On the nature of fully-charmed four-quark exotic state $X(6900)$ from its photoproduction off nuclei

Technical Summary: On the Nature of Fully-Charmed Four-Quark Exotic State X(6900) from its Photoproduction off Nuclei

Problem Statement
Since the discovery of the X(3872) and subsequent observations of exotic hadrons, the internal structure of these states remains a subject of intense debate. Specifically, the fully-charmed tetraquark candidate X(6900), observed by LHCb, ATLAS, and CMS in the $J/\psi J/\psi$ invariant mass spectrum, lacks a definitive structural interpretation. Theoretical proposals range from compact diquark-antidiquark tetraquarks ($cc\bar{c}\bar{c}$) to loosely bound hadronic molecules (e.g., $J/\psi\psi(3770)$) or mixtures of both. While high-energy proton-proton collisions have established the existence of X(6900), they offer limited clarity on its intrinsic nature due to complex initial-state interactions. This paper investigates whether near-threshold inclusive photoproduction of X(6900) off nuclear targets can provide observables sensitive enough to distinguish between these competing structural models.

Methodology
The study employs a collision model based on the nuclear spectral function to calculate the inclusive photoproduction of X(6900) off nuclei ($^{12}\text{C}$ and $^{184}\text{W}$) at near-threshold photon energies (30–40 GeV). The framework considers direct photon-nucleon interactions ($\gamma N \to X(6900) N$) and accounts for three distinct scenarios for the X(6900) internal structure:

1. Compact Tetraquark: A tightly bound $2S$-wave $cc\bar{c}\bar{c}$ state with a radius of $\sim 0.575$ fm.
2. Hadronic Molecule: A loosely bound $J/\psi\psi(3770)$ state.
3. Hybrid State: A 50/50 mixture of the compact and molecular components.

Key theoretical inputs include:

• Production Cross Section: The free-space cross section $\sigma_{\gamma p \to X(6900)p}$ is estimated by scaling results from an $\omega$-meson exchange model (originally calculated for a $0^{++}$ state) using a spin-state counting rule ($5/1$ ratio for $2^{++}$ vs. $0^{++}$).
• Absorption Cross Section ($\sigma_{X(6900)N}$): This is the critical parameter distinguishing the scenarios.
  • For the compact state, it is approximated geometrically ($\pi r^2$), yielding $\sim 10$ mb.
  • For the molecular state, it is treated as the sum of constituent charmonium-nucleon cross sections ($\sigma_{J/\psi N} + \sigma_{\psi(3770)N}$), yielding $\sim 21.2$ mb.
  • For the hybrid state, it is an incoherent probability-weighted sum, yielding $\sim 15.6$ mb.
• Nuclear Effects: The model incorporates nuclear spectral functions to account for nucleon binding energy and Fermi motion. It assumes negligible medium modifications for the outgoing high-momentum X(6900) and nucleons, and neglects X(6900) decay within the nucleus due to its long mean free path relative to nuclear radii.

Key Contributions and Results
The paper calculates several observables to test the sensitivity of photoproduction to the X(6900) structure:

1. Excitation Functions: The total production cross sections on $^{12}\text{C}$ and $^{184}\text{W}$ are calculated for photon energies up to 40 GeV.

   • Results indicate that the influence of the target nucleon Fermi motion is significant only at subthreshold energies.
   • For the heavy nucleus $^{184}\text{W}$, the cross sections show experimentally measurable differences of $\sim 20\text{--}30\%$ between the compact and molecular scenarios. For $^{12}\text{C}$, the difference is smaller ($\sim 10\%$) but potentially accessible with high-precision experiments.
   • Estimated event rates for future electron-ion colliders (EIC and EicC) suggest thousands to hundreds of thousands of detectable events per year, assuming realistic detection efficiencies.

2. Momentum Differential Cross Sections: Calculated for laboratory polar angles of $0^\circ\text{--}5^\circ$ at $E_\gamma = 35$ GeV.

   • The momentum distributions exhibit sizeable variations depending on the structural scenario, particularly for the heavy $^{184}\text{W}$ target.
   • Absolute cross sections in the central momentum region ($33\text{--}33.5$ GeV/c) are predicted to be $\sim 10\text{--}20$ nb/(GeV/c) for $^{12}\text{C}$ and $\sim 100\text{--}200$ nb/(GeV/c) for $^{184}\text{W}$.

3. Transparency Ratios ($S_A$ and $T_A$):

   • The transparency ratio $S_A$ (nuclear cross section normalized to $A$ times the free proton cross section) and $T_A$ (normalized to $^{12}\text{C}$) are analyzed as functions of the mass number $A$ and photon energy.
   • $S_A$ and $T_A$ show strong dependence on the assumed internal structure. For heavy nuclei (e.g., $^{208}\text{Pb}$, $^{238}\text{U}$), the molecular scenario predicts significantly lower transparency (values $\sim 0.2$ for $S_A$) compared to the compact scenario due to the larger absorption cross section.
   • Variations between scenarios are measurable even for medium-mass nuclei (e.g., $^{40}\text{Ca}$), with differences of $\sim 14\text{--}17\%$ in $S_A$.

Significance and Claims
The paper concludes that the observables calculated—specifically the excitation functions, momentum differential cross sections, and transparency ratios—reveal a definite sensitivity to the intrinsic structure of the X(6900). The author asserts that these theoretical predictions provide a viable pathway to determine the nature of the X(6900) (compact tetraquark, molecule, or hybrid) through comparison with future experimental data. The proposed measurements are specifically targeted for upcoming facilities, namely the Electron-Ion Collider (EIC) in the United States and the Electron-ion Collider in China (EicC), where near-threshold photoproduction and electroproduction on nuclear targets can be performed. The study emphasizes that such experiments would allow for the discrimination between the proposed structural scenarios, addressing a fundamental gap in the understanding of fully-heavy tetraquark states.