⚛️ phenomenology

Nuclear transverse momentum dependent gluon density at low $x$ and inclusive soft hadron production in proton-lead collisions at LHC

This paper presents a modified quark-gluon string model incorporating nuclear-transverse momentum dependent gluon densities to successfully describe inclusive soft hadron production in proton-lead collisions at LHC energies, demonstrating superior agreement with CMS, ATLAS, and ALICE data at low transverse momentum compared to other theoretical predictions.

Original authors: A. V. Lipatov, G. I. Lykasov, M. A. Malyshev

Published 2026-03-20

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CC BY 4.0

Original authors: A. V. Lipatov, G. I. Lykasov, M. A. Malyshev

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 the Large Hadron Collider (LHC) as the world's most powerful "particle smasher." Scientists fire protons (tiny building blocks of matter) at each other at nearly the speed of light to see what happens when they crash. But sometimes, instead of smashing two single protons together, they smash a single proton into a massive "lead nucleus," which is like a tiny, dense ball made of 208 protons and neutrons stuck together.

This paper is about trying to understand exactly what happens when that single proton hits the giant lead ball, specifically looking at the "soft" debris (pions and kaons) that flies out at low speeds.

Here is the story of the paper, broken down into simple concepts:

1. The Problem: The "Free Nucleon" Myth

For a long time, physicists thought a lead nucleus was just a bag of 208 independent marbles (nucleons) floating around. They thought if you smashed a proton into the lead ball, it was just like smashing it into 208 separate protons, one after another.

The Reality: It's not that simple. When you pack 208 marbles into a tiny space, they start to interact with each other. They get crowded, they shadow each other, and the whole system behaves differently than the sum of its parts. This is called the "nuclear environment." The paper tries to figure out exactly how that environment changes the crash.

2. The Tool: A New "String" Model

The authors use a theory called the Modified Quark-Gluon String Model (QGSM).

The Analogy: Imagine the proton isn't a solid ball, but a bundle of rubber bands (strings) made of quarks and gluons. When two protons collide, these rubber bands stretch and snap. When they snap, they create new particles (like pions and kaons) out of the energy, much like snapping a stretched rubber band creates a "pop" and a bit of heat.
The Modification: The old version of this model worked great for simple proton-proton crashes but failed when things got crowded (low energy, high density). The authors updated the model to include "soft gluons" (fuzzy, low-energy energy packets) that act like a thick fog inside the proton.

3. The Secret Ingredient: The "Foggy" Gluon Map

The core of this paper is a new way of mapping the "gluon fog" inside a nucleus.

The Concept: In a single proton, the gluons are spread out in a specific pattern. But in a lead nucleus, because the space is so crowded, the gluons behave differently. They seem to follow a rule called "Geometric Scaling."
The Metaphor: Imagine you have a map of a city (the proton). Now, imagine you have a map of a massive, crowded metropolis (the lead nucleus). The authors realized that if you zoom out on the city map and shrink the metropolis map to the same size, the traffic patterns (gluon density) look surprisingly similar. They used this "zooming" trick to predict how the gluons behave inside the heavy lead nucleus without having to reinvent the wheel.

4. The Experiment: Predicting the Debris

The authors used their new model to predict what the "debris" (pions and kaons) would look like after a proton hits a lead nucleus at the LHC. They focused on particles moving slowly (low momentum), which are the hardest to predict because they are influenced by the "fog" of the nucleus.

They compared their predictions to real data collected by three giant detectors at the LHC: CMS, ATLAS, and ALICE.

5. The Results: A Better Fit

The Competition: Other scientists use computer programs (like EPOS, HIJING, and AMPT) to simulate these crashes. Some of these programs get the results wrong, predicting that the debris flies out too fast or in the wrong amounts.
The Victory: The authors' new model, using the "Geometric Scaling" trick for the gluon fog, matched the real experimental data almost perfectly.
- For Pions (the most common debris), their model fit the data better than the other computer programs.
- For Kaons (a slightly heavier type of debris), their model was the best at matching the data, outperforming all the other tools.

The Bottom Line

This paper is a success story of "connecting the dots." The authors took a theory that worked well for simple crashes, added a clever mathematical trick to account for the crowded nature of a heavy nucleus, and found that it perfectly predicts what happens in real-world experiments.

In everyday terms: They figured out how to accurately predict the spray of sparks when a single bullet hits a dense brick wall, proving that the "fog" inside the wall changes the way the sparks fly, and their new map of that fog is the most accurate one we have. This helps scientists understand the fundamental rules of how matter is built and how it behaves under extreme pressure.

1. Problem Statement

The paper addresses the challenge of describing inclusive soft hadron production (specifically pions and kaons) in proton-lead (pA) collisions at the Large Hadron Collider (LHC) energies ( $\sqrt{s} = 5.02$ TeV).

The Gap: While proton-proton (pp) collisions are well-described by standard models, nuclear collisions involve complex modifications to parton distribution functions (nPDFs) due to nuclear effects (shadowing, anti-shadowing, EMC effects).
Specific Limitation: Existing Monte Carlo event generators (like hijing and ampt) show substantial deviations from LHC data in the low transverse momentum ( $p_T$ ) region. While epos lhc performs well, the authors aim to provide a theoretical framework based on Transverse Momentum Dependent (TMD) gluon densities that offers a self-consistent description of both HERA and LHC data without relying solely on phenomenological event generators.
Focus: The study focuses on the low- $x$ (Bjorken scaling variable) and low-scale ( $Q^2 \leq Q_s^2$ ) regime, where non-perturbative effects and gluon saturation dominate.

2. Methodology

The authors extend the Modified Quark-Gluon String Model (QGSM) to pA interactions, incorporating a specific nuclear modification of the gluon density.

A. Theoretical Framework: Modified QGSM

Standard QGSM: Describes pp interactions via the exchange of colorless strings formed between quarks/diquarks. It works well at large $x$ but fails at low $x$ .
Modification: The model is extended to include non-perturbative (soft) gluons that split into $q\bar{q}$ pairs. This is governed by a TMD (unintegrated) gluon density, $f_g(x, k_T^2)$ .
Cross-Section Formula: The inclusive hadron production cross-section is calculated as a convolution of the interaction amplitude and fragmentation functions:
$\frac{d^3\sigma}{d^2p_T dy} = |F_{pp}|^2 \otimes (D_{q/qq \to h} + D_{g \to h})$
At low $p_T$ and LHC energies, the gluon contribution dominates, allowing the authors to neglect quark/diquark contributions for the pA calculations.

B. Nuclear Modification via Geometrical Scaling

To apply the model to nuclei (Lead, A=208), the authors utilize the property of Geometrical Scaling, observed in HERA data:

Scaling Variable: The cross-sections depend on a single variable $\tau = Q^2 / Q_s^2(x)$ , where $Q_s(x)$ is the saturation scale.
Nuclear Saturation Scale: The saturation scale for a nucleus $A$ is modified relative to a proton:
$Q_{s,A}^2(x) = Q_s^2(x) \left( \frac{A \pi R_p^2}{\pi R_A^2} \right)^{1/\delta}$
This modification absorbs the nuclear radius dependence ( $R_A$ ) into the saturation scale.
TMD Gluon Density: The proton TMD gluon density (parameterized with specific functions of $x$ and $k_T$ ) is extended to nuclei by replacing $Q_s^2(x)$ with $Q_{s,A}^2(x)$ and applying a normalization factor $R_A^2/R_p^2$ .

C. Fragmentation and Fitting

The model uses fragmentation functions for gluons to pions ( $\pi$ ) and kaons ( $K$ ).
The slope parameters ( $B_g^\pi$ and $B_g^K$ ) in the fragmentation functions are treated as free parameters and fitted to LHC data to account for potential jet quenching (softening of spectra) in nuclear collisions.

3. Key Contributions

Extension of Modified QGSM to pA: Successfully adapted a model previously successful in pp collisions to proton-nucleus systems by integrating nuclear TMD gluon densities.
Application of Geometrical Scaling: Demonstrated that the geometrical scaling property, originally derived for deep inelastic scattering (DIS) on nuclei, can be effectively used to modify the TMD gluon density for hadron production in pA collisions.
Independent Parameterization: Unlike previous studies where kaon and pion fragmentation slopes were assumed identical, this work independently determined the slope parameters $B_g^\pi$ and $B_g^K$ for pA collisions.
Validation against Multiple Experiments: The model was tested against data from three major LHC collaborations: CMS, ATLAS, and ALICE.

4. Results

The authors compared their theoretical predictions with experimental data for $p_T < 1$ GeV at $\sqrt{s} = 5.02$ TeV.

Pion ( $\pi^\pm$ ) Production:
- The model achieved a best fit with a slope parameter $B_g^\pi = 9.1^{+0.2}_{-0.5}$ .
- This value is consistent with the pp collision value ( $8.7 \pm 0.34$ ) within uncertainties.
- The results show good agreement with CMS, ATLAS, and ALICE data.
- Deviation: A slight deviation was observed at $p_T \sim 1$ GeV (suggesting a need for QCD evolution effects) and at very low $p_T < 0.2$ GeV (suggesting a potential lack of quark/diquark contributions in the model).
Kaon ( $K$ ) Production:
- The model determined a distinct slope parameter $B_g^K = 4.8^{+0.4}_{-0.3}$ .
- This resulted in an excellent agreement with the available CMS data.
Comparison with Monte Carlo Generators:
- epos lhc: The authors' results are very close to epos lhc predictions and both agree well with data.
- hijing & ampt: These generators predicted steeper $p_T$ distributions and significantly lower average $p_T$ values compared to the data.
- Conclusion: The proposed TMD-based approach provides a better description of low- $p_T$ data than hijing and ampt, and is comparable to or slightly better than epos lhc in specific kinematic regions.

5. Significance

Validation of nTMD Gluon Density: The successful description of pA data validates the extension of the TMD gluon density to nuclei using geometrical scaling. It confirms that this approach is consistent with collider data at low $x$ and low hard scales ( $\sim 1$ GeV).
Non-Perturbative Dynamics: The study highlights the importance of non-perturbative gluon dynamics and saturation effects in understanding soft hadron production in nuclear environments.
Future Collider Physics: The results provide crucial constraints for nuclear parton distribution functions (nPDFs), which are essential for interpreting future data from the LHC, RHIC, and upcoming colliders (FCC-he, EicC, CEPC).
Model Improvement: By showing that a modified string model with TMD gluons outperforms standard Monte Carlo generators in the soft sector, the paper suggests a more robust theoretical tool for analyzing nuclear effects in high-energy collisions.