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Spectral analysis and decay mechanisms of $1^{- +}$ hybrid states in light meson sector

Fu-Yuan Zhang^1,*, Qi Huang^2,†, and Li-Ming Wang ^1,‡

¹Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, China
²Department of Physics, Nanjing Normal University, Nanjing 210023, People’s Republic of China

^*Contact author: fyzhang1110@163.com
^†Contact author: 06289@njnu.edu.cn
^‡Contact author: lmwang@ysu.edu.cn

Phys. Rev. D 113, 014002 – Published 2 January, 2026

DOI: https://doi.org/10.1103/1wkx-3s2l

Abstract

The exploration of exotic mesons, which transcend the conventional quark-antiquark framework, is pivotal for advancing our understanding of QCD and the strong interaction. Among these, states possessing the quantum numbers $J^{P C} = 1^{- +}$ , such as $π_{1} (1600)$ , $π_{1} (2015)$ , and the recently discovered $η_{1} (1855)$ , have attracted significant attention due to their potential hybrid nature, which involves gluonic excitations. However, the physical interpretation of these states remains contentious, primarily due to inconsistencies between theoretical predictions and experimental observations regarding their decay widths and mass spectra. To address these challenges, we employ a potential model inspired by SU(3) lattice gauge theory to calculate the masses of light-flavor $1^{- +}$ hybrid states and utilize the constituent gluon model to analyze their strong decay properties. Our mass calculations suggest that while the $π_{1} (1600)$ and $η_{1} (1855)$ masses are in close agreement with the predicted hybrid states, the corresponding decay widths for $π_{1} (1600)$ and $π_{1} (2015)$ do not support their classification as hybrids, implying alternative interpretations, such as tetraquark or molecular configurations. Conversely, the $η_{1} (1855)$ is consistent with a mixed hybrid state composed of $(u \bar{u} + d \bar{d}) g / \sqrt{2}$ and $s \bar{s} g$ components, with mixing angles ranging from 17.7° to 84.2°. Additionally, we predict the masses and decay widths of yet-unobserved isoscalar and excited hybrid states, providing valuable targets for future experimental searches. This study not only clarifies the hybrid nature of certain exotic mesons but also contributes to the systematic classification of hybrid states within the meson nonet, thereby enhancing our comprehension of exotic hadronic states.

Physics Subject Headings (PhySH)

Hybrid mesons

Article Text

References (77)

G. ’t Hooft and M. J. G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B44, 189 (1972).
H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Advantages of the color octet gluon picture, Phys. Lett. 47B, 365 (1973).
D. J. Gross and F. Wilczek, Ultraviolet behavior of nonabelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
F. Gross, E. Klempt, S. J. Brodsky et al., 50 years of quantum chromodynamics, Eur. Phys. J. C 83, 1125 (2023).
K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
P. G. O. Freund and Y. Nambu, Dynamics in the Zweig-Iizuka rule and a new vector meson below $2 GeV / c^{2}$ , Phys. Rev. Lett. 34, 1645 (1975).
G. S. Adams et al. (E852 Collaboration), Observation of a new $J^{P C} = 1^{- +}$ exotic state in the reaction $π^{-} p \to π^{+} π^{-} π^{-} p$ at 18 GeV/c, Phys. Rev. Lett. 81, 5760 (1998).
J. Kuhn et al. (E852 Collaboration), Exotic meson production in the $f_{1} (1285) π^{-}$ system observed in the reaction $π^{-} p \to η π^{+} π^{-} π^{-} p$ at $18 GeV / c$ , Phys. Lett. B 595, 109 (2004).
M. Ablikim et al. (BESIII Collaboration), Observation of an isoscalar resonance with exotic $J^{P C} = 1^{- +}$ quantum numbers in $J / ψ \to γ η η^{'}$ , Phys. Rev. Lett. 129, 192002 (2022); 130, 159901(E) (2023).
M. Ablikim et al. (BESIII Collaboration), Partial wave analysis of $J / ψ \to γ η η^{'}$ , Phys. Rev. D 106, 072012 (2022); 107, 079901(E) (2023).
D. Alde et al. (IHEP-Brussels-Los Alamos-Annecy(LAPP) Collaboration), Evidence for a $1^{- +}$ exotic meson, Phys. Lett. B 205, 397 (1988).
H. Aoyagi, S. Fukui, T. Hasegawa, N. Hayashi, N. Horikawa, J. Iizuka, S. Inaba, S. Ishimoto, Y. Ishizaki, T. Iwata et al., Study of the $η π^{-}$ system in the $π^{-} p$ reaction at $6.3 GeV / c$ , Phys. Lett. B 314, 246 (1993).
D. R. Thompson et al. (E852 Collaboration), Evidence for exotic meson production in the reaction $π^{-} p \to η π^{-} p$ at 18 GeV/c, Phys. Rev. Lett. 79, 1630 (1997).
V. Dorofeev et al. (VES Collaboration), The $J^{P C} = 1^{- +}$ hunting season at VES, AIP Conf. Proc. 619, 143 (2002).
A. Abele et al. (Crystal Barrel Collaboration), Exotic $η π$ state in $\bar{p} d$ annihilation at rest into $π^{-} π^{0} η p_{spectator}$ , Phys. Lett. B 423, 175 (1998).
C. A. Meyer and Y. Van Haarlem, The status of exotic-quantum-number mesons, Phys. Rev. C 82, 025208 (2010).
C. A. Baker, C. J. Batty, K. Braune, D. V. Bugg, N. Djaoshvili, W. Dünnweber, M. A. Faessler, F. Meyer-Wildhagen, L. Montanet I. Uman et al., Confirmation of $a_{0} (1450)$ and $π_{1} (1600)$ in $\bar{p} p \to ω π^{+} π^{-} π^{0}$ at rest, Phys. Lett. B 563, 140 (2003).
Y. A. Khokhlov (VES Collaboration), Study of X(1600) $1^{- +}$ hybrid, Nucl. Phys. A663, 596 (2000).
M. Alekseev et al. (COMPASS Collaboration), Observation of a $J^{P C} = 1^{- +}$ exotic resonance in diffractive dissociation of $190 GeV / c$ $π^{-}$ into $π^{-} π^{-} π^{+}$ , Phys. Rev. Lett. 104, 241803 (2010).
G. S. Adams et al. (CLEO Collaboration), Amplitude analyses of the decays $χ_{c 1} \to η π^{+} π^{-}$ and $χ_{c 1} \to η^{'} π^{+} π^{-}$ , Phys. Rev. D 84, 112009 (2011).
S. Narison, Gluonia, scalar and hybrid mesons in QCD, Nucl. Phys. A675, 54C (2000).
V. Shastry, C. S. Fischer, and F. Giacosa, The phenomenology of the exotic hybrid nonet with $π_{1} (1600)$ and $η_{1} (1855)$ , Phys. Lett. B 834, 137478 (2022).
B. Chen, S. Q. Luo, and X. Liu, Constructing the $J^{P C} = 1^{- +}$ light flavor hybrid nonet with the newly observed $η_{1} (1855)$ , Phys. Rev. D 108, 054034 (2023).
F. Iddir, A. Le Yaouanc, L. Oliver, O. Pene, J. C. Raynal, and S. Ono, $q \bar{q} g$ hybrid and $q q \bar{q} \bar{q}$ diquonium interpretation of gams $1^{- +}$ resonance, Phys. Lett. B 205, 564 (1988).
A. Benhamida and L. Semlala, Hybrid meson interpretation of the exotic resonance $π_{1} (1600)$ , Adv. High Energy Phys. 2020, 9105240 (2020).
W. I. Eshraim, C. S. Fischer, F. Giacosa, and D. Parganlija, Hybrid phenomenology in a chiral approach, Eur. Phys. J. Plus 135, 945 (2020).
H. X. Chen, A. Hosaka, and S. L. Zhu, The $I^{G} J^{P C} = 1^{-} 1^{- +}$ tetraquark states, Phys. Rev. D 78, 054017 (2008).
S. Narison, $1^{- +}$ light exotic mesons in QCD, Phys. Lett. B 675, 319 (2009).
M. Lu et al. (E852 Collaboration), Exotic meson decay to $ω π^{0} π^{-}$ , Phys. Rev. Lett. 94, 032002 (2005).
J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Toward the excited meson spectrum of dynamical QCD, Phys. Rev. D 82, 034508 (2010).
B. D. Wan, S. Q. Zhang, and C. F. Qiao, Possible structure of newly found exotic state $η_{1} (1855)$ , Phys. Rev. D 106, 074003 (2022).
X. K. Dong, Y. H. Lin, and B. S. Zou, Interpretation of the $η_{1}$ (1855) as a $K {\bar{K}}_{1} (1400) + c . c .$ molecule, Sci. China Phys. Mech. Astron. 65, 261011 (2022).
F. Yang, H. Q. Zhu, and Y. Huang, Analysis of the $η_{1} (1855)$ as a $K {\bar{K}}_{1} (1400)$ molecular state, Nucl. Phys. A1030, 122571 (2023).
M. J. Yan, J. M. Dias, A. Guevara, F. K. Guo, and B. S. Zou, On the $η_{1} (1855)$ , $π_{1} (1400)$ , and $π_{1} (1600)$ as dynamically generated states and their SU(3) partners, Universe 9, 109 (2023).
L. Qiu and Q. Zhao, Towards the establishment of the light $J^{P (C)} = 1^{- (+)}$ hybrid nonet, Chin. Phys. C 46, 051001 (2022).
H. X. Chen, N. Su, and S. L. Zhu, QCD axial anomaly enhances the $η η^{'}$ decay of the hybrid candidate $η_{1} (1855)$ , Chin. Phys. Lett. 39, 051201 (2022).
A. V. Anisovich, V. V. Anisovich, and A. V. Sarantsev, Systematics of $q \bar{q}$ states in the $(n, M^{2})$ and $(J, M^{2})$ planes, Phys. Rev. D 62, 051502 (2000).
C. R. Allton et al. (UKQCD Collaboration), Light hadron spectroscopy with $O (a)$ improved dynamical fermions, Phys. Rev. D 60, 034507 (1999).
F. Karbstein, M. Wagner, and M. Weber, Determination of $Λ_{\bar{MS}}^{(n_{f} = 2)}$ and analytic parametrization of the static quark-antiquark potential, Phys. Rev. D 98, 114506 (2018).
S. Capitani, O. Philipsen, C. Reisinger, C. Riehl, and M. Wagner, Precision computation of hybrid static potentials in SU(3) lattice gauge theory, Phys. Rev. D 99, 034502 (2019).
C. Schlosser and M. Wagner, Hybrid static potentials in SU(3) lattice gauge theory at small quark-antiquark separations, Phys. Rev. D 105, 054503 (2022).
S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
P. R. Page, E. S. Swanson, and A. P. Szczepaniak, Hybrid meson decay phenomenology, Phys. Rev. D 59, 034016 (1999).
N. Isgur, R. Kokoski, and J. Paton, Gluonic excitations of mesons: Why they are missing and where to find them, Phys. Rev. Lett. 54, 869 (1985).
F. E. Close and P. R. Page, The production and decay of hybrid mesons by flux tube breaking, Nucl. Phys. B443, 233 (1995).
T. Barnes, F. E. Close, and E. S. Swanson, Hybrid and conventional mesons in the flux tube model: Numerical studies and their phenomenological implications, Phys. Rev. D 52, 5242 (1995).
A. Le Yaouanc, L. Oliver, O. Pene, J. C. Raynal, and S. Ono, $q \bar{q} g$ hybrid mesons in $ψ \to γ +$ hadrons, Z. Phys. C 28, 309 (1985).
S. Ishida, H. Sawazaki, M. Oda, and K. Yamada, Decay properties of hybrid mesons with a massive constituent gluon and search for their candidates, Phys. Rev. D 47, 179 (1993).
Y. S. Kalashnikova, Exotic hybrids and their nonexotic counterparts, Z. Phys. C 62, 323 (1994).
E. S. Swanson and A. P. Szczepaniak, Decays of hybrid mesons, Phys. Rev. D 56, 5692 (1997).
F. Iddir and A. S. Safir, The decay of the observed $J^{P C} = 1^{- +} (1400)$ and $J^{P C} = 1^{- +} (1600)$ hybrid candidates, Phys. Lett. B 507, 183 (2001).
G. J. Ding and M. L. Yan, A candidate for $1^{- -}$ strangeonium hybrid, Phys. Lett. B 650, 390 (2007).
F. Iddir and L. Semlala, Hybrid states from constituent glue model, Int. J. Mod. Phys. A 23, 5229 (2008).
C. Farina, H. Garcia Tecocoatzi, A. Giachino, E. Santopinto, and E. S. Swanson, Heavy hybrid decays in a constituent gluon model, Phys. Rev. D 102, 014023 (2020).
M. Tanimoto, Decay patterns of $q \bar{q} g$ hybrid mesons, Phys. Lett. 116B, 198 (1982).
M. Tanimoto, Decay of an exotic $q \bar{q} g$ hybrid meson, Phys. Rev. D 27, 2648 (1983).
F. De Viron and J. Govaerts, Some decay modes of $1^{- +}$ hybrid mesons, Phys. Rev. Lett. 53, 2207 (1984).
S. L. Zhu, Masses and decay widths of heavy hybrid mesons, Phys. Rev. D 60, 014008 (1999).
A. l. Zhang and T. G. Steele, Decays of the $\hat{ρ} (1^{- +})$ exotic hybrid and $η - η^{'}$ mixing, Phys. Rev. D 65, 114013 (2002).
H. X. Chen, Z. X. Cai, P. Z. Huang, and S. L. Zhu, Decay properties of the $1^{- +}$ hybrid state, Phys. Rev. D 83, 014006 (2011).
P. Z. Huang, H. X. Chen, and S. L. Zhu, Strong decay patterns of the $1^{- +}$ exotic hybrid mesons, Phys. Rev. D 83, 014021 (2011).
Z. R. Huang, H. Y. Jin, T. G. Steele, and Z. F. Zhang, Revisiting the $b_{1} π$ and $ρ π$ decay modes of the $1^{- +}$ light hybrid state with light-cone QCD sum rules, Phys. Rev. D 94, 054037 (2016).
C. McNeile et al. (UKQCD Collaboration), Hybrid meson decay from the lattice, Phys. Rev. D 65, 094505 (2002).
C. McNeile and C. Michael (UKQCD Collaboration), Decay width of light quark hybrid meson from the lattice, Phys. Rev. D 73, 074506 (2006).
A. J. Woss et al. (Hadron Spectrum Collaboration), Decays of an exotic $1^{- +}$ hybrid meson resonance in QCD, Phys. Rev. D 103, 054502 (2021).
E. G. S. Luna, Diffraction and an infrared finite gluon propagator, Braz. J. Phys. 37, 84 (2007).
A. A. Natale, Phenomenology of infrared finite gluon propagator and coupling constant, Braz. J. Phys. 37, 306 (2007).
A. C. Aguilar, D. Binosi, J. Papavassiliou, and J. Rodriguez-Quintero, Non-perturbative comparison of QCD effective charges, Phys. Rev. D 80, 085018 (2009).
F. E. Close and P. R. Page, The photoproduction of hybrid mesons from CEBAF to HERA, Phys. Rev. D 52, 1706 (1995).
J. J. Dudek et al. (Hadron Spectrum Collaboration), Toward the excited isoscalar meson spectrum from lattice QCD, Phys. Rev. D 88, 094505 (2013).
Z. X. Ma, Q. Huang, L. M. Wang, X. H. Hu, Y. Tan, J. He, and H. X. Huang, Proper constituent gluon mass as the final piece to construct hybrid mesons, Phys. Rev. D 112, L111503 (2025).
R. Kokoski and N. Isgur, Meson decays by flux tube breaking, Phys. Rev. D 35, 907 (1987).
L. M. Wang, J. Z. Wang, S. Q. Luo, J. He, and X. Liu, Studying $X (2100)$ hadronic decays and predicting its pion and kaon induced productions, Phys. Rev. D 101, 034021 (2020).
S. Godfrey and N. Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
E. S. Swanson, Light hybrid meson mixing and phenomenology, Phys. Rev. D 107, 074028 (2023).
V. Shastry and F. Giacosa, Radiative production and decays of the exotic $η_{1}^{'} (1855)$ and its siblings, Nucl. Phys. A1037, 122683 (2023).

Phys. Rev. D 113, 014002– Published 2 January, 2026

Vol. 113, Iss. 1 — 1 January 2026

Received 10 March 2025
Accepted 11 December 2025

DOI: https://doi.org/10.1103/1wkx-3s2l

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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Spectral analysis and decay mechanisms of $1^{- +}$ hybrid states in light meson sector

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Spectral analysis and decay mechanisms of 1−+ hybrid states in light meson sector

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Physics Subject Headings (PhySH)

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References (77)

Spectral analysis and decay mechanisms of $1^{- +}$ hybrid states in light meson sector