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$\eta_c\eta_c$ and $J/\psi J/\psi$ scatterings from lattice QCD

This lattice QCD study investigates $S$ -wave $\eta_c\eta_c$ and $J/\psi J/\psi$ scatterings at unphysical pion masses, revealing that quark rearrangement dominates the interactions and supporting the $2^{++}$ assignment of the $X(6600)$ resonance while suggesting a near-threshold scalar structure in the $0^{++}$ channel that may correspond to the $X(6200)$ .

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

Published 2026-08-19

📖 6 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 family of fundamental particles called quarks. Usually, these quarks are bound together in tight groups of two or three, forming familiar particles like protons and neutrons, or the heavier cousins known as charmonium, which consist of a charm quark and its antimatter partner. For decades, physicists believed these were the only stable ways quarks could arrange themselves. However, in recent years, massive particle colliders have begun to spot strange, short-lived structures that appear to contain four charm quarks at once. These "tetraquarks" challenge our understanding of how the strong force, the glue that holds the universe together, operates when more than the usual number of quarks are present. The big question is whether these new particles are simply four quarks stuck together in a tight knot, or if they are actually two separate pairs of quarks interacting with each other, like two dancers moving in sync but not touching.

A team of researchers in China has taken a fresh look at this puzzle by simulating the behavior of these four-quark systems on a supercomputer. Instead of smashing particles together in a collider, they used a method called lattice quantum chromodynamics, which breaks space and time into a tiny grid to calculate how quarks interact from first principles. They focused on two specific scenarios involving pairs of charmonium particles: one where the particles are in a state of zero spin, and another where they have a spin of two. By running these simulations at different energy levels, they mapped out how these pairs attract or repel one another, effectively creating a map of the forces at play in this exotic realm.

The researchers discovered that the interaction between these pairs is dominated by a process called quark rearrangement. Imagine two pairs of dancers, each pair holding hands. In a standard interaction, they might just bump into each other. But in this quantum world, the dancers can swap partners mid-movement, with a quark from the first pair exchanging places with a quark from the second. The study found that this swapping is the primary driver of the forces between the particles, far outweighing the effects of other potential interactions. When the particles were in the zero-spin state, this rearrangement created a strong repulsive force for the $\eta_c\eta_c$ pair, pushing them apart, while the $J/\psi J/\psi$ pair experienced an attractive force, pulling them together. However, in the spin-two state, the same rearrangement mechanism produced a repulsive force for the $J/\psi J/\psi$ pair near the threshold.

This difference in behavior led to the discovery of a new, broad resonance in the spin-two channel. A resonance is a fleeting state that exists for a brief moment before decaying, similar to a drumbeat that rings out and fades. The researchers calculated that this resonance has a mass of approximately 6.54 GeV and a width of about 0.55 GeV. These numbers align remarkably well with a mysterious structure recently spotted by the ATLAS and CMS experiments at the Large Hadron Collider, known as X(6600) or X(6400). The study provides strong theoretical support for the idea that this observed particle has a spin of two, a conclusion that matches the latest experimental analysis of how the particle decays.

In the zero-spin channel, the results were more subtle. The simulations suggested an attractive force for the $J/\psi J/\psi$ system that could allow for a "virtual state" to exist just below the energy threshold where the two particles would normally separate. This virtual state is not a stable particle that can be caught in a detector, but rather a fleeting fluctuation in the energy of the system. The researchers noted that this feature is consistent with another structure observed by the LHCb experiment, called X(6200). However, they cautioned that because their simulations were performed with heavier-than-realistic masses for the light quarks, they could not pinpoint the exact nature of this state with absolute certainty. The presence of other complex effects, such as the exchange of lighter particles, makes it difficult to determine if a true bound state exists or if it is merely a virtual fluctuation.

The study also clarified why these four-quark systems behave the way they do. By analyzing the mathematical structure of the quark interactions, the team showed that the difference between the repulsive zero-spin state (for $\eta_c\eta_c$ ) and the attractive zero-spin state (for $J/\psi J/\psi$ ), as well as the repulsive spin-two state, arises from the specific way the quarks rearrange themselves. This mechanism, known as Fierz rearrangement, acts like a rulebook for how the quarks can swap partners, and the rules dictate whether the result is a push or a pull. The researchers found that this quark-swapping effect is so powerful that it overshadows other potential forces, such as the exchange of gluons or light mesons, at the energy levels they studied.

While the findings offer a compelling explanation for the new particles seen in experiments, the authors are careful to note the limitations of their work. Their simulations were conducted on a relatively coarse grid and used pion masses that are heavier than those found in nature. These factors introduce uncertainties that could shift the precise values of the masses and widths they calculated. Furthermore, the small size of the simulated space means that long-range forces might not be fully captured. The team emphasizes that future studies with finer grids and lighter quark masses will be necessary to confirm these results and to fully understand the long-range dynamics that might be at play.

Despite these caveats, the work represents a significant step forward in decoding the behavior of fully-charmed tetraquarks. By isolating the specific forces at work and linking them directly to the structures seen in collider data, the researchers have provided a clearer picture of how nature assembles these exotic particles. The confirmation that the X(6600) likely has a spin of two, and the identification of the quark rearrangement as the dominant force, brings us closer to understanding the full spectrum of matter that can exist under the influence of the strong force. As experimental data continues to accumulate, these theoretical insights will serve as a crucial guide for interpreting the complex signals emerging from the world's most powerful particle accelerators.

Technical Summary: $\eta_c\eta_c$ and $J/\psi J/\psi$ Scattering from Lattice QCD

Problem Statement
Following experimental observations by LHCb, ATLAS, and CMS of narrow and broad structures in the $J/\psi J/\psi$ invariant mass spectrum (labeled $X(6200)$ , $X(6400)$ , $X(6600)$ , $X(6900)$ , and $X(7100)$ ), there is a critical need to understand the underlying dynamics of fully-charmed tetraquark ( $T_{cc\bar{c}\bar{c}}$ ) states. Specifically, the nature of the interaction between two charmonia ( $\eta_c\eta_c$ and $J/\psi J/\psi$ ) and the quantum numbers ( $J^{PC}$ ) of these observed structures remain theoretically debated. While phenomenological models attribute these dynamics to pomeron exchanges or light meson exchanges, a first-principles investigation using Lattice QCD is required to determine the scattering properties and potential bound or resonant states in the $0^{++}$ and $2^{++}$ channels up to a center-of-mass energy of 6.6 GeV.

Methodology
The authors performed $N_f=2$ lattice QCD calculations using anisotropic gauge ensembles at two unphysical pion masses ( $m_\pi \approx 420$ MeV and $250$ MeV) with a spatial lattice spacing of $a_s \approx 0.136$ fm.

Lattice Setup: Two distinct spatial volumes ( $12^3 \times 96$ and $16^3 \times 128$ ) were used for each pion mass to extract finite-volume energy levels (FVELs).
Operator Construction: The study utilized the distillation method with Laplacian Heaviside (LHS) smearing to handle quark fields. A comprehensive set of meson-meson interpolating operators was constructed for the $\eta_c\eta_c$ and $J/\psi J/\psi$ systems, including $S$ -wave ( $1S_0$ ) and $D$ -wave ( $1D_2, 5S_2$ ) configurations.
Correlation Analysis: Large correlation matrices were computed and analyzed using the Generalized Eigenvalue Problem (GEVP) to extract energy levels. The study explicitly checked for coupled-channel effects between $\eta_c\eta_c$ and $J/\psi J/\psi$ , as well as between $S$ - and $D$ -waves.
Scattering Formalism: Finite-volume energy levels were mapped to infinite-volume scattering amplitudes using Lüscher's formalism. Single-channel analyses were performed for the $0^{++}$ ( $1S_0$ ) and $2^{++}$ ( $5S_2$ ) channels. The scattering amplitudes were parametrized using Effective Range Expansion (ERE) and K-matrix formalisms to identify pole structures (bound states, virtual states, or resonances).
Systematic Checks: The authors rigorously checked the dependence of results on the number of eigenvectors ( $N_V$ ) in the distillation subspace to ensure reliable extraction of high-energy states and verified the negligible coupling between $\eta_c\eta_c$ and $J/\psi J/\psi$ channels.

Key Contributions and Results

Negligible Coupled-Channel Effects: The study found that the coupled-channel mixing between $\eta_c\eta_c$ and $J/\psi J/\psi$ is negligible in both the $0^{++}$ and $2^{++}$ channels. Consequently, single-channel analyses were sufficient to describe the scattering properties.
Dominance of Quark Rearrangement: The interactions in these dicharmonium systems are dominated by the quark rearrangement (QR) effect rather than direct gluonic exchange or light-hadron exchange.
- $0^{++}$ Channel: The $\eta_c\eta_c$ system exhibits a repulsive interaction (scattering length $a_0 \approx -0.12$ fm), while the $J/\psi J/\psi$ system shows an attractive interaction ( $a_0 \approx 0.20\text{--}0.25$ fm). This difference is explained by Fierz rearrangement: the QR term is repulsive for $\eta_c\eta_c$ but attractive for $J/\psi J/\psi$ due to spin and color factors.
- $2^{++}$ Channel: The $5S_2$ $J/\psi J/\psi$ system exhibits a repulsive interaction near the threshold ( $a_0 \approx -0.16$ fm), contrasting with the attraction in the $0^{++}$ channel.
$0^{++}$ $J/\psi J/\psi$ Scattering and $X(6200)$ : The attractive interaction in the $1S_0$ $J/\psi J/\psi$ channel suggests the existence of a near-threshold virtual state with a binding energy of approximately 28–38 MeV below the threshold. The authors note that while left-hand cut effects prevent a definitive pole determination, this virtual state is compatible with the phenomenological $X(6200)$ structure. The results support a $0^{++}$ assignment for $X(6200)$ if it exists.
$2^{++}$ $J/\psi J/\psi$ Scattering and $X(6600)$ : In the $5S_2$ $5 S_{2}$ channel, the scattering amplitude exhibits a Castillejo-Dalitz-Dyson (CDD) zero at $\sqrt{s} \approx 6.45$ $s \approx 6.45$ GeV. More significantly, a resonance pole is identified at:
- $m_R = 6.543(10) - i\,0.548(34)/2$ GeV ( $m_\pi \approx 420$ MeV)
- $m_R = 6.538(13) - i\,0.537(56)/2$ GeV ( $m_\pi \approx 250$ MeV)
  This resonance corresponds to the broad structure $X(6600)$ (or $X(6400)$ ) observed by ATLAS and CMS. The results support the $2^{++}$ quantum number assignment for this state, aligning with recent CMS angular distribution analyses that disfavor $0^{++}$ at 95% confidence.

Significance and Claims
The paper claims to provide the first lattice QCD investigation of dicharmonium scattering in the $0^{++}$ and $2^{++}$ channels up to 6.6 GeV. Its primary significance lies in:

Dynamical Insight: Establishing that quark rearrangement, driven by one-gluon exchange mechanisms and Fierz transformations, is the dominant force governing near-threshold dicharmonium interactions, rather than light-meson exchanges.
Experimental Corroboration: Providing theoretical support for the $2^{++}$ assignment of the $X(6600)$ resonance and the potential $0^{++}$ nature of the $X(6200)$ structure, consistent with recent experimental refinements.
Methodological Validation: Demonstrating the feasibility of extracting high-energy scattering states in heavy-quark systems using distillation and LHS smearing, while highlighting the limitations imposed by momentum suppression and finite-volume effects.

Limitations and Modesty
The authors explicitly state that their conclusions are based on unphysical pion masses and a single, relatively coarse lattice spacing ( $a \approx 0.136$ fm). They note that:

The precise pole positions are subject to uncertainties from left-hand cut effects (due to light-hadron exchange) which are not fully resolved at these pion masses.
Systematic uncertainties related to finite volume and discretization are not fully controlled.
Future studies with physical pion masses and finer lattices are necessary to fully explore the long-range light-hadron dynamics and confirm the pole positions.
The paper does not claim to definitively prove the existence of these states as tetraquarks but rather characterizes the scattering interactions that could lead to such structures.

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