⚛️ nuclear theory

Medium modification of pseudoscalar meson structure: Distribution amplitudes and parton distribution functions

This study utilizes a light-front quark model incorporating quark-meson coupling effects to demonstrate that the internal structure of pseudoscalar mesons, specifically their distribution amplitudes and parton distribution functions, undergoes significant medium-induced modifications in nuclear matter, characterized by suppressed intermediate- $x$ distributions and enhanced Gegenbauer coefficients that qualitatively resemble the EMC effect.

Original authors: Muhammad Ridwan, Parada Tobel Paraduan Hutauruk, Ahmad Jafar Arifi, Kazuo Tsushima

Published 2026-10-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Ridwan, Parada Tobel Paraduan Hutauruk, Ahmad Jafar Arifi, Kazuo Tsushima

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

Inside the heart of every atom, protons and neutrons are not solid, indivisible spheres but complex, bustling cities made of even smaller particles called quarks. These quarks are bound together by a powerful force that acts like a stretchy rubber band, never letting them escape. While scientists have spent decades mapping out the internal structure of protons and neutrons, the particles that orbit them—mesons—have remained somewhat mysterious. Mesons are short-lived pairs of a quark and an antiquark, and understanding how they are built is crucial for decoding the fundamental rules that govern all matter. However, a complete picture has been missing because we cannot easily trap a meson to study it, and we know that the environment inside a dense atomic nucleus might change how these particles behave, much like how a person's stride might change when walking through deep water versus on dry land.

A team of researchers has now taken a significant step toward filling this gap by simulating how the internal structure of mesons changes when they are squeezed inside the dense matter of an atomic nucleus. Using a sophisticated computer model that treats quarks as moving particles on a specific timeline, the scientists examined four types of mesons: the pion, the kaon, and two heavier versions containing charm and bottom quarks. They compared how these particles look when floating freely in empty space against how they appear when surrounded by the intense pressure of nuclear matter. The study reveals that the environment inside a nucleus does not just slightly nudge these particles; it fundamentally reshapes their internal architecture, pushing their constituent parts into new arrangements that were previously difficult to predict.

The researchers focused on two key ways to describe a meson's interior. The first is a map of how the momentum is shared between the quark and the antiquark, known as a distribution amplitude. The second is a probability map showing where a quark is likely to be found carrying a specific fraction of the meson's total speed, known as a parton distribution function. By running their simulations at different levels of nuclear density, they discovered a striking pattern. In the vacuum of empty space, the momentum is shared relatively evenly, with a single peak in the middle of the distribution. However, as the nuclear density increases, this central peak flattens and shrinks, while the edges of the distribution grow taller. This means that inside a nucleus, the quarks are less likely to share the momentum equally and more likely to carry either very little or very much of the total speed, leaving the middle ground largely empty.

This reshaping has profound implications for our understanding of the "EMC effect," a long-standing puzzle in physics where the internal structure of protons and neutrons is observed to change when they are bound inside a nucleus. The researchers found that their simulated mesons showed a similar suppression of momentum in the middle range, suggesting that the same underlying mechanism might be at work for both mesons and nucleons. The study indicates that the change is driven by the modification of the quarks themselves as they interact with the surrounding nuclear fields. As the density of the nuclear matter rises, the effective mass of the light quarks decreases, which in turn alters how they move and bind together. This effect is most dramatic for the lighter pions and kaons, while the heavier mesons containing charm or bottom quarks show a much more muted response, as the heavy quarks are less influenced by the surrounding environment.

To ensure their findings were robust, the team tested their model using two different mathematical shapes to describe the quarks' movement: one that drops off sharply at high speeds and another that fades more gradually. While the exact numbers varied slightly between these two approaches, the overall story remained the same. The internal structure of the mesons consistently broadened and shifted as the nuclear density increased. The researchers also checked their results against data from lattice quantum chromodynamics, a different and highly respected method of calculation, and found that their predictions for the free-space particles matched well. This agreement gives them confidence that their model is capturing the real physics of the situation.

The study concludes that the medium inside a nucleus acts as a powerful lens, distorting the view of the meson's internal structure. The suppression of the middle range of momentum and the enhancement at the edges are not just minor adjustments but substantial modifications that redefine the particle. These findings provide a unified description of how hadrons behave in both empty space and dense matter, offering a potential key to unlocking the mystery of the EMC effect. By showing that the internal structure of mesons is fluid and responsive to their environment, the work suggests that the matter inside atomic nuclei is far more dynamic and interconnected than previously thought, with the behavior of individual quarks being inextricably linked to the density of the world around them.

Technical Summary: Medium Modification of Pseudoscalar Meson Structure

Problem Statement
While the internal structure of hadrons in free space (vacuum) is encoded in nonperturbative distributions such as distribution amplitudes (DAs) and parton distribution functions (PDFs), their behavior within a nuclear medium remains comparatively unexplored. Experimental evidence, most notably the European Muon Collaboration (EMC) effect, indicates that nucleon structure is significantly modified in nuclei, yet the origin of this phenomenon remains a long-standing problem. A fully consistent theoretical framework that simultaneously describes hadron properties and internal structure in both vacuum and nuclear matter, grounded in quark degrees of freedom, has been lacking. This work addresses the need to investigate the in-medium modifications of pseudoscalar mesons ( $\pi^\pm, K^\pm, D^\pm, B^\pm$ ), specifically focusing on how nuclear matter alters their DAs and PDFs.

Methodology
The authors employ a hybrid theoretical approach combining the Light-Front Quark Model (LFQM) with the Quark-Meson Coupling (QMC) model.

LFQM Framework: The meson state is treated as a bound state of constituent quarks and antiquarks. The study utilizes two distinct phenomenological wave function ansätze to assess model dependence: a Gaussian wave function and a power-law (PL) wave function. The radial wave functions are constructed to satisfy rotational invariance via a Jacobian factor derived from the $k_z \to x$ variable transformation.
QMC Input: In-medium quark properties are obtained from the QMC model, which describes nuclear matter using a relativistic mean-field approximation. In this model, light quarks ( $u, d$ ) interact with mean meson fields ( $\sigma, \omega$ ) within a MIT bag, leading to a density-dependent effective quark mass ( $m_q^* = m_q - V_{q\sigma}$ ). Heavy quarks ( $s, c, b$ ) are assumed to remain unaffected by the nuclear medium as they do not couple to the scalar $\sigma$ field.
Evolution: The calculated distributions are evolved to higher energy scales using the Efremov-Radyushkin-Brodsky-Lepage (ERBL) equation for DAs and the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations for PDFs. Calculations are performed at renormalization scales $\mu^2 = 4$ and $27 \text{ GeV}^2$ .
Observables: The study computes DAs, PDFs, Gegenbauer coefficients ( $a_n$ ), $\xi$ -moments, and Mellin moments ( $\langle x^n \rangle$ ) for various nuclear densities ( $\rho/\rho_0 = 0, 0.5, 1.0$ ).

Key Contributions and Results

Unified Description: The work provides a unified description of meson structure in vacuum and nuclear matter using quark degrees of freedom, bridging the gap between free-space properties and in-medium modifications.
Light-Front Wave Function (LFWF) Modification: As nuclear density increases, the LFWFs undergo a structural transition. In vacuum, they exhibit a single-peak structure; in nuclear matter, they develop a two-peak structure near the endpoints ( $x \to 0, 1$ ). This is driven by the reduction of the in-medium effective light-quark mass.
Suppression at Intermediate $x$ : The modification of the LFWFs leads to a substantial suppression of both DAs and PDFs in the intermediate- $x$ region ( $x \approx 0.2\text{--}0.8$ ) and an enhancement near the endpoints. This behavior is observed for both Gaussian and PL wave functions, though the PL function exhibits a slower falloff at high momentum.
Gegenbauer Coefficients:
- For the pion ( $\pi^\pm$ ), the second Gegenbauer coefficient ( $a_2^\pi$ ) increases significantly with nuclear density (e.g., from 0.096 in vacuum to 0.232 at normal nuclear density for the Gaussian case), indicating a broader DA.
- For the kaon ( $K^\pm$ ), the asymmetry in quark content leads to non-vanishing odd coefficients. The coefficients show moderate density dependence, with $a_1^K$ increasing slightly.
- For heavy-light mesons ( $D^\pm, B^\pm$ ), medium modifications are progressively suppressed due to the large mass of the heavy quark, which dominates the longitudinal momentum distribution.
PDF Ratios and the EMC Effect: The ratio of in-medium to vacuum PDFs ( $R_{\rho/0}$ ) for the pion exhibits a suppression at intermediate $x$ and enhancement at endpoints. This qualitative behavior resembles the EMC effect observed in nucleons. The study highlights that QCD evolution shifts the location and modifies the magnitude of this suppression but preserves the qualitative trend.
Validation: The calculated free-space DAs and PDFs for pions and kaons show reasonable agreement with available lattice QCD results and experimental data (e.g., Drell-Yan measurements), validating the framework.

Significance and Claims
The paper claims that the observed suppression of meson DAs and PDFs at intermediate $x$ in nuclear matter, driven by modified quark properties, provides a possible connection between in-medium hadron structure and the mechanisms underlying the EMC effect. By demonstrating that QCD evolution plays a crucial role in shaping these medium-modified distributions, the authors suggest that the interplay between quark dynamics and nuclear many-body effects is essential for understanding hadron structure in nuclei.

The study emphasizes that while direct experimental measurement of in-medium meson distributions remains challenging, the theoretical predictions presented here offer a benchmark for future facilities such as the Electron-Ion Collider (EIC), AMBER/COMPASS++, J-PARC, and EicC. The work underscores the necessity of self-consistent frameworks that treat hadron structure and nuclear matter on the same footing to deepen the understanding of Quantum Chromodynamics (QCD) in dense environments.

More like this