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Multiplicity dependence of the size of the common hadron emission source in pp collisions at the LHC

This paper presents a femtoscopic analysis of proton-proton correlations in minimum bias $pp$ collisions at $\sqrt{s}=13.6$ TeV using a high-luminosity ALICE dataset, which enables the first simultaneous measurement of the hadron-emitting source size's dependence on multiplicity and transverse mass, revealing a distinct multiplicity trend compared to Pb-Pb collisions and demonstrating the robustness of the results across various nucleon-nucleon interaction models.

Original authors: ALICE Collaboration

Published 2026-06-29

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CC BY 4.0

Original authors: ALICE Collaboration

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 two tiny, high-speed cars crashing into each other in a dark room. When they smash, they don't just break apart; they spray out a shower of smaller, invisible marbles (particles). Physicists want to know: How close together were these marbles when they were first born?

This paper from the ALICE collaboration at CERN is like a high-tech detective story trying to answer that question. They are studying "pp collisions," which are head-on crashes between two protons (the building blocks of atoms) at the Large Hadron Collider.

Here is the breakdown of what they did and found, using simple analogies:

1. The "Femto-Scope" Camera

The scientists use a technique called femtotoscopy. Think of this as a super-powerful camera that doesn't take pictures of the crash itself, but instead looks at the echoes of the crash.

When two particles are born very close together, they "talk" to each other through invisible forces (like magnets or gravity, but much stronger and stranger). By measuring how often these particles arrive at the detector together versus apart, the scientists can work backward to figure out the size of the "birth room" where they were created. It's like hearing two people shout in a cave; by how their voices echo, you can guess how big the cave is.

2. The New "High-Definition" Dataset

In the past, the ALICE detector was like a camera with a slow shutter speed. It could only take a few snapshots of these crashes.

The Upgrade: In 2022, the detector got a massive upgrade. It switched to "continuous readout," meaning it can now record a constant stream of data, like a high-definition video camera instead of a slow shutter.
The Result: They collected a huge amount of data (19.3 pb⁻¹ of "integrated luminosity," which is just a fancy way of saying "a lot of crash data"). This allowed them to look at the crashes in much finer detail than ever before.

3. The Two Main Discoveries

A. The "Crowded Room" vs. "Empty Room" Effect (Multiplicity)
The scientists looked at crashes that produced different numbers of particles.

High Multiplicity: Imagine a crash that sprays out a huge crowd of particles (a "crowded room").
Low Multiplicity: Imagine a crash that only sprays out a few particles (an "empty room").

The Finding: They found that in the "crowded rooms," the particles were born further apart from each other. In the "empty rooms," they were born much closer together. This makes sense intuitively: if you have more stuff exploding, the "room" needs to be bigger to hold it all. However, the paper notes that the relationship between the crowd size and the room size is different in these tiny proton crashes compared to massive collisions of heavy atoms (like Lead-Lead crashes).

B. The "Heavy vs. Light" Effect (Transverse Mass)
They also looked at how "heavy" the pairs of particles were moving.

The Finding: The heavier the pair of particles (in terms of their combined energy and mass), the smaller the "birth room" appears to be. It's as if the heavier particles are squeezed into a tighter space.

4. Checking the "Rulebook" (Interaction Models)

To figure out the size of the room, the scientists had to use a theoretical "rulebook" to understand how protons push and pull on each other. There are a few different rulebooks (models) that physicists use.

The Test: They ran their data through several different, state-of-the-art rulebooks.
The Result: No matter which rulebook they used, they got the same answer for the size of the room. This proves their measurement is robust. It's like measuring a table with a ruler, a tape measure, and a laser; if they all say the table is 2 meters long, you can be very confident in that number.

5. The Big Picture: Why Compare to Heavy Crashes?

The paper compares these tiny proton crashes to massive crashes of Lead nuclei (Pb-Pb).

The Analogy: Imagine comparing a small kitchen explosion (proton-proton) to a massive warehouse explosion (Lead-Lead).
The Surprise: In the warehouse explosions, the size of the explosion grows very predictably with the amount of debris. In the kitchen explosions, the size grows, but the rate at which it grows is different.
The Implication: This difference suggests that the physics governing these tiny proton crashes might be more complex than just a simple "explosion." It hints that even in these tiny systems, there might be some collective behavior (like a fluid flowing) that we don't fully understand yet.

Summary

This paper is a milestone because, for the first time, the ALICE team could look at these tiny proton crashes with enough data to see how the "birth room" size changes based on how many particles are created and how heavy they are. They proved their measurements are solid regardless of the theoretical models used, and they found that the rules for these tiny crashes are slightly different from the rules for the massive crashes of heavy atoms. This helps physicists refine their understanding of how matter behaves at the smallest scales.

Technical Summary: Multiplicity Dependence of the Size of the Common Hadron Emission Source in pp Collisions at the LHC

Problem and Motivation
Femtoscopic analysis serves as a precision tool for studying the strong interaction among hadrons, particularly in ultra-relativistic proton-proton (pp) collisions where the small system size (emission distances of $\sim 1.5-2$ fm) enhances sensitivity to short-range interaction components. While previous studies by the ALICE Collaboration and others have successfully characterized particle emission sources using $\pi$ - $\pi$ , $K$ - $p$ , and baryon-baryon correlations, a comprehensive understanding of the source size dependence on both transverse mass ( $m_T$ ) and event multiplicity in pp collisions has been limited.

Previous measurements were primarily differential in $m_T$ but integrated over multiplicity, often restricted to high-multiplicity events (average charged tracks $\approx 30$ ). The origin of the observed $m_T$ scaling of the source size in pp collisions remains debated; it is unclear whether this behavior stems from collective effects (such as radial flow) or other mechanisms, as standard event generators fail to quantitatively reproduce the scaling. Furthermore, precise knowledge of the femtoscopic source size is a prerequisite for using femtoscopy to study residual strong interactions and for coalescence models predicting (anti)nuclei yields in cosmic-ray interactions and dark matter searches. This paper addresses the gap by extending femtoscopic measurements to the low-multiplicity regime and simultaneously analyzing the dependence on $m_T$ and event multiplicity.

Methodology
The analysis utilizes a minimum-bias (MB) dataset of pp collisions at a center-of-mass energy of $\sqrt{s} = 13.6$ TeV, recorded by the upgraded ALICE detector in 2022 with an integrated luminosity of $19.3 \text{ pb}^{-1}$ . The detector upgrade enabled continuous readout, facilitating the collection of this extensive dataset.

Event Selection and Multiplicity Classes: Events were selected based on signals from the Fast Interaction Trigger (FT0) and a reconstructed primary vertex. Events were classified into three multiplicity classes (0–10%, 10–50%, and 50–100%) based on the FT0 amplitude. The analysis specifically focused on events containing at least one reconstructed (anti)proton–(anti)proton pair, resulting in average charged-particle densities ( $\langle dN_{ch}/d\eta \rangle_{|\eta|<0.5}$ ) of 23.95, 16.72, and 9.73 for the respective classes, extending the reach to significantly lower multiplicities (down to $\approx 9.7$ tracks) compared to previous studies.
Particle Identification and Tracking: Protons and antiprotons were identified using the Time Projection Chamber (TPC) and Time-Of-Flight (TOF) detectors, with a combined significance cut ( $n\sigma_{combined} < 3$ ). Tracks were required to have transverse momentum ( $p_T$ ) between 0.5 and 2.2 GeV/c. Primary particle selection was enforced via strict Distance of Closest Approach (DCA) cuts to the primary vertex, suppressing secondary contributions from weak decays and material interactions.
Correlation Function Measurement: The correlation function $C(k^*)$ was constructed as the ratio of same-event (SE) to mixed-event (ME) pair distributions, where $k^*$ is the relative momentum in the pair rest frame. Systematic uncertainties were evaluated by varying selection criteria (e.g., $p_T$ , track quality, PID cuts) and generating 44 variation sets.
Theoretical Modeling: The correlation functions were fitted using the Koonin–Pratt equation. Two source models were employed:
1. A single Gaussian source parametrization.
2. The Resonance Source Model (RSM), which accounts for a primordial Gaussian core ( $r_{core}$ ) and a halo of exponential tails arising from short-lived resonance decays.
  The strong interaction between protons was modeled using the Norfolk potential (NV2-II A), with comparisons made against the Argonne v18 and N4LO chiral effective field theory potentials to assess theoretical uncertainties. The fit included contributions from genuine proton-proton pairs, feed-down from $\Lambda$ and $\Sigma^+$ decays, and misidentified particles, parameterized via $\lambda$ coefficients derived from data-driven purity and feed-down fraction calculations.

Key Results

Source Size Extraction: The effective source radius ( $r_0$ ) and core radius ( $r_{core}$ ) were extracted for seven $m_T$ intervals across the three multiplicity classes.
$m_T$ Dependence: A clear power-law dependence of the source size on the average transverse mass ( $\langle m_T \rangle$ ) was observed for all multiplicity classes, consistent with previous high-multiplicity measurements. The source size decreases as $m_T$ increases.
Multiplicity Dependence: A hierarchy in source size was established: events with higher average multiplicities exhibit larger source radii. This trend holds across all measured $m_T$ intervals.
Model Robustness: The extracted radii showed consistency across different nucleon-nucleon interaction models (Argonne v18, Norfolk, and N4LO). The differences in wave functions at short distances ( $r < 1.2$ fm) between these models were found to have a negligible impact on the correlation function and extracted radii, as the dominant contribution arises from pair separations where the wave functions are identical.
Comparison with Pb–Pb Collisions: When comparing the multiplicity dependence of the source size in pp collisions to recent measurements in Pb–Pb collisions at $\sqrt{s} = 5.02$ TeV, a marked difference in slope was observed. While both systems show a linear scaling of the source radius with the cube root of the charged-particle multiplicity (suggesting a volume scaling), the slope is significantly steeper in Pb–Pb collisions than in pp collisions.

Significance and Claims
The paper claims that this work provides the first simultaneous measurement of the femtoscopic source size in pp collisions as a function of both transverse mass and event multiplicity, extending the analysis into the low-multiplicity regime. The results demonstrate the robustness of the measurement against variations in the assumed nucleon-nucleon interaction models.

The authors state that the observed differences in the multiplicity dependence between pp and Pb–Pb systems provide new insights into the system-size dependence of particle emission dynamics. Specifically, the steeper slope in Pb–Pb collisions compared to pp collisions suggests a stronger dependence on multiplicity in heavy-ion systems, potentially linked to the formation of a quark-gluon plasma (QGP), although the paper notes that no established theoretical threshold currently links this behavior directly to QGP formation in pp collisions.

The measured source sizes provide necessary input for coalescence models used to estimate (anti)nuclei yields, which are crucial for cosmic-ray physics and dark matter searches. Furthermore, the differential nature of this measurement bridges the gap between femtoscopic studies in small (pp) and large (Pb–Pb) systems, offering constraints for future theoretical modeling of collective effects and particle production mechanisms in small collision systems.