Associated production of $J/\psi$ mesons and photons in the Parton Reggeization Approach and the double parton scattering model

This paper investigates the associated production of $J/\psi$ mesons and photons in proton-proton collisions using the Parton Reggeization Approach, demonstrating that double parton scattering significantly dominates single parton scattering and that theoretical predictions are highly sensitive to the choice of hadronization model.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle accelerator, smashing two beams of protons together at nearly the speed of light. When these protons collide, they don't just bounce off each other; they shatter into a chaotic shower of smaller particles. Among the debris, physicists are looking for a very specific "treasure hunt": a J/ψ meson (a heavy, short-lived particle made of a charm quark and an anti-charm quark) appearing at the same time as a photon (a particle of light).

This paper is a theoretical investigation into how these two particles are created together. The authors, Lev Alimov and Vladimir Saleev, are trying to figure out the "recipe" for this event.

Here is the breakdown of their findings using simple analogies:

1. The Two Ways to Cook the Meal: SPS vs. DPS

The paper compares two different mechanisms for creating this particle pair. Think of it like ordering food at a restaurant.

• Single Parton Scattering (SPS): The "Single Chef" Scenario.
  Imagine one chef (a single interaction between parts of the protons) trying to cook both the J/ψ and the photon in one go. It's a complex, high-pressure task. The chef has to juggle many ingredients at once to get the perfect dish.

  • The Paper's Finding: In the world of high-energy physics, this "single chef" method is possible, but it's actually quite rare for this specific dish.

• Double Parton Scattering (DPS): The "Two Chefs" Scenario.
  Imagine the protons are actually two separate kitchens inside one building. In a DPS event, two different chefs work independently at the same time. One chef cooks the J/ψ, and the other chef, in a completely separate part of the kitchen, cooks the photon. They don't talk to each other; they just happen to finish their dishes at the exact same moment.

  • The Paper's Finding: This is the big surprise! The authors found that the "Two Chefs" method (DPS) is much more common than the "Single Chef" method (SPS). In fact, the DPS contribution is significantly larger. It's like realizing that most of the time, when you see a J/ψ and a photon together, they weren't cooked by the same person; they were just two separate events that happened to coincide.

2. The "Glue" Problem: How do we turn raw ingredients into the final dish?

Once the raw ingredients (quarks) are created, they need to be "glued" together to form the J/ψ meson. This process is called hadronization. Since we can't calculate this glue perfectly using standard math, physicists use two different "cookbooks" (models) to guess how it works:

• NRQCD (Non-Relativistic QCD): This is like a detailed, complex recipe book that accounts for every tiny movement of the ingredients. It assumes the J/ψ is formed through specific, precise steps.
• ICEM (Improved Color Evaporation Model): This is a simpler, more "fuzzy" approach. It assumes that if the ingredients have the right amount of energy, they will eventually turn into a J/ψ, regardless of the exact steps, as long as they stay within a certain energy range.

The Twist: The authors found that the choice of cookbook matters a lot.

• If you use the NRQCD recipe, you predict a huge number of J/ψ + photon events.
• If you use the ICEM recipe, you predict far fewer events.
  Even though both recipes work well when predicting single J/ψ particles, they give very different answers when predicting the pair production. This tells us that our understanding of the "glue" is still a bit shaky.

3. The "Pocket Formula"

To calculate the "Two Chefs" (DPS) scenario, the authors used a standard formula they call the "pocket formula."

• The Analogy: Imagine you want to know how often two people in a crowded room will bump into each other. You take the probability of Person A bumping into someone, multiply it by the probability of Person B bumping into someone, and divide by a "crowdedness factor" (called $\sigma_{eff}$).
• The authors used data from previous experiments to set this "crowdedness factor" and applied it to their new J/ψ + photon calculations.

4. The Main Takeaway

The authors ran their calculations for the LHC (operating at 13 TeV energy) and looked at the results in two ways:

1. Central Region: Looking straight at the collision point.
2. Forward Region: Looking at the sides of the collision.

The Conclusion:
No matter which "cookbook" (NRQCD or ICEM) you use, the Double Parton Scattering (DPS) is the dominant way these particles are produced. The "Single Chef" (SPS) method is actually the minority player here.

However, the total number of events predicted changes wildly depending on which cookbook you trust. This suggests that while we know how these particles are likely being made (two separate collisions), we still need to refine our understanding of exactly how the heavy quarks turn into the final J/ψ particle.

In a nutshell:
The paper says, "We looked at how J/ψ mesons and photons are born together in proton collisions. We found that they are usually born from two separate, independent collisions happening at once (DPS), not one big complex collision (SPS). However, the exact number of these events depends heavily on which theoretical model we use to describe how the particles stick together."

Technical Summary

1. Problem Statement

The paper addresses the theoretical description of the associated production of $J/\psi$ mesons and direct photons ($J/\psi + \gamma$) in proton-proton collisions at the Large Hadron Collider (LHC) with a center-of-mass energy of $\sqrt{s} = 13$ TeV.

While inclusive production of single $J/\psi$ mesons and photons is well-studied, the associated production channel remains experimentally unexplored. Theoretical challenges include:

• Kinematic Coverage: Standard collinear parton models (CPM) are limited to large transverse momenta ($p_T \gg m_\psi$), while Transverse-Momentum-Dependent (TMD) factorization is restricted to small $p_T$. A unified framework covering the full experimental $p_T$ range is needed.
• Hadronization Models: There is significant uncertainty regarding non-perturbative hadronization mechanisms. The two primary frameworks are Non-Relativistic QCD (NRQCD) and the Improved Color Evaporation Model (ICEM). Previous studies suggested these models yield vastly different cross-section predictions for associated production.
• Scattering Mechanisms: It is unclear whether Single Parton Scattering (SPS) or Double Parton Scattering (DPS) dominates the $J/\psi + \gamma$ production channel, a question critical for constraining Parton Distribution Functions (PDFs) and understanding multi-parton interactions.

2. Methodology

The authors employ a multi-faceted theoretical approach:

• Framework: The calculations are performed within the Parton Reggeization Approach (PRA), a gauge-invariant version of high-energy ($k_T$) factorization. This allows for the description of differential cross-sections across the full range of transverse momenta using Reggeized gluons and quarks.
• Hadronization Models: Two distinct models are used to describe the transition of a $c\bar{c}$ pair into a charmonium state:
  1. NRQCD: Expands the cross-section over Fock states (color-singlet and color-octet). Long-Distance Matrix Elements (LDMEs) are fitted to experimental data.
  2. ICEM: Assumes charmonium arises from $c\bar{c}$ pairs with invariant masses between the $J/\psi$ mass and the $D\bar{D}$ threshold, governed by a single effective parameter ($F_{J/\psi}$).
• Parameter Fixing:
  • LDMEs: Color-octet LDMEs for NRQCD were fitted to CMS and ATLAS data for single $J/\psi$ and $\psi(2S)$ production.
  • ICEM Parameter: Set to $F_{J/\psi} = 0.02$ based on previous literature.
  • DPS Effective Cross-section: $\sigma_{\text{eff}}$ is fixed at $11.0 \pm 0.2$ mb, derived from fits to other associated production channels ($J/\psi J/\psi$, $J/\psi \Upsilon$, etc.) within the PRA.
• Calculations:
  • SPS: Computed using the PRA formula involving the convolution of unintegrated PDFs (uPDFs) and hard-scattering amplitudes.
  • DPS: Calculated using the standard "pocket formula": $\sigma_{\text{DPS}} = \frac{\sigma_{\text{SPS}}(J/\psi) \times \sigma_{\text{SPS}}(\gamma)}{\sigma_{\text{eff}}}$.
• Kinematic Regions: Predictions are made for central rapidity ($|y| < 3$) and forward rapidity ($2.0 < y < 4.5$) regions, with $p_T^\psi < 20$ GeV and $p_T^\gamma > 5$ GeV.

3. Key Contributions

• First Theoretical Prediction: This work provides the first theoretical predictions for differential cross-sections and correlation spectra for associated $J/\psi + \gamma$ production at LHC energies ($\sqrt{s}=13$ TeV).
• Unified Factorization: Demonstrates the applicability of the PRA (based on high-energy factorization) to associated production, bridging the gap between low-$p_T$ and high-$p_T$ regimes where traditional collinear or pure TMD approaches struggle.
• Model Comparison: A rigorous comparison of NRQCD and ICEM within the same high-energy framework, highlighting the sensitivity of predictions to the hadronization model choice.
• DPS Dominance: Establishes that the Double Parton Scattering mechanism is the dominant production mode for this specific final state.

4. Results

• DPS vs. SPS: The study finds that the DPS contribution significantly exceeds the SPS contribution across all calculated differential cross-sections (transverse momenta, rapidities, azimuthal angle differences $\Delta\phi$, pair invariant mass $M$, etc.). This confirms the dominant role of DPS in associated heavy quarkonium and photon production.
• Hadronization Sensitivity: The theoretical predictions are strongly sensitive to the choice of hadronization model:
  • The NRQCD model yields substantially larger cross-sections than the ICEM.
  • While both models describe single $J/\psi$ data well, they diverge significantly in the associated production channel.
• Kinematic Distributions:
  • In the SPS model, quark-antiquark annihilation and color-octet contributions are negligible at low $p_T$ ($p_T^\psi < 20$ GeV).
  • The DPS contribution dominates the correlation spectra, particularly in the rapidity difference ($\Delta y$) and azimuthal angle difference ($\Delta \phi$) distributions.
• Uncertainties: Theoretical uncertainties due to the choice of the hard scale (varied by a factor of 2) were quantified and shown as shaded bands in the figures.

5. Significance

• Experimental Guidance: The paper provides crucial benchmarks for future experimental searches at the LHC (CMS and ATLAS) for the $J/\psi + \gamma$ channel.
• Mechanism Validation: It reinforces the hypothesis that DPS is a dominant mechanism in processes involving heavy quarkonium and other particles, offering a new probe for studying multi-parton interactions and the effective cross-section $\sigma_{\text{eff}}$.
• Model Discrimination: The large discrepancy between NRQCD and ICEM predictions for the $J/\psi + \gamma$ channel suggests that experimental measurements of this process could serve as a powerful discriminator to validate or rule out specific hadronization models.
• PDF Constraints: By constraining the production mechanisms, these results contribute to refining Transverse-Momentum-Dependent (TMD) gluon PDFs, which are essential for precision physics at high energies.

In conclusion, the authors demonstrate that while the choice of hadronization model (NRQCD vs. ICEM) drastically affects the absolute magnitude of the cross-section, the Double Parton Scattering mechanism is the primary driver of associated $J/\psi + \gamma$ production at the LHC, regardless of the model used.