Nature of $K^*(1680)$ and $q\bar{q}$-hybrid mixing as the SU(3) partner of $η_{1}(1855)$ in the strange sector

This study investigates the $K^*(1680)$ state using flux-tube and quark pair creation models, concluding that its anomalous decay pattern cannot be explained by a conventional $q\bar{q}$ scenario but instead provides strong evidence for $q\bar{q}$-hybrid mixing in the strange sector, offering guidance for future experimental searches at BESIII, LHCb, and Belle-II.

The Gist

Imagine the subatomic world as a massive, bustling construction site. For decades, physicists have had a very reliable blueprint called the Quark Model. According to this blueprint, most particles (hadrons) are built in two simple ways:

1. Mesons: A "brick" (quark) glued to an "anti-brick" (antiquark).
2. Baryons: Three bricks stuck together.

But, just like in any big construction project, there are rumors of "ghosts" or "exotic structures" that don't fit the standard blueprint. One of these rumors is the Hybrid Meson. Think of a hybrid not just as a brick and an anti-brick, but as a brick and an anti-brick holding a glowing, vibrating energy rope (a gluon) between them. This extra rope gives the particle special powers (quantum numbers) that a normal brick-and-anti-brick pair simply cannot have.

The Mystery of the "K-star"

The paper focuses on a specific particle called K(1680)*. For a long time, scientists thought this was just a standard "brick-and-anti-brick" particle, specifically one that was excited (like a guitar string vibrating in a higher note).

However, when they looked at how K*(1680) breaks apart (decays) into smaller pieces, the blueprint didn't match the reality.

• The Problem: If K*(1680) were a normal particle, it should break apart in a specific ratio of flavors (like breaking a chocolate bar into specific pieces). But the experiments showed it was breaking apart in a weird, unexpected way. It was like a cake that, when sliced, gave you more frosting than the recipe said it should.

The New Theory: A "Ghost" in the Machine

The authors of this paper propose a solution: K(1680) isn't a pure standard particle; it's a mix.*

Imagine K*(1680) as a cocktail.

• The Main Ingredient (90%): A standard quark-antiquark pair (the "brick" and "anti-brick").
• The Secret Ingredient (10%): A tiny bit of that exotic "gluon rope" (the hybrid).

Even though the hybrid part is small, it acts like a spice that completely changes the flavor of the drink. The paper argues that this small "hybrid spice" is exactly what's needed to explain why the particle breaks apart in the weird way we see in the lab.

How They Tested It

The researchers used two different "kitchen tools" (theoretical models) to simulate the particle's behavior:

1. The "Pair Creation" Model (QPC): This assumes the particle breaks apart by popping a new pair of bricks out of the vacuum. This works great for normal particles.
2. The "Flux Tube" Model (FT): This assumes the particle breaks apart because the "energy rope" snaps. This is the tool for hybrids.

They tried to fit the K*(1680) data using only the first tool (normal particle). It failed. The numbers didn't add up.
Then, they mixed in the second tool (hybrid). Success! By adjusting the "mixing angle" (how much hybrid is in the mix), they could perfectly recreate the experimental data.

The Big Picture: The "Sibling" Connection

The paper connects this to a recent discovery by the BESIII experiment: a particle called η1(1855). This particle was confirmed to be a pure hybrid (a brick, an anti-brick, and a gluon rope).

In the world of particle physics, particles come in families (multiplets), like a set of siblings. If you find one sibling (η1), you expect to find the others.

• The Prediction: If η1(1855) is the "isoscalar" sibling, there should be a "strange" sibling with a similar structure.
• The Candidate: The authors suggest that K(1680)* is that strange sibling. It's the "cousin" of the hybrid family that got mixed up with a normal family member.

Why Does This Matter?

1. It Solves a Puzzle: It explains why K*(1680) behaves strangely.
2. It Hints at Another Mystery: The paper suggests that another particle, K(1410), might be the other side of this mix. If K(1680) is mostly normal with a little hybrid, maybe K*(1410) is mostly hybrid with a little normal. This would explain why K*(1410) is so light and weird, which the standard blueprint can't explain.
3. Future Treasure Hunts: This gives scientists at big labs (like BESIII, LHCb, and Belle-II) a specific map. They now know exactly what to look for to find more of these "hybrid" particles and confirm that the "gluon rope" theory is real.

In a Nutshell

Think of the K*(1680) as a chameleon. For years, we thought it was just a green lizard (a normal particle). But it kept changing colors in ways green lizards shouldn't. This paper says, "Ah, it's not just a green lizard; it's a green lizard wearing a tiny, invisible purple cloak (the hybrid)." That tiny cloak is enough to make it look different, and finding it proves that the "purple cloak" (hybrids) actually exists in nature.

Technical Summary

Here is a detailed technical summary of the paper "Nature of K∗(1680) and q¯q-hybrid mixing as the SU(3) partner of η1(1855) in the strange sector" by Samee Ullah et al.

1. Problem Statement

The paper addresses the long-standing puzzle regarding the nature of the $K^*(1680)$ meson.

• Context: Following the discovery of the exotic isoscalar hybrid candidate $\eta_1(1855)$ (with quantum numbers $J^{PC}=1^{-+}$) by the BESIII collaboration, theoretical interest has surged in identifying its SU(3) flavor partners. The strange isospin-1/2 partner of the $1^{-+}$ hybrid nonet is expected to exist in the mass range of 1.6–1.9 GeV.
• The Candidate: The $K^*(1680)$ is the only known strange vector meson in this mass region.
• The Conflict:
  • If $K^*(1680)$ is a conventional quark-antiquark ($q\bar{q}$) state (specifically a $1^3D_1$or$3^3S_1$state), its strong decay patterns (branching ratios to$K\pi$,$K\eta$,$K^*\pi$, etc.) calculated using the **Quark Pair Creation (QPC)** or$^3P_0$ model do not match experimental data.
  • Specifically, the ratio of decays to $K\eta$ vs. $K\pi$ and the absolute widths of $K^*\pi$ and $K\rho$ channels are inconsistent with pure $q\bar{q}$ predictions.
  • Furthermore, the charge conjugation parity ($C$) is not conserved in the strange sector, allowing mixing between conventional $q\bar{q}$ states ($1^{--}$) and hybrid states ($1^{-+}$).
• Objective: To determine if $K^*(1680)$ can be explained as a physical state resulting from the mixing of a conventional $q\bar{q}$ configuration and a $q\bar{q}g$ hybrid configuration.

2. Methodology

The authors employ a combined theoretical framework utilizing the Flux-Tube (FT) Model and the Quark Pair Creation (QPC) model.

• Theoretical Framework:

  • Hybrid Decay Mechanisms: The decay of a hybrid state is modeled via two distinct modes in the FT model:
    1. Collinear Mode: The constituent gluon moves along the $q-\bar{q}$ axis. This mode is dynamically equivalent to the QPC model (breaking the flux tube by creating a $q\bar{q}$ pair).
    2. Transverse Mode: The gluon moves transversely. This mode generates unique decay amplitudes, particularly for channels involving flavor-singlet hadrons (like $\eta, \eta', \omega, \phi$), which are often suppressed in pure $q\bar{q}$ decays by the OZI rule but enhanced in hybrids.
  • Mixing Formalism: The physical state $|K^*(1680)\rangle$ is defined as a superposition:
    $$|K^*(1680)\rangle = \cos\zeta |q\bar{q}(1^3D_1)\rangle + \sin\zeta |q\bar{q}g\rangle$$
    where $\zeta$ is the mixing angle.
  • Amplitude Construction: The total decay amplitude is the coherent sum of the $q\bar{q}$ contribution (calculated via QPC) and the hybrid contribution (calculated via FT, parameterized by collinear and transverse couplings).
    $$M_{tot} = \cos\zeta M_{QPC} + \sin\zeta M_{FT}$$
  • Coupling Parameters: The collinear hybrid coupling is linked to the QPC coupling but scaled by a factor $\gamma_2$ (assumed $\gamma_2 < \gamma_1$ due to off-shell effects). The relative strength of the transverse mode is defined by $\delta = g_2/g_1$.

• Numerical Analysis:

  • The authors fix standard QPC parameters (constituent quark masses, harmonic oscillator strengths, mixing angle for $\eta-\eta'$).
  • They treat the mixing angle $\zeta$, the transverse coupling ratio $\delta$, and the hybrid coupling strength $\gamma_2$ as free parameters to be fitted against experimental branching ratios and total widths.
  • They test pure $q\bar{q}$ scenarios ($2^3S_1$,$3^3S_1$,$1^3D_1$) and$S-D$ mixing scenarios before introducing the hybrid mixing.

3. Key Contributions

• Demonstration of QPC Failure: The study quantitatively demonstrates that no pure $q\bar{q}$ assignment (radial excitation $3^3S_1$or orbital excitation$1^3D_1$) can simultaneously reproduce the experimental partial widths for$K\pi$,$K\eta$,$K^*\pi$, and$K\rho$. Specifically, pure$1^3D_1$overpredicts the$K\eta$ width by an order of magnitude.
• Identification of Hybrid Mixing: The paper provides strong evidence that the decay pattern of $K^*(1680)$ is best described by a small but non-negligible mixing between the $1^3D_1$$q\bar{q}$state and the$1^{-+}$ hybrid state.
• Quantification of Mixing: The authors extract a specific mixing angle and coupling parameters that resolve the discrepancies between theory and experiment.
• Phenomenological Implications: The study links the nature of $K^*(1680)$ to the lower mass state $K^*(1410)$, suggesting that $K^*(1410)$ might be the lower mass partner in the same mixing scheme, dominated by the hybrid component.

4. Results

• Pure $q\bar{q}$ Scenarios:

  • **$2^3S_1$:** Fails to explain$K\eta$and$K\pi$ widths.
  • $3^3S_1$: Yields couplings too small to account for observed widths.
  • **$1^3D_1$:** Matches$K\pi$and$K^*\pi$reasonably well but predicts a$K\eta$width ($\sim 71$MeV) far exceeding experimental data ($\sim 4.5$ MeV).
  • $S-D$ Mixing: Cannot resolve the $K\eta$ discrepancy without contradicting other channels.

• $q\bar{q}$-Hybrid Mixing Results:

  • Best Fit Parameters:
    • Mixing angle: $\zeta \approx 7.15^\circ \pm 0.76^\circ$.
    • Transverse coupling ratio: $\delta \approx 1.2$.
    • Hybrid coupling strength: $\gamma_2 = 7.0$ (compared to $\gamma_1 = 10.4$ for pure $q\bar{q}$).
  • Decay Widths: With this mixing, the destructive interference in the $K\eta$ channel (due to the opposite sign of spin factors in $1^3D_1$vs.$1^3S_1$-like hybrid amplitudes) significantly reduces the$K\eta$width to match experimental data. Simultaneously, the$K^*\pi$and$K\rho$ widths are enhanced to match observations.
  • $\chi^2$ Improvement: The mixing scenario yields a $\chi^2 \approx 0.49$, a significant improvement over the pure $q\bar{q}$ scenarios (which have no valid $\chi^2$ fit).

• Physical Interpretation:

  • $K^*(1680)$ is predominantly a **conventional $1^3D_1$$q\bar{q}$state** ($\cos^2\zeta \approx 98%$).
  • However, the $\sim 2\%$ hybrid component is crucial for the correct decay phenomenology.
  • The mass difference between the physical state and the pure $q\bar{q}$ state is small, implying a large transition matrix element ($\Delta \approx 135$ MeV) between the pure $q\bar{q}$ and pure hybrid states.

5. Significance

• Validation of Hybrid Nonet: The results support the existence of the light hybrid nonet, identifying $K^*(1680)$ as the strange partner of the $\eta_1(1855)$ and $\pi_1(1600)$.
• Resolution of Spectroscopy Puzzles: The study offers a unified explanation for the anomalous properties of both $K^*(1680)$ and the low-mass $K^*(1410)$, suggesting the latter may be the lower mass eigenstate dominated by the hybrid component.
• Experimental Guidance: The paper provides specific predictions for the decay patterns of strange hybrid states, guiding future searches at BESIII, LHCb, and Belle-II. It emphasizes that even a small hybrid admixture in a dominant $q\bar{q}$ state can drastically alter decay branching ratios, a critical factor for identifying exotic hadrons in experimental data.
• Theoretical Insight: It highlights the necessity of including both collinear and transverse flux-tube modes to accurately describe hybrid decays, particularly those involving flavor-singlet mesons.