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| CMS-BPH-24-002 ; CERN-EP-2025-118 | ||
| Determination of the spin and parity of all-charm tetraquarks | ||
| CMS Collaboration | ||
| 9 June 2025 | ||
| Nature 648 (2025) 58 | ||
| Abstract: The traditional quark model [1, 2] accounts for the existence of baryons, such as protons and neutrons, which consist of three quarks, as well as mesons, composed of a quark-antiquark pair. Only recently has substantial evidence started to accumulate for exotic states composed of four or five quarks and antiquarks [3]. The exact nature of their internal structure remains uncertain [4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. This paper reports the first measurement of quantum numbers of the recently discovered family of three all-charm tetraquarks [30, 31, 32], using data collected by the CMS experiment at the Large Hadron Collider from 2016 to 2018 [33, 34]. The angular analysis techniques developed for the discovery and characterization of the Higgs boson [35, 36, 37] have been applied to the new exotic states. Here we show that the quantum numbers for parity $ P $ and charge conjugation $ C $ symmetries are found to be $ + $1. The spin $ J $ of these exotic states is determined to be consistent with 2$ \hbar $, while 0$ \hbar $ and 1$ \hbar $ are excluded at 95% and 99% confidence level, respectively. The $ J^{PC}= $ 2$^{++} $ assignment implies particular configurations of constituent spins and orbital angular momenta, which constrain the possible internal structure of these tetraquarks. | ||
| Links: e-print arXiv:2506.07944 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; | ||
| Figures | |
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Figure 1:
Candidates for all-charm tetraquarks. The $ \mathrm{J}/\psi \mathrm{J}/\psi\to\mu^{+}\mu^{-}\mu^{+}\mu^{-} $ invariant mass $ m_{4\mu} $ spectrum shows the three exotic states, X(6600), X(6900), and X(7100). Parameterizations of these states are displayed both individually and as a combined signal that includes quantum-mechanical interference (denoted by ``Total signal"). The full model [32] incorporates both signal and background components, with the background originating from di-$ \mathrm{J}/\psi $ production, including contributions from nonresonant production and an enhancement near the kinematic threshold of 6.2 GeV. |
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Figure 2:
{Internal structure models for the particle $ \mathrm{X} $.} The particle $ \mathrm{X} $, composed of $ {\mathrm{c}}{\mathrm{c}}{\overline{\mathrm{c}}}{\overline{\mathrm{c}}} $, is shown at rest. Two models of the internal structure of $ \mathrm{X} $ are presented: a tightly-bound tetraquark (upper) and a loosely-bound molecule of two mesons (lower). The colours assigned to individual quarks or quark pairs denote possible colour charge assignments in strong interactions, where attractive forces are mediated by gluon exchange (shown as wavy lines) and meson exchange (shown as a solid pair of arrows). The $ \mathrm{X} $ decays into two $ \mathrm{J}/\psi $ mesons with spin projections $ \lambda_i $ along their respective directions of motion; each meson then decays into a $ \mu^{+}\mu^{-} $ pair. The polar and azimuthal angles $ \mathbf{\Omega}_{i}=(\theta_{i}, \Phi_{i}) $ describe the direction of the $ \mu^{-} $ relative to the $ z_i $ axis, which is defined to point opposite to the $ \mathrm{X} $ direction in the centre-of-mass frame of the corresponding $ \mathrm{J}/\psi $ meson, for $ i = $ 1 and 2. |
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Figure 3:
Analysis of angular distributions. Left: Distributions of $ \mathcal{D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical apart from systematic uncertainties arising from polarization effects. Right: Normalized distributions of the test statistic $ q=-2{\ln(\mathcal{L}_{ 0 ^{-}}/\mathcal{L}_{2^{+} _m})} $ from pseudoexperiments generated under the 2 $ _m^{+} $ (blue/right) and $ 0^{-} $ (orange/left) hypotheses, with the arrow indicating the observed value $ q_\text{obs} $. |
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Figure 3-a:
Analysis of angular distributions. Left: Distributions of $ \mathcal{D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical apart from systematic uncertainties arising from polarization effects. Right: Normalized distributions of the test statistic $ q=-2{\ln(\mathcal{L}_{ 0 ^{-}}/\mathcal{L}_{2^{+} _m})} $ from pseudoexperiments generated under the 2 $ _m^{+} $ (blue/right) and $ 0^{-} $ (orange/left) hypotheses, with the arrow indicating the observed value $ q_\text{obs} $. |
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Figure 3-b:
Analysis of angular distributions. Left: Distributions of $ \mathcal{D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical apart from systematic uncertainties arising from polarization effects. Right: Normalized distributions of the test statistic $ q=-2{\ln(\mathcal{L}_{ 0 ^{-}}/\mathcal{L}_{2^{+} _m})} $ from pseudoexperiments generated under the 2 $ _m^{+} $ (blue/right) and $ 0^{-} $ (orange/left) hypotheses, with the arrow indicating the observed value $ q_\text{obs} $. |
png pdf |
Figure 4:
Summary of statistical tests. Distributions of the test statistic $ q $ for various $ J_i^{P} $ hypotheses tested against the 2 $ _m^{+} $ model. The observed $ q_\text{obs} $ values are indicated by the black dots. The expected median and the 68.3%, 95.4%, and 99.7% confidence level regions for the 2 $ _m^{+} $ model (blue/left) and for each of the alternative $ J_i^{P} $ hypotheses (orange/right) are shown. The first entry corresponding to $ 0^- $ reflects the information shown in Fig. 3 (right). For $ 0^+ $ and $ 2^- $ models, eleven points correspond to varying fractions in the mixture of the two structures of interaction. |
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Figure 5:
Angular observables. The production and decay of a resonance $ \mathrm{X} $ in proton collisions $ \mathrm{p}\mathrm{p}\to \mathrm{X} \to \mathrm{V}_1\mathrm{V}_2 \to 4\mu $ define the angular observables in the centre-of-mass frames of the corresponding particles [46, 35], where the $ \mathrm{V}_1 $ and $ \mathrm{V}_2 $ refer to the $ \mathrm{J}/\psi $ mesons. The $ z $ axis approximates the proton beam line, while the $ z' $ axis corresponds to the direction of the four-muon system. |
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Figure 6:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-a:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-b:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-c:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-d:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-e:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-f:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 6-g:
Angular distributions. Distribution of the decay angles: $ \Phi $ (upper row), $ \cos\theta_1, \cos\theta_2 $ (second row), production angles defined with respect to axis $ z $: $ \Phi_1, \cos\theta^* $ (third row), and defined with respect to axis $ z' $: $ \Phi_1', \cos\theta^{\prime *} $ (lower row) in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five $ J^P_i $ signal models. The background is subtracted from the data, based on the expected distributions, and systematic uncertainties are not incorporated in these plots. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
png pdf |
Figure 7:
Optimal observables. Distributions of $ \mathcal{D}_{ij} $ that are optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (upper left), 0 $ _{h}^+ $ (upper right), $ 1^- $ (lower left), and $ 1^+ $ (lower right) models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical, apart from systematic uncertainties arising from polarization effects. The lower panels display the ratios of the data and of the model predictions to the mean expectations from the 2 $ _{m}^+ $ model. |
png pdf |
Figure 7-a:
Optimal observables. Distributions of $ \mathcal{D}_{ij} $ that are optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (upper left), 0 $ _{h}^+ $ (upper right), $ 1^- $ (lower left), and $ 1^+ $ (lower right) models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical, apart from systematic uncertainties arising from polarization effects. The lower panels display the ratios of the data and of the model predictions to the mean expectations from the 2 $ _{m}^+ $ model. |
png pdf |
Figure 7-b:
Optimal observables. Distributions of $ \mathcal{D}_{ij} $ that are optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (upper left), 0 $ _{h}^+ $ (upper right), $ 1^- $ (lower left), and $ 1^+ $ (lower right) models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical, apart from systematic uncertainties arising from polarization effects. The lower panels display the ratios of the data and of the model predictions to the mean expectations from the 2 $ _{m}^+ $ model. |
png pdf |
Figure 7-c:
Optimal observables. Distributions of $ \mathcal{D}_{ij} $ that are optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (upper left), 0 $ _{h}^+ $ (upper right), $ 1^- $ (lower left), and $ 1^+ $ (lower right) models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical, apart from systematic uncertainties arising from polarization effects. The lower panels display the ratios of the data and of the model predictions to the mean expectations from the 2 $ _{m}^+ $ model. |
png pdf |
Figure 7-d:
Optimal observables. Distributions of $ \mathcal{D}_{ij} $ that are optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (upper left), 0 $ _{h}^+ $ (upper right), $ 1^- $ (lower left), and $ 1^+ $ (lower right) models in the range 6.2 $ < m_{4\mu} < $ 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared to the experimental data points with error bars, with uncertainty bands representing post-fit model uncertainties, which are partially correlated with the data. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical, apart from systematic uncertainties arising from polarization effects. The lower panels display the ratios of the data and of the model predictions to the mean expectations from the 2 $ _{m}^+ $ model. |
| Tables | |
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Table 1:
Quantum numbers. The possible assignments of quantum numbers $ J^{PC} $, the $ J_i^{P} $ models considered, and the contributing amplitudes in the decay $ \mathrm{X}\to \mathrm{J}/\psi \mathrm{J}/\psi $ are presented. |
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Table 2:
Summary of statistical tests. The $ p $-value and the associated $ Z $-score are shown for alternative models $ J_i^{P} $, tested against the 2 $ _m^{+} $ model. A higher Z-score implies that the model is less compatible with observation. |
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Table 3:
Results from hypothesis test for pairs of spin-parity models. This is an extended version of Table 2. The expected $ p $-value is presented based on the assumption of the scenario of 2 $ _m^+ $. Results with $ Z > $ 5 have been derived through Gaussian extrapolation. |
| Summary |
| The study of tetraquark states has attracted considerable interest due to its potential to provide insight into the structure of the hadronic matter that makes up the world around us [4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In this paper, we have presented the first measurement of the quantum numbers for a recently discovered family of three all-charm tetraquarks, based on data collected by the CMS experiment at the LHC. Our results, summarized in Table 2, favour a $ J^{PC}=$ 2$^{++} $ assignment. For a system of two constituents, as represented by either model in Fig. 2, an orbital angular momentum of $ L = $ 1 is excluded by the requirement $ P = + $1, while higher orbital excitations with $ L \geq $ 2 are energetically disfavoured. This makes the $ S $-wave ($ L = $ 0) configuration the most likely. In a molecular scenario, the quark-antiquark pairs are not required to be spin-1 mesons, making a $ J = $ 2 configuration less likely. This picture agreed with existing data for all other well-established tetraquark candidates with measured spin, such as the X(3872) and $ \mathrm{Z}_{\mathrm{c}}(3900)^{+} $, all of which have $ J < $ 2 [3]. In contrast, a tightly-bound $ \mathrm{c}\mathrm{c}\overline{\mathrm{c}}\overline{\mathrm{c}} $ tetraquark with a diquark--antidiquark configuration requires both diquarks to be in spin-1 states, which restricts the total spin to $ J = $ 0 or $ J = $ 2, making $ J = $ 2 a natural choice. This spin-1 diquark requirement, however, does not apply to tetraquark candidates with mixed-flavour quark content, a consideration relevant to all previously observed candidates, which contained both heavy and light quarks [3]. Other theoretical considerations also favour $ J = $ 2 for the tightly-bound tetraquark states [18, 28]. This advancement in understanding exotic hadrons was enabled by the study of all-heavy tetraquarks and it brings us closer to uncovering their internal structure. Although our findings do not definitively distinguish between tightly-bound tetraquark and meson-meson molecular models, they provide constraints on the possible internal structures and favour the tightly-bound scenario. |
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