⚛️ phenomenology

Pion structure in magnetized nuclear medium at finite temperature

This paper investigates how magnetic fields, baryon density, temperature, and isospin asymmetry modify the pion's parton distribution functions and electromagnetic form factors in a nuclear medium by employing a unified framework that combines the chiral SU(3) quark mean field model with the light-cone quark model.

Original authors: Fathimath Shifa N K, Navpreet Kaur, Abi Jebarson A, Arvind Kumar, Suneel Dutt, Harleen Dahiya

Published 2026-09-30

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CC BY 4.0

Original authors: Fathimath Shifa N K, Navpreet Kaur, Abi Jebarson A, Arvind Kumar, Suneel Dutt, Harleen Dahiya

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

Deep inside the heart of matter, where protons and neutrons live, there exists a constant, invisible pressure. This pressure comes from the surrounding environment, whether it is the crushing density inside a neutron star or the searing heat of a collision between heavy atomic nuclei. Scientists have long known that when particles are trapped in such extreme conditions, their internal nature changes. They do not remain the same rigid objects they are in empty space; instead, their structure shifts, their mass can fluctuate, and the way they hold together is altered by the chaos around them. To understand the universe, from the first moments after the Big Bang to the cores of dead stars, physicists must map out exactly how these particles behave when they are not alone. One of the most fundamental particles in this story is the pion, a lightweight messenger that carries the force holding atomic nuclei together. By studying how pions change when squeezed by intense heat, dense matter, and powerful magnetic fields, researchers can uncover the hidden rules that govern the building blocks of our reality.

A team of researchers at the National Institute of Technology in Jalandhar, India, has taken a fresh look at this problem, focusing specifically on the pion in a "hot magnetized nuclear medium." In their study, they simulated a environment where the temperature is high, the density of matter is significant, and a magnetic field is present, mimicking conditions found in heavy-ion collisions or the interiors of exotic stars. They did not just observe the pion in isolation; they examined how its internal parts, known as quarks, distribute their momentum and how the particle's electric charge is arranged when subjected to these extreme forces. To do this, they built a unified theoretical model that connects two different ways of looking at matter: one that describes how quarks interact with the surrounding sea of particles, and another that describes how those quarks move and bind together inside the pion itself.

The researchers found that the behavior of the pion depends heavily on the density of the surrounding matter and whether they account for the "Dirac sea," a concept representing the vacuum of space filled with virtual particles that constantly pop in and out of existence. When they included the effects of this quantum vacuum, the results changed dramatically. In environments with low to moderate density, such as the vacuum of space or the surface of a nucleus, increasing the strength of the magnetic field made the pion's internal structure more compact. The quarks inside became heavier and held together more tightly, causing the particle to behave as if it were more solid and less spread out. This phenomenon, known as magnetic catalysis, suggests that the magnetic field strengthens the bonds within the pion, making it harder to break apart.

However, the story flips when the density of the surrounding matter becomes very high. In simulations where the density was twice that of normal nuclear matter, the effect of the magnetic field reversed. Instead of making the pion tighter, the magnetic field caused the internal structure to loosen and spread out. The quarks became lighter, and the bonds holding the pion together weakened. This behavior, called inverse magnetic catalysis, indicates that in the crushing pressure of dense matter, the magnetic field actually works against the binding forces, making the pion more fragile. The researchers also discovered that temperature plays a significant role in this balance. As the heat increased, the sensitivity of the pion to the magnetic field grew, particularly in dense environments, suggesting that heat and density compete to determine how the particle responds to magnetic forces.

Another key discovery was the role of the vacuum itself. When the researchers ignored the contributions of the Dirac sea, the magnetic field had almost no effect on the pion's structure; the particle seemed indifferent to the magnetic environment. But once they included the vacuum effects, the magnetic field became a powerful driver of change. This highlights that the empty space around the pion is not truly empty; it is an active participant that mediates how external forces like magnetism influence the particle. The study also looked at the difference between symmetric matter, where protons and neutrons are present in equal numbers, and asymmetric matter, where one type dominates. They found that while the overall trends remained similar, the specific details of how the pion changed were slightly different depending on this balance, showing that the composition of the surrounding medium fine-tunes the particle's response.

Ultimately, this work provides a detailed map of how the pion's internal landscape shifts under extreme conditions. The researchers showed that the pion is not a static object but a dynamic system that reshapes itself in response to its environment. In low-density, high-magnetic-field scenarios, it becomes more robust and compact. In high-density scenarios, it becomes more diffuse and fragile. These findings are crucial for interpreting data from future experiments at facilities like the Electron-Ion Collider and for understanding the physics of neutron stars, where such extreme conditions naturally occur. By revealing how the vacuum and density compete to alter the fundamental structure of matter, this study offers a clearer picture of the forces that shape the universe at its most extreme limits.

Technical Summary: Pion Structure in Magnetized Nuclear Medium at Finite Temperature

Problem Statement
The modification of hadron properties within a surrounding medium is a central topic in contemporary high-energy physics, linked to the partial restoration of chiral symmetry. While the in-medium modification of nucleon structure functions (the EMC effect) is experimentally established, similar modifications for mesons, particularly the pion, remain less explored despite their fundamental role as Goldstone bosons of spontaneous chiral symmetry breaking. The pion's internal structure, characterized by parton distribution functions (PDFs) and electromagnetic form factors (EMFFs), is expected to be altered by extreme conditions such as finite baryon density, temperature, isospin asymmetry, and strong magnetic fields. These conditions are relevant to relativistic heavy-ion collisions (RHIC, LHC) and astrophysical objects like neutron stars and magnetars. A systematic investigation of the pion structure in hot, magnetized nuclear matter, specifically accounting for vacuum polarization effects, has not been previously addressed within a unified theoretical framework.

Methodology
The authors employ a unified framework combining the Chiral SU(3) Quark Mean Field (CQMF) model and the Light-Cone Quark Model (LCQM).

CQMF Model: This model describes the medium at the constituent quark level, incorporating scalar ( $\sigma, \zeta, \delta$ ) and vector ( $\omega, \rho$ ) meson fields, along with a scalar dilaton field ( $\chi$ ) to account for scale invariance breaking. The model is extended to include external magnetic fields at the nucleon level via an effective Lagrangian that accounts for the anomalous magnetic moments (AMMs) of nucleons and Landau level quantization. Crucially, the study includes contributions from the Dirac Sea (DS) to the thermodynamic potential to account for vacuum polarization effects, which are often neglected in conventional approaches. The scalar fields are determined self-consistently by minimizing the thermodynamic potential $\Omega$ with respect to the fields, considering finite temperature ( $T$ ), baryon density ( $\rho_B$ ), isospin asymmetry ( $\eta$ ), and magnetic field strength ( $eB$ ).
LCQM: The in-medium effective masses of quarks ( $m^*_q$ ) derived from the CQMF model serve as inputs for the Light-Cone Quark Model. The pion is treated as a leading Fock state ( $|q\bar{q}\rangle$ ). The light-cone wave function (LCWF) is constructed using a spin component and a momentum-space component (Brodsky-Huang-Lepage framework) dependent on the effective quark mass.
Observables: Using the medium-modified LCWFs, the authors calculate the unpolarized parton distribution functions (PDFs) and the electromagnetic form factors (EMFFs) for the pion. The study systematically compares scenarios with and without Dirac Sea contributions to isolate the role of vacuum polarization.

Key Contributions

Unified Framework: The paper establishes a self-consistent approach linking medium-modified constituent quark masses (via CQMF) to relativistic light-front descriptions of hadrons (via LCQM) under the combined influence of magnetic fields, temperature, and density.
Inclusion of Dirac Sea (DS): A primary contribution is the systematic investigation of magnetized Dirac Sea contributions. The study demonstrates that neglecting DS effects leads to a near-constant response of scalar fields and quark masses to magnetic fields, whereas including DS reveals significant non-linear modifications.
Magnetic Catalysis (MC) and Inverse Magnetic Catalysis (IMC): The work provides a detailed mapping of MC and IMC phenomena within the pion structure. It identifies that at low densities ( $\rho_B \leq \rho_0$ ), the system exhibits Magnetic Catalysis (enhancement of scalar condensates and quark masses with $B$ ), while at high densities ( $\rho_B = 2\rho_0$ ), it exhibits Inverse Magnetic Catalysis (suppression of these quantities with increasing $B$ ).
Isospin Asymmetry and Temperature: The study quantifies the interplay between isospin asymmetry and temperature, noting that while density is the dominant factor, thermal effects can enhance sensitivity to isospin asymmetry, particularly in the context of EMFFs.

Results

Scalar Fields and Quark Masses:
- With DS included, the scalar fields $\sigma$ and $\zeta$ show a pronounced dependence on the magnetic field. At low densities, their magnitudes increase with $B$ (MC), while at $\rho_B = 2\rho_0$ , they decrease with $B$ (IMC).
- The effective quark masses ( $m^*_u, m^*_d$ ) follow the trend of the scalar fields. At low densities, $m^*_q$ increases with $B$ ; at high densities, $m^*_q$ decreases.
- Without DS contributions, the magnetic field induces negligible changes in scalar fields and quark masses, highlighting the critical role of vacuum polarization.
Light-Cone Wave Functions (LCWFs):
- At low densities ( $\rho_B = 0, \rho_0$ ), the LCWF peak around $x \sim 0.5$ enhances with increasing magnetic field, indicating stronger binding and more symmetric momentum sharing.
- At high density ( $\rho_B = 2\rho_0$ ), the peak amplitude is suppressed, and the distribution broadens, reflecting the partial restoration of chiral symmetry and weaker binding due to reduced quark masses.
Parton Distribution Functions (PDFs):
- PDFs mirror the LCWF behavior. At low densities, the amplitude at $x \sim 0.5$ increases with $B$ . At high density, the amplitude decreases as $B$ increases.
- Isospin asymmetry ( $\eta = 0.3$ ) introduces minor differences compared to symmetric matter, which become slightly more visible at higher temperatures.
Electromagnetic Form Factors (EMFFs):
- EMFFs increase with magnetic field strength at low densities, suggesting a more compact charge distribution.
- Conversely, at high density ( $\rho_B = 2\rho_0$ ), EMFFs decrease with increasing magnetic field.
- Temperature effects are generally less prominent than density effects but show a slight separation in distributions at saturation density.

Significance
The paper claims that the inclusion of Dirac Sea contributions is essential for a realistic description of pion structure in magnetized media. The results highlight that the pion's internal structure is highly sensitive to the surrounding medium's thermodynamic conditions. The observed transitions between Magnetic Catalysis and Inverse Magnetic Catalysis significantly alter the pion's binding and spatial extent. These findings provide valuable theoretical insights for interpreting future experimental data from facilities such as FAIR (PANDA, CBM), RHIC, and the Electron-Ion Collider (EIC), where medium-induced modifications of hadrons are expected to be probed. The framework offers a basis for understanding the behavior of strongly interacting matter under extreme conditions found in heavy-ion collisions and compact astrophysical objects.

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