Special Issue: Chiral Symmetry in Physics

The study of symmetry principles has consistently provided excellent guidance in the search to understand the fundamental laws of nature. Symmetries underpin our theoretical frameworks, define conservation laws, and guide the classification of particles and interactions. Among these, chiral symmetry occupies a position of particular significance. It provides profound insights into many areas of physics, notably field theory and elementary particle physics. Rooted in the structure of quantum chromodynamics (QCD), chiral symmetry and its spontaneous (i.e., dynamical) breaking are essential for explaining a considerable portion of particle physics data [1] and understanding a wide array of physical phenomena, from the many properties of hadrons to the collective behavior of matter under extreme conditions.

This Special Issue brings together a diverse set of contributions that explore both the established and emerging roles of chiral symmetry in modern theoretical physics. It assembles seven outstanding papers—five original research articles and two reviews—that employ a range of theoretical tools, from perturbative and nonperturbative field theory methods to lattice QCD simulations and effective field theories. These works collectively offer a broad and insightful survey of how chiral symmetry, and its dynamical breaking or partial restoration, manifest across various physical systems and energy scales. The Table of Contents lists the articles in the chronological order of their acceptance. The first five are research articles; the final two are reviews. Each paper highlights a different aspect of the evolving role of chiral symmetry in modern physics, and together they underscore the enduring relevance of symmetry principles in shaping theoretical inquiry.

• Selected Advances in Chiral Symmetry Research

The papers in this Special Issue present some of the key modern developments in our understanding of chiral symmetry. Its spontaneous, viz. dynamical, breaking in quantum chromodynamics (QCD) leads to emergent phenomena in hadron and nuclear physics.

Craig Roberts’ exploration of Empirical Consequences of Emergent Mass [2] has attracted significant attention and won the Symmetry 2020 Best Paper Award [3] It demonstrates how the relatively simple Lagrangian of QCD generates the remarkable complexity observed in nuclear physics through the mechanism of emergent mass. As Roberts eloquently explains, this emergent mass may be QCD’s self-stabilizing mechanism, with observable consequences ranging from pion parton distributions to nucleon electromagnetic form factors. This work provides a crucial framework for understanding how abstract symmetry principles manifest themselves in measurable physical properties. Some subsequent results and related considerations are presented in, e.g., [4,5,6,7] and references therein.

Ulf-G. Meißner’s highly cited work Two-Pole Structures in QCD: Facts, Not Fantasy! [8] challenges conventional wisdom by revealing that certain states in the hadron spectrum—as listed in The Review of Particle Physics [1]—actually represent two distinct states in the complex plane. Beginning with the $\Lambda \left(1405\right)$ and extending to excited charm mesons, Meißner’s analysis employs unitarized chiral perturbation theory to demonstrate how this two-pole structure emerges from fundamental symmetry considerations. Here, progress requires an interplay between theoretical methods, experimental data, and lattice QCD simulations. To date, almost a dozen such two-pole structures have now been identified, with those presented in [9,10,11,12,13,14] being the most recent, establishing a new class of players in hadron spectroscopy. This work exemplifies how careful, symmetry-guided analysis can uncover subtleties missed by more conventional approaches. Related considerations and results are presented in, e.g., refs. [15,16,17] and references therein.

In a complementary vein, Mannque Rho’s investigation titled Multifarious Roles of Hidden Chiral-Scale Symmetry [18] has made substantial contributions to our understanding of how the axial current coupling constant impacts nuclear processes. Rho demonstrates that the long-standing puzzle of the “quenched” $g_A$ in nuclear superallowed Gamow–Teller transitions actually encodes the emergence of chiral-scale symmetry hidden in QCD. This insight connects phenomena across vastly different energy scales—from neutrinoless double-beta decays to the equation of state in massive compact stars—illustrating the unifying power of symmetry principles. More recent work on “quenched” $g_A$ can be found in [19,20,21].

The article by David Blaschke, Denis Devyatyarov, and Olaf Kaczmarek, titled Quark Cluster Expansion Model for Interpreting Finite-T Lattice QCD Thermodynamics [22], provides a unified approach to understanding the thermodynamics of hadron–quark–gluon matter. Their generalized Beth-Uhlenbeck approach, with its ansatz for hadronic phase shifts, reveals how the transition from a hadron resonance gas to a quark–gluon plasma occurs in a narrow temperature window of 150–185 MeV, with the Mott dissociation of hadrons triggered by the restoration of chiral symmetry. Their work exemplifies how symmetry considerations can illuminate phase transitions in strongly interacting matter. This approach has triggered follow-up research, as reported in [23], where the ansatz for the hadronic phase shifts has been qualitatively improved and the composition of the hadron–quark–gluon matter has been evaluated, and in [24], where the Beth-Uhlenbeck approach has been extended to the entire QCD phase diagram and the location of the critical end point has been discussed, while also admitting its absence.

Gergely Fejős’ contribution, titled Perturbative RG Analysis of the Condensate Dependence of the Axial Anomaly [25], employs renormalization group techniques to investigate the thermal behavior of ’t Hooft’s determinant coupling in the three-flavor linear sigma model. To this end, the standard ’t Hooft term is extended by a dimension-5 anomaly operator, resulting in a condensate-dependent effective anomaly coupling. His finding—that mesonic fluctuations enhance the anomaly strength as the chiral condensate decreases at high temperatures—offers critical insight into the intricate relationship between chiral symmetry breaking and the $U_A\left(1\right)$ anomaly. Applying formalism to the nuclear liquid–gas transition reveals that the partial restoration of chiral symmetry at the transition point leads to an approximately 20% increase in the effective $U_A\left(1\right)$ coupling. This contrasts with many other predictions of a weakening axial anomaly at high temperatures (for example, [26,27,28,29,30], including those based on lattice QCD, e.g., [31,32], and references therein). Hence, this underscores the importance of mesonic fluctuations and highlights the need for further theoretical and numerical investigations into the nonperturbative dynamics of $U_A\left(1\right)$ symmetry. Related considerations and results are presented in [33,34,35,36].

Especially striking in its bridging of theoretical domains and synthesis of seemingly disparate fields is the mini-review by Ma and Rho, titled Dichotomy of Baryons as Quantum Hall Droplets and Skyrmions in Compact-Star Matter [37]. Their exploration of the possible domain-wall structure of compressed baryonic matter in massive compact stars brings together concepts from condensed matter physics, nuclear theory, and astrophysics. By incorporating hidden symmetries—flavor local symmetry and scale symmetry—into an effective nuclear field theory, they offer a fresh perspective on the structure of nuclear matter across density regimes. Their framework suggests a hadron–quark duality in baryonic matter that could revolutionize our approach to nuclear dynamics, from ordinary nuclear matter to the extreme densities found in compact stars. Some examples of papers outlining the broader context of bridging these different fields, and/or presenting recent related results, are [38,39,40,41,42,43,44,45,46] and references therein. In particular, [21] points out the crucial link between “quenched” $g_A$, as explained in Ref. [18], and the pseudo-conformal sound velocity in dense compact stars.

The comprehensive pedagogical review Renormalization Theory and Chiral Gauge Theories in Dimensional Regularization with Non-Anticommuting $\gamma$₅ [47] by Dominik Stöckinger, Amon Ilakovac and collaborators provides an invaluable resource for researchers working with chiral gauge theories, which are the theoretical foundation of our understanding of fundamental interactions of elementary particles. Their detailed exposition of how to handle the spurious breaking of gauge invariance and determine symmetry-restoring counterterms offers both theoretical foundations and practical guidance. The paper’s methodical presentation of the BPHZ renormalization, Slavnov–Taylor identities, and BRST formalism makes complex techniques accessible to a broader audience. In particular, it contains the first published pedagogical exposition of the Breitenlohner–Maison–’t Hooft–Veltman (BMHV) scheme of dimensional regularization applied to the renormalization of chiral gauge theories. For related research and results, see [48,49,50,51] and references therein.

• Summary

Collectively, the contributions to this Special Issue underscore the central role that chiral symmetry and its breaking play across diverse domains of physics—from the substructure of hadrons to the interior of dense astrophysical objects. These papers illustrate the richness of the theoretical landscape and highlight the evolving interplay between analytical methods, numerical simulations, and experimental data. As new tools and perspectives continue to emerge—especially in areas such as lattice QCD, functional renormalization group methods, and holographic duality—the study of chiral symmetry remains a fertile and dynamic field of research. Its unifying nature not only connects very different energy scales but also bridges conceptual frameworks, from quantum field theory to condensed matter analogies. Pedagogical expositions of complex topics are especially valuable for young researchers entering the field. This Special Issue includes such contributions, with the final article in particular offering a clear and comprehensive review of foundational methods in chiral gauge theories. It is our hope that this Special Issue will serve as a resource for both active researchers and those new to the field.