Pith. sign in

REVIEW 3 major objections 5 minor 87 references

Effective field theories for dark matter pairs in the early universe: Debye mass effects

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read During freeze-out, bound-state formation and dissociation must use Debye-resummed photon propagators; fixed-order NLO overestimates the rates by up to a factor of five.

desk verdict Genuinely new Debye-resummed bound-state rates in the m_D ~ ΔE regime, with clean scale separation, but the headline benchmarks sit at the edge of the assumed T ≫ m_D hierarchy, so the few-percent numbers are indicative rather than controlled. read the letter →

arxiv 2501.03327 v2 pith:5FI7RSA3 submitted 2025-01-06 hep-ph astro-ph.COhep-th

Simone Biondini , Nora Brambilla , Andrii Dashko , Gramos Qerimi , Antonio Vairo This is my paper · ORCID
classification hep-phastro-ph.COhep-th
keywords darkmatterrelicdensitybound-stateformationDebyemasshardthermalloopresummationnon-relativisticeffectivefieldtheoryfreeze-outLandaudampingphoton
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

This paper tries to establish that, for heavy fermionic dark matter in a U(1) dark sector with light fermionic bath particles, the rates controlling bound-state formation, dissociation, and bound-to-bound transitions during freeze-out cannot be reliably computed order by order when the temperature is much larger than the emitted-photon energy. In that regime the thermal photon propagator develops a Debye mass, and the paper shows that resumming this mass through hard-thermal-loop propagators changes the thermally averaged cross sections and widths: the fixed-order next-to-leading-order calculation overestimates them by up to a factor of five near freeze-out. The consequence is quantitative: using Debye-resummed rates reduces the predicted correction to the dark matter relic density from 3.7-6.0% to 2.5-3.5% for the 1S ground state alone, and from 7.3-11.0% to 4.8-6.3% when n <= 2 bound states and transitions are included. A sympathetic reader would care because these are the percent-level shifts that decide whether a given dark matter mass and coupling point is consistent with the observed relic abundance.

What carries the argument

The load-bearing object is the Debye-resummed dark photon propagator, obtained by hard-thermal-loop resummation of the longitudinal and transverse polarization tensors. In the real-time Schwinger-Keldysh formalism the heavy-pair self-energy splits into contributions from the scale $T$, the Debye scale $m_D$, and the binding-energy scale $\Delta E$; the resummed propagator makes the infrared divergences and renormalization-scale dependence cancel between the scales, yielding finite expressions. In the general case $m_D \sim \Delta E$ the result is carried by the two integrals $X_l$ and $X_t$, which encode longitudinal Landau-damping scattering, transverse photo-emission, and transverse bath-particle scattering channels.

What would settle it

Compute the bound-state formation rate from the full retarded photon propagator without expanding in $q_0/T$ or imposing $T \gg m_D$, at $m_D/T \approx 0.92$; agreement with eq. (3.43) at the claimed few-percent level would confirm the resummation, while a significant deviation would show the weakly-coupled hierarchy is the weak point.

Watch

Extended reading notes

Core claim

The paper's central claim is that in the scale hierarchy $M \gg \sqrt{MT} \gg M\alpha \gg T \gg m_D$, with no assumed ordering between the Debye mass $m_D$ and the Coulombic binding energy $M\alpha^2$, the correct bound-state formation cross section, dissociation width, and bound-to-bound transition widths are the Debye-resummed ones in eqs. (3.43), (3.49) and their transition analogues. These expressions interpolate between the limits $m_D \gg \Delta E$ and $\Delta E \gg m_D$ through two finite numerical integrals $X_l(\Delta E/m_D)$ and $X_t(\Delta E/m_D)$. When the rates are thermally averaged and inserted in the effective Boltzmann equation, the fixed-order NLO calculation is found to overestimate the individual rates by up to a factor of five near freeze-out, yet the residual effect on the relic density is only 2.5--3.5% for the 1S state and 4.8--6.3% for $n \leq 2$ with bound-to-bound transitions, because bound-state formation and dissociation enter the effective cross section largely as a ratio.

Load-bearing premise

The argument rests on the assumed hierarchy $T \gg m_D$, which keeps the dark plasma weakly coupled; at the benchmark $n_f=2$, $\alpha(2M)=0.1$ the ratio is $m_D/T \approx 0.92$, so the hierarchy is only marginal and the quoted percent corrections could shift if the plasma is closer to strongly coupled.

Editorial extensions

If this is right

  • At temperatures near freeze-out, the fixed-order next-to-leading-order bound-state formation cross section and dissociation width exceed the Debye-resummed results by a factor of up to five, so earlier NLO-based depletion estimates are systematically high.
  • With Debye resummation the relic-density correction is 2.5--3.5% for the 1S state alone and 4.8--6.3% including $n \leq 2$ states with transitions, roughly half the fixed-order NLO values.
  • In ionization equilibrium the enhancements of bound-state formation and dissociation largely cancel in the effective cross section, which is why the net relic-density effect is at the few-percent level rather than the order-of-magnitude level of the individual rates.
  • The resummation has a larger impact on the 2S and 2P excited states than on the ground state, so any treatment including a full tower of bound states should use the Debye-resummed rates.

Reading between the lines

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

  • Inference: the same Debye-resummed machinery should apply to non-abelian dark sectors and to coannihilating colored states, where a gluon Debye mass arises without extra light fermions; the closed-form integrals here provide a template, but the non-abelian case needs its own calculation.
  • Inference: because fixed-order NLO and Debye-resummed corrections meet at the boundary between the $T \gg \Delta E$ and $T \lesssim \Delta E$ regimes, the clean interpolation used over the whole $v_{\rm rel}$ range deserves a dedicated matched expansion; the paper shows the numerical impact is below 0.1% on the relic density, but a systematic all-order matching would remove the residual ambiguity.
  • Inference: since the companion recoil corrections [40] have the opposite sign, combining the two calculations could push the total correction below the 1% observational accuracy of the relic density, which would make the benchmark predictions testable by future precision measurements.
  • Inference: a direct comparison with full non-equilibrium quantum-field-theory calculations of the rates in the region where $m_D \sim \Delta E$ would test whether the factor-of-five gap between the NLO and resummed results is quantitatively correct.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a thermal EFT treatment of bound-state formation, dissociation, and bound-to-bound transitions for heavy Dirac fermion dark matter interacting with a dark photon and nf light dark fermions. It assumes the hierarchy M ≫ sqrt(MT) ≫ Mα ≫ T and T ≫ m_D, and analyzes the regimes m_D ≫ ΔE, ΔE ≫ m_D, and m_D ∼ ΔE, deriving closed-form or numerically integrable expressions for the rates after HTL-resumming the dark photon propagator. The main quantitative result is that the fixed-order NLO treatment overestimates the inelastic rates by up to a factor of five near freeze-out and therefore over-predicts dark matter depletion, while Debye-resummed rates give relic-density corrections of 2.5–3.5% (1S only) and 4.8–6.3% (n ≤ 2 with transitions) versus 3.7–6.0% and 7.3–11.0% at NLO. The calculation is first-principles, with explicit verification of scale cancellation in the key self-energy result, and it goes beyond earlier work by treating the m_D ∼ ΔE window without an assumed hierarchy.

Significance. If the quantitative claims are controlled, this is an important step for thermal DM freeze-out computations: it provides the first pNRQED-based treatment of the Debye-mass scale in the regime where m_D ∼ ΔE, gives explicit closed forms suitable for numerical evaluation (eqs. (3.30), (3.37), (3.43), (3.49)), and demonstrates that the fixed-order NLO treatment can misestimate depletion. The derivation is systematic and scale-cancellation is checked; the paper also explicitly includes excited states and bound-to-bound transitions, and it makes a falsifiable comparison of resummed versus fixed-order relic densities. The main weakness is that the quantitative reach of the headline benchmark numbers is limited by the strength of the hierarchy T ≫ m_D, as detailed in the major comments.

major comments (3)
  1. [§2 (eq. (2.3)), §3.1.4 (eqs. (3.43), (3.49)), §5, Figs. 11–12] The central quantitative comparison relies on the assumption (2.3), T ≫ m_D, which is stated to hold weakly for the benchmarks. For n_f = 2 and α(2M) = 0.1, one has m_D/T = sqrt(4π n_f α/3) ≈ 0.92, so (m_D/T)^2 ≈ 0.84 and the HTL expansion parameter is of order one. The resummed propagators (B.31) and (B.32) are leading-order in this expansion, so neglected corrections are formally O(1) with respect to the very effects that produce the quoted 4.8–6.3% versus 7.3–11.0% difference in the relic density. The authors themselves exclude T ∼ m_D in §3.2 and §5. A revision should either restrict the quantitative claims to benchmark points with a clearly small m_D/T (for example m_D/T ≲ 0.3), or provide an estimate of the next-order thermal corrections and demonstrate numerically that the quoted percentages are stable under them. This is a load-bearing issue because it affects the central numerical message, not just the presentation.
  2. [§3.2, paragraph after eq. (3.43)] The sentence 'The NLO terms in the second lines of eqs. (3.41), (3.42) and (3.43) are suppressed by nf α/π with respect to the LO radiative photon emission' is not correct as stated. In eq. (3.41), for example, the second line contains (m_D/(2ΔE_p^n))^2 times a bracket of order one with no explicit nf α/π factor. In the regime T ≫ m_D ≫ ΔE this term scales as (m_D/ΔE)^2 relative to the LO term and can be parametrically large, as the Landau-damping contribution in figure 5 indeed shows. The claim of suppression should be restricted to the vacuum-polarization term in the first line of these equations, or reworded to distinguish the parametric enhancement (~(T/ΔE)^2) from the coupling suppression. As written, the sentence contradicts the paper's own numerical finding that bath-particle scattering dominates at high temperatures.
  3. [§5, abstract, and figure captions] The quantitative boundary of the quoted numbers is not stated consistently. The abstract and §3.2 quote a factor of up to five overestimation of the rates, while the conclusions say 'a factor of order three'. Moreover, §5 states that the parameter choices satisfy 'T > m_D' at all times, but the formal hierarchy used is T ≫ m_D, and for n_f = 2 the initial value is already very close to T ≈ m_D. Please state the precise quantitative criterion used to define the validity range of the quoted numbers, including the effect of the one-loop running coupling on m_D/T at the lower temperatures shown in the figures.
minor comments (5)
  1. [§3.2, final paragraph] The condition written as 'sqrt(π n_f α) ∼ 1' differs from the hierarchy condition (2.3), which is sqrt(4π n_f α/3) ≪ 1; include the factor 4/3 for consistency.
  2. [Eq. (3.10)] The expression '−2/3 α r_i (ΔE)^2 r_i T' is hard to parse because r_i and ΔE are operators; please insert brackets or state explicitly that this is a matrix element between scattering and bound states.
  3. [§3.2, figure 4 discussion] The statement that using (3.43) over the whole vrel range versus splitting the integral changes the result by at most 4% is useful, but the text does not explain why the splitting effect becomes smaller in the effective cross section; a one-sentence explanation of the compensation in eq. (4.2) would help.
  4. [Appendix C, eq. (C.16)] The function X1(ΔE_p^n, μ) is defined only by reference to (C.12); since X2 is written explicitly in (C.17), it would be clearer to display X1 explicitly as well.
  5. [§1 and §2] The statement that the hierarchy (2.2) holds 'to a good extent' at freeze-out is overly optimistic for α(2M)=0.1, where Mα/T ≈ 2.5 at T ≈ M/25; a brief comment on the size of the resulting Mα/T suppression would be appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Debye-resummed rates and relic-density shifts are computed from the Lagrangian (2.1) via thermal EFT, not fitted, and no central claim reduces by construction to its inputs.

full rationale

The derivation chain is self-contained and first-principles. The Debye mass is computed from the one-loop retarded polarization tensor in appendix B, m_D^2 = n_f g^2 T^2/3, rather than inserted by hand, and the resummed propagators (B.31) and (B.32) follow from standard hard-thermal-loop resummation of that polarization. The central rate formulas (3.43) and (3.49) are obtained by evaluating the two-loop pNRQED_DM self-energy (3.3) with those propagators and combining the scale-T contribution (3.8) with the m_D ~ Delta_E longitudinal and transverse integrals (3.32) and (3.36), giving (3.40); the cross section and width then follow by the optical theorem. The NLO comparison in appendix C is a separate fixed-order calculation, so the factor-of-five difference between NLO and resummed rates is an internal consistency check, not a fitted parameter renamed as a prediction. The relic density is solved from the Boltzmann equation (4.1) using these rates as functions of the input parameters M, alpha, and n_f, and is not matched to the Planck value; the paper reports relative shifts from LO rates rather than claiming a parameter extraction. The only self-citations are to [39,40] for the pNRQED_DM framework and to [55,79,85,86] for standard HTL polarization results; these are external, reproducible thermal-field-theory results and do not assume the target bound-state rates or relic-density outcome. The concern that the benchmark n_f = 2, alpha = 0.1 corresponds to T/m_D ~ 1.09 and therefore sits at the edge of the stated hierarchy T >> m_D is a validity or convergence issue about the size of higher-order HTL corrections, not a circularity; the paper explicitly flags the strongly coupled T ~ m_D regime as outside its scope. No step in the derivation equates an output to an input by definition.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the scale hierarchy (2.2)-(2.3), the weak-coupling assumption, and standard thermal field theory techniques. No parameters are fitted to the relic density; M, n_f, and α(2M) are scanned benchmarks. No new particles, forces, or conserved quantities are introduced; the Debye mass is derived, not postulated.

free parameters (3)
  • α(2M) = 0.05 to 0.1, run at one loop
    Dark gauge coupling at the hard scale; controls annihilation, bound states, and the Debye mass. It is a benchmark scan parameter, not fitted to data.
  • n_f = 1 and 2
    Number of massless light dark fermions in the bath; sets the Debye mass and the annihilation channels through Im[d_v]. Scanned as a model input.
  • M = 1 to 10 TeV (10 TeV for yield plots)
    Dark matter mass; enters the hierarchy, the Hubble rate, and the kinetic equilibrium distribution. Scanned as a benchmark, not fitted.
assumptions (8)
  • domain assumption Scale hierarchy M ≫ sqrt(MT) ≫ Mα ≫ T ≫ m_D, eqs. (2.2)-(2.3)
    Justifies pNRQED_DM and the separation of scales; the T ≫ m_D part is only marginally satisfied for the largest benchmarks, which the authors acknowledge.
  • domain assumption Weak coupling, α ≪ 1 and sqrt(π n_f α) ≪ 1
    Required for the plasma to be weakly coupled and for HTL resummation to apply; the paper restricts to α ≤ 0.1 and n_f ≤ 2 but for n_f = 2, α = 0.1, sqrt(π n_f α) ≈ 0.79, near the boundary.
  • domain assumption Light dark fermions are massless (m_i ≈ 0) and relativistic (m_i ≪ T)
    Stated in section 2 and used throughout the polarization tensor calculation in appendix B.
  • domain assumption Dark sector and SM share a common temperature; portal interactions are neglected in the rates
    Assumed in section 2 based on small kinetic mixing, following ref. [66].
  • domain assumption Heavy DM pairs are kinetically equilibrated with a Maxwell-Boltzmann distribution
    Used in the thermal average in eq. (3.44), following ref. [9].
  • domain assumption Single effective Boltzmann equation is valid when bound states are close to equilibrium, H ≪ Γ
    Used in section 4, following ref. [82], with the extension to bound-to-bound transitions from ref. [33].
  • standard math Real-time Schwinger-Keldysh formalism and Kobes-Semenoff cutting rules
    Standard thermal field theory machinery used in section 3 and appendices A and C.
  • domain assumption Decoupling of heavy dark matter fields from the dark photon polarization
    Invoked in appendix A so that only light fermions enter the one-loop polarization tensor.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effective field theories for dark matter pairs in the early universe: Debye mass effects." pith.science (2026). https://pith.science/paper/5FI7RSA3

@misc{pith2026250103327,
  author       = {Pith},
  title        = {Pith review of: Effective field theories for dark matter pairs in the early universe: Debye mass effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FI7RSA3}},
  note         = {Machine review of arXiv:2501.03327}
}
abstract

In some scenarios for the early universe, non-relativistic thermal dark matter chemically decouples from the thermal environment once the temperature drops well below the dark matter mass. The value at which the energy density freezes out depends on the underlying model. In a simple setting, we provide a comprehensive study of heavy fermionic dark matter interacting with the light degrees of freedom of a dark thermal sector whose temperature $T$ decreases from an initial value close to the freeze-out temperature. Different temperatures imply different hierarchies of energy scales. By exploiting the methods of non-relativistic effective field theories at finite $T$, we systematically determine the thermal and in-vacuum interaction rates. In particular, we address the impact of the Debye mass on the observables and ultimately on the dark matter relic abundance. We numerically compare the corrections to the present energy density originating from the resummation of Debye mass effects with the corrections coming from a next-to-leading order treatment of the bath-particle interactions. We observe that the fixed-order calculation of the inelastic heavy-light scattering at high temperatures provides a larger dark matter depletion, and hence an undersized yield for given benchmark points in the parameter space, with respect to the calculation where Debye mass effects are resummed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

87 extracted references · 22 canonical work pages

  1. [1]

    Zwicky,Die Rotverschiebung von extragalaktischen Nebeln, Helv

    F. Zwicky,Die Rotverschiebung von extragalaktischen Nebeln, Helv. Phys. Acta6 (1933) 110

    1933

  2. [2]

    Rubin, N

    V.C. Rubin, N. Thonnard and W.K. Ford, Jr.,Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/, Astrophys. J. 238 (1980) 471

    1980

  3. [3]

    Clowe, M

    D. Clowe, M. Bradac, A.H. Gonzalez, M. Markevitch, S.W. Randall, C. Jones et al.,A direct empirical proof of the existence of dark matter, Astrophys. J. Lett.648 (2006) L109 [astro-ph/0608407]

    arXiv 2006

  4. [4]

    Feng,Dark Matter Candidates from Particle Physics and Methods of Detection, Ann

    J.L. Feng,Dark Matter Candidates from Particle Physics and Methods of Detection, Ann. Rev. Astron. Astrophys.48 (2010) 495 [1003.0904]

    arXiv 2010

  5. [5]

    Bertone and D

    G. Bertone and D. Hooper,History of dark matter, Rev. Mod. Phys.90 (2018) 045002 [1605.04909]

    arXiv 2018

  6. [6]

    Planck collaboration, Planck 2018 results. I. Overview and the cosmological legacy of Planck, Astron. Astrophys.641 (2020) A1 [1807.06205]

    arXiv 2020

  7. [7]

    Roszkowski, E.M

    L. Roszkowski, E.M. Sessolo and S. Trojanowski,WIMP dark matter candidates and searches—current status and future prospects, Rept. Prog. Phys.81 (2018) 066201 [1707.06277]

    arXiv 2018

  8. [8]

    Bertone,The moment of truth for WIMP Dark Matter, Nature 468 (2010) 389 [1011.3532]

    G. Bertone,The moment of truth for WIMP Dark Matter, Nature 468 (2010) 389 [1011.3532]

    arXiv 2010

Show all 87 references
  1. [9]

    Gondolo and G

    P. Gondolo and G. Gelmini,Cosmic abundances of stable particles: Improved analysis, Nucl. Phys. B 360 (1991) 145

    1991

  2. [10]

    Griest and D

    K. Griest and D. Seckel,Three exceptions in the calculation of relic abundances, Phys. Rev. D 43 (1991) 3191

    1991

  3. [11]

    Sommerfeld,Über die Beugung und Bremsung der Elektronen, Annalen Phys

    A. Sommerfeld,Über die Beugung und Bremsung der Elektronen, Annalen Phys. 403 (1931) 257

    1931

  4. [12]

    Hisano, S

    J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito,Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center, Phys. Rev. D71 (2005) 063528 [hep-ph/0412403]

    2005 arXiv

  5. [13]

    J.L. Feng, M. Kaplinghat, H. Tu and H.-B. Yu,Hidden Charged Dark Matter, JCAP 0907 (2009) 004 [0905.3039]

    2009 arXiv

  6. [14]

    von Harling and K

    B. von Harling and K. Petraki,Bound-state formation for thermal relic dark matter and unitarity, JCAP 1412 (2014) 033 [1407.7874]

    2014 arXiv

  7. [15]

    Beneke, C

    M. Beneke, C. Hellmann and P. Ruiz-Femenia,Heavy neutralino relic abundance with Sommerfeld enhancements - a study of pMSSM scenarios, JHEP 03 (2015) 162 [1411.6930]

    2015 arXiv

  8. [16]

    Petraki, M

    K. Petraki, M. Postma and J. de Vries,Radiative bound-state-formation cross-sections for dark matter interacting via a Yukawa potential, JHEP 04 (2017) 077 [1611.01394]

    2017 arXiv

  9. [17]

    Kim and M

    S. Kim and M. Laine,On thermal corrections to near-threshold annihilation, Journal of Cosmology and Astroparticle Physics2017 (2017) 013

    2017

  10. [18]

    Beneke, A

    M. Beneke, A. Bharucha, F. Dighera, C. Hellmann, A. Hryczuk, S. Recksiegel et al.,Relic density of wino-like dark matter in the MSSM, JHEP 03 (2016) 119 [1601.04718]. – 50 –

    2016 arXiv

  11. [19]

    Mitridate, M

    A. Mitridate, M. Redi, J. Smirnov and A. Strumia,Cosmological Implications of Dark Matter Bound States, JCAP 05 (2017) 006 [1702.01141]

    2017 arXiv

  12. [20]

    Biondini and M

    S. Biondini and M. Laine,Re-derived overclosure bound for the inert doublet model, JHEP 08 (2017) 047 [1706.01894]

    2017 arXiv

  13. [21]

    Harz and K

    J. Harz and K. Petraki,Radiative bound-state formation in unbroken perturbative non-Abelian theories and implications for dark matter, JHEP 07 (2018) 096 [1805.01200]

    2018 arXiv

  14. [22]

    Biondini and M

    S. Biondini and M. Laine,Thermal dark matter co-annihilating with a strongly interacting scalar, JHEP 04 (2018) 072 [1801.05821]

    2018 arXiv

  15. [23]

    Biondini,Bound-state effects for dark matter with Higgs-like mediators, JHEP 06 (2018) 104 [1805.00353]

    S. Biondini,Bound-state effects for dark matter with Higgs-like mediators, JHEP 06 (2018) 104 [1805.00353]

    2018 arXiv

  16. [24]

    Biondini and S

    S. Biondini and S. Vogl,Coloured coannihilations: Dark matter phenomenology meets non-relativistic EFTs, JHEP 02 (2019) 016 [1811.02581]

    2019 arXiv

  17. [25]

    Binder, L

    T. Binder, L. Covi and K. Mukaida,Dark Matter Sommerfeld-enhanced annihilation and Bound-state decay at finite temperature, Phys. Rev. D98 (2018) 115023 [1808.06472]

    2018 arXiv

  18. [26]

    Oncala and K

    R. Oncala and K. Petraki,Dark matter bound states via emission of scalar mediators, JHEP 01 (2019) 070 [1808.04854]

    2019 arXiv

  19. [27]

    Oncala and K

    R. Oncala and K. Petraki,Dark matter bound state formation via emission of a charged scalar, JHEP 02 (2020) 036 [1911.02605]

    2020 arXiv

  20. [28]

    Harz and K

    J. Harz and K. Petraki,Higgs-mediated bound states in dark-matter models, JHEP 04 (2019) 130 [1901.10030]

    2019 arXiv

  21. [29]

    Binder, K

    T. Binder, K. Mukaida and K. Petraki,Rapid bound-state formation of Dark Matter in the Early Universe, Phys. Rev. Lett.124 (2020) 161102 [1910.11288]

    2020 arXiv

  22. [30]

    Biondini, S

    S. Biondini, S. Kim and M. Laine,Non-relativistic susceptibility and a dark matter application, JCAP 10 (2019) 078 [1908.07541]

    2019 arXiv

  23. [31]

    Biondini and S

    S. Biondini and S. Vogl,Scalar dark matter coannihilating with a coloured fermion, JHEP 11 (2019) 147 [1907.05766]

    2019 arXiv

  24. [32]

    Binder, B

    T. Binder, B. Blobel, J. Harz and K. Mukaida,Dark matter bound-state formation at higher order: a non-equilibrium quantum field theory approach, JHEP 09 (2020) 086 [2002.07145]

    2020 arXiv

  25. [33]

    Garny and J

    M. Garny and J. Heisig,Bound-state effects on dark matter coannihilation: Pushing the boundaries of conversion-driven freeze-out, Phys. Rev. D105 (2022) 055004 [2112.01499]

    2022 arXiv

  26. [34]

    Biondini and V

    S. Biondini and V. Shtabovenko,Bound-state formation, dissociation and decays of darkonium with potential non-relativistic Yukawa theory for scalar and pseudoscalar mediators, JHEP 03 (2022) 172 [2112.10145]

    2022 arXiv

  27. [35]

    Binder, A

    T. Binder, A. Filimonova, K. Petraki and G. White,Saha equilibrium for metastable bound states and dark matter freeze-out, Phys. Lett. B833 (2022) 137323 [2112.00042]

    2022 arXiv

  28. [36]

    Biondini and V

    S. Biondini and V. Shtabovenko,Non-relativistic and potential non-relativistic effective field theories for scalar mediators, JHEP 08 (2021) 114 [2106.06472]

    2021 arXiv

  29. [37]

    Binder, M

    T. Binder, M. Garny, J. Heisig, S. Lederer and K. Urban,Excited bound states and their role in dark matter production, Phys. Rev. D108 (2023) 095030 [2308.01336]

    2023 arXiv

  30. [38]

    Biondini, J

    S. Biondini, J. Bollig and S. Vogl,Indirect detection of dark matter with (pseudo)-scalar interactions, JHEP 04 (2024) 050 [2308.14594]. – 51 –

    2024 arXiv

  31. [39]

    Biondini, N

    S. Biondini, N. Brambilla, G. Qerimi and A. Vairo,Effective field theories for dark matter pairs in the early universe: cross sections and widths, JHEP 07 (2023) 006 [2304.00113]

    2023 arXiv

  32. [40]

    Biondini, N

    S. Biondini, N. Brambilla, G. Qerimi and A. Vairo,Effective field theories for dark matter pairs in the early universe: center-of-mass recoil effects, JHEP 07 (2024) 021 [2402.12787]

    2024 arXiv

  33. [41]

    Lahanas, D.V

    A.B. Lahanas, D.V. Nanopoulos and V.C. Spanos,Neutralino relic density in a universe with nonvanishing cosmological constant, Phys. Rev. D62 (2000) 023515 [hep-ph/9909497]

    2000 arXiv

  34. [42]

    Baer and J

    H. Baer and J. O’Farrill,Probing neutralino resonance annihilation via indirect detection of dark matter, JCAP 04 (2004) 005 [hep-ph/0312350]

    2004 arXiv

  35. [43]

    Cirelli, N

    M. Cirelli, N. Fornengo and A. Strumia,Minimal dark matter, Nucl. Phys. B 753 (2006) 178 [hep-ph/0512090]

    2006 arXiv

  36. [44]

    Beneke, C

    M. Beneke, C. Hellmann and P. Ruiz-Femenia,Non-relativistic pair annihilation of nearly mass degenerate neutralinos and charginos I. General framework and S-wave annihilation, JHEP 03 (2013) 148 [1210.7928]

    2013 arXiv

  37. [45]

    Carone, S

    C.D. Carone, S. Chaurasia and T.V.B. Claringbold,Dark sector portal with vectorlike leptons and flavor sequestering, Phys. Rev. D99 (2019) 015009 [1807.05288]

    2019 arXiv

  38. [46]

    Edsjo and P

    J. Edsjo and P. Gondolo,Neutralino relic density including coannihilations, Phys. Rev. D56 (1997) 1879 [hep-ph/9704361]

    1997 arXiv

  39. [47]

    Ellis, K.A

    J.R. Ellis, K.A. Olive and Y. Santoso,Calculations of neutralino stop coannihilation in the CMSSM, Astropart. Phys.18 (2003) 395 [hep-ph/0112113]

    2003 arXiv

  40. [48]

    Berggren, A

    M. Berggren, A. Cakir, D. Krücker, J. List, I.A. Melzer-Pellmann, B. Safarzadeh Samani et al.,Non-simplified SUSY: eτ -coannihilation at LHC and ILC, Eur. Phys. J. C76 (2016) 183 [1508.04383]

    2016 arXiv

  41. [49]

    Garny, A

    M. Garny, A. Ibarra and S. Vogl,Signatures of Majorana dark matter with t-channel mediators, Int. J. Mod. Phys. D24 (2015) 1530019 [1503.01500]

    2015 arXiv

  42. [50]

    Arina, B

    C. Arina, B. Fuks and L. Mantani,A universal framework for t-channel dark matter models, Eur. Phys. J. C80 (2020) 409 [2001.05024]

    2020 arXiv

  43. [51]

    Braaten and R.D

    E. Braaten and R.D. Pisarski,Simple effective Lagrangian for hard thermal loops, Phys. Rev. D 45 (1992) R1827

    1992

  44. [52]

    Laine, O

    M. Laine, O. Philipsen, P. Romatschke and M. Tassler,Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300]

    2007 arXiv

  45. [53]

    Laine, O

    M. Laine, O. Philipsen and M. Tassler,Thermal imaginary part of a real-time static potential from classical lattice gauge theory simulations, JHEP 09 (2007) 066 [0707.2458]

    2007 arXiv

  46. [54]

    Beraudo, J.P

    A. Beraudo, J.P. Blaizot and C. Ratti,Real and imaginary-time Q anti-Q correlators in a thermal medium, Nucl. Phys. A806 (2008) 312 [0712.4394]

    2008 arXiv

  47. [55]

    Brambilla, J

    N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky,Static quark-antiquark pairs at finite temperature, Phys. Rev. D78 (2008) 014017 [0804.0993]

    2008 arXiv

  48. [56]

    Escobedo and J

    M.A. Escobedo and J. Soto,Non-relativistic bound states at finite temperature (I): The Hydrogen atom, Phys. Rev. A78 (2008) 032520 [0804.0691]

    2008 arXiv

  49. [57]

    Escobedo and J

    M.A. Escobedo and J. Soto,Non-relativistic bound states at finite temperature (II): the muonic hydrogen, Phys. Rev. A82 (2010) 042506 [1008.0254]. – 52 –

    2010 arXiv

  50. [58]

    Brambilla, A

    N. Brambilla, A. Pineda, J. Soto and A. Vairo,Effective field theories for heavy quarkonium, Rev. Mod. Phys.77 (2005) 1423 [hep-ph/0410047]

    2005 arXiv

  51. [59]

    Feldman, B

    D. Feldman, B. Kors and P. Nath,Extra-weakly Interacting Dark Matter, Phys. Rev. D75 (2007) 023503 [hep-ph/0610133]

    2007 arXiv

  52. [60]

    Fayet,U-boson production in e+ e- annihilations, psi and Upsilon decays, and Light Dark Matter, Phys

    P. Fayet,U-boson production in e+ e- annihilations, psi and Upsilon decays, and Light Dark Matter, Phys. Rev. D75 (2007) 115017 [hep-ph/0702176]

    2007 arXiv

  53. [61]

    Goodsell, J

    M. Goodsell, J. Jaeckel, J. Redondo and A. Ringwald,Naturally Light Hidden Photons in LARGE Volume String Compactifications, JHEP 11 (2009) 027 [0909.0515]

    2009 arXiv

  54. [62]

    Morrissey, D

    D.E. Morrissey, D. Poland and K.M. Zurek,Abelian Hidden Sectors at a GeV, JHEP 07 (2009) 050 [0904.2567]

    2009 arXiv

  55. [63]

    Andreas, M.D

    S. Andreas, M.D. Goodsell and A. Ringwald,Dark matter and dark forces from a supersymmetric hidden sector, Phys. Rev. D87 (2013) 025007 [1109.2869]

    2013 arXiv

  56. [64]

    Holdom,Two U(1)’s and Epsilon Charge Shifts, Phys

    B. Holdom,Two U(1)’s and Epsilon Charge Shifts, Phys. Lett. B166 (1986) 196

    1986

  57. [65]

    Foot and X.-G

    R. Foot and X.-G. He,Comment on Z Z-prime mixing in extended gauge theories, Phys. Lett. B 267 (1991) 509

    1991

  58. [66]

    Evans, S

    J.A. Evans, S. Gori and J. Shelton,Looking for the WIMP Next Door, JHEP 02 (2018) 100 [1712.03974]

    2018 arXiv

  59. [67]

    Caswell and G.P

    W.E. Caswell and G.P. Lepage,Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories, Phys. Lett. 167B (1986) 437

    1986

  60. [68]

    Bodwin, E

    G.T. Bodwin, E. Braaten and G.P. Lepage,Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51 (1995) 1125 [hep-ph/9407339]

    1995 arXiv

  61. [69]

    Braaten,Introduction to the NRQCD factorization approach to heavy quarkonium, in3rd International Workshop on Particle Physics Phenomenology, 11, 1996 [hep-ph/9702225]

    E. Braaten,Introduction to the NRQCD factorization approach to heavy quarkonium, in3rd International Workshop on Particle Physics Phenomenology, 11, 1996 [hep-ph/9702225]

    1996 arXiv

  62. [70]

    Petrelli, M

    A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M.L. Mangano,NLO production and decay of quarkonium, Nucl. Phys. B 514 (1998) 245 [hep-ph/9707223]

    1998 arXiv

  63. [71]

    Pineda and J

    A. Pineda and J. Soto,Matching at one loop for the four quark operators in NRQCD, Phys. Rev. D58 (1998) 114011 [hep-ph/9802365]

    1998 arXiv

  64. [72]

    Vairo,A Theoretical review of heavy quarkonium inclusive decays, Mod

    A. Vairo,A Theoretical review of heavy quarkonium inclusive decays, Mod. Phys. Lett. A19 (2004) 253 [hep-ph/0311303]

    2004 arXiv

  65. [73]

    Brambilla, A

    N. Brambilla, A. Pineda, J. Soto and A. Vairo,The p mΛQCD scale in heavy quarkonium, Phys. Lett. B580 (2004) 60 [hep-ph/0307159]

    2004 arXiv

  66. [74]

    Pineda and J

    A. Pineda and J. Soto,Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. Proc. Suppl.64 (1998) 428 [hep-ph/9707481]

    1998 arXiv

  67. [75]

    Pineda and J

    A. Pineda and J. Soto,The Lamb shift in dimensional regularization, Phys. Lett. B420 (1998) 391 [hep-ph/9711292]

    1998 arXiv

  68. [76]

    Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    M.L. Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    2011 doi

  69. [77]

    Laine and A

    M. Laine and A. Vuorinen,Basics of Thermal Field Theory, vol. 925, Springer (2016), 10.1007/978-3-319-31933-9, [1701.01554]

    2016 arXiv

  70. [78]

    Ghiglieri, A

    J. Ghiglieri, A. Kurkela, M. Strickland and A. Vuorinen,Perturbative Thermal QCD: Formalism and Applications, Phys. Rept. 880 (2020) 1 [2002.10188]. – 53 –

    2020 arXiv

  71. [79]

    Brambilla, M.A

    N. Brambilla, M.A. Escobedo, J. Ghiglieri, J. Soto and A. Vairo,Heavy Quarkonium in a weakly-coupled quark-gluon plasma below the melting temperature, JHEP 09 (2010) 038 [1007.4156]

    2010 arXiv

  72. [80]

    Binder, K

    T. Binder, K. Mukaida, B. Scheihing-Hitschfeld and X. Yao,Non-Abelian electric field correlator at NLO for dark matter relic abundance and quarkonium transport, JHEP 01 (2022) 137 [2107.03945]

    2022 arXiv

  73. [81]

    Burnier, M

    Y. Burnier, M. Laine, J. Langelage and L. Mether,Colour-electric spectral function at next-to-leading order, JHEP 08 (2010) 094 [1006.0867]

    2010 arXiv

  74. [82]

    Ellis, F

    J. Ellis, F. Luo and K.A. Olive,Gluino Coannihilation Revisited, JHEP 09 (2015) 127 [1503.07142]

    2015 arXiv

  75. [83]

    Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022) 083C01

    2022

  76. [84]

    Carignano, C

    S. Carignano, C. Manuel and J. Soto,Power corrections to the HTL effective Lagrangian of QED, Phys. Lett. B780 (2018) 308 [1712.07949]

    2018 arXiv

  77. [85]

    Brambilla, M.A

    N. Brambilla, M.A. Escobedo, J. Ghiglieri and A. Vairo,Thermal width and quarkonium dissociation by inelastic parton scattering, JHEP 05 (2013) 130 [1303.6097]

    2013 arXiv

  78. [86]

    Carrington, D.-f

    M.E. Carrington, D.-f. Hou and M.H. Thoma,Equilibrium and nonequilibrium hard thermal loop resummation in the real time formalism, Eur. Phys. J. C7 (1999) 347 [hep-ph/9708363]

    1999 arXiv

  79. [87]

    Kobes and G.W

    R.L. Kobes and G.W. Semenoff,Discontinuities of Green Functions in Field Theory at Finite Temperature and Density, Nucl. Phys. B 260 (1985) 714. – 54 –

    1985

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.