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Unitarity violation and restoration in radiative bound-state formation

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Marcos M. Flores, Kalliopi Petraki

High Energy Physics - Phenomenology · Submitted 23 Feb 2026 · 43 references

Pith's verdict: A genuine derivation of unitarity violation in monopole BSF, but the restoration result relies on an unbenchmarked truncation of the W matrix and on the authors' own prior framework.

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arXiv ID2602.20243 (version 2)
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CitationKalliopi Petraki, Marcos M. Flores. "Unitarity violation and restoration in radiative bound-state formation." arXiv:2602.20243 (2026). https://pith.science/paper/WLEBLULB
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The paper

The paper claims

For radiative bound-state formation via charged-scalar emission, standard cross sections violate partial-wave unitarity at low velocities for arbitrarily small couplings, and the violation is cured by resumming the absorptive self-energy of

The reading

The paper establishes that standard non-relativistic computations of radiative bound-state formation severely violate unitarity when the emitted boson carries charge, and that the violation is only apparent: resumming the inelastic contributions to the incoming-state self-energy restores unitarity. It derives Kramers-like partial-wave formulae showing that the summed cross sections grow monotonically at low velocities for scattering-to-bound coupling ratios λ≤1, so the violation occurs for arbitrarily small radiative couplings. It then applies a previously developed unitarization scheme to monopole bound-state formation, including the singularities of the amplitudes in the complex momentum plane. The result is that unitarized inclusive cross sections saturate at an O(1) fraction of the partial-wave unitarity ceiling—about one quarter—at low velocities, rather than diverging. This matters because bound-state formation is central to dark-matter freeze-out, indirect detection, and self-interactions.

What carries the argument

The central object is the separable, non-local anti-Hermitian potential V_{A,ℓ}(r',r) = -i Σ_n ν_{nℓ}(r')ν*_{nℓ}(r), constructed from the irreducible BSF amplitudes, which is added to the Schrödinger equation to resum absorptive self-energy insertions. The accompanying regularization matrix N_ℓ = 1 + M_BSF,unreg M†_BSF,unreg + i W_ℓ encodes both on-shell depletion and the dispersive mixing with off-shell bound states; W_ℓ is computed from the singularities of the BSF amplitudes in the complex momentum plane, and it is these singularities that tame the low-velocity growth.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is twofold. First, for monopole BSF via emission of a charged scalar, the inclusive partial-wave cross-section summed over all bound levels accessible from a given incoming partial wave grows monotonically at low velocities whenever the scattering-to-bound coupling ratio λ=α_S/α_B≤1; for λ=0 it grows as 4α_rad ζ_B ln(ζ_B/b_ℓ), and for 0<λ<1 as 4α_rad ζ_B times order-one λ-dependent factors, so the unitarity bound is violated for arbitrarily small α_rad once ζ_B is large enough. Second, resumming the BSF contributions to the incoming-state self-energy—implemented via the non-local anti-Hermitian potential of Refs. [2,15] and the associated regul

Load-bearing premise

The unitarization scheme itself—the anti-Hermitian potential derived from irreducible inelastic amplitudes and the complex-momentum continuation used to compute W_ℓ—is assumed complete and unique; if the analyticity or convergence conditions fail, or if other ultrasoft channels such as bremsstrahlung substantially alter the self-energy, the claimed restoration of unitarity would not follow.

Editorial extensions

If this is right

  • For λ≤1, existing BSF cross sections are invalid at low velocities and must be unitarized before use in dark-matter freeze-out, indirect detection, or self-interaction studies.
  • The unitarized inclusive BSF cross-section saturates at about one quarter of the partial-wave unitarity limit for perturbative couplings, over a wide range of coupling strengths.
  • Off-diagonal elements of W_ℓ are essential: truncating to diagonal elements only reduces but does not eliminate the low-velocity growth.
  • For λ>1, excited-level contributions are exponentially suppressed, so unitarity violation does not arise and unitarization changes little.
  • Bremsstrahlung and other inelastic channels sharing the same initial state should be treated on the same footing as BSF, since resumming one affects the others.

Reading between the lines

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

  • The same singularity-driven mechanism may apply to dipole transitions and non-Abelian emission, because the monopole case is presented as parametrically representative; a direct partial-wave calculation for dipole BSF would test this.
  • The universal saturation near the 1/4 inelastic ceiling suggests that, at sufficiently low velocities, long-range attractive capture into bound states becomes geometrically limited, which could simplify freeze-out rate estimates.
  • A combined BSF-plus-bremsstrahlung resummation may show both channels collectively saturating unitarity, with observable consequences for the photon and neutrino spectra in indirect detection.
  • The need to sum bound levels up to n~10ζ_B/√ln(ζ_B) at low velocities indicates that practical implementations may require analytic or truncated resummations beyond brute-force summation.
Original abstract

State-of-the-art calculations motivated by dark matter exhibit severe violation of partial-wave unitarity in the non-relativistic regime in radiative bound-state-formation processes. It has been recently shown, in a model-independent fashion, that unitarity is restored by the proper resummation of the inelastic contributions to the self-energy of the incoming state. In this work, we first derive Kramers-like formulae for individual partial waves, demonstrating that existing calculations of bound-state formation severely violate unitarity. We then discuss how unitarity is restored through the resummation of the absorptive contributions to the incoming-state self-energy, generated by bound-state formation processes, taking into account their analytic structure in the complex momentum plane. Our results can be generalized to a variety of theories and employed in phenomenological studies, such as dark-matter freeze-out, indirect detection and self-interactions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,371 input tokens.

What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

State-of-the-art calculations motivated by dark matter exhibit severe violation of partial-wave unitarity in the non-relativistic regime in radiative bound-state-formation processes. It has been recently shown, in a model-independent fashion, that unitarity is restored by the proper resummation of the inelastic contributions to the self-energy of the incoming state. In this work, we first derive Kramers-like formulae for individual partial waves, demonstrating that existing calculations of bound-state formation severely violate unitarity. We then discuss how unitarity is restored through the resummation of the absorptive contributions to the incoming-state self-energy, generated by bound-state formation processes, taking into account their analytic structure in the complex momentum plane. Our results can be generalized to a variety of theories and employed in phenomenological studies, such as dark-matter freeze-out, indirect detection and self-interactions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 State-of-the-art calculations motivated by dark matter exhibit severe violation of partial-wave unitarity in the non-relativistic regime in radiative bound-state-formation processes. background 0.96 0.11 0.07 asserted 1.00
s1 It has been recently shown, in a model-independent fashion, that unitarity is restored by the proper resummation of the inelastic contributions to the self-energy of the incoming state. background 1.00 0.05 0.22 asserted 1.00
s2 In this work, we first derive Kramers-like formulae for individual partial waves, demonstrating that existing calculations of bound-state formation severely violate unitarity. contribution 0.88 0.95 0.17 asserted 1.00
s3 We then discuss how unitarity is restored through the resummation of the absorptive contributions to the incoming-state self-energy, generated by bound-state formation processes, taking into account their analytic structure in the complex momentum plane. contribution 0.61 0.83 0.47 asserted 1.00
s4 Our results can be generalized to a variety of theories and employed in phenomenological studies, such as dark-matter freeze-out, indirect detection and self-interactions. interpretation 0.62 0.44 0.10 asserted 0.56

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
  • relations to other papers
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not run
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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The 2PI kernels generating long- and finite-range potentials for
Figure 2. Bound-state formation via emission of a charged scalar,
Figure 3. The factors rnℓ ∝ σℓ(k; n)/σU ℓ (k) vs ζB, for αS = 0 and different values of n, ℓ, as denoted in the labels. We compare the original expression (3.4), with the analytic approximation (3.5). While the phase of the oscillations is different, both curves are enveloped by the same function. Larger n implies more oscillations, higher maximum value of rnℓ and larger ζB value at which this maximum occurs. 10-1 1 10 102 10-5 10-3 10-1 10 103 10-2 10-1 1 10 102 10-5 10-3 10-1 10 103
Figure 4. rnℓ vs n/ζB, for αS = 0 and ℓ = 0. Left: We compare the original expression (3.4), with the analytic approximation (3.5). (The curve corresponding to Eq. (3.4) is not smooth as it connects the discrete, physical values of n.) In the range p bℓζB < n < ζB (encompassed within the vertical gray lines), rnℓ ∝ 1/n, while for n > ζB, rnℓ ∝ 1/n 3 [cf. Eq. (3.10)]. Right: The oscillations increase with ζB. 16
Figure 5. rℓ = P∞ n=1+ℓ rnℓ vs ζB, for different values of ℓ. At ζB ≫ 1 + ℓ, rℓ depends only mildly on ℓ. We compare rℓ calculated numerically using the full expression (3.4) for rnℓ, with the analytic approximation (3.14). The growth of rℓ with ζB implies that unitarity is violated at sufficiently low velocities, for arbitrarily low couplings, αrad [cf. Eq. (3.2)]. 17
Figure 6. rnℓ vs ζB, for ℓ = 0, and different values of λ = αS/αB and n, as denoted in the labels. The asymptotic behavior at low velocities and large n, ζB, |ζS| ≫ n ≫ ℓ + 1 does not depend on ℓ [cf. Eq. (3.26)]. (This does not hold for λ = 0.) Upper row: λ ⩽ 0; the excited levels contribute at lower velocities and at increasing strength. The Sommerfeld suppression for λ < 0 gives rise to a sharp cutoff at low velocities. Lower row: λ > 0. Capture into excited levels is significant for λ ≲ 1, but becomes exponentially suppressed for λ ≫ 1. (For λ = 1, see Footnote 8.) The Sommerfeld…
Figure 7. rℓ as a function of ζB, shown for different values of λ = αS/αB and ℓ, as indicated by the labels (see
Figure 8. Representative examples of Wnn′ ℓ for αS = 0, defined in Eq. (4.30). The Wnn′ ℓ elements tend to constant values at large and low momenta, given by Eqs. (4.41). that the residue of a function f(x) at an isolated pole of order n + 1 at x = x0 is Resx=x0 f(x) = 1 n! lim x→x0 d n dx n h (x − x0) n+1 f(x) i . (4.31) Considering this, we deduce from Eq. (4.30) that Wnn′ ℓ are ratios of polynomials in ζB
Figure 9. ℓ = 0 and αS = 0. Left panels: Unregulated inclusive BSF cross-section normalized to the unitarity limit, yℓ,unreg, and regularization parameters, wℓ and ˜wℓ, vs ζB = αB/vrel. Right pan￾els: Inclusive BSF cross-sections normalized to the unitarity limit. We compare the unregulated version (3.2), with the cross-sections regulated according to the prescription (4.21) that neglects the singularities of the BSF amplitudes in the complex momentum plane, and the full formulas (4.20) that take into account the singularities. Top, middle, bottom rows: αrad = 0.01, 0.05, 1, re￾spectively.…
Figure 10. ℓ = 0, αS = 0 and αrad = 0.1. BSF cross-sections normalized to the unitarity cross-section, for individual bound levels, n = 1 (upper left), n = 3 (upper right), n = 5 (lower left), n = 10 (lower right). We compare the unregulated cross-sections (3.2) (solid blue lines) with the regulated ones under the prescription (4.21) than neglects singularities (dashed blue lines) and under the full prescription (4.20) that accounts for singularities (solid red lines). See Section 4.3.4 for discussion of results. 37
Figure 11. Same as in Fig
Figure 12. Same as in Fig
Figure 13. For ℓ = 0 and different values of λ = κS/κB shown in the legend, we examine the low-velocity behavior (ζB = 100) with respect to the coupling αrad. Left panel: The unregulated inclusive BSF cross-sections normalized to the unitarity limit, yℓ,unreg, and the regularization pa￾rameters wℓ and ˜wℓ. Right panel: The regulated inclusive BSF cross-sections normalized to the unitarity limit, with and without accounting for the singularities of the BSF amplitudes. See Sec￾tions 4.3.4 and 4.4 for discussion of results. 40

Figures reproduced from arXiv:2602.20243 by the authors.

References
How the literature uses it

2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Compares with it 1 paper
  • Refs. [ref] both require a hierarchy of scales between the range/energy scale of the annihilation process, and the range/energy scale of the long-range potential; the work of Ref. [ref] (extended in Ref. [ref] ) does not impose this requirement and thus generalizes to situations where there is no large hierarchy, e.g. where the “annihilation” corresponds to capture into a bound state with radius comparable to the range of the potential [ref] .

    On the equivalence of unitarization prescriptions for the Sommerfeld enhancement · Introduction · 0.81

Cites it as background 1 paper

What it cites · 68 sentences without a confident label

Builds on 3 papers
Uses the method or tool of 2 papers
Compares with 1 paper
Cites as support 3 papers
  • For DM freeze-out, the severe unitarity violation can seemingly prevent freeze-out [ref] ; resumming BSF restores unitarity and ensures freeze-out, yielding reliable predictions for the parameters of the theory [ref] .

    Excited bound states and their role in dark matter production · Introduction · 0.97

  • For DM freeze-out, the severe unitarity violation can seemingly prevent freeze-out [ref] ; resumming BSF restores unitarity and ensures freeze-out, yielding reliable predictions for the parameters of the theory [ref] .

    2505.20443 · Introduction · 0.97

  • In addition, the overlaps of wavefunctions of different potentials exhibit non-monotonic dependence on the momenta, and can become very large, even if only for a finite momentum range [ref] .

    Dark matter bound state formation via emission of a charged scalar · Models · 0.97

Cites as background 2 papers
  • The light species, $ϱ$, $φ$ and $V^μ$, redshift as radiation until late times, and, depending on their relative mass hierarchies, may decay into each other and/or into SM particles, thereby eventually attaining negligible relic densities; we refer to [ref] for more detailed phenomenological considerations.

    Dark matter bound state formation via emission of a charged scalar · Abelian model with doubly-charged scalar mediator · 0.98

  • Such interactions arise in supersymmetric and non-supersymmetric frameworks that have long served as archetypical models for WIMP DM (see e.g. [ref] for a classification and related phenomenological considerations).

    On Minimal Dark Matter coupled to the Higgs · Higgs-portal models · 1.00

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