Exclusive $D \bar{D}$ pair production with low invariant mass in ultraperipheral Pb-Pb collisions at the LHC

This paper presents theoretical predictions for exclusive low-invariant-mass $D\bar{D}$ pair production in ultraperipheral Pb-Pb collisions at the LHC, incorporating continuum and resonant $\chi_{c0,2}(2P)$ contributions to calculate cross sections and differential distributions relevant for ALICE, ATLAS, CMS, and LHCb experiments.

The Gist

The Big Picture: A High-Speed Dance of Light and Heavy Metals

Imagine two massive, heavy trains (Lead nuclei, or Pb) speeding toward each other on parallel tracks at the Large Hadron Collider (LHC). Usually, if these trains crash head-on, it's a total disaster—a fiery explosion of debris.

But in this experiment, the physicists are interested in a very specific, delicate scenario called an Ultraperipheral Collision (UPC). Imagine the trains pass each other so closely that their wheels almost touch, but they don't actually crash. They just "whiz" past one another.

Because these trains are so heavy and moving so fast, they are surrounded by a massive cloud of invisible "light" (photons). As the trains pass, these clouds of light collide. It's like two powerful spotlights from the passing trains hitting each other in mid-air.

The Goal: The scientists want to see what happens when these two beams of light smash together to create a pair of heavy particles called D-mesons (specifically a "D" and an anti-D). It's like two flashlights colliding and suddenly creating a pair of heavy bowling balls out of thin air.

The Mystery: What's Hiding in the Middle?

When light beams smash together to create these particle pairs, they can do it in two ways:

1. The "Smooth" Way (Continuum): The light just turns directly into the particles, like water flowing smoothly into a river.
2. The "Bumpy" Way (Resonances): The light first creates a temporary, unstable "middleman" particle (a resonance) that quickly falls apart into the final pair. Think of this like a trampoline: you jump on it, it stretches (the middleman), and then you bounce off.

The paper focuses on two specific "middlemen" particles: $\chi_{c0}(3860)$ and $\chi_{c2}(3930)$.

• The Analogy: Imagine you are trying to identify a specific type of bird by the sound of its wings flapping. You hear a flapping sound. Is it just the wind blowing through leaves (the smooth way), or is it a specific, rare bird landing on a branch (the resonance)?
• The Science: The authors suspect these "middleman" particles are actually excited versions of "charmonium" (a family of heavy particles made of a charm quark and an anti-charm quark). They are like the "first excited state" of a guitar string—a higher note than the usual sound.

What Did They Do?

The authors built a complex mathematical model (a "simulation") to predict exactly what would happen in these near-miss train collisions at the LHC.

1. Checking the Recipe: First, they checked their model against data from previous experiments (Belle and BaBar) where electrons and positrons collided. They found their model worked well, correctly predicting how often these particles are made and at what angles.
2. The Prediction: They then used this model to predict what the LHC (specifically the ALICE, ATLAS, CMS, and LHCb detectors) would see if they ran the heavy lead-ion collisions.

The Results: What to Expect

Here are the key takeaways from their calculations:

• It's Happening: They predict that this process happens quite often. For every billion lead-ion collisions, they expect to see hundreds of thousands of these specific particle pairs being created.
• The "Back-to-Back" Clue: This is the most important part. When these particles are created by light colliding with light, they fly apart in exactly opposite directions (like two skaters pushing off each other).
  • Why this matters: Other processes (like a "glitch" where a train hits a wall and spits out debris) tend to scatter particles in random directions. If the detectors see the particles flying perfectly back-to-back, it's a smoking gun that proves they came from the "light collision" method, not a crash.
• Finding the Rare Birds: By looking closely at the mass of the particle pairs, they hope to see "bumps" or peaks in the data. These bumps would confirm the existence of those mysterious "middleman" particles ($\chi_{c0}$ and $\chi_{c2}$) and help physicists understand the internal structure of heavy quarks.

Why Should We Care?

Think of the universe as a giant puzzle. We know the pieces (quarks), but we don't fully understand how they fit together to make the heavier, excited versions of themselves.

This paper is essentially a map for the LHC detectors. It tells the experimentalists:

"Hey, look for these specific particles flying in opposite directions with these specific energies. If you find them, you'll have discovered new information about how the heavy building blocks of our universe vibrate and interact."

Summary in One Sentence

The authors calculated that when two heavy lead nuclei zoom past each other at the LHC, their surrounding light fields can collide to create pairs of heavy particles, and by studying how these pairs fly apart, we might finally solve the mystery of some of nature's most elusive "middleman" particles.

Technical Summary

1. Problem Statement

The paper addresses the theoretical prediction of exclusive $D\bar{D}$ meson pair production (where $D$ represents $D^0$ or $D^+$) in ultraperipheral heavy-ion collisions (UPCs) of Lead (Pb) nuclei at the Large Hadron Collider (LHC).

• Context: Previous studies (e.g., Ref. [1]) focused on high invariant masses ($M_{D\bar{D}} > 4$ GeV) and high transverse momenta ($p_{t,D} > 1$ GeV), yielding small cross-sections.
• Gap: There is a need to understand the production mechanism in the low invariant mass region ($M_{D\bar{D}} < 4.3$ GeV) and at low transverse momenta ($p_{t,D} \lesssim 1$ GeV). This region is crucial for identifying potential resonant structures, specifically the first excited P-wave charmonia states $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, which are candidates for the observed $\chi_{c0}(3860)$ and $\chi_{c2}(3930)$ resonances.
• Challenge: Theoretical models must account for the interference between non-resonant continuum mechanisms and resonant $s$-channel contributions to accurately describe experimental data from $e^+e^-$ collisions (Belle and BaBar) before extrapolating to nuclear collisions.

2. Methodology

The authors employ a two-step theoretical framework combining the Equivalent Photon Approximation (EPA) for nuclear collisions and a specific amplitude model for the elementary photon-photon subprocess.

A. Nuclear Cross Section Calculation (EPA)

• Framework: The total nuclear cross section for $PbPb \to PbPb D\bar{D}$ is calculated using the impact-parameter dependent Equivalent Photon Approximation.
• Formula: The cross section is integrated over photon energies ($\omega$), impact parameters ($b$), and the invariant mass of the $\gamma\gamma$ system ($W_{\gamma\gamma}$).
• Survival Probability: A sharp cut-off model is used for the absorption (survival) factor $S^2_{abs}(b)$, ensuring only ultraperipheral collisions (where nuclei do not break up) are considered ($b > R_1 + R_2 \approx 14$ fm).
• Photon Flux: Calculated using the nuclear charge form factor (monopole type) for Lead ($Z=82, A=208$).

B. Elementary Subprocess ($\gamma\gamma \to D\bar{D}$)

The core of the calculation relies on the model developed in Ref. [5], which describes the $\gamma\gamma \to D\bar{D}$ amplitude as a coherent sum of:

1. Continuum Contributions (Non-resonant):
   • $D^0\bar{D}^0$ channel: Dominated by $t$- and $u$-channel exchanges of the vector meson $D^*(2007)^0$.
   • $D^+D^-$ channel: Dominated by $t$- and $u$-channel exchanges of charged $D^\pm$ mesons.
   • Note: $D^*(2010)^\pm$ exchanges are found to be negligible.
2. Resonant Contributions ($s$-channel):
   • Intermediate states: $\chi_{c0}(3860)$ and $\chi_{c2}(3930)$, interpreted as the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ charmonia.
   • Vertices: Calculated using the NRQCD (Non-Relativistic QCD) limit and light-front approaches. The coupling constants are derived from decay widths and fitted to Belle/BaBar data.
   • Form Factors: Off-shell effects are handled via form factors (e.g., exponential form factors for $D^*$ exchanges).

C. Kinematic Cuts

Predictions are provided for experimental cuts relevant to ALICE, ATLAS, CMS, and LHCb:

• Pseudorapidity: $|\eta_D| < 2.5$ (and other ranges).
• Transverse momentum: $p_{t,D} > 0.2$ GeV and $0.5$ GeV.
• Invariant mass: $M_{D\bar{D}} < 4.3$ GeV.

3. Key Contributions

• Extension to Low Mass/Momentum: The study extends previous theoretical work to the low $p_t$ and low invariant mass regime, which is experimentally accessible and theoretically rich due to resonance structures.
• Comprehensive Amplitude Model: The paper integrates the continuum mechanisms (meson exchanges) with the resonant $\chi_{c0,2}(2P)$ contributions, including their interference effects, which are shown to be significant (especially for the neutral channel).
• First Nuclear Predictions: It provides the first quantitative predictions for exclusive $D\bar{D}$ production in Pb-Pb UPCs at $\sqrt{s_{NN}} = 5.02$ TeV and the upcoming 5.36 TeV, specifically tailored for LHC detector acceptance.
• Background Separation Strategy: The authors propose that the back-to-back correlation (acoplanarity $A_{co} \approx 0$) of the charm mesons can be used to distinguish this exclusive $\gamma\gamma$ process from background processes like diffractive photoproduction (photon-gluon fusion), where hadronization destroys the back-to-back correlation.

4. Results

• Elementary Cross Sections: The calculated $\gamma\gamma \to D\bar{D}$ cross sections agree well with Belle and BaBar data in the $M_{D\bar{D}} < 4.3$ GeV region when including both continuum and resonant terms. The continuum contribution is dominant near the threshold, while resonances shape the distribution at higher masses.
• Nuclear Cross Sections (at $\sqrt{s_{NN}} = 5.02$ TeV):
  • $D^0\bar{D}^0$ Channel: With cuts $|\eta_D| < 2.5$, $p_{t,D} > 0.2$ GeV, and $M_{D\bar{D}} < 4.3$ GeV, the cross section is estimated at 99.6 – 131.5 $\mu$b. The range reflects uncertainty in the off-shell form factor parameter ($\Lambda_{D^*}$).
  • $D^+D^-$ Channel: Under the same cuts, the cross section is 29.3 $\mu$b.
  • Resonant Contributions: The resonant parts alone contribute significantly:
    • $\chi_{c0}(3860) \to D^0\bar{D}^0$: ~3.9 $\mu$b.
    • $\chi_{c2}(3930) \to D^0\bar{D}^0$: ~15.2 $\mu$b.
• Distributions:
  • The invariant mass distribution shows clear structures corresponding to the resonances.
  • The transverse momentum ($p_t$) distribution peaks at low values ($< 1$ GeV), confirming the necessity of low-$p_t$ cuts to maximize the signal.
  • The rapidity difference ($y_{diff}$) is peaked at zero, indicating the back-to-back nature of the exclusive process.
• Energy Dependence: Results for $\sqrt{s_{NN}} = 5.36$ TeV show a slight increase in cross sections (e.g., $D^0\bar{D}^0$ rises to ~103–136 $\mu$b).

5. Significance

• Discovery Potential: The calculated cross sections (tens to hundreds of microbarns) are sufficiently large to be measurable by LHC experiments (ALICE, ATLAS, CMS, LHCb). This opens a new window for studying exotic charmonium-like states and the nature of the $\chi_{c0}(3860)$ and $\chi_{c2}(3930)$.
• Testing QCD Dynamics: The ability to measure these processes at low $p_t$ allows for a rigorous test of the interplay between non-perturbative continuum mechanisms (meson exchanges) and perturbative/resonant QCD dynamics.
• Experimental Guidance: The paper provides specific kinematic cuts and differential distributions (mass, $p_t$, rapidity) that experimentalists can use to isolate the exclusive signal from the inclusive background (photon-gluon fusion), which typically produces broader azimuthal correlations.
• Future Physics: The results serve as a baseline for future studies of higher-excited P-wave charmonia and potentially other exotic hadronic states in heavy-ion collisions.