Fully-strange tetraquarks: fall-apart decays and experimental candidates

This paper presents a systematic analysis of fall-apart decays for fully-strange tetraquark states, suggesting that most have narrow widths and proposing that the newly observed $X(2300)$ and $X(2500)$ resonances correspond to specific low-lying $1S$-wave and $1P$-wave tetraquark candidates, respectively, which can be further investigated through their dominant decay channels.

The Gist

Imagine the universe is a giant, cosmic LEGO set. For decades, physicists have been building structures using specific rules: you can snap two bricks together to make a "meson" (like a proton's cousin), or three bricks to make a "baryon" (like a proton or neutron). This is the standard "Quark Model," and it worked perfectly for a long time.

But recently, scientists have started finding weird, exotic LEGO creations that don't follow the old rules. They are finding structures made of four bricks stuck together. These are called tetraquarks.

This paper is like a detective's guidebook for finding a very specific, rare type of these four-brick structures: the "Fully-Strange" Tetraquark.

The Cast of Characters: The "Strange" Bricks

In our LEGO analogy, there are different types of bricks:

• Up and Down bricks: These are common and make up most of the matter around us.
• Strange bricks: These are heavier, rarer, and a bit more "exotic."

A "Fully-Strange" tetraquark is a structure made of four strange bricks (two particles and two anti-particles). Because they are made entirely of these heavy, exotic bricks, they are very hard to find and very unstable. They are like a house of cards made of heavy gold bricks; they want to fall apart immediately.

The Main Event: The "Fall-Apart" Dance

The paper focuses on how these structures break apart. The authors call this "fall-apart decay."

Imagine you have a tetraquark (4 bricks). It's so unstable that it doesn't just slowly crumble; it instantly snaps back into two pairs of bricks (two mesons).

• The Analogy: Think of a tetraquark as a clumsy dancer holding hands with three other dancers. Suddenly, the music stops, and they instantly split into two couples.
• The Goal: The paper calculates exactly how they split. Do they split into two heavy couples? Two light couples? Or a mix?

The authors used a complex mathematical model (a "quark-exchange model") to predict the speed and direction of this split. They found that for most of these strange tetraquarks, the "fall-apart" happens relatively slowly (in physics terms), giving them a narrow "width" (a measure of how long they live before breaking).

The Detective Work: Matching Clues to Suspects

The paper isn't just theory; it's trying to match these predictions with real-life sightings at particle accelerators like BESIII (a giant microscope in China that smashes particles together).

Here are the "suspects" the paper is investigating:

1. The X(2300) Suspect:

   • The Clue: Scientists recently saw a new particle called X(2300). It has a specific "spin" and "charge" (like a unique fingerprint).
   • The Theory: The paper says, "Hey! This looks exactly like our predicted '1S-wave' tetraquark made of four strange bricks!"
   • The Verdict: It's a strong match. The mass and the way it decays fit the prediction perfectly.

2. The X(2500) Suspect:

   • The Clue: Another particle, X(2500), was seen earlier.
   • The Theory: This one might be a different type of tetraquark (a "1P-wave" state).
   • The Verdict: It's a likely candidate, though the math is a bit more complex.

3. The "Impossible" Suspect (The 2++ State):

   • The Mystery: There are some particles listed in the official encyclopedia of particles (like the f2(2300)) that some people thought were tetraquarks.
   • The Twist: The paper says, "Nope, these can't be our tetraquarks."
   • Why? The math shows that for a specific type of tetraquark, breaking into a certain pair of particles is forbidden. It's like trying to push a square peg into a round hole; the laws of physics (specifically, symmetry) cancel out the possibility completely. If a particle does break into that pair, it can't be the tetraquark we are looking for.

The Treasure Map: Where to Look Next

The most exciting part of the paper is the "Treasure Map" for future experiments. The authors say: "If you want to find these hidden four-brick structures, look in these specific channels."

They predict that these tetraquarks love to break apart into specific combinations of particles, such as:

• Two phi mesons (a pair of strange particles).
• A phi and an eta (a mix of strange and light particles).
• Combinations involving h1 or f2 particles.

The Analogy: Imagine you are looking for a lost dog. You don't just wander aimlessly. You know the dog loves to run toward the park and the bakery. So, you go check those two spots first.

• The Paper's Advice: "Don't just look everywhere. Go check the phi-phi and eta-phi channels at the BESIII and Belle-II experiments. That's where the fully-strange tetraquarks are most likely to show up."

Why Does This Matter?

Finding these particles is like finding a new species of animal. It proves that the "rules" of how matter is built are more flexible than we thought.

• If we find them, it confirms that nature allows for these complex, four-part structures made entirely of "strange" matter.
• It helps us understand the "glue" (the strong force) that holds the universe together.
• It might even help us understand why the universe is made of matter and not just energy.

Summary in a Nutshell

This paper is a theoretical hunting guide.

1. The Hunt: We are looking for rare, four-particle structures made entirely of "strange" quarks.
2. The Method: We calculated exactly how they should break apart (fall-apart decay).
3. The Findings: Most of them break apart slowly (narrow width).
4. The Matches: The recently discovered X(2300) and X(2500) look very much like our predicted tetraquarks.
5. The Exclusion: Some other known particles (like the f2(2300)) are likely not tetraquarks because the math says they can't break apart in the way observed.
6. The Next Step: Go to the particle accelerators and look specifically for these particles breaking into phi-phi or eta-phi pairs. If you find them, you've found a new building block of the universe!

Technical Summary

1. Problem Statement

The search for genuine exotic hadrons beyond the conventional quark model is a central goal in high-energy physics. While fully-charmed tetraquarks ($T_{cccc}$) have recently been observed (e.g., $X(6900)$), the existence of fully-strange tetraquark states ($T_{ss\bar{s}\bar{s}}$) remains a subject of active investigation.

• The Challenge: Several resonances (e.g., $X(2300)$, $X(2500)$, $f_0(2200)$, $f_2(2340)$) have been observed in experiments (BESIII, LHCb, etc.) that could potentially be $T_{ss\bar{s}\bar{s}}$ states. However, their nature is ambiguous because their masses can sometimes be explained by conventional $s\bar{s}$ mesons, glueballs, or molecular states.
• The Gap: While theoretical mass spectra for $T_{ss\bar{s}\bar{s}}$ states have been calculated using various models (QCD sum rules, quark models), there is a lack of systematic studies on their decay properties, specifically fall-apart decays. Without decay width predictions and branching ratios, it is difficult to distinguish these exotic states from conventional hadrons or confirm their tetraquark nature experimentally.

2. Methodology

The authors perform a systematic analysis of the fall-apart decay mechanisms for $1S$, $1P$, and $2S$-wave $T_{ss\bar{s}\bar{s}}$ states.

• Mass and Wave Functions: They utilize the mass spectra and wave functions of the $T_{ss\bar{s}\bar{s}}$ states calculated in their previous work [7] using a nonrelativistic quark model.
• Decay Framework: The decay amplitudes are calculated using a quark-exchange model.
  • Mechanism: The fall-apart decay is induced by quark-quark interactions (part of the Hamiltonian) causing quark rearrangement between the initial tetraquark and the final meson pair.
  • Formalism: The decay amplitude $\mathcal{M}(A \to BC)$ is derived from the matrix element of the interaction potential $V_{ij}$ between the inner quarks of the final hadrons.
  • Partial Widths: The partial decay width $\Gamma$ is calculated using the standard formula involving the phase space factor and the squared amplitude, accounting for statistical factors for identical particles.
• Input Parameters:
  • Final state meson wave functions are modeled using a Single Harmonic Oscillator (SHO) form.
  • SHO parameters ($\beta$) are determined by fitting the root-mean-square radii of $s\bar{s}$ states from potential model calculations.
  • The $\eta$ and $\eta'$ mesons are treated as mixed states of $n\bar{n}$ and $s\bar{s}$ with a mixing angle $\phi_P = 41.2^\circ$.
  • Masses for well-established mesons are taken from PDG; others are taken from potential model predictions.

3. Key Contributions

• Systematic Decay Analysis: This is the first comprehensive study providing partial decay widths and branching ratios for a wide range of $T_{ss\bar{s}\bar{s}}$ states ($1S, 1P, 2S$) across various $J^{PC}$ quantum numbers.
• Identification of Experimental Candidates: The authors correlate their theoretical predictions with recently observed resonances ($X(2300)$, $X(2500)$) and established resonances ($f_0(2200)$, $f_2(2340)$, $\phi(2170)$) to propose or refute their tetraquark nature.
• Discovery of Dynamical Suppression: The paper identifies a specific dynamical cancellation mechanism that forbids the decay of the $1S$ state $T(4s)^{2++}(2378)$ into the $\phi\phi$ channel, a crucial insight for interpreting experimental data.
• Experimental Roadmap: The authors propose specific dominant decay channels (e.g., $\phi\phi$, $\phi\phi(1680)$, $\eta^{(\prime)}h_1(1415)$, $\phi f'_2(1525)$) as optimal search modes for future experiments at BESIII and Belle-II.

4. Key Results

A. General Characteristics

• Most fully-strange tetraquark states exhibit narrow fall-apart decay widths of the order of $O(10)$ MeV.
• Exceptions occur for highly excited states where specific channels open up, leading to broader widths (up to $\sim 200-700$ MeV).

B. Specific State Analysis and Experimental Candidates

1. $1S$-wave States:

• $T(4s)^{0++}(2218)$: Predicted width $\Gamma \approx 19$ MeV. Dominant channels: $\eta\eta'$, $\eta'\eta'$, $\phi\phi$.
  • Candidate: Consistent with the scalar resonance $f_0(2200)$ and structures observed in $J/\psi \to \gamma \phi\phi$ around 2.2 GeV.
• $T(4s)^{1+-}(2323)$: Predicted width $\Gamma \approx 8$ MeV. Dominant channels: $\eta\phi$ and $\eta'\phi$ (ratio $\approx 1:1.5$).
  • Candidate: Strongly favors the newly observed $X(2300)$ at BESIII. The mass, quantum numbers ($1^{+-}$), and decay modes align well, whereas the conventional $s\bar{s}$ interpretation ($h_1(3P)$) fails on mass and width.
• $T(4s)^{2++}(2378)$: Predicted width $\Gamma \approx 1$ MeV.
  • Crucial Finding: The decay into $\phi\phi$ is strictly forbidden due to exact cancellation of dynamical transition matrix elements arising from wavefunction symmetry.
  • Implication: This rules out $T(4s)^{2++}(2378)$ as the candidate for $f_2(2300)$ or $f_2(2340)$, which are observed in the $\phi\phi$ channel with broad widths. These resonances may instead be glueballs.

2. $1P$-wave States:

• $T(4s)^{0-+}(2481)$: Predicted width $\Gamma \approx 15$ MeV. Dominant channels: $\phi\phi$, $\eta f_0(1370)$, $\eta' f_0(1370)$.
  • Candidate: Consistent with the $X(2500)$ observed in $J/\psi \to \gamma \phi\phi$. The mass and $0^{-+}$ assignment match, though the experimental width is broader than the pure fall-apart prediction (suggesting other decay mechanisms or mixing).
• $T(4s)^{1--}$ States: The lowest state $T(4s)^{1--}(2455)$ has a narrow width ($\approx 9$ MeV) dominated by $\phi f_0(1370)$.
  • Candidate: May correspond to the vague peak $X(2400)$ seen in $\phi f_0(980)$ spectra.
  • Note: The well-known $\phi(2170)$ is not a candidate for the lowest $1--$ tetraquark, as its mass is $\sim 270$ MeV too low compared to the prediction.

3. $2S$-wave States:

• These states are generally heavier ($2.8 - 3.2$ GeV) and have broader widths due to the opening of channels like $\phi\phi(1680)$ and $h_1(1415)h_1(1415)$.
• Specific candidates are proposed for $0^{++}$, $2^{++}$, and $1^{+-}$ states, with $\phi\phi(1680)$ and $\eta^{(\prime)}h_1(1415)$ being key search channels.

5. Significance

• Validation of Exotic Hadrons: The study provides a robust theoretical framework to validate the $T_{ss\bar{s}\bar{s}}$ nature of observed resonances like $X(2300)$ and $X(2500)$, moving beyond mass-only comparisons to include decay dynamics.
• Refutation of Conventional Assignments: By demonstrating the forbidden $\phi\phi$ decay for the $2^{++}$ state and the mass mismatch for $\phi(2170)$, the paper helps eliminate incorrect assignments, guiding the community toward alternative interpretations (e.g., glueballs or hybrids) for those specific resonances.
• Experimental Guidance: The paper offers a clear "search list" for experimentalists. It identifies that while some states are narrow and easy to spot, others require specific high-energy channels (like $\phi\phi(1680)$ or $\eta h_1$) to be distinguished from background.
• Theoretical Insight: The identification of the exact cancellation mechanism in the $2^{++}$ state highlights the importance of wavefunction symmetry in multiquark dynamics, a feature that suppresses rearrangement decays in $ss\bar{s}\bar{s}$ systems generally.

In conclusion, this work bridges the gap between theoretical mass predictions and experimental observables for fully-strange tetraquarks, offering a definitive guide for future searches at facilities like BESIII and Belle-II.