Role of $a_0(1710)$ in the $J/\psi\to\rho^+\rho^-\omega$ and $J/\psi\to\gamma\rho^0\omega$ reactions

This paper predicts that the scalar meson $a_0(1710)$, dynamically generated through final-state interactions, will manifest as distinct peaks in the invariant mass distributions of $\rho\omega$ pairs in both the strong decay $J/\psi\to\rho^+\rho^-\omega$ and the radiative decay $J/\psi\to\gamma\rho^0\omega$, offering a promising avenue for future experimental verification and precise characterization by facilities like BESIII, Belle II, and STCF.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. In this dance, particles called mesons are the dancers, constantly pairing up, spinning, and sometimes crashing into one another to create new, temporary formations.

This paper is a theoretical investigation into a specific, somewhat mysterious dancer named $a_0(1710)$.

The Mystery of the "Ghost" Dancer

For a long time, physicists have known about most of the dancers on this floor. They fit neatly into a standard rulebook (the "quark-antiquark" model). But the $a_0(1710)$ is a bit of an outlier. It's a "scalar meson," which is a fancy way of saying it's a specific type of particle that is hard to pin down.

Think of the $a_0(1710)$ like a ghost that has only recently been spotted by security cameras (experiments like BABAR, BESIII, and LHCb). We know it exists because we see a blur in the footage, but we don't agree on exactly how heavy it is or how long it stays visible (its "width"). Some cameras say it weighs 1704 units, others say 1817. It's a bit of a mess.

The Theory: A Molecular Dance

The authors of this paper propose a specific theory about how this ghost is formed. They suggest that the $a_0(1710)$ isn't a single, solid dancer. Instead, it's a molecular structure—a temporary partnership formed when two other dancers, specifically vector mesons (like $K^*$ and $\bar{K}^*$), crash into each other and stick together for a split second.

It's like two people bumping into each other in a crowded room and, for a brief moment, holding hands and spinning as a single unit before letting go.

The Experiment: The $J/\psi$ Party

To find this ghost, the authors looked at a very specific party: the decay of a particle called $J/\psi$.

• The Strong Decay ($J/\psi \to \rho^+\rho^-\omega$): Imagine the $J/\psi$ exploding into three particles. The authors calculated that if you watch the dance of the $\rho$ and $\omega$ particles, you should see a distinct "bump" or peak in the data around 1.8 GeV (a specific energy level). This bump is the signature of the $a_0(1710)$ forming.
• The Radiative Decay ($J/\psi \to \gamma\rho^0\omega$): This is similar, but one of the particles is replaced by a photon (light). The authors argue this is an even "cleaner" party. Because there is less background noise (fewer other dancers interfering), the ghost's signature should be even clearer here.

The Results: A Clear Signal

The authors ran complex mathematical simulations (using a framework called "chiral unitary approach") to see what would happen if this molecular theory were true.

1. The Peak: In both types of decays, their calculations showed a clear, distinct peak in the mass distribution around 1.8 GeV.
2. Stability: They tested their theory with different assumptions (changing the "weights" of the dance moves). No matter how they tweaked the parameters, that peak remained. It didn't disappear; it was a robust feature of the dance.
3. Feasibility: They calculated that these events happen frequently enough (with a high "branching ratio") that current and future particle detectors (like BESIII, Belle II, and the planned Super Tau-Charm Facility) should be able to see them easily.

The Bottom Line

The paper claims that if you go to the $J/\psi$ decay experiments and look closely at the energy of the $\rho$ and $\omega$ particles, you will see a clear "mountain" in the data. This mountain is the $a_0(1710)$ resonance, dynamically generated by the interaction of other particles.

By finding this peak, scientists hope to finally agree on the exact weight and size of this elusive particle, solving the mystery of its structure once and for all. The authors are essentially handing the experimentalists a map, saying, "Look right here, around 1.8 GeV, and you'll find the ghost."

Technical Summary

Technical Summary: Role of $a_0(1710)$ in the $J/\psi \to \rho^+\rho^-\omega$ and $J/\psi \to \gamma\rho^0\omega$ Reactions

Problem Statement
The internal structure of light scalar mesons, particularly the $a_0(1710)$, remains a subject of active debate. While the conventional quark-antiquark ($q\bar{q}$) picture explains many mesons, it struggles with states exhibiting exotic properties or large decay widths. The $a_0(1710)$, the isospin partner of the $f_0(1710)$, has recently received direct experimental support from the BABAR, BESIII, and LHCb collaborations. However, experimental determinations of its mass and width vary significantly (ranging from $\sim 1704$ to $1817$ MeV), complicating the interpretation of its nature. Theoretical frameworks, such as the chiral unitary approach using the local hidden gauge formalism, predict the $a_0(1710)$ as a dynamically generated scalar meson arising from S-wave vector-vector ($VV$) interactions, specifically coupling strongly to $K^*\bar{K}^*$, $\rho\omega$, and $\rho\phi$ channels. This paper investigates the feasibility of observing this resonance in charmonium decays to clarify its properties and production mechanisms.

Methodology
The authors employ a theoretical framework based on the heavy quark spin symmetry and the local hidden gauge approach. The $J/\psi$ meson is treated as an SU(3) flavor singlet ($c\bar{c}$). The study focuses on two decay processes:

1. Strong Decay: $J/\psi \to \rho^+\rho^-\omega$
2. Radiative Decay: $J/\psi \to \gamma\rho^0\omega$

The theoretical formalism constructs invariant amplitudes by contracting scalar terms with three vector fields, utilizing the vector meson matrix $V$. The amplitudes are decomposed into three independent structures: $\langle VVV \rangle$, $\langle VV \rangle \langle V \rangle$, and $\langle V \rangle \langle V \rangle \langle V \rangle$, weighted by parameters $A$, $B$, and $C$. Heavy quark spin symmetry suggests that structures with fewer traces dominate.

The calculation includes both tree-level mechanisms and final-state interactions (FSI). The FSI, specifically the S-wave rescattering of vector-vector pairs ($K^*\bar{K}^*$, $\rho\omega$, $\rho\phi$), is responsible for the dynamical generation of the $a_0(1710)$ resonance. The transition amplitudes incorporate loop functions ($G$) regularized with a cutoff ($q_{max} = 960$ MeV) and resonance propagators. For the radiative decay, the Vector Meson Dominance (VMD) model is used to couple the primary $VVV$ vertex to the final photon.

To constrain the unknown production weights ($A$ and $B$), the authors perform multiple fits using experimental branching ratios for various $J/\psi$ radiative decays ($J/\psi \to \gamma\rho\rho$, $\gamma\omega\omega$, $\gamma\phi\phi$, $\gamma K^*\bar{K}^*$) and upper bounds for $\gamma\rho\phi$ and $\gamma\rho\omega$. Four distinct fitting scenarios (Fit 1 through Fit 4) are explored to account for uncertainties and experimental limits.

Key Contributions and Results

• Dynamical Generation: The study confirms that the $a_0(1710)$ emerges dynamically in these reactions through S-wave $VV$ final-state interactions, consistent with previous theoretical predictions where the state couples strongly to $K^*\bar{K}^*$, $\rho\omega$, and $\rho\phi$.
• Invariant Mass Distributions: The primary result is the prediction of a clear, pronounced peak structure in the invariant mass distributions of the $\rho\omega$ system:
  • In the strong decay $J/\psi \to \rho^+\rho^-\omega$, a distinct peak appears around 1.8 GeV in the $\rho^+\omega$ (and $\rho^-\omega$) invariant mass spectrum.
  • In the radiative decay $J/\psi \to \gamma\rho^0\omega$, a similarly distinct peak is predicted in the $\rho^0\omega$ invariant mass distribution.
• Stability of the Signal: The analysis demonstrates that the position and prominence of the $a_0(1710)$ peak are remarkably stable against variations in the fitting parameters $A$ and $B$. While the overall normalization of the decay rates changes across different fits, the line shape of the resonance remains consistent, rising clearly above a smooth background generated by tree-level and non-resonant interactions.
• Branching Ratios:
  • The branching ratio for the strong decay $J/\psi \to \rho^+\rho^-\omega$ is found to be consistently large, typically exceeding 20%, indicating it is a prominent channel in $J/\psi$ decays.
  • The branching ratio for the radiative decay $J/\psi \to \gamma\rho^0\omega$ is estimated to be in the range of $(5 \sim 15) \times 10^{-4}$. While uncertainties are large, these values are within the reach of current experimental capabilities.
• Fit Sensitivity: The authors note that while the tree-level approximation used to constrain $A$ and $B$ has limitations (e.g., neglecting strong $\rho\rho$ interactions that generate other resonances), the resulting predictions for the $a_0(1710)$ signal are robust. The discrepancies in fitting certain channels (like $\gamma\rho\rho$) highlight the need for improved measurements of rates like $J/\psi \to \gamma\rho\phi$ and $J/\psi \to \gamma\rho\omega$.

Significance and Claims
The paper claims that the $J/\psi \to \rho^+\rho^-\omega$ and $J/\psi \to \gamma\rho^0\omega$ reactions serve as valuable windows for observing the $a_0(1710)$ resonance. The authors assert that the predicted peak structures are sufficiently clear to be observed in future experimental measurements by the BESIII and Belle II Collaborations, as well as the planned Super Tau-Charm Facility (STCF).

The significance of these observations lies in two areas:

1. Confirmation: Detecting these peaks would confirm the presence of the $a_0(1710)$ in these specific decay channels, supporting the theoretical picture of it being a dynamically generated molecular state from vector-vector interactions.
2. Precision: Observing these processes would allow for the extraction of more precise values for the mass and width of the $a_0(1710)$, addressing the current lack of consensus in experimental determinations. The radiative decay, in particular, is highlighted as providing a "clean environment" where the signal is well-separated from the background.

The authors maintain a modest tone regarding the absolute rates, acknowledging that the tree-level determination of parameters introduces uncertainties. However, they emphasize that the qualitative feature—the existence of a clear resonance peak—is a robust prediction independent of the precise values of the production weights.