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REVIEW 2 major objections 6 minor 35 references

Excited bound-state formation obeys an exact rescaling law, so one reference table covers arbitrary mass and coupling.

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2026-08-03 13:34 UTC pith:UGNGK35M

load-bearing objection Exact rescaling law is real and useful; the high-x extrapolation and reproducibility need work before I'd trust the full coverage. the 2 major comments →

arxiv 2512.23812 v2 pith:UGNGK35M submitted 2025-12-29 hep-ph

BSFfast: Rapid computation of bound-state effects on annihilation in the early Universe

Tobias Binder , Mathias Garny , Jan Heisig , Stefan Lederer This is my paper
classification hep-ph
keywords bound-state formationeffective annihilation cross sectionrescaling relationsexcited bound statesdark matter freeze-outsuperWIMPdipole transitionsearly Universe
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to make bound-state formation (BSF) effects on early-Universe annihilation cheap enough for repeated use inside Boltzmann solvers. It claims that, when the gauge coupling is frozen, the full thermally averaged effective BSF cross section — including all excited bound states and radiative bound-to-bound transitions — obeys an exact two-parameter rescaling law, so a single reference table in temperature yields the cross section for any particle mass and coupling strength. For theories with running couplings, such as Standard Model QCD, the paper provides precomputed two-dimensional tables plus an approximate rescaling prescription evaluated at the potential scale, validated to about 15% on one model class. The result matters because highly excited bound states can dominate the effective annihilation rate at late times, and previously including them required computationally expensive on-the-fly evaluations that made parameter scans prohibitive.

Core claim

The central claim is Eq. (39): under frozen couplings, ⟨σv⟩_eff,BSF^rescaled(x; m, α) = (m0 α / (m α0))^2 × ⟨σv⟩_eff,BSF(x (α/α0)^2; m0, α0). Because the ionization, bound-to-bound transition, and decay rates entering the bound-state network all scale linearly with mass, the depletion efficiencies are mass-independent; and because a coupling rescaling r = α'/α shifts the thermal variable as x' = x r^2 while multiplying all rates by a common factor r^5 that cancels in the efficiency ratios, the temperature dependence also collapses. The paper presents this as an exact analytic result for dipole-mediated transitions, not a numerical fit. It further shows that, for dark QCD with constant coupli

What carries the argument

The load-bearing identity is the rescaling law of Eq. (39) for the effective thermally averaged BSF cross section, together with the transition-network depletion efficiencies R_i of Eq. (4). The law holds because all rates entering R_i — ionization, decay, and radiative bound-to-bound transitions — share the same mass and coupling scalings, so the ratios that define R_i are invariant under the combined rescaling (m, α) → (m', α') once x is rescaled by (α'/α)^2. Only the overall α^2 prefactor of the dipole cross section survives. For non-Abelian models the dipole matrix elements distinguish three effective couplings — emission, scattering-state potential, and bound-state potential — and the p

Load-bearing premise

The dark-QCD coverage rests on the unverified assumption that evaluating the frozen-coupling rescaling law at a coupling taken at the scale sqrt(mT) reproduces the full running-coupling result to roughly 15% everywhere in the dark-QCD parameter plane, not only in the single model class shown.

What would settle it

For any frozen-coupling model, compute the full effective BSF cross section directly at three masses and three couplings without using Eq. (39), plot (m0 α/(m α0))^2 ⟨σv⟩_eff,BSF(x;m,α) against x (α/α0)^2, and check whether all curves collapse onto one universal line; any spread beyond integration error falsifies the exact rescaling claim. For the running-coupling approximation, repeat the Fig. 2 comparison between full two-dimensional tables and Eq. (42) for a dark-QCD model with different particle content; agreement much worse than the stated ~15% would falsify the empirical dQCD coverage.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Dark-matter relic-density calculations can include excited bound states up to n = 100 at negligible runtime, removing the computational bottleneck that previously made such studies prohibitive.
  • For frozen-coupling models (dark QED, dark QCD), a single one-dimensional table in x exactly covers arbitrary masses and couplings, and the same table can be used to chart unitarity-violating regions of the parameter plane.
  • For Standard Model QCD models, two-dimensional tables in mass and temperature provide fast evaluation of the effective BSF cross section for coloured mediators in t-channel dark matter and superWIMP setups.
  • The superWIMP illustration shows that including excited bound states shifts the predicted dark-matter mass by roughly an order of magnitude for late-decaying mediators, and changes which parts of the parameter space are excluded by Lyman-α constraints on warm dark matter.
  • The provided interpolation interfaces make repeated evaluation fast enough for integration into existing Boltzmann solvers, turning previously impractical parameter scans into routine computations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because Eq. (39) is exact at frozen coupling, any direct full computation that disagrees with the rescaled table would signal an error or a missing process in the rate network, making the rescaling a strong internal consistency check for BSF implementations.
  • The running-coupling approximation of Eq. (42) has been validated on only one model class; testing it against full two-dimensional tables for other dark-QCD particle content would show whether the O(1) constant C needs re-optimisation or whether the approximation breaks down.
  • The same argument that produces the rescaling law — factorising the cross section as α^2 f(α/v) times a function of x v^2 — suggests the law could extend to other radiative 2-to-2 capture processes, while higher-multipole transitions would break it; the size of the breaking could be estimated from the same parametric structure.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper presents BSFfast, a lightweight numerical tool that provides precomputed, tabulated values of the effective bound-state formation cross section ⟨σv⟩_eff,BSF for non-relativistic X-X̄ pairs with long-range U(1) or SU(3) interactions. The covered models include SM QCD and QED as well as dark QCD/QED, for scalar and fermionic constituents, with excited bound states up to principal quantum number n=100 and, where applicable, the full network of radiative bound-to-bound transitions (Sec. 4.1, Table 1). The central new analytic result is the exact rescaling law of Sec. 3.1: under frozen couplings, ⟨σv⟩_eff,BSF(x; m, α) = (m0 α/(m α0))² ⟨σv⟩_eff,BSF(x(α/α0)²; m0, α0) (Eq. (39)), so that a single one-dimensional reference table suffices for arbitrary mass and coupling in the dark-QED/QCD models. For SM QCD, where the coupling runs, the paper relies on direct two-dimensional tabulation in (x, m); Sec. 3.2 additionally proposes an empirical frozen-coupling approximation (Eq. (42)) as an illustration. The tool ships with C, Python, and Mathematica interpolation interfaces and is applied to a superWIMP scenario with a coloured mediator (Sec. 5), reproducing the qualitative behaviour of Ref. [15] at a greatly reduced computational cost.

Significance. If its advertised accuracy holds, BSFfast addresses a genuine bottleneck: including excited bound states and transition networks in Boltzmann solvers is numerically expensive, and the rescaling identities (Eqs. (28), (38), (39)) are exact, parameter-free analytic results of independent value. I checked the derivation of Sec. 3.1: with (σv)^i_BSF ∝ α² I_R and the scaling I_R(v; α') = r^{-5} I_R(v/r; α), ω(v; α') = r² ω(v/r; α), the thermal average satisfies Eq. (32); all rates scale as r⁵, so the efficiency factors R_i cancel and Eq. (39) is internally consistent under the stated dipole/frozen-coupling assumptions. The n=100 convergence check within the tabulated x range, the conservative unitarity-warning design (Sec. 4.3), the public code, and the three-language interfaces are concrete strengths. The main weakness is that the claimed coverage beyond x=10⁶ rests on an unvalidated extrapolation (footnote 4), which affects the arbitrary-α coverage for α > α0 and the superWIMP application. The closed-form rate expressions are taken from prior literature, so there is no circularity in the derivation.

major comments (2)
  1. [Sec. 4.2 (footnote 4); Sec. 3.1; Sec. 5] The extrapolation ⟨σv⟩ ≃ a x^b for x > 10^6 is introduced as 'controlled asymptotic behaviour' and is nowhere validated. It is load-bearing: (i) for dQED/dQCD with α > α0, Eq. (39) maps a requested x to x' = x(α/α0)^2 > 10^6 in the reference table, so a large part of the advertised arbitrary-α coverage is the extrapolated result; (ii) the superWIMP illustration (Sec. 5) uses m = 5×10^6–10^7 GeV and T down to ~1 GeV, i.e. x ≈ 5×10^6–10^7, beyond the tabulated range, and the n ≤ 100 convergence check is only stated for x ≤ 10^6. Since n up to 4000 is already computed for Fig. 5, a direct validation of the extrapolation (e.g., n ≤ 4000 at x up to 10^7, or a benchmark against the on-the-fly computation of Ref. [15]) is feasible and should be reported, or the advertised coverage restricted to x ≤ 10^6.
  2. [Sec. 5, Fig. 6] No quantitative benchmark against the full computation is shown for the tool's output. Sec. 5 states that the superWIMP results 'qualitatively' agree with Ref. [15], and Fig. 6 uses BSFfast interpolations (including the extrapolation region) to draw relic-density contours and Lyman-α statements. Since the authors have access to the on-the-fly machinery of Ref. [15], a direct comparison of ⟨σv⟩_eff,BSF and final Ω_χ h² for at least the bottom-philic benchmark (Q = −1/3, m_q̃ = 5×10^6 GeV) would establish the precision of the tables, interpolation, and extrapolation as a whole. The current text asserts accuracy rather than demonstrating it.
minor comments (6)
  1. [Sec. 3.2 / Sec. 4.1 / Sec. 6] Clarify the status of Eq. (42). The released QCD tables are, per Sec. 6, direct 2D tabulations, and dQCD/dQED use constant couplings (Sec. 2.2); Eq. (42) with the tunable O(1) constant C is illustrated on one model class (Fig. 2, ≲15%). The sentence in Sec. 4.1 on 'an approximate correction to restore running effects' should state explicitly that this is not part of the shipped tables.
  2. [Table 1] Last row: 'dQED-SnoTr and dQED-SnoTrare' contains a typo; presumably dQED-FnoTr is meant. Also, for the QCD rows the 'rescaling parameters' column could note that the coupling is fixed by the SM rather than a free parameter.
  3. [Sec. 4.2] The grid density (number of points in x and m) and the interpolation error are not stated; a sentence quantifying the linear-interpolation accuracy in log-log space would support the 'accurate' description.
  4. [Sec. 4.3, Fig. 5] The dQCD unitarity-boundary fit log10 α = −0.166 − 0.251 log10(1/v) − 0.250 log10 ρ is used to trigger warnings, but the fit range and scatter are not described; give the provenance and estimated uncertainty of this fit.
  5. [Sec. 2.1 / Sec. 5] The quasi-equilibrium premise (rates ≫ H) behind Eq. (1) is not checked in the superWIMP application; a quick estimate at T ~ 1 GeV and m ~ 5×10^6–10^7 GeV would confirm the premise holds along the plotted curves.
  6. [Sec. 4.2, footnote 4] The cross-reference 'see Sec. 4.3' for the extrapolation is off — Sec. 4.3 discusses unitarity, not the extrapolation; fix the reference or add the promised discussion.

Circularity Check

0 steps flagged

No significant circularity: Eq. (39) is an exact rescaling identity derived from the stated rates; the only empirical element (Eq. 42) is illustrative and not the basis of the main tables.

full rationale

The central result, Eq. (39), is a rescaling identity, not a fit masquerading as a prediction. Section 3.1 derives it by explicit power counting: the decay, transition, and ionization rates are proportional to m, the depletion factors R_i depend only on rate ratios (so the m-dependence factors out in Eq. (28)), and for frozen couplings the BSF cross section has the parametric form (σv)^i_BSF = α^2 f_i(α/v) (Eq. (29)). The thermal average is then invariant under x' = x r^2, v' = v/r, α' = r α, giving Eq. (38), and Eq. (39) follows by composing the mass and coupling rescalings. No parameter is fitted to the quantity being predicted; the 'prediction' for arbitrary (m, α) is the same physical function evaluated at the rescaled point, so the reduction to a single reference table is the content of the theorem, not a circularity. The closed-form rates in Sec. 2.2 are stated explicitly in the paper, even though they originate in Refs. [13,15,17]; the new rescaling derivation does not hide an ansatz behind a citation. The empirical running-coupling approximation of Sec. 3.2, Eq. (42), with C 'which can be optimised to the problem at hand', is validated against a full QCD computation in Fig. 2 and is explicitly not the basis of the tabulated SM-QCD models, which use direct running-coupling evaluation, nor of the dQCD models, which are defined with constant α. Thus it is not a fitted input called prediction. The main validation gap is the x > 10^6 extrapolation in Sec. 4.2 ('due to the controlled asymptotic behaviour towards large x ... we provide extrapolation of the form ⟨σv⟩_eff,BSF ≃ a × x^b beyond x = 10^6'), which is asserted without a numerical comparison to n > 100 computations and is used in the superWIMP benchmarks with m ~ 5e6–1e7 GeV; this is an accuracy/robustness limitation, not a circularity, because the extrapolation is not fitted to the same cross-section values it purports to replace. Likewise, the quasi-equilibrium premise behind Eq. (1) is an inherited physical assumption, not a circular input. Self-citations are used as sources of published formulas and previous applications, but the load-bearing new step—the rescaling theorem—is derived in-line from displayed expressions rather than reduced to those citations. Therefore the paper is self-contained with respect to its claimed rescaling result.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles or forces; the model classes (dQED/dQCD, SM QCD/QED mediators) are existing scenarios. The central claim rests on: (1) an inherited quasi-equilibrium effective-description framework, (2) the standard dipole/Coulomb closed-form rates, (3) the new exact rescaling identities (clean, parameter-free), and (4) an empirical running-coupling approximation with a hand-chosen O(1) scale C plus two fitted extrapolations (x > 10^6 tail; dQCD unitarity warnings). Items (1)-(2) are standard domain assumptions; item (4) is the genuinely fragile content the reader pays for.

free parameters (3)
  • C (scale-setting parameter in the running-coupling approximation) = O(1), unconstrained ('can be optimised to the problem at hand')
    Sec. 3.2, Eq. (42): dQCD approximate rescaling evaluates the frozen-coupling table at α(C μ_p) with μ_p = √(mT). Validated on one model class (Fig. 2, ~15%), but C is not fixed by any systematic procedure; it is the main hand-chosen parameter of the approximate mode.
  • Extrapolation coefficients (a, b) for x > 10^6 = not quoted; fitted per model
    Sec. 4.2: 'we provide extrapolation of the form ⟨σv⟩_eff,BSF ≃ a × x^b beyond x = 10^6... approximates the contributions of bound states n > 100'. These coefficients are fitted to the tabulated tail and enter tool output beyond the table.
  • dQCD unitarity-warning contour fit = log10(α) = −0.166 − 0.251 log10(1/v) − 0.250 log10(ρ)
    Sec. 4.3: a linear fit in double-log space to the computed unitarity-violation boundary (computed with n up to 4000), extrapolated to lower v, used to raise warnings inside BSFfast for dQCD models. Affects warnings only, not the cross-section values.
axioms (6)
  • domain assumption Bound-state processes are fast compared to Hubble expansion in the epoch of interest, so the X-abundance evolution reduces to a single Boltzmann equation with effective cross section ⟨σv⟩_eff (Eq. 1).
    Sec. 2.1, Eq. (1): the effective-description framework inherited from Refs. [6,13,16]. If rates were comparable to or below H, the network of individual bound-state abundances would need to be evolved instead.
  • domain assumption BSF, decay and transition rates are computed in the dipole/Coulomb approximation with potentials −α_s/r and −α_b/r (Eq. 13), including SU(N) color factors C_F and C_A/2.
    Sec. 2.2, Eqs. (9)-(17): all tabulated cross sections use these closed forms; the entire tool inherits the accuracy of this non-relativistic potential-model approximation.
  • standard math The Milne relation for ionization (Eq. 7) and detailed balance for excitations (Eq. 19) hold, with the radiation bath in equilibrium.
    Sec. 2.1-2.2: the Boltzmann factors and the relation Γ_ion ↔ ⟨σv⟩_BSF close the network; requires kinetic/chemical equilibrium of the unbound-pair distribution.
  • domain assumption S=1 (spin-triplet) bound-state contributions are suppressed and are neglected except for QED mediator models, where the photon s-channel decay makes them comparable.
    Sec. 2.2, footnote 2: inclusion of S=1 states for the QED class relies on the decay-rate relation Eq. (5.4) of Ref. [21]; for all other models the S=1 suppression is assumed without explicit computation.
  • ad hoc to paper Running-coupling results are approximately captured by the frozen-coupling rescaling evaluated at the potential scale: ⟨σv⟩(x;m) ≈ ⟨σv⟩_rescaled(x;m, α(C√(mT))) (Eq. 42).
    Sec. 3.2, Eq. (42) and Fig. 2: stated as an empirical approximation validated on one model class at the ~15% level; this is the mechanism by which one-dimensional tables cover dQCD models with running-like effects. It is the weakest load-bearing premise of the dQCD coverage.
  • ad hoc to paper Partial-wave unitarity bounds (Eq. 43) define where the tabulated perturbative results are trustworthy, with the dQCD violation boundary extrapolated by a linear fit.
    Sec. 4.3: the tool's reliability domain is set by the unitarity analysis, and the dQCD warning region depends on the extrapolated fit log10(α) = −0.166 − 0.251 log10(1/v) − 0.250 log10(ρ).

pith-pipeline@v1.3.0-alltime-deepseek · 19307 in / 21522 out tokens · 184982 ms · 2026-08-03T13:34:06.608537+00:00 · methodology

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read the original abstract

Bound-state formation (BSF) can have a large impact on annihilation of new physics particles with long-range interactions in the early Universe. In particular, the inclusion of excited bound states has been found to strongly reduce the dark matter abundance and qualitatively modify the associated freeze-out dynamics. While these effects can be captured by an effective annihilation cross section, its explicit computation is numerically expensive and therefore impractical for repeated use in Boltzmann solvers or parameter scans. In this work we present BSFfast, a lightweight numerical tool that provides precomputed, tabulated effective BSF cross sections for a wide class of phenomenologically relevant models, including highly excited bound states and, where applicable, the full network of radiative bound-to-bound transitions. We exploit rescaling relations of the cross section to efficiently cover models with additional free parameters and provide fast interpolation routines in Mathematica, python and C for use in Boltzmann solvers. As an illustration, we apply BSFfast to a superWIMP scenario with a colored mediator, demonstrating that the tool enables phenomenological studies that would otherwise be computationally prohibitive. The code is publicly available on GitHub.

Figures

Figures reproduced from arXiv: 2512.23812 by Jan Heisig, Mathias Garny, Stefan Lederer, Tobias Binder.

Figure 1
Figure 1. Figure 1: Bound state contribution ⟨σv⟩eff,BSF to the thermally averaged effective cross section for a scalar X with quantum numbers identical to those of the right-handed up-type quarks (‘stop’-like, see Tab. 1) provided by BSFfast. The blue line shows the full result when taking bound state formation/ionization, bound state decays and transitions among bound states into account. The black line corresponds to the c… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of QCD-S models and approximately rescaled dQCD-S models (darker lines) where the SU(3) gauge coupling at every x is chosen at the potential scale αs = αs( √ 2mT). Upper panel: m2 ⟨σv⟩ eff,BSF plotted over temperature for different masses (different colours). The black dashed line shows the result for setting α = 0.05 at all x which approximates αQCD(107 GeV) = 0.0460. Note that choosing a diffe… view at source ↗
Figure 3
Figure 3. Figure 3: Ratio of the summed s-wave BSF cross section to the corresponding unitarity bound, (σv) ℓ=0 BSF/(σv) ℓ=0 uni , for SM QCD, plotted as a function of inverse velocity. The vertical gray line indicates the virial velocity corresponding to the highest x included in our tables, v = p 6/xmax. The coloured region, where the curves begin to fan up at low velocities, indicates the onset of deviations between two pr… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the effective thermally averaged cross section [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Unitarity violation in dark QCD) with constant coupling. Problematic and [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relic density contours in parameter space of decay width and dark matter [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

discussion (0)

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