Open Access

$D_{s 1} (2460)$ and other open-charm $1^{+}$ states in relativistic chiral effective field theory

Ze-Rui Liang^1,2, Qi-Chao Xiao ^2,3, Zhi-Hui Guo⁴, and De-Liang Yao ^2,3,*

¹College of Physics and Hebei Key Laboratory of Photophysics Research and Application and Physics Postdoctoral Research Station at Hebei Normal University, Hebei Normal University, Shijiazhuang, Hebei 050024, China
²School of Physics and Electronics, Hunan University, 410082 Changsha, China
³Hunan Provincial Key Laboratory of High-Energy Scale Physics and Applications, Hunan University, 410082 Changsha, China
⁴College of Physics and Hebei Key Laboratory of Photophysics Research and Application, Hebei Normal University, Shijiazhuang, Hebei 050024, China

^*Contact author: yaodeliang@hnu.edu.cn

Phys. Rev. D 114, 074027 – Published 14 October, 2026

DOI: https://doi.org/10.1103/gl5b-hvbm

Abstract

We derive the pertinent chiral potentials for charmed vector meson interactions with light pseudoscalar bosons in a relativistic U(3) chiral effective field theory up to next-to-leading order. Predictions for the $S$ - and $P$ -wave scattering lengths are obtained for all the relevant elastic channels. A comparison with the most recent—and currently the sole—lattice QCD data on the $S$ -wave $I = 1 / 2$ $D^{*} π$ scattering length at a pion mass of 391 MeV reveals good agreement, thereby validating the estimation of low-energy constants via heavy quark spin symmetry. Within the relativistic formalism, we confirm that the $D_{s 1} (2460)$ can be identified with a bound-state pole, while the $D_{1} (2430)$ corresponds to the interplay of two poles: a lower one on the second Riemann sheet and a higher one on the third Riemann sheet. We show that the $D_{s 1} (2460)$ and the lower $D_{1} (2430)$ pole originate from the same flavor SU(3) triplet, whereas the higher $D_{1} (2430)$ pole belongs to the SU(3) sextet. All these states are not of $\bar{q} q$ nature, as they flow to complex infinity in the large $N_{C}$ limit. Our results provide quantitative benchmarks for future lattice QCD and experimental studies, particularly those using femtoscopy.

Physics Subject Headings (PhySH)

Article Text

References (95)

B. Aubert et al. (BABAR Collaboration), Observation of a narrow meson decaying to $D_{s}^{+} π^{0}$ at a mass of 2.32- $GeV / c^{2}$ , Phys. Rev. Lett. 90, 242001 (2003).
D. Besson et al. (CLEO Collaboration), Observation of a narrow resonance of mass $2.46 GeV / c^{2}$ decaying to $D_{s}^{* +} π^{0}$ and confirmation of the $D_{s J}^{*} (2317)$ state, Phys. Rev. D 68, 032002 (2003); 75, 119908(E) (2007).
S. Godfrey and N. Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
S. Godfrey and R. Kokoski, The properties of $P$ wave mesons with one heavy quark, Phys. Rev. D 43, 1679 (1991).
H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, A review of the open charm and open bottom systems, Rep. Prog. Phys. 80, 076201 (2017).
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).
L. Meng, B. Wang, G.-J. Wang, and S.-L. Zhu, Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules, Phys. Rep. 1019, 1 (2023).
M.-Z. Liu, Y.-W. Pan, Z.-W. Liu, T.-W. Wu, J.-X. Lu, and L.-S. Geng, Three ways to decipher the nature of exotic hadrons: Multiplets, three-body hadronic molecules, and correlation functions, Phys. Rep. 1108, 1 (2025).
Z.-G. Wang, Review of the QCD sum rules for exotic states, Front. Phys. (Beijing) 21, 016300 (2026).
J. M. Link et al. (FOCUS Collaboration), Measurement of masses and widths of excited charm mesons $D_{2}^{*}$ and evidence for broad states, Phys. Lett. B 586, 11 (2004).
K. Abe et al. (Belle Collaboration), Study of $B^{-} \to D^{* * 0} π^{-}$ ( $D^{* * 0} \to D^{(*) +} π^{-}$ ) decays, Phys. Rev. D 69, 112002 (2004).
J. R. Peláez, From controversy to precision on the sigma meson: A review on the status of the non-ordinary $f_{0} (500)$ resonance, Phys. Rep. 658, 1 (2016).
D.-L. Yao, L.-Y. Dai, H.-Q. Zheng, and Z.-Y. Zhou, A review on partial-wave dynamics with chiral effective field theory and dispersion relation, Rep. Prog. Phys. 84, 076201 (2021).
M.-L. Du, F.-K. Guo, C. Hanhart, B. Kubis, and U.-G. Meißner, Where is the lightest charmed scalar meson?, Phys. Rev. Lett. 126, 192001 (2021).
M.-L. Du, F.-K. Guo, and U.-G. Meißner, Implications of chiral symmetry on $S$ -wave pionic resonances and the scalar charmed mesons, Phys. Rev. D 99, 114002 (2019).
G. Burdman and J. F. Donoghue, Union of chiral and heavy quark symmetries, Phys. Lett. B 280, 287 (1992).
T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin, and H.-L. Yu, Heavy quark symmetry and chiral dynamics, Phys. Rev. D 46, 1148 (1992); 55, 5851(E) (1997).
M. B. Wise, Chiral perturbation theory for hadrons containing a heavy quark, Phys. Rev. D 45, R2188 (1992).
J. Gasser and U. G. Meißner, Chiral expansion of pion form factors beyond one loop, Nucl. Phys. B357, 90 (1991).
A. Dobado and J. R. Peláez, The inverse amplitude method in chiral perturbation theory, Phys. Rev. D 56, 3057 (1997).
J. A. Oller and E. Oset, Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the $σ$ , $f_{0} (980)$ , $a_{0} (980)$ scalar mesons, Nucl. Phys. A620, 438 (1997); A652, 407(E) (1999).
J. A. Oller and E. Oset, N/D description of two meson amplitudes and chiral symmetry, Phys. Rev. D 60, 074023 (1999).
J. A. Oller, Coupled-channel approach in hadron–hadron scattering, Prog. Part. Nucl. Phys. 110, 103728 (2020).
S. Weinberg, Phenomenological Lagrangians, Physica A (Amsterdam) 96, 327 (1979).
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (N.Y.) 158, 142 (1984).
J. Gasser and H. Leutwyler, Chiral perturbation theory: Expansions in the mass of the strange quark, Nucl. Phys. B250, 465 (1985).
E. E. Kolomeitsev and M. F. M. Lutz, On heavy light meson resonances and chiral symmetry, Phys. Lett. B 582, 39 (2004).
F.-K. Guo, P.-N. Shen, H.-C. Chiang, R.-G. Ping, and B.-S. Zou, Dynamically generated $0^{+}$ heavy mesons in a heavy chiral unitary approach, Phys. Lett. B 641, 278 (2006).
D. Gamermann, E. Oset, D. Strottman, and M. J. Vicente Vacas, Dynamically generated open and hidden charm meson systems, Phys. Rev. D 76, 074016 (2007).
M. Altenbuchinger, L. S. Geng, and W. Weise, Scattering lengths of Nambu-Goldstone bosons off $D$ mesons and dynamically generated heavy-light mesons, Phys. Rev. D 89, 014026 (2014).
M. Altenbuchinger and L.-S. Geng, Off-shell effects on the interaction of Nambu-Goldstone bosons and $D$ mesons, Phys. Rev. D 89, 054008 (2014).
Z.-H. Guo, U.-G. Meißner, and D.-L. Yao, New insights into the $D_{s 0}^{*} (2317)$ and other charm scalar mesons, Phys. Rev. D 92, 094008 (2015).
M. Albaladejo, P. Fernández-Soler, F.-K. Guo, and J. Nieves, Two-pole structure of the $D_{0}^{*} (2400)$ , Phys. Lett. B 767, 465 (2017).
M.-L. Du, M. Albaladejo, P. Fernández-Soler, F.-K. Guo, C. Hanhart, U.-G. Meißner, J. Nieves, and D.-L. Yao, Towards a new paradigm for heavy-light meson spectroscopy, Phys. Rev. D 98, 094018 (2018).
J. Luo, B. Wu, P.-P. Shi, and M.-L. Du, Analysis of the $D_{0}^{*} (2300)$ resonance from lattice QCD under chiral symmetry, Phys. Rev. D 114, 014066 (2026).
Z. Zhuang, F. G. Domínguez, and R. Molina, Revealing the $D_{0}^{*} (2300)$ two-pole structure from lattice data and the SU(3) limit, arXiv:2606.03826.
U.-G. Meißner, Two-pole structures in QCD: Facts, not fantasy!, Symmetry 12, 981 (2020).
F.-K. Guo, C. Hanhart, and U.-G. Meißner, Interactions between heavy mesons and Goldstone bosons from chiral dynamics, Eur. Phys. J. A 40, 171 (2009).
L. Liu, K. Orginos, F.-K. Guo, C. Hanhart, and U.-G. Meißner, Interactions of charmed mesons with light pseudoscalar mesons from lattice QCD and implications on the nature of the $D_{s 0}^{*} (2317)$ , Phys. Rev. D 87, 014508 (2013).
Y.-R. Liu, X. Liu, and S.-L. Zhu, Light pseudoscalar meson and heavy meson scattering lengths, Phys. Rev. D 79, 094026 (2009).
L. S. Geng, N. Kaiser, J. Martin-Camalich, and W. Weise, Low-energy interactions of Nambu-Goldstone bosons with $D$ mesons in covariant chiral perturbation theory, Phys. Rev. D 82, 054022 (2010).
D.-L. Yao, M.-L. Du, F.-K. Guo, and U.-G. Meißner, One-loop analysis of the interactions between charmed mesons and Goldstone bosons, J. High Energy Phys. 11 (2015) 058.
M.-L. Du, F.-K. Guo, U.-G. Meißner, and D.-L. Yao, Study of open-charm $0^{+}$ states in unitarized chiral effective theory with one-loop potentials, Eur. Phys. J. C 77, 728 (2017).
B.-L. Huang, Z.-Y. Lin, and S.-L. Zhu, Light pseudoscalar meson and heavy meson scattering lengths to $O (p^{4})$ in heavy meson chiral perturbation theory, Phys. Rev. D 105, 036016 (2022).
M. F. M. Lutz, X.-Y. Guo, Y. Heo, and C. L. Korpa, Coupled-channel dynamics with chiral long-range forces in the open-charm sector of QCD, Phys. Rev. D 106, 114038 (2022).
T. Isken, X.-Y. Guo, Y. Heo, C. L. Korpa, and M. F. M. Lutz, Triangle and box diagrams in coupled-channel systems from the chiral Lagrangian, Phys. Rev. D 109, 034032 (2024).
S.-Z. Jiang, Y.-R. Liu, and Q.-H. Yang, Chiral Lagrangians for mesons with a single heavy quark, Phys. Rev. D 99, 074018 (2019).
D. Mohler, C. B. Lang, L. Leskovec, S. Prelovsek, and R. M. Woloshyn, $D_{s 0}^{*} (2317)$ meson and $D$ -meson-kaon scattering from lattice QCD, Phys. Rev. Lett. 111, 222001 (2013).
D. Mohler, S. Prelovsek, and R. M. Woloshyn, $D π$ scattering and $D$ meson resonances from lattice QCD, Phys. Rev. D 87, 034501 (2013).
G. Moir, M. Peardon, S. M. Ryan, C. E. Thomas, and D. J. Wilson, Coupled-channel $D π$ , $D η$ and $D_{s} \bar{K}$ scattering from lattice QCD, J. High Energy Phys. 10 (2016) 011.
G. S. Bali, S. Collins, A. Cox, and A. Schäfer, Masses and decay constants of the $D_{s 0}^{*} (2317)$ and $D_{s 1} (2460)$ from $N_{f} = 2$ lattice QCD close to the physical point, Phys. Rev. D 96, 074501 (2017).
G. K. C. Cheung, C. E. Thomas, D. J. Wilson, G. Moir, M. Peardon, and S. M. Ryan (Hadron Spectrum Collaboration), $D K$ $I = 0$ , $D \bar{K}$ $I = 0$ , 1 scattering and the $D_{s 0}^{*} (2317)$ from lattice QCD, J. High Energy Phys. 02 (2021) 100.
H. Yan, C. Liu, L. Liu, Y. Meng, and H. Xing, Pion mass dependence in $D π$ scattering and the $D_{0}^{*} (2300)$ resonance from lattice QCD, Phys. Rev. D 111, 014503 (2025).
M.-L. Du, F.-K. Guo, U.-G. Meißner, and D.-L. Yao, Aspects of the low-energy constants in the chiral Lagrangian for charmed mesons, Phys. Rev. D 94, 094037 (2016).
D. L. Yao, P. Fernández-Soler, M. Albaladejo, F. K. Guo, and J. Nieves, Heavy-to-light scalar form factors from Muskhelishvili–Omnès dispersion relations, Eur. Phys. J. C 78, 310 (2018).
N. Lang and D. J. Wilson (Hadron Spectrum Collaboration), Axial-vector $D_{1}$ hadrons in $D^{*} π$ scattering from QCD, Phys. Rev. Lett. 129, 252001 (2022).
N. Lang and D. J. Wilson (Hadron Spectrum Collaboration), $D_{1}$ and $D_{2}$ resonances in coupled-channel scattering amplitudes from lattice QCD, J. High Energy Phys. 07 (2025) 060.
N. Isgur and M. B. Wise, Spectroscopy with heavy quark symmetry, Phys. Rev. Lett. 66, 1130 (1991).
S. Acharya et al. (ALICE Collaboration), Studying the interaction between charm and light-flavor mesons, Phys. Rev. D 110, 032004 (2024).
S. Pratt, Pion interferometry of quark-gluon plasma, Phys. Rev. D 33, 1314 (1986).
J. Hofmann and M. F. M. Lutz, Open charm meson resonances with negative strangeness, Nucl. Phys. A733, 142 (2004).
F.-K. Guo, P.-N. Shen, and H.-C. Chiang, Dynamically generated $1^{+}$ heavy mesons, Phys. Lett. B 647, 133 (2007).
D. Gamermann and E. Oset, Axial resonances in the open and hidden charm sectors, Eur. Phys. J. A 33, 119 (2007).
X.-Y. Guo, Y. Heo, and M. F. M. Lutz, On chiral excitations with exotic quantum numbers, Phys. Lett. B 791, 86 (2019).
E. Witten, Current algebra theorems for the U(1) Goldstone boson, Nucl. Phys. B156, 269 (1979).
G. Veneziano, U(1) without instantons, Nucl. Phys. B159, 213 (1979).
S. R. Coleman and E. Witten, Chiral symmetry breakdown in large N chromodynamics, Phys. Rev. Lett. 45, 100 (1980).
A. V. Manohar, Large N QCD, in Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions (1998), pp. 1091–1169, arXiv:hep-ph/9802419.
J. Oller and U. G. Meißner, Chiral dynamics in the presence of bound states: Kaon nucleon interactions revisited, Phys. Lett. B 500, 263 (2001).
F. Takahashi et al. (Particle Data Group Collaboration), Review of particle physics, Int. J. Mod. Phys. A 41, 2630011 (2026).
M. Gell-Mann, R. J. Oakes, and B. Renner, Behavior of current divergences under $SU (3) \times SU (3)$ , Phys. Rev. 175, 2195 (1968).
M. Jacob and G. C. Wick, On the general theory of collisions for particles with spin, Ann. Phys. (N.Y.) 7, 404 (1959).
A. D. Martin and T. D. Spearman, Elementary Particle Theory (North-Holland Publishing Co., Amsterdam, 1970).
Z.-H. Guo, L. Liu, U.-G. Meißner, J. A. Oller, and A. Rusetsky, Towards a precise determination of the scattering amplitudes of the charmed and light-flavor pseudoscalar mesons, Eur. Phys. J. C 79, 13 (2019).
Z.-W. Liu, Y.-R. Liu, X. Liu, and S.-L. Zhu, The Pseudoscalar meson and heavy vector meson scattering lengths, Phys. Rev. D 84, 034002 (2011).
J. Nieves and E. Ruiz Arriola, Bethe-Salpeter approach for meson-meson scattering in chiral perturbation theory, Phys. Lett. B 455, 30 (1999).
J. Nieves and E. Ruiz Arriola, Bethe-Salpeter approach for unitarized chiral perturbation theory, Nucl. Phys. A679, 57 (2000).
J. Nieves and E. Ruiz Arriola, Bethe-Salpeter approach for the $P_{33}$ elastic pion-nucleon scattering in heavy baryon chiral perturbation theory, Phys. Rev. D 63, 076001 (2001).
P. C. Bruns, M. Mai, and U. G. Meissner, Chiral dynamics of the $S_{11} (1535)$ and $S_{11} (1650)$ resonances revisited, Phys. Lett. B 697, 254 (2011).
J.-M. Xie, Z.-W. Liu, J.-X. Lu, H. Liang, and L.-S. Geng, Off-shell chiral dynamics in the $Λ (1405)$ resonance and $K^{-} p / π^{\pm} Σ^{\mp}$ femtoscopic correlations, Phys. Rev. D 114, 014046 (2026).
L. Gayer, N. Lang, S. M. Ryan, D. Tims, C. E. Thomas, and D. J. Wilson (Hadron Spectrum Collaboration), Isospin- $1 / 2$ $D π$ scattering and the lightest $D_{0}^{*}$ resonance from lattice QCD, J. High Energy Phys. 07 (2021) 123.
X.-K. Guo, Z.-H. Guo, J. A. Oller, and J. J. Sanz-Cillero, Scrutinizing the $η − η^{'}$ mixing, masses and pseudoscalar decay constants in the framework of U(3) chiral effective field theory, J. High Energy Phys. 06 (2015) 175.
Z.-H. Zhang and F.-K. Guo, Classification of coupled-channel near-threshold structures, Phys. Lett. B 863, 139387 (2025).
R. J. Eden and J. R. Taylor, Poles and shadow poles in the many-channel S matrix, Phys. Rev. 133, B1575 (1964).
A. Asokan, M.-N. Tang, F.-K. Guo, C. Hanhart, Y. Kamiya, and U.-G. Meißner, Can the two-pole structure of the $D_{0}^{*} (2300)$ be understood from recent lattice data?, Eur. Phys. J. C 83, 850 (2023).
M. L. Du, F. K. Guo, C. Hanhart, F. Herren, B. Kubis, and R. van Tonder, Discovering the $D_{0}^{*} (2100)$ in B semileptonic decays, Eur. Phys. J. C 85, 1289 (2025).
J. R. Peláez, On the nature of light scalar mesons from their large $N_{c}$ behavior, Phys. Rev. Lett. 92, 102001 (2004).
Z. Xiao and H. Q. Zheng, Analyticity and the $N_{c}$ counting rule of $S$ matrix poles, Mod. Phys. Lett. A 22, 55 (2007).
J. R. Peláez and G. Rios, Nature of the $f_{0} (600)$ from its $N_{c}$ dependence at two loops in unitarized chiral perturbation theory, Phys. Rev. Lett. 97, 242002 (2006).
Z.-H. Guo, $N_{c}$ evolution of the light meson resonances, Eur. Phys. J. Special Topics 230, 1659 (2021).
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B72, 461 (1974).
E. Witten, Baryons in the $1 / N$ expansion, Nucl. Phys. B160, 57 (1979).
J. R. Peláez, Light scalars as tetraquarks or two-meson states from large $N_{c}$ and unitarized chiral perturbation theory, Mod. Phys. Lett. A 19, 2879 (2004).
Z.-H. Guo, J. A. Oller, and J. Ruiz de Elvira, Chiral dynamics in form factors, spectral-function sum rules, meson-meson scattering and semi-local duality, Phys. Rev. D 86, 054006 (2012).
Z.-H. Guo and J. A. Oller, Resonances from meson-meson scattering in U(3) CHPT, Phys. Rev. D 84, 034005 (2011).

Phys. Rev. D 114, 074027– Published 14 October, 2026

Vol. 114, Iss. 7 — 1 October 2026

Received 5 July 2026
Accepted 24 September 2026

DOI: https://doi.org/10.1103/gl5b-hvbm

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.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

$D_{s 1} (2460)$ and other open-charm $1^{+}$ states in relativistic chiral effective field theory

Abstract

Physics Subject Headings (PhySH)

Article Text

References (95)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Ds1(2460) and other open-charm 1+ states in relativistic chiral effective field theory

Abstract

Physics Subject Headings (PhySH)

Article Text

References (95)

$D_{s 1} (2460)$ and other open-charm $1^{+}$ states in relativistic chiral effective field theory