Particles

32 pages, 2308 KB

Open AccessArticle

by Francisco Barranco, Gregory Potel and Enrico Vigezzi

Particles 2026, 9(2), 66; https://doi.org/10.3390/particles9020066 - 22 Jun 2026

Viewed by 599

We present a review of theoretical studies of the structure and reactions of N = 7 and N = 8 nuclei in the vicinity of ¹¹Li, carried out within a framework based on Nuclear Field Theory. The coupling of valence nucleons to [...] Read more.

We present a review of theoretical studies of the structure and reactions of N = 7 and N = 8 nuclei in the vicinity of ¹¹Li, carried out within a framework based on Nuclear Field Theory. The coupling of valence nucleons to low-lying surface vibrations of the spherical core plays a central role, giving rise to self-energy processes that renormalize single-particle states and transfer form factors, as well as to an induced pairing interaction arising from the exchange of collective vibrations, which renormalizes the bare pairing force. Excitation spectra and cross sections for one- and two-nucleon transfer reactions populating states in the quasi-continuum are calculated and compared with available experimental data. Collective excitations in the particle-particle channel are investigated, with particular emphasis on Giant Pairing Vibrations and on their damping mechanisms arising from coupling to more complex configurations and continuum states. Comparisons with other theoretical schemes are also presented. We conclude that a coherent understanding of experimental data requires the detailed consideration of particle-vibration coupling effects. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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29 pages, 5546 KB

Open AccessReview

The Charging-Up Phenomenon in Gas Electron Multiplier Detector

by Sayak Chatterjee, Supriya Das and Saikat Biswas

Particles 2026, 9(2), 65; https://doi.org/10.3390/particles9020065 - 17 Jun 2026

Viewed by 1469

Abstract

Gas Electron Multiplier (GEM) detectors have become an indispensable component of modern tracking systems. The heart of a GEM detector is a thin polyimide foil (∼50 µm) clad with copper (∼5 µm) on both sides and containing an array of regularly spaced holes [...] Read more.

Gas Electron Multiplier (GEM) detectors have become an indispensable component of modern tracking systems. The heart of a GEM detector is a thin polyimide foil (∼50 µm) clad with copper (∼5 µm) on both sides and containing an array of regularly spaced holes (typically diameter of ∼70 µm and pitch of ∼140 µm) fabricated using photolithographic techniques. The presence of the dielectric substrate (polyimide) within the amplification region introduces a time dependent response when the detector is exposed to external irradiation, a phenomenon commonly referred to as the charging-up effect. This effect arises from the accumulation of charge on the insulating polyimide surfaces, leading to a gradual modification of the local electric field configuration inside the GEM holes and, consequently, a variation in the detector gain over time. The charging-up behaviour has been systematically investigated for triple GEM chamber prototypes using an Fe-55 radioactive source (5.9 keV X-rays) with an activity of ∼20 mCi. The characteristic charging-up time constant has been extracted, and its dependence on detector gain and irradiation rate has been examined. In addition, the uniformity of detector performance in terms of count rate, gain, and energy resolution has been studied both before and after the charging-up process. In this review article, the experimental setup, data acquisition methodology, and analysis procedures developed and carried out by our group are summarised. The key findings reported by other groups, relevant Monte Carlo simulation efforts, and future outlook for the charging-up investigation on GEM based detectors are also discussed in this article. The investigations and their outcomes reviewed here provide valuable insight into the charging-up dynamics of GEM detectors and their dependence on operational parameters. Full article

(This article belongs to the Section Experimental Physics and Instrumentation)

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20 pages, 15638 KB

Open AccessArticle

Quark Deconfinement Phase Transition in Hot Neutron-Star Matter: Effects of Neutrino Trapping

by Grigor Alaverdyan and Ani Alaverdyan

Particles 2026, 9(2), 64; https://doi.org/10.3390/particles9020064 - 8 Jun 2026

Cited by 1 | Viewed by 2024

Abstract

We study the effect of trapped neutrinos on the properties of the deconfinement phase transition from hot

β

-equilibrated, electrically neutral hadronic matter to quark matter. To describe the thermodynamic properties of hot hadronic matter, an extended relativistic mean field (RMF) theory is [...] Read more.

We study the effect of trapped neutrinos on the properties of the deconfinement phase transition from hot

β

-equilibrated, electrically neutral hadronic matter to quark matter. To describe the thermodynamic properties of hot hadronic matter, an extended relativistic mean field (RMF) theory is used, which also incorporates the isovector–Lorentz-scalar

δ

-meson effective field. The three-flavor quark phase is described within the framework of the local Nambu–Jona-Lasinio (NJL) model. It was assumed that the surface tension at the quark-hadron interface is so strong that the phase transition occurs according to Maxwell’s construction. The thermodynamic properties of the quark and hadronic phases were calculated for both neutrino-trapped and neutrino-transparent regimes at various temperatures ranging from 0 to 100 MeV and baryon number densities from 0 to 1.8

{fm}^{- 3}

. The impact of trapped neutrinos on the thermodynamic properties of the coexistence state has been investigated. It has been demonstrated that the baryon chemical potential in the coexistence state decreases as temperature increases. The critical endpoint parameters in the

T - n_{B}

plane of the phase diagram were obtained for the case of trapped neutrinos (74 MeV; 0.269

{fm}^{- 3}

) and for the case of the absence of neutrinos (75.6 MeV; 0.255

{fm}^{- 3}

). Full article

(This article belongs to the Special Issue Selected Papers from “The Modern Physics of Compact Stars and Relativistic Gravity 2025”)

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23 pages, 418 KB

Open AccessArticle

Ionization Potential Depression in Degenerate Plasmas and Pauli Blocking of Multi-Electron Ions

by Gerd Röpke

Particles 2026, 9(2), 63; https://doi.org/10.3390/particles9020063 - 1 Jun 2026

Viewed by 761

Abstract

The composition of partially ionised plasmas is investigated for densities and temperatures at which the free electrons are degenerate. Based on a quantum statistical approach, the effect of Pauli blocking is addressed. Specifically, one- and two-electron ions are studied. Approximations for deriving an [...] Read more.

The composition of partially ionised plasmas is investigated for densities and temperatures at which the free electrons are degenerate. Based on a quantum statistical approach, the effect of Pauli blocking is addressed. Specifically, one- and two-electron ions are studied. Approximations for deriving an in-medium Schrödinger equation for the ionization potential are indicated. New results regarding the degree of ionisation and the Mott effect are presented. Standard codes for plasma properties do not take Pauli blocking effects into account and are therefore unable to explain the experiments in the high-density regime, where the electrons are degenerate. Full article

(This article belongs to the Special Issue Particles and Plasmas in Strong Fields, Part 1)

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20 pages, 5604 KB

Open AccessArticle

Some Predictions on Behavior of the Nuclear Matter in Nuclear Collisions at FAIR-GSI Energies

by Nicolae George Țuțuraș, Alexandru Jipa, Dănuț Argintaru, Oana Ristea, Marius Călin, Cătălin Ristea, Ionel Lazanu, Tiberiu Eșanu, Adam Jinaru and Murat Ablai

Particles 2026, 9(2), 62; https://doi.org/10.3390/particles9020062 - 26 May 2026

Viewed by 914

Abstract

In order to describe the heavy ion collision dynamics which implies the formation of hot and very dense nuclear matter in the overlapping region of the two colliding nuclei, we used simulated numerical calculations for FAIR available energies. We used the anti- [...] Read more.

In order to describe the heavy ion collision dynamics which implies the formation of hot and very dense nuclear matter in the overlapping region of the two colliding nuclei, we used simulated numerical calculations for FAIR available energies. We used the anti-

k_{T}

jet-detection algorithm for highlighting the main directions of flow in Au-Au collisions at CBM energies, thus obtaining structures of the events depending on the number of flow streams. The jet-finder algorithm identified domains in the y-

ψ

(rapidity-azimuthal angle) plane, where the number of charged particles, momenta and energy take higher values compared to other areas of this plane. The anisotropic flow coefficients

v_{n}

may offer information about the pressure gradients in the early stages of the collision and about the high-density nuclear matter properties. The observation of

K^{+}

mesons in heavy ion collisions is of interest since

K^{+}

mesons, due to their strangeness, have a mean free path that exceeds the dimensions of the “fireball”. In the numerical calculations the interval of rapidity

0 < y < 0.8

is highlighted, for which the fluctuations of the antiparticle to particle ratio excitation functions show non-monotonic behavior in the 10–13 A GeV energy interval. Full article

(This article belongs to the Section Nuclear and Hadronic Theory)

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16 pages, 585 KB

Open AccessArticle

Isentropic Hybrid Stars in the Nambu–Jona-Lasinio Model: Effects of Neutrino Trapping

by Andrea Sabatucci and Armen Sedrakian

Particles 2026, 9(2), 61; https://doi.org/10.3390/particles9020061 - 26 May 2026

Viewed by 966

Abstract

Binary neutron star mergers and proto-neutron stars provide unique environments where dense matter is hot, lepton-rich, and potentially undergoes a transition from hadronic to deconfined quark matter. We investigate the thermodynamics and stellar properties of hybrid matter under such conditions. The hadronic phase [...] Read more.

Binary neutron star mergers and proto-neutron stars provide unique environments where dense matter is hot, lepton-rich, and potentially undergoes a transition from hadronic to deconfined quark matter. We investigate the thermodynamics and stellar properties of hybrid matter under such conditions. The hadronic phase is described within a covariant density functional framework, while the quark phase is modeled using a Nambu–Jona-Lasinio (NJL) model that includes repulsive vector interactions, the axial

U_{A} (1)

-breaking ’t Hooft determinant interaction, and two-flavor color-superconducting (2SC) pairing. The phase transition between hadronic and quark matter is constructed using a mixed-phase prescription that enforces baryon and lepton number conservation, allowing us to follow thermodynamic trajectories at fixed entropy per baryon and a fixed lepton fraction. We analyze the phase structure of dense matter at a finite temperature and study the composition of the hadronic, mixed, and quark phases in both neutrino-trapped and neutrino-free regimes. Our results show that neutrino trapping significantly modifies the particle composition and shifts the onset of deconfinement to higher densities. The mixed phase exhibits a density-dependent pressure due to the presence of multiple conserved charges. Using the resulting equations of state, we compute static stellar configurations and examine the influence of the temperature and lepton content on the mass–radius relation in hybrid stars. Hot, neutrino-rich configurations are found to have larger radii and slightly higher maximum masses than their cold counterparts. As the star cools and deleptonizes, its radius contracts at an approximately constant baryonic mass, potentially triggering changes in the internal phase structure. These results highlight the roles of color superconductivity, lepton trapping, and thermal effects in shaping the structure and evolution of hybrid stars in transient astrophysical environments. Full article

(This article belongs to the Special Issue Selected Papers from “The Modern Physics of Compact Stars and Relativistic Gravity 2025”)

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11 pages, 2048 KB

Open AccessReview

One-Neutron Halo Nuclei in the Mass Region of 15 ≲ A ≲ 50 from Microscopic Structure to Reaction Observables

by Shisheng Zhang, Jialin An, Qi Lu and Kaiyuan Zhang

Particles 2026, 9(2), 60; https://doi.org/10.3390/particles9020060 - 20 May 2026

Viewed by 915

Abstract

We briefly review our recent theoretical progress on one-neutron (

1 n

) halo nuclei in the mass region of 15 ≲ A ≲ 50 from microscopic structure to reaction observables, by combining the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) and its [...] Read more.

We briefly review our recent theoretical progress on one-neutron (

1 n

) halo nuclei in the mass region of 15 ≲ A ≲ 50 from microscopic structure to reaction observables, by combining the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) and its triaxial extension (TRHBc) with the Glauber reaction model. In such an effective scheme, we first succeed in reproducing the enhanced reaction cross sections and narrow longitudinal momentum distributions of both the heaviest

1 n

p-wave halo nucleus ³⁷Mg and the heavier ³¹Ne on a carbon target, which are loosely bound and well-deformed systems with dominant p-wave configurations of the valence neutron. To test its capability for the lighter halo cases, we select neutron-rich carbon isotopes as examples. It turns out that the DRHBc + Glauber approaches are still valid for the s-wave halo in ¹⁵C, while a better description of the ground state for ¹⁹C requires the inclusion of exchange terms and tensor forces via the deformed relativistic Hartree–Fock–Bogoliubov (D-RHFB) model.Finally, these approaches are applied to search for heavier

1 n

halo candidates. It was suggested that ^40,42Al are promising candidates as

1 n

p-wave triaxial halo nuclei and ^43,45Si as

1 n

p-wave axial halo nuclei with prominent shape decoupling from the oblate core. Our studies cast a new light on future experimental measurements for new halo nuclei. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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59 pages, 3209 KB

Open AccessArticle

Gauge Symmetry Beyond Perturbation Theory: BRST and Anti-BRST Structure, Background Fields, and Infrared Dynamics of Yang–Mills Theory

by Daniele Binosi

Particles 2026, 9(2), 59; https://doi.org/10.3390/particles9020059 - 19 May 2026

Cited by 1 | Viewed by 620

Abstract

We present a pedagogical and self-contained account of the functional formulation of non-Abelian gauge theories, aimed at the construction of a process-independent effective charge for Yang–Mills theory. Starting from the path integral quantization of gauge fields, we review gauge fixing and the emergence [...] Read more.

We present a pedagogical and self-contained account of the functional formulation of non-Abelian gauge theories, aimed at the construction of a process-independent effective charge for Yang–Mills theory. Starting from the path integral quantization of gauge fields, we review gauge fixing and the emergence of Faddeev–Popov ghosts, illustrating how gauge invariance is preserved at the quantum level through Becchi–Rouet–Stora–Tyutin (BRST) symmetry. We then develop the BRST and anti-BRST formalisms and show how their simultaneous implementation leads to powerful functional identities that severely constrain the ghost and gluon sectors. Background field gauges are introduced as a natural framework in which these symmetries manifest themselves through Abelian-like Ward identities, allowing for a transparent separation between quantum and background degrees of freedom. This structure makes it possible to define renormalization group-invariant combinations of Green functions that generalize the QED effective charge to the non-Abelian case. The resulting effective charge is shown to be unique, gauge-invariant, and process-independent, providing a unified description of the theory from the ultraviolet down to the infrared. The interplay between functional identities, Dyson–Schwinger equations, and lattice results is discussed in detail, highlighting how dynamical mass generation and infrared saturation naturally emerge within this framework. Full article

(This article belongs to the Special Issue Strong QCD and Hadron Structure)

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44 pages, 12613 KB

Open AccessArticle

Quantum Theory of a Single Photon in an Arbitrary Medium

by Ashot S. Gevorkyan, Aleksandr V. Bogdanov and Vladimir V. Mareev

Particles 2026, 9(2), 58; https://doi.org/10.3390/particles9020058 - 18 May 2026

Viewed by 1093

Abstract

The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group

S U (2) \otimes U (1)

using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial [...] Read more.

The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group

S U (2) \otimes U (1)

using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates. Full article

(This article belongs to the Special Issue Selected Papers from “The Modern Physics of Compact Stars and Relativistic Gravity 2025”)

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26 pages, 11942 KB

Open AccessArticle

Halo Nuclei from Ab Initio Nuclear Theory

by Petr Navrátil, Sofia Quaglioni, Guillaume Hupin, Michael Gennari and Kostas Kravvaris

Particles 2026, 9(2), 57; https://doi.org/10.3390/particles9020057 - 14 May 2026

Cited by 5 | Viewed by 1055

Abstract

A realistic description of halo nuclei, characterized by low-lying breakup thresholds, requires a proper treatment of continuum effects. We have developed an ab initio approach, the No-Core Shell Model with Continuum (NCSMC), capable of describing both bound and unbound states in light nuclei [...] Read more.

A realistic description of halo nuclei, characterized by low-lying breakup thresholds, requires a proper treatment of continuum effects. We have developed an ab initio approach, the No-Core Shell Model with Continuum (NCSMC), capable of describing both bound and unbound states in light nuclei in a unified way. With chiral two- and three-nucleon interactions as the only input, we can predict the structure and dynamics of halo and other light nuclei and, by comparing to available experimental data, test the quality of chiral nuclear forces. We review NCSMC calculations of weakly bound states and resonances of the exotic halo nuclei ⁶He, ⁸B, ¹¹Be, and ¹⁵C. For the latter, we discuss its production in the capture reaction ¹⁴C(n,

γ

)¹⁵C. We highlight the challenges of a description of ⁶He as a Borromean n-n-⁴He system. Finally, we present our calculations of excited states in ¹⁰Be exhibiting a one-neutron halo structure and a large scale No-Core Shell Model investigation of ¹¹Li as a precursor of a full n-n-⁹Li NCSMC study. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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9 pages, 365 KB

Open AccessArticle

Generalized Beth–Uhlenbeck Approach to the (2+1)-Dimensional Gross–Neveu Model

by Biplab Mahato and David Blaschke

Particles 2026, 9(2), 56; https://doi.org/10.3390/particles9020056 - 14 May 2026

Cited by 1 | Viewed by 621

Abstract

We study the thermodynamics of the (2+1)-dimensional Gross–Neveu model inspired from graphene. We focus on the entropy density of the Gaussian fluctuation beyond the mean field. The full in-medium, momentum-dependent evaluation reveals that the fluctuations give a substantial contribution, even comparable to that [...] Read more.

We study the thermodynamics of the (2+1)-dimensional Gross–Neveu model inspired from graphene. We focus on the entropy density of the Gaussian fluctuation beyond the mean field. The full in-medium, momentum-dependent evaluation reveals that the fluctuations give a substantial contribution, even comparable to that of the mean field. We argue that the back-reaction from the fluctuations to the mean field should be included, which reduces the contribution mainly coming from the Landau-damping region. To treat this self-consistently, we use the generalized version of the Beth–Uhlenbeck approach for the entropy density. Compared with the standard Beth–Uhlenbeck formulation, the generalized version suppresses the low-energy contributions while preserving the bound-state effects. To illustrate this, we consider the respective contributions of the bound excitons and unbound fermions to the total entropy. This shows a sharper crossover between the degrees of freedom compared to the standard Beth–Uhlenbeck approach. This behavior is consistent with Mott-transition physics in two-dimensional materials. Full article

(This article belongs to the Special Issue Particles and Plasmas in Strong Fields, Part 1)

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12 pages, 1160 KB

Open AccessArticle

A Meson Binding State Model in Two-Dimensional Space

by Yasushi Muraki and Shoichi Shibata

Particles 2026, 9(2), 55; https://doi.org/10.3390/particles9020055 - 13 May 2026

Viewed by 624

Abstract

In this paper, we first show that the four masses of the Upsilon binding states—namely,

Y (1 S)

through

Y (4 S)

—composed of bottom quarks and anti-bottom quarks follow logarithmic spacing. The correlation coefficient R between the experimental [...] Read more.

In this paper, we first show that the four masses of the Upsilon binding states—namely,

Y (1 S)

through

Y (4 S)

—composed of bottom quarks and anti-bottom quarks follow logarithmic spacing. The correlation coefficient R between the experimental values and the straight line is 0.99997, indicating an extremely good fit. When the three peaks—

Y (5 S)

,

Y (6 S)

, and

Y (7 S)

, considered higher Upsilon binding states, as indicated in the recent Belle experiment—are added and plotted on the line of logarithmic spacing, the correlation coefficient R between the straight line and the experimental values for these seven “binding state levels” is 0.9998. If this line is extended to higher masses, an eighth peak in the cross-section is expected at a mass of (11,119 ± 10) MeV. In other words, it is predicted that the peak in the cross-section created by

Y (8 S)

will be found in the Belle experiment in the future. Next, we consider why meson binding states are represented by a line with logarithmic spacing. An example is the electric field generated by a charge on a two-dimensional plane; the electric field created by a charge placed on the plane is expressed as

F \sim 1 / r

, and the potential energy is expressed as

V \sim log (r)

. Therefore, we solve the two-dimensional Schrödinger equation numerically under

F \sim 1 / r

and compare the results with the three-dimensional solution. We find that differences appear in the energy levels of the

J = 0

η_{c}

meson. Specifically, it is shown that the mass of the

η_{c}

meson is closer to the value obtained by solving the two-dimensional Schrödinger equation. Based on this

η_{c}

meson series, we predict the existence of a new

η_{c}

meson with a mass of 3955 MeV. Full article

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15 pages, 2968 KB

Open AccessArticle

Effects of Isovector Spin–Orbit Interaction on the Charge-Weak Form Factor Difference in ⁴⁸Ca, ²⁰⁸Pb, ⁹⁰Zr and ⁶²Ni

by Tong-Gang Yue, Zhen Zhang and Lie-Wen Chen

Particles 2026, 9(2), 54; https://doi.org/10.3390/particles9020054 - 12 May 2026

Viewed by 570

Abstract

The nucleon spin–orbit interaction is a cornerstone of nuclear structure theory, yet its isospin dependence remains insufficiently constrained within modern nuclear energy density functional (EDF) theory. It was recently shown that, within the framework of extended Skyrme EDFs, the charge-weak form factor difference [...] Read more.

The nucleon spin–orbit interaction is a cornerstone of nuclear structure theory, yet its isospin dependence remains insufficiently constrained within modern nuclear energy density functional (EDF) theory. It was recently shown that, within the framework of extended Skyrme EDFs, the charge-weak form factor difference

Δ F_{CW}

in ⁴⁸Ca exhibits remarkable sensitivity to the effective isovector spin–orbit (IVSO) interaction, whereas

Δ F_{CW}

in ²⁰⁸Pb is much less sensitive to this channel. Extending this analysis to other nuclei, we find that ⁹⁰Zr, with its ten spin–orbit unpaired

1 g_{9 / 2}

neutrons, displays a

Δ F_{CW}

sensitivity to the IVSO strength similar to that of ⁴⁸Ca, arising from modifications to the central mean-field potential rather than the one-body spin–orbit potential. In contrast, ⁶²Ni, like ²⁰⁸Pb, remains largely insensitive to the IVSO interaction. This structure-driven distinction suggests an experimental strategy: future parity-violating electron scattering measurements, e.g., the MREX experiment at the MESA facility, on ⁴⁸Ca and ⁹⁰Zr would help constrain the effective IVSO strength, while measurements on ²⁰⁸Pb and ⁶²Ni can provide a cleaner probe of the density dependence of the symmetry energy with reduced IVSO sensitivity. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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12 pages, 1496 KB

Open AccessArticle

The Absolute Stability and Mass Constraints of Strange Stars in the MIT Bag Model

by Hasmik Shahinyan, Tigran Sargsyan and Arsen Babajanyan

Particles 2026, 9(2), 53; https://doi.org/10.3390/particles9020053 - 12 May 2026

Viewed by 535

Abstract

The primary objective of this study is a comprehensive investigation of the self-bound properties of strange quark matter (SQM), which is hypothesized to represent the absolute ground state of superdense strongly interacting matter. An analysis is performed within the framework of the MIT [...] Read more.

The primary objective of this study is a comprehensive investigation of the self-bound properties of strange quark matter (SQM), which is hypothesized to represent the absolute ground state of superdense strongly interacting matter. An analysis is performed within the framework of the MIT bag model, including first-order perturbative QCD corrections and the finite strange quark mass. By systematically varying the vacuum pressure (bag constant,

B

) and the strong coupling constant (

α_{c}

) over a broad parameter space, while assuming a finite strange quark mass (

m_{s} \neq 0

), we explicitly compute the thermodynamic characteristics of the system including pressure, energy density, baryon number density, and the chemical potentials of quarks and charge-neutralizing electrons under conditions of

β

-equilibrium and global charge neutrality. Particular emphasis is placed on determining the minimum energy per baryon, which serves as the criterion for absolute stability. For parameter sets satisfying the self-binding condition, the integral properties of strange stars are derived via the numerical integration of the Tolman–Oppenheimer–Volkoff equations. The resulting mass–radius and mass–central density relations are analyzed, yielding the maximum stellar masses in the range

(1.9 - 2.4) M_{⊙}

. This study identifies the regions in the space of phenomenological parameters that allow for pure self-bound strange stars and demonstrates the sensitivity of stability and stellar properties to the underlying bag model parameters. Full article

(This article belongs to the Special Issue Selected Papers from “The Modern Physics of Compact Stars and Relativistic Gravity 2025”)

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Graphical abstract

24 pages, 2939 KB

Open AccessArticle

Getting a Handle on Correlation Functions

by Gernot Eichmann

Particles 2026, 9(2), 52; https://doi.org/10.3390/particles9020052 - 12 May 2026

Viewed by 619

Abstract

The central objects in a quantum field theory are its n-point correlation functions and matrix elements. Their structure is determined by Lorentz invariance and leads to tensor decompositions, the Lorentz-invariant coefficient functions of which encode the physics of the process. For growing [...] Read more.

The central objects in a quantum field theory are its n-point correlation functions and matrix elements. Their structure is determined by Lorentz invariance and leads to tensor decompositions, the Lorentz-invariant coefficient functions of which encode the physics of the process. For growing n, the complexity of these objects may increase considerably and make it challenging to deal with them. Here, we give a pedagogical introduction to the topic and provide some tools to manage this complexity, and we will show how symmetries can be used as organizing principles. Full article

(This article belongs to the Special Issue Strong QCD and Hadron Structure)

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18 pages, 3587 KB

Open AccessArticle

Controlling Proton Acceleration with Advanced Gold Nanoantennas in a Kinetic Plasma Environment

by Konstantin Zsukovszki and Istvan Papp

Particles 2026, 9(2), 51; https://doi.org/10.3390/particles9020051 - 11 May 2026

Viewed by 714

Abstract

Metallic nanoantennas are promising structures for enhancing energy transfer in high-intensity laser–matter interactions, especially in nanoplasmonic-assisted fusion. Under ultrashort laser pulses, they generate strong localized fields, modify ionization dynamics, and significantly affect charge acceleration in dense media. In this work, we present a [...] Read more.

Metallic nanoantennas are promising structures for enhancing energy transfer in high-intensity laser–matter interactions, especially in nanoplasmonic-assisted fusion. Under ultrashort laser pulses, they generate strong localized fields, modify ionization dynamics, and significantly affect charge acceleration in dense media. In this work, we present a comprehensive particle-in-cell (PIC) study of gold nanoantennas of various geometries—dipoles, planar crosses, three-dimensional crosses, and Yagi-inspired planar structures—irradiated by near-infrared femtosecond pulses at intensities at a range of ~4 × 10¹⁷–4 × 10¹⁸ W/cm². The antenna structures are embedded in a dense hydrogen-rich medium, allowing us to follow electron emission, gold ionization, and proton acceleration self-consistently. Crossed and Yagi-type geometries exhibit more robust resonant behavior than dipoles, with higher field localization and greatly reduced sensitivity to incident polarization. The proton energies increase to ~200 keV at 4 × 10¹⁷ W/cm², and saturate around ~300 keV at a higher intensity >~4 × 10¹⁸ W/cm², dependent on the geometry. This happens largely due to a rapid loss of conduction electrons from the gold structures. Our results highlight Yagi-based and cross-based nanoantennas as promising resonant dopes for laser-driven energy coupling and point toward optimized multi-arm architectures for future nanofusion-target engineering applications. Full article

(This article belongs to the Special Issue Particles and Plasmas in Strong Fields, Part 1)

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17 pages, 15566 KB

Open AccessReview

A Halo: Triggering a New Era of Nuclear Correlations

by Hiroyuki Sagawa, Xiao Lu and Shan-Gui Zhou

Particles 2026, 9(2), 50; https://doi.org/10.3390/particles9020050 - 9 May 2026

Viewed by 556

Abstract

In this contribution to the Halo-40 Proceedings, we discuss two topics regarding halo phenomena. The first is the pairing anti-halo effect on the neutron radius of halo nuclei and the restoration of the halo due to the cancellation between the anti-halo effect and [...] Read more.

In this contribution to the Halo-40 Proceedings, we discuss two topics regarding halo phenomena. The first is the pairing anti-halo effect on the neutron radius of halo nuclei and the restoration of the halo due to the cancellation between the anti-halo effect and the continuum coupling; the second is the soft dipole excitation of deformed halo nuclei. We demonstrate the importance of Hartree–Fock–Bogoliubov and relativistic Hartree–Bogoliubov theory in a continuum for properly taking into account the halo nature of extended wave functions in the calculations of neutron radii as well as the soft dipole excitations of halo nuclei. It is shown that the anti-halo effect is very sensitive to the continuum coupling induced by Bogoliubov-type quasi-particles, which largely cancels the anti-halo effect on the neutron radius. The soft dipole excitations of deformed halo nuclei ³¹Ne and ³⁷Mg are discussed within the deformed Woods–Saxon model. We point out that the sharp peak just above the threshold in the dipole response is created by the halo effect, and its strength can be used to identify the magnitude of deformation and the halo configuration in the Nilsson-level scheme. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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16 pages, 1822 KB

Open AccessArticle

Beaming of Polarized Radiation in Subcritical X-Ray Pulsars

by Ivan D. Markozov, Alexander Y. Potekhin, Alexander D. Kaminker and Alexander A. Mushtukov

Particles 2026, 9(2), 49; https://doi.org/10.3390/particles9020049 - 5 May 2026

Cited by 1 | Viewed by 565

Abstract

Radiation of X-ray pulsars is powered by accretion on the neutron star surface from a binary companion under the influence of a strong magnetic field. We study the beaming of this radiation in the case of subcritical X-ray pulsars, where it is formed [...] Read more.

Radiation of X-ray pulsars is powered by accretion on the neutron star surface from a binary companion under the influence of a strong magnetic field. We study the beaming of this radiation in the case of subcritical X-ray pulsars, where it is formed in the accretion channel close to the neutron star surface. We solve equations of the hydrodynamics and radiative transfer of two coupled polarization modes in the accretion channel numerically, taking into account resonant Compton scattering and vacuum polarization. The beaming patterns are obtained for different accretion rates, photon energies, and polarizations, as well as for different models of the neutron star surface radiation. The calculated beaming patterns are converted into light curves for both the intensity and polarization, taking into account the effects of General Relativity. These beaming patterns and light curves are found to be strongly affected by the resonant Compton scattering for photon energies comparable with the electron cyclotron energy. In particular, the angular redistribution of radiation near the cyclotron resonance may reduce the light-curve modulation amplitude, which is consistent with observational indications of a suppressed pulsed fraction at these energies. Full article

(This article belongs to the Special Issue Selected Papers from “The Modern Physics of Compact Stars and Relativistic Gravity 2025”)

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22 pages, 3923 KB

Open AccessReview

Nuclear Exotic Structures, Exotic Decays and Near-Barrier Reactions

by Cheng Yin, Chengjian Lin, Lei Yang, Feng Yang, Huiming Jia, Nanru Ma, Peiwei Wen and Tianpeng Luo

Particles 2026, 9(2), 48; https://doi.org/10.3390/particles9020048 - 1 May 2026

Viewed by 674

Abstract

The reaction dynamics of weakly-bound nuclear systems at near-barrier energies is a compelling topic in nuclear physics. This review summarizes decades of experimental work by the Nuclear Reaction Group at the China Institute of Atomic Energy. Using transfer reactions with the distorted wave [...] Read more.

The reaction dynamics of weakly-bound nuclear systems at near-barrier energies is a compelling topic in nuclear physics. This review summarizes decades of experimental work by the Nuclear Reaction Group at the China Institute of Atomic Energy. Using transfer reactions with the distorted wave born approximation and asymptotic normalization coefficient analyses, we confirm the first excited neutron halo (¹³C) on the

β

-stability line and identified new halo states in ¹²B. Total reaction cross-section measurements revealed proton halo nuclei

{}^{27}P

and

{}^{29}S

, with core enlargement observed in

{}^{27}P

and

{}^{28}P

. We established conditions for halo formation and delineated the proton halo existence region. In two-proton emission studies, we observed

{}^{2}{He}

cluster emission from highly excited

{}^{17, 18}{Ne}

and

{}^{28, 29}S

, with

{}^{29}S

being the second such case internationally. In

β

-delayed decay, we discovered

β

2p emission in

{}^{22}{Si}

and determined its mass, observing isospin-symmetry breaking in

{}^{20}{Mg}

,

{}^{22}{Si}

, and

{}^{27}S

. Decay schemes for

{}^{27}S

and

{}^{26}P

addressed the

{}^{26}{Al}

abundance problem. For nuclear interactions, we investigated the

{}^{6}{He}

optical potential, finding the dispersion relation inapplicable for

{}^{6}{He}

+

{}^{209}{Bi}

, and developed notch and Bayesian methods to constrain uncertainties. For unstable nuclei, the proton drip-line systems ⁸B and ¹⁷F have been intensively studied via complete kinematics measurements of the ⁸B + ¹²⁰Sn and ¹⁷F + ⁵⁸Ni reactions, respectively. The results show that elastic breakup dominates for proton-halo

{}^{8}B

, while inelastic breakup prevails for

{}^{17}F

, with proton-rich nuclei exhibiting lower breakup probabilities than neutron-halo nuclei due to Coulomb effects. Fusion studies revealed sub-barrier enhancement in

{}^{17}F

+

{}^{58}{Ni}

from continuum couplings. We propose direct fusion–evaporation measurements with deflection systems integrated with breakup detection to disentangle complete and incomplete fusion channels. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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13 pages, 1547 KB

Open AccessArticle

Lifetime Measurements—A Powerful Tool to Study Nuclear Structure

by Dimitar Tonev, Galina D. Dimitrova, Anguel Demerdjiev, Giovanni De Gregorio, Giacomo de Angelis, Elena Geleva, Nikolay Goutev, Nikolay N. Petrov, Ivaylo Pantaleev and Lilianna Panteleev-Simeonova

Particles 2026, 9(2), 47; https://doi.org/10.3390/particles9020047 - 1 May 2026

Viewed by 1029

Abstract

Advanced Doppler-shift methods for the calculation of the γ-ray lineshape registered in recoil-distance Doppler-shift and Doppler-shift attenuation methods are presented, emphasizing the case using a gate set on the shifted part of a direct feeding transition. For the precise description of the γ-ray [...] Read more.

Advanced Doppler-shift methods for the calculation of the γ-ray lineshape registered in recoil-distance Doppler-shift and Doppler-shift attenuation methods are presented, emphasizing the case using a gate set on the shifted part of a direct feeding transition. For the precise description of the γ-ray lineshape, the process of evaporation of light particles from the compound nucleus has to be taken into account in the case of heavy ion-induced fusion-evaporation reactions. In addition, the impact of different approaches for calculating stopping powers is investigated in the process of the lifetime determinations. In the RDDS experiments, the γ-emission during the slowing down in the stopper is discussed in detail. Applications of the new procedures are demonstrated in two experiments: the first one is a plunger experiment performed in order to check for chirality in the ¹³⁴Pr nucleus and the second one is a DSAM experiment conducted to test the isospin symmetry in ³¹P and ³¹S mirror nuclei. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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16 pages, 644 KB

Open AccessReview

Radiative Decays of Hadronic Molecules: From Confusion to Inspiration

by Feng-Kun Guo, Christoph Hanhart and Alexey Nefediev

Particles 2026, 9(2), 46; https://doi.org/10.3390/particles9020046 - 28 Apr 2026

Viewed by 691

Abstract

Radiative decays of hadronic states provide an essential source of information that can facilitate deciphering their nature and properties. However, a lot of confusion concerning radiative decays of hadronic molecules and their interpretation can be found in the literature. In this paper, we [...] Read more.

Radiative decays of hadronic states provide an essential source of information that can facilitate deciphering their nature and properties. However, a lot of confusion concerning radiative decays of hadronic molecules and their interpretation can be found in the literature. In this paper, we briefly review several types of such decays and pinpoint similarities and essential differences between them. In particular, we emphasise the crucial role played by the hierarchy of the scales relevant to the studied system and the resulting necessity of employing an approach that considers them appropriately. We illustrate the situation with several instructive examples. Full article

(This article belongs to the Special Issue Strong QCD and Hadron Structure)

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43 pages, 1772 KB

Open AccessReview

Symmetry- Preserving Contact Interaction Approaches: An Overview of Meson and Diquark Form Factors

by Laura Xiomara Gutiérrez-Guerrero and Roger José Hernández-Pinto

Particles 2026, 9(2), 45; https://doi.org/10.3390/particles9020045 - 24 Apr 2026

Cited by 2 | Viewed by 800

Abstract

We present an updated overview of the symmetry-preserving contact interaction model in hadronic physics, which was developed a little over a decade ago to describe the mass spectrum and internal structure of mesons and diquarks composed of light and heavy quarks. Over the [...] Read more.

We present an updated overview of the symmetry-preserving contact interaction model in hadronic physics, which was developed a little over a decade ago to describe the mass spectrum and internal structure of mesons and diquarks composed of light and heavy quarks. Over the years, the contact interaction model has evolved into a framework capable of treating both ground and excited states, providing a simple yet consistent approach to nonperturbative QCD. In this review, we examine the mass spectrum and elastic form factors of forty mesons with different spins and parities, together with their corresponding diquark partners. Importantly, we update the comparison of contact interaction predictions using recent results from the literature, offering a fresh perspective on the model’s performance, strengths, and limitations. The analysis presented here refines previous conclusions and supports the contact interaction model as a practical tool for hadron structure studies, with potential applications to baryons and multiquark states. We also present comparisons with other theoretical models and approaches, including lattice quantum chromodynamics, and comment on future prospects in view of ongoing and planned experimental programs regarding hadron structure. In particular, forthcoming measurements at FAIR together with future studies at Jefferson Lab and the Electron Ion Collider are expected to provide key insights into hadron structure, with FAIR offering indirect constraints via hadron spectroscopy, hadronic interactions, and in-medium properties; high-precision data on meson structure and form factors from Jefferson Lab and the Electron Ion Collider will provide valuable benchmarks with which to confront predictions based on the contact interaction model. Full article

(This article belongs to the Special Issue Strong QCD and Hadron Structure)

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17 pages, 450 KB

Open AccessReview

A Solution of the Scalar Nonet Mass Puzzle

by Mihail Chizhov, Emanuil Chizhov, Daniela Kirilova and Momchil Naydenov

Particles 2026, 9(2), 44; https://doi.org/10.3390/particles9020044 - 23 Apr 2026

Viewed by 503

Abstract

We present a short review dedicated to low-lying meson states. We present all meson nonets, which consist from up, down and strange light quarks. We consider the scalar nonet as a basic nonet. We work in the framework of the massless Nambu–Jona-Lasinio [...] Read more.

We present a short review dedicated to low-lying meson states. We present all meson nonets, which consist from up, down and strange light quarks. We consider the scalar nonet as a basic nonet. We work in the framework of the massless Nambu–Jona-Lasinio

U_{R} (3) \times U_{L} (3)

quark model. The collective meson states are described through initially bare quark–antiquark pairs, whose condensates lead simultaneously to spontaneous breaking of the chiral and the flavour symmetry. After quantisation and the spontaneous breaking of the chiral symmetry, when quarks obtain constituent nonzero masses, they become dressed. We present an explanation of the inverse mass hierarchy of the low-lying nonet of the scalar mesons. The proposed explanation is based on symmetry principles. It is shown that, due to the flavour symmetry breaking, two isodoublets of

K_{0}^{*} (700)

mesons play the role of Goldstone bosons. It is also proven that there exists a solution with almost degenerate masses of the

a_{0} (980)

and

f_{0} (980)

mesons and a zero mass of the

f_{0} (500)

meson. Short description of the physical properties of other meson nonets is provided. In particular unique mass relations among the different nonets, which are experimentally confirmed, are presented. Full article

(This article belongs to the Special Issue Selected Papers from the 14th International Conference on New Frontiers in Physics (ICNFP 2025))

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16 pages, 1112 KB

Open AccessArticle

Nuclear Binding Energies from Composite-Knot Ropelength: A Topological Model That Mirrors Quantum-Mechanical Phenomenology

by Thomas Riedel

Particles 2026, 9(2), 43; https://doi.org/10.3390/particles9020043 - 22 Apr 2026

Viewed by 1188

Abstract

We report a curious numerical observation: If atomic nuclei are modelled as connect-sums of threefoil knots with alternating chirality, the ropelength of the composite knot—a purely geometric quantity requiring no quantum mechanics—tracks the experimental binding-energy curve from hydrogen to uranium. A two-parameter fit [...] Read more.

We report a curious numerical observation: If atomic nuclei are modelled as connect-sums of threefoil knots with alternating chirality, the ropelength of the composite knot—a purely geometric quantity requiring no quantum mechanics—tracks the experimental binding-energy curve from hydrogen to uranium. A two-parameter fit to 50 nuclei gives

R^{2} = 0.9998

(coefficient of determination; 1 = perfect fit) and

RMS = 6.9 MeV

(root-mean-square deviation between model and experiment), comparable to the five-parameter Bethe–Weizsäcker formula (

RMS = 8.3 MeV

) at less than half the parameter count. Out-of-sample predictions for

{}^{244}{Pu}

and

{}^{252}{Cf}

, not used in the fit, are accurate to

0.4 MeV

and

8.4 MeV

, respectively. What makes the observation worth reporting is not the fit itself, but the range of nuclear phenomenology that emerges uninstructed from the topology: saturation, surface energy, isospin pairing, odd-even staggering, and geometric analogues of nuclear isomers all appear as consequences of the connect-sum construction, without additional assumptions. We catalogue these correspondences, assess which are structural and which may be coincidental, and identify concrete numerical tests that would distinguish the two possibilities. Full article

(This article belongs to the Section Nuclear and Hadronic Theory)

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21 pages, 6527 KB

Open AccessArticle

Poincaré Invariance and the Unruh Effect

by Alexandre Deur, Stanley J. Brodsky, Craig D. Roberts and Balša Terzić

Particles 2026, 9(2), 42; https://doi.org/10.3390/particles9020042 - 22 Apr 2026

Viewed by 1184

Abstract

In quantum field theory, the vacuum is popularly considered to be a complex medium populated with virtual particle + antiparticle pairs. To an observer experiencing uniform acceleration, it is generally held that these virtual particles become real, appearing as a gas at a [...] Read more.

In quantum field theory, the vacuum is popularly considered to be a complex medium populated with virtual particle + antiparticle pairs. To an observer experiencing uniform acceleration, it is generally held that these virtual particles become real, appearing as a gas at a temperature that grows with the acceleration. This is the Unruh effect. However, it has been shown that vacuum complexity is an artifact produced by treating quantum field theory in a manner that does not manifestly enforce causality. Choosing a quantization approach that patently enforces causality, the quantum field theory vacuum is barren, bereft even of virtual particles. We show that acceleration has no effect on a trivial vacuum; hence, there is no Unruh effect in such a treatment of quantum field theory. Since the standard calculations suggesting an Unruh effect are formally consistent, insofar as they have been completed, there must be a canceling contribution that is omitted in the usual analyses. We argue that it is the dynamical action of conventional Lorentz transformations on the structure of an Unruh detector. Full article

(This article belongs to the Section Quantum Field Theory and Quantum Gravity)

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25 pages, 639 KB

Open AccessArticle

Observational Diagnostics of a Parametrized Deceleration Parameter in FLRW Cosmology

by Bhupendra Kumar Shukla, Deger Sofuoğlu, Aroonkumar Beesham, Rishi Kumar Tiwari and Mfanafuthi Siyabonga Msweli

Particles 2026, 9(2), 41; https://doi.org/10.3390/particles9020041 - 20 Apr 2026

Viewed by 1499

Abstract

The evolution of the deceleration parameter

q (z)

plays a crucial role in understanding the dynamics of dark energy within the framework of modern cosmology. In this study, we perform a parametric reconstruction of

q (z)

in a spatially [...] Read more.

The evolution of the deceleration parameter

q (z)

plays a crucial role in understanding the dynamics of dark energy within the framework of modern cosmology. In this study, we perform a parametric reconstruction of

q (z)

in a spatially flat Friedmann–Robertson–Walker (FLRW) Universe composed of radiation, pressureless dark matter, and dark energy. We consider a physically motivated form of

q (z)

that effectively describes the transition of the Universe from a decelerating to an accelerating expansion phase. This parametrization is incorporated into the Friedmann equations to derive the corresponding Hubble parameter, which is then confronted with a comprehensive set of observational data, including Hubble parameter measurements

H (z)

, Type Ia supernovae (SNIa), and Baryon Acoustic Oscillations (BAO) data. Employing the Markov Chain Monte Carlo (MCMC) approach, we constrain the model parameters using the combined

H (z) + S N I a + B A O

dataset. The best-fit parameters are subsequently used to reconstruct the cosmographic quantities, such as the deceleration, jerk, and snap parameters, which provide deeper insight into the expansion history of the Universe. Finally, a comparative analysis with the standard

Λ

CDM model is carried out to assess the compatibility and effectiveness of the proposed parametrization. Full article

(This article belongs to the Section Astroparticle Physics and Cosmology)

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15 pages, 972 KB

Open AccessArticle

β Decay of ²⁰Na

by Qiang Wang, You-Bao Wang, Jun Su, Zhi-Yu Han, B. Alex Brown, Li-Hua Chen, Zi-Qiang Chen, Bao-Qun Cui, Bo Dai, Tao Ge, Xin-Yue Li, Yun-Ju Li, Zhi-Hong Li, Gang Lian, Yin-Long Lyu, Rui-Gang Ma, Tian-Li Ma, Xie Ma, Ying-Jun Ma, Yi Su, Bing Tang, Chun-Guang Wang, Hong-Yi Wu, Fu-Rong Xu, Sheng-Quan Yan, Sheng Zeng, Hao Zhang, Yun Zheng, Chao Zhou, Yang-Ping Shen, Bing Guo, Tian-Jue Zhang and Wei-Ping Liu Show full author list Hide full author list

Particles 2026, 9(2), 40; https://doi.org/10.3390/particles9020040 - 17 Apr 2026

Viewed by 796

Abstract

²⁰Na is a well-known

β

-delayed

α

emitter, owing to the large decay energy of ²⁰Na above the α + ¹⁶O threshold in the

A = 5 α

daughter nucleus ²⁰Ne. In this work, the decay property of ²⁰ [...] Read more.

²⁰Na is a well-known

β

-delayed

α

emitter, owing to the large decay energy of ²⁰Na above the α + ¹⁶O threshold in the

A = 5 α

daughter nucleus ²⁰Ne. In this work, the decay property of ²⁰Na is investigated in detail via the β-γ β-α and β-γ-α coincidence spectroscopy. As the day-one experiment of the Beijing Rare Isotope Facility (BRIF), the intense ²⁰Na beam was produced using the Isotope Separator On Line (ISOL) technique through the 100 MeV proton bombarding a stack of MgO as a thick target. Specific interest was focused on the exotic decay mode of ²⁰Na; the previously reported low-energy α lines at 713 and 846 keV were confirmed, and several weak β-γ-α decay sequences were clearly identified for the first time, thanks to the strong resolving power of α-γ coincidence spectroscopy. The decay properties of ²⁰Na are compared to the shell model calculation, which agree reasonably well with the allowed β transition strengths and subsequent electro-magnetic transitions with the use of the sd shell-model space with the USDB interaction. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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10 pages, 321 KB

Open AccessArticle

Leading Low-Temperature Correction to the Heisenberg–Euler Lagrangian

by Felix Karbstein

Particles 2026, 9(2), 39; https://doi.org/10.3390/particles9020039 - 15 Apr 2026

Viewed by 817

Abstract

In this article, we show that the well-known leading low-temperature correction to the Heisenberg–Euler Lagrangian in a constant electromagnetic field arising at two loops can be efficiently extracted from its one-loop zero-temperature analogue. Resorting to the real-time formalism of equilibrium quantum field theory [...] Read more.

In this article, we show that the well-known leading low-temperature correction to the Heisenberg–Euler Lagrangian in a constant electromagnetic field arising at two loops can be efficiently extracted from its one-loop zero-temperature analogue. Resorting to the real-time formalism of equilibrium quantum field theory that explicitly separates out the zero-temperature contribution from the finite-temperature corrections, the determination becomes essentially trivial. In essence, it only requires taking derivatives of the Heisenberg–Euler Lagrangian at one loop and zero temperature for the field strength. As a bonus, we then effectively dress the low-temperature contribution at two loops by one-particle reducible tadpole structures. This generates a subset of higher-loop contributions to the Heisenberg–Euler Lagrangian in the limit of low temperatures. We extract their leading strong-field behavior at a given loop order, and finally resum these to all loop orders. Full article

(This article belongs to the Special Issue Particles and Plasmas in Strong Fields, Part 1)

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23 pages, 14334 KB

Open AccessReview

Recent Developments in and Applications of the Relativistic Chiral Nuclear Force

by Li-Sheng Geng, Jun-Xu Lu, Qing-Yu Zhai, Zhi-Wei Liu and Shi-Hang Shen

Particles 2026, 9(2), 38; https://doi.org/10.3390/particles9020038 - 4 Apr 2026

Cited by 1 | Viewed by 988

Abstract

The nuclear force is central to our understanding of complex nuclear phenomena and to the applications of nuclear techniques. The non-perturbative nature of low-energy strong interaction and color confinement have provided an ab initio understanding of nuclear force, a challenge for almost a [...] Read more.

The nuclear force is central to our understanding of complex nuclear phenomena and to the applications of nuclear techniques. The non-perturbative nature of low-energy strong interaction and color confinement have provided an ab initio understanding of nuclear force, a challenge for almost a century, since the pioneering work of Yukawa. Since 1990, chiral effective field theory (ChEFT) has become the de facto standard for describing nuclear interactions; most prior studies employed heavy-baryon chiral perturbation theory. Only recently, there have been successful attempts to construct a chiral nuclear force employing covariant baryon chiral perturbation theory. In this work, we review recent developments and applications of relativistic chiral nuclear forces. We first elaborate on the necessity of relativistic/covariant theories, then present the construction of the first high-precision relativistic chiral nuclear force up to next-to-next-to-leading order (NNLO), and discuss the ongoing progress in higher-order nucleon–nucleon (NN) and n-d scattering, as well as their applications in nuclear matter, finite nuclei, and hypernuclear systems. Finally, we summarize the achievements and outline the future outlook of this research field. Full article

(This article belongs to the Special Issue Selected Papers from the International Symposium Commemorating the 40th Anniversary of Halo Nuclei)

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28 pages, 794 KB

Open AccessArticle

Emergent Higgs Field and the Schwarzschild Black Hole

by Dragana Pilipović

Particles 2026, 9(2), 37; https://doi.org/10.3390/particles9020037 - 3 Apr 2026

Viewed by 2262

Abstract

The derivations presented in this paper suggest an intimate relationship between geometry and the electroweak sector at the Planck scale. A Lorentz-invariant maximally symmetric stochastically perturbed spacetime transformed to spherical coordinates reveals an emergent Schwarzschild metric, entirely a statistical structure of stochastic spacetime. [...] Read more.

The derivations presented in this paper suggest an intimate relationship between geometry and the electroweak sector at the Planck scale. A Lorentz-invariant maximally symmetric stochastically perturbed spacetime transformed to spherical coordinates reveals an emergent Schwarzschild metric, entirely a statistical structure of stochastic spacetime. Similarly, the transition from a maximally symmetric universe with a complex

S U (2)

scalar doublet

ϕ

, comprising four independent real scalar fields with a zero vacuum expectation value (VEV), to spherical coordinates at the Planck scale reveals the spontaneously broken electroweak (EW) sector. Working in the unitarity gauge, the resulting EW potential can be simultaneously mapped in space at the Planck scale and across the EW sector. In space, the resulting EW potential includes a deep well within the Schwarzschild sphere and a shallow well just outside corresponding to an accretion disk. The same potential mapped in the EW space provides an entire family of possible sombrero hat potentials with fourth-order coupling specific to a point in space. At the minimum points of the potential in space, inside the Schwarzschild sphere and at the accretion disk, the

λ

corresponding to the Standard Model (SM) fourth-order coupling is instead derived as

\frac{λ}{5}

. The factor of

\frac{1}{5}

is a simple consequence of the conservation of the EW VEV and the fact that the SM formulation of the EW potential does not account for situations where the perturbations in

ϕ

dominate. A more general formulation of the EW potential restores the SM quartic coupling and preserves

λ

in space. An emergent Higgs field inside the Schwarzschild black hole is found to directly relate to the stochastic spacetime fields normalized by the Schwarzschild radius. The corresponding Higgs vacuum has both a ground and excited state and the possibility of both positive and negative vacuum entropy. Finally, the scalar-field VEV degeneracy in EW space of the metastable Higgs vacuum appears instead differentiated in space with possible probability, tunneling, and entropy implications. Full article

(This article belongs to the Section Phenomenology and Physics Beyond the Standard Model)

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