Common femtoscopic hadron-emission source in pp collisions at the LHC

By modeling resonance decay effects on pion and kaon-proton correlations in 13 TeV pp collisions, this study demonstrates that a universal, collectively scaling primordial emission source exists for all hadron species in small collision systems at the LHC.

The Gist

Imagine you are at a massive, chaotic concert where thousands of people (particles) are rushing out of the venue after the show ends. You want to figure out exactly where everyone started standing inside the building before the doors opened. But there's a catch: you can't see the inside, and the crowd is moving so fast that by the time you spot them outside, they've already drifted apart.

This is essentially what physicists at CERN's ALICE experiment are trying to do, but instead of concert-goers, they are studying subatomic particles (like pions and kaons) created in high-speed collisions of protons.

Here is a simple breakdown of what this paper is about, using some everyday analogies:

1. The "Femtoscopy" Camera

The scientists use a technique called femtoscopy. Think of this as a super-powered, microscopic camera that doesn't take pictures of light, but of time and distance.

When two identical particles (like two pions) are born from the same collision, they have a special quantum connection (called Bose-Einstein correlation). It's like they are "twins" that prefer to stick together. By measuring how close they are to each other when they fly apart, the scientists can work backward to calculate the size of the "room" (the source) they were born in.

2. The Problem: The "Ghost" Particles

In a proton-proton collision, most particles aren't born directly from the initial crash. Instead, they are the "children" of short-lived, unstable particles called resonances.

• The Analogy: Imagine the initial collision is a firework exploding. The bright sparks you see immediately are the "primordial" particles. But many of the sparks you see a split second later are actually pieces of other fireworks that exploded after the first one.
• The Issue: If you just look at the final sparks, you might think the explosion happened in a huge, messy area. But the real explosion (the primordial source) might have been tiny and compact. The "children" (resonances) travel a bit before decaying, making the whole group look bigger and more spread out than it actually is.

3. The Solution: The "Resonance Source Model"

The ALICE team created a new mathematical model (the Resonance Source Model) to act like a detective.

• They know the "family tree" of the particles. They know which particles came directly from the crash and which ones are "grandchildren" from decaying resonances.
• They use this knowledge to mathematically "peel back" the layers. They strip away the extra distance traveled by the resonance decays to reveal the primordial source—the actual size of the "room" where the particles were born.

4. The Big Discovery: A "Universal" Source

Once they peeled back the layers for different types of particles (pions, kaons, and protons), they found something amazing:

• The Shape: The "room" where the particles are born isn't a perfect sphere (Gaussian). Because of the resonance decays, the effective source looks like an exponential curve (like a bell that tapers off slowly).
• The Scaling: They found that the size of this "room" changes depending on how heavy and fast the particles are moving (a property called transverse mass, $m_T$).
  • The Analogy: Imagine a crowd leaving a building. If you look at the light, heavy people (high mass), they seem to come from a smaller, tighter area. If you look at the light, fast people (low mass), they seem to come from a larger area.
  • The Result: Whether they looked at pions (mesons) or protons (baryons), the "room size" followed the exact same rule.

5. Why This Matters

This is a huge deal because it suggests that all hadrons (the family of particles that includes protons, neutrons, and pions) in these tiny proton collisions come from a single, common source.

• The "Collective" Effect: It implies that even in a tiny collision (just two protons hitting each other), the particles behave as if they are part of a collective fluid, similar to what happens in massive collisions between heavy atomic nuclei (like gold or lead).
• The Future: Now that they have a reliable map of this "source," they can use it to study rare particles (like those containing strange or charm quarks) with much higher precision. It's like finally having a clear map of a city; now you can easily find the rare shops you were looking for.

Summary

In short, the ALICE team figured out how to filter out the "noise" of decaying particles to see the true, tiny origin point of particle creation in proton collisions. They discovered that despite the chaos, there is a hidden, universal order: all particles seem to emerge from the same "birthplace," and the size of that birthplace changes in a predictable way based on the particles' speed and weight. This brings us one step closer to understanding how matter behaves under extreme conditions.

Technical Summary

1. Problem and Motivation

Femtoscopic studies utilize Bose-Einstein correlations (BEC) and final-state interactions (FSI) to probe the space-time geometry of the particle-emitting source in high-energy collisions. While well-established in heavy-ion collisions (where the source is typically Gaussian and exhibits $m_T$ scaling due to radial flow), applying these techniques to small collision systems (proton-proton, $pp$) is challenging.

Key Challenges:

• Resonance Dominance: In $pp$ collisions, the majority of pions are not primordial (produced directly in the collision) but stem from the decay of short-lived resonances. These decays significantly distort the effective source profile, making it difficult to extract the properties of the primordial source.
• Source Modeling: Previous analyses often assumed simple parameterizations (e.g., exponential or Cauchy distributions) for the effective source without explicitly modeling the resonance contribution. This limits the precision of extracting the primordial source size.
• Universality: It remains an open question whether a "common emission source" exists for all hadron species (mesons and baryons) in small systems, and whether the observed $m_T$ scaling (source size decreasing with increasing transverse mass) is a universal collective phenomenon or an artifact of specific particle production mechanisms.

2. Methodology

The study analyzes $pp$ collisions at $\sqrt{s} = 13$ TeV using data from the ALICE detector. The analysis focuses on same-sign pion pairs ($\pi^+\pi^+$, $\pi^-\pi^-$) and kaon-proton pairs ($K^\pm p$).

Data Selection:

• Events: Minimum Bias (MB) and High Multiplicity (HM, top 0.17% of inelastic events) events.
• Particle Identification (PID): Tracks are identified using the Time Projection Chamber (TPC) and Time-Of-Flight (TOF) detectors. Strict cuts on $n\sigma$ (deviation from expected energy loss/velocity) ensure high purity (>99%).
• Background Suppression:
  • Feed-down: Tracks from weak decays are suppressed by requiring a small Distance of Closest Approach (DCA) to the primary vertex.
  • Mini-jets: A transverse sphericity cut ($S_T > 0.7$) is applied to MB events to select spherical events and suppress jet-like correlations.
• Correlation Function: The correlation function $C(k^*)$ is constructed as the ratio of same-event pairs to mixed-event pairs, where $k^*$ is the relative momentum in the pair rest frame.

Modeling Framework (Resonance Source Model - RSM):

• Theoretical Basis: The study employs the Resonance Source Model (RSM) to disentangle the primordial source from resonance contributions.
• Source Function ($S(r^*)$): Modeled as a convolution of:
  1. A Gaussian core ($r_{core}$): Represents the primordial emission source.
  2. An Exponential tail: Represents the "halo" of particles from short-lived resonance decays.
• Resonance Abundances: Calculated using the THERMAL-FIST package based on a canonical Statistical Hadronization Model (SHM).
• Decay Kinematics: Extracted from the EPOS event generator (version 3.117), which includes hydrodynamics and a hadronic afterburner (UrQMD).
• Fitting Tool: The CATS (Correlation Analysis Tool using the Schrödinger Equation) framework is used to numerically solve the Schrödinger equation for the two-particle wave function, incorporating Coulomb and strong interactions.
• Fit Parameters: The fit extracts the Gaussian core radius ($r_{core}$) and the $\lambda$ parameter (incoherence factor), accounting for purity and feed-down fractions.

3. Key Contributions

• Explicit Resonance Modeling: This is the first application of the RSM to extract the primordial source size for meson-meson ($\pi$-$\pi$) and meson-baryon ($K$-$p$) pairs in $pp$ collisions. Previous works often treated the effective source as a single entity.
• Validation of Gaussian Primordial Source: The analysis demonstrates that assuming a Gaussian distribution for the primordial source is compatible with the data, provided the exponential tails from resonances are explicitly modeled.
• Extension to Meson-Baryon Sector: The methodology is successfully extended to $K$-$p$ pairs, allowing for a direct comparison between meson-meson and meson-baryon source sizes.
• Differential Analysis: The study provides results across multiple multiplicity classes (MB) and transverse mass ($m_T$) intervals, offering high differential precision.

4. Key Results

• Effective Source Profile: The effective source (primordial + resonances) is found to be approximately exponential, consistent with previous LHC measurements. However, the underlying primordial source is Gaussian.
• $\lambda$ Parameters: The incoherence parameter $\lambda_{gen}$ (genuine pairs) is calculated to be approximately 0.66 for $\pi$-$\pi$ and 0.65–0.76 for $K$-$p$, corrected for long-lived resonance feed-down.
• $m_T$ Scaling:
  • The extracted Gaussian core radius ($r_{core}$) for $\pi$-$\pi$ and $K$-$p$ pairs exhibits a clear $m_T$ scaling: the source size decreases as the transverse mass increases.
  • This scaling is consistent with the $m_T$ scaling previously observed for baryon-baryon ($p$-$p$, $p$-$\Lambda$) pairs in the same dataset.
• Saturation at Low $m_T$:
  • For $\pi$-$\pi$ pairs at low $m_T$ ($< 0.6$ GeV/$c^2$), the $r_{core}$ reaches a saturation regime and becomes independent of $m_T$.
  • This suggests that at low $m_T$, the region of homogeneity spans the entire physical extension of the hadronization hypersurface, preventing further growth of the apparent source size.
• Universality: The agreement of $r_{core}$ scaling across different particle species (pions, kaons, protons, lambdas) strongly supports the hypothesis of a common emission source for all hadrons in small $pp$ systems.

5. Significance

• Confirmation of Collective Behavior: The observation of universal $m_T$ scaling across mesons and baryons in small $pp$ systems provides robust evidence for collective phenomena (likely radial flow) even in the smallest collision systems, challenging the notion that such effects are exclusive to heavy-ion collisions.
• Precision Source Characterization: By explicitly modeling resonance contributions, the paper establishes a rigorous method to determine the primordial source function. This allows for high-precision extraction of source parameters, which is crucial for interpreting final-state interactions.
• Future Applications: The validated framework enables the determination of source functions for rare hadron pairs (e.g., strange or charmed particles) where statistics are low. This grants access to the properties of final-state interactions for these less abundant particles, potentially constraining chiral effective field theories and lattice QCD predictions.
• Theoretical Input: The data provides critical constraints for transport models and effective source models (like CECA) that aim to describe the spatial extension of particle emission in small systems and the nature of the hadronization hypersurface.

In conclusion, this work solidifies the concept of a universal hadron emission source in high-multiplicity $pp$ collisions at the LHC, demonstrating that the interplay between primordial Gaussian emission and resonance halos can be quantitatively disentangled to reveal collective dynamics.