Determination of the spin and parity of all-charm tetraquarks

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 antiquarks³. The exact nature of their internal structure remains uncertain^4-29. Here we report the first measurement of quantum numbers of the recently discovered family of three all-charm tetraquarks^30-32, using data collected by the CMS experiment at the Large Hadron Collider from 2016 to 2018 (refs. ^33,34). The angular analysis techniques developed for the discovery and characterization of the Higgs boson^35-37 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 2ħ, while 0ħ and 1ħ are excluded at 95% and 99% confidence levels, 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.

Fig. 1. Candidates for all-charm tetraquarks.

The J/Ψ J/Ψ → μ⁺μ⁻μ⁺μ⁻ invariant mass m_4μ spectrum shows the three exotic states, X(6600), X(6900) and X(7100). Parameterizations of these states are shown both individually and as a combined signal that includes quantum-mechanical interference (denoted by Total signal). The full model incorporates both signal and background components, with the background originating from di-J/Ψ production, including contributions from nonresonant production and an enhancement near the kinematic threshold of 6.2 GeV.

Fig. 2. Internal structure models for the particle X.

The particle X, composed of $\mathrm{c}\mathrm{c}\overline{\mathrm{c}}\overline{\mathrm{c}}$, is shown at rest. Two models of the internal structure of X are presented: a tightly bound tetraquark (top) and a loosely bound molecule of two mesons (bottom). The colours assigned to individual quarks or quark pairs denote possible colour charge assignments in strong interactions, in which attractive forces are mediated by gluon exchange (shown as wavy lines) and meson exchange (shown as a solid pair of arrows). The X decays into two J/Ψ mesons with spin projections λ_i along their respective directions of motion; each meson then decays into a μ⁺μ⁻ pair. The polar and azimuthal angles Ω_i = (θ_i, Φ_i) describe the direction of the μ⁻ relative to the z_i-axis, which is defined to point opposite to the X-direction in the centre-of-mass frame of the corresponding J/Ψ meson, for i = 1 and 2.

Fig. 3. Analysis of angular distributions.

a, Distributions of ${\mathcal{D}}_{2_{\mathrm{m}}^{+}{0}^{-}}$ for the 0⁻, $2_{\mathrm{m}}^{-}$, and $2_{\mathrm{m}}^{+}$ models in the range 6.2 < m_4μ < 8.0 GeV. Distributions for signal only (dashed) and for signal plus background (solid and dash-dot-dotted) models are compared with 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 $2_{\mathrm{m}}^{-}$ distributions are identical apart from systematic uncertainties arising from polarization effects. b, Normalized distributions of the test statistic $q=-2\ln \left({\mathcal{L}}_{0^{-}}/{\mathcal{L}}_{2_{\mathrm{m}}^{+}}\right)$ from pseudo-experiments generated under the $2_{\mathrm{m}}^{+}$ (blue, right) and 0⁻ (orange, left) hypotheses, with the arrow indicating the observed value q_obs.

Fig. 4. Summary of statistical tests.

Distributions of the test statistic q for various $J_i^P$ hypotheses tested against the $2_{\mathrm{m}}^{+}$ model. The observed q_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 $2_{\mathrm{m}}^{+}$ 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. 3b. For 0⁺ and 2⁻ models, 11 points correspond to varying fractions in the mixture of the two structures of interaction.

Extended Data Fig. 1. Angular observables.

The production and decay of a resonance X in proton collisions pp → X → V₁V₂ → 4μ define the angular observables in the centre-of-mass frames of the corresponding particles^,, where the V₁ and V₂ refer to the J/Ψ mesons. The z axis approximates the proton beam line, while the $z\prime$ axis corresponds to the direction of the four-muon system.

Extended Data Fig. 2. Angular distributions.

Distribution of the decay angles: Φ (upper row), $\cos {\theta }_1,\cos {\theta }_2$ (second row), production angles defined with respect to axis z: ${\mathrm{\Phi }}_1,\cos {\theta }^{*}$ (third row), and defined with respect to axis $z\prime$: ${\mathrm{\Phi }}_1^{\prime },\cos {\theta }^{\prime *}$ (lower row) in the range 6.2 < m_4μ < 8.0 GeV presented together with the five $J_i^P$ 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 $2_h^{-}$.

Extended Data Fig. 3. 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_4μ < 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.