Interpretation of $Υ(11020)$ as an $S$-Wave $B_1\bar{B}$--$B_1\bar{B}^*$ Molecular State

This paper proposes that the $\Upsilon(11020)$ resonance is an $S$-wave $B_1\bar{B}$--$B_1\bar{B}^*$ molecular state and supports this hypothesis by calculating its strong decay widths, which reveal a dominant $B_s^*\bar{B}^*$ channel and distinctive patterns in multi-pion transitions that serve as experimental signatures for heavy-quark symmetry.

The Gist

Imagine the universe of subatomic particles as a massive, bustling construction site. For decades, physicists have been building a "standard model" of how these tiny bricks (quarks) fit together. Usually, they fit in predictable patterns: two bricks make a meson, three make a baryon. But recently, workers have found some strange, oddly shaped structures that don't fit the blueprints. These are called "exotic states," and they might be built differently—perhaps as loose clusters of particles held together by a weak force, much like a molecule in chemistry.

This paper is a detective story about one specific particle: the $\Upsilon(11020)$.

The Mystery: A Particle Out of Place

For a long time, scientists thought the $\Upsilon(11020)$ was a standard "bottomonium" particle. Think of this like a heavy-duty dumbbell made of two heavy weights (a bottom quark and its anti-quark) glued tightly together.

However, this particle has been acting suspicious. When it breaks apart (decays), it doesn't follow the rules expected of a standard dumbbell. Instead, it seems to be taking a detour through a specific, strange intermediate step involving other particles called $Z_b$. It's as if a standard car, when driving, suddenly decided to take a detour through a specific, narrow alleyway that only a very specific type of vehicle could use.

The Hypothesis: A "Molecular" Partnership

The authors, Qing Lu, Cai Cheng, and Yin Huang, propose a new theory: The $\Upsilon(11020)$ isn't a glued-together dumbbell; it's a "molecule."

In this scenario, the particle is actually a loose partnership between two different heavy mesons (specifically $B_1$ and $\bar{B}$).

• The Analogy: Imagine a standard car is a solid block of metal. A "molecular" particle is like two cars parked very close together, holding hands with a weak magnetic force. They aren't fused into one solid block; they are a team that can easily drift apart.
• The Connection: The authors suggest this particle is the "heavy cousin" of a known particle called $X(3872)$, which is already known to be a molecular state. Heavy-Quark Symmetry (a rule of physics) predicts that if one cousin exists, the other should too.

The Investigation: Testing the Theory

To prove this, the authors didn't just guess; they built a detailed mathematical simulation (a "virtual lab").

1. The Setup: They used a set of rules (Effective Lagrangians) that describe how these heavy particles talk to each other.
2. The Calibration: They adjusted the "knobs" on their simulation (specifically the strength of the connection between the particles) until the simulation matched the real-world data we already have. They looked at two specific real-world events:
   • How often the particle turns into an electron and a positron ($e^+e^-$).
   • How often it turns into a specific mix of pions and a bottomonium state ($\pi\pi\pi\chi_b$).
3. The Result: When they tuned their simulation to match these real events, the math worked perfectly only if they assumed the particle was indeed a $B_1\bar{B}$ molecule, with the $B_1\bar{B}$ part making up about 75% of the whole thing.

The Prediction: What to Look For

If this theory is right, the $\Upsilon(11020)$ should behave in very specific ways that are different from a standard particle. The authors calculated exactly what these "fingerprints" would look like:

• The "Silent" Channels: If you look for the particle turning into certain combinations of pions and other bottomonium states (like $\pi\pi\Upsilon$), the signal should be incredibly faint—almost invisible (measured in electron-volts, which is tiny).
• The "Loud" Channels: In contrast, if you look for it turning into three pions and a specific particle called $\chi_b$, the signal should be much louder (measured in Mega-electron-volts).
• The Hidden Treasure: The authors predict the particle's favorite way to decay is into a pair of strange-bottom mesons ($B^*_s\bar{B}^*_s$). However, this channel has never been seen in experiments yet.

The Conclusion

The paper argues that the $\Upsilon(11020)$ is likely a "molecular" state—a loose team of heavy particles rather than a solid block.

• Why it matters: If future experiments (like those at the LHCb facility) go looking for these specific "fingerprints" (the loud three-pion signal and the silent two-pion signal) and find them, it will confirm that this particle is a molecule.
• The Big Picture: This would be a major victory for "Heavy-Quark Symmetry," proving that nature builds these exotic molecular structures in the heavy-quark world just as it does in the light-quark world. It would also solve the mystery of why this particle has been acting so weirdly compared to its siblings.

In short, the authors have built a mathematical case that the $\Upsilon(11020)$ is a molecular team player, and they have provided a specific "shopping list" of decay patterns for experimentalists to check off to confirm the theory.

Technical Summary

Technical Summary: Interpretation of $\Upsilon(11020)$ as an S-Wave $B_1 \bar{B} - B_1 \bar{B}^*$ Molecular State

Problem Statement
Heavy-quark symmetry (HQS) predicts the existence of molecular partners for known heavy-light meson systems. While the $D_1 \bar{D}$ molecular state has candidates such as $Z_2^+(4250)$ and $Y(4260)$, its heavy-quark-flavor partner, the $B_1 \bar{B}$ molecule, has not been experimentally confirmed. The $\Upsilon(11020)$ (often identified as the $\Upsilon(6S)$ state) exhibits decay properties that deviate from the conventional quarkonium picture, particularly its decay into $\Upsilon(nS)\pi\pi$ and $h_b(kP)\pi\pi$ proceeding almost exclusively through intermediate $Z_b^{(\prime)}$ states. This paper investigates whether the $\Upsilon(11020)$ can be interpreted as an S-wave $B_1 \bar{B} - B_1 \bar{B}^*$ molecular state, dominated by the $B_1 \bar{B}$ component, and whether this interpretation resolves the discrepancies between its observed decay patterns and standard quark model predictions.

Methodology
The authors employ an effective field theory approach based on hadronic degrees of freedom to calculate the strong decay widths of the $\Upsilon(11020)$ under the molecular hypothesis.

• Theoretical Framework: The study utilizes effective Lagrangians derived from heavy-quark symmetry and chiral perturbation theory (ChPT). The interaction vertices include $\Upsilon(11020)B_1 \bar{B}^{(*)}$, $B_1 B^* \pi$, $Z_b^{(\prime)} B \bar{B}^*$, and various couplings involving light vector mesons ($\rho, \omega$) and pseudoscalars.
• Formalism: The decay amplitudes are calculated using Feynman diagrams involving hadronic loops (for two-body decays like $B^{(*)} \bar{B}^{(*)}$ and three-body decays like $\pi\pi\Upsilon(nS)$) and tree-level diagrams (for three-body decays like $B^* \pi \bar{B}^{(*)}$).
• Regulator and Couplings: A Gaussian-type correlation function $\Phi(p_E^2/\Lambda^2)$ is introduced to account for the finite size of the molecule and ensure ultraviolet finiteness. The cutoff parameter $\Lambda$ and the coupling constants $g_{\Upsilon(11020)B_1 \bar{B}^{(*)}}$ are determined via the compositeness condition, which requires the wave function renormalization constant to vanish.
• Fitting Procedure: The model parameters are constrained by fitting the calculated partial widths to experimental data for $\Upsilon(11020) \to e^+e^-$ and $\Upsilon(11020) \to \pi\pi\pi\chi_{bJ}$. The coupling $g_{TH}/\Lambda_\chi$ is fixed using the experimental width of $B_1(5721) \to B^* \pi$.

Key Results

1. Molecular Composition: The fitting procedure yields a range of valid parameters where the $\Upsilon(11020)$ is predominantly a $B_1 \bar{B}$ molecular state. Specifically, for a cutoff $\Lambda = 0.613$ GeV, the $B_1 \bar{B}$ component ($X_{B_1 \bar{B}}$) is approximately 75.4%, with the remainder being a $B_1 \bar{B}^*$ component.
2. Total Decay Width: Within this molecular framework, the calculated total decay width agrees well with the experimental value of $24$ MeV. The total width is found to be highly sensitive to the molecular composition and the cutoff parameter.
3. Dominant Decay Channels:
   • The dominant decay channel is predicted to be $B_s^* \bar{B}_s^*$, with a partial width of 10.573 MeV. This channel has not yet been observed experimentally.
   • The three-body decay $B^* \pi \bar{B}^{(*)}$ (induced by tree-level diagrams) contributes 5.573 MeV.
   • In contrast to conventional quark model predictions (which favor $B\bar{B}$ or $B^*\bar{B}^*$), the $B_s \bar{B}_s$ channel is suppressed in this model (0.418 MeV).
4. Rare Decay Channels:
   • The transitions $\Upsilon(11020) \to \pi\pi\Upsilon(nS)$ and $\Upsilon(11020) \to \pi\pi h_b(nP)$, proceeding via intermediate $Z_b^{(\prime)}$ states, are calculated to have extremely small partial widths, ranging from 8 to 162 eV. This is significantly lower than the keV-scale widths predicted by conventional quark models.
   • The channel $\Upsilon(11020) \to \pi\pi\pi\chi_{bJ}$ is significantly enhanced. The $\pi\pi\pi\chi_{b1}$ width is 0.167 MeV, and the unobserved $\pi\pi\pi\chi_{b0}$ channel is predicted to be 0.754 keV.

Significance and Claims
The paper claims that the distinctive decay pattern of the $\Upsilon(11020)$—specifically the dominance of the $B_s^* \bar{B}_s^*$ channel, the suppression of $\pi\pi\Upsilon(nS)$ and $\pi\pi h_b(nP)$ modes, and the enhancement of $\pi\pi\pi\chi_{bJ}$ modes—serves as a unique experimental signature for its molecular nature.

The authors argue that if these specific decay patterns are confirmed experimentally, they would provide strong evidence that the $\Upsilon(11020)$ is an S-wave $B_1 \bar{B} - B_1 \bar{B}^*$ molecule rather than a conventional $b\bar{b}$ quarkonium state. Furthermore, confirming this interpretation would validate the applicability of heavy-quark symmetry in predicting the existence of the $B_1 \bar{B}$ partner to the $D_1 \bar{D}$ system. The study suggests that these signatures could be probed at facilities such as LHCb.