⚛️ phenomenology

How cross section fluctuations affect multiplicity and geometry in pA collisions

This paper presents a new Monte Carlo Glauber model for pA collisions utilizing KMR/SHRiMPS cross sections, demonstrating that their inherent impact parameter dependence and long-tailed wounded nucleon distributions effectively describe multiplicity distributions and enhance spatial anisotropy compared to other models.

Original authors: Chiara Le Roux

Published 2026-02-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chiara Le Roux

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand what happens when a single, tiny billiard ball (a proton) smashes into a large, fluffy cluster of billiard balls glued together (a nucleus). Physicists call this a "proton-nucleus" or pA collision.

For decades, scientists have used a tool called the Glauber model to predict the outcome of these crashes. Think of this model as a simulation game where you drop the single ball onto the cluster and count how many balls in the cluster get "hit" or "wounded."

However, the old versions of this game had a problem. They were too rigid. They assumed that every time a ball hit another, the result was predictable and uniform, like a perfect circle of ink spreading on paper. In reality, nature is messy. Sometimes the hit is a glancing blow; sometimes it's a massive explosion. The old models failed to capture the extreme "tails" of the data—meaning they couldn't explain the rare events where the crash was either surprisingly small or surprisingly huge.

The New Approach: A "Shape-Shifting" Ball

In this paper, the author, Chiara Le Roux, introduces a new, smarter way to run this simulation. Instead of using a rigid, unchanging ball, she uses a shape-shifting ball based on a complex theory called the KMR/SHRiMPS model.

Here is the core idea using a simple analogy:

The Old Way (The Black Disk):
Imagine the proton is a solid, black rubber disk. If it touches a nucleon (a ball in the cluster), it hits. If it misses, it doesn't. The size of the disk is fixed. This is simple, but it misses the nuance of real life.

The New Way (The KMR/SHRiMPS Model):
Imagine the proton is a cloud of smoke that can change its density and shape.

Fluctuations: Sometimes the cloud is dense and heavy; other times, it's thin and wispy. This represents the fact that the proton isn't a solid object but a jumble of smaller particles (quarks and gluons) that move around.
The "Impact Parameter": This is just a fancy word for "how close the center of the proton gets to the center of the target." In the new model, the chance of hitting a target ball depends on exactly where the cloud overlaps, not just a simple "yes or no" circle.

What Did the Simulation Show?

The author ran thousands of these virtual crashes and compared the new "cloud" model against the old "solid disk" models. Here are the two main discoveries:

1. The "Long Tail" of Wounded Nucleons
When you count how many balls get hit (called "wounded nucleons"), the old models struggled with the extremes. They couldn't explain why sometimes very few balls get hit, or why sometimes very many get hit.

The Result: The new "cloud" model naturally produces these extreme results. Because the proton cloud can be very dense in some moments and very sparse in others, it creates a "long tail" in the data. It successfully mimics the rare, wild crashes that the old models missed.

2. The Shape of the Crash (Geometry)
This is perhaps the most surprising finding. When the proton hits the nucleus, the wounded nucleons don't just form a random pile; they form a specific shape (like an oval or a teardrop). Physicists need to know this shape because it determines how the "soup" of particles flows afterward.

The Result: The new model creates a much more "lopsided" or "anisotropic" shape than the old models.
The Analogy: Imagine dropping a drop of water on a sponge.
- The Old Model drops a perfect sphere of water. The wet spot is a perfect circle.
- The New Model drops a drop of water that is already squashed and irregular. The wet spot is a weird, stretched-out shape.
- The author found that this "squashed" nature of the proton (due to its internal fluctuations) makes the resulting pattern of wounded nucleons much more irregular. This is crucial because if you get the initial shape wrong, your predictions for how the particles fly apart later will also be wrong.

The Bottom Line

The paper argues that to truly understand these high-energy collisions, we cannot treat protons as simple, solid balls. We must treat them as fluctuating clouds where the "size" and "shape" of the collision change from moment to moment.

The new model, which calculates these fluctuations directly from the physics of the collision rather than guessing, does a better job of predicting:

How many particles get hit (especially the rare, extreme cases).
What shape the collision zone takes (which is vital for understanding the initial state of the crash).

The author concludes that for a complete picture of these collisions, we need to account for both the random size changes of the proton (integrated cross-section fluctuations) and the specific way those changes happen across the face of the proton (non-trivial impact parameter dependence). The new model handles both, making it a more accurate tool for physicists studying the fundamental building blocks of the universe.

Technical Summary: How Cross Section Fluctuations Affect Multiplicity and Geometry in pA Collisions

Problem Statement
In the phenomenology of relativistic heavy ion collisions, Glauber models are essential tools for estimating the number of binary proton-proton interactions and the geometric configuration of the initial state. These estimates are critical for scaling observables to compare proton-nucleus (pA) and nucleus-nucleus (AA) collisions with proton-proton (pp) data. However, standard Monte Carlo Glauber models often systematically underestimate the tails of the charged particle multiplicity distribution (specifically the number of wounded nucleons, $N_W$ ) in pA collisions. Furthermore, the interpretation of final-state anisotropies relies heavily on the initial geometric fluctuations defined by the Glauber model. While previous studies have shown that adding cross-section fluctuations improves the description of multiplicity tails, a gap remains in fully describing these distributions and their impact on initial-state geometry.

Methodology
The author develops a new Monte Carlo Glauber model for pA collisions that incorporates hadronic cross sections calculated within the KMR model, as implemented in the SHRiMPS minimum bias module of the SHERPA event generator.

The KMR/SHRiMPS Approach: Unlike models that impose a specific impact parameter ( $b$ ) dependence, the KMR model derives the $b$ -dependent differential cross sections directly from the same calculation that yields the integrated cross section. This is achieved through a multi-channel eikonal formalism where hadrons are treated as superpositions of Good-Walker (GW) states (specifically the proton and its excited state $N^*(1440)$ ). The eikonals are computed via evolution equations involving parton densities and absorptive corrections, resulting in non-trivial impact parameter profiles without ad-hoc assumptions.
Comparison Models: The new model is benchmarked against two other approaches:
1. Black Disk (BD) Model: A non-fluctuating model with a fixed interaction radius.
2. Glauber-Gribov (GG) Model: A model that includes fluctuations in the integrated cross section (and thus the interaction radius) but retains a trivial (step-function) impact parameter dependence.
Simulation Setup: The study simulates pPb collisions at $\sqrt{s} = 5020$ GeV. Nucleons are distributed according to a Woods-Saxon distribution. The impact parameter is sampled, and the probability of interaction between the projectile proton and target nucleons is determined by the respective model's cross-section profile.

Key Contributions

Unified Cross Section Implementation: The paper introduces a Glauber implementation where the impact parameter dependence is intrinsic to the cross-section calculation (KMR/SHRiMPS), avoiding the need for separate assumptions about the spatial profile of the interaction.
Fluctuation Analysis: The study isolates the effects of two types of fluctuations:
- Fluctuations in the integrated cross section (present in GG and SH models).
- Fluctuations arising from the non-trivial impact parameter dependence (unique to the SH model).
Geometry and Multiplicity Correlation: The work explicitly connects the nature of cross-section fluctuations to both the multiplicity distribution ( $N_W$ ) and the spatial eccentricity ( $\varepsilon_n$ ) of the wounded nucleon distribution.

Results

Multiplicity Distributions ( $N_W$ ):
- Models incorporating cross-section fluctuations (GG and SH) successfully extend the tails of the $N_W$ distribution compared to the standard Black Disk model.
- The GG model produces the longest tails for large $N_W$ because it can fluctuate to very large interaction radii, wounding all nucleons within that radius.
- The SH model also produces a long tail but is less extreme than GG for large $N_W$ . However, the SH model exhibits a distinct "double-peaked" structure when the impact parameter is fixed at $b=0$ , a feature arising from the discrete fluctuation between GW states.
- The SH model's ability to generate large $N_W$ tails is attributed to its non-trivial $b$ -dependence allowing interactions at larger distances, even if the integrated cross section is not as large as the extreme fluctuations in the GG model.
Geometric Eccentricities ( $\varepsilon_n$ ):
- The Black Disk and GG models yield very similar eccentricity distributions ( $\varepsilon_2, \varepsilon_3, \varepsilon_4$ ), indicating that fluctuations in the integrated cross section alone contribute only marginally to geometric anisotropy compared to nucleon position fluctuations.
- In contrast, the SH model predicts significantly larger mean eccentricities (up to ~20% higher) and broader distributions for all orders of $n$ .
- This enhancement persists even when the impact parameter is fixed ( $b=0$ ), demonstrating that the non-trivial impact parameter dependence of the SH cross sections is a dominant source of initial-state geometric fluctuations.

Significance and Claims
The paper concludes that a complete description of the initial state in pA collisions requires accounting for both integrated cross-section fluctuations and non-trivial impact parameter dependence.

Integrated cross-section fluctuations are identified as essential for describing the shape of the multiplicity distribution (specifically the tails).
Non-trivial impact parameter dependence is shown to be extremely important for defining the geometry (eccentricity) of the wounded nucleon distribution, a factor crucial for hydrodynamic simulations of the initial state.
The KMR/SHRiMPS model is presented as a unified framework that naturally incorporates both effects. The author notes that the generalization of this approach to A+A collisions is straightforward.

The New Approach: A "Shape-Shifting" Ball

What Did the Simulation Show?

The Bottom Line

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