Initial-state geometry and multiplicity distributions in pp and pPb collisions

This study utilizes Monte Carlo simulations based on the Color Glass Condensate formalism to demonstrate that multiplicity distributions in proton-proton and proton-lead collisions can distinguish between different initial-state proton geometries, specifically testing the three-quark Y-shape baryon junction model, while highlighting the critical importance of accounting for intrinsic saturation scale fluctuations.

The Gist

Imagine you are trying to figure out the shape of a mysterious, invisible object by smashing two of them together at incredible speeds and counting how many tiny shrapnel pieces fly out. That is essentially what this paper is about, but instead of invisible objects, physicists are studying protons (the tiny building blocks inside atoms) and instead of shrapnel, they are counting particles created in high-energy collisions at the Large Hadron Collider (LHC).

Here is a simple breakdown of their investigation, using some everyday analogies.

1. The Big Question: What does a proton really look like?

We know protons are made of three "quarks" (like three tiny marbles) held together by "gluons" (like invisible rubber bands). But how are they arranged?

• The Old Idea: For a long time, physicists thought the quarks were just scattered randomly inside a fuzzy cloud, like raisins in a muffin.
• The New Idea (The Baryon Junction): This paper investigates a specific theory called the Baryon Junction. Imagine the three quarks are the corners of a triangle, and the gluons connect them in a "Y" shape (like a three-pronged fork or a tripod). The center of the "Y" is a special point where everything meets.

The authors want to know: Is the proton a fuzzy cloud, or is it a distinct "Y" shape?

2. The Experiment: Smashing and Counting

To test this, they simulate two types of collisions:

1. Proton vs. Proton (pp): Two small marbles hitting each other.
2. Proton vs. Lead (pPb): A tiny marble hitting a giant, heavy bowling ball (the Lead nucleus).

They use a super-computer simulation (a "Monte Carlo event generator") to play out these collisions millions of times. They input different shapes for the proton (the "Y" shape vs. the fuzzy cloud) and see which one produces a particle count that matches what real scientists (the ALICE experiment) actually measured.

3. The Key Analogy: The Flashlight and the Wall

Why is the Proton vs. Lead collision so important? The authors use a brilliant analogy in the paper:

• Proton vs. Proton: Imagine two flashlights shining at each other. Because they are the same size, they overlap in the middle. It's hard to tell if one flashlight has a weird shape because the other one is blocking the view. The shapes get "blurred" together.
• Proton vs. Lead: Imagine shining a tiny, shaped flashlight (the proton) onto a giant, flat white wall (the Lead nucleus). The wall is so big that the tiny flashlight's shape is perfectly projected onto it. You can clearly see the "Y" shape or the "Cloud" shape on the wall.

The Result: The paper finds that the Proton vs. Lead collisions are much better at revealing the proton's true shape. In these collisions, the data strongly suggests the proton looks like the "Y" shape (Baryon Junction).

4. The Secret Ingredient: "Intrinsic Fluctuations"

There was a catch. Even with the right shape, the simulation didn't match the data perfectly. They needed to add a "secret ingredient" called intrinsic fluctuations.

Think of it like baking cookies.

• The Shape: You decide to make star-shaped cookies (the Baryon Junction).
• The Fluctuations: But sometimes, the dough is a bit wetter, or the oven is a few degrees hotter. This means one star might be huge and crispy, while another is tiny and soft.

The authors found that to explain the rare, massive explosions of particles (the "high multiplicity" events), they had to assume that the "glue" holding the proton together (the saturation scale) isn't perfectly steady. It jitters and fluctuates. When they added this "jitter" to their "Y" shape model, the results matched the real-world data almost perfectly.

5. The Conclusion

• In Proton-Proton collisions: The data is a bit ambiguous. A fuzzy cloud model works okay, but the "Y" shape works better for the biggest, most energetic crashes.
• In Proton-Lead collisions: The "Y" shape (Baryon Junction) is the clear winner. The data strongly supports the idea that protons have this specific geometric structure.

The Bottom Line:
This paper doesn't say, "We have 100% proof the proton is a Y." Instead, it says, "If the proton is a Y, and if we account for natural jitters in the system, our computer models match the real world perfectly."

It's a strong hint that the "Y" shape is real. The authors suggest that the future Electron-Ion Collider (a new, even more powerful microscope) will be the place to finally take a clear photo and confirm this shape once and for all.

Technical Summary

1. Problem Statement

The fundamental geometric structure of the proton remains a significant open question in high-energy physics. While the proton is known to consist of quarks and gluons (partons), the specific spatial arrangement of these constituents is debated.

• The Hypothesis: The paper investigates the Baryon Junction (BJ) model, originally proposed by Rossi and Veneziano. This model suggests that the three valence quarks of a nucleon are connected by a Y-shaped gluon string (flux tube) meeting at a central Fermat point, rather than being distributed as independent constituent quarks or a smooth Gaussian cloud.
• The Challenge: Previous phenomenological successes of the BJ model (e.g., in baryon rapidity distributions and diffractive $J/\psi$ production) are not definitive proof, as other models can mimic these results. Furthermore, at high energies (LHC), quantum evolution (DGLAP/BFKL) might "wash out" the initial geometric Y-shape.
• The Goal: To determine if the initial-state geometry of the proton (specifically the Y-shape vs. Gaussian vs. Hard-sphere) leaves a distinct imprint on charged particle multiplicity distributions in proton-proton ($pp$) and proton-lead ($p\text{Pb}$) collisions at the LHC.

2. Methodology

The authors employ a Monte Carlo event generator, MC-KLN, which implements the $k_T$-factorization formalism of the Color Glass Condensate (CGC) effective theory.

A. Initial State Geometries

Four distinct spatial distributions for the nucleon matter were simulated as inputs:

1. Hard-Sphere: A uniform disk with a sharp edge, defined by a collision criterion based on transverse area ($\sigma_0 = 42$ mb).
2. Gaussian: A smooth, bell-shaped transverse profile without sub-nucleonic structure, characterized by a width parameter $B_p$.
3. Baryon Junction 1 (BJ1): An analytical "Y-shape" model. Three quarks are placed at the vertices of an equilateral triangle, with gluon distributions (hotspots) located halfway between the center and the quarks.
4. Baryon Junction 2 (BJ2): A numerical "Y-shape" model. Quark positions are drawn from a 3D distribution, a Fermat point is calculated to minimize distances, and gluon flux tubes are filled with Gaussians.

B. Collision Simulation

• Formalism: The single-inclusive gluon production cross-section is calculated using $k_T$-factorization with KLN unintegrated gluon distributions (UGD).
• Saturation Scale ($Q_s$): The saturation scale is modulated by the local thickness function $T(r_\perp)$ of the projectile and target.
• Fluctuations: Crucially, the study includes intrinsic fluctuations of the saturation scale. These are modeled as log-normal deviations from the mean saturation scale, necessary to reproduce the tails of multiplicity distributions (high-multiplicity events).
• Hadronization: The conversion from partons to final-state hadrons is handled via parton-hadron duality, assuming a constant proportionality factor between produced gluons and detected charged particles.
• Target Nucleus: For $p\text{Pb}$ collisions, the lead nucleus is modeled using a Woods-Saxon profile with 208 nucleons, each possessing one of the four tested internal geometries.

C. Data Comparison

The simulated multiplicity distributions ($P(N_{ch})$) were compared against recent ALICE experimental data (2024) for:

• $pp$ collisions: $\sqrt{s} = 2.76, 5.02, 7, 8, 13$ TeV.
• $p\text{Pb}$ collisions: $\sqrt{s} = 5.02, 8.16$ TeV.
• Kinematic Range: $0.15 < p_T < 10$ GeV/c and $|\eta| < 0.8$.
• Scaling: Results were analyzed using KNO scaling variables ($z = N_{ch}/\langle N_{ch} \rangle$).

3. Key Contributions

1. Systematic Geometry Comparison: The first comprehensive application of MC-KLN to compare Hard-sphere, Gaussian, and two distinct Baryon Junction models (analytical and numerical) simultaneously against LHC multiplicity data.
2. Sensitivity Analysis: Demonstrated that $p\text{Pb}$ collisions are significantly more sensitive to the initial proton geometry than $pp$ collisions. In $pp$ collisions, the overlap of two similarly sized protons often obscures the specific internal structure (except in ultra-central collisions), whereas in $p\text{Pb}$, the small proton acts as a probe scanning the large lead nucleus, preserving the proton's shape in the overlap region.
3. Role of Intrinsic Fluctuations: Confirmed that intrinsic saturation scale fluctuations are essential. Models relying solely on geometric fluctuations fail to describe the high-multiplicity tails of the distribution.
4. Differentiation of BJ Models: Showed that while BJ1 and BJ2 are difficult to distinguish in $pp$ collisions, they yield significantly different results in $p\text{Pb}$ collisions, allowing for model discrimination.

4. Key Results

• $pp$ Collisions:
  • The Gaussian initial state provides the best overall description of the multiplicity distribution across most of the range.
  • However, in the high-multiplicity region ($z = N_{ch}/\langle N_{ch} \rangle \geq 4$), the Baryon Junction models (BJ1 and BJ2) provide a better fit to the data than the Gaussian or Hard-sphere models.
• $p\text{Pb}$ Collisions:
  • The data are best described by the BJ1 (analytical Y-shape) initial condition.
  • The BJ1 model significantly outperforms the Gaussian and Hard-sphere models.
  • BJ1 and BJ2 are distinguishable in this system, with BJ1 being clearly favored.
• Impact of Fluctuations:
  • Figures 4b and 4d in the paper explicitly show that without intrinsic fluctuations, the model fails to reproduce the high-multiplicity tail. Including these fluctuations (log-normal distribution of $Q_s$) is crucial for agreement with data.
• KNO Scaling:
  • The study confirms that KNO scaling holds reasonably well for the $p\text{Pb}$ data and the BJ1 model across different energies ($\sqrt{s} = 5.02$ and $8.16$ TeV) and pseudorapidity bins.

5. Significance and Conclusion

• Evidence for Baryon Junction: While the existence of the baryon junction is not yet definitively proven, this work provides strong phenomenological evidence that the Y-shaped configuration (specifically BJ1) is a superior description of the proton's initial state in high-energy $p\text{Pb}$ collisions compared to standard Gaussian or Hard-sphere models.
• Methodological Insight: The paper establishes that $p\text{Pb}$ collisions are a more powerful tool for probing the internal geometric structure of the proton than $pp$ collisions due to the "probe" nature of the proton against the large nucleus.
• Future Directions: The authors conclude that while current LHC data favors the BJ model in specific regimes, further confirmation is needed. They propose that the upcoming Electron-Ion Collider (EIC) will be the ideal facility to unambiguously confirm the baryon junction structure through high-precision measurements of diffractive processes and geometric deformations.

In summary, the paper argues that the proton's "Y-shape" geometry is not washed out by quantum evolution at LHC energies and manifests clearly in multiplicity distributions, particularly when the proton collides with a heavy nucleus, provided that intrinsic saturation scale fluctuations are properly accounted for.