Determination of the spin and parity of all-charm tetraquarks

The traditional quark model 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. The exact nature of their internal structure remains uncertain. This paper reports the first measurement of quantum numbers of the recently discovered family of three all-charm tetraquarks, using data collected by the CMS experiment at the Large Hadron Collider from 2016 to 2018. The angular analysis techniques developed for the discovery and characterization of the Higgs boson 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 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.

💡 Research Summary

The CMS Collaboration presents the first determination of the quantum numbers of the three all‑charm tetraquark candidates X(6600), X(6900) and X(7100) observed in the di‑J/ψ final state. Using the full 2016‑2018 data set corresponding to an integrated luminosity of 135 fb⁻¹, 8 651 events with two reconstructed J/ψ→μ⁺μ⁻ decays were selected. The analysis exploits the angular correlations of the four muons (the helicity angles θ₁, θ₂ of the two J/ψ mesons and the angle Φ between their decay planes) to discriminate among possible spin‑parity‑charge‑conjugation assignments.

Theoretical considerations restrict the allowed J^{PC} values to six possibilities (0^{−+}, 0^{++}, 1^{−+}, 1^{++}, 2^{−+}, 2^{++}) once the C‑parity of the final state (C=+1) is imposed. For each hypothesis a set of decay amplitudes A_{λ₁λ₂} is defined, respecting angular‑momentum conservation, Bose symmetry of the two identical J/ψ mesons, and the transformation properties under P and C. Table 1 of the paper enumerates the amplitude structures (e.g. minimal “m” and higher‑complexity “h” models) and the corresponding contributions to the angular distribution.

A detailed Monte‑Carlo simulation of the CMS detector, including trigger, reconstruction efficiencies and resolution effects, is used to map the theoretical angular probability density functions P_i(θ₁,θ₂,Φ,m_{4μ}) onto observable distributions. Background from non‑resonant di‑J/ψ production and threshold enhancements is modeled from data sidebands and simulated samples. An unbinned maximum‑likelihood fit is performed simultaneously in the three‑dimensional angular space and the invariant mass m_{4μ}, allowing for interference between the two amplitude components of a given J^{PC} hypothesis.

The likelihood comparison shows a clear preference for the J^{PC}=2^{++} hypothesis. The fit yields a spin‑2 assignment with parity P=+1 and charge‑conjugation C=+1. Alternative hypotheses are strongly disfavoured: J=0 (scalar) is excluded at the 95 % confidence level, while J=1 (vector) is excluded at the 99 % confidence level. The data also rule out configurations with C=−1, eliminating the 1^{−+} and 2^{−+} possibilities.

From a phenomenological standpoint, the 2^{++} result implies an orbital angular momentum L=0 (since P=(−1)^L=+1) and a symmetric spin coupling of the two charm‑charm diquark and two charm‑anticharm antidiquark pairs, each in a spin‑1 configuration. This is precisely the pattern expected for a compact, tightly‑bound diquark–antidiquark tetraquark rather than a loosely bound “molecular” state of two charmonia, which would more naturally allow J=0 or 1. Moreover, the three observed resonances exhibit a regular spacing consistent with a radial Regge trajectory, supporting the interpretation that they are radial excitations of the same underlying configuration.

The paper concludes that the CMS angular analysis provides the first definitive spin‑parity measurement for all‑charm tetraquarks, establishing J^{PC}=2^{++} and strongly favoring a compact tetraquark picture. It also outlines future directions: studying alternative decay channels (e.g. η_cη_c, χ_cχ_c), combining results with LHCb and ATLAS, and improving statistical precision to probe the internal dynamics of these exotic QCD bound states.