Compositeness of near-threshold states with repulsive Coulomb interaction combined with short-range potential

Tetsuo Hyodo; Tomona Kinugawa

REVIEW 4 major objections 4 minor 25 references

Compositeness of near-threshold states with repulsive Coulomb interaction combined with short-range potential

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Repulsive Coulomb interaction makes a shallow bound state turn directly into a resonance, with the compositeness fixed by three scattering observables.

desk verdict A concise, model-based extension of compositeness to repulsive Coulomb systems; the new formula is plausible, but the paper omits its derivation and the effective-range truncation is load-bearing. read the letter →

arxiv 2507.22399 v1 pith:MW4WELNN submitted 2025-07-30 hep-ph nucl-th

Tomona Kinugawa , Tetsuo Hyodo This is my paper · ORCID

classification hep-phnucl-th

keywords compositenesshadronicmoleculesCoulombinteractioneffectiverangeexpansionnear-thresholdstatesresonancesboundlow-energyuniversality

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 asks what happens to the internal structure of near-threshold states when a repulsive Coulomb force acts alongside a short-range interaction, and answers in terms of the compositeness—the probability that the state is a hadronic molecule rather than an elementary object. Using a model in which the pole momentum is fixed by the Coulomb scattering length $a_s$, the Coulomb effective range $r_e$, and the Bohr radius $a_B$, the authors show that a bound state turns directly into a resonance as the scattering length is varied, with no virtual-state stage in between, in contrast to the pure short-range case. They also show that the compositeness of these near-threshold eigenstates is determined solely by those three observables. The practical consequence is that, when $|r_e|$ is much smaller than $a_B$, both shallow bound states and near-threshold resonances become composite dominant, so Coulomb repulsion does not erase the molecular character of a state that is near threshold.

What carries the argument

The central object is the near-threshold pole condition, the Coulomb-modified effective range expansion truncated at order $k^2$: $-1/a_s + (r_e/2)k^2 - i k - (2/a_B)[\log(-i a_B k) + \psi(1 + i/(a_B k))] = 0$, with $\psi$ the digamma function. The logarithmic and digamma terms encode the long-range Coulomb physics and create the branch cut that changes the pole trajectory; solving this equation gives the eigenmomentum $k_h$. The compositeness is then obtained from $X = [1 - r_e/R]^{-1}$, where $R$ is built from $k_h$ and $a_B$ with the trigamma function $\psi_1$. This two-step machinery converts three scattering observables into a statement about whether the state is molecular or elementary.

What would settle it

A precise extraction of the $\alpha\alpha$ scattering length and effective range would show whether the pole moves straight from threshold into the fourth quadrant of the complex momentum plane; observing a virtual-state branch between the bound and resonance regions would contradict the pole condition Eq. (4).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Coulomb-modified effective range expansion has a branch cut on the negative imaginary axis of the complex momentum plane because of the $\log(-i a_B k)$ term. As the inverse Coulomb scattering length $1/a_s$ crosses zero, the bound-state pole moves directly into the fourth quadrant ($\mathrm{Re}\, k > 0$, $\mathrm{Im}\, k < 0$) and becomes a resonance, never passing through a virtual state on the negative imaginary axis. The compositeness formula $X = [1 - r_e/R]^{-1}$ then expresses the molecular content of the state through the same three observables, with $a_s$ entering through the pole condition. When the effective range is far smaller than the Bohr radius, $X$ approaches unity for both shallow bound states and near-threshold resonances, indicating a remnant of low-energy universality under the probabilistic interpretation of complex resonance compositeness.

Load-bearing premise

The load-bearing premise is that the Coulomb-modified effective range expansion can be cut at the $k^2$ term, with shape parameters and the bare-state energy dependence negligible, and that the adopted scheme for interpreting complex resonance compositeness as probabilities is valid.

Editorial extensions

If this is right

For a system such as $\alpha\alpha$ scattering or a charged hadron pair, a near-threshold resonance can be as composite as a shallow bound state once $|r_e| \ll a_B$, so the molecular fraction follows from measured scattering parameters.
The absence of a virtual-state stage between bound state and resonance is a direct Coulomb signature; in pure short-range systems, by contrast, the pole normally moves through the virtual-state branch before becoming a resonance.
Compositeness near threshold can be extracted from the Coulomb scattering length, Coulomb effective range, and Bohr radius alone, without reconstructing the scattering wave function.
When $|r_e|$ is not very small compared with $a_B$, the near-threshold state can have small compositeness, so Coulomb effects can make a state look more elementary even though it sits close to threshold.

Reading between the lines

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

The paper does not spell it out, but the same three-observable parametrization should be extendable to attractive Coulomb plus short-range systems, where the branch structure of the logarithm and digamma terms differs and the bound-to-resonance path could be different.
One concrete testable extension would be to fit the pole condition to precision $\alpha\alpha$ (8Be) phase shifts and check whether the pole trajectory really bypasses the virtual state; seeing a virtual-state segment would force a modification of the expansion.
The condition $|r_e| \ll a_B$ could serve as a data-driven rule for judging whether an observed near-threshold resonance is molecular, before any microscopic model is built.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is Eq. (6), which expresses the compositeness of a Coulomb-modified eigenstate in terms of the Coulomb scattering length, effective range, and Bohr radius, and the observation that repulsive Coulomb makes a bound state turn directly into a resonance, bypassing the virtual state. That qualitative point is made cleanly, and the numerical trajectory for r_e/a_B = -0.1 backs it up. The paper is honest about the regime: the near-threshold Coulomb-modified effective range expansion is taken from the cited literature, and the result is consistent with the EFT-based universality discussion in Refs. [17,22].

The main weakness is that Eq. (6) is not derived. The text goes from Eq. (5) to Eq. (6) with 'From Eq. (5)...', but no steps are shown. For a proceedings this is short, but since Eq. (6) is the central new output, a referee will want at least an appendix. The deeper caveat is that the whole analysis rests on the pole condition Eq. (4), which is the Coulomb-modified effective range expansion truncated at O(k^2). The stress-test note is right: no estimate of the next shape parameter is given, and because X comes from the energy derivative of the inverse amplitude, a P k^4 term would shift the pole and change X. The claim that compositeness is fixed by just three observables is therefore true within the two-parameter model, not as a model-independent consequence. 'Near-threshold' language does not by itself control the omitted term.

The probabilistic interpretation of complex resonance compositeness is taken from the authors' own Ref. [15]. That is not a flaw by itself, but it is an interpretive scheme, not yet independently validated. The lack of data comparison is not a concern for a model study of this length.

All told, this is a solid and useful proceedings contribution. People working on charged near-threshold states (T_cc, 8Be, etc.) and anyone extending compositeness formulas to Coulomb systems will cite it. I would ask the authors to add the derivation of Eq. (6) and a sentence on the ERE truncation before publication. If submitted as a journal paper, it deserves a serious referee; as a proceedings, it is acceptable with those additions. I would bring it to reading group and would cite it.

Referee Report

4 major / 4 minor

Summary. This paper studies the compositeness of near-threshold two-body states with a repulsive Coulomb interaction plus a short-range potential. The authors use a model in which a discrete state couples to the scattering channel; after renormalization, the pole condition is the Coulomb-modified effective range expansion Eq. (4), depending on the Coulomb scattering length a_s, the Coulomb effective range r_e, and the Bohr radius a_B. They state a compact formula, Eq. (6), for the compositeness X in terms of these quantities and solve Eq. (4) numerically for r_e/a_B = -0.1. They find that as 1/a_s is varied, the bound state moves directly into a resonance, bypassing the virtual state, and that in the near-threshold window X is large for both bound states and resonances. The paper interprets this as a remnant of low-energy universality when |r_e| is much smaller than a_B.

Significance. Within the effective-range truncation, Eq. (6) would be a useful closed-form relation between compositeness and three scattering observables for Coulombic systems, with potential applications to hadronic molecules and nuclear near-threshold states such as 8Be. The branch-cut structure of Eq. (4) gives a clean qualitative reason why virtual states are absent in the repulsive Coulomb case. The paper is not circular: Eq. (6) is derived from the model, though the derivation is not shown, and the claim about large X is a falsifiable prediction of the two-parameter model rather than an assumption. The main weaknesses are that the central formula and the general claims are supported only by a single numerical trajectory and an uncontrolled truncation of the effective-range expansion.

major comments (4)

[Sec. 2, Eq. (6)] The central compositeness formula Eq. (6) is presented without derivation. Since the paper's main claim is that X can be expressed solely in terms of a_s, r_e, and a_B, the manuscript should show explicitly how Eq. (6) follows from Eqs. (2), (4), and (5), including the elimination of the bare parameters A and nu_0 in favor of a_s and r_e. As it stands, the reader cannot check whether the formula is exact within the model or contains an additional approximation.
[Sec. 2, Eq. (4)] The pole condition is a Coulomb-modified effective range expansion truncated at O(k^2). The compositeness in Eq. (6) is obtained from the energy derivative of the inverse amplitude, so an omitted shape-parameter term (e.g., P k^4) contributes to X at the same order of sensitivity. No estimate or power-counting argument is given for the neglect of such terms in the near-threshold window |a_B/a_s| < |a_B/r_e|. Therefore the claim that X is determined solely by (a_s, r_e, a_B) and the conclusion that resonances are composite-dominant for |r_e| << a_B are established only within the two-parameter model. Please add the next-order term and show stability, or state explicitly the EFT order at which the result is claimed.
[Sec. 3, Fig. 1] The pole trajectory and the probabilities X, Y, and Z are shown for a single value r_e/a_B = -0.1. Because the abstract and Sec. 4 make a general statement for |r_e| << |a_B|, the authors should either provide an analytic argument that the results are independent of the particular value of r_e/a_B in that regime or show additional trajectories (e.g., r_e/a_B = -0.01 and -0.5) to support the generality. The text '|<a_B/a_B| region' and the 'virtical dotted line' should also be corrected and clarified.
[Sec. 3] The probabilistic interpretation of resonances uses the scheme of Ref. [15] without describing it. Since the conclusion that near-threshold resonances are composite-dominant is based on the probabilities X and Y for complex poles, the authors should summarize the scheme and indicate why it applies to the repulsive-Coulomb case. Currently this is a separate interpretive input that is not validated within the manuscript.

minor comments (4)

[Sec. 1] The word 'compositneess' is a typo and should read 'compositeness'.
[Fig. 1 caption] The caption contains the typo 'solif line' and should read 'solid line'; the variable labels for Y and Z in the right panel should also be checked for readability.
[Sec. 3, text near Fig. 1] The phrase '|<a_B/a_B| region' is nonsensical; it should presumably be '|a_B/a_s| < 1' or '|a_B/a_s| < |a_B/r_e|', and the vertical dotted line should be defined precisely in the caption.
[Eq. (6)] The trigamma function psi_1 is not defined in the text; please define it explicitly as the derivative of the digamma function, for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (6) is derived algebraically from the pole condition and the compositeness formula; self-citations supply standard relations and an interpretive scheme, not fitted inputs.

full rationale

The derivation chain is self-contained and non-circular. The model self-energy Eq. (2) is taken from Domcke [17], and the pole condition Eq. (4) is a standard reparameterization of the model in terms of the Coulomb scattering length, Coulomb effective range, and Bohr radius, citing Bethe and EFT derivations [16,18-21]. The compositeness formula Eq. (5) is cited from the authors' own Refs. [8,24], but it is a standard relation between compositeness and the energy derivative of the self-energy; it is not equivalent to the paper's target result. Equation (6) is then obtained algebraically from Eqs. (4) and (5), with no parameter fitted to the compositeness that is subsequently reported. The pole trajectory and the large-X behavior in the near-threshold region follow from solving Eq. (4), not from assuming the conclusion. The only notable self-citation is the probabilistic scheme for resonances adopted from Ref. [15]; the paper explicitly states that it 'adopt[s]' this scheme, and it is used for interpretation of the complex resonance compositeness rather than as an input to Eq. (6). This is an interpretive assumption and a possible correctness risk, but it does not make the central derivation circular. The claims are conditioned on the O(k^2) truncation of the effective-range expansion in Eq. (4), which is a model-dependence concern rather than a circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central results rest on the Coulomb-modified effective range expansion, the self-energy definition of compositeness, and the same-authors' scheme for probabilistic interpretation of resonance compositeness. The paper introduces no free parameters fitted to data; the only hand-chosen number is the illustrative ratio r_e/a_B = -0.1.

free parameters (1)

r_e/a_B (illustrative ratio) = -0.1
The numerical pole trajectory and probability plots in Fig. 1 are computed at this fixed ratio. The qualitative claim is stated for |r_e| much smaller than a_B, so this specific value is an illustrative choice rather than a fit to data.

assumptions (4)

domain assumption The pole condition Eq. (4) determines all eigenmomenta for the Coulomb plus short-range system in the near-threshold region.
This is the Coulomb-modified effective range expansion from Refs. [16-21], truncated at O(k^2); the whole paper starts from this equation.
domain assumption The compositeness X is given by the energy derivative of the self-energy, Eq. (5), following Refs. [8,24].
The relation defines compositeness in this model and is used without derivation to obtain Eq. (6).
domain assumption For resonances, the complex compositeness X can be converted into real probabilities X,Y,Z using the scheme of Ref. [15] by the same authors.
The probabilistic interpretation of resonance compositeness is adopted without independent evidence; the right panel of Fig. 1 depends on it.
standard math Standard pole classification on the complex k plane and the Schwarz reflection principle apply.
Used to identify bound states, resonances, and virtual states and to restrict Fig. 1 to the right half-plane.

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Cite this review

Pith. "Pith review of Compositeness of near-threshold states with repulsive Coulomb interaction combined with short-range potential." pith.science (2026). https://pith.science/paper/MW4WELNN

@misc{pith2026250722399,
  author       = {Pith},
  title        = {Pith review of: Compositeness of near-threshold states with repulsive Coulomb interaction combined with short-range potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MW4WELNN}},
  note         = {Machine review of arXiv:2507.22399}
}

read the original abstract

We investigate the internal structure of near-threshold states in a system with a repulsive Coulomb interaction combined with a short-range potential, using the compositeness. We construct a model in which the eigenmomentum is expressed in terms of three observables: the Coulomb scattering length, the Coulomb effective range, and the Bohr radius. In the presence of the Coulomb interaction, a bound state directly goes into a resonance as parameters are varied, bypassing a virtual state, in contrast to the case with only the short-range interaction. We show that the compositeness of near-threshold states can be expressed solely in terms of these observables. When the magnitude of the Coulomb effective range is much smaller than that of the Bohr radius, both shallow bound states and near-threshold resonances exhibit common structures with large compositeness, reflecting the remnant of the low-energy universality.

Figures

Figures reproduced from arXiv: 2507.22399 by the authors.

**Figure 1.** Left panel: The pole trajectory by varying 𝑎𝐵/𝑎𝑠 with negative 𝑟𝑒/𝑎𝐵 = −0.1. Right panel: The probabilities X (solif line), Y (dotted line), and Z (dashed line) as functions of 𝑎𝐵/𝑎𝑠 with 𝑟𝑒/𝑎𝐵 = −0.1. The shaded area shows the |𝑎𝐵/𝑎𝑠 | < |𝑎𝐵/𝑟𝑒 | region, and the virtical dotted lines correspond to |𝑎𝐵/𝑎𝑠 | = 1. 4. Summary In this study, we investigate the nature of near-threshold states in a system with repulsive C… view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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