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Scalar and tensor structures in $J/\psi J/\psi$ scattering from lattice QCD

This lattice QCD study determines $J/\psi J/\psi$ scattering amplitudes up to 6.6 GeV, revealing an attractive interaction in the scalar channel that may correspond to the $X(6200)$ and a tensor resonance in the ${}^5S_2$ channel with parameters compatible with the $X(6600)$ , attributing these distinct behaviors to dominant quark rearrangement effects.

Original authors: Geng Li, Chunjiang Shi, Ying Chen, Wei Sun

Published 2026-08-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Geng Li, Chunjiang Shi, Ying Chen, Wei Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, matter is built from a small set of fundamental particles called quarks, which are held together by a force so strong it is often described as the glue of the universe. Usually, these quarks pair up in twos or threes to form familiar particles like protons and neutrons. However, under the right conditions, they can also clump together in larger groups, creating exotic states of matter that have only recently begun to be observed. Among the most intriguing of these are fully charmed tetraquarks, which are made entirely of four heavy quarks. When these particles collide, they can scatter off one another, and by studying how they bounce apart, physicists can map out the invisible forces at play. This is crucial because the patterns of these collisions might reveal new, stable structures of matter that have never been seen before, helping scientists understand the rules that govern the most extreme forms of matter in the universe.

A team of researchers has now taken a deep dive into these interactions by simulating the collision of two specific heavy particles, known as J/psi mesons, inside a virtual grid of space and time. Using a powerful computational method called lattice quantum chromodynamics, they recreated the conditions of the early universe to see how these particles behave when they come close together. The study focused on two distinct ways the particles can interact based on their spin, a property that determines how they rotate. In one scenario, the particles interact in a way that pulls them together, creating an attractive force. In the other, they push each other away. The simulations, which were run with two different settings to ensure the results were robust, revealed that this attractive force is strong enough to potentially hold a new, short-lived particle together right at the edge of where the two particles would normally just touch. This hypothetical structure could be the elusive X(6200), a particle that experimentalists have been searching for in data from high-energy colliders.

While the attractive interaction suggested a possible new particle, the repulsive interaction told a different story. In the channel where the particles pushed each other away, the researchers found no sign of a simple bound state. Instead, the data pointed to the existence of a resonance, a fleeting, unstable particle that appears at higher energies before quickly falling apart. This resonance has a mass and width that align remarkably well with a particle recently observed by the ATLAS and CMS experiments, known as X(6600). The study suggests that this particle has a specific quantum nature, identified as a tensor state, which matches the latest experimental classifications. The researchers also discovered that the difference between the attractive and repulsive forces is not due to a complex exchange of other particles, but rather to a fundamental rearrangement of the quarks themselves. As the two heavy particles approach, their internal quarks swap partners in a way that either binds them or pushes them apart, depending on their spin alignment.

The findings offer a compelling explanation for the broad structures seen in recent experimental data, linking the theoretical behavior of quarks to the actual particles detected in detectors. The simulations indicate that the attractive force in the first scenario allows for a near-threshold structure, while the repulsive force in the second scenario creates a resonance that sits slightly higher in energy. Although the study was conducted with slightly heavier-than-natural quark masses, the consistency between the two simulation settings gives the results significant weight. The work does not claim to have definitively proven the existence of these particles, as the complex nature of the forces involved makes precise calculations difficult, but it provides strong evidence that such structures are a natural consequence of the laws of physics. By isolating the specific mechanisms of quark rearrangement, the researchers have provided a clear picture of why these particles behave the way they do, offering a solid foundation for future experiments to confirm the existence of these exotic forms of matter.

Technical Summary: Scalar and Tensor Structures in $J/\psi J/\psi$ Scattering from Lattice QCD

Problem Statement
Following the experimental observation of the $X(6900)$ state and related structures in the di- $J/\psi$ spectrum by LHCb, ATLAS, and CMS, there is a pressing need to understand the dynamics governing the interaction between two charmonium states. While phenomenological models suggest the existence of fully charmed tetraquarks ( $T_{cc\bar{c}\bar{c}}$ ) and various resonances (e.g., $X(6200)$ , $X(6600)$ , $X(7100)$ ), theoretical interpretations often rely on unverified assumptions regarding pomeron or light meson exchanges. Specifically, the nature of the near-threshold structures and the spin-dependent interactions in the $J/\psi J/\psi$ system remain to be clarified from first principles. This work aims to explore the $J/\psi J/\psi$ interactions in the scalar ( $1S_0$ , $0^{++}$ ) and tensor ( $5S_2$ , $2^{++}$ ) channels using lattice Quantum Chromodynamics (QCD) to determine scattering amplitudes and pole structures without coupled-channel complications below 6.6 GeV.

Methodology
The study employs $N_f=2$ lattice QCD simulations performed on anisotropic lattices with a spatial lattice spacing $a_s \approx 0.136$ fm and an aspect ratio $\xi = a_s/a_t \approx 5.0$ . Calculations are conducted at two unphysical pion masses, $m_\pi \approx 420$ MeV and $250$ MeV, across two lattice volumes ( $L_s = 12a_s$ and $16a_s$ ). The valence charm quark action is tuned to reproduce the spin-averaged $1S$ charmonium mass.

The finite-volume energies (FVEs) of the $J/\psi J/\psi$ systems are extracted from correlation functions constructed using the distillation method. The operators include both direct diagrams and quark rearrangement (QR) diagrams (Fig. 1). The analysis focuses on the energy region below 6.6 GeV, where the $J/\psi J/\psi$ channel is effectively decoupled from the $\eta_c \eta_c$ channel, allowing for the application of the single-channel Lüscher formalism. Scattering amplitudes are derived from the phase shifts, and pole structures are analyzed using effective range expansions (ERE) for the $1S_0$ channel and a $K$ -matrix parametrization for the $5S_2$ channel.

Key Results

$1S_0$ ( $0^{++}$ ) Channel:
The analysis reveals a near-threshold attractive interaction. The phase shift data up to $k^2 \approx 1.2$ GeV $^2$ are well-described by an effective range expansion with positive scattering lengths ( $a_0 \approx 0.25$ fm at $m_\pi \approx 420$ MeV). While the attractive interaction suggests the possibility of a near-threshold scalar structure (potentially corresponding to the phenomenologically reported $X(6200)$ ), the authors note that a precise pole determination is hindered by the left-hand cut (LHC) effect arising from light meson ( $\sigma$ or two-pion) exchange. Consequently, the study concludes that the interplay of the LHC and attraction permits the existence of a scalar structure near the threshold but does not definitively confirm a bound or virtual state pole.
$5S_2$ ( $2^{++}$ ) Channel:
In contrast, the $5S_2$ channel exhibits a near-threshold repulsive interaction, indicated by a negative scattering length ( $a_0 \approx -0.16$ fm). However, the scattering amplitude displays a resonance at higher energies. The $K$ -matrix analysis identifies a Castillejo-Dalitz-Dyson (CDD) zero at $\sqrt{s}_{CDD} \approx 6.45$ GeV. Solving the pole equation in the complex plane yields a tensor resonance with parameters:
- $m_\pi \approx 420$ MeV: $(m_R, \Gamma_R) = (6.543(10), 0.548(34))$ GeV.
- $m_\pi \approx 250$ MeV: $(m_R, \Gamma_R) = (6.538(13), 0.537(56))$ GeV.
  These results are compatible with the properties of the $X(6600)$ (or $X(6400)$ ) reported by ATLAS and CMS and support the $2^{++}$ quantum number assignment proposed in recent CMS studies.
Dynamical Origin:
The distinct behaviors of the two channels are attributed to the dominance of quark rearrangement effects. By analyzing the ratio of quark rearrangement contributions ( $Q(t)$ ) to the total correlation function, the study finds that the QR effect is attractive in the $1S_0$ channel and repulsive in the $5S_2$ channel. This is quantitatively supported by the relation $\langle 5S_2 | H_{QR} | 5S_2 \rangle / \langle 1S_0 | H_{QR} | 1S_0 \rangle \approx -2$ , derived from the spin wave functions of the initial states.

Significance and Claims
The paper claims to provide the first lattice QCD determination of $J/\psi J/\psi$ scattering amplitudes in both scalar and tensor channels up to 6.6 GeV. The primary contributions are:

Confirmation of Spin-Dependent Dynamics: The study demonstrates that the interaction between two charmonia is strongly spin-dependent, with attraction in the $0^{++}$ channel and repulsion in the $2^{++}$ channel near threshold, driven primarily by quark rearrangement.
Support for Experimental Observations: The calculated tensor resonance parameters align with the experimental $X(6600)$ signal, providing a theoretical foundation for its existence as a $2^{++}$ tetraquark candidate.
Constraint on Scalar States: While not confirming a specific pole, the results support the phenomenological hypothesis of a near-threshold scalar structure ( $X(6200)$ ) in the $0^{++}$ channel, while highlighting the limitations imposed by left-hand cuts in lattice analyses.

The authors modestly note that systematic uncertainties arising from unphysical pion masses, finite lattice spacing, and finite volume effects require further investigation in future studies with more sophisticated setups. Additionally, the theoretical origin of the CDD zero and the specific mechanism of the tensor resonance formation remain open questions for future theoretical work.

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