All-heavy tetraquarks with different flavors

Using a nonrelativistic potential quark model with the explicitly correlated Gaussian method, this study predicts the mass spectra and narrow fall-apart decay widths of various all-heavy tetraquarks with different flavors ($bb\bar{b}\bar{c}$, $cc\bar{c}\bar{b}$, $bb\bar{c}\bar{c}$, and $bc\bar{b}\bar{c}$), suggesting they are promising candidates for experimental detection at the LHC through channels like $\Upsilon J/\psi$ and $B_c$ meson pairs.

The Gist

Imagine the universe is a giant, bustling construction site. Usually, the "bricks" that build matter are called quarks. Most of the time, these bricks come in pairs (like a proton and an electron holding hands) or triplets (like three friends forming a tight circle).

But sometimes, nature gets creative and builds structures with four bricks at once. These are called tetraquarks.

This paper is about a very special, very heavy, and very rare type of tetraquark. Let's break down what the scientists did, using some everyday analogies.

1. The "Heavyweight Champions" of the Construction Site

Most tetraquarks are made of a mix of light and heavy bricks. But the scientists in this paper are interested in the "Heavyweight Champions." They are looking at structures made entirely of the heaviest bricks available: Charm and Bottom quarks.

Think of it like building a house.

• Normal houses might use a mix of wood, brick, and steel.
• These scientists are asking: "What happens if we build a house using only the heaviest, most expensive steel beams?"

They studied four specific combinations of these heavy bricks:

• bb̄b̄c: Three bottom bricks and one charm brick.
• cc̄c̄b: Three charm bricks and one bottom brick.
• bb̄c̄c: Two bottom and two charm bricks (mixed).
• bc̄b̄c: A specific mix of bottom and charm bricks.

2. The "Super-Computer" Calculation

In the past, scientists tried to guess the weight (mass) of these heavy structures using rough sketches. It was like trying to guess the weight of a car by looking at a stick-figure drawing.

In this paper, the team used a much more powerful tool called the Explicitly Correlated Gaussian (ECG) method.

• The Analogy: Imagine you are trying to find the perfect shape for a cloud of smoke. A rough sketch just says "it's round." The ECG method is like using a high-speed camera and a super-computer to track every single molecule of smoke, seeing exactly how they wiggle, bump, and stick together.
• The Result: Because they used this "high-definition" method, they found that these heavy structures are actually lighter than previously thought (by about 30 to 100 units of weight). They also found that the different versions of these structures are arranged in a slightly different order than we guessed before.

3. The "Explosive" but "Quiet" Decay

Once they calculated the weight, they asked: "How long do these things last?"

In the world of particles, if a structure is unstable, it "falls apart" instantly into smaller pieces. This is called a fall-apart decay.

• The Analogy: Imagine a house of cards. If you blow on it, it collapses immediately. That's a "broad" decay (it happens fast and is messy).
• The Surprise: The scientists found that these heavy tetraquarks are surprisingly stable. They don't collapse instantly. Instead, they are like a well-built stone fortress. When they do fall apart, they do so very slowly and quietly.
• The Numbers: They predicted these structures would last long enough to be seen in experiments, with a "decay width" (a measure of how fast they break) ranging from a tiny fraction of a unit to just a few units. In particle physics, that's considered very narrow and very stable.

4. Where to Look for Them? (The "Treasure Hunt")

The paper ends with a "Where to look" guide for experimentalists at the LHC (the Large Hadron Collider, a giant particle accelerator in Europe).

Since these heavy structures are so rare, you can't just look for them anywhere. You have to look for the specific "footprints" they leave behind when they finally fall apart.

• The Footprints: The scientists calculated that these tetraquarks are most likely to break apart into specific pairs of particles, such as:
  • A J/ψ (a heavy charm particle) and a Bc (a heavy bottom-charm particle).
  • An Υ (a heavy bottom particle) and a Bc.
• The Advice: If you are an experimentalist at the LHC, don't just look at everything. Focus your search on these specific pairs. If you see a "bump" in the data at the specific weights the scientists calculated (around 16 GeV, 9.7 GeV, 12.9 GeV, etc.), you might have found a new, exotic form of matter!

Summary

• What they did: They used a super-precise mathematical method to calculate the exact weight and behavior of four types of "all-heavy" tetraquarks.
• What they found: These structures are real, they are compact (tight little balls), and they are surprisingly stable (they don't fall apart instantly).
• Why it matters: This gives experimentalists a clear "Wanted Poster" with the exact weights and the specific "footprints" to look for. If the LHC finds them, it will confirm that nature can build these incredibly heavy, four-brick structures, opening a new chapter in our understanding of the universe's building blocks.

Technical Summary

1. Problem Statement

The paper addresses the theoretical characterization of all-heavy tetraquarks containing different flavors of heavy quarks (charm $c$ and bottom $b$). While the existence of fully heavy tetraquarks like $cc\bar{c}\bar{c}$ (e.g., $X(6900)$) and searches for $bb\bar{b}\bar{b}$ have been active topics, systems with mixed flavors ($bb\bar{b}\bar{c}$, $cc\bar{c}\bar{b}$, $bb\bar{c}\bar{c}$, and $bc\bar{b}\bar{c}$) remain less explored experimentally and theoretically.

Previous theoretical studies by the same group (2019) and others suffered from model dependencies and incomplete trial wave functions. Specifically, earlier treatments often approximated oscillator parameters as mass-independent or used simplified Gaussian bases that failed to fully account for the correlations between non-identical quarks. This led to potential inaccuracies in mass predictions and an incomplete understanding of the internal structure (color/spin configurations) of these states.

2. Methodology

The authors employ a rigorous nonrelativistic potential quark model framework enhanced by the Explicitly Correlated Gaussian (ECG) method.

• Hamiltonian: The system is described by a four-body Hamiltonian including quark masses, kinetic energy, and effective inter-quark potentials. The potential $V_{ij}$ includes:
  • Confinement (linear term).
  • One-gluon exchange (Coulomb-like term).
  • Spin-spin interaction (hyperfine splitting).
  • Color-magnetic interactions.
• Wave Function Construction:
  • Spatial Part: Instead of simple harmonic oscillators, the authors use Explicitly Correlated Gaussians (ECG). The spatial wave function is expanded as a linear combination of Gaussian basis functions with variational parameters ($\alpha_{ij}$) that depend on the specific quark masses and correlations.
  • Completeness: A key improvement over previous work is the inclusion of cross terms in the Gaussian basis (e.g., $\xi_1 \cdot \xi_2$). This allows the wave function to capture higher partial wave contributions arising from the non-identical nature of the quarks (e.g., $b$ vs. $c$), ensuring the trial function is complete and satisfies the Pauli principle correctly for mixed-flavor systems.
  • Variational Method: The parameters are optimized using the stochastic variational method to minimize the energy eigenvalues.
• Decay Calculation:
  • Fall-apart Decays: The authors evaluate the decay widths into two-meson final states (e.g., $\Upsilon B_c$) using a quark-exchange model.
  • Amplitude: The decay amplitude is calculated via the matrix element of the inter-quark potential between the initial tetraquark state and the final meson pair states.
  • Final States: The final mesons are approximated using Single Harmonic Oscillator (SHO) wave functions, with parameters fitted to reproduce the root-mean-square radii calculated from the same potential model.

3. Key Contributions

• Refined Mass Spectra: The study provides the first precise mass spectra for the $1S$-wave states of $bb\bar{b}\bar{c}$, $cc\bar{c}\bar{b}$, $bb\bar{c}\bar{c}$, and $bc\bar{b}\bar{c}$ using the ECG method, correcting previous approximations.
• Wave Function Completeness: By explicitly treating the correlations between non-identical quarks, the authors demonstrate that previous models underestimated the binding energy (predicting masses too high) and misidentified the dominant color-spin configurations for certain states.
• Decay Width Predictions: The paper systematically calculates the fall-apart decay widths for all predicted $1S$ states, identifying optimal experimental search channels.
• Comparison with Other Models: The results are benchmarked against various approaches (QCD sum rules, complex scaling, diquark models), highlighting the stability and reliability of the ECG approach for few-body heavy systems.

4. Key Results

A. Mass Spectra (1S States)

The calculated mass ranges (in GeV) are:

• $bb\bar{b}\bar{c}$: $\sim 16.06 - 16.14$
• $cc\bar{c}\bar{b}$: $\sim 9.65 - 9.74$
• $bb\bar{c}\bar{c}$: $\sim 12.89 - 12.94$
• $bc\bar{b}\bar{c}$: $\sim 12.75 - 12.99$

Observations:

• Compared to the authors' 2019 work, the masses are shifted downward by 30–100 MeV due to the more complete wave function.
• All predicted states are compact (RMS radii $\approx 0.27 - 0.51$ fm) and lie significantly above the threshold for dissociation into two ground-state mesons.
• Configuration Mixing: The dominant color configurations ($\bar{3} \otimes 3$ vs. $6 \otimes \bar{6}$) for specific states (e.g., the $0^+$ states in $bb\bar{b}\bar{c}$) are found to be different from previous predictions, altering the ordering of the spectrum.

B. Decay Properties

• Narrow Widths: All predicted $1S$ states are narrow resonances with fall-apart decay widths ranging from a few tenths of an MeV to several MeV. This contradicts some QCD sum rule predictions that suggested broad widths ($\sim 100$ MeV).
• Optimal Decay Channels:
  • $bb\bar{b}\bar{c}$: Best channels are $\Upsilon B_c^*$ and $\eta_b B_c^*$.
  • $cc\bar{c}\bar{b}$: Best channels are $J/\psi B_c^*$ and $\eta_c B_c$.
  • $bb\bar{c}\bar{c}$: Best channels are $B_c B_c$ and $B_c^* B_c^*$.
  • $bc\bar{b}\bar{c}$: Best channels include $\Upsilon J/\psi$, $\eta_b \eta_c$, and $B_c^+ B_c^-$.
• Specific Highlights:
  • The $0^+$ state of $bb\bar{b}\bar{c}$ at 16.064 GeV has a total width of $\sim 1.25$ MeV, with a dominant branching ratio to $\Upsilon B_c^*$.
  • The $bc\bar{b}\bar{c}$ system shows a rich spectrum of $0^{++}$, $1^{+-}$, $1^{++}$, and $2^{++}$ states, with the $2^{++}$ state at 12.860 GeV being a promising candidate for observation via $\Upsilon J/\psi$.

5. Significance

• Experimental Guidance: The paper provides specific mass predictions and narrow decay widths, suggesting that these tetraquarks are viable candidates for discovery at the LHC (specifically in the $pp$ collision data from LHCb, CMS, and ATLAS). The identification of optimal decay channels (e.g., $\Upsilon J/\psi$, $B_c B_c$) offers concrete targets for experimentalists.
• Theoretical Validation: The study validates the ECG method as a superior tool for solving few-body heavy quark problems compared to simpler variational approaches, resolving discrepancies in mass splittings and configuration mixing found in earlier literature.
• Nature of Exotic Hadrons: The confirmation that these states are compact and narrow supports the interpretation of all-heavy tetraquarks as genuine compact multiquark states rather than loosely bound meson molecules, reinforcing the validity of the diquark-antidiquark or compact tetraquark picture in QCD.