Pith. sign in

REVIEW 5 minor 3 cited by

A Practical and Consistent Parametrization of Dark Matter Self-Interactions

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Dark matter's self-scattering reduces to a two-parameter formula.

desk verdict A practical, honest paper that makes the effective-range two-parameter language the default for SIDM velocity dependence; the numerics check out and the limitations are disclosed, so it deserves a real referee. read the letter →

arxiv 1908.06067 v3 pith:QNTWNE4O submitted 2019-08-16 hep-ph astro-ph.HE

Xiaoyong Chu , Camilo Garcia-Cely , Hitoshi Murayama This is my paper · ORCID
classification hep-phastro-ph.HE
keywords effectiverangetheorydarkmatterself-interactionsscatteringlengthvelocity-dependentcrosssectionYukawapotentialresonantSIDM
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 the velocity-dependent cross section for dark matter self-interactions in halos can be described, without committing to a specific particle model, by two parameters: the S-wave scattering length $a$ and the effective range $r_e$, together with the dark matter mass $m$. This matters because dwarf galaxies seem to require a self-interaction cross section of order $1$--$10\,\mathrm{cm^2/g}$ at velocities around $10\,\mathrm{km/s}$, while galaxy clusters bound it to below about $1\,\mathrm{cm^2/g}$ at velocities near $1000\,\mathrm{km/s}$, and translating this velocity dependence into model parameters has usually been done case by case. The paper shows that the same two-parameter formula reproduces the full numerical cross sections for Yukawa forces, for Breit-Wigner resonances, and for scattering through bound states or virtual levels. If right, this gives astrophysical simulations and model comparisons a common, minimal description of self-interacting dark matter.

What carries the argument

The load-bearing object is the effective-range expansion of the phase shift, $$ $k^{{2\ell+1}}$\cot\delta_\ell(k) \simeq -\frac{1}{a_\$ell^{{2\ell+1}}$} + \frac{1}{2} r_{e,\ell}^{2\ell-1} $k^{2}$, $$ truncated at order $k^2$; for S-wave scattering this produces Eq. (4), and written in terms of the relative velocity it becomes Eq. (8). The expansion is justified because $k^{2\ell+1}\cot\delta_\ell(k)$ is analytic at $k=0$ for finite-range interactions, so low-energy scattering depends only on the scattering length and effective range. The paper also derives a first-order differential equation for the phase shift, which makes the numerical extraction of $a$ and $r_e$ straightforward, and uses the amplitude poles $k_\pm = (i/a)(1 \pm \sqrt{1-2r_e/a})$ to connect those two parameters to bound states, virtual levels, and resonances. For cases where the minimal formula fails, a version with a background phase $e^{2ikR}$ describes antiresonances and sharp resonances.

What would settle it

Compute the exact S-wave cross section for an attractive Yukawa potential with $m_\phi=1\,\mathrm{MeV}$, $m=3\,\mathrm{GeV}$, and $\alpha=0.04$ at $v=1000\,\mathrm{km/s}$, and compare it with Eq. (8) using $a$ and $r_e$ extracted from low velocities; agreement would support the universality claim, while the mismatch in this classical regime would show where the two-parameter formula stops working.

Watch

Extended reading notes

Core claim

The central claim is that the effective-range approximation for non-relativistic scattering works for dark matter self-interactions across the astrophysically relevant velocity range. Concretely, the S-wave cross section is $$ \$\sigma$(v) = 4\pi $a^{2}$ \left( \left(1 - \frac{1}{8}\frac{r_e}{a}(m a v)^2\right)^2 + \frac{1}{4}(m a v)^2 \right)^{-1}, $$ where $v$ is the relative velocity, $m$ the dark matter mass, $a$ the scattering length, and $r_e$ the effective range; only $ma$ and the ratio $r_e/a$ control the velocity dependence. The paper verifies this formula against numerical solutions of the Schrödinger equation for attractive and repulsive Yukawa potentials, shows that it reproduces a Breit-Wigner resonance with an energy-dependent width, and traces its poles in the complex momentum plane to bound states, virtual levels, or resonances. Fitting the formula to semi-analytically extracted halo cross sections gives benchmark parameters such as $a=19.2\,\mathrm{fm}$, $r_e=0.01\,\mathrm{fm}$, and $m=14.9\,\mathrm{GeV}$ for the best-fit case.

Load-bearing premise

The formula assumes the dark-matter force has a short range, so that at every velocity the collision looks point-like and S-wave scattering dominates; for a light mediator whose range rivals the particle's de Broglie wavelength, the approximation breaks down.

Editorial extensions

If this is right

  • Astrophysical simulations can implement self-interacting dark matter by specifying $(a, r_e, m)$ instead of a full particle model, making different scenarios directly comparable in the same simulation.
  • For $\sigma/m \sim 10\,\mathrm{cm^2/g}$ at dwarf scales, dark matter masses below a few GeV are excluded by cluster observations unless the effective-range parameters are tuned to suppress high-velocity scattering.
  • The velocity dependence of the cross section determines only the ratio $r_e/a$ and the product $ma$; the signs of $a$ and $r_e$ cannot be fixed by $\sigma(v)$, so the underlying intermediate state (bound, virtual, or resonant) is not observable from the scattering data alone.
  • The formalism extends to subleading inelastic processes: a complex scattering length relates annihilation and elastic cross sections by $\sigma_{\mathrm{an},0}(k) \simeq \sigma_0(k) |\mathrm{Im}\,a|/(\mathrm{Re}\,a)^2$.

Reading between the lines

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

  • A measurement of $\sigma$ at two well-separated velocities (say dwarf and cluster scales) would fix $a$ and $r_e$; a third measurement would test whether nature's self-interactions are truly effective-range-like or require the improved antiresonance formula of Section V.
  • If future data show a cross section rising toward cluster scales, that would select $r_e/a > 1$ and hence a narrow-resonance interpretation; a flat cross section would point to contact-like models—a diagnostic that does not require building a UV-complete theory.
  • The classical-regime breakdown for light mediators implies that any model with mediator mass of a few MeV must be treated with the full partial-wave sum once cluster-scale velocities matter; the two-parameter formula is not a universal fit for such models.
  • The benchmark fits suggest that current data do not strongly constrain $r_e/a$, so upcoming simulations exploring sub-halo dynamics inside the Milky Way may be the first place where the effective range actually becomes measurable.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. This paper proposes the effective-range expansion (ERE) as a practical, model-independent parametrization of the velocity-dependent S-wave dark matter self-interaction cross section. The central formula, Eq. (8), expresses σ(v) in terms of the scattering length a, the effective range r_e, and the DM mass m, and the authors benchmark this approximation against exact numerical Schrödinger solutions for attractive and repulsive Yukawa potentials (Figs. 1, 8, 9), finding excellent agreement outside the classical regime. They then use the parametrization to derive astrophysical constraints (Figs. 4 and 5), fit the parameters to inferred halo cross sections (Fig. 6), interpret a and r_e in terms of concrete models (contact interaction, light mediator, bound/virtual states, resonances, SIMPs), and propose extensions for antiresonances, sharp resonances, and inelastic scattering (Sec. V). The paper is explicit about the domain of validity of the ERE, identifying the classical regime and antiresonances as places where the two-parameter formula fails.

Significance. If the claims hold, the paper provides a valuable model-independent tool for SIDM simulations and phenomenological studies, analogous to the role of the effective-range expansion in nuclear physics. Its strengths include: (i) the central formula is derived from the standard analytic expansion of k cot δ, not fitted to numerical data; (ii) the approximation is validated against exact quantum-mechanical results for both signs of the Yukawa potential; (iii) the limitations are disclosed rather than hidden, with concrete extensions in Sec. V; (iv) the insight that only |a| and r_e/a can be constrained by velocity-dependent cross sections is stated and demonstrated. The astrophysical fits in Fig. 6 are appropriately hedged as illustrative. The paper does not oversell the parametrization; it explicitly notes where it fails.

minor comments (5)
  1. [Sec. IV.C, Eq. (17)] Equation (17) as printed is dimensionally inconsistent: the left-hand side k has units of inverse length, while the right-hand side (i a/2)(1 ± sqrt(1 - 2 r_e/a)) has units of length. The correct solution of the pole condition k cot δ = i k is k = (i/r_e)(1 ± sqrt(1 - 2 r_e/a)), and the subsequent deuteron binding-energy check in Eq. (18) and Fig. 10 are consistent with the correct formula. Please correct Eq. (17) and any related text.
  2. [Sec. II, Fig. 1] The paper clearly acknowledges the classical regime (m_φ ≲ 5 MeV for the benchmark) where the effective-range approximation fails, but since the abstract and introduction emphasize the broad applicability of the two-parameter formula, consider adding an explicit caveat in the abstract that the parametrization applies to short-range interactions with kR ≪ 1.
  3. [Sec. III.B, Fig. 6] The fitting procedure that produces the benchmark parameter sets S1–S4 is not fully described; please state the likelihood or χ² definition and the velocity distribution parameters (e.g., v0 for each halo class) so that the fit is reproducible.
  4. [Sec. IV.B, Eq. (15)] In the Born regime, Eq. (15) gives r_e = 4/(m α), which is negative for a repulsive Yukawa potential (α < 0). The text does not comment on this sign, and since Eq. (8) depends only on r_e/a, negative values are physically acceptable; a short remark would avoid confusion.
  5. [Sec. V.A, Eq. (21)] In the improved antiresonance formula, the parameters a and r_e are not the standard scattering length and effective range, as the text notes; however, using different symbols (e.g., ã and r̃_e) would prevent readers from confusing them with the parameters of Sec. II.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective-range parametrization is a standard low-energy expansion benchmarked against numerical quantum mechanics, and the astrophysical fit is an application, not an input.

full rationale

The central claim, Eqs. (4) and (8), follows algebraically from the standard effective-range expansion k cot δ0 = -1/a + (1/2) r_e k^2 stated in Eq. (3), with a and r_e defined by the low-momentum expansion of the phase shift (Eq. (A6)-(A7)). No target cross section is fed into the derivation of the formula; the two-parameter velocity dependence is a consequence of the expansion, not a fit to the astrophysical data that Eq. (8) is later applied to. The paper explicitly benchmarks the approximation against numerically solved Schrödinger cross sections for Yukawa potentials in Figs. 8 and 9, and it discloses the classical-regime and antiresonance failures (Sec. II and Sec. V), which further shows that the validity claim is delimited rather than manufactured. The fit in Fig. 6 is an application of the parametrization to observational extractions and is labeled as a fit, not as a prediction from first principles. The self-citation [20] is used only for mapping narrow-resonance phenomenology and appears in a remark about extensions; the central effective-range formalism is supported by the independent, classical Bethe/Blatt-Jackson derivation reproduced in Appendix A, and its numerical validation is performed in this paper. No load-bearing argument reduces to a self-citation or to the fitted values, so the honest finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the standard quantum-mechanical effective-range expansion and on the domain assumptions that the interaction is short-ranged, elastic, S-wave dominated, and that velocities follow a Maxwell-Boltzmann distribution. The only numbers fitted to astrophysical data are a, r_e, and m in the benchmark fits of Fig. 6; these are parameters of the proposed description rather than inputs smuggled into the derivation.

free parameters (4)
  • Scattering length a = Benchmark values: S1 = 19.2 fm, S2 = 25.6 fm, S3 = 3.8 fm, S4 = 37.4 fm
    In the effective-range parametrization, a is one of two parameters describing the S-wave cross section. In the astrophysical application (Sec. III.B, Fig. 6), a is fit to inferred halo cross sections, so it is a free parameter of the data fit; in the model interpretations (Sec. IV) it is computed from potential parameters.
  • Effective range r_e = Benchmark values: S1 = 0.01 fm, S2 = 256.1 fm, S3 = -57 fm, S4 = -748.9 fm
    Second parameter of the effective-range expansion; fit to data in Fig. 6. The sign and magnitude determine velocity dependence via Eq. (8).
  • Dark matter mass m = Benchmark values: S1 = 14.9 GeV, S2 = 9.2 GeV, S3 = 1 GeV, S4 = 15.7 GeV
    Dark matter mass enters the reduced mass and the conversion from cross section to cross section per unit mass; treated as a free parameter in the fits of Fig. 6.
  • Hard-sphere radius R (antiresonance improvement) = First antiresonance of Yukawa: a approximately R = -0.85, r_e = 13.4 (units of m_phi^{-1})
    Introduced in Eq. (21) to improve the effective-range approximation at antiresonances; for the example in Fig. 11, R is chosen to match the exact result.
assumptions (7)
  • standard math Non-relativistic Schrodinger equation with a finite-range potential governs dark matter self-scattering.
    Used throughout Section II and Appendix A; justified because halo velocities are below about 10^{-2} c (Sec. II, footnote 2).
  • standard math The phase shift quantity k^{2l+1} cot delta_l is analytic at k = 0 and expands as Eq. (3), with higher-order terms negligible.
    Effective-range expansion (Bethe 1949, Blatt and Jackson 1949); the paper assumes kR << 1. Accuracy is validated by numerical comparison in Figs. 8 and 9.
  • domain assumption S-wave dominates the scattering in the velocity and parameter regions considered.
    Used to write Eq. (4) and the astrophysical cross sections; high partial waves become relevant in the classical regime, which the paper excludes (Sec. II).
  • domain assumption Interactions are elastic in halos, with inelastic processes subleading.
    Stated in Sec. II: unless stated otherwise, the authors assume inelastic processes are relatively weaker; inelastic scattering is deferred to Sec. V.C.
  • domain assumption Dark matter velocities in halos follow a Maxwell-Boltzmann distribution with cutoff, Eq. (10).
    Used for velocity averaging in Sec. III.B and Appendix B; the paper notes the distribution choice changes results only mildly.
  • domain assumption Observational constraints from galaxy clusters (sigma/m < 0.2-1 cm^2/g) and dwarf scales (1-10 cm^2/g) are external inputs.
    Used to derive astrophysical exclusion in Figs. 4-5 and benchmark fits in Fig. 6; sourced from refs [11-16].
  • domain assumption In the Born regime, perturbative expansion of the phase shift is valid (alpha m << m_phi).
    Used in Sec. IV.B.a to derive a = -m alpha / m_phi^2 and r_e = 4 / (m alpha); an external quantum mechanics result.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Practical and Consistent Parametrization of Dark Matter Self-Interactions." pith.science (2026). https://pith.science/paper/QNTWNE4O

@misc{pith2026190806067,
  author       = {Pith},
  title        = {Pith review of: A Practical and Consistent Parametrization of Dark Matter Self-Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNTWNE4O}},
  note         = {Machine review of arXiv:1908.06067}
}
read the original abstract

Self-interacting dark matter has been proposed to explain the apparent mass deficit in astrophysical small-scale halos, while observations from galaxy clusters suggest that the corresponding cross section depends on the velocity. Accounting for this is often believed to be highly model-dependent with studies mostly focusing on scenarios with light mediators. Based on the effective-range formalism, in this work we point out a model-independent approach which accurately approximates the velocity dependence of the self-interaction cross section with only two parameters. We illustrate how this parameterization can be simultaneously interpreted in various well-motivated scenarios, including self-interactions induced by Yukawa forces, Breit-Wigner resonances and bound states. We investigate the astrophysical implications and discuss how the approximation can be improved in certain special regimes where it works poorly.

Figures

Figures reproduced from arXiv: 1908.06067 by the authors.

Figure 1
Figure 1. Comparison of the numerical S-wave cross section per unit mass (solid) against the effective-range approximation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Self-scattering cross section as a function of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Contours of the cross section per unit of mass at cluster scales ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: Contours of σ/m within the range of 1 cm2 /g–10 cm2 /g at dwarf scales (v = 10 km/s). The gray areas represent the exclusion limit from cluster-scale observables (v = 2000 km/s), and are extended to the gray dashed curves if one requires σ/m . 0.2 cm2 /g at cluster sca…
Figure 6
Figure 6. Figure 6: Fit of DM self-interaction cross sections at vari [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Left: S-wave scattering length as a function of αm/mφ for the Yukawa (solid) and the Hulth´en (dashed) potentials. The case of a repulsive force (α < 0) is shown in gray. For the attractive case (α > 0), as αm/mφ increases, a different color is chosen after the phase s…
Figure 8
Figure 8. Figure 8: S-wave scattering cross section for the repulsive Yukawa potential in Eq. (5). The red solid lines are the nu￾merical results, while the dashed gray lines are given by the corresponding effective-range approximation. where |a| → ∞, and the bound states that are formed …
Figure 9
Figure 9. Figure 9: S-wave scattering cross section for the attractive Yukawa potential in Eq. (5). The orange solid lines are the numerical results, while the dashed gray lines are the corresponding effective-range approximation. Vertical cyan lines correspond to the first resonance at α…
Figure 10
Figure 10. Figure 10: The real and imaginary parts of k pole + in units of 1/|a| as a function of re/a for both a > 0 (left) and a < 0 (right). The corresponding physical states are labelled in texts. Note that in left panel the pole at re/a > 1/2 is unphysical (see footnote 9). states is …
Figure 12
Figure 12. Figure 12: Potential in Eq. (23), which exhibits unstable [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Same as Fig. 11 but for the potential of Eq. (23). The exact result is the solid orange line, the dashed line is the [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: Contours of the ratio of hσvi and σ(hvi)hvi as a function of amhvi and re/a. 3. The Hulth´en Potential In the main text, it has been mentioned that the Hulth´en potential V (r) = ±αδe−δr/(1 − e −δr) (A19) approximates well the Yukawa potential if one sets p δ = 2ζ(3)m…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Dark Neutrons as Dark Matter: Collisions in Halos and Direct Detection from Dark CP Violation

    hep-ph 2026-07 conditional novelty 7.0 of 10

    A non-zero topological angle in a confining dark sector induces CP-violating pion-baryon couplings that naturally generate velocity-dependent dark matter self-interactions and dark electric dipole moments for direct d...

  2. Pion dark matter in a $\theta$ vacuum: a thermal relic with sharp velocity-dependent self-interactions

    hep-ph 2025-08 conditional novelty 6.0 of 10

    A QCD-like dark sector with a theta vacuum and a dark photon portal can simultaneously reproduce the dark matter relic abundance and produce the sharply velocity-dependent self-interactions inferred from small-scale s...

  3. Effect of Cosmic Neutrino Background on the Dark Matter Self-interaction via Neutrino force

    hep-ph 2025-09 conditional novelty 5.0 of 10

    The cosmic neutrino background screens the neutrino-pair force between scalar dark matter particles for m_chi below the CnuB temperature, reshaping SIDM parameter space and eliminating Sommerfeld enhancement.

Reference graph

Works this paper leans on

84 extracted references · 16 canonical work pages · cited by 3 Pith papers

  1. [1]

    A simple method to find the phase shift 14

  2. [2]

    Then, Eqs

    The S-wave case Let us take 𝓁 = 0 and introduce uk(r) = rRk, 0(r). Then, Eqs. (A1) and (A2) read ( d2 dr2 +k2−mV (r) ) uk(r) = 0, (A12) and uk(0) = 0, uk(r)→ψk(r) = sin (kr +δ0) sinδ0 at r→∞ . (A13) Here we have chosen a convenient normalization factor for uk. In the following we will find it useful to employ 10 Simple changes of variable on Eq. (A4) allow...

  3. [3]

    practically no information could be obtained, from classical scattering experiments, on the shape of the potential

    The Hulth´ en Potential 16 B. Velocity-averaged cross sections 16 ∗Electronic address: xiaoyong.chu@oeaw.ac.at †Electronic address: camilo.garcia.cely@desy.de ‡Electronic address: hitoshi@berkeley.edu, hitoshi.murayama@ipmu.jp, Hamamatsu Professor I. INTRODUCTION In contrast to the ordinary substances found on Earth, more than three quarters of the matter...

    arXiv 1908

  4. [4]

    [35] presented a system- atic method for solving Eq

    A simple method to find the phase shift In the SIDM context, Ref. [35] presented a system- atic method for solving Eq. (A1). Here we would like to point out a simpler possibility that will not only pro- vide a powerful method to solve for the phase shift but will also allow us to define the scattering length and the effective range. Let us first define t𝓁,k(r)...

  5. [5]

    The tentative non-vanishing values of σ/m –extracted from observational data at various scales– can also be fit in terms of the effective-range parameters. See Fig. 6. Our results suggest that more precise measurements and 14 better extractions are needed to identify or constrain such parameters. Nevertheless, since the value ofre/a only af- fects the cross...

  6. [6]

    The Hulth´ en Potential In the main text, it has been mentioned that the Hulth´ en potential V (r) =±αδe−δr/(1−e−δr) (A19) approximates well the Yukawa potential if one sets δ =√ 2ζ(3)mφ, where α gives the coupling and mφ is the mediator mass of the Yukawa potential [78]. The advan- tage of employing the Hulth´ en potential is that its cor- responding Sch...

    work page

  7. [7]

    Aghanim et al

    N. Aghanim et al. (Planck) (2018), 1807.06209

    arXiv 2018

  8. [8]

    D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000), astro-ph/9909386

    arXiv 2000

Show all 84 references
  1. [9]

    Tulin and H.-B

    S. Tulin and H.-B. Yu, Phys. Rept. 730, 1 (2018), 1705.02358

    2018 arXiv

  2. [10]

    J. S. Bullock and M. Boylan-Kolchin, Ann. Rev. Astron. Astrophys. 55, 343 (2017), 1707.04256

    2017 arXiv

  3. [11]

    R. Dave, D. N. Spergel, P. J. Steinhardt, and B. D. Wan- delt, Astrophys. J. 547, 574 (2001), astro-ph/0006218

    2001 arXiv

  4. [12]

    Vogelsberger, J

    M. Vogelsberger, J. Zavala, and A. Loeb, Mon. Not. Roy. Astron. Soc. 423, 3740 (2012), 1201.5892

    2012 arXiv

  5. [13]

    Rocha, A

    M. Rocha, A. H. G. Peter, J. S. Bullock, M. Kaplinghat, S. Garrison-Kimmel, J. Onorbe, and L. A. Moustakas, Mon. Not. Roy. Astron. Soc. 430, 81 (2013), 1208.3025

    2013 arXiv

  6. [14]

    A. H. G. Peter, M. Rocha, J. S. Bullock, and M. Kapling- hat, Mon. Not. Roy. Astron. Soc. 430, 105 (2013), 1208.3026

    2013 arXiv

  7. [15]

    O. D. Elbert, J. S. Bullock, S. Garrison-Kimmel, M. Rocha, J. Oorbe, and A. H. G. Peter, Mon. Not. Roy. Astron. Soc. 453, 29 (2015), 1412.1477

    2015 arXiv

  8. [16]

    A. B. Fry, F. Governato, A. Pontzen, T. Quinn, M. Trem- mel, L. Anderson, H. Menon, A. M. Brooks, and J. Wad- 17 sley, Mon. Not. Roy. Astron. Soc. 452, 1468 (2015), 1501.00497

    2015 arXiv

  9. [17]

    S. W. Randall, M. Markevitch, D. Clowe, A. H. Gon- zalez, and M. Bradac, Astrophys. J. 679, 1173 (2008), 0704.0261

    2008 arXiv

  10. [18]

    Kaplinghat, S

    M. Kaplinghat, S. Tulin, and H.-B. Yu, Phys. Rev. Lett. 116, 041302 (2016), 1508.03339

    2016 arXiv

  11. [19]

    Robertson, R

    A. Robertson, R. Massey, and V. Eke, Mon. Not. Roy. Astron. Soc. 465, 569 (2017), 1605.04307

    2017 arXiv

  12. [20]

    Bondarenko, A

    K. Bondarenko, A. Boyarsky, T. Bringmann, and A. Sokolenko, JCAP 1804, 049 (2018), 1712.06602

    2018 arXiv

  13. [21]

    O. D. Elbert, J. S. Bullock, M. Kaplinghat, S. Garrison- Kimmel, A. S. Graus, and M. Rocha, ApJ 853, 109 (2018), 1609.08626

    2018 arXiv

  14. [22]

    Harvey, A

    D. Harvey, A. Robertson, R. Massey, and I. G. McCarthy (2018), 1812.06981

    2018 arXiv

  15. [23]

    J. L. Feng, M. Kaplinghat, and H.-B. Yu, Phys. Rev. Lett. 104, 151301 (2010), 0911.0422

    2010 arXiv

  16. [24]

    M. R. Buckley and P. J. Fox, Phys. Rev. D81, 083522 (2010), 0911.3898

    2010 arXiv

  17. [25]

    Tulin, H.-B

    S. Tulin, H.-B. Yu, and K. M. Zurek, Phys. Rev. Lett. 110, 111301 (2013), 1210.0900

    2013 arXiv

  18. [26]

    X. Chu, C. Garcia-Cely, and H. Murayama, Phys. Rev. Lett. 122, 071103 (2019), 1810.04709

    2019 arXiv

  19. [27]

    Ibe and H.-b

    M. Ibe and H.-b. Yu, Phys. Lett. B692, 70 (2010), 0912.5425

    2010 arXiv

  20. [28]

    Duch and B

    M. Duch and B. Grzadkowski, JHEP 09, 159 (2017), 1705.10777

    2017 arXiv

  21. [29]

    J. D. March-Russell and S. M. West, Phys. Lett. B676, 133 (2009), 0812.0559

    2009 arXiv

  22. [30]

    Braaten and H

    E. Braaten and H. W. Hammer, Phys. Rev. D88, 063511 (2013), 1303.4682

    2013 arXiv

  23. [31]

    J. M. Cline, Z. Liu, G. Moore, and W. Xue, Phys. Rev. D90, 015023 (2014), 1312.3325

    2014 arXiv

  24. [32]

    Braaten, D

    E. Braaten, D. Kang, and R. Laha, JHEP 11, 084 (2018), 1806.00609

    2018 arXiv

  25. [33]

    Mahbubani, M

    R. Mahbubani, M. Redi, and A. Tesi (2019), 1908.00538

    2019 arXiv

  26. [34]

    H. A. Bethe, Phys. Rev. 76, 38 (1949)

    1949

  27. [35]

    J. M. Blatt and J. D. Jackson, Phys. Rev. 76, 18 (1949)

    1949

  28. [36]

    H. P. Noyes, Ann. Rev. Nucl. Part. Sci. 22, 465 (1972)

    1972

  29. [37]

    V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Phys. Rev. C48, 792 (1993)

    1993

  30. [38]

    T. F. O’Malley, L. Spruch, and L. Rosenberg, Journal of Mathematical Physics 2, 491 (1961)

    1961

  31. [39]

    R. O. Berger and L. Spruch, Phys. Rev. 138, B1106 (1965)

    1965

  32. [40]

    van Haeringen and L

    H. van Haeringen and L. P. Kok, Phys. Rev. A26, 1218 (1982)

    1982

  33. [41]

    Tulin, H.-B

    S. Tulin, H.-B. Yu, and K. M. Zurek, Phys. Rev. D87, 115007 (2013), 1302.3898

    2013 arXiv

  34. [42]

    A. B. Newman, T. Treu, R. S. Ellis, and D. J. Sand, Astrophys. J. 765, 25 (2013), 1209.1392

    2013 arXiv

  35. [43]

    Kuzio de Naray, S

    R. Kuzio de Naray, S. S. McGaugh, and W. J. G. de Blok, Astrophys. J. 676, 920 (2008), 0712.0860

    2008 arXiv

  36. [44]

    S.-H. Oh, W. J. G. de Blok, E. Brinks, F. Walter, and R. C. Kennicutt, Jr, Astron. J. 141, 193 (2011), 1011.0899

    2011 arXiv

  37. [45]

    Valli and H.-B

    M. Valli and H.-B. Yu, Nat. Astron. 2, 907 (2018), 1711.03502

    2018 arXiv

  38. [46]

    Sokolenko, K

    A. Sokolenko, K. Bondarenko, T. Brinckmann, J. Zavala, M. Vogelsberger, T. Bringmann, and A. Boyarsky, JCAP 1812, 038 (2018), 1806.11539

    2018 arXiv

  39. [47]

    Kummer, M

    J. Kummer, M. Brggen, K. Dolag, F. Kahlhoefer, and K. Schmidt-Hoberg (2019), 1902.02330

    2019 arXiv

  40. [48]

    Kaplinghat, M

    M. Kaplinghat, M. Valli, and H.-B. Yu (2019), 1904.04939

    2019 arXiv

  41. [49]

    Sameie, H.-B

    O. Sameie, H.-B. Yu, L. V. Sales, M. Vogelsberger, and J. Zavala (2019), 1904.07872

    2019 arXiv

  42. [50]

    Kahlhoefer, M

    F. Kahlhoefer, M. Kaplinghat, T. R. Slatyer, and C.-L. Wu (2019), 1904.10539

    2019 arXiv

  43. [51]

    A. D. Dolgov, Yad. Fiz. 31, 1522 (1980)

    1980

  44. [52]

    E. D. Carlson, M. E. Machacek, and L. J. Hall, Astro- phys. J. 398, 43 (1992)

    1992

  45. [53]

    Hochberg, E

    Y. Hochberg, E. Kuflik, T. Volansky, and J. G. Wacker, Phys. Rev. Lett. 113, 171301 (2014), 1402.5143

    2014 arXiv

  46. [54]

    L. D. Landau and E. M. Lifshits, Quantum Mechanics , vol. v.3 of Course of Theoretical Physics (Butterworth- Heinemann, Oxford, 1991), ISBN 9780750635394

    1991

  47. [55]

    A. G. Sitenko and P. J. Shepherd, Lectures in scattering theory, Internat. Ser. Mono. Natural Philos. (Pergamon, Oxford, 1971), trans. from the Russian, URL https:// cds.cern.ch/record/102667

    1971

  48. [56]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. 137, B672 (1965)

    1965

  49. [57]

    Hochberg, E

    Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky, and J. G. Wacker, Phys. Rev. Lett. 115, 021301 (2015), 1411.3727

    2015 arXiv

  50. [58]

    Yamanaka, S

    N. Yamanaka, S. Fujibayashi, S. Gongyo, and H. Iida (2014), 1411.2172

    2014 arXiv

  51. [59]

    H. M. Lee and M.-S. Seo, Phys. Lett. B748, 316 (2015), 1504.00745

    2015 arXiv

  52. [60]

    Hansen, K

    M. Hansen, K. Langæble, and F. Sannino, Phys. Rev. D92, 075036 (2015), 1507.01590

    2015 arXiv

  53. [61]

    Bernal and X

    N. Bernal and X. Chu, JCAP 1601, 006 (2016), 1510.08527

    2016 arXiv

  54. [62]

    Bernal, C

    N. Bernal, C. Garcia-Cely, and R. Rosenfeld, JCAP 1504, 012 (2015), 1501.01973

    2015 arXiv

  55. [63]

    Bernal, X

    N. Bernal, X. Chu, C. Garcia-Cely, T. Hambye, and B. Zaldivar, JCAP 1603, 018 (2016), 1510.08063

    2016 arXiv

  56. [64]

    Choi and H

    S.-M. Choi and H. M. Lee, JHEP 09, 063 (2015), 1505.00960

    2015 arXiv

  57. [65]

    Hochberg, E

    Y. Hochberg, E. Kuflik, and H. Murayama, JHEP 05, 090 (2016), 1512.07917

    2016 arXiv

  58. [66]

    Kuflik, M

    E. Kuflik, M. Perelstein, N. R.-L. Lorier, and Y.-D. Tsai, Phys. Rev. Lett. 116, 221302 (2016), 1512.04545

    2016 arXiv

  59. [67]

    Choi and H

    S.-M. Choi and H. M. Lee, Phys. Lett. B758, 47 (2016), 1601.03566

    2016 arXiv

  60. [68]

    Choi, Y.-J

    S.-M. Choi, Y.-J. Kang, and H. M. Lee, JHEP 12, 099 (2016), 1610.04748

    2016 arXiv

  61. [69]

    Pappadopulo, J

    D. Pappadopulo, J. T. Ruderman, and G. Trevisan, Phys. Rev. D94, 035005 (2016), 1602.04219

    2016 arXiv

  62. [70]

    Farina, D

    M. Farina, D. Pappadopulo, J. T. Ruderman, and G. Tre- visan, JHEP 12, 039 (2016), 1607.03108

    2016 arXiv

  63. [71]

    U. K. Dey, T. N. Maity, and T. S. Ray (2016), 1612.09074

    2016 arXiv

  64. [72]

    Cline, H

    J. Cline, H. Liu, T. Slatyer, and W. Xue (2017), 1702.07716

    2017 arXiv

  65. [73]

    S.-M. Choi, H. M. Lee, and M.-S. Seo, JHEP 04, 154 (2017), 1702.07860

    2017 arXiv

  66. [74]

    S.-M. Choi, Y. Hochberg, E. Kuflik, H. M. Lee, Y. Mam- brini, H. Murayama, and M. Pierre (2017), 1707.01434

    2017 arXiv

  67. [75]

    Chu and C

    X. Chu and C. Garcia-Cely, Phys. Rev. D96, 103519 (2017), 1708.06764

    2017 arXiv

  68. [76]

    S.-M. Choi, H. M. Lee, P. Ko, and A. Natale, Phys. Rev. D98, 015034 (2018), 1801.07726

    2018 arXiv

  69. [77]

    Berlin, N

    A. Berlin, N. Blinov, S. Gori, P. Schuster, and N. Toro, Phys. Rev. D97, 055033 (2018), 1801.05805

    2018 arXiv

  70. [78]

    S.-M. Choi, H. M. Lee, Y. Mambrini, and M. Pierre (2019), 1904.04109. 18

    2019 arXiv

  71. [79]

    Bhattacharya, P

    S. Bhattacharya, P. Ghosh, and S. Verma (2019), 1904.07562

    2019 arXiv

  72. [80]

    McGehee, H

    R. McGehee, H. Murayama, and Y.-D. Tsai, in prepara- tion (2019)

    2019

  73. [81]

    K. Blum, R. Sato, and T. R. Slatyer, JCAP 1606, 021 (2016), 1603.01383

    2016 arXiv

  74. [82]

    Mitroy, Phys

    J. Mitroy, Phys. Rev. A 66, 022716 (2002)

    2002

  75. [83]

    D. R. Phillips and T. D. Cohen, Phys. Lett. B390, 7 (1997), nucl-th/9607048

    1997 arXiv

  76. [84]

    Cassel, J

    S. Cassel, J. Phys. G37, 105009 (2010), 0903.5307

    2010 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.