(2)_L$ singlet $S$, both odd under the stabilizing $Z_2$ -- change the nature of the vector-like lepton dark matter candidate. Through one-loop diagrams involving bottom quarks and down-type leptoquarks, the neutral component $f^0$ of the doublet receives a Majorana mass, so the single Dirac state splits into two pseudo-Dirac mass eigenstates $f_1^0$ and $f_2^0$. The lighter state is the dark matter; the splitting is typically above 250 keV across the parameter scan, which is enough to make the $Z$-mediated inelastic transition $f_1^0\to f_2^0$ kinematically forbidden at LUX-ZEPLIN. The same leptoquark couplings open new annihilation and co-annihilation channels that lower the relic density to the Planck-observed value for dark matter masses from about 1 to 2.5 TeV. The parameter space that survives relic-density and indirect-detection constraints also lies below the current LZ and H.E.S.S. limits, with part of it accessible to the next generation of direct-detection experiments.

Load-bearing premise

The direct-detection evasion rests on the loop-generated pseudo-Dirac splitting being genuinely larger than about 250 keV in the claimed parameter region; if the one-loop Majorana-mass calculation has the wrong sign, wrong mixing factors, or wrong loop kinematics, the splitting could be far smaller and the $Z$-mediated scattering would reappear.

Editorial extensions

If this is right

The minimal pure doublet VLL model is excluded through $Z$-mediated scattering; with the two inert scalar leptoquarks, the model becomes viable for $m_{f^0_1}$ between roughly 1 and 2.5 TeV.
The loop-induced splitting is typically above 250 keV, putting the model in the inelastic-dark-matter regime and closing the LUX-ZEPLIN scattering channel.
The new leptoquark-mediated (co)annihilation channels, dominated by co-annihilations among the nearly degenerate dark-sector states, bring the relic density into agreement with the Planck measurement over a wide parameter range.
The viable parameter space also satisfies the H.E.S.S. bound on the $W^+W^-$ annihilation channel.
A substantial part of the predicted spin-independent cross-section is below the current LZ limit but within the projected reach of DARWIN, the upgraded LZ, and XENONnT.

Reading between the lines

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

The mechanism should generalize: any $Z_2$-odd scalar coupling a vector-like doublet fermion to quarks can generate the same pseudo-Dirac splitting, so the core idea is not tied to the particular leptoquark charges chosen here.
Because the splitting scales as $y_a y_b \sin2\theta$ and is suppressed by the bottom-quark mass, the model effectively predicts strong third-generation quark couplings for the leptoquarks, which could be probed through associated top/bottom plus missing-energy events at the LHC.
If the splitting sits only slightly above 250 keV, the heavier pseudo-Dirac state could be excited by up-scattering in direct-detection experiments, producing a distinctive inelastic signature rather than the usual elastic recoil; current and future detectors could search for that signature directly.

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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

Colleague,

You should know two things before deciding what to do with this paper. First, the physics idea is legitimate: inert scalar leptoquarks radiatively split the neutral vector-like lepton into pseudo-Dirac states, suppressing Z-mediated direct detection, while the new Yukawa channels help fix the relic density. That specific combination is not in the literature and is worth taking seriously. Second, the printed formula for the splitting, Eq. (32), is not something you can trust as written. The index sum over U_RS(i,i) U^dagger_RS(i,j) with the loop function depending only on i is not a Wick-contracted amplitude. At theta=0 it would give B0(m1)+B0(m2), but with no S–R_d mixing the two Yukawa vertices cannot be connected by a propagator; the term should vanish. The expected structure is sin2theta [B0(m1^2)-B0(m2^2)], which can differ from Eq. (32) by a large factor. This matters because Eq. (32) is the only source of the >250 keV pseudo-Dirac splitting that lets the model evade LUX-ZEPLIN.

What the paper does well: the setup is simple, the phenomenological survey is broad (relic density, direct and indirect detection, T parameter, h->gamma gamma, flavour assumptions), and the numerical pipeline uses standard public codes (SARAH, SPheno, MicrOmegas). The qualitative mechanism is sound, and the parameter scan is honestly described rather than hidden.

The soft spots are concentrated in the loop section. Besides the index issue, there is no subtraction scheme for the UV-divergent B0, so the numerical size of m_L is undefined absent a renormalization convention. Eq. (37) with m_L<0 gives a negative 'lighter' eigenvalue unless an absolute value is intended. And the scan choices (ya,b > 0.5, LQ mass splitting fixed at 128 GeV) are made with the direct-detection constraint in view, so the claim of a large viable parameter space is partly built into the priors. No code or benchmark points are shipped, so the printed formula cannot be checked against their implementation.

If Eq. (32) is a typo and the numerical implementation is correct, the central claim survives. If it is not, the DD evasion threshold is not established. This is exactly what a referee should pin down. I would send it to review and ask for a corrected derivation of the loop amplitude, a stated renormalization scheme, and benchmark points. The idea deserves referee time; the paper as printed needs revision.

Referee Report

4 major / 6 minor

Summary. The paper extends the minimal SU(2)_L doublet vector-like lepton (VLL) dark matter model by adding two inert scalar leptoquarks, an SU(2) singlet S and an SU(2) doublet R, both odd under the stabilizing Z2 symmetry. The authors argue that one-loop diagrams involving bottom quarks and the down-type leptoquark mass eigenstates induce a Majorana mass for the neutral VLL component, splitting it into two pseudo-Dirac states; if the splitting exceeds roughly 250 keV, the Z-mediated elastic scattering that excludes the pure VLL model becomes kinematically forbidden. They further show that new (co)annihilation channels involving leptoquarks allow the correct relic abundance for DM masses roughly in the 1-2.5 TeV range, while the predicted spin-independent direct detection cross sections remain below the LUX-ZEPLIN bound. The numerical analysis uses SARAH, SPheno, CalcHEP and micrOMEGAs and applies constraints from vacuum stability, perturbativity, the T parameter, h->γγ, flavour physics and LHC searches.

Significance. If correct, the paper is significant: it offers a radiative, triplet-vev-free mechanism for converting a Z-coupled Dirac VLL into a pseudo-Dirac dark matter candidate, reviving a well-motivated minimal dark matter setup in the TeV range and connecting it to leptoquark and LHC missing-transverse-energy searches. The paper is also useful for its broad treatment of constraints (EWPO, h->γγ, flavour, collider bounds) and for the decomposition of the relic density into Type-I/II/III channels. Credit is due for using standard public codes and for presenting a falsifiable phenomenological profile, including future DARWIN/LZ/XENONnT projections. However, the printed one-loop Majorana-mass formula that anchors the direct-detection evasion claim is not a valid Wick contraction, and no benchmark points or public model files are supplied, so the numerical results cannot currently be verified from the manuscript.

major comments (4)

[Sec. 5, Eqs. (31) and (32)] The mixing factor in Eqs. (31) and (32), sum_{i,j} U_RS(i,i) U^dagger_RS(i,j) B(... , m_Sigma_i^2), is not a Wick-contracted amplitude: the loop function depends only on i, so summing over j is just a row sum of U^dagger_RS and has no physical meaning. The correct projection of the two Yukawa vertices onto the physical LQ mass eigenstates should produce a factor proportional to U_{iS} U_{iR_d}, i.e. a sin(2theta) [B(m_Sigma_2^2) - B(m_Sigma_1^2)] structure. In particular, for theta -> 0 the printed expression does not vanish, whereas the physical Majorana mass must vanish because y_a and y_b couple to two different unmixed fields and cannot be connected by a single scalar propagator. Since the splitting in Fig. 7(B) and the entire LUX-ZEPLIN evasion argument in Sec. 6.3 are driven by Eq. (32), this is a load-bearing issue; please provide a proper derivation, correct the printed formulas, and confirm that the numerical scans were based on the corrected expression.
[Sec. 5, Eqs. (32) and (37)] Even after the mixing structure is corrected, the expression mL ~ 6 m_b y_a y_b^dagger ... B0(m_b^2, 0, m_Sigma_i^2) is not numerically meaningful as written: B0 is UV divergent and no subtraction scheme is specified, and the choice of external momentum p^2 = m_b^2 rather than a scale tied to the DM mass is not justified. The statement in the text that mL is negative is also incomplete, because inserting mL < 0 into Eq. (37) gives a negative value for m_{f^0_1}; physical masses require an absolute value or an explicit phase convention, and the mass splitting shown in Fig. 7(B) must be defined consistently with that convention. Please supply the renormalized loop expression and demonstrate that the >250 keV splitting, and hence the direct-detection evasion, survives.
[Sec. 6, numerical pipeline] The numerical pipeline is described only in general terms ('SARAH ... SPheno ... CalcHEP ... micrOMEGAs') and no benchmark points are provided. Because the printed one-loop formula is inconsistent as written, the reader cannot tell whether Figs. 7, 11, 13 and 15 were generated from Eq. (32) or from a different, correct implementation. Please provide at least one benchmark table with the Lagrangian parameters (m_f, y_a, y_b, y_c, sin(theta), m_Sigma_{d1}, m_Sigma_{d2} - m_Sigma_{d1}) and the corresponding outputs (m_{f^0_1}, Delta m0, Omega h^2, sigma_SI, <sigma v>), and ideally make the model files public. This is necessary to validate the central claims.
[Sec. 6, scan ranges and Eq. (32)] The scan is restricted to 0.5 <= y_a,b <= 1.5 and 0.5 <= y_c <= 1, and the text motivates these lower bounds by the direct-detection requirement that Delta m0 ~ y_a y_b sin(2theta) exceed the inelastic-scattering threshold. This means the DD-evading region is imposed as a prior rather than emerging as a prediction. Please show explicitly, with the corrected mL formula, how large the allowed parameter region is for smaller Yukawas (e.g., y ~ 0.1-0.3) and how much of the parent parameter space is excluded by LUX-ZEPLIN; otherwise the claim of a 'large parameter space' should be appropriately qualified.

minor comments (6)

[Sec. 4, Eq. (12)] The vacuum stability conditions as printed, e.g. lambda5 = lambda5 + sqrt(lambda1 + lambda2), appear to be a typographical corruption of the standard conditions lambda5 + 2 sqrt(lambda1 lambda2) > 0 (and similarly for lambda7 and lambda4); please correct these expressions and ensure that the conditions used in the numerical analysis are the correct ones.
[Sec. 6.1, Eq. (39)] The formula for <sigma v>_eff contains a parenthesis error in the last term: the factors (1 + Delta chi_i)^{3/2} and (1 + Delta chi_j)^{3/2} should be written with explicit brackets, as the current expression is ambiguous.
[Sec. 6.1, Fig. 12] The decomposition of the effective annihilation rate into Type-I, Type-II and Type-III processes is shown in Fig. 12, but the text does not explain how this decomposition was performed within micrOMEGAs; a short description of the procedure would improve reproducibility.
[References] References [55] and [59] are incomplete (no arXiv number or journal/volume information); please update them.
[Sec. 5, Eq. (30)] In Eq. (30), the Higgs-leptoquark coupling g_Sigma_{d1,2} is expressed in terms of the dimensionful parameter mu_RS with no explicit mass-dimensional factor; please clarify the normalization used in Eq. (24) so that the loop amplitude is dimensionless.
[Sec. 5, Fig. 7(B)] The horizontal threshold at 250 keV that is central to the direct-detection argument is mentioned in the text but is not marked in Fig. 7(B); adding it would make the viability condition immediately visible.

Circularity Check

1 steps flagged · score 4.0 of 10

Direct-detection viability is partly constructed through the y>0.5 scan cut, but the loop-induced pseudo-Dirac splitting and relic/ID computations remain independent.

fitted input called prediction [Section 6 (scan parameter ranges) and Section 6.3 (Direct Detection)]
"In addition, the Yukawa couplings ya,b,c are constrained to be above 0.25 from DD observations. This is because the DD constraint is dependent on the mass splitting ∆m0 between the two light pseudo-Dirac states, which in turn is proportional to the Yukawa couplings (∆m0∝ yayb sin 2θ) ... We therefore restrict ourselves to Yukawa couplings greater than 0.5 in our analysis. ... We find that the entire parameter space consistent with relic density and ID constraints remains below the present exclusion limits."

The scan cut is taken directly from the DD constraint: the paper requires Δm0>250 keV to kinematically forbid Z-mediated scattering and notes Δm0∝y_a y_b sin2θ, so y_a,b,c are restricted to values that guarantee this condition. With that cut imposed, the Z-mediated contribution that excluded the pure VLL is switched off by construction, and the later statement that the scanned parameter space lies below LZ partly restates the input cut rather than predicting it. The circularity is only partial because the LQ-mediated tree-level DD diagrams are computed numerically and independently of this cut, and the relic-density and indirect-detection constraints are not encoded in the same way.

full rationale

The central loop calculation, the one-loop Majorana mass in Eq. (32), the Dirac splitting in Eq. (31), and the resulting pseudo-Dirac spectrum, is a self-contained field-theory computation and is not equivalent to any fitted observable. The relic-density and indirect-detection results are obtained with SARAH/SPheno/CalcHEP/micrOMEGAs and checked against external Planck and H.E.S.S. bounds, so those parts are not circular. There is no load-bearing self-citation chain: the pseudo-Dirac/inelastic-DM mechanism is imported from external references, and the authors' own earlier work is not used to forbid alternatives. The genuinely circular element is the scan design: the Yukawa couplings are restricted to y>0.5 explicitly because the DD constraint requires Δm0>250 keV and Δm0∝y_a y_b sin2θ, and the LQ-DM mass gap is chosen large enough to suppress the LQ-mediated diagrams; the claim that the scanned parameter space lies below LZ therefore partly restates these input cuts. The LQ-mediated DD contribution is still an independent numerical result, and the relic-density consistency is not forced, so the overall score is a moderate 4 rather than a 6 or higher. Separate correctness concerns about the index structure of Eq. (32) and the sign handling of mL in Eq. (37) are technical risks, not circularity, and are not counted in this score.

Assumptions & free parameters 7 free parameters · 4 assumptions · 3 invented entities

The model's central claims depend on several free parameters that are scanned rather than predicted: the VLL mass, three Yukawa couplings, the LQ mixing angle, and the LQ mass spectrum. The main physics assumptions are the Z2 stabilization, third-generation-only LQ couplings, one-loop dominance, and the fidelity of the public-code implementation. The new fields are experimentally searchable, so they are not unfalsifiable, but they are postulated for this model.

free parameters (7)

VLL tree-level mass m_f = scanned as mDM in 1000-2500 GeV
Input mass for the dark matter candidate; relic density and direct detection are computed as functions of it.
Yukawa coupling y_a = scanned 0.5-1.5
Controls VLL-LQ interactions, enters the Majorana mass and annihilation cross sections; values below 0.5 are excluded by the paper's direct detection requirement.
Yukawa coupling y_b = scanned 0.5-1.5
Enters the one-loop Majorana mass mL proportionally to m_b y_a y_b; chosen large enough to make the pseudo-Dirac splitting exceed the DD threshold.
Yukawa coupling y_c = scanned 0.5-1
Enters the Dirac mass splitting and co-annihilation processes; scanned over a smaller range than y_a and y_b.
LQ mixing angle theta = 0 to 45 degrees (sin 2theta up to 1)
Determines the down-type LQ mass eigenstate mixing and appears in the loop formulas and the Higgs diphoton amplitude.
Lightest down-type LQ mass m_Sigma_d1 = mDM + 350 GeV to mDM + 1000 GeV
The LQ-DM mass gap controls Boltzmann suppression for co-annihilation and the mediator suppression in direct detection.
Down-type LQ mass splitting m_Sigma_d2 - m_Sigma_d1 = 128 GeV
Benchmark value chosen near the maximal T-parameter limit (about 130 GeV at maximal mixing) and fixed independent of the mixing angle.

assumptions (4)

domain assumption An unbroken Z2 symmetry makes the VLL and both leptoquarks odd while all SM fields are even.
Introduced in Section 2 and used throughout; it stabilizes the neutral VLL as dark matter and makes the leptoquarks inert rather than ordinary leptoquarks.
domain assumption The leptoquarks couple only to third-generation quarks.
Stated in Section 4 as the way to avoid neutral meson oscillation constraints; this choice shapes the relic density final states and the collider phenomenology.
standard math One-loop radiative corrections dominate the mass splitting, with higher-order corrections negligible.
The Majorana and Dirac splittings are computed with Passarino-Veltman B0/B1 functions and loop diagrams; no resummation or higher-loop estimate is given.
domain assumption The SARAH, SPheno, CalcHEP, and micrOMEGAs numerical implementation faithfully encodes the Lagrangian in Eqs. (5) and (6).
The paper reports results from this pipeline but provides no public model files, benchmark tables, or validation against the analytic formulas.

invented entities (3)

Vector-like lepton doublet f independent evidence
purpose: Dark matter candidate; its neutral component f1^0 is the stable DM particle.
A new fermion field with electroweak pair production and missing-energy signatures at colliders; searchable but not yet observed.
Scalar leptoquark doublet R independent evidence
purpose: Provides y_b and y_c interactions, contributes to Dirac and Majorana mass corrections, and opens co-annihilation channels.
A colored scalar with LHC pair production, decays into quarks plus missing energy, and effects on h to gamma gamma; no direct evidence yet.
Scalar leptoquark singlet S independent evidence
purpose: Provides the fermion-number-violating y_a interaction and mixes with R to form the down-type LQ mass eigenstates.
Searchable through pair production and missing-energy signatures at the LHC; no direct evidence yet.

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Cite this review

Pith. "Pith review of Exploring Fermionic Dark Matter in the Presence of Scalar Leptoquarks." pith.science (2026). https://pith.science/paper/7CEYKN2Z

@misc{pith2026250902744,
  author       = {Pith},
  title        = {Pith review of: Exploring Fermionic Dark Matter in the Presence of Scalar Leptoquarks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CEYKN2Z}},
  note         = {Machine review of arXiv:2509.02744}
}

abstract

We study an extension of the vector-like lepton dark matter model by introducing scalar leptoquarks that modify the properties of the vector-like lepton dark matter candidate, helping it evade stringent direct detection constraints. In the minimal setup, the neutral component of a pure $SU(2)$ doublet vector-like lepton fails to simultaneously account for the observed relic abundance while remaining consistent with current direct detection limits. We show that the addition of the scalar leptoquarks can induce corrections to the mass of the vector-like lepton dark matter, splitting it into two non-degenerate pseudo-Dirac states. This mass splitting can help evade the direct detection bounds naturally. In addition, the extended setup also opens up a larger parameter space of the model that can accommodate the correct relic density.

Figures

Figures reproduced from arXiv: 2509.02744 by the authors.

**Figure 2.** Plot of DM relic density (A) and spin-independent direct detection cross-section (B) with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Variation of ∆T with the scalar LQ mass spectrum, assuming maximal mixing (sin 2θ = 1) between the down-type states. The black line denotes the experimental upper bound on ∆T. From the colour bar in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Panel (A): Variation of the loop amplitude for the Higgs diphoton decay width arising [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Feynman diagrams contributing to mass gap of charged and neutral component of the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Feynman diagram contributing to the Majorana mass term of the neutral component of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: The figure on the left (A) shows the variation in mass splitting between the charged [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Illustrating the Feynman diagrams for DM self-annihilation processes. These will be [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: Illustrating the Feynman diagrams for DM co-annihilation processes where only one of [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: Illustrating the Feynman diagrams for annihilation processes between DS particles where [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

**Figure 11.** Figure 11: Scatter plots of the DM relic density as a function of its mass. In panel (A), the color [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: The panel (A), (B) and (C) shows the fractional contribution of the Type-II, III and I [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: Scatter plot of the velocity-weighted annihilation cross-section [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

**Figure 14.** Figure 14: Tree level Feynman diagrams contributing to direct detection cross-section. [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]

**Figure 15.** Figure 15: Scatter plot of left of spin-independent direct detection cross-section of DM with its [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]

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