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Non-relativistic susceptibility and a dark matter application

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Pith's one-line read This paper argues that in a stop-like dark matter model, the susceptibility correction $\hat p_2$ does not efficiently regularize the late-time freeze-out abundance when $2\Delta M < \Delta E$, so the viable domain remains controlled by…

desk verdict New lattice measurement of p̂2 and a plausible negative result for its role in freeze-out; the quantitative uncertainty is real but does not overturn the main conclusion. read the letter →

arxiv 1908.07541 v2 pith:K2MAHA4M submitted 2019-08-20 hep-ph hep-lat

S. Biondini , Seyong Kim , M. Laine This is my paper · ORCID
classification hep-phhep-lat
keywords darkmatterfreeze-outnon-relativisticsusceptibilitySahaequationSommerfeldenhancementboundstateslatticeQCDMajoranasingletcoannihilation
topics 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 asks whether the non-relativistic susceptibility correction $\hat p_2$, which implements Saha-type ionization equilibrium in the modified Lee-Weinberg equation, can regularize late-time dark matter freeze-out when bound states enhance annihilation. Studying a Majorana singlet dark matter particle co-annihilating with a strongly interacting stop-like scalar, the authors conclude that it cannot: whenever the mass splitting obeys $2\Delta M < \Delta E$, the thermally averaged Sommerfeld factor grows faster than $\hat p_2$ and drives the abundance to very small values before $\hat p_2$ becomes relevant. An order-of-magnitude estimate based on a single Coulomb-like bound state, with binding energy $\Delta E = \alpha^2 M/4$, is supported by a non-perturbative lattice measurement, giving confidence in the extrapolation to the cosmological regime. If this conclusion holds, the viable domain of such models remains controlled by the mass splitting $\Delta M$, not by the susceptibility. The result also clarifies where the inclusive chemical-potential framework is reliable and where resolved bound-state dynamics are necessary.

What carries the argument

The central object is the susceptibility-normalized coefficient $\hat p_2$, defined through $p_2 = \hat p_2 n_{\rm eq}^2 T$, which closes the modified Lee-Weinberg equation by expressing the chemical potential in terms of the density through $e^{\beta\mu} n_{\rm eq} = 2n/(1+\sqrt{1+8\hat p_2 n})$. Analytically it is estimated as $T^3\hat p_2 \simeq (2/N_c^2)(\pi T/M)^{3/2}(e^{\beta\Delta E}-1)$, assuming one Coulomb-like bound state of the stop-antistop system with binding energy $\Delta E = \alpha^2 M/4$. Non-perturbatively it is measured from the disconnected heavy-propagator correlator in eq. (5.1), using non-relativistic lattice QCD; the connected part cancels, and the gauge-field lines linking the two propagators are what carry the bound-state physics. The comparison between the two estimates, with vacuum-like coupling at low temperature and thermal coupling at high temperature, is what licenses using the analytic form in cosmology.

What would settle it

Compute $T^3\hat p_2$ at the couplings and values of $z = M/T$ relevant to cosmology, either through the full three-loop perturbative evaluation of eq. (4.7) or through lattice simulations at smaller coupling; if for $M \sim 1\text{--}20$ TeV and $z \sim 10^3$ it exceeds $10^{11}$, then $8\hat p_2 n$ can reach order unity at the yields of interest and the regularization would be effective, contrary to the paper's conclusion.

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Extended reading notes

Core claim

The paper's central claim is that the coefficient $\hat p_2$, defined through $p_2 = \hat p_2 \, n_{\rm eq}^2 T$ and entering the number density through $e^{\beta\mu} n_{\rm eq} = 2n/(1+\sqrt{1+8\hat p_2 n})$, does not provide an efficient regularization of the late-time abundance in the stop-like model. The mechanism is quantitative: for $\hat p_2$ to matter one needs $8\hat p_2 n \gtrsim 1$, which for $Y \simeq 10^{-13}$ requires $T^3\hat p_2 \gg 10^{11}$; such values are reached only at $z \equiv M/T$ so large that the Sommerfeld factor $\bar S_3$ has already exceeded $10^{10}$ and pulled the yield down. The authors establish this with three ingredients: an order-of-magnitude estimate of $\hat p_2$ from a Coulomb-like ground state, a lattice measurement in large-coupling QCD that agrees with the estimate, and an explicit integration of the modified rate equation for $M = 1\ldots 500$ TeV. The stated conclusion is that in practice the regularization by $\hat p_2$ is insufficient to make the system viable if $2\Delta M < \Delta E$, leaving $\Delta M$ as the only possible equilibrium regulator.

Load-bearing premise

The load-bearing premise is that a single Coulomb-like ground state with binding energy $\Delta E = \alpha^2 M/4$ dominates the susceptibility, making the estimate exponentially sensitive to the coupling; the lattice test is performed at large coupling and moderate $M/T$, far from the small-coupling cosmological regime, so a much larger true $\hat p_2$ could overturn the conclusion.

Editorial extensions

If this is right

  • If the paper is right, models with strongly interacting mediators must satisfy $2\Delta M > \Delta E$ to avoid unacceptably large late-time annihilation, since $\hat p_2$ cannot do the job.
  • The viable dark matter mass range extends at least to the multi-TeV domain, as previously found, but only when such a mass splitting is present.
  • At late times the ratio $\bar S_3/(T^3\hat p_2) \simeq (M\alpha/T)^3 \gg 1$, so the suppression mechanism is generic rather than accidental in this model.
  • The lattice test at large coupling supports using the analytic $\hat p_2$ estimate at small coupling, meaning the negative conclusion is not an artifact of the extrapolation, within the uncertainty of the estimate.

Reading between the lines

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

  • If additional bound states or excited states contribute to $\hat p_2$ more than assumed, the estimate could be systematically low; a full three-loop calculation and lower-coupling lattice data would be the direct test.
  • The same ratio $\bar S_3/(T^3\hat p_2)\sim (M\alpha/T)^3$ suggests that in models with weaker Sommerfeld enhancement or stronger binding, $\hat p_2$ could still provide the regularizing effect the authors find absent here.
  • The lattice method for measuring the disconnected correlator could be applied to other non-relativistic dark sectors, giving a non-perturbative route to chemical equilibration rates that does not require enumerating bound states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the susceptibility coefficient p̂2 that enters the modified Lee-Weinberg equation (1.2) when chemical potentials are eliminated in favour of the number density. Section 2 derives the Saha-type relation (2.12) from a fugacity expansion truncated at two-particle states. Section 3 gives an order-of-magnitude estimate, Eq. (3.9), in which the two-particle spectral sum is replaced by a single Coulomb-like ground state with binding energy ΔE = α²M/4, while octet and excited states are neglected. Section 4 formulates p̂2 as a non-perturbative lattice observable, Eq. (4.7), and Section 5 presents a single NRQCD lattice measurement at M_kin = 14Λ, comparing it with the analytic estimate at both vacuum and thermal coupling choices. Section 6 applies the same analytic p̂2 to a Majorana singlet dark matter model with a strongly interacting stop-like mediator, integrating Eq. (1.2) down to late times. The principal conclusion is that the susceptibility-induced regularization is ineffective: the combination 8 p̂2 n remains far below unity in the regime where the abundance Y ≈ 10⁻¹³ is set, so the viable parameter space continues to be controlled by the mass splitting ΔM rather than by p̂2. The paper closes by noting that the analytic value of p̂2 is only an order-of-magnitude estimate and that a full perturbative calculation would be of three-loop order.

Significance. The paper addresses a genuinely open technical question in thermal dark matter computations: whether the Saha-type susceptibility, which is in principle present in Eq. (1.2), can regularize the late-time growth of Sommerfeld-enhanced annihilation. The authors provide a concrete non-perturbative lattice definition of the susceptibility, Eq. (5.1), and take the first step toward measuring it with NRQCD, which is a meaningful methodological contribution. They also formulate a general low-temperature ratio S̄3/(T³ p̂2) ≃ (Mα/T)³ that gives a plausible reason for the inefficiency of the p̂2 regularization, and they explicitly flag the limitations of their estimate in Section 7. If the central conclusion holds, it rules out a potentially attractive solution to the overclose problem of coannihilating dark matter and sharpens the role of the mass-splitting regulator. The main limitation is that the central claim relies on an order-of-magnitude analytic estimate that is exponentially sensitive to α and is validated by only one lattice point with statistical-only errors and up to ~50% systematic uncertainty; the general ratio argument mitigates but does not fully remove this concern.

major comments (3)
  1. [Sec. 3, Eq. (3.9); Sec. 6, Eq. (6.4)] The central no-regularization conclusion rests on the single-ground-state estimate (3.9). The authors correctly stress that this is an order-of-magnitude estimate and that it is exponentially sensitive to α, but the manuscript does not quantify how much T³ p̂2 could change if a more complete treatment of the two-particle spectral sum were adopted, including excited Coulomb states, octet degrees of freedom, or a modestly different effective binding energy. Since the integration in Section 6 uses this same estimate, the agreement shown in Fig. 1 does not independently validate the late-time comparison on which the conclusion is based. I request a robustness scan: for the target yield Y ≈ 10⁻¹³ and the criterion 8 p̂2 n ≫ 1 from Eq. (6.4), please determine how large T³ p̂2 would need to be at the relevant z for the p̂2 regularization to affect the final abundance, and then state whether any plausible modification of Eq. (3.9) — for example, a multiplicity factor of 10–100 or a shift in ΔE within the uncertainty of α — can achieve that value before S̄3 drives Y below the observed value.
  2. [Sec. 5, Fig. 1] The lattice test is limited to a single rest mass M_kin = 14Λ, in a regime where α ≳ 0.3, far from the cosmological value α ≈ 0.1 used in Section 6. The errors quoted are statistical only, and the authors themselves note that systematic uncertainties could be as large as ~50%. Moreover, the two analytic curves labeled 'vacuum coupling' and 'thermal coupling' are scaled versions of the same estimate (3.9), so the comparison in Fig. 1 is a test of the temperature dependence of a single-parameter form rather than an independent probe of the absolute normalization of T³ p̂2. I ask the authors to state explicitly how the central claim would be affected if T³ p̂2 were larger than Eq. (3.9) by a factor of 10 or 100 at the z values of interest (z ≈ 10³–10⁴), and to separate, in Fig. 1 or its caption, the systematic uncertainty in the lattice point from the spread between the two coupling prescriptions.
  3. [Sec. 7] The general argument that S̄3/(T³ p̂2) ≃ (Mα/T)³ ≫ 1 is presented as the reason why p̂2 cannot compensate for S̄3. This ratio is derived from the same ground-state approximations, Eqs. (6.2) and (6.3), that underlie the quantitative estimate, so it inherits the uncertainties listed above. If the true bound-state spectrum supplies a large multiplicity factor to p̂2 but a smaller one to S̄3, the ratio could be shifted substantially. Please clarify to what extent this argument is parametric (i.e., would remain valid for any spectral sum with the same leading exponential behaviour) and to what extent it depends on the single-ground-state approximation.
minor comments (4)
  1. [Sec. 7] The word 'estime' in the final paragraph should be 'estimate'.
  2. [Sec. 3, Eq. (3.9)] The notation d²_s appears without explicit definition of the spin degeneracy factor in Eq. (3.8); please spell out that d_s ≡ 2s + 1 at the first use of c_{θ,η} → d²_s.
  3. [Sec. 5, Fig. 1] The caption states that 'the errors of the lattice results are statistical only', but the figure does not show error bars. I recommend adding explicit error bars (or the systematic band) to the figure, since the quoted ~50% uncertainty is essential to the calibration claim.
  4. [Sec. 6, Eq. (6.4)] The combination 8 p̂2 n is introduced abruptly; a sentence explaining that this is the quantity controlling the correction term in Eq. (2.12) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: p̂₂ is derived, lattice-checked, and used in explicit equations; no prediction reduces to its inputs.

full rationale

The central object p̂₂ is defined through the susceptibility (eqs. (2.10)/(2.11)), estimated from a fugacity expansion in Sec. 3, and measured non-perturbatively on the lattice via eq. (5.1); the analytic estimate is compared with, not fitted to, the lattice data (Fig. 1). The dark-matter conclusion follows by combining explicit formulas (eqs. (6.2)–(6.4)): the criterion 8p̂₂n ≫ 1 is evaluated with the analytic p̂₂ and the Sommerfeld factor taken from prior work, with no free parameter tuned to the final yield. The use of same-author results for the Sommerfeld factor and lattice setup is normal scientific continuity; those results do not presuppose the no-stabilization conclusion. The paper itself flags the order-of-magnitude character of eq. (3.9) in Sec. 7, which is a robustness caveat, not a circular step. No equation reduces to its inputs by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard quantum statistical mechanics and a specific dark matter model. The main ad hoc assumptions are the single-Coulomb-bound-state dominance for p̂2 and the validity of extrapolating the lattice calibration to small coupling. No new particles or entities are invented.

free parameters (2)
  • coupling renormalization scale = vacuum scale e^{-γ_E}/a at low T, thermal scale μ~πT at high T, crossover at z≈250-600
    The value of α in ΔE and p̂2 depends on an arbitrary renormalization scale; the paper interpolates between two choices based on qualitative lattice comparison, affecting the numerical magnitude of p̂2 in fig. 2.
  • binding energy ΔE = α²M/4
    Assumed Coulomb ground-state binding energy; its magnitude is exponentially sensitive to α, so this is effectively a parameter that determines p̂2.
assumptions (5)
  • domain assumption The system is dilute (T ≪ M), so a Boltzmann/fugacity expansion truncated at p2 is valid.
    Stated in footnote 2 and used throughout sec. 3 to derive p̂2.
  • domain assumption The dark matter model consists of a Majorana singlet fermion and a strongly interacting scalar mediator with a specified mass splitting ΔM.
    The model is introduced in sec. 6 and used for the freeze-out application.
  • ad hoc to paper The susceptibility is dominated by a single Coulomb-like ground state in the (1,1) sector; octet and higher bound states are exponentially suppressed.
    Used to arrive at eq. (3.9), the central analytic estimate of p̂2.
  • domain assumption Kinetic and ionization equilibrium are maintained in the dark sector down to the temperatures considered.
    Noted in the caption of fig. 3 and in sec. 6; required for the modified Lee-Weinberg equation to apply.
  • ad hoc to paper The lattice study at large coupling can validate the analytic estimate for small coupling.
    The extrapolation from the lattice regime (large α, moderate T) to the cosmological regime (small α) is stated in sec. 5 but not tested.

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Pith. "Pith review of Non-relativistic susceptibility and a dark matter application." pith.science (2026). https://pith.science/paper/K2MAHA4M

@misc{pith2026190807541,
  author       = {Pith},
  title        = {Pith review of: Non-relativistic susceptibility and a dark matter application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2MAHA4M}},
  note         = {Machine review of arXiv:1908.07541}
}
read the original abstract

When thermal rate equations are derived for the evolution of slow variables, it is often practical to parametrize the right-hand side with chemical potentials. To close the system, the chemical potentials are subsequently re-expressed in terms of the slow variables, which involves the consideration of a "susceptibility". Here we study a non-relativistic situation in which chemical potentials are large compared with the temperature, as is relevant for late-time pair annihilations in dark matter freeze-out. An order-of-magnitude estimate and a lattice simulation are presented for a susceptibility dominated by bound states of stop-like mediators. After this "calibration", the formalism is applied to a model with Majorana singlet dark matter, confirming that masses up to the multi-TeV domain are viable in the presence of sufficient (though not beyond a limit) mass degeneracy in the dark sector.

Figures

Figures reproduced from arXiv: 1908.07541 by the authors.

Figure 1
Figure 1. Comparison of an order-of-magnitude estimate of T 3pˆ2 from sec. 3 and a lattice estimate from sec. 5. The dashed line shows a temperature at which confinement sets in. The errors of the lattice results are statistical only; systematic uncertainties could be as large as ∼ 50%. In any case, based on this test, a vacuum-like coupling performs best at low temperatures, whereas towards high temperatures the slope seen i… view at source ↗
Figure 2
Figure 2. Values of the thermally averaged Sommerfeld factor S¯ 3 (cf. eq. (6.3)) and the rescaled susceptibility T 3 p˜2 (cf. eq. (6.2)) for M = 1...500 TeV. We have now included ˆp2 in the dynamics described by eq. (1.2), by solving for e βµ neq from eq. (2.12). The presence of the neutral field implies that neq ≃ 2  MT 2π 3 2 e −βM  1 + Nc e −β∆MT  , (6.1) where the thermally modified mass difference ∆MT is given in eq… view at source ↗
Figure 3
Figure 3. Examples of a solution, for M = 500 TeV, with tree-level annihilation rates (“tree-level”) and after including a thermally averaged Sommerfeld factor (“with S¯ 3”) and a susceptibility (“with S¯ 3 and ˆp2 ”). The symbols y, λ3 , h refer to couplings defined in ref. [25], whose precise values have little impact on the general pattern. This plot assumes that kinetic/ionization equilibrium is maintained in the dark sec… view at source ↗

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