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arxiv: 2604.17813 · v1 · submitted 2026-04-20 · ✦ hep-ph · nucl-th

Recognition: unknown

Compositeness of near-threshold eigenstates with Coulomb plus short-range interactions

Tomona Kinugawa , Tetsuo Hyodo
Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:52 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords compositenessnear-threshold statesCoulomb interactionweak-binding relationeffective field theoryexotic hadronsscattering length
0
0 comments X

The pith

For near-threshold states with Coulomb and short-range interactions, compositeness is determined solely by the Coulomb scattering length, effective range, and Bohr radius.

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

The paper shows that the internal makeup of near-threshold bound or resonant states in two-body systems can be quantified by their compositeness, which measures the probability that the state consists of the two free particles. Using an effective field theory approach, it derives a formula that expresses this compositeness using only three quantities when the state is close to the threshold energy: the Coulomb-modified scattering length, the effective range, and the Bohr radius of the Coulomb force. A reader would care because this offers a practical way to assess whether states like certain exotic hadrons or light nuclei are mostly composite without solving the full dynamics, and it reveals how the long-range Coulomb force changes the behavior compared to short-range only cases. The analysis also demonstrates that resonances can inherit composite character from their connection to bound states.

Core claim

The compositeness for near-threshold s-wave eigenstates is expressed in terms of the energy derivative of the self-energy, which reduces to a formula involving only the Coulomb scattering length, Coulomb effective range, and Bohr radius. This generalizes the weak-binding relation to systems with Coulomb interactions. Numerical results indicate that a relatively strong Coulomb interaction removes the enhancement of compositeness near threshold seen in short-range cases, while a weak Coulomb interaction allows a remnant of short-range universality where bound states are composite dominant, and resonances remain composite due to their continuous connection to the bound-state regime.

What carries the argument

The generalized weak-binding relation expressed through the Coulomb scattering length, effective range, and Bohr radius, derived from the energy derivative of the self-energy in the effective field theory.

If this is right

  • Near-threshold eigenstates' compositeness can be estimated model-independently from scattering data and the Bohr radius.
  • Strong Coulomb interactions suppress the increase in compositeness as the state approaches threshold.
  • Weak Coulomb interactions preserve the tendency for near-threshold bound states to be composite.
  • Resonances with Coulomb forces are composite dominant because they connect continuously to bound states.
  • The formalism applies directly to analyzing the structure of exotic hadrons and atomic nuclei.

Where Pith is reading between the lines

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

  • This approach could guide interpretations of experimental data on charged particle systems near thresholds, such as in heavy ion collisions or precision spectroscopy.
  • It highlights the need to include long-range effects in compositeness calculations to avoid overestimating or underestimating the composite nature in charged systems.
  • The pole trajectories suggest that similar relations might hold for other long-range potentials beyond Coulomb.
  • Applications to more particles or higher waves could test the robustness of the three-parameter dependence.

Load-bearing premise

The nonrelativistic effective field theory framework allows the compositeness to be written in terms of the energy derivative of the self-energy even when the Coulomb interaction is non-separable.

What would settle it

If a computed or measured compositeness for a near-threshold state with known values of the Coulomb scattering length, effective range, and Bohr radius does not match the derived formula, the relation would be falsified.

Figures

Figures reproduced from arXiv: 2604.17813 by Tetsuo Hyodo, Tomona Kinugawa.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagram of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Pole trajectories in the complex momentum plane for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Imaginary part of the resonance eigenmomentum [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Pole trajectories in the complex momentum plane for the repulsive Coulomb plus short-range system with [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Pole trajectories in different Riemann sheets; (a) in the クーロ 有効レ eigenenergy [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The real and imaginary parts of the eigenenergy as functions of the inverse scattering length [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The real and imaginary parts of the compositeness [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Compositeness in the interpretation schemes introduced in Sec. II D as functions of the inverse scattering length [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Pole trajectories in the complex momentum plane [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Compositeness [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Compositeness in the interpretation schemes intro [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Same as Fig. 7 but with the positive effective range [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Same as Fig. 10 but with the positive effective range [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Same as Fig. 11 but with the positive effective range [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
read the original abstract

We investigate the internal structure of near-threshold $s$-wave eigenstates in a two-body system with Coulomb plus short-range interactions. Using a nonrelativistic effective field theory, we derive the expression for the compositeness in terms of the energy derivative of the self-energy, which is applicable to the present system with the non-separable Coulomb interaction. For near-threshold states, the compositeness can be written solely in terms of the Coulomb scattering length, the Coulomb effective range, and the Bohr radius, providing the weak-binding relation in the presence of the Coulomb interaction. We numerically study the pole trajectories and the compositeness and find that the Coulomb interaction qualitatively modifies the threshold behavior of the poles and the internal structure of the eigenstates. We show that when the Coulomb interaction is relatively strong, the enhancement of the compositeness near the threshold is absent, in contrast to purely short-range interactions. On the other hand, for a weak Coulomb interaction, a remnant of short-range universality survives, and near-threshold bound states tend to be composite dominant. Furthermore, even resonances are dominated by the composite component in the presence of the Coulomb interaction, owing to their continuous connection to the bound-state regime. We apply the formalism to realistic systems with near-threshold eigenstates, including exotic hadrons and nuclei.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper claims to derive, within a nonrelativistic effective field theory framework, an expression for the compositeness of near-threshold s-wave eigenstates in systems with Coulomb plus short-range interactions. This expression is given in terms of the energy derivative of the self-energy and, for near-threshold states, reduces to a weak-binding relation depending only on the Coulomb scattering length, the Coulomb effective range, and the Bohr radius. Numerical investigations of pole trajectories demonstrate that the Coulomb interaction alters the threshold behavior of the poles and the internal structure of the states, with strong Coulomb suppressing compositeness enhancement and weak Coulomb preserving some short-range universality. Resonances are found to be composite-dominated due to their connection to bound states. The formalism is applied to exotic hadrons and nuclei.

Significance. This result extends the concept of compositeness and weak-binding relations to include long-range Coulomb interactions, which is significant for the study of near-threshold states in nuclear and particle physics. The parameter-free nature of the relation in terms of independently measurable quantities is a strength. The numerical findings provide concrete examples of how Coulomb modifies the composite vs. elementary nature of states, offering potential guidance for interpreting data on exotic hadrons and nuclear bound states or resonances.

major comments (1)
  1. [EFT derivation] The derivation that the compositeness reduces solely to a function of a_C, r_C, and a_B relies on the self-energy derivative isolating the composite probability. However, the manuscript should explicitly demonstrate that this holds even when the short-range interaction has a range comparable to the Bohr radius, as non-separability could introduce surviving momentum dependence in the loop. The current numerical tests do not isolate this regime, which is load-bearing for the universality claim.
minor comments (3)
  1. [Abstract] The abstract mentions 'non-separable Coulomb interaction' but a brief clarification on what 'non-separable' means in this context would help readers.
  2. [Numerical study] Figures showing pole trajectories would benefit from including the corresponding compositeness values as a function of the parameters for direct comparison.
  3. [Applications section] When applying to realistic systems, the specific values of the Coulomb scattering length and effective range used for the examples should be provided in a table to allow readers to reproduce the compositeness calculations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comment on the EFT derivation. We address the point below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [EFT derivation] The derivation that the compositeness reduces solely to a function of a_C, r_C, and a_B relies on the self-energy derivative isolating the composite probability. However, the manuscript should explicitly demonstrate that this holds even when the short-range interaction has a range comparable to the Bohr radius, as non-separability could introduce surviving momentum dependence in the loop. The current numerical tests do not isolate this regime, which is load-bearing for the universality claim.

    Authors: The general expression for the compositeness in terms of the energy derivative of the self-energy follows from the normalization of the eigenstate wave function in the presence of the non-separable Coulomb interaction, as derived in Sec. II of the manuscript; this step is independent of the detailed form of the short-range potential. The subsequent reduction to a function of a_C, r_C, and a_B employs the effective-range expansion of the short-range phase shift, which is the standard low-energy parameterization in the EFT. This expansion is valid for near-threshold states whose typical momentum is small compared with the inverse range of the short-range force. When the short-range range becomes comparable to the Bohr radius, higher-order terms in the short-range expansion may appear, but they enter only at sub-leading order in the weak-binding relation and do not alter the leading dependence on a_C, r_C, and a_B. The numerical examples in the manuscript use a separable short-range interaction for computational simplicity, yet the analytic result itself is formulated within the EFT and therefore applies more generally. To make this regime of validity explicit, we will add a clarifying paragraph in Sec. III and a brief remark in the discussion of the numerical results. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper starts from the standard nonrelativistic EFT Lagrangian with Coulomb plus contact short-range terms, defines the self-energy loop integral (non-separable due to the long-range Coulomb propagator), and derives the compositeness as X = [1 - dSigma/dE]^{-1} evaluated at the pole. For near-threshold poles it then substitutes the Coulomb effective-range expansion of the loop function, yielding an algebraic expression in a_C, r_C and a_B. These low-energy constants are independently measurable from scattering data and are not fitted to the compositeness itself; the reduction is a controlled threshold expansion, not a tautology. No load-bearing step relies on a self-citation whose content is merely renamed or assumed without external verification. Numerical pole trajectories are presented as illustrations, not as the source of the analytic relation. The central claim therefore adds independent content beyond its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard nonrelativistic EFT assumptions and the definition of compositeness, with scattering parameters serving as inputs rather than new fitted constants invented by the paper.

free parameters (2)
  • Coulomb scattering length
    Characterizes low-energy scattering in the presence of Coulomb; treated as an input parameter in the derived expression.
  • Coulomb effective range
    Second parameter in the effective range expansion with Coulomb; input to the near-threshold compositeness formula.
axioms (2)
  • domain assumption Nonrelativistic effective field theory applies to the two-body system with Coulomb plus short-range interactions.
    Invoked to derive the self-energy and compositeness expression applicable to non-separable Coulomb.
  • domain assumption Compositeness is given by the energy derivative of the self-energy.
    Standard definition in the field used to obtain the expression for the present system.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Compositeness of near-threshold states in charged hadronic systems

    hep-ph 2026-04 unverdicted novelty 7.0

    Derives compositeness expression for near-threshold states with Coulomb interactions and applies it to pp, alpha-alpha, Omega-Omega, and other systems.

Reference graph

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