Studies of beauty hadron and non-prompt charm hadron production in pp collisions at $\sqrt{s}$=13 TeV within a transport model approach

This study employs a refined AMPT transport model with adjusted beauty quark mass and flavor-specific coalescence parameters to successfully reproduce LHCb and ALICE data on beauty and non-prompt charm hadron production in 13 TeV $pp$ collisions, offering new insights into heavy quark dynamics and hadronization mechanisms in small collision systems.

The Gist

Imagine you are a detective trying to solve a mystery inside a tiny, high-speed particle collider. The mystery involves "heavy" particles called beauty quarks and charm quarks. These are like the "heavyweights" of the subatomic world.

The scientists in this paper are trying to figure out how these heavyweights behave when they crash into each other at the LHC (Large Hadron Collider) in Switzerland. Specifically, they want to know: Do they stick together to form "families" (baryons) or stay as pairs (mesons), and does the size of the crash crowd change how they behave?

Here is the breakdown of their investigation, explained simply:

1. The Problem: The "Heavy" Mystery

In the world of particle physics, there's a rulebook (called Quantum Chromodynamics) that predicts how these particles should act. Usually, the rulebook says: "It doesn't matter where you crash; the particles will break apart and reassemble the same way."

But recent experiments showed something weird. When heavy particles crash in a crowded room (a high-energy collision), they seem to form more "families" (baryons) than the rulebook predicted. It's like if you threw a bunch of Lego bricks into a room, and in a small room, they mostly built towers, but in a huge room, they mostly built pairs. The scientists wanted to understand why this happens.

2. The Tool: A Digital Sandbox (The AMPT Model)

The researchers used a super-computer simulation called AMPT. Think of this as a "flight simulator" for particle collisions.

• The Setup: They programmed the simulator to crash protons together at 13 TeV (a massive amount of energy).
• The Glitch: The simulator's default settings were wrong. It was predicting too many beauty particles, like a video game that spawns too many enemies.
• The Fix: The scientists tweaked two main knobs in their simulation:
  1. The Weight Knob: They made the "beauty quark" slightly heavier in the simulation. This is like adding a little extra weight to a bowling ball so it doesn't roll as fast or as far, which matched the real-world data better.
  2. The "Stickiness" Knob: They adjusted a parameter called the coalescence parameter. Imagine this as a "magnet strength" setting. If the magnet is strong, particles are more likely to stick together to form a family (baryon). If it's weak, they stay as pairs (meson). They tuned this magnet specifically for beauty quarks to match real experiments.

3. The Detective Work: Using "Non-Prompt" Clues

Here is the tricky part. Beauty particles are very hard to catch directly because they decay (fall apart) almost instantly. It's like trying to photograph a ghost.

So, the scientists used a clever trick: The "Grandchild" Strategy.

• Beauty Quarks decay into Charm Quarks (the "children").
• These Charm Quarks then form Charm Hadrons (the "grandchildren").
• The scientists call these "non-prompt" charm hadrons. They are the "footprints" left behind by the beauty quarks.

By studying these "grandchildren," the scientists could indirectly figure out what the "parents" (beauty quarks) were doing. It's like figuring out how a parent behaved by watching how their child acts.

4. The Big Discovery: The "Crowd" Effect

The team looked at how the particles behaved in collisions with different numbers of people (multiplicity).

• Low Crowd: When few particles are created, the beauty quarks mostly act like loners, forming pairs (mesons).
• High Crowd: When the collision is chaotic and crowded, the beauty quarks are more likely to "stick" to their neighbors and form families (baryons).

The Analogy: Imagine a dance floor.

• If the dance floor is empty, dancers (particles) just pair up with their partners and leave.
• If the dance floor is packed, dancers bump into everyone. They might grab a third person and form a group dance (a baryon) because there's so much energy and contact.

The simulation showed that the "stickiness" (coalescence) is what causes this group formation in crowded collisions.

5. Why This Matters

This study is a big deal because it connects two different ways of thinking about the universe:

1. The "Hard" Physics: How particles are created in a violent crash.
2. The "Soft" Physics: How they settle down and form matter.

By proving that the "stickiness" of particles changes based on how crowded the collision is, the scientists have created a unified framework. This helps us understand not just beauty quarks, but also how the very first moments of the universe (the Big Bang) might have worked, where everything was incredibly dense and hot.

In Summary:
The scientists fixed their computer simulation to match reality, used the "footprints" of beauty particles to solve a mystery, and discovered that crowded environments make heavy particles more likely to stick together and form families. This helps us understand the fundamental rules of how matter is built in our universe.

Technical Summary

1. Problem Statement

High-energy proton-proton ($pp$) collisions at the LHC provide a unique environment to study Quantum Chromodynamics (QCD). While heavy quark production is calculable via perturbative QCD (pQCD), recent experimental data (particularly from ALICE and LHCb) have challenged the assumption of universal hadronization.

• The Discrepancy: Experiments show a significant enhancement in heavy-flavor baryon-to-meson ratios (e.g., $\Lambda_c/D^0$ and $\Lambda_b/B^0$) in $pp$ collisions compared to $e^+e^-$ collisions, suggesting that hadronization is sensitive to the collision environment (multiplicity).
• The Challenge: Direct measurements of beauty hadrons are difficult at low transverse momentum ($p_T$). However, non-prompt charm hadrons (charm hadrons originating from beauty decays) serve as an indirect but powerful probe of beauty dynamics.
• The Gap: There is a need for a unified theoretical framework that can simultaneously describe the initial hard production of beauty quarks, their transport through the partonic medium, and their subsequent hadronization (via coalescence or fragmentation) to explain the observed multiplicity-dependent trends in small collision systems.

2. Methodology

The authors employ the A Multi-Phase Transport (AMPT) model (string melting version) coupled with PYTHIA8 for initial conditions. The study focuses on $\sqrt{s} = 13$ TeV $pp$ collisions.

Key Modifications to the AMPT Framework:
To achieve agreement with experimental data, two critical refinements were introduced:

1. Effective Beauty Quark Mass Tuning:
   • The default PYTHIA8 (Monash tune) overestimates the $b\bar{b}$ production cross-section at 13 TeV because it is a leading-order generator lacking higher-order pQCD corrections.
   • The authors increased the effective beauty quark mass ($m_b$) during the initial production stage from the default $4.8$ GeV/$c^2$ to $6.6$ GeV/$c^2$.
   • This adjustment suppresses the total yield and mimics the higher-order suppression effects found in state-of-the-art calculations like FONLL (Fixed-Order Next-to-Leading Log), bringing the initial $b\bar{b}$ cross-section into agreement with ALICE and LHCb data. The mass is reset to the default value for subsequent transport and hadronization.
2. Flavor-Specific Coalescence Parameter:
   • The AMPT model uses a spatial coalescence mechanism where quarks combine based on proximity. The probability of forming a baryon vs. a meson is controlled by a parameter $r_{BM}$.
   • A specific parameter $r^b_{BM} = 1.2$ was introduced for beauty quarks (tuned to reproduce LHCb $\Lambda_b/B^0$ ratios). This is distinct from the charm parameter ($r^c_{BM} = 1.4$) used in previous works, acknowledging flavor-dependent hadronization dynamics.

Simulation Workflow:

• Initial State: PYTHIA8 generates $b\bar{b}$ pairs with the tuned mass.
• Partonic Evolution: The Zhang's Parton Cascade (ZPC) model simulates parton scatterings with a cross-section $\sigma = 0.15$ mb.
• Hadronization: Quarks hadronize via a naive spatial coalescence algorithm.
• Decay: Beauty hadrons decay into non-prompt charm hadrons, which are then tracked.

3. Key Contributions

• Unified Framework: Established a consistent AMPT framework that successfully describes both beauty hadron production and non-prompt charm production in small systems ($pp$ collisions).
• Phenomenological Tuning: Demonstrated that adjusting the initial production mass in a leading-order generator can effectively emulate higher-order QCD corrections for cross-section normalization.
• Flavor-Dependent Dynamics: Provided evidence that beauty and charm quarks may have different coalescence dynamics ($r^b_{BM} \neq r^c_{BM}$), challenging the assumption of universal hadronization parameters across heavy flavors.
• Non-Prompt Probes: Validated non-prompt charm hadrons as sensitive observables for disentangling the interplay between initial production, coalescence, and decay kinematics.

4. Key Results

A. Beauty Hadron Production

• Cross-Sections: The tuned model (with $m_b = 6.6$ GeV/$c^2$) reproduces the $p_T$-differential cross-sections for $B^0, B^+, B_s$, and $\Lambda_b^0$ at both mid-rapidity ($|y|<0.5$) and forward rapidity ($2<y<4.5$), matching ALICE and LHCb data.
• $\Lambda_b^0/B^0$ Ratio: The model successfully reproduces the LHCb measurement of the $\Lambda_b^0/B^0$ ratio as a function of $p_T$ and event multiplicity.
  • A mild enhancement at low $p_T$ ($\approx 2$ GeV/$c$) is observed, attributed to the kinematic effects of the coalescence mechanism (vector sum of three quarks vs. two).
  • The ratio increases with charged-particle multiplicity, consistent with the transition from string fragmentation (low density) to coalescence (high density).

B. Non-Prompt Charm Hadron Production

• Spectra: The model describes the $p_T$ spectra of non-prompt $D^0, D^+, D_s^+$, and $\Lambda_c^+$ reasonably well, though it slightly underestimates non-prompt $\Lambda_c^+$ at low $p_T$.
• Ratios ($D^+/D^0$ and $D_s^+/(D^0+D^+)$): The model reproduces the isospin symmetry ($D^+/D^0 \approx 0.5$) and strangeness enhancement trends.
• $\Lambda_c^+/D^0$ Ratios:
  • Prompt vs. Non-Prompt: While prompt $\Lambda_c^+/D^0$ shows a strong enhancement at low $p_T$, the non-prompt ratio is flatter and lower at low $p_T$. This is due to the "smearing" effect of beauty decay kinematics, which dilutes the baryon enhancement generated by coalescence.
  • Multiplicity Dependence: Both prompt and non-prompt ratios increase with multiplicity, but the prompt ratio rises more steeply.

C. Non-Prompt to Prompt Ratios

• $p_T$ Dependence: The model predicts a relatively flat non-prompt-to-prompt ratio with $p_T$, whereas data shows a rising trend. The authors attribute this to the increased $m_b$ shifting beauty production to lower $p_T$, making the parent beauty spectra and daughter charm spectra more similar in shape.
• Multiplicity Dependence (Crucial Finding):
  • The ratio of non-prompt to prompt charm hadrons is highly sensitive to the beauty coalescence parameter $r^b_{BM}$.
  • Mechanism: Non-prompt charm mesons primarily come from beauty meson decays, while non-prompt charm baryons come from beauty baryon decays.
  • Sensitivity:
    • If $r^b_{BM}$ is small (suppressing beauty baryons), the non-prompt-to-prompt meson ratio rises steeply with multiplicity, while the baryon ratio drops.
    • If $r^b_{BM}$ is large (enhancing beauty baryons), the meson ratio flattens, and the baryon ratio rises.
  • The tuned value $r^b_{BM} = 1.2$ yields a multiplicity dependence consistent with ALICE data, confirming that the relative growth of baryons vs. mesons is driven by flavor-dependent coalescence.

5. Significance and Conclusion

This work establishes a unified transport framework for heavy quark dynamics in small collision systems. The study demonstrates that:

1. Non-prompt charm hadrons are not just background but are essential probes for understanding beauty quark hadronization, especially in low-$p_T$ regions where direct beauty measurements are scarce.
2. Flavor-dependent coalescence is a necessary ingredient to describe experimental data; beauty and charm quarks do not hadronize with identical probabilities.
3. The multiplicity dependence of non-prompt-to-prompt ratios serves as a powerful constraint on theoretical models, bridging the gap between microscopic transport approaches (like AMPT) and statistical hadronization models.

The findings suggest that future high-precision measurements of non-prompt baryon-to-meson ratios as a function of event activity are critical for understanding the transition from dilute to dense QCD matter and the thermalization of heavy quarks in high-multiplicity $pp$ collisions.