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Multiple Soft Scatterings in Scalar Dark Matter Freeze-In

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Including multiple soft plasma scatterings changes scalar dark-matter freeze-in relic densities by up to 27 percent.

desk verdict First scalar LPM equation combined with 1PI-resummed freeze-in rate; the 1-27% effect is real but conditional on a switch-off whose uncertainty at small splitting is comparable to the effect—and the authors say so. read the letter →

arxiv 2506.11185 v1 pith:HGAQHBU2 submitted 2025-06-12 hep-ph

Mathias Becker , Maria Jose Fernandez Lozano , Julia Harz , Carlos Tamarit This is my paper
classification hep-ph
keywords scalardarkmatterfreeze-inLandau-Pomeranchuk-Migdaleffect1PIresummationthermalfieldtheoryrelicdensitymultiplesoftscatteringFIMP
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 improves the calculation of scalar dark matter production by freeze-in, the process in which dark matter accumulates from rare decays and scatterings without ever reaching thermal equilibrium. It claims that a previously missing piece of the production rate, the Landau-Pomeranchuk-Migdal (LPM) effect of coherent multiple soft scatterings in the plasma, is a leading-order contribution of size $g^2 T$ and must be added to the earlier calculation based on one-particle-irreducible (1PI) resummed propagators. The paper derives for the first time an integral equation for the LPM rate of a scalar particle, Eq. (3.11), and combines it with the 1PI result. The resulting relic density is 1 to 27 percent larger than the 1PI-only result, with the largest enhancement for larger gauge couplings and smaller mass splittings. The result matters because most freeze-in predictions so far omitted this leading-order effect, and the paper calibrates simpler approximations against the improved rate.

What carries the argument

The load-bearing object is the scalar LPM integral equation, Eq. (3.11), for the function $\chi(\vec{k}_\perp)$, together with the spectral self-energy Eq. (3.10) that it feeds. The equation balances the pole-distance term $\epsilon(\vec{k}_\perp)$, which contains the vacuum and thermal masses of the mediator and the Standard-Model fermion, against a collision integral with kernel $K_i(\vec{q}_\perp)=1/\vec{q}_\perp^2 - 1/(\vec{q}_\perp^2 + m_{D i}^2)$, so that arbitrarily many soft scatterings are summed at the same parametric order $g^2 T$ as the one-loop rate. The companion switch-off function $f_\zeta(m_{F,0}/T)$, defined from the derivative of a fermionic thermal loop function, suppresses the LPM contribution as the temperature approaches the vacuum mass scale, where the collinear scaling $p_\perp\sim gT$ used in the derivation fails.

What would settle it

Compute the LPM production rate with the physical vacuum masses of the mediator and dark matter kept in the $T\sim M$ regime, without invoking a switch-off function; if the resulting relic density differs from the thermal-function result by more than the 30 percent spread among switch-off schemes, the paper's quoted 1 to 27 percent LPM contribution would not hold in that regime. This is precisely the calculation the paper identifies as the missing step.

Watch

Extended reading notes

Core claim

The central claim is that the LPM effect contributes at leading order to scalar dark matter production and therefore changes the predicted relic density. The paper's new object is Eq. (3.11), the scalar analogue of the LPM resummation: an integral equation for $\chi(\vec{k}_\perp)$ in which the free-propagation energy difference $\epsilon(\vec{k}_\perp)$ is balanced against a collision term integrating over soft transverse gauge-boson momenta with a Debye-screened kernel. The spectral self-energy built from $\chi$ in Eq. (3.10) resums an infinite ladder of soft scatterings between the two fermion lines of the production diagram. Because this LPM derivation assumes collinear ultra-relativistic kinematics, the paper attaches a phenomenological switch-off function $f_\zeta(m_{F,0}/T)$ and subtracts the LPM Born limit to avoid double counting the decay. For five charge assignments, over $G\in[0.4,1.6]$ and $\delta\in[0.1,10]$, the LPM contribution increases the relic density by 1 to 27 percent, and the paper reports that the choice of switch-off scheme introduces up to 30 percent uncertainty for small mass splittings.

Load-bearing premise

The load-bearing premise is that the phenomenological switch-off function used to turn off the LPM contribution at low temperature reliably interpolates between the ultra-relativistic regime, where the LPM calculation is valid, and the non-relativistic regime, where the 1PI rate should take over, even though the paper states that no available switch-off method is strictly valid in the $T\sim M$ region most relevant for freeze-in and that the choice among schemes changes the final relic density by up to 30 percent for small mass splittings.

Editorial extensions

If this is right

  • Scalar singlet dark matter produced from gauge-charged fermions should have its freeze-in relic density revised upward by 1 to 27 percent, with the correction largest for strongly coupled, nearly degenerate dark sectors.
  • The scalar LPM integral equation provides a reusable tool for any scalar production process mediated by fermions in a plasma, not only dark matter freeze-in.
  • Common semi-classical treatments are now benchmarked against the improved rate: vacuum-mass decays stay within about 10 percent for large mass splittings, while thermal-mass decays plus scatterings can deviate by roughly -30 to +25 percent.
  • The HTL-resummed 1PI rate without LPM happens to match the full result to about 10 percent across the parameter scan, because its overestimate is offset by the missing LPM enhancement.
  • The LPM rate depends on the specific gauge charges of the mediator, unlike the 1PI one-loop rate, so relic-density predictions must be made separately for each realization such as $q_L$, $e_R$, or $d_R$.

Reading between the lines

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

  • If the switch-off ambiguity is the dominant residual uncertainty, then freeze-in relic densities in the small-splitting, large-coupling region should be quoted as a band rather than a single number.
  • A full LPM calculation that keeps the physical mediator and dark matter masses in the $T\sim M$ regime, which the paper leaves to future work, could move the central relic density outside the quoted 1 to 27 percent range.
  • The thermal switch-off function, tied to the scalar thermal mass, might also apply to other scalar self-energy processes such as axion-like particle production from a thermal plasma.
  • The model dependence through gauge charges suggests that the same resummation could be used to rank mediator representations by production efficiency before scanning couplings.
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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 extends the 1PI-resummed freeze-in calculation for scalar dark matter of Ref. [1] by including the Landau-Pomeranchuk-Migdal (LPM) effect. It derives a new integral equation for the LPM rate of a scalar particle, Eq. (3.11), computes its Born limit, and combines the LPM rate with the previous 1PI rate using a phenomenological switch-off function f_zeta defined in Eq. (3.25). On this basis the authors report that the LPM contribution changes the scalar DM relic density by 1--27%, with larger effects for larger gauge couplings and smaller mass splittings, and they compare their results with HTL-based and semi-classical Boltzmann approaches.

Significance. If the central numerical claim is accepted, the paper would be a useful step beyond the 1PI-resummed treatment of Ref. [1], providing the first explicit LPM resummation equation for a scalar FIMP and a detailed comparison of common approximations. The paper is transparent about the ad hoc nature of the switch-off and quantifies its uncertainty by comparing three schemes. It also provides fit functions and a scan over model realizations, which are valuable for phenomenological use. However, the central quantitative result, the 1--27% range in Sec. 4.2, is conditioned on a switch-off function whose theoretical status is not derived, and the paper's own comparison shows that the associated uncertainty is comparable to the claimed effect. The derivation of the LPM equation is sketched rather than fully self-contained, and the regime T ~ m_F0 where freeze-in is most sensitive is precisely the regime where the collinear LPM approximation is not valid.

major comments (3)
  1. [Sec. 3.4, Eq. (3.25); Sec. 4.2, Fig. 10] The switch-off function f_zeta is defined as the normalized derivative of the fermionic thermal loop function, which controls the real part of the scalar self-energy in the static limit, yet it is used to suppress the imaginary part that constitutes the LPM rate. No derivation or justification is given for why this real-part suppression is an adequate proxy for the LPM imaginary-part suppression. The paper's own comparison (Sec. 3.4, Fig. 5) shows that switching between f_zeta, f_kappa and the integrand-level interpolation changes the final relic density by about 30% at delta=0.1 and about 9% at delta=10, while the claimed LPM effect in the same regions is 27% and 8%. Therefore the headline range in Sec. 4.2 is not a robust measurement of the LPM contribution; it is a statement conditioned on one specific interpolation, and the central quantitative claim should either be accompanied by an explicit systematic uncertainty band or be reformulated as a range that accounts for the switch-off ambiguity.
  2. [Sec. 3.1, Eq. (3.11); Sec. 3.4] The LPM integral equation is derived under the power-counting assumption p_parallel ~ T, p_perp ~ gT, and masses ~ gT, as stated in Eq. (3.2). This assumption is not satisfied when m_F0 ~ T, and the freeze-in integrand is non-negligible in exactly this intermediate regime, as the paper itself emphasizes at the beginning of Sec. 1 and in Sec. 3.4. The paper concedes that none of the three switch-off schemes is strictly valid in the T ~ m_F0 region most relevant for freeze-in. Without a controlled treatment of that regime, the quantitative impact on the relic density, including the claimed 8--27% effects, is not established to the precision that the abstract and conclusion suggest. The authors should clearly restrict the domain of validity of the numerical results or supply an additional systematic check, such as a comparison with a calculation that retains vacuum masses in the LPM kernel, even in a simplified model.
  3. [Sec. 3.2, Eq. (3.18); Sec. 3.3, Eq. (3.22)] The Born-limit subtraction in Eq. (3.22) is an essential step: it removes the collinear decay contribution from the LPM rate to avoid double counting with the 1PI rate. The paper states in Sec. 3.3 that the LPM rate and its Born limit coincide in the non-relativistic regime, and treats this as a consistency check. However, both quantities are computed from the same collinear approximation, so this agreement does not independently validate the subtraction in the transition region. The authors should clarify what independent check, if any, exists for the subtraction at intermediate temperatures, and how sensitive the final relic density is to the exact form of the subtraction as opposed to the switch-off function.
minor comments (4)
  1. [Sec. 2] In the sentence 'such as the LMP effect' the acronym should be 'LPM', not 'LMP'.
  2. [Sec. 3.3] In the text following Eq. (3.22), the switch-off function is written as f(m_F,0F/T); this appears to be a typo for f(m_F,0/T).
  3. [Fig. 5] The caption states that the plots consider a mediator of type d_R with G=1.2 and mass splitting delta=10, but the left and right panels correspond to different mass splittings (delta=0.1 and delta=10, respectively, as discussed in the text). The caption should be clarified.
  4. [Sec. 3.4, Sec. 5] The smooth interpolation method is attributed to Ref. [34] in Sec. 3.4 and to Ref. [56] in Sec. 5; please verify that these citations refer to the correct and distinct works, and make the attribution consistent.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the scalar LPM rate is derived from standard collinear thermal-field-theory ingredients, the switch-off function is an openly acknowledged proxy with quantified uncertainty, and the 1PI baseline from prior same-author work is an independent published input.

full rationale

The central new result, Eq. (3.11), is obtained by adapting the established Besak-Bödeker LPM recursion relation to a scalar external state, with the hard-vertex factor in Eq. (3.5) computed from explicit fermionic spinors in the collinear limit. The inputs are standard thermal propagators, Debye masses, and distribution functions; no target relic-density quantity is used to define the integral equation. The subsequent rate expression Eq. (3.10) is a direct quadrature over the solved function χ, and the Born limit in Eq. (3.18) is an independent check obtained by dropping the collision kernel, matching the decay kinematics of the model rather than importing the final result. The combination in Eq. (3.22) is a physical double-counting subtraction: γ_LPM includes the collinear decay Born piece, which is already contained in γ_1PI, so it is removed. The switch-off function f_ζ of Eq. (3.25) is admittedly a proxy: the paper states that the derivative of the thermal loop function is used 'as a proxy for the suppression of its imaginary part' and explicitly warns that none of the three interpolation schemes is strictly valid at T ~ m_F0. This is a transparency about approximation uncertainty, not a self-definitional reduction; the spread among switch-off choices (about 30% at small splittings) is quantified in Sec. 3.4 rather than hidden. The 1PI baseline is taken from Ref. [1], which shares three authors with this paper; however, that citation is an independently published, checkable calculation and serves as the benchmark being extended, not as the justification for the new LPM term. The linear fit in Eq. (4.1) is a compact fit to the authors' own computed LPM rates and is not relabeled as an independent prediction. No step in the chain equates the derived rate to an input assumption or to a fitted parameter by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The calculation is built on standard finite-temperature field theory, the LPM power counting of Ref. [28], and the recursion relation of Ref. [36]. Its quantitative outcome additionally depends on a hand-picked switch-off function and on the same authors' earlier 1PI rate. There are no new physical particles or forces, but there is one functional freedom with a direct impact on the headline number.

free parameters (2)
  • switch-off function f_zeta(m_F,0/T)
    Phenomenological function (Eq. 3.25) chosen as the most conservative among three options; the final LPM contribution to the relic density depends on this choice, with up to 30 percent variation at small mass splittings (Sec. 3.4).
  • LPM rate fit coefficients a and b = a in [-4.0e-5, -8.1e-8], b in [4.7e-4, 6.7e-4] (Table 1)
    Fit parameters in Eq. (4.1) for gamma_LPM/(y_DM^2 T^4) at z=0.01 for five mediator realizations; used to compress numerical data, not derived from first principles.
assumptions (4)
  • domain assumption LPM power-counting hierarchy p_parallel ~ T, p_perp ~ gT, p^2 ~ g^2 T^2 (Eq. 3.2)
    The collinear nearly-light-like kinematics justify resumming ladder diagrams; it fails when vacuum masses become comparable to T, which is why a switch-off is needed (Secs. 3.1 and 3.3).
  • domain assumption Validity of adapting the fermionic LPM recursion relation of Ref. [36] to a scalar external leg
    Equation (3.11) is the scalar analog obtained by replacing spinor structures with hard-vertex factors; the paper provides no independent derivation from first principles beyond this adaptation.
  • domain assumption The 1PI-resummed one-loop rate of Ref. [1] is the correct baseline
    The paper uses gamma_1PI from the same authors' earlier work as the non-LPM contribution; residual gauge dependence is reported at the percent level but not eliminated.
  • ad hoc to paper Born-limit subtraction and switch-off prescription in Eq. (3.22)
    The subtraction of gamma_LPM_Born and multiplication by f_zeta are designed to avoid double counting and to remove the unphysical low-temperature growth; the paper states this is phenomenological and not derived.

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Pith. "Pith review of Multiple Soft Scatterings in Scalar Dark Matter Freeze-In." pith.science (2026). https://pith.science/paper/HGAQHBU2

@misc{pith2026250611185,
  author       = {Pith},
  title        = {Pith review of: Multiple Soft Scatterings in Scalar Dark Matter Freeze-In},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGAQHBU2}},
  note         = {Machine review of arXiv:2506.11185}
}
abstract

We present an improved calculation of the freeze-in production rate for scalar dark matter (DM) from a gauge-charged parent particle via a renormalizable interaction. Building on the previously developed 1PI-resummed framework to accurately capture the relevant regime $T \sim M$, we expand the analysis to include the Landau-Pomeranchuk-Migdal (LPM) effect, which contributes at leading order $g^2 T$ to the interaction rate in the ultra-relativistic limit. To this end, we derive an equation for the LPM rate of a scalar particle for the first time and combine it with the previous 1PI results, providing a new state-of-the art calculation. In contrast to the 1PI results, the LPM treatment neglects vacuum mass scales such that a phenomenological switch-off function between the ultra-relativistic and non-relativistic regime is required. We propose a new function motivated by a thermal loop contribution and compare it to other approaches in the literature, quantifying the resulting uncertainty of this method. Depending on the gauge coupling and mass splitting between DM and mediator particles, the LPM effect contributes between 1% and 27% to the relic density, with the impact increasing for larger gauge couplings and smaller mass splittings. Additionally, we compare our results to commonly used semi-classical Boltzmann approaches. For instance, when these include decays and scatterings regulated with thermal masses, we find deviations ranging from -30% to +20% depending on the mass splitting. Finally, we compare to results based on hard-thermal-loop (HTL) approximations.

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Reviewed August 7, 2026 · model on record in the stance chip above.