Effects of the initial-state geometry on D-meson production in pp and pPb collisions

Using a Monte Carlo event generator with $k_T$-factorization, the study demonstrates that the observed stronger-than-linear growth of D-meson yields in high-multiplicity pp and pPb collisions is reproduced by various initial-state spatial distributions, indicating that this observable is insufficient for distinguishing the specific matter geometry within the proton.

The Gist

Imagine you are a detective trying to figure out the shape of a mysterious, invisible object. You can't see the object itself, but you can watch what happens when you smash it into other things.

This paper is about a team of physicists trying to figure out the shape of the matter inside a proton (a tiny particle that makes up atoms). Specifically, they are asking: Is the proton a smooth, round ball, or does it have a weird, three-pronged "Y" shape inside?

Here is the story of their investigation, broken down simply:

1. The Mystery: The "Y" Shape vs. The Ball

For a long time, scientists thought protons were just simple, round balls of stuff. But recent theories suggest something more exotic: the Baryon Junction.

Think of a proton like a three-legged stool or a Y-shaped kite.

• The Old View: The proton is a smooth, fluffy cloud (like a Gaussian distribution).
• The New Theory: The proton has three "legs" (valence quarks) connected by a string in the middle, forming a "Y" shape.

The scientists wanted to know: Which one is it?

2. The Experiment: Smashing Cars at Different Speeds

To test this, they used the Large Hadron Collider (LHC) to smash protons into other protons (pp) and protons into giant lead nuclei (pPb).

They didn't just look at one crash; they looked at thousands. They noticed something interesting:

• When the crashes were "gentle" (low multiplicity), everything looked normal.
• But when the crashes were chaotic and crowded (high multiplicity), the production of a specific particle called the D-meson (think of it as a "smoke ring" left behind by the crash) grew much faster than expected.

It was like saying: "If I hit a car gently, it makes a small dent. If I hit it really hard, it doesn't just make a bigger dent; it explodes into pieces!"

3. The Simulation: The Virtual Crash Test

Since they can't see inside the proton, the authors built a virtual reality simulator (a Monte Carlo event generator).

They programmed the simulator with four different "blueprints" for the proton's interior:

1. Hard-Sphere: A solid, rigid ball.
2. Gaussian: A soft, fuzzy cloud.
3. BJ1 (Analytical): A mathematical "Y" shape.
4. BJ2 (Numerical): A computer-generated "Y" shape.

They ran millions of virtual crashes, changing the "density" of the crash (how crowded the particles were) to see which blueprint produced the "smoke rings" (D-mesons) that matched the real-world data.

4. The Twist: The "Y" Shape Didn't Win

Here is the surprising result: All four blueprints worked.

Whether the proton was a hard ball, a soft cloud, or a "Y" shape, the simulator could reproduce the real-world data perfectly. The "smoke rings" grew at the same rate in the simulation as they did in the real experiment, regardless of the proton's shape.

Why did this happen?
The scientists realized that the "chaos" of the crash (the high multiplicity) was so intense that it washed out the subtle differences between the shapes. It's like trying to tell the difference between a square and a round table by looking at the shadows they cast during a massive, blinding lightning storm. The storm is so bright, you can't see the shape of the table anymore.

5. The Conclusion: Not the Right Tool for the Job

The authors concluded that while their theory about how particles are created is correct, this specific measurement (D-mesons in high-multiplicity crashes) is not a good magnifying glass for seeing the proton's shape.

The data is too "noisy" (the error bars are too big) in the high-energy region to tell the difference between a "Y" shape and a ball.

The Takeaway

• The Goal: Find the shape of the proton.
• The Method: Smashing particles and counting the debris.
• The Result: The debris count matched the theory, but it matched every shape theory equally well.
• The Lesson: We need better data (more crashes, clearer pictures) from future runs of the LHC to finally solve the mystery of whether the proton is a ball or a "Y".

In short: They built a great map of the territory, but the fog was too thick to see if the mountain was shaped like a cone or a pyramid. They need a clearer day to find out.

Technical Summary

1. Problem Statement

The internal spatial distribution of matter (specifically gluons) within the proton remains a critical open question in high-energy physics. While recent measurements have characterized the proton's mass and scalar radius, the specific shape of the matter distribution is still debated. A leading theoretical hypothesis is the Baryon Junction (BJ) model, where valence quarks are connected by a "Y-shaped" gluon string (string junction) to a Fermat point. This configuration implies that baryon number can be carried by sea quarks, potentially explaining baryon number asymmetry in mid-rapidity regions.

Previous studies suggested that the BJ model enhances $J/\psi$ relative yields in high-multiplicity collisions. However, it is unclear if D-meson production exhibits similar sensitivity to these initial-state geometries. The core problem addressed is whether the normalized D-meson yield as a function of charged-particle multiplicity can serve as a discriminator to distinguish between different proton spatial configurations (Hard-sphere, Gaussian, and Baryon Junction models).

2. Methodology

The authors utilized the MC-KLN event generator, a Monte Carlo framework that combines Glauber initial conditions with $k_T$-factorization formalism and KLN (Kowalski-Levin-Nikolov) unintegrated gluon distributions (UGDs).

• Initial State Geometries: The study simulated four distinct nucleon spatial distributions:

  1. Hard-sphere: Uniform density within a radius.
  2. Gaussian: Standard Gaussian density profile.
  3. BJ1 (Analytical Baryon Junction): Based on an analytical "Y" string configuration.
  4. BJ2 (Numerical Baryon Junction): Based on numerical lattice QCD-inspired configurations.
  • For Lead (Pb) nuclei, nucleon centers were positioned according to a Woods-Saxon profile, combined with the four proton geometries.

• Cross-Section Calculation: The gluon production cross-section was calculated using the $k_T$-factorization formula:
  $$\frac{d\sigma}{dy d^2p_T d^2r_\perp} \propto \int d^2k_T \, \alpha_s \, \phi_p \left(\frac{|p_T + k_T|}{2}, x_1; b\right) \phi_T \left(\frac{|p_T - k_T|}{2}, x_2; R-b\right)$$
  where $\phi$ represents the KLN UGD, dependent on the saturation scale $Q_s$.

• Fluctuations: To model high-multiplicity events, intrinsic fluctuations in the saturation scale ($Q_s$) were introduced via a log-normal distribution. The width of this distribution ($\sigma$) was treated as a free parameter tuned for each geometry.

• Hadronization and Observables:

  • Charged Multiplicity: Calculated via parton-hadron duality (proportional to gluon production).
  • D-Meson Production: The gluon spectrum was convoluted with the KKKS08 fragmentation function ($g \to D$) at a factorization scale $\mu = m_D = 1.8$ GeV.
  • Observable: The study focused on the relative yield of D-mesons ($D^0, D^+, D^{*+}$), defined as the yield per charged particle normalized by its minimum-bias average value, plotted against the normalized charged-particle multiplicity.

3. Key Contributions

• Extension of MC-KLN: The authors extended previous multiplicity distribution studies to include heavy-flavor (D-meson) production in both proton-proton (pp) and proton-lead (pPb) collisions.
• Systematic Geometry Comparison: They provided a systematic comparison of how four distinct proton spatial models (including two variations of the Baryon Junction) affect D-meson production in the high-multiplicity regime.
• Quantitative Constraints: The study quantified the sensitivity of D-meson yields to initial geometry, determining the necessary statistical precision required to distinguish between models.

4. Results

The simulations were compared against experimental data from the LHC (pp at $\sqrt{s}=7$ TeV and pPb at $\sqrt{s}=5.02$ TeV) for transverse momentum ranges $1 < p_T < 2$ GeV/c and $2 < p_T < 4$ GeV/c.

• Low Multiplicity: All four initial-state geometries produced results consistent with experimental data in the low-multiplicity region ($\langle dN_{ch}/d\eta \rangle / \langle dN_{ch}/d\eta \rangle \lesssim 2$).
• High Multiplicity Divergence: As multiplicity increased ($\gtrsim 3$), the theoretical curves began to diverge:
  • The Hard-sphere and Gaussian models showed distinct behaviors from the BJ models.
  • The BJ1 and BJ2 models remained closely clustered.
• Data Comparison: While the models predicted different slopes for the normalized D-meson yield in the high-multiplicity region, the experimental data possessed large error bars in this specific regime. Consequently, all four models could statistically reproduce the data within uncertainties.
• Parameter Sensitivity: The models required different fluctuation parameters ($\sigma$) to fit the data (e.g., Hard-sphere $\sigma \approx 1.15$ vs. BJ2 $\sigma \approx 0.35$ in pp collisions), indicating that the fluctuation strength compensates for geometric differences.

5. Significance and Conclusion

• Limitation of Current Data: The primary conclusion is that D-meson relative yields are currently not a sufficient observable to constrain the detailed spatial distribution of matter (or distinguish between Hard-sphere, Gaussian, and Baryon Junction models) in the proton. The large experimental uncertainties in the high-multiplicity region mask the theoretical differences between the geometries.
• Future Prospects: The authors emphasize that the formalism successfully describes the data using gluon production and intrinsic fluctuations. However, to definitively test the Baryon Junction hypothesis or other geometric models, higher statistics data from future LHC runs are required to reduce experimental error bars in the high-multiplicity tail.
• Theoretical Validation: The work validates the MC-KLN framework's ability to describe heavy-flavor production across different collision systems and multiplicities, provided that intrinsic fluctuations in the saturation scale are properly accounted for.