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11 pages, 1117 KiB

Open AccessArticle

The Inhomogeneous Road to Chiral Symmetry Breaking: A Ginzburg–Landau–Langevin Analysis

by Theo F. Motta and Gastão Krein

Symmetry 2025, 17(4), 568; https://doi.org/10.3390/sym17040568 - 9 Apr 2025

Viewed by 197

Abstract

We investigate the time evolution of the quark condensate toward a chiral symmetry broken phase in hot and dense quark matter using a field-theoretic quark model with nonlocal chiral-invariant four-fermion coupling. By purposely selecting a parameter set in which inhomogeneous phases are energetically [...] Read more.

We investigate the time evolution of the quark condensate toward a chiral symmetry broken phase in hot and dense quark matter using a field-theoretic quark model with nonlocal chiral-invariant four-fermion coupling. By purposely selecting a parameter set in which inhomogeneous phases are energetically disfavored, we nonetheless observe the emergence of metastable patterned configurations that appear to persist for remarkably long timescales. These findings suggest that even when not fully stable, inhomogeneous phases may play a significant role in the dynamics of chiral symmetry breaking and restoration. To gain deeper insight into these phenomena, we also analyze the impact of the dimensionality of coordinate space on both the formation and stability of inhomogeneous chiral condensates. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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22 pages, 771 KiB

Open AccessArticle

Effects of Quark Core Sizes of Baryons in Neutron Star Matter

by Wolfgang Bentz and Ian C. Cloët

Symmetry 2025, 17(4), 505; https://doi.org/10.3390/sym17040505 - 26 Mar 2025

Viewed by 175

Abstract

We describe the quark substructure of hadrons and the equation of state of high-density neutron star matter by using the Nambu–Jona-Lasinio (NJL) model, which is an effective quark theory based on QCD. The interaction between quarks fully respects the chiral and flavor symmetries. [...] Read more.

We describe the quark substructure of hadrons and the equation of state of high-density neutron star matter by using the Nambu–Jona-Lasinio (NJL) model, which is an effective quark theory based on QCD. The interaction between quarks fully respects the chiral and flavor symmetries. Guided by the success of various low-energy theorems, we assume that the explicit breaking of these symmetries occurs only via the current quark masses, and all other symmetry breakings are of dynamical nature. In order to take into account the effects of the finite quark core sizes of the baryons on the equation of state, we make use of an excluded volume framework that respects thermodynamic consistency. The effects generated by the swelling quark cores generally act repulsively and lead to an increase in the pressure with increasing baryon density. On the other hand, in neutron star matter, these effects also lead to a decrease in the density window where hyperons appear because it becomes energetically more favorable to convert the faster moving nucleons into hyperons. Our quantitative analysis shows that the net effect of the excluded volume is too small to solve the long-standing “hyperon puzzle”, which is posed by the large observed masses of neutron stars. Thus, the puzzle persists in a relativistic effective quark theory which takes into account the short-range repulsion between baryons caused by their finite and swelling quark core sizes in a phenomenological way. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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18 pages, 1047 KiB

Open AccessArticle

Influence of the Effective Mass on the Properties of Nuclear Matter at Finite Density and Temperature

by Hajime Togashi, Debashree Sen, Hana Gil and Chang Ho Hyun

Symmetry 2025, 17(3), 445; https://doi.org/10.3390/sym17030445 - 17 Mar 2025

Viewed by 213

Abstract

The significance of the chiral symmetry restoration is studied by considering the role of the modification of the nucleon mass in nuclear medium at finite density and temperature. Using the Korea-IBS-Daegu-SKKU density functional theory, we can create models that have an identical nuclear [...] Read more.

The significance of the chiral symmetry restoration is studied by considering the role of the modification of the nucleon mass in nuclear medium at finite density and temperature. Using the Korea-IBS-Daegu-SKKU density functional theory, we can create models that have an identical nuclear matter equation of state but different isoscalar and isovector effective masses at zero temperature. The effect of the effective mass becomes transparent at non-zero temperatures, and it becomes more important as temperature increases. The role of the effective mass is examined thoroughly by calculating the dependence of thermodynamic variables such as free energy, internal energy, entropy, pressure and chemical potential on density, temperature and proton fraction. We find that sensitivity to the isoscalar effective mass is several times larger than that of the isovector effective mass, so the uncertainties arising from the effective mass are dominated by the isoscalar effective mass. In the analysis of the relative uncertainty, we obtain that the maximum uncertainty is less than 2% for free energy, internal energy and chemical potential, but it amounts to 20% for pressure. Entropy shows a behavior completely different from the other four variables that the uncertainty is about 40% at the saturation density and increases monotonically as density increases. The effect of the uncertainty to properties of physical systems is investigated with the proto-neutron star. It is shown that temperature depends strongly on the effective mass at a given density, and substantial swelling of the radius occurs due to the finite temperature. The equation of state is stiffer with smaller isoscalar effective mass, so the effect of the effective mass appears clearly in the mass–radius relation of the proto-neutron star, where a larger radius corresponds to a smaller effective mass. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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42 pages, 565 KiB

Open AccessArticle

From QCD Phenomenology to Nuclear Physics Phenomenology: The Chiral Confining Model

by Guy Chanfray, Magda Ericson, Hubert Hansen, Jérôme Margueron and Marco Martini

Symmetry 2025, 17(2), 313; https://doi.org/10.3390/sym17020313 - 19 Feb 2025

Viewed by 382

Abstract

We present a theoretical framework that allows one to make an explicit connection between the phenomenology of QCD, namely the properties of the gluon correlator and Wilson loops, and a particular relativistic model for the description of nuclear matter and neutron stars: the [...] Read more.

We present a theoretical framework that allows one to make an explicit connection between the phenomenology of QCD, namely the properties of the gluon correlator and Wilson loops, and a particular relativistic model for the description of nuclear matter and neutron stars: the chiral confining model. Starting with the field correlator method, which explicitly and simultaneously incorporates confinement and chiral symmetry breaking, we describe how to obtain the response of the composite nucleon to the nuclear scalar field, as well as the relative role of confinement and chiral symmetry breaking in in-medium nucleon mass evolution, thereby generating the three-body forces needed for the saturation mechanism. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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24 pages, 579 KiB

Open AccessArticle

Chiral Symmetry in Dense Matter with Meson Condensation

by Takumi Muto, Toshiki Maruyama and Toshitaka Tatsumi

Symmetry 2025, 17(2), 270; https://doi.org/10.3390/sym17020270 - 10 Feb 2025

Viewed by 579

Abstract

Kaon condensation in hyperon-mixed matter [(Y+K) phase], which may be realized in neutron stars, is discussed on the basis of chiral symmetry. With the use of the effective chiral Lagrangian for kaon–baryon and kaon–kaon interactions; coupled with the relativistic [...] Read more.

Kaon condensation in hyperon-mixed matter [(Y+K) phase], which may be realized in neutron stars, is discussed on the basis of chiral symmetry. With the use of the effective chiral Lagrangian for kaon–baryon and kaon–kaon interactions; coupled with the relativistic mean field theory and universal three-baryon repulsive interaction, we clarify the effects of the s-wave kaon–baryon scalar interaction simulated by the kaon–baryon sigma terms and vector interaction (Tomozawa–Weinberg term) on kaon properties in hyperon-mixed matter, the onset density of kaon condensation, and the equation of state with the (Y+K) phase. In particular, the quark condensates in the (Y+K) phase are obtained, and their relevance to chiral symmetry restoration is discussed. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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12 pages, 426 KiB

Open AccessArticle

Gauge Covariance of the Gap Equation: From the Rainbow Truncation to Gauge Symmetry Constraints

by Bruno El-Bennich

Symmetry 2025, 17(1), 110; https://doi.org/10.3390/sym17010110 - 12 Jan 2025

Viewed by 803

Abstract

The gauge covariance of the quark gap equation is compared for the case of three different quark–gluon vertices: the bare vertex, a Ball–Chiu-like vertex constrained by the corresponding Slavnov–Taylor identity, and a full vertex including the transverse components derived from transverse Slavnov–Taylor identities. [...] Read more.

The gauge covariance of the quark gap equation is compared for the case of three different quark–gluon vertices: the bare vertex, a Ball–Chiu-like vertex constrained by the corresponding Slavnov–Taylor identity, and a full vertex including the transverse components derived from transverse Slavnov–Taylor identities. The covariance properties are verified with the chiral quark condensate and the pion decay constant in the chiral limit. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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Review

Jump to: Research

46 pages, 1576 KiB

Open AccessReview

Electroweak Form Factors of Baryons in Dense Nuclear Matter

by G. Ramalho, K. Tsushima and Myung-Ki Cheoun

Symmetry 2025, 17(5), 681; https://doi.org/10.3390/sym17050681 (registering DOI) - 29 Apr 2025

Abstract

There is evidence that the properties of hadrons are modified in a nuclear medium. Information about the medium modifications of the internal structure of hadrons is fundamental for the study of dense nuclear matter and high-energy processes, including heavy-ion and nucleus–nucleus collisions. At [...] Read more.

There is evidence that the properties of hadrons are modified in a nuclear medium. Information about the medium modifications of the internal structure of hadrons is fundamental for the study of dense nuclear matter and high-energy processes, including heavy-ion and nucleus–nucleus collisions. At the moment, however, empirical information about medium modifications of hadrons is limited; therefore, theoretical studies are essential for progress in the field. In the present work, we review theoretical studies of the electromagnetic and axial form factors of octet baryons in symmetric nuclear matter. The calculations are based on a model that takes into account the degrees of freedom revealed in experimental studies of low and intermediate square transfer momentum

q^{2} = - Q^{2}

: valence quarks and meson cloud excitations of baryon cores. The formalism combines a covariant constituent quark model, developed for a free space (vacuum) with the quark–meson coupling model for extension to the nuclear medium. We conclude that the nuclear medium modifies the baryon properties differently according to the flavor content of the baryons and the medium density. The effects of the medium increase with density and are stronger (quenched or enhanced) for light baryons than for heavy baryons. In particular, the in-medium neutrino–nucleon and antineutrino–nucleon cross-sections are reduced compared to the values in free space. The proposed formalism can be extended to densities above the normal nuclear density and applied to neutrino–hyperon and antineutrino–hyperon scattering in dense nuclear matter. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

13 pages, 1143 KiB

Open AccessReview

Reflections on Chiral Symmetry Within QCD

by Anthony Thomas

Symmetry 2025, 17(4), 512; https://doi.org/10.3390/sym17040512 - 28 Mar 2025

Viewed by 187

Abstract

The fact that chiral symmetry is a crucial feature of the strong force was realized before the discovery of quantum chromodynamics. However, the full power it exerts on the structure of the nucleon only became apparent afterwards. We present a high-level and somewhat [...] Read more.

The fact that chiral symmetry is a crucial feature of the strong force was realized before the discovery of quantum chromodynamics. However, the full power it exerts on the structure of the nucleon only became apparent afterwards. We present a high-level and somewhat personal overview of its role in almost every aspect of the proton structure, from its mass and spin to the asymmetry of its antimatter content and its strange quark content. The lessons learned from studying the proton are also vital with respect to the modern challenge of the nature of baryon-excited states. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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45 pages, 4574 KiB

Open AccessReview

Chiral Effective Model of Cold and Dense Two-Color QCD: The Linear Sigma Model Approach

by Daiki Suenaga

Symmetry 2025, 17(1), 124; https://doi.org/10.3390/sym17010124 - 15 Jan 2025

Cited by 1 | Viewed by 777

Abstract

This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC₂D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC₂D, both [...] Read more.

This review is devoted to summarizing recent developments of the linear sigma model (LSM) in cold and dense two-color QCD (QC₂D), in which lattice simulations are straightforwardly applicable thanks to the disappearance of the sign problem. In QC₂D, both theoretical and numerical studies derive the presence of the so-called baryon superfluid phase at a sufficiently large chemical potential (

μ_{q}

), where diquark condensates govern the ground state. The hadron mass spectrum simulated in this phase shows that the mass of an iso-singlet (

I = 0

) and

0^{-}

state is remarkably reduced, but such a mode cannot be described by the chiral perturbation theory. Motivated by this fact, I have invented a LSM constructed upon the linear representation of chiral symmetry, more precisely Pauli–Gürsey symmetry. It is shown that my LSM successfully reproduces the low-lying hadron mass spectrum in a broad range of

μ_{q}

simulated on the lattice. As applications of the LSM, topological susceptibility and sound velocity in cold and dense QC₂D are evaluated to compare with the lattice results. Additionally, the generalized Gell–Mann–Oakes–Renner relation and hardon mass spectrum in the presence of a diquark source are analyzed. I also introduce an extended version of the LSM incorporating spin-1 hadrons. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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26 pages, 596 KiB

Open AccessReview

Axial-Vector and Tensor Spin Polarization and Chiral Restoration in Quark Matter

by Tomoyuki Maruyama and Toshitaka Tatsumi

Symmetry 2024, 16(12), 1642; https://doi.org/10.3390/sym16121642 - 11 Dec 2024

Viewed by 774

Abstract

We study spontaneous the spin polarization of quark matter with flavor

S U (2)

symmetry at zero temperature in the NJL model. In a relativistic framework, there are two types of spin–spin interactions: axial vector (AV) and tensor (T), which accordingly [...] Read more.

We study spontaneous the spin polarization of quark matter with flavor

S U (2)

symmetry at zero temperature in the NJL model. In a relativistic framework, there are two types of spin–spin interactions: axial vector (AV) and tensor (T), which accordingly give rise to different types of spin-polarized materials. When the spin–spin interaction is sufficiently strong, the spin-polarized phase emerges within a specific density region. As the spin–spin interaction becomes stronger, this phase extends over a higher-density region beyond the critical density of chiral restoration in normal quark matter. We show that the spin-polarized phase leads to another kind of spontaneous chiral symmetry breaking phase. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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19 pages, 645 KiB

Open AccessReview

Extraordinary Nature of the Nucleon Scalar Charge and Its Densities as a Signal of Nontrivial Vacuum Structure of QCD

by Masashi Wakamatsu

Symmetry 2024, 16(11), 1481; https://doi.org/10.3390/sym16111481 - 6 Nov 2024

Viewed by 1232

Abstract

It is widely known that the nucleon scalar charge is proportional to the pion–nucleon sigma term as one of the important low-energy observables of QCD. Especially interesting to us is the physics of the nucleon scalar charge densities. This comes from the fact [...] Read more.

It is widely known that the nucleon scalar charge is proportional to the pion–nucleon sigma term as one of the important low-energy observables of QCD. Especially interesting to us is the physics of the nucleon scalar charge densities. This comes from the fact that the corresponding operator has the same quantum number as the physical vacuum. It indicates unusual behavior of the nucleon scalar density as a function of the distance r from the nucleon center. Namely, it would not be reduced down to zero at the spatial infinity but rather approach some nonzero constant corresponding to the vacuum quark condensate. Naturally, this unique nature of the nucleon scalar density in the position space also affects the corresponding density in the momentum space, i.e., the corresponding parton distribution function (PDF) as a function of the Bjorken variable x. This PDF is known as the chiral-odd twist-3 PDF

e (x)

. We argue that

e (x)

is likely to have a delta-function-type singularity at

x = 0

and that the appearance of this singularity can be interpreted as a signal of the nontrivial vacuum structure of QCD. Full article

(This article belongs to the Special Issue Chiral Symmetry, and Restoration in Nuclear Dense Matter)

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