Open Access

Is the $3 S − 2 D$ mixing strong for the charmonia $ψ (4040)$ and $ψ (4160)$ ?

Zi-Long Man, Si-Qiang Luo, and Xiang Liu ^*

School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China; Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Key Laboratory of Quantum Theory and Applications of MoE, Gansu Provincial Research Center for Basic Disciplines of Quantum Physics, Lanzhou University, Lanzhou 730000, China; MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China, and Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China

^*Contact author: xiangliu@lzu.edu.cn

Phys. Rev. D 112, 074025 – Published 23 October, 2025

DOI: https://doi.org/10.1103/76fd-njsy

Abstract

In this work, we revisit the $3 S − 2 D$ mixing scheme for the charmonia $ψ (4040)$ and $ψ (4160)$ . We introduce a coupled-channel mechanism—distinct from the tensor-force contribution in potential models, which alone is insufficient to induce significant mixing—to describe the mixing between these states. Our analysis yields mixing angles of $θ_{1} = 7 °$ and $θ_{2} = 10 °$ , inconsistent with the larger angle inferred from experimental data, such as the dielectronic widths of the $ψ (4040)$ and $ψ (4160)$ . We discuss possible origins of this discrepancy and emphasize the need for future experiments to resolve it. Precise measurements of the resonance parameters and dielectronic decay widths, via both inclusive and exclusive processes, will be crucial in clarifying this issue.

Physics Subject Headings (PhySH)

Quark model

Article Text

References (36)

R. Brandelik et al. (DASP Collaboration), Total cross-section for hadron production by $e^{+} e^{-}$ annihilation at center-of-mass energies between 3.6 GeV and 5.2 GeV, Phys. Lett. 76B, 361 (1978).
J. L. Rosner, Charmless final states and $S − D$ wave mixing in the $ψ^{''}$ , Phys. Rev. D 64, 094002 (2001).
Z.-L. Man, S.-Q. Luo, Z.-Y. Bai, and X. Liu, Coupled-channel study of $4 S − 3 D$ mixing dynamics in $ψ (4220)$ and $ψ (4380)$ , Phys. Lett. B 868, 139644 (2025).
K.-T. Chao, Interpretations for the $X (4160)$ observed in the double charm production at B factories, Phys. Lett. B 661, 348 (2008).
B.-Q. Li and K.-T. Chao, Higher charmonia and $X$ , $Y$ , $Z$ states with screened potential, Phys. Rev. D 79, 094004 (2009).
A. M. Badalian, B. L. G. Bakker, and I. V. Danilkin, The $S − D$ mixing and di-electron widths of higher charmonium $1^{- -}$ states, Phys. At. Nucl. 72, 638 (2009).
M. N. Anwar, Y. Lu, and B.-S. Zou, Modeling charmonium- $η$ decays of $J^{P C} = 1^{- -}$ higher charmonia, Phys. Rev. D 95, 114031 (2017).
Y. Yang, Y. Guo, J. Sun, N. Wang, Q. Chang, and G. Lu, $B_{u} \to ψ M$ decays and $S − D$ wave mixing effects, Chin. Phys. C 42, 113102 (2018).
J.-Z. Wang and X. Liu, Confirming the existence of a new higher charmonium $ψ (4500)$ by the newly released data of $e^{+} e^{-} \to K^{+} K^{-} J / ψ$ , Phys. Rev. D 107, 054016 (2023).
C. A. Bokade and Bhaghyesh, Conventional and XYZ charmonium states in a relativistic screened potential model, Phys. Rev. D 111, 014030 (2025).
Z. Zhao, K. Xu, A. Limphirat, W. Sreethawong, N. Tagsinsit, A. Kaewsnod, X. Liu, K. Khosonthongkee, S. Cheedket, and Y. Yan, Mass spectrum of $1^{- -}$ heavy quarkonium, Phys. Rev. D 109, 016012 (2024).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
K. K. Seth, Alternative analysis of the R measurements: Resonance parameters of the higher vector states of charmonium, Phys. Rev. D 72, 017501 (2005).
M. Ablikim et al. (BES Collaboration), Determination of the $ψ (3770)$ , $ψ (4040)$ , $ψ (4160)$ and $ψ (4415)$ resonance parameters, eConf C 070805, 02 (2007).
H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rep. 639, 1 (2016).
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).
Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Pentaquark and tetraquark states, Prog. Part. Nucl. Phys. 107, 237 (2019).
M. Mai, U.-G. Meißner, and C. Urbach, Towards a theory of hadron resonances, Phys. Rep. 1001, 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).
A. De Rujula, H. Georgi, and S. L. Glashow, Hadron masses in a gauge theory, Phys. Rev. D 12, 147 (1975).
S. Godfrey and N. Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
A. Le Yaouanc, L. Oliver, O. Pene, and J. C. Raynal, Naive quark pair creation model of strong interaction vertices, Phys. Rev. D 8, 2223 (1973).
A. Le Yaouanc, L. Oliver, O. Pene, and J. C. Raynal, Naive quark pair creation model and baryon decays, Phys. Rev. D 9, 1415 (1974).
M. R. Pennington and D. J. Wilson, Decay channels and charmonium mass-shifts, Phys. Rev. D 76, 077502 (2007).
M.-X. Duan, S.-Q. Luo, X. Liu, and T. Matsuki, Possibility of charmoniumlike state $X (3915)$ as $χ_{c 0} (2 P)$ state, Phys. Rev. D 101, 054029 (2020).
M.-X. Duan and X. Liu, Where are $3 P$ and higher $P$ -wave states in the charmonium family?, Phys. Rev. D 104, 074010 (2021).
Q. Deng, R.-H. Ni, Q. Li, and X.-H. Zhong, Charmonia in an unquenched quark model, Phys. Rev. D 110, 056034 (2024).
Z.-Y. Zhou and Z. Xiao, Comprehending heavy charmonia and their decays by hadron loop effects, Eur. Phys. J. A 50, 165 (2014).
T.-C. Peng, Z.-Y. Bai, J.-Z. Wang, and X. Liu, Reevaluating the $ψ (4160)$ resonance parameter using $B^{+} \to K^{+} μ^{+} μ^{-}$ data in the context of unquenched charmonium spectroscopy, Phys. Rev. D 111, 054023 (2025).
R.-H. Ni, Q. Deng, J.-J. Wu, and X.-H. Zhong, Bottomonia in an unquenched quark model, Phys. Rev. D 111, 114027 (2025).
X. H. Mo, C. Z. Yuan, and P. Wang, On the leptonic partial widths of the excited $ψ$ states, Phys. Rev. D 82, 077501 (2010).
R. Aaij et al. (LHCb Collaboration), Observation of a resonance in $B^{+} \to K^{+} μ^{+} μ^{-}$ decays at low recoil, Phys. Rev. Lett. 111, 112003 (2013).
M. Ablikim et al. (BESIII Collaboration), Precise measurement of the $e^{+} e^{-} \to D_{s}^{* +} D_{s}^{* -}$ cross sections at center-of-mass energies from threshold to 4.95 GeV, Phys. Rev. Lett. 131, 151903 (2023).
W. Kwong, P. B. Mackenzie, R. Rosenfeld, and J. L. Rosner, Quarkonium annihilation rates, Phys. Rev. D 37, 3210 (1988).
J.-Z. Wang, D.-Y. Chen, X. Liu, and T. Matsuki, Constructing $J / ψ$ family with updated data of charmoniumlike $Y$ states, Phys. Rev. D 99, 114003 (2019).
I. Balossino, F. Cossio, R. Farinelli, and L. Lavezzi, The CGEM-IT: An upgrade for the BESIII experiment, Symmetry 14, 905 (2022).

Phys. Rev. D 112, 074025– Published 23 October, 2025

Vol. 112, Iss. 7 — 1 October 2025

Received 26 July 2025
Accepted 1 October 2025

DOI: https://doi.org/10.1103/76fd-njsy

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

Is the $3 S − 2 D$ mixing strong for the charmonia $ψ (4040)$ and $ψ (4160)$ ?

Abstract

Physics Subject Headings (PhySH)

Article Text

References (36)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Is the 3S−2D mixing strong for the charmonia ψ(4040) and ψ(4160)?

Abstract

Physics Subject Headings (PhySH)

Article Text

References (36)

Is the $3 S − 2 D$ mixing strong for the charmonia $ψ (4040)$ and $ψ (4160)$ ?