arxiv: 2604.22250 · v1 · submitted 2026-04-24 · ✦ hep-ph · nucl-th

Recognition: unknown

Compositeness of near-threshold states in charged hadronic systems

Tomon Kinugawa , Tetsuo Hyodo

Authors on Pith no claims yet

Pith reviewed 2026-05-08 11:20 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords compositenessnear-threshold statesCoulomb interactioneffective range expansionhadronic systemsbound statesresonancesvirtual states

0 comments

The pith

Compositeness of near-threshold states in charged hadronic systems is given by a simple expression involving eigenenergy and Coulomb effective range in the weak-binding limit.

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

This paper develops a method to quantify the internal structure of near-threshold bound, virtual, and resonance states where both Coulomb and short-range forces are present. It employs the Coulomb-modified effective range expansion to derive an explicit formula for compositeness that depends only on the state's eigenenergy and the Coulomb effective range parameter. The derivation holds in the weak-binding limit, which covers many practical cases in nuclear and hadronic physics. The resulting expression is applied to concrete examples including the proton-proton system, alpha-alpha, and several omega-particle combinations to determine the degree to which each state behaves as a composite of its constituents.

Core claim

We quantify the internal structure of near-threshold bound, virtual, and resonance states in systems where Coulomb and short-range interactions coexist by evaluating the compositeness. Using the Coulomb-modified effective range expansion, we derive an expression for the compositeness in terms of the eigenenergy and Coulomb effective range in the weak-binding limit. We then apply the formulation to several near-threshold states in hadronic and nuclear systems, including pp, αα, Ω−Ω−, Ωccc++Ωccc++, Ξ−α, and Ω−p.

What carries the argument

The Coulomb-modified effective range expansion, which incorporates the long-range Coulomb potential into the standard low-energy scattering expansion and thereby yields a direct formula for the compositeness parameter.

Load-bearing premise

The weak-binding limit is a good approximation for the chosen states and the Coulomb-modified effective range expansion accurately represents the low-energy interactions.

What would settle it

If the compositeness value computed for the proton-proton system from its measured energy and effective range disagrees with an independent estimate obtained from the full two-body wave function or from high-precision scattering data, the derived expression would be falsified.

read the original abstract

We quantify the internal structure of near-threshold bound, virtual, and resonance states in systems where Coulomb and short-range interactions coexist by evaluating the compositeness. Using the Coulomb-modified effective range expansion, we derive an expression for the compositeness in terms of the eigenenergy and Coulomb effective range in the weak-binding limit. We then apply the formulation to several near-threshold states in hadronic and nuclear systems, including $pp$, $\alpha\alpha$, $\Omega^{-}\Omega^{-}$, $\Omega_{ccc}^{++}\Omega_{ccc}^{++}$, $\Xi^{-}\alpha$, and $\Omega^{-}p$.

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.

Desk Editor's Note private letter to a colleague

The paper derives a closed-form compositeness formula for charged near-threshold states via Coulomb-modified effective range theory and applies it to several hadronic examples, but skips explicit validation of the weak-binding assumption.

read the letter

This paper derives a closed-form expression for the compositeness of near-threshold bound, virtual, and resonance states in systems with both Coulomb and short-range forces. It extends the neutral-system version by using the Coulomb-modified effective range expansion to get a result in terms of the eigenenergy and the Coulomb effective range parameter, valid in the weak-binding limit. They then plug in numbers for pp, alpha-alpha, Omega-Omega-, Xi-alpha, Omega-p, and a couple of resonance cases. That is the actual new piece: a usable formula where none existed before for the charged case. The derivation follows standard steps from effective range theory without obvious circularity or invented entities, and the applications are to real systems that come up in nuclear and hadron physics. The paper does a clean job keeping the focus on a practical recurring question—how composite these near-threshold states actually are. The numbers it produces could help people interpret scattering data or lattice results on light nuclei and exotic hadrons. The main soft spot is the weak-binding limit itself. The formula requires the binding momentum to be much smaller than the inverse range of the short-range force, yet the paper applies it to the listed states without reporting explicit checks on that hierarchy or estimating truncation error from higher terms in the expansion. For resonances or states where the energy scale sits closer to the Coulomb scale, the analytic continuation could carry uncontrolled errors. That makes the concrete results for those systems preliminary rather than definitive. This is for researchers in hadronic and nuclear effective theory who already use compositeness criteria and need the charged extension. A reader working on threshold phenomena in charged systems will get direct value from the formula and the examples. It deserves a serious referee because the core derivation is new and grounded enough to be worth checking, even if the applications would benefit from stronger validation of the approximation. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The paper derives an expression for the compositeness of near-threshold bound, virtual, and resonance states in charged hadronic systems from the Coulomb-modified effective range expansion in the weak-binding limit, giving compositeness in terms of the eigenenergy and Coulomb effective range parameter. It then applies the result to several systems including pp, αα, Ω^{-}Ω^{-}, Ξ^{-}α, Ω^{-}p, and resonance cases.

Significance. If the derivation is valid and the weak-binding limit applies, the work supplies a practical, parameter-free tool for quantifying molecular versus compact structure in systems with both Coulomb and short-range forces, extending standard effective-range theory without introducing ad-hoc parameters. This is useful for interpreting exotic hadrons and nuclear states. The approach builds directly on established ERE methods, which is a clear strength.

major comments (1)

Applications section: the formula is applied to pp, αα, Ω^{-}Ω^{-}, Ξ^{-}α, Ω^{-}p and resonance cases without explicit verification that the binding momentum |k| = sqrt(2μ|E|) satisfies |k| ≪ inverse range of the short-range force (typically ~m_π). The derivation assumes this hierarchy for the weak-binding limit; absence of |k| values, truncation-error estimates, or checks for cases where |E| is comparable to the Coulomb scale makes the reported compositeness values for these states unreliable.

minor comments (1)

Abstract: the list of states mixes bound, virtual, and resonance cases; adding a brief clause on which are bound versus resonant would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comment. We address the concern below and will revise the manuscript to incorporate explicit verification of the weak-binding limit.

read point-by-point responses

Referee: Applications section: the formula is applied to pp, αα, Ω^{-}Ω^{-}, Ξ^{-}α, Ω^{-}p and resonance cases without explicit verification that the binding momentum |k| = sqrt(2μ|E|) satisfies |k| ≪ inverse range of the short-range force (typically ~m_π). The derivation assumes this hierarchy for the weak-binding limit; absence of |k| values, truncation-error estimates, or checks for cases where |E| is comparable to the Coulomb scale makes the reported compositeness values for these states unreliable.

Authors: We agree that the absence of explicit checks for the weak-binding limit in the applications section is a valid concern and that including such verification will make the reported compositeness values more reliable and transparent. In the revised manuscript we will add a dedicated paragraph and accompanying table in the applications section. This will list the binding momentum |k| = sqrt(2μ|E|) for each system (pp, αα, Ω^{-}Ω^{-}, Ξ^{-}α, Ω^{-}p, Ω_{ccc}^{++}Ω_{ccc}^{++}, and the resonance cases), compare |k| to the inverse range of the short-range force (∼ m_π ≈ 140 MeV), provide estimates of truncation errors from higher-order terms in the Coulomb-modified effective-range expansion, and discuss proximity to the Coulomb scale. Preliminary evaluation shows that the hierarchy |k| ≪ m_π is satisfied for the systems considered (e.g., |k| ∼ few MeV for pp), but we will document the numbers and error estimates explicitly so readers can judge the applicability directly. revision: yes

Circularity Check

0 steps flagged

Derivation of compositeness from Coulomb-modified ERE is self-contained and independent of target values

full rationale

The paper derives an expression for compositeness X in the weak-binding limit directly from the Coulomb-modified effective range expansion, writing X in terms of the eigenenergy E and the Coulomb effective range parameter r_C. This follows from standard low-energy scattering theory (the ERE pole condition and residue relations) without any reduction of X to a fitted input or self-referential definition. No load-bearing step equates the output compositeness to a parameter fitted from the same states; external values of E (from experiment or lattice) and r_C are inserted after the derivation. Self-citations, if present, are not invoked to justify uniqueness or to smuggle an ansatz. The central claim therefore remains non-circular, though its accuracy depends on the validity of the weak-binding approximation (a separate correctness issue).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central result rests on the validity of the Coulomb-modified effective range expansion and the weak-binding approximation; no free parameters or new entities are introduced in the abstract description.

axioms (2)

domain assumption Coulomb-modified effective range expansion accurately describes low-energy scattering with both Coulomb and short-range forces
Invoked to derive the compositeness expression
domain assumption Weak-binding limit is applicable to the studied states
Required for the simplified expression in terms of eigenenergy and effective range

pith-pipeline@v0.9.0 · 5388 in / 1256 out tokens · 42992 ms · 2026-05-08T11:20:09.084872+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Hosaka, T

A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai, S. Yasui, Exotic hadrons with heavy flavors:𝑋,𝑌,𝑍, and related states, PTEP 2016 (2016) 062C01

2016
[2]

Thomas, A

N.Brambilla,S.Eidelman,C.Hanhart,A.Nefediev,C.-P.Shen,C.E. Thomas, A. Vairo, C.-Z. Yuan, The𝑋𝑌 𝑍states: experimental and theoretical status and perspectives, Phys. Rept. 873 (2020) 1–154

2020
[3]

F.-K.Guo,C.Hanhart,U.-G.Meißner,Q.Wang,Q.Zhao,B.-S.Zou, Hadronic molecules, Rev. Mod. Phys. 90 (2018) 015004. [Erratum: Rev.Mod.Phys. 94, 029901 (2022)]

2018
[4]

Weinberg, Evidence That the Deuteron Is Not an Elementary Particle, Phys

S. Weinberg, Evidence That the Deuteron Is Not an Elementary Particle, Phys. Rev. 137 (1965) B672–B678

1965
[5]

Hyodo, Structure and compositeness of hadron resonances, Int

T. Hyodo, Structure and compositeness of hadron resonances, Int. J. Mod. Phys. A 28 (2013) 1330045

2013
[6]

U.vanKolck,Weinberg’sCompositeness,Symmetry14(2022)1884

2022
[7]

Kinugawa, T

T. Kinugawa, T. Hyodo, Structure of exotic hadrons by a weak- bindingrelationwithfinite-rangecorrection, Phys.Rev.C106(2022) 015205

2022
[8]

Kinugawa, T

T. Kinugawa, T. Hyodo, Compositeness of hadrons, nuclei, and atomic systems, Eur. Phys. J. A 61 (2025) 154

2025
[9]

Braaten, H

E. Braaten, H. W. Hammer, Universality in few-body systems with large scattering length, Phys. Rept. 428 (2006) 259–390

2006
[10]

Naidon, S

P. Naidon, S. Endo, Efimov Physics: a review, Rept. Prog. Phys. 80 (2017) 056001

2017
[11]

C 90 (2014) 055208

T.Hyodo, Hadronmassscalingnearthes-wavethreshold, Phys.Rev. C 90 (2014) 055208

2014
[12]

Hyodo, Structure of Near-Threshold s-Wave Resonances, Phys

T. Hyodo, Structure of Near-Threshold s-Wave Resonances, Phys. Rev. Lett. 111 (2013) 132002

2013
[13]

I.Matuschek,V.Baru,F.-K.Guo,C.Hanhart, Onthenatureofnear- threshold bound and virtual states, Eur. Phys. J. A 57 (2021) 101

2021
[14]

Kinugawa, T

T. Kinugawa, T. Hyodo, Compositeness of Tcc and X(3872) by considering decay and coupled-channels effects, Phys. Rev. C 109 (2024) 045205. T. Kinugawa, T. Hyodo:Preprint submitted to ElsevierPage 3 of 4 Compositeness of near-threshold states in charged hadronic systems

2024
[15]

Kinugawa, T

T. Kinugawa, T. Hyodo, Compositeness of near-threshold𝑠-wave resonances (2024)

2024
[16]

R.Higa,H.W.Hammer,U.vanKolck, alphaalphaScatteringinHalo Effective Field Theory, Nucl. Phys. A 809 (2008) 171–188

2008
[17]

Y. Lyu, H. Tong, T. Sugiura, S. Aoki, T. Doi, T. Hatsuda, J. Meng, T. Miyamoto, Dibaryon with Highest Charm Number near Unitarity from Lattice QCD, Phys. Rev. Lett. 127 (2021) 072003

2021
[18]

Hiyama, M

E. Hiyama, M. Isaka, T. Doi, T. Hatsuda, Probing theΞN interaction throughinversionofspin-doubletsinΞN𝛼𝛼nuclei, Phys.Rev.C106 (2022) 064318

2022
[19]

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

T. Kinugawa, T. Hyodo, Compositeness of near-threshold eigen- stateswithCoulombplusshort-rangeinteractions, arXiv:2604.17813 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[20]

H. A. Bethe, Theory of the Effective Range in Nuclear Scattering, Phys. Rev. 76 (1949) 38–50

1949
[21]

Domcke, Analytic theory of resonances and bound states near Coulomb thresholds, J

W. Domcke, Analytic theory of resonances and bound states near Coulomb thresholds, J. Phys. B: Atom. Mol. Phys. 16 (1983)

1983
[22]

X. Kong, F. Ravndal, Proton proton scattering lengths from effective fieldtheory, Phys.Lett.B450(1999)320–324.[Erratum:Phys.Lett.B 458, 565–565 (1999)]

1999
[23]

X. Kong, F. Ravndal, Coulomb effects in low-energy proton proton scattering, Nucl. Phys. A 665 (2000) 137–163

2000
[24]

S.-i. Ando, J. W. Shin, C. H. Hyun, S. W. Hong, Low energy proton- proton scattering in effective field theory, Phys. Rev. C 76 (2007) 064001

2007
[25]

H.W.Hammer,R.Higa, AModelStudyofDiscreteScaleInvariance and Long-Range Interactions, Eur. Phys. J. A 37 (2008) 193–200

2008
[26]

Mochizuki, Y

S. Mochizuki, Y. Nishida, Universal bound states and resonances with Coulomb plus short-range potentials, Phys. Rev. C 110 (2024) 064001

2024
[27]

Gongyo, et al., Most Strange Dibaryon from Lattice QCD, Phys

S. Gongyo, et al., Most Strange Dibaryon from Lattice QCD, Phys. Rev. Lett. 120 (2018) 212001

2018
[28]

Iritani, et al.,𝑁Ωdibaryon from lattice QCD near the physical point, Phys

T. Iritani, et al.,𝑁Ωdibaryon from lattice QCD near the physical point, Phys. Lett. B 792 (2019) 284–289

2019
[29]

Navas, et al., Review of particle physics, Phys

S. Navas, et al., Review of particle physics, Phys. Rev. D 110 (2024) 030001

2024
[30]

P. J. Mohr, D. B. Newell, B. N. Taylor, E. Tiesinga, CODATA recommended values of the fundamental physical constants: 2022*, Rev. Mod. Phys. 97 (2025) 025002

2022
[31]

Lyu, private communication

Y. Lyu, private communication
[32]

Kamiya, private communication

Y. Kamiya, private communication
[33]

R.B.Wiringa,S.C.Pieper,J.Carlson,V.R.Pandharipande,Quantum Monte Carlo calculations of A = 8 nuclei, Phys. Rev. C 62 (2000) 014001

2000
[34]

Otsuka, T

T. Otsuka, T. Abe, T. Yoshida, Y. Tsunoda, N. Shimizu, N. Itagaki, Y. Utsuno, J. Vary, P. Maris, H. Ueno,𝛼-Clustering in atomic nucleifromfirstprincipleswithstatisticallearningandtheHoylestate character, Nature Commun. 13 (2022) 2234. T. Kinugawa, T. Hyodo:Preprint submitted to ElsevierPage 4 of 4

2022