Criterion for the Existence of the $G(3900)$ Resonance

Technical Summary: Criterion for the Existence of the G(3900) Resonance

Problem Statement
The existence and nature of the $G(3900)$ structure, observed near 3900 MeV in the $e^+e^- \to D\bar{D}$ process, have remained a subject of significant debate. While the BESIII collaboration confirmed the structure in 2024 with an enlarged dataset, its interpretation is contested. Some theoretical frameworks propose it as a genuine hadronic state, specifically a $P$-wave $D\bar{D}^*$ molecular state. Conversely, alternative explanations suggest the observed enhancement arises from interference effects, threshold openings (specifically $D\bar{D}^*$), or nodal structures in the $\psi(3S)$ radial wave function, rather than a distinct resonance. Current analyses of the invariant mass spectrum alone have proven insufficient to conclusively distinguish between a genuine resonance and non-resonant interference phenomena.

Methodology
The authors propose a novel criterion to resolve this ambiguity by analyzing the differential cross section of the $e^+e^- \to D\bar{D}$ process as a function of the transverse momentum ($P_\perp$) of the final-state $D$ meson, rather than solely relying on the invariant mass spectrum.

1. Theoretical Framework: The study employs a conventional effective Lagrangian approach. The reaction amplitude includes contributions from intermediate charmonium resonances ($\psi(3686)$, $\psi(3770)$, $\psi(4040)$, $\psi(4160)$) and the $G(3900)$, as well as the opening of the $D\bar{D}^*$ threshold.
2. Kinematic Analysis: The authors derive the normalized differential cross section $\sigma^{-1} d\sigma/d\mu$ (where $\mu = P_\perp/P$). They highlight that if a genuine resonance exists, the kinematic boundary of the decay produces a characteristic Jacobian peak at a specific transverse momentum value ($P_{\perp}^{max}$), determined uniquely by the resonance mass.
3. Model Fitting: The authors perform comprehensive fits to existing experimental data for both $e^+e^- \to D\bar{D}$ and $e^+e^- \to D\bar{D}^*$. They test multiple models:
   • Models including only known charmonia ($\psi(4040)$, $\psi(4160)$).
   • Models including a background term to simulate interference.
   • Models including the $G(3900)$ as a genuine resonance.
   • Variations in coupling constants and cutoff parameters ($\Lambda$) to assess model dependence.
4. Simulation: Using amplitudes fitted to experimental data, the authors simulate the transverse momentum distribution for two scenarios: one where $G(3900)$ is treated as a genuine resonance and one where it is excluded (attributing the signal to interference).

Key Results

• Jacobian Peak as a Signature: The analysis reveals that if $G(3900)$ is a genuine resonance, a prominent Jacobian peak appears in the transverse momentum distribution at $P_{\perp}^{max} \approx 521$ MeV (for $D^0\bar{D}^0$) and $504$ MeV (for $D^+D^-$). Crucially, this peak is a kinematic feature independent of the scattering amplitude's magnitude.
• Differentiation from Interference: When the $G(3900)$ contribution is removed and the enhancement is modeled solely through interference and threshold effects, the Jacobian peak disappears. This provides a direct method to distinguish a true resonance from kinematic or interference artifacts.
• Model Dependence of Decay Widths: The authors find that the interpretation of $G(3900)$ as a $D\bar{D}^*$ molecule is highly sensitive to the specific fit parameters.
  • In the "Full" model (A1), the ratio of decay widths suggests $G(3900)$ contributes negligibly to $e^+e^- \to D\bar{D}^*$, which contradicts the molecular hypothesis.
  • In the "A2" model (incorporating a new particle and background), the decay widths allow for a molecular interpretation.
  • This indicates that fitting the invariant mass spectrum alone cannot uniquely determine the internal structure (molecular vs. compact) of the state due to strong parameter dependence.
• Fit Quality: Models including the $G(3900)$ resonance (A1 and A2) provide excellent descriptions of the $e^+e^- \to D\bar{D}$ and $e^+e^- \to D\bar{D}^*$ data ($\chi^2/dof \approx 1.12$), comparable to or better than models relying solely on interference and background terms.

Significance and Claims
The paper claims to offer a robust, model-independent criterion for establishing the existence of the $G(3900)$ resonance. By shifting the focus from the invariant mass spectrum to the transverse momentum distribution, the authors demonstrate that the presence of a Jacobian peak serves as direct evidence for a genuine resonant state, a feature that interference effects alone cannot replicate.

While the authors acknowledge that the specific classification of $G(3900)$ as a $P$-wave molecular state remains ambiguous due to parameter dependence in decay width ratios, they assert that the existence of the resonance itself can be confirmed via the Jacobian peak. The authors conclude that this strategy—analyzing transverse momentum distributions to identify kinematic peaks—can be extended to other hadronic systems, providing a powerful tool to identify genuine resonances amidst complex interference backgrounds in hadron spectroscopy.