Competition of $χ_{c}(2P)$ quarkonia and continuum in $e^+ e^- \to e^+ e^- D \bar{D}$

This paper analyzes the production of $D\bar{D}$ pairs in $e^+e^-$ collisions via continuum mechanisms and $\chi_c$ resonances, concluding that the observed 3.8 GeV bump in the $D^0\bar{D}^0$ channel is likely of continuum origin rather than a broad $\chi_{c0}(3860)$ resonance, while also providing branching fraction estimates and cross-section predictions for the $\chi_{c2}(3930)$ state relevant to Belle II experiments.

The Gist

Imagine you are a detective trying to solve a mystery at a high-energy particle collider. The "crime scene" is a collision between an electron and a positron (matter and antimatter twins). When they smash together, they don't just disappear; they create a flash of energy that sometimes turns into pairs of heavy particles called D mesons (specifically, a D and an anti-D).

The mystery? Scientists have been seeing a strange "bump" or hump in the data around a specific energy level (3.8 GeV). They weren't sure what was causing it. Was it a brand-new, broad, fuzzy particle (a resonance) named $\chi_{c0}(3860)$? Or was it just a chaotic background noise (a continuum) caused by standard particle interactions?

This paper, written by a team of physicists from Poland, acts as a forensic analysis to figure out what's really going on. Here is the breakdown in simple terms:

1. The Two Suspects: The "Fuzzy Blob" vs. The "Background Noise"

The authors are investigating two main ways these D meson pairs can be created:

• Suspect A: The Resonance (The "Fuzzy Blob")
  Imagine a drum being hit. It vibrates at a specific frequency, creating a clear, distinct note. In particle physics, this is a resonance. The scientists are looking for a specific excited state of a "charmonium" particle (a heavy particle made of a charm quark and an anti-charm quark). They suspect the bump at 3.8 GeV might be a broad, fuzzy version of this particle, called $\chi_{c0}(3860)$.
• Suspect B: The Continuum (The "Background Noise")
  Now imagine a crowd of people shuffling around randomly. There is no single note, just a general hum. This is the continuum. It happens when particles exchange other particles (like virtual "D-star" mesons) in a way that creates a smooth, rising curve of data rather than a sharp peak.

2. The Investigation: Neutral vs. Charged

The team looked at two different "crime scenes":

• The Neutral Case ($D^0\bar{D}^0$): This is like a pair of identical twins.
• The Charged Case ($D^+D^-$): This is like a pair of siblings with opposite charges.

The Big Discovery:
When they looked at the Neutral data, they saw a big bump at 3.8 GeV. But when they looked at the Charged data, that bump was almost invisible.

The Analogy:
If a specific "Fuzzy Blob" particle (the resonance) existed and was responsible for the bump, it should have shown up clearly in both the Neutral and Charged cases, just like a loud drumbeat would be heard in both rooms.
However, because the bump was huge in the Neutral room but silent in the Charged room, the authors concluded: It's probably not a new particle. It's more likely just the "Background Noise" (the continuum) behaving strangely in the neutral case.

3. The "D-Exchange" Mechanism

How does this background noise happen?
Think of the colliding particles as two people throwing balls at each other.

• In the Neutral case, they are throwing "D-star" balls (heavy, vector particles). These balls can be exchanged easily, creating a strong signal (the bump).
• In the Charged case, the rules of the game make it much harder to throw these specific balls. The "exchange" is weak, so the signal is tiny.

The authors calculated that the "D-star" exchange explains the Neutral bump perfectly without needing to invent a new particle.

4. The Second Mystery: The $\chi_{c2}(3930)$

While debunking the 3.8 GeV bump, the team also looked at a second, clearer signal at 3.93 GeV. This one is likely a real particle, a "Fuzzy Blob" called $\chi_{c2}(3930)$ (which they think is the excited version of a charmonium state, $\chi_{c2}(2P)$).

They used a complex mathematical model (like a recipe) involving different "potential models" (different ways to describe how quarks hold hands). They compared their recipe's predictions to real data from the BaBar and Belle experiments.

The Result:
Their calculations matched the real-world data very well. They were able to estimate how often this particle decays into D mesons. It's like saying, "If we bake this specific cake (the particle), about 58% of the time it will turn into this specific topping (D mesons)."

5. The Conclusion

• The 3.8 GeV Bump: It's likely not a new particle. It's just a "traffic jam" of standard particle exchanges (continuum) that looks like a bump. The fact that it disappears in the charged channel is the smoking gun.
• The 3.93 GeV Bump: This is a real particle ($\chi_{c2}(3930)$). The authors have successfully calculated its properties and confirmed it fits the data.

Why Does This Matter?

In the world of particle physics, it's easy to get excited about a new "bump" in the data and think you've found a new fundamental particle. This paper is a reality check. It shows that sometimes, what looks like a new discovery is just the complex, messy background of known physics interacting in a specific way.

The authors are essentially saying: "Don't call the police (look for a new particle) just yet. It's probably just the neighbors shuffling around (continuum). However, we did find a very solid new house (the $\chi_{c2}$) at the next address."

This work helps future experiments, like Belle II, know exactly what to look for and what to ignore, saving time and resources in the hunt for the truly exotic particles of the universe.

Technical Summary

1. Problem Statement

The paper addresses the nature of excited $P$-wave charmonium states, specifically the candidates for $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, observed in two-photon fusion processes ($e^+e^- \to e^+e^- \gamma\gamma \to e^+e^- D\bar{D}$).

• The $\chi_{c2}(3930)$: Identified as a candidate for $\chi_{c2}(2P)$, observed as a resonance in both $D^0\bar{D}^0$ and $D^+D^-$ channels.
• The $\chi_{c0}(3860)$ vs. Continuum: A broad enhancement is observed around $M_{D\bar{D}} \approx 3.8$ GeV in the neutral $D^0\bar{D}^0$ channel. The central problem is determining whether this enhancement corresponds to a broad resonance ($\chi_{c0}(2P)$, potentially $X(3860)$) or if it arises from non-resonant continuum mechanisms.
• Discrepancy: Experimental data (Belle and BaBar) show a significant enhancement in the neutral channel but a much weaker or absent effect in the charged $D^+D^-$ channel. The authors aim to explain this asymmetry and quantify the contributions of resonant vs. continuum processes.

2. Methodology

The authors perform a theoretical calculation of the exclusive reaction $e^+e^- \to e^+e^- D\bar{D}$ at a center-of-mass energy $\sqrt{s} = 10.54$ GeV (Belle II kinematics). The calculation involves an 8-dimensional phase space integration of the fully differential cross-section.

Key Theoretical Components:

• Continuum Mechanism:
  • Neutral Channel ($D^0\bar{D}^0$): Modeled via $t$- and $u$-channel exchanges of the vector meson $D^{*0}(2007)$. The $D^{*0}D^0\gamma$ coupling is derived from the radiative decay width. The $D^*$ propagator is treated using a Reggeized approach with a specific "square-root" trajectory to account for high-energy behavior. Off-shell form factors are introduced with a cutoff parameter $\Lambda_{D^*}$.
  • Charged Channel ($D^+D^-$): Modeled via $t$- and $u$-channel exchanges of pseudoscalar $D^\pm$ mesons plus a contact term. The contribution from $D^{*\pm}$ exchanges is calculated but found to be negligible due to the small $D^* \to D\gamma$ branching ratio.
• Resonant Mechanism:
  • $\chi_{c2}(3930)$: Treated as a tensor meson ($J^{PC}=2^{++}$) produced via $\gamma\gamma \to \chi_{c2} \to D\bar{D}$. The calculation assumes dominance of the helicity-2 component. The two-photon width ($\Gamma_{\gamma\gamma}$) is calculated using the Light-Front Quark Model (NRQCD limit) for various potential models (Cornell, Buchmüller-Tye, etc.).
  • $\chi_{c0}(3860)$: Treated as a scalar meson ($J^{PC}=0^{++}$). Two scenarios for its total width ($\Gamma_{\chi_{c0}}$) are tested: a narrow width ($50$ MeV) and the PDG-reported broad width ($200$ MeV).
• Form Factors and Couplings:
  • Coupling constants ($g_{D^*D\gamma}$, $g_{\chi_{c}D\bar{D}}$) are constrained by experimental decay widths or fitted to data.
  • Off-shell effects are handled via exponential or monopole form factors.

3. Key Contributions

• Continuum vs. Resonance Separation: The authors provide a rigorous separation of the continuum background from the resonant signal for both neutral and charged channels, demonstrating that the continuum contribution is significantly larger for $D^0\bar{D}^0$ than for $D^+D^-$.
• Origin of the 3.8 GeV Bump: The study concludes that the broad bump at $3.8$ GeV observed in the $D^0\bar{D}^0$ channel is primarily of continuum origin (mediated by $D^{*0}$ exchange) rather than a broad $\chi_{c0}(2P)$ resonance. The lack of a similar enhancement in the $D^+D^-$ channel supports this, as the continuum mechanism for charged mesons is suppressed.
• Refined Branching Fractions: By comparing their model (including continuum interference) with BaBar data, the authors derive updated branching fractions for $\chi_{c2}(3930) \to D\bar{D}$.
• Interference Effects: The paper highlights significant interference between the $\chi_{c0}(3860)$ resonance and the $D^0\bar{D}^0$ continuum, which affects the line shape near the threshold.

4. Results

• Continuum Dominance in Neutral Channel: The calculated continuum cross-section for $D^0\bar{D}^0$ is large, peaking around $3.8$ GeV. In contrast, the $D^+D^-$ continuum is small. This successfully reproduces the experimental asymmetry without invoking a broad resonance in the charged channel.
• $\chi_{c2}(3930)$ Parameters:
  • Using the Buchmüller-Tye potential, the authors find $\Gamma_{\gamma\gamma} \times \mathcal{B}(\chi_{c2} \to D\bar{D}) = 0.32 \pm 0.07$ keV.
  • This yields a branching fraction $\mathcal{B}(\chi_{c2} \to D\bar{D}) = 0.58 \pm 0.13$ (assuming $\Gamma_{\gamma\gamma} = 0.544$ keV).
  • These results are consistent with Belle and BaBar experimental values within uncertainties.
• $\chi_{c0}(3860)$ Analysis:
  • A broad resonance with $\Gamma \sim 200$ MeV (PDG value) fails to describe the near-threshold data well, particularly for the $D^+D^-$ channel, and implies an unphysically small branching fraction to $D\bar{D}$.
  • A narrower width scenario ($\Gamma \approx 50$ MeV) provides a better fit to the data, suggesting the PDG average width may be overestimated or the state's nature is more complex.
• Kinematic Distributions: The paper provides realistic predictions for differential distributions (transverse momentum $p_T$, momentum transfer $t$, invariant mass) for Belle II, showing that cuts on $p_T$ significantly reduce the cross-section and limit momentum transfers.

5. Significance

• Resolution of the "Bump" Puzzle: The work offers a compelling alternative explanation for the $3.8$ GeV enhancement, attributing it to $D^{*0}$ exchange continuum rather than a new broad resonance. This challenges the interpretation of the $X(3860)$ as a pure $\chi_{c0}(2P)$ state.
• Precision for Future Experiments: The detailed differential distributions and integrated cross-sections provided are essential for the Belle II experiment to optimize selection cuts and distinguish between resonant and non-resonant contributions in future high-statistics data.
• Theoretical Constraints: The study constrains the two-photon widths and branching fractions of excited charmonia, providing benchmarks for potential models (Cornell vs. Buchmüller-Tye) and Light-Front QCD calculations.
• Methodological Advance: The application of Reggeized propagators for $D^*$ mesons in the continuum calculation provides a more robust framework for describing high-energy exclusive processes involving open charm.

In summary, the paper argues that the observed structures in $e^+e^- \to e^+e^- D\bar{D}$ are a complex interplay of a well-defined $\chi_{c2}(3930)$ resonance and a dominant, asymmetric continuum background, rather than a single broad $\chi_{c0}(2P)$ resonance.