⚛️ phenomenology

A Colour-Casimir Adjacency Matrix Approach to Fully-Heavy Tetraquarks

This paper introduces a phenomenological framework for fully-heavy tetraquark spectroscopy using a colour-Casimir adjacency matrix approach, which corrects the mass ordering flaws of previous Laplacian-based models and predicts an all-bottom ground state at 18.7–18.9 GeV while highlighting the necessity of long-range molecular dynamics to explain the under-binding of the $T_{cc}^+$ state.

Original authors: M. Monemzadeh, N. Tazimi

Published 2026-09-11

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CC0 1.0

Original authors: M. Monemzadeh, N. Tazimi

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

The universe is built from a handful of fundamental particles that stick together to form everything we see. Among these, quarks are the smallest known building blocks, usually found in pairs or triplets, bound tightly by a force so strong it has no name in everyday language, only a mathematical one. For decades, physicists have searched for exotic combinations where four quarks might bind together, a configuration known as a tetraquark. While some of these strange particles have been spotted, the most elusive and fascinating ones are the "fully-heavy" varieties, where all four quarks are the heaviest types available: either charm or bottom. Because these particles are so massive, they move slowly enough that scientists can treat them like tiny, heavy billiard balls rather than wild, relativistic waves, making them a perfect laboratory for testing how the strong force works when pushed to its limits. The question driving this research is simple yet profound: if you take four of these heavy quarks and force them into a single cluster, what mass will they have, and how will they arrange themselves?

A team of researchers from the University of Kashan has tackled this problem by abandoning the complex, multi-variable equations usually required to describe such systems. Instead, they turned to a branch of mathematics called spectral graph theory, which studies the properties of networks. In their model, the four quarks are represented as four points, or vertices, on a graph, and the invisible force holding them together is represented by the lines, or edges, connecting them. The strength of the connection between any two quarks is determined not by distance or speed, but by their "color charge," a property of quarks that functions somewhat like electric charge but comes in three types. The researchers calculated the specific strength of these connections based on the rules of quantum mechanics and arranged them into a single mathematical object known as an adjacency matrix. By solving this matrix, they could predict the energy levels, and therefore the masses, of the resulting particle without needing to guess at dozens of unknown variables.

The study began by testing this method against the all-charm system, where the four quarks are all charm quarks. Recent experiments at the Large Hadron Collider have identified three distinct structures in this sector, named X(6900), X(7100), and a more tentative signal called X(7200). The researchers used the known masses of these three particles to calibrate their model, essentially tuning the three adjustable knobs in their equation until the predictions matched the experimental data perfectly. This calibration yielded a specific mixing angle that describes how the quarks combine, an energy scale for the force between them, and an effective mass for the charm quark within this cluster. Crucially, the team corrected a flaw in an earlier version of their work where a mathematical step had accidentally reversed the logic of the model, making the most attractive force appear as the heaviest state. By removing this error and working directly with the original matrix, they ensured that the most strongly bound configuration correctly corresponded to the lightest mass, aligning with physical intuition.

Once the model was calibrated using the charm quarks, the researchers applied it to a completely different system without making any further adjustments: the all-bottom tetraquark, where every quark is a bottom quark. Because the mathematical rules for how these particles interact are identical regardless of whether they are charm or bottom, the same three numbers derived from the charm sector were used to predict the mass of the bottom version. The result was a precise prediction that the ground state of this all-bottom particle should lie between 18.7 and 18.9 billion electron volts, depending on the exact mass of the bottom quark used. This range aligns with the consensus of other theoretical methods, yet it was achieved here without using any bottom-quark data to tune the model. This serves as a genuine test of the framework, suggesting that the underlying logic of the color-force network holds true across different types of heavy matter.

The model was also applied to a different kind of particle, the Tcc+, which contains two charm quarks and two light antiquarks. Here, the researchers found a significant discrepancy. When they calculated the mass of this particle using only the short-range forces between the four quarks, the result was about 93 million electron volts heavier than the mass actually measured by experiments. This gap is not a failure of the model but a revealing diagnostic. It indicates that the simple picture of four quarks stuck together in a tight cluster is missing a crucial piece of the puzzle. The missing 93 million electron volts of binding energy must come from a different mechanism, likely the long-range attraction between the quarks acting as a pair of mesons, a phenomenon known as molecular dynamics. The model successfully quantifies exactly how much of the particle's stability comes from the tight cluster versus the looser, long-range molecular bond.

The paper also addresses a tension with other interpretations of the data. Some experimental analyses suggest that the three charm structures are simply different vibrational levels of the same object, much like the notes on a guitar string. The researchers' model, however, describes these states as different arrangements of the quarks' internal color charges, with no vibrational component. The model cannot accommodate a fourth, lighter state that some other theories predict, suggesting that if such a state exists, it cannot be explained by this specific color-mixing mechanism alone. The authors are careful to note that their work is a phenomenological tool—a way to organize and understand the data—rather than a derivation from first principles. They have shown that a static, discrete network of color forces can reproduce the known spectrum of heavy particles and provide a quantitative measure of the long-range forces at play in lighter systems. While the model does not yet explain every detail of the data, it offers a clear, parameter-efficient framework that successfully predicts the behavior of unseen particles and quantifies the limits of the compact-quark picture.

Technical Summary: A Colour-Casimir Adjacency Matrix Approach to Fully-Heavy Tetraquarks

Problem Statement
The spectroscopy of fully-heavy tetraquarks (systems composed of four heavy valence quarks, such as $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ ) presents a unique theoretical challenge. While the non-relativistic approximation remains valid due to the heavy quark masses, the colour algebra is richer than in ordinary mesons, and traditional phenomenological approaches (potential models, QCD sum rules, lattice QCD) often introduce a large number of free parameters (constituent masses, string tensions, coupling constants, cut-offs). Furthermore, recent experimental data from LHCb, ATLAS, and CMS regarding the all-charm sector ( $X(6900)$ , $X(7100)$ , and tentative $X(7200)$ ) has sparked debate between competing interpretations: a dynamical picture of radial excitations of aligned spin-1 diquarks versus a static picture of discrete colour-configuration mixing. This paper seeks to explore the latter via a parameter-economical framework based on the spectral theory of weighted graphs, while correcting a structural inconsistency found in a previous iteration of the model.

Methodology
The authors propose a phenomenological framework where the four valence partons of a tetraquark are identified as the vertices of a complete graph $K_4$ . The edges of this graph are weighted by the colour-Casimir factors $\langle \lambda_i \cdot \lambda_j \rangle$ associated with the two available colour-singlet diquark–antidiquark clustering channels: $\bar{3} \otimes 3$ and $6 \otimes \bar{6}$ .

Adjacency Matrix Construction: Two explicit adjacency matrices, $A_{\bar{3}}$ and $A_6$ , are constructed based on the SU(3) $_c$ Casimir factors for intra-cluster (diquark/antidiquark) and inter-cluster (quark–antiquark) interactions.
Mixed Spectrum: A physical state is modeled as a coherent superposition of these two colour channels, controlled by a mixing angle $\alpha$ (where $\beta = \sqrt{1-\alpha^2}$ ). The physical adjacency matrix is defined as $A(\alpha) = \alpha A_{\bar{3}} + \beta A_6$ .
Direct Diagonalisation vs. Laplacian: A critical methodological correction is introduced. An earlier version of this framework passed the adjacency matrix through a graph Laplacian ( $L = D - A$ ) to ensure a non-negative spectrum. The authors demonstrate analytically that for the symmetric graphs relevant here, the Laplacian transformation inverts the physical ordering: it assigns the lightest mass to the least attractive colour configuration (the most repulsive eigenvalue of $A$ ). The current work abandons the Laplacian step and directly diagonalises the colour-weighted adjacency matrix $A(\alpha)$ .
Mass Formula: The physical masses are derived via a linear map:
$M_k = M_{\text{thr}} + \gamma \lambda_k(\alpha)$
where $M_{\text{thr}} = 4m_Q$ is the sum of constituent quark masses, $\gamma$ is an overall energy scale, and $\lambda_k(\alpha)$ are the eigenvalues of $A(\alpha)$ . This formulation allows for both attractive (mass lowering) and repulsive (mass raising) shifts relative to the free threshold, consistent with colour-hyperfine splitting.

Key Contributions

Correction of Formalism: The paper identifies and rectifies a fundamental flaw in the previous Laplacian-based construction, showing that direct diagonalisation of the adjacency matrix preserves the correct physical correspondence where "more colour attraction" implies "lower mass."
Analytical Tractability: Due to the shared eigenbasis of $A_{\bar{3}}$ and $A_6$ (a consequence of the graph's $Z_2 \times Z_2$ permutation symmetry), the spectrum of the mixed matrix $A(\alpha)$ is derived in closed form without numerical diagonalisation.
Parameter Economy: The model utilizes only three adjustable parameters per flavour sector: the mixing angle $\alpha$ , the energy scale $\gamma$ , and the effective constituent mass $m_Q$ .

Results

All-Charm Calibration ( $cc\bar{c}\bar{c}$ ):
Using the masses of $X(6900)$ , $X(7100)$ , and the tentative $X(7200)$ as inputs, the model performs an exact solve (zero residual degrees of freedom) to fix the parameters:
- $\alpha = 0.740$ (implying $\beta^2 \approx 45\%$ hidden-colour admixture).
- $\gamma = 23.2$ MeV.
- $m_c = 1.77$ GeV.
  The authors explicitly state that this calibration is not a statistical fit but a construction of the model's parameters. The fitted charm mass is compatible with literature values but disfavors significantly lower values (e.g., 1.5 GeV) which would require a negative $\gamma$ .
All-Bottom Prediction ( $bb\bar{b}\bar{b}$ ):
Applying the fixed parameters ( $\alpha, \gamma$ ) from the charm sector to the bottom sector without retuning, the model predicts the ground state mass to be in the range 18.7–18.9 GeV for constituent bottom masses ( $m_b$ ) in the standard literature range (4.70–4.78 GeV). This aligns with theoretical consensus ranges but remains a prediction awaiting experimental confirmation.
Diagnostic for $T_{cc}^+$ ( $cc\bar{u}\bar{d}$ ):
The same colour matrices are applied to the doubly-charmed tetraquark $T_{cc}^+$ , assuming the colour algebra depends only on the clustering representation, not the light quark flavours.
- The model predicts a mass of 3967.5 MeV.
- The experimental mass is 3874.8 MeV (just below the $D^0 D^{*+}$ threshold).
- The discrepancy is $\Delta M \approx 93$ MeV.
  The authors interpret this gap quantitatively: the compact, short-distance colour-Casimir mechanism alone under-binds the state by ~93 MeV. This missing binding energy is attributed to long-range molecular dynamics (meson-exchange) between the $D$ and $D^*$ mesons, which are outside the scope of the static four-parton colour graph.

Significance and Claims
The paper modestly frames its significance not as a definitive solution to the full tetraquark spectrum, but as a specific diagnostic tool with clear limitations:

Predictive Content: The primary value lies in the two cross-sector applications (all-bottom prediction and $T_{cc}^+$ diagnostic) derived from parameters fixed solely by the charm sector. These serve as internal consistency checks rather than independent statistical fits.
Limitations: The model is static and discrete; it possesses no radial quantum numbers. Consequently, it cannot accommodate the $X(6600)$ state if interpreted as a radial excitation, nor can it explain the linear mass-squared pattern observed by CMS without invoking a different physical mechanism. The authors acknowledge a tension with the "radial excitation of aligned spin-1 diquarks" interpretation favored by recent CMS data.
Falsifiability: The model is currently over-constrained in the charm sector (zero degrees of freedom). To become a falsifiable, over-constrained model, it requires additional independent data, such as confirmed spin-parity assignments (which the current colour-only model cannot predict) or the definitive confirmation or exclusion of the $X(7200)$ state.
Physical Insight: The $T_{cc}^+$ diagnostic provides a quantitative, physically interpretable measure of the importance of long-range dynamics, distinguishing them from compact colour interactions.

In conclusion, the paper offers a corrected, analytically transparent framework for exploring static colour-configuration mixing in fully-heavy tetraquarks. It successfully reproduces the all-charm spectrum by construction, predicts an all-bottom ground state consistent with theoretical expectations, and quantifies the deficit of compact colour binding in the $T_{cc}^+$ system, thereby isolating the contribution of molecular dynamics.

Technical Summary: A Colour-Casimir Adjacency Matrix Approach to Fully-Heavy Tetraquarks

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