Light Scalar Top Quarks and Supersymmetric Dark Matter

A. Djouadi (LPMT Montpellier); C. Boehm; M. Drees (TU Munchen)

REVIEW

Light Scalar Top Quarks and Supersymmetric Dark Matter

C. Boehm, A. Djouadi (LPMT Montpellier), M. Drees (TU Munchen)

High Energy Physics - Phenomenology · Submitted 26 Nov 1999 · 21 references

Pith's verdict: A real, stop-specific result — co-annihilation does suppress the bino relic density — with a quantitative bound softer than the abstract admits.

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arXiv IDhep-ph/9911496 (version 2)

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CitationA. Djouadi (LPMT Montpellier), C. Boehm, M. Drees (TU Munchen). "Light Scalar Top Quarks and Supersymmetric Dark Matter." arXiv:hep-ph/9911496 (1999). https://pith.science/paper/4GGJEJPG

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Editor's note

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The thing worth knowing: this is the paper that put stop–neutralino co-annihilation on the map, and two pieces of it hold up. First, the cross sections for the channels in eq. (8) — χt̃ → tg, tH, bH⁺; t̃t̃ → tt; t̃t̃* → gg, HH, bb̄, tt̄ — are computed twice, by trace techniques and by helicity amplitudes. That is a genuine internal check, not a gesture. Second, it makes a point specific to the stop that sleptons don't have: because the stop's dominant decay at small δm is the flavor-changing t̃ → cχ, the 2↔1 equilibration condition (eq. 7) needs a mixing-suppressed coupling ǫ > 5·10⁻⁵ rather than the automatic equilibrium you get for staus. That is the part I would cite, and it is the part that survives scrutiny. The companion point — that the relic-density bound does not guarantee LHC discovery, contra Wells — is a useful corrective and correctly argued from the δm → 0 limit. The central claim, δm ≳ 10–30 GeV for mχ ≲ 300 GeV, is qualitatively robust: to kill it you would need co-annihilation to fail by orders of magnitude, not by a factor of two. But the quantitative number is softer than the text implies. The stress-test arithmetic is right — a factor f in σ_eff shifts δm by T_F ln f ≈ 10 GeV at mχ = 300, comparable to the quoted 9–19 GeV — so 'fairly accurate' is doing more work than it should. The direction of the factor-2 is not stated; near-threshold QCD usually enhances stop annihilation, which would push the bound up, but they should say which way they expect it to go. The abstract is the weakest part: abstract:s2 states the bound unconditionally, while the body makes it conditional on ǫ > 5·10⁻⁵ or mχ > M_W. A reader of the abstract alone would not know the bound can vanish. That is a fixable overstatement, not a flaw in the calculation. Minor: the δm axis is scanned by varying A0 between 2.5m0 and 3.2m0, so the contours are a one-parameter family rather than a general scan, and the cross-section expressions are deferred to a later paper, so the figures cannot be checked from this letter. Who it is for: SUSY dark matter phenomenologists and the stop search community. It is a well-posed letter with a clear result and honest flagging of its two weak points. Send it out.

Referee report

Significance

The paper addresses a real and, at the time, under-appreciated loophole in the standard naturalness-plus-cosmology argument that LHC must find superparticles. The two independent cross-section calculations are a genuine internal check and are reported honestly, as is the leading-order S-wave truncation and the resulting factor-of-two uncertainty. The parameter choices (μ = −2M₂, m₀ = 2M₂, heavy Higgs in Fig. 1) are deliberately conservative in the sense of minimizing the co-annihilation cross section, which strengthens the lower bound on δm. The prediction is falsifiable in both directions: a light stop nearly degenerate with the neutralino would exclude the neutralino as the dark matter, and the multi-TeV window is in principle testable by the absence of a cosmologically required superpartner spectrum at the LHC. The result is a useful, citable contribution to the co-annihilation literature.

Major objections (4)

[Abstract vs. Eq. (7) and the paragraph following it] The abstract states the δm bound unconditionally ('has to exceed ~10 to 30 GeV'), whereas the body makes it explicitly conditional on the stop remaining in relative chemical equilibrium with the neutralino down to T_F, which requires either a flavour-changing coupling ǫ > 5·10⁻⁵ (Eq. 7) or a neutralino heavy enough that χ̃₁⁰ + W⁺ ↔ t̃₁ + b̄ is fast. If neither holds, co-annihilation is absent, Ω_χ̃h² reverts to the larger standard bino value, and small δm is viable. The abstract should carry this condition, or at minimum the sentence in the body that introduces the assumption should be flagged as a premise of the bound rather than a detail.
[Paragraph after Eq. (8) and the discussion of Fig. 1] The manuscript argues that the exponential dependence of σ_eff on δm (Eqs. 5–6) makes the inferred bounds 'fairly accurate' despite a factor-of-two uncertainty in the co-annihilation cross sections. That argument is incomplete: a multiplicative error f in σ_eff translates into an additive shift T_F ln f in δm, which with T_F ≈ m_χ̃₁⁰/20 and f = 2 is about 10 GeV at m_χ̃₁⁰ = 300 GeV. This is comparable to the upper end of the quoted '9 to 19 GeV' range in Fig. 1 and to the '20 GeV' statement for Fig. 2. The text should state the shift explicitly rather than describing the bound as fairly accurate, and the abstract's '10 to 30 GeV' should be qualified accordingly.
[Fig. 1 caption and the choice of A₀] The δm axis in Figs. 1 and 2 is realized by varying A₀ between about 2.5m₀ and 3.2m₀, as the caption states. This means δm is not an independent coordinate: the stop mixing angle, and hence the t̃₁–χ̃₁⁰ coupling that drives the co-annihilation cross sections, varies along the same axis. The contours therefore conflate the kinematic Boltzmann suppression with a variation of the relevant matrix element. The authors should either demonstrate that the coupling variation is small over the plotted range or state explicitly how much of the contour shape is attributable to it.
[Discussion of the multi-TeV window (abstract:s4) and the paragraph before the conclusions] The claim that δm → 0 permits m_χ̃₁⁰ up to 4–6 TeV rests on the same leading-order S-wave cross sections, evaluated in a regime where the t̃₁ is non-relativistic and strongly interacting. The manuscript acknowledges the need for finite-temperature resummation to cancel Coulomb singularities, but the robustness argument offered ('even if we over-estimated them by a factor of 2, the bound would only be reduced by √2') addresses only a multiplicative suppression, not a Sommerfeld or bound-state enhancement, which is the more likely sign of the neglected correction at small velocity. Since this is the paper's second headline claim, the authors should state whether the neglected corrections could enhance rather than suppress σ_eff, and what that would do to the quoted 4–6 TeV ceiling.

Minor objections (8)

[Abstract] The token 'lsim' appears in the abstract in place of the intended relation symbol; this should be corrected in the published version.
[Reference [8]] The Griest–Seckel reference is given as 'Phys. Rev. D443, 3191 (1991)'; the volume should read D43.
[Reference [21]] The journal name is given as 'Eur. J. Phys. C10'; the correct abbreviation is 'Eur. Phys. J. C10'.
[Paragraph discussing Tevatron sensitivity, after Fig. 1] The sentence stating that the required splitting is 'still not high enough for current t̃₁ search strategies at the Tevatron [14] to be sensitive' cites [14], but the preceding sentence on e⁺e⁻ searches cites [13, 19]; the citation pairing should be checked for consistency with the sentence that follows Fig. 2, which cites [13, 20] for Tevatron sensitivity.
[Fig. 2 caption] The caption refers to exclusion by 'Higgs boson searches at LEP' without a reference; a citation should be supplied for the heavy dotted line.
[Paragraph introducing the mSUGRA variant] The statement that the authors 'do not know of any model that predicts m_t̃₁ ≃ m_χ̃₁⁰' would be strengthened by naming the specific mSUGRA parameter regions in which near-degeneracy arises, since the paper later invokes mSUGRA as the motivating framework.
[Eq. (7) and the accompanying text] The estimate ǫ² e^{−δm/T_F} ≫ α e^{−m_χ̃₁⁰/T_F} with α ∼ 0.01 is stated without derivation; a sentence on where the α arises (phase space versus coupling) would help the reader assess the numerical threshold ǫ > 5·10⁻⁵.
[Eq. (5)] The multiplicity factors g_ij are introduced only schematically; a short definition, or a pointer to where they are specified, would make the effective cross section reproducible from the text alone.

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The paper

The paper claims

If the lightest stop quark sits within about 10 GeV of the neutralino, the neutralino is too scarce to be the dark matter — unless its mass runs into the multi-TeV range.

The reading

The paper asks whether the lightest neutralino, taken as the lightest supersymmetric particle, can account for the cosmological dark matter when its mass happens to lie close to that of the lighter scalar top quark. When two superparticles are nearly degenerate in mass, they keep converting into one another in the early universe, so the total number of superparticles is governed by the summed annihilation rate rather than by neutralino pair annihilation alone; this co-annihilation can suppress the relic density by a large factor. Working in a supergravity-motivated framework with a bino-like neutralino, the authors compute the cross sections for stop–neutralino and stop–antistop annihilation into tops, gluons, Higgs bosons and bottoms, then solve the Boltzmann equation for the summed superparticle density. The payoff is a two-sided statement: for neutralino masses below roughly 300 GeV the stop must lie at least about 10 to 30 GeV heavier, or the neutralino is too scarce to form galactic halos; conversely, at near-zero splitting the relic density stays acceptable up to neutralino masses of several TeV. The second half matters because it undercuts the common claim that the relic-density bound guarantees the Large Hadron Collider will see supersymmetry.

What carries the argument

The load-bearing object is the effective annihilation cross section $��sigma_{�rm eff}$ of eqs. (5)–(6), which sums neutralino–neutralino, neutralino–stop and stop–antistop channels weighted by multiplicity factors and by the temperature-dependent relative Boltzmann factor $B_{�tilde P} = (m_{�tilde P}/m_{�tilde��chi_1^0})^{3/2} e^{-(m_{�tilde P}-m_{�tilde��chi_1^0})/T}$. Because the conversion reactions $�tilde��chi_1^0 + X �leftrightarrow �tilde P + Y$ are linear rather than quadratic in the Boltzmann suppression, the stop stays in relative chemical equilibrium with the neutralino long after ordinary annihilation has frozen out, so the relic density is set by the summed density $n_{�rm SUSY}$ through a single Boltzmann equation. The exponential sensitivity of $B_{�tilde P}$ to $��delta m$ is what converts a relic-density bound into a GeV-scale bound on a particle mass difference. The authors compute the underlying cross sections twice, once by trace techniques with the non-relativistic limit taken at the end and once by helicity amplitudes with the limit taken at the start, and they keep the top quark mass finite, which the slepton co-annihilation literature did not need to do.

Extended reading notes

Core claim

The central claim is that co-annihilation with the lighter stop eigenstate sets a lower bound on the stop–neutralino mass splitting, and simultaneously opens an escape hatch to very heavy neutralinos. For $m_{�tilde��chi_1^0} �lesssim 300$ GeV, requiring $��Omega_{�tilde��chi} h^2 > 0.025$ — the level needed for the neutralino to form galactic halos — forces $��delta m = m_{�tilde t_1} - m_{�tilde��chi_1^0}$ above about 9 to 19 GeV in the moderate-$�tan�beta$, heavy-Higgs scenario, and above about 20 GeV in the large-$�tan�beta$, light-Higgs scenario; demanding the favoured range $0.1 �lesssim ��Omega_{�tilde��chi} h^2 �lesssim 0.2$ pushes the splitting to 11–33 GeV and above about 40 GeV respectively. Scenarios with smaller splittings, precisely those in which the stop is hardest to see at colliders because its decay releases little visible energy, are therefore disfavoured cosmologically. In the opposite direction, as $��delta m �to 0$ the same co-annihilation keeps the relic density under the conservative bound $��Omega_{�tilde��chi} h^2 �le 0.5$ for neutralino masses up to about 4 TeV in the first scenario and about 6 TeV in the second, corresponding to gluino masses of 20 to 30 TeV, far beyond LHC reach.

Load-bearing premise

The whole result rests on the stop and the neutralino staying in relative chemical equilibrium down to freeze-out, which for stops is not automatic: it requires either a flavour-changing stop–charm–neutralino coupling larger than about $5�times 10^{-5}$ or a neutralino heavy enough that $�tilde��chi_1^0 + W^+ �to �tilde t_1 + �bar b$ proceeds briskly. If neither holds, the stop decouples early, co-annihilation never operates, and both the lower bound on the splitting and the multi-TeV window disappear.

Editorial extensions

If this is right

For neutralino masses in the natural range, a stop within about 10 GeV of the neutralino is excluded as a dark-matter scenario, so a collider signal of such a nearly degenerate stop would itself rule out the neutralino as the dark matter.
The relic-density upper bound does not guarantee LHC discovery of supersymmetry: with a near-degenerate stop, bino-like neutralinos as heavy as 4 to 6 TeV survive cosmology, implying gluino masses of 20 to 30 TeV.
In the large-$�tan�beta$, light-Higgs case the required splitting rises to above about 20 GeV for the halo bound and above about 40 GeV for $��Omega_{�tilde��chi} h^2 �sim 0.1$, which is large enough for Tevatron stop searches to have sensitivity.
In the moderate-$�tan�beta$ case the cosmologically allowed splitting is large enough for conventional $e^+e^-$ stop search strategies to be efficient but still too small for the Tevatron, so the two collider programmes are not equally matched to the cosmological window.

Reading between the lines

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

The argument reverses the usual direction of inference: instead of using collider data to test a dark-matter model, a null search for a stop nearly degenerate with the neutralino becomes a cosmological measurement, and a positive signal becomes a falsification of the neutralino as dark matter.
If the flavour-changing coupling is really the pivot, then measuring the stop's decay into a charm quark and a neutralino — its branching ratio or its lifetime — at a future lepton collider would calibrate directly whether co-annihilation was ever active, turning a soft mixing angle into a cosmological parameter.
The quoted GeV bounds are robust because they ride an exponential, but the multi-TeV ceiling scales only as the square root of the cross-section error; a finite-temperature QCD treatment that changed the stop cross sections by a factor of a few would move the ceiling by a factor of order one and leave the lower bound essentially intact.
The same logic should transfer to any coloured co-annihilator — a light sbottom or a near-degenerate gluino — with the stop merely the most likely degenerate partner in supergravity-inspired spectra, so the qualitative conclusion is broader than the specific channel computed here.

Original abstract

A stable neutralino $\chi_1^0$, assumed to be the lightest supersymmetric particle, is a favored particle physics candidate for the cosmological Dark Matter. We study co-annihilation of the lightest neutralino with the lighter scalar top quark $\tilde{t}_1$. We show that for natural values of the neutralino mass, $\lsim 300$ GeV, the $\chi_1^0 - \tilde{t}_1$ mass difference has to exceed $\sim 10$ to 30 GeV if $\chi_1^0$, is to contribute significantly to the Dark Matter. Scenarios with smaller mass splitting, where $\tilde{t}_1$ is quite difficult to detect at collider experiments, are thus cosmologically disfavored. On the other hand, for small $\tilde{t}_1 - \chi_1^0$ mass splitting, we show that co--annihilation allows very large neutralino masses, $m_{\chi_1^0} \sim 5$ TeV, without ``overclosing'' the Universe.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,546 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A stable neutralino $\chi_1^0$, assumed to be the lightest supersymmetric particle, is a favored particle physics candidate for the cosmological Dark Matter. We study co-annihilation of the lightest neutralino with the lighter scalar top quark $\tilde{t}_1$. We show that for natural values of the neutralino mass, $\lsim 300$ GeV, the $\chi_1^0 - \tilde{t}_1$ mass difference has to exceed $\sim 10$ to 30 GeV if $\chi_1^0$, is to contribute significantly to the Dark Matter. Scenarios with smaller mass splitting, where $\tilde{t}_1$ is quite difficult to detect at collider experiments, are thus cosmologically disfavored. On the other hand, for small $\tilde{t}_1 - \chi_1^0$ mass splitting, we show that co--annihilation allows very large neutralino masses, $m_{\chi_1^0} \sim 5$ TeV, without ``overclosing'' the Universe.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A stable neutralino $\chi_1^0$, assumed to be the lightest supersymmetric particle, is a favored particle physics candidate for the cosmological Dark Matter.	background 1.00	0.05	0.03	asserted 0.52
s1	We study co-annihilation of the lightest neutralino with the lighter scalar top quark $\tilde{t}_1$.	contribution 1.00	0.87	0.04	asserted 0.78
s2	We show that for natural values of the neutralino mass, $\lsim 300$ GeV, the $\chi_1^0 - \tilde{t}_1$ mass difference has to exceed $\sim 10$ to 30 GeV if $\chi_1^0$, is to contribute significantly to the Dark Matter.	result 1.00	0.84	0.50	asserted 0.99
s3	Scenarios with smaller mass splitting, where $\tilde{t}_1$ is quite difficult to detect at collider experiments, are thus cosmologically disfavored.	interpretation 0.98	0.54	0.26	asserted 0.96
s4	On the other hand, for small $\tilde{t}_1 - \chi_1^0$ mass splitting, we show that co--annihilation allows very large neutralino masses, $m_{\chi_1^0} \sim 5$ TeV, without ``overclosing'' the Universe.	result 1.00	0.92	0.21	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from simulation or computation 0.49.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Contours of constant Ωχ˜h 2 = 0.5 (solid), 0.1 (dashed) and 0.025 (dotted) in the (mχ˜ 0 1 , δm) plane, where δm = mt˜1 − mχ˜ 0 1 . We took µ, m0 and mA to be fixed multiples of M2 ≃ 2mχ˜ 0 1 , as indicated, whereas tanβ = 10 has been kept fixed. The parameter A0 varies between about 2.5m0 and 3.2m0, with larger A0 values corresponding to smaller values of δm. So far we have focused on LSP masses in the range favored by naturalness arguments. It is sometimes claimed [18] that the upper bound on Ωχ˜h 2 implies that LHC experiments must find superparticles if the MSSM is correct and…

**Figure 2.** As in Fig. 1, except that we took a large value of tan

Figures reproduced from arXiv:hep-ph/9911496 by the authors.

References

Works cited (21)

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Revisiting the Higgs Mass and Dark Matter in the CMSSM
hep-ph 2012-02 conditional novelty 4.0 of 10

Within the CMSSM, a 125 GeV Higgs boson would leave only the large-tan-beta, A0 > 0 stau-coannihilation and H/A funnel strips as viable thermal dark matter, with spin-independent scattering below about 10^-9 pb.
Light Stop Searches at the LHC with Monojet Events
hep-ph 2012-01 conditional novelty 4.0 of 10

With the full Standard Model background, detector simulation, and optimized cuts, stop pair production in association with one hard jet gives a 5 sigma reach up to about 290 GeV in stop mass for small stop-neutralino ...
Dark Matter Detection in the Light of Recent Experimental Results
hep-ph 2003-09 accept novelty 2.0 of 10

A 2003 review finds that universal supergravity neutralinos scatter below then-current detector sensitivity, while non-universal scalar masses can reach the DAMA region.

How the literature uses it

61 arXiv papers cite this in 66 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 54 sentences without a confident label

Cites it as support 1 paper

Note also that our generic requirement of a small mass gap between the lightest stop and the lightest neutralino supports efficient coannihilation in the neutralino relic density calculation [ref] .

Measuring Flavor Mixing with Minimal Flavor Violation at the LHC · Conclusions · 0.96

Cites it as background 11 papers

There have been analyses in the literature of the cases where the coannihilating particle is a squark, specifically the lighter stop squark [ref] , and also the case of the gluino [ref] .

Gluino Coannihilation Revisited · Introduction · 0.95
Examples of the former include hypersurfaces in the SUSY parameter space where the LSP is almost degenerate in mass with some next-to-lightest SUSY particle (NLSP), such as the lighter stau $1$ [ref] , stop $$ [ref] or chargino $1$ [ref] , or where $1$ is almost half the mass of a boson such as a heavy Higgs $H/A$ [ref] , a light Higgs $h$ or $Z$ [ref] , in which case rapid direct-channel annihilation may bring the CDM density into the allowed range.

Supersymmetric Dark Matter after LHC Run 1 · Introduction · 0.82
The stop coannihilation strip in the cMSSM, for example, has been studied in much detail by many authors [ref] , with new limits resulting in particular after the discovery [ref] of the Higgs boson.

Leading QCD Corrections for Indirect Dark Matter Searches: a Fresh Look · Relic Abundance · 0.86

Discussion

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