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| CMS-PAS-BPH-24-002 | ||
| Spin and symmetry properties of all-charm tetraquarks | ||
| CMS Collaboration | ||
| 8 April 2025 | ||
| Abstract: The traditional quark model accounts for the existence of baryons, like protons and neutrons, which consist of three quarks, as well as mesons, composed of a quark and antiquark pair. Only recently has substantial evidence started to accumulate for exotic states composed of four or five quarks or antiquarks. In this study, the CMS Collaboration investigates the recently discovered family of three tetraquark candidates composed of four charm quarks and antiquarks. The exact nature of their internal structure remains uncertain. They could either be tightly bound states of true tetraquarks, similar to quarks bound within protons and neutrons, or molecules composed of two familiar mesons, loosely bound like protons and neutrons in a nucleus, with other potential configurations still being considered. Angular analysis techniques for decay products, developed for the discovery and characterization of the Higgs boson, are now being applied to the new exotic states. The quantum numbers for parity $ P $ and charge conjugation $ C $ symmetries are found to be +1. The spin $ J $ of these exotic states is most consistent with $ J=2\hbar $, a value that is uncommon for such particles, while the $ J=0\hbar $ and 1 $ \hbar $ are excluded at 95% and 99% confidence level, respectively. The $ J^{PC}=2^{++} $ quantum numbers match the expected values for tetraquarks with specific configurations of spin and angular momenta of its components, which helps in narrowing down the tetraquark's internal structure. | ||
| Links: CDS record (PDF) ; CADI line (restricted) ; | ||
| Figures | |
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Figure 1:
Candidates for all-charm tetraquarks. The mass spectrum of the three exotic states, $ \mathrm{X(6600)} $, $ \mathrm{X(6900)} $, and $ \mathrm{X(7100)} $, displayed individually and as a combined signal that includes quantum-mechanical interference, observed against the background of a pair of $ \mathrm{J}/\psi $ mesons, including contributions from non-resonant and threshold production [15]. |
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Figure 2:
The concept of analyzing the internal structure of exotic particles. The particle $ \mathrm{X} $, composed of four charm quarks and antiquarks, is shown at rest. Two models of the internal structure of $ \mathrm{X} $ are presented: a tightly-bound tetraquark (top) and a loosely-bound molecule of two mesons (bottom). The colors of the quarks in the diagram represent possible color charge assignments in strong interactions, with curly lines indicating gluon exchange and arrows representing meson exchange. The $ \mathrm{X} $ splits into two mesons, $ \mathrm{J}/\psi $, each displayed in its own rest frame. Inside each meson, the quark-antiquark pair annihilates, producing a muon pair, $ \mu^+\mu^- $. The polar angles $ {\Omega}_{1}=(\theta_{1}, \Phi_{1}) $ and $ {\Omega}_{2}=(\theta_{2}, \Phi_{2}) $ define the directions of the $ \mu^- $ relative to the $ \mathrm{J}/\psi $ directions, aligned with the opposite $ z_1 $ and $ z_2 $ axes. |
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Figure 3:
Analysis of angular distributions. On the left: Distributions of $ {\cal D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models, shown for signal-only (dashed) and including background (solid), compared to the experimental data (points with error bars). The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical. On the right: Distributions of the test statistic $ q $ for the 2 $ _m^{+} $ (blue) and $ 0^{-} $ (orange) models. The black arrow indicates the observed $ q_\text{obs} $ value in the experiment. |
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Figure 3-a:
Analysis of angular distributions. On the left: Distributions of $ {\cal D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models, shown for signal-only (dashed) and including background (solid), compared to the experimental data (points with error bars). The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical. On the right: Distributions of the test statistic $ q $ for the 2 $ _m^{+} $ (blue) and $ 0^{-} $ (orange) models. The black arrow indicates the observed $ q_\text{obs} $ value in the experiment. |
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Figure 3-b:
Analysis of angular distributions. On the left: Distributions of $ {\cal D}_{2_m^{+}0^{-}} $ for the $ 0^{-} $, 2 $ _m^{-} $, and 2 $ _m^{+} $ models, shown for signal-only (dashed) and including background (solid), compared to the experimental data (points with error bars). The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical. On the right: Distributions of the test statistic $ q $ for the 2 $ _m^{+} $ (blue) and $ 0^{-} $ (orange) models. The black arrow indicates the observed $ q_\text{obs} $ value in the experiment. |
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Figure 4:
Summary of statistical tests. Distributions of the test statistic $ q $ for various $ J_{\mathrm{X}}^{PC} $ 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) and for the alternative $ J_{\mathrm{X}}^{PC} $ hypotheses (orange) are shown. The first entry corresponding to $ 0^- $ reflects the information shown in Fig. 3 (right), while the same information is also captured for the other models. For $ 0^+ $ and $ 2^- $, eleven points correspond to varying fractions in the mixture of the two tensor 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{J}/\psi\mathrm{J}/\psi \to 4\mu $ define the angular observables in the center-of-mass frames of the corresponding particles [18,19]. The motion of the four-muon system within the laboratory frame leads to appearance of non-collinear proton collisions and defines axis $ z^\prime $, while axis $ z $ approximates the proton beam line. |
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Figure 6-a:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-b:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-c:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-d:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-e:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-f:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 6-g:
Angular distributions. Distribution of the decay angles: $ \Phi $ (top), $ \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^\prime $: $ \Phi_1^\prime, \cos\theta^{\prime *} $ (bottom) in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV presented together with the five signal models shown. Several $ J^P $ models are presented, with the background subtracted from the data based on the expected distributions. The $ 0^{-} $ and 2 $ _m^{-} $ distributions are identical, as are those of $ 1^{-} $ and 2 $ _h^{-} $. |
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Figure 7:
Optimal observables. Distributions of the discriminants $ {\cal D}_{ij} $ optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (top-left), 0 $ _{h}^+ $ (top-right), $ 1^- $ (bottom-left), and $ 1^+ $ (bottom-right) model in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical. Dashed histograms indicate expectations for the signal only, whereas solid histograms include background as well. |
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Figure 7-a:
Optimal observables. Distributions of the discriminants $ {\cal D}_{ij} $ optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (top-left), 0 $ _{h}^+ $ (top-right), $ 1^- $ (bottom-left), and $ 1^+ $ (bottom-right) model in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical. Dashed histograms indicate expectations for the signal only, whereas solid histograms include background as well. |
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Figure 7-b:
Optimal observables. Distributions of the discriminants $ {\cal D}_{ij} $ optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (top-left), 0 $ _{h}^+ $ (top-right), $ 1^- $ (bottom-left), and $ 1^+ $ (bottom-right) model in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical. Dashed histograms indicate expectations for the signal only, whereas solid histograms include background as well. |
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Figure 7-c:
Optimal observables. Distributions of the discriminants $ {\cal D}_{ij} $ optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (top-left), 0 $ _{h}^+ $ (top-right), $ 1^- $ (bottom-left), and $ 1^+ $ (bottom-right) model in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical. Dashed histograms indicate expectations for the signal only, whereas solid histograms include background as well. |
png pdf |
Figure 7-d:
Optimal observables. Distributions of the discriminants $ {\cal D}_{ij} $ optimal for separating the 2 $ _{m}^+ $ model against the 0 $ _{m}^+ $ (top-left), 0 $ _{h}^+ $ (top-right), $ 1^- $ (bottom-left), and $ 1^+ $ (bottom-right) model in the mass range 6.2 $ < m_{4\mu} < $ 8.0 GeV. The $ 1^{-} $ and 2 $ _h^{-} $ distributions are identical. Dashed histograms indicate expectations for the signal only, whereas solid histograms include background as well. |
| Tables | |
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Table 1:
Quantum numbers. The possible assignments of quantum numbers, the 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 probability (p-value) and Z-score (which measures how many standard deviations the observed test statistic $ q $ deviates from the mean of the modeled distribution) for an alternative model $ J_{\mathrm{X}}^{P} $, tested against the 2 $ _m^{+} $ model. The $ J_{\mathrm{mix}}^P $ models represent the mixed scenarios with minimal separation against 2 $ _m^{+} $. |
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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 $ J^P_{\mathrm{X}}=2_m^+ $. Results with a Z-score exceeding 5 have been derived through Gaussian extrapolation. |
| Summary |
| The spins of the four quarks and antiquarks forming the tetraquarks must combine to give $J_{\mathrm{X}}=2\hbar$, while the orbital angular momenta between any of the components within the tetraquark state must be $L=0\hbar$. While this reasoning does not unambiguously distinguish between the tightly-bound tetraquark and molecular models, it significantly helps in narrowing down the internal structure to the possible configurations. |
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