{"id":"d2d2e64b-5c69-4f04-a9ec-50e5e726fd12","arxiv_id":"2512.23812","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A public tool with new exact rescaling identities makes bound-state-formation cross sections with up to 100 excited states fast enough for routine dark-matter Boltzmann-solver scans.","lead":"BSFfast ships precomputed tables and fast interpolation for bound-state-formation effects on dark matter annihilation, so Boltzmann solvers can include highly excited bound states without expensive on-the-fly integrals. New analytic rescaling laws let a single reference table cover wide ranges of particle mass and gauge coupling with exact scaling for constant couplings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"High-x extrapolation beyond x=10^6 is asserted but not validated, and both arbitrary-coupling rescaling and the superWIMP application depend on it.","rationale":"The reader's verdict of CONDITIONAL is appropriate, but not because of the dQCD approximation identified as the weakest assumption. The dQCD tables are based on frozen couplings and therefore use the exact rescaling law, Eq. (39), not the empirical Eq. (42). The more serious gap is the unvalidated high-x extrapolation, which the paper itself introduces in Sec. 4.2 without a convergence check or an independent comparison. This extrapolation is load-bearing for two reasons: it extends the reference table beyond x = 10^6, where n <= 100 is no longer converged, and the superWIMP application in Sec. 5 uses mediator masses and temperatures that push x beyond 10^6. The central analytical result, Eq. (39), is sound for the frozen-coupling model; the risk is in the numerical implementation's asymptotic approximation. The proposed check would settle whether the extrapolation is accurate. If it fails, the coverage claims for large alpha and the superWIMP contours would need to be restricted or re-derived, which is a condition on acceptance rather than a reason to reject the paper outright.","tokens_in":19961,"tokens_out":22524,"duration_ms":190196,"concrete_test":"Take a representative frozen-coupling model (e.g., dQCD-S with m0 = 10^4 GeV, alpha0 = 0.1) and compute <sigma v>_eff,BSF directly with n_max = 4000 at x = 2e6, 5e6, and 1e7, using the Sec. 2.2 formulas. Then compare to BSFfast's a*x^b extrapolation at the same x. If relative deviations exceed ~5%, the arbitrary-alpha rescaling for alpha > alpha0 and the superWIMP contours in Fig. 6 (which reach x ~ 1e7 at T ~ 1 GeV) are unreliable outside the tabulated range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The exact rescaling law, Eq. (39), is internally consistent for fixed couplings: the r^5 cancellation in the rates and the invariance of the Bose-enhancement factor under x' = x r^2, v' = v/r, and alpha' = r alpha all check out. The frozen-coupling table is therefore a sound idea. The least-secure link is the bridge from the tabulated range to the claimed coverage. BSFfast tabulates x in [10, 10^6] with n <= 100, but Section 4.2 introduces an extrapolation <sigma v> ~ a x^b beyond x = 10^6, described only as 'controlled asymptotic behaviour' and said to approximate n > 100 contributions. No comparison against higher-n computations is shown. This matters because, for alpha > alpha0, Eq. (39) maps a requested x to x' = x (alpha/alpha0)^2 > 10^6 in the reference table, so the 'arbitrary alpha' coverage inherits the unvalidated extrapolation. It also matters for the headline superWIMP application (Sec. 5): with mediator masses ~5e6-1e7 GeV and decay temperatures potentially near 1 GeV, x = m/T reaches ~5e6-1e7, beyond the table. The reader's stated dQCD concern is less compelling: dQCD models are defined with constant alpha_dQCD (Sec. 2.2), so their coverage uses the exact Eq. (39), not the empirical Eq. (42); Eq. (42) is an illustration for approximating running QCD, not the basis of the dQCD tables.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20269,"tokens_out":28334,"duration_ms":230228,"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":[{"comment":"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.","section":"Sec. 4.2 (footnote 4); Sec. 3.1; Sec. 5"},{"comment":"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.","section":"Sec. 5, Fig. 6"}],"minor_comments":[{"comment":"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.","section":"Sec. 3.2 / Sec. 4.1 / Sec. 6"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"Sec. 4.2"},{"comment":"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.","section":"Sec. 4.3, Fig. 5"},{"comment":"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.","section":"Sec. 2.1 / Sec. 5"},{"comment":"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.","section":"Sec. 4.2, footnote 4"}],"recommendation":"major_revision","confidential_remarks":"The core analytic content (Eqs. (28), (38), (39)) is sound and I verified it independently. The publication decision should hinge on the validation of the extrapolation and on an end-to-end benchmark; both are feasible with the machinery already in the paper (n up to 4000 for unitarity, and the authors' own on-the-fly code from Ref. [15]). I recommend major_revision rather than reject because the load-bearing issue is a missing validation/qualification, not an error in the derivation. I would also suggest the editor ask for benchmark numbers (e.g., max relative error in ⟨σv⟩_eff,BSF in the tabulated and extrapolated ranges, and a comparison of Ω_χ h² with Ref. [15] for the bottom-philic case)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: Eq. (39) is the real contribution here. Under the stated frozen-coupling assumptions, the effective BSF cross section obeys an exact rescaling law. I checked the algebra independently — the r^2 factor from BSF cross sections, the r^5 factor from ionization and transition rates, and the cancellation in the efficiency factors R_i all work out. That is a genuinely new analytic result, and it turns what would be a multi-dimensional tabulation into a one-dimensional table in x. The tool is also a solid piece of engineering: bound states up to n=100, convergence checked at the percent level, and wrappers in three languages. The superWIMP application shows order-of-magnitude shifts and makes a convincing case for why the tool matters.\n\nThe main soft spot is the extrapolation beyond x=10^6. Section 4.2 asserts 'controlled asymptotic behaviour' and uses a power-law fit ⟨σv⟩ ~ a x^b, but the paper never compares that fit against higher-n computations. This is not a marginal detail. For alpha > alpha0, Eq. (39) maps to x' = x (alpha/alpha0)^2, which can exceed 10^6, so the arbitrary-alpha coverage inherits the unvalidated extrapolation. The superWIMP application itself reaches x ~ 5e6–1e7. I'd want a plot showing n=100 vs n=300 at several x values, plus a stability check of the fit, before trusting the large-alpha/large-x claims.\n\nOne concern from our reader I'd push back on: the dQCD coverage does not rest on the empirical Eq. (42). dQCD is defined with constant alpha_dQCD (Sec. 2.2), so it uses the exact Eq. (39). Equation (42) is an illustration for SM QCD, and for those models the paper does not rely on rescaling anyway.\n\nThe remaining issues are smaller. There's no commit hash or shipped tables in the manuscript, a reproducibility gap for a code paper. There's a typo in Table 1 ('dQED-SnoTr' repeated). And the conclusion's 'at least an order of magnitude' unitarity margin is in mild tension with Fig. 3, where a few masses approach the 10% line at low v. All fixable.\n\nThis is for anyone doing relic-density calculations for long-range interactions, t-channel mediators, or superWIMP scenarios, and it deserves a serious referee. My recommendation: accept after the extrapolation is validated and the repository is pinned.","headline":"Exact rescaling law is real and useful; the high-x extrapolation and reproducibility need work before I'd trust the full coverage.","tokens_in":20865,"tokens_out":6139,"would_cite":true,"duration_ms":52136,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Excited bound-state formation obeys an exact rescaling law, so one reference table covers arbitrary mass and coupling.","keywords":["bound-state formation","effective annihilation cross section","rescaling relations","excited bound states","dark matter freeze-out","superWIMP","dipole transitions","early Universe"],"falsifier":"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.","tokens_in":19657,"feed_emoji":"⚛️","tokens_out":6937,"duration_ms":61520,"temperature":0.7,"pith_summary":"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.","feed_headline":"One table now covers bound-state effects for any mass and coupling","feed_subtitle":"Exact rescaling identities make a single expensive run cover the whole mass–coupling plane, speeding up dark-matter scans.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Exact rescaling law makes bound-state scans one-run wonders","One bound-state calculation now covers all masses and couplings","Exact scaling identity speeds dark-matter bound-state scans","BSFfast: exact rescaling turns expensive scans into a single run","Masses and couplings? One table does it all for bound states"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact rescaling law makes bound-state scans one-run wonders","One bound-state calculation now covers all masses and couplings","Exact scaling identity speeds dark-matter bound-state scans","BSFfast: exact rescaling turns expensive scans into a single run","Masses and couplings? One table does it all for bound states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2316,"prompt_tokens":768,"completion_tokens":1548,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":1462}},"tokens_in":512,"tokens_out":1548,"duration_ms":12778,"temperature":1.0,"reasoning_tokens":1462,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:34:06.608537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}