Inclusive $J/ψ$ productions in pp collisions at $\sqrt{s}=$ 5.02, 7, and 13 TeV with the PACIAE model

This study utilizes the PACIAE 4.0 model, which incorporates both color-singlet and color-octet NRQCD contributions alongside cluster collapse and $b$-hadron decays, to successfully simulate inclusive $J/\psi$ production in proton-proton collisions at 5.02, 7, and 13 TeV and provide a quantitative analysis of the relative contributions and rescattering effects across various production mechanisms.

The Gist

Imagine you are a detective trying to solve a mystery inside a tiny, high-speed particle collider. The mystery? How a specific, heavy particle called the J/ψ (pronounced "Jay-psi") is born when two protons smash into each other at nearly the speed of light.

This paper is the detective's report. The team used a sophisticated computer simulation called PACIAE 4.0 to recreate these collisions and figure out exactly how the J/ψ is made.

Here is the story of their findings, broken down into simple concepts and analogies.

1. The Setting: The High-Speed Crash

Think of a proton-proton collision like two freight trains smashing into each other at 13 trillion electron volts (a mind-boggling amount of energy). When they crash, they don't just break apart; they explode into a chaotic shower of smaller particles, including "charm" and "anti-charm" quarks.

The J/ψ is a special "couple" made of one charm quark and one anti-charm quark holding hands tightly. The big question is: How do they find each other and hold hands in the chaos?

2. The Three Ways the Couple Gets Together

The researchers found that the J/ψ isn't made in just one way. They identified three main "matchmaking" methods:

• The "Direct Match" (NRQCD): This is the most common method (about 75–85% of the time). It's like two people meeting at a busy party and immediately deciding to dance. In physics terms, the charm quarks are created by a hard collision and then settle down into a J/ψ. The paper looked at two types of "dance styles": the "Color-Singlet" (a perfect, formal dance) and the "Color-Octet" (a more chaotic, energetic dance where they need to shed some extra energy first).
• The "Clump Collapse" (Cluster Collapse): Sometimes, the charm quarks are born so close together in the chaotic explosion that they just snap together instantly, like two magnets snapping shut before they can even move apart. This happens about 6–10% of the time.
• The "Grandparent" (Non-Prompt): This is the slowest method. Sometimes, a much heavier particle (containing a "bottom" quark) is created first. This heavy particle decays (dies) and leaves behind a J/ψ as its "child." This happens about 10–16% of the time, depending on the energy.

3. The Energy Effect: Turning Up the Heat

The team ran simulations at three different energy levels (5.02, 7, and 13 TeV), which is like turning up the volume on a speaker.

• What they found: As they turned up the energy, the "Grandparent" method (Non-Prompt) became more popular. Why? Because making heavy bottom quarks gets easier at higher energies, just like it's easier to buy a luxury car when you have more money.
• The Shift: Because the "Grandparent" method grew faster, the percentage of J/ψs made by the "Direct Match" (NRQCD) actually went down slightly, even though the total number of J/ψs increased.

4. The Location Effect: Middle vs. The Edge

The researchers looked at particles flying out from the middle of the collision versus those flying out to the edges (forward rapidity).

• The Middle: This is the chaotic center of the party. Here, all three methods happen, but the "Direct Match" still wins.
• The Edge: This is like the VIP section at the edge of the room. Here, the "Direct Match" and "Clump Collapse" become even more dominant. The "Grandparent" method gets suppressed because the conditions at the edge aren't quite right for making those heavy bottom quarks.

5. The Family Tree: Who is the Parent?

The "Direct Match" method isn't just one thing. It's a family tree.

• Sometimes the J/ψ is born directly.
• Sometimes it's born from the decay of heavier cousins like $\psi(2S)$ or $\chi_c$ particles.
• The Discovery: At low speeds (low momentum), the J/ψ mostly comes from the $\chi_{c2}$ cousin. But at high speeds (high momentum), the $\chi_{c1}$ cousin takes over. It's like a relay race where different runners take the baton depending on how fast the race is going.

6. The "Traffic Jam" Effect (Rescattering)

This is a crucial part of the study. After the J/ψ is born, it has to travel through a crowd of other particles.

• Partonic Rescattering (Before the party ends): The charm quarks bump into other quarks before they form the J/ψ. The team found this has almost zero effect on the final count. It's like people bumping into each other in a hallway before entering a room; it doesn't change who ends up inside.
• Hadronic Rescattering (After the party starts): Once the J/ψ is formed, it travels through a crowd of other particles (pions, protons, etc.). Here, the J/ψ can crash into them and break apart (like a fragile vase hitting a wall).
• The Result: This "traffic jam" destroys about 8% of the J/ψs. It selectively wipes out the ones made by the "Direct Match" and "Clump Collapse" methods, but it doesn't touch the "Grandparent" ones (because those are born later, after the crowd has thinned out).

The Big Picture

This paper is a major step forward because it combines all these different "matchmaking" methods into one complete picture.

• Before: Scientists looked at these methods separately or used older, less accurate models.
• Now: They have a model that accounts for the chaos of the collision, the different ways the particle forms, and the traffic it faces afterward.

In summary: The J/ψ is a resilient particle. It is mostly born from a direct, high-energy handshake between quarks, but it also has a significant "family tree" of parents. As we crank up the energy of the collider, we see more "Grandparent" births, and as the particle tries to escape the collision zone, about 8% of them get knocked out by the crowd. This detailed understanding helps scientists better interpret what happens in even bigger collisions, like those between heavy atomic nuclei, which are used to study the very first moments of the universe.

Technical Summary

1. Problem Statement

The production of $J/\psi$ mesons in proton-proton (pp) collisions serves as a critical baseline for understanding heavy-ion collisions, where $J/\psi$ suppression is a signature of Quark-Gluon Plasma (QGP) formation. However, disentangling "hot" nuclear matter effects (QGP) from "cold" nuclear matter effects (e.g., nuclear PDF modifications) requires a precise, quantitative understanding of $J/\psi$ production mechanisms in the absence of a nucleus.

Previous theoretical studies often relied on simplified models (e.g., Color-Singlet Model only) or older Monte Carlo generators (PYTHIA 6.4) that lacked comprehensive treatment of:

• Color-Octet mechanisms within the Non-Relativistic QCD (NRQCD) framework.
• Feed-down contributions from excited charmonium states ($\psi(2S)$, $\chi_{cJ}$) and $b$-hadron decays.
• Dynamic rescattering effects (both partonic and hadronic) that occur before and after hadronization.

This study aims to provide a comprehensive, quantitative analysis of inclusive $J/\psi$ production across a range of LHC energies (5.02, 7, and 13 TeV) and rapidities (mid and forward), incorporating all major production mechanisms and rescattering dynamics.

2. Methodology

The authors utilized the PACIAE 4.0 model, an extension of the PYTHIA 8.3 event generator. The methodology involves a four-stage sequential simulation of ultra-relativistic pp collisions:

1. Parton Initialization: Based on PYTHIA 8.3 (Monash tune), generating initial partons, including multiple parton interactions (MPIs) and color reconnections (CR).
2. Partonic Rescattering (PRS): Before hadronization, the model simulates $2 \to 2$ hard and soft parton-parton scatterings using leading-order pQCD cross-sections, modified by an empirical $K$-factor (adjusted to 0.68–0.75 to match data).
3. Hadronization:
   • NRQCD Framework: Includes both Color-Singlet and Color-Octet contributions for prompt $J/\psi$ production.
   • Cluster Collapse: A mechanism where a $c\bar{c}$ pair forms a bound state directly if they are close in phase space and the string potential is insufficient to produce a light quark pair.
   • Feed-down: Includes decays from heavier charmonia ($\psi(2S)$, $\chi_{c0,1,2}$).
   • Non-prompt: Includes $J/\psi$ from weak decays of $b$-hadrons.
4. Hadronic Rescattering (HRS): After hadronization, the model simulates elastic and inelastic collisions between $J/\psi$ and hadrons (nucleons, pions, rhos). Key inelastic channels include $J/\psi + N/\pi/\rho \to D + \Lambda_c/\Sigma_c$, which can destroy the $J/\psi$.

Simulation Parameters:

• Energies: $\sqrt{s} = 5.02, 7, 13$ TeV.
• Event Statistics: 400 million inelastic non-diffractive events generated per energy.
• Comparison: Results were compared against experimental data from the ALICE collaboration for transverse momentum ($p_T$) differential cross-sections at mid-rapidity and forward rapidity.

3. Key Contributions

• Comprehensive Mechanism Integration: Unlike previous work by the same group (which considered only Color-Singlet), this study fully integrates Color-Octet NRQCD channels, Cluster Collapse, and Non-prompt ($b$-hadron) contributions.
• Quantitative Decomposition: The paper provides a detailed breakdown of the relative fractions of different production sources (NRQCD, cluster collapse, non-prompt) and their dependence on collision energy ($\sqrt{s}$) and rapidity ($y$).
• Sub-channel Analysis: It quantifies the specific contributions of direct production versus feed-down from $\psi(2S)$, $\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ within the NRQCD sector.
• Rescattering Impact: It offers a systematic, quantitative estimate of how Partonic Rescattering (PRS) and Hadronic Rescattering (HRS) modify the final $J/\psi$ yield, distinguishing their effects on prompt vs. non-prompt sources.

4. Key Results

A. Agreement with Experimental Data

The PACIAE 4.0 model successfully reproduces the experimental $p_T$-differential cross-sections for inclusive $J/\psi$ at both mid-rapidity and forward rapidity across all three energies. The discrepancy between the model and data is generally within 30%.

B. Relative Contributions to Inclusive Yield

• Dominance of NRQCD: The NRQCD channel is the dominant source of $J/\psi$ production.
• Energy Dependence: As collision energy increases:
  • The relative contribution of NRQCD decreases (from ~85% at 5.02 TeV to ~74% at 13 TeV at mid-rapidity).
  • The contributions from Cluster Collapse and Non-prompt ($b$-hadron) processes increase. This is attributed to the faster energy scaling of $b\bar{b}$ production compared to $c\bar{c}$.
• Rapidity Dependence:
  • At forward rapidity, the NRQCD and Cluster Collapse fractions are enhanced compared to mid-rapidity.
  • The Non-prompt fraction is suppressed at forward rapidity due to the scarcity of large-$x$ partons required for heavy $b$-quark production.

C. NRQCD Sub-channel Breakdown

• Low $p_T$: Dominated by Direct Production and $\chi_{c2}$ feed-down.
• High $p_T$: Dominated by Direct Production and $\chi_{c1}$ feed-down.
• Small Contributions: Feed-down from $\psi(2S)$ (6-8%) and $\chi_{c0}$ (2-3%) are minor.
• Energy Evolution: The fraction of direct production saturates around 13 TeV. The $\chi_{c1}$ and $\psi(2S)$ fractions grow with energy (favoring color-octet fragmentation), while $\chi_{c0}$ and $\chi_{c2}$ fractions decline.

D. Rescattering Effects

• Partonic Rescattering (PRS): Has a negligible effect on the final $J/\psi$ yield (ratios close to 100%). This is because the model does not include $J/\psi$-parton scattering channels (e.g., $J/\psi + g \to c + \bar{c}$).
• Hadronic Rescattering (HRS): Causes a significant suppression (~8-9%) of the prompt $J/\psi$ yield (NRQCD and Cluster Collapse) due to inelastic absorption ($J/\psi + \pi/N \to D + \Lambda_c$).
• Non-prompt Stability: The non-prompt yield remains largely unaffected by HRS because the model does not simulate rescattering for $b$-hadrons before their decay.

5. Significance

• Theoretical Advancement: This work represents a significant step forward in modeling inclusive $J/\psi$ production by unifying perturbative (NRQCD) and non-perturbative (Cluster Collapse, Rescattering) mechanisms within a single dynamic framework.
• Baseline for Heavy-Ion Physics: The quantitative breakdown of production mechanisms and the identification of HRS suppression effects provide a more accurate "cold" baseline. This is essential for isolating genuine QGP suppression signals in nucleus-nucleus (AA) collisions.
• Constraining NRQCD Parameters: The observed energy and rapidity dependencies of the various feed-down components offer new observational grounds to constrain the Long-Distance Matrix Elements (LDMEs) in NRQCD, particularly for distinguishing between different $\chi_{cJ}$ states.
• Model Validation: The success of PACIAE 4.0 in reproducing LHC data validates the inclusion of both color-octet mechanisms and hadronic rescattering, suggesting these are crucial for a complete description of heavy quarkonium phenomenology in pp collisions.