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Compositeness and wave function of shallow bound states in relation to scattering observables

Ibuki Terashima ^* and Tetsuo Hyodo ^†,‡

Department of Physics, Tokyo Metropolitan University, Hachioji 192-0397, Japan

^*Contact author: terashima-ibuki@ed.tmu.ac.jp
^†Contact author: hyodo@rcnp.osaka-u.ac.jp
^‡Present address: Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan.

Phys. Rev. C 113, 035202 – Published 3 March, 2026

DOI: https://doi.org/10.1103/9yk7-1rrc

Abstract

We study the internal structure of exotic hadrons, especially focusing on the relation between the compositeness and physical observables. Defined as the probability of finding hadronic molecular components in the wave function, compositeness serves as a quantitative measure of the internal structure of exotic hadrons. We utilize the coupled-channel potential model incorporating both quark and hadron degrees of freedom, which naturally generate the “bare state” responsible for the elementary component as the bound state in the quark channel. The behavior of the compositeness under the variation of the model parameters is investigated by using the $X (3872)$ as an example. In particular, we analyze the associated scattering phase shifts and the bound-state wave functions to discuss the relation between the compositeness and the scattering observables for a shallow bound state. As a phenomenological application of the present framework, the compositeness of the $X (3872), T_{c c} (3875), D_{s 0} (2317)$ , and $D_{s 1} (2460)$ is discussed.

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S. K. Choi et al. (Belle Collaboration), Observation of a narrow charmonium-like state in exclusive $B^{\pm} \to K^{\pm} π^{+} π^{-} J / ψ$ decays, Phys. Rev. Lett. 91, 262001 (2003).
R. Aaij et al. (LHCb Collaboration), Observation of $J / ψ p$ resonances consistent with pentaquark states in $Λ_{b}^{0} \to J / ψ K^{-} p$ decays, Phys. Rev. Lett. 115, 072001 (2015).
R. Aaij et al. (LHCb Collaboration), Observation of a narrow pentaquark state, $P_{c} {(4312)}^{+}$ , and of two-peak structure of the $P_{c} {(4450)}^{+}$ , Phys. Rev. Lett. 122, 222001 (2019).
R. Aaij et al. (LHCb Collaboration), Observation of an exotic narrow doubly charmed tetraquark, Nat. Phys. 18, 751 (2022).
R. Aaij et al. (LHCb Collaboration), Study of the doubly charmed tetraquark $T_{c c}^{+}$ , Nat. Commun. 13, 3351 (2022).
A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai, and S. Yasui, Exotic hadrons with heavy flavors: $X, Y, Z$ , and related states, Prog. Theor. Exp. Phys. 2016, 062C01 (2016).
F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018); 94, 029901(E) (2022).
N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, The $X Y Z$ states: Experimental and theoretical status and perspectives, Phys. Rep. 873, 1 (2020).
T. Hyodo and M. Niiyama, QCD and the strange baryon spectrum, Prog. Part. Nucl. Phys. 120, 103868 (2021).
S. Weinberg, Evidence that the deuteron is not an elementary particle, Phys. Rev. 137, B672 (1965).
V. Baru, J. Haidenbauer, C. Hanhart, Y. Kalashnikova, and A. E. Kudryavtsev, Evidence that the $a_{0} (980)$ and $f_{0} (980)$ are not elementary particles, Phys. Lett. B 586, 53 (2004).
T. Hyodo, D. Jido, and A. Hosaka, Compositeness of dynamically generated states in a chiral unitary approach, Phys. Rev. C 85, 015201 (2012).
T. Hyodo, Structure and compositeness of hadron resonances, Int. J. Mod. Phys. A 28, 1330045 (2013).
Y. Kamiya and T. Hyodo, Structure of near-threshold quasibound states, Phys. Rev. C 93, 035203 (2016).
Y. Kamiya and T. Hyodo, Generalized weak-binding relations of compositeness in effective field theory, Prog. Theor. Exp. Phys. 2017, 023D02 (2017).
J. A. Oller, New results from a number operator interpretation of the compositeness of bound and resonant states, Ann. Phys. 396, 429 (2018).
U. van Kolck, Weinberg's compositeness, Symmetry 14, 1884 (2022).
T. Kinugawa and T. Hyodo, Compositeness of hadrons, nuclei, and atomic systems, Eur. Phys. J. 61, 154 (2025).
Y. Li, F.-K. Guo, J.-Y. Pang, and J.-J. Wu, Generalization of Weinberg's compositeness relations, Phys. Rev. D 105, L071502 (2022).
M. Albaladejo and J. Nieves, Compositeness of S-wave weakly-bound states from next-to-leading order Weinberg's relations, Eur. Phys. J. C 82, 724 (2022).
T. Kinugawa and T. Hyodo, Structure of exotic hadrons by a weak-binding relation with finite-range correction, Phys. Rev. C 106, 015205 (2022).
H. Sazdjian, The interplay between compact and molecular structures in tetraquarks, Symmetry 14, 515 (2022).
R. F. Lebed and S. R. Martinez, Diabatic representation of exotic hadrons in the dynamical diquark model, Phys. Rev. D 106, 074007 (2022).
T. Kinugawa and T. Hyodo, Compositeness of $T_{c c}$ and $X (3872)$ by considering decay and coupled-channels effects, Phys. Rev. C 109, 045205 (2024).
J. Song, L. R. Dai, and E. Oset, Evolution of compact states to molecular ones with coupled channels: The case of the $X$ (3872), Phys. Rev. D 108, 114017 (2023).
I. Terashima and T. Hyodo, Hadron-hadron potentials coupled to quark degrees of freedom for exotic hadrons, Phys. Rev. C 108, 035204 (2023).
I. Terashima and T. Hyodo, Properties of $X (3872)$ from hadronic potentials coupled to quarks, in 16th Conference on Quark Confinement and the Hadron Spectrum PoS (QCHSC24) (2025), Vol. 483, p. 134.
H. Feshbach, Unified theory of nuclear reactions, Ann. Phys. 5, 357 (1958).
H. Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. 19, 287 (1962).
Y. Yamaguchi, Two nucleon problem when the potential is nonlocal but separable. I, Phys. Rev. 95, 1628 (1954).
Y. Yamaguchi and Y. Yamaguchi, Two nucleon problem when the potential is nonlocal but separable. II, Phys. Rev. 95, 1635 (1954).
S. Aoki and K. Yazaki, Derivative expansion in the HAL QCD method for a separable potential, Prog. Theor. Exp. Phys. 2022, 033B04 (2022).
M. Takizawa and S. Takeuchi, $X$ (3872) as a hybrid state of charmonium and the hadronic molecule, Prog. Theor. Exp. Phys. 2013, 903D01 (2013).
T. Hyodo, D. Jido, and A. Hosaka, Origin of the resonances in the chiral unitary approach, Phys. Rev. C 78, 025203 (2008).
K. Miyahara, T. Hyodo, and W. Weise, Construction of a local $\bar{K} N - π Σ - π Λ$ potential and composition of the $Λ (1405)$ , Phys. Rev. C 98, 025201 (2018).
J. Formánek, R. J. Lombard, and J. Mareš, Wave equations with energy-dependent potentials, Czech. J. Phys. 54, 289 (2004).
K. Miyahara and T. Hyodo, Structure of $Λ$ (1405) and construction of $\bar{K} N$ local potential based on chiral SU(3) dynamics, Phys. Rev. C 93, 015201 (2016).
S. Navas et al. (Particle Data Group Collaboration), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
S. Godfrey and N. Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
E. Braaten and H. W. Hammer, Universality in few-body systems with large scattering length, Phys. Rep. 428, 259 (2006).
P. Naidon and S. Endo, Efimov physics: A review, Rep. Prog. Phys. 80, 056001 (2017).
T. Hyodo, Hadron mass scaling near the $s$ -wave threshold, Phys. Rev. C 90, 055208 (2014).
J. Song, L. R. Dai, and E. Oset, How much is the compositeness of a bound state constrained by $a$ and $r_{0}$ ? The role of the interaction range, Eur. Phys. J. A 58, 133 (2022).
M. Karliner and J. L. Rosner, Discovery of doubly-charmed $Ξ_{c c}$ baryon implies a stable ( $b b \bar{u} \bar{d}$ ) tetraquark, Phys. Rev. Lett. 119, 202001 (2017).
V. Baru, X.-K. Dong, M.-L. Du, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Effective range expansion for narrow near-threshold resonances, Phys. Lett. B 833, 137290 (2022).
A. Martínez Torres, E. Oset, S. Prelovsek, and A. Ramos, Reanalysis of lattice QCD spectra leading to the $D_{s 0}^{*} (2317)$ and $D_{s 1}^{*} (2460)$ , J. High Energy Phys. 05 (2015) 153.
G.-J. Wang, Z. Yang, J.-J. Wu, M. Oka, and S.-L. Zhu, New insight into the exotic states strongly coupled with the $D D^{*}$ from the $T_{c c}^{+}$ , Sci. Bull. 69, 3036 (2024).
M.-L. Du, V. Baru, X.-K. Dong, A. Filin, F.-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, Coupled-channel approach to $T_{c c}^{+}$ including three-body effects, Phys. Rev. D 105, 014024 (2022).
L. R. Dai, J. Song, and E. Oset, Evolution of genuine states to molecular ones: The $T_{c c} (3875)$ case, Phys. Lett. B 846, 138200 (2023); 864, 139424 (2025).

Phys. Rev. C 113, 035202– Published 3 March, 2026

Vol. 113, Iss. 3 — March 2026

Received 26 May 2025
Revised 27 October 2025
Accepted 3 December 2025

DOI: https://doi.org/10.1103/9yk7-1rrc

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP³.

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Compositeness and wave function of shallow bound states in relation to scattering observables

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