⚛️ nuclear theory

Effects of in-medium $NN$ inelastic cross sections and the high-momentum tail of nucleon momentum distributions on pion production in heavy-ion collisions

This study demonstrates that simultaneously incorporating in-medium modifications of nucleon-nucleon inelastic cross sections and short-range-correlation-induced high-momentum tails into the UrQMD transport model successfully reproduces experimental pion yields and ratios in intermediate-energy heavy-ion collisions, thereby highlighting the critical role of both factors in accurately modeling pion production.

Original authors: Pengcheng Li, A. B. Larionov, Yongjia Wang, Gaochan Yong, Qingfeng Li

Published 2026-09-23

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CC0 1.0

Original authors: Pengcheng Li, A. B. Larionov, Yongjia Wang, Gaochan Yong, Qingfeng Li

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Deep inside the heart of an atom, protons and neutrons are packed together so tightly that they behave less like individual particles and more like a single, dense fluid. When scientists smash heavy atomic nuclei together at incredibly high speeds, they recreate conditions similar to those found in the core of a collapsing star or the birth of a neutron star. These collisions compress matter to densities far beyond what exists in normal atoms, creating a fleeting, super-dense environment where the rules of nuclear physics are pushed to their limits. To understand how this matter behaves under such extreme pressure, researchers look for specific signals left behind after the crash. One of the most valuable signals is the production of pions, which are short-lived particles created when the colliding protons and neutrons interact violently. By studying how many pions are made and what types they are, scientists can infer the properties of the nuclear matter that existed for a split second during the collision, including how the force of the nuclear symmetry energy behaves at high densities.

A recent study published by researchers from China and Russia takes a closer look at the factors that determine how many of these pions are created. The team used a sophisticated computer simulation to model collisions between gold nuclei at intermediate energies, a range where the collisions are energetic enough to create new particles but not so violent that they completely shatter the nuclei into quarks. The researchers focused on two specific ingredients that had often been treated separately in previous studies: how the surrounding dense environment changes the likelihood of particles colliding, and the fact that some protons and neutrons inside the nucleus are already moving much faster than expected before the crash even begins.

In the dense soup of a heavy-ion collision, the probability of two particles hitting each other and transforming into a new state is not the same as it is in empty space. The researchers incorporated calculations showing that in this crowded environment, the chance of a proton and neutron colliding to form a short-lived resonance, which then decays into a pion, is actually reduced. This "in-medium" effect acts like a brake, suppressing the total number of pions produced because the dense matter makes it harder for the necessary collisions to happen. However, the story has a second chapter. Inside the nucleus, a small fraction of nucleons are part of tight, short-range pairs that give them a "high-momentum tail," meaning they possess significantly more speed than their neighbors. When the collision starts, these fast-moving nucleons carry extra energy into the crash, effectively boosting the available energy for individual collisions and making it easier to create the heavy resonances needed to produce pions.

The team ran simulations of gold-on-gold collisions at an energy of 1.23 GeV per nucleon, testing different combinations of these effects. They found that if they only accounted for the dense environment slowing down the collisions, the model predicted far fewer pions than what is actually observed in experiments. Conversely, if they only included the fast-moving nucleons, the model produced too many. The breakthrough came when they combined both effects. The suppression caused by the dense environment and the enhancement caused by the fast-moving nucleons worked against each other, balancing out to produce a result that matched experimental data from the HADES and FOPI collaborations remarkably well. This balance was crucial not just for the total number of pions, but also for the ratio of negatively charged pions to positively charged ones, a key indicator of the neutron-to-proton composition in the dense matter.

The study also examined how these factors influenced the flow of particles as they moved away from the collision point. The researchers discovered that the collective motion of protons at lower speeds was not very sensitive to these specific effects, suggesting that this part of the signal is dominated by the overall pressure of the nuclear matter. However, the fast-moving nucleons did leave a clearer mark on protons with higher speeds, indicating that the initial high-momentum components of the nucleus play a significant role in the early, violent stages of the collision. The difference in the flow of negatively and positively charged pions, a potential probe for the symmetry energy, showed little sensitivity to these specific corrections, suggesting that this observable remains a robust tool for studying the nuclear force, provided the complex details of particle interactions are handled correctly in future models.

Ultimately, this work demonstrates that to accurately describe the birth of pions in heavy-ion collisions, scientists cannot rely on a single factor. They must consistently include both the way the dense medium suppresses particle production and the way the initial high-speed motion of nucleons enhances it. By weaving these two competing effects together, the researchers were able to reproduce the complex patterns seen in real-world experiments, offering a clearer window into the behavior of matter at the highest densities achievable in the laboratory. This consistency is a vital step toward using pion production as a precise tool to map the equation of state for nuclear matter, helping to solve long-standing puzzles about the structure of neutron stars and the dynamics of the universe's most extreme environments.

Technical Summary: Effects of In-Medium NN Inelastic Cross Sections and High-Momentum Tails on Pion Production

Problem Statement
Pion production in intermediate-energy heavy-ion collisions (HICs) serves as a critical probe for the nuclear equation of state (EoS) and isospin-dependent reaction dynamics, particularly near the production threshold. However, extracting reliable information on the high-density symmetry energy from pion observables remains challenging due to persistent discrepancies between transport-model calculations and experimental data. Current models often struggle to simultaneously describe pion yields, spectra, and charged-pion ratios ( $\pi^-/\pi^+$ ) across broad beam-energy ranges. This difficulty stems from the complex interplay of various factors, including mean-field potentials, resonance dynamics, and initial nuclear conditions. Specifically, two key ingredients have often been treated separately or omitted: the in-medium modification of nucleon-nucleon (NN) inelastic cross sections (specifically $NN \to N\Delta$ ) and the high-momentum tail (HMT) of the nucleon momentum distribution induced by short-range correlations (SRCs).

Methodology
To investigate the combined influence of these factors, the authors implemented two specific modifications into the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) transport model for simulating Au+Au collisions at intermediate energies (e.g., 1.23A GeV):

In-Medium NN Inelastic Cross Sections: The model incorporates in-medium corrections to the $NN \to N\Delta$ cross sections calculated previously within the relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) transport theory. These corrections account for density dependence and threshold shifts arising from scalar and vector self-energies of baryons in dense matter. The cross section is parametrized as a function of center-of-mass energy ( $\sqrt{s}$ ) and density ( $\rho$ ), where the energy-dependent part is fitted to calculations at isospin asymmetry $\alpha = 0$ and scaled by a simplified density-dependent factor, while isospin dependence is introduced via Clebsch–Gordan factors and threshold shifts.
High-Momentum Tail (HMT): The initialization of the nuclear system was modified to include an SRC-induced HMT. Instead of a standard Fermi-gas distribution, the nucleon momentum distribution was extended up to a cutoff of $\lambda k_F$ (where $\lambda = 2.2$ ), with 20% of nucleons populating the high-momentum region following a $k^{-4}$ tail. To maintain nuclear stability and avoid artificial changes to the EoS, the Skyrme-type density-dependent parameters were refitted after introducing the HMT.

Key Contributions and Results
The study systematically analyzed the separate and combined effects of these two mechanisms on the dynamical evolution and final observables of HICs:

Dynamical Evolution: The inclusion of in-medium reduced $NN \to N\Delta$ cross sections suppresses the production of $\Delta$ resonances in dense matter, leading to reduced stopping power and a lower peak baryon density (approx. 2.75 $\rho_0$ vs. 2.95 $\rho_0$ in the baseline). Conversely, the HMT enhances early-stage collisions by increasing the available energy in individual NN interactions, thereby promoting $NN \to N\Delta$ reactions and partially restoring the central density (to approx. 2.90 $\rho_0$ ).
Pion Multiplicity and Spectra: The in-medium cross section modification strongly suppresses total pion yields (reducing $\pi^-$ and $\pi^+$ multiplicities by ~33% and ~39%, respectively). The HMT counteracts this suppression by increasing the probability of subthreshold particle production, partially recovering the pion yields.
Comparison with Experiment:
- HADES Data (1.23A GeV): Calculations using only in-medium elastic corrections overestimate pion rapidity distributions. Adding the in-medium inelastic correction suppresses yields too much, falling below data. The simultaneous inclusion of the HMT restores the yields, bringing the calculated rapidity distributions, transverse momentum spectra, and centrality dependence into reasonable agreement with HADES measurements.
- FOPI Data: The combined model successfully reproduces the beam-energy dependence of charged-pion multiplicities and the $\pi^-/\pi^+$ ratio measured by FOPI.
Collective Flows: The proton elliptic flow ( $v_2$ ) at low transverse momentum ( $p_T$ ) shows little sensitivity to these effects, suggesting it is dominated by bulk compression dynamics. However, at higher $p_T$ , the HMT leads to larger $v_2$ values. The difference in directed flow between charged pions ( $\Delta v_1 = v_1(\pi^-) - v_1(\pi^+)$ ) shows weak sensitivity to these specific modifications, implying it may remain a robust observable for constraining the isovector mean field, provided charge-dependent in-medium cross sections are treated more completely in the future.

Significance and Claims
The paper claims that a consistent description of pion production near the threshold in heavy-ion collisions requires the simultaneous inclusion of both in-medium reaction cross sections and SRC-induced high-momentum components. The results demonstrate that these two effects act in opposite directions: the in-medium modification suppresses pion production by reducing resonance formation probability, while the HMT enhances it by increasing the available collision energy. Neglecting either factor leads to significant discrepancies with experimental data from HADES and FOPI. The authors conclude that future transport-model studies aiming to extract the high-density nuclear EoS must treat these ingredients consistently to avoid misinterpreting pion observables.

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