Special Issue: Symmetry in Hadron Physics

Symmetry is a key with which it is possible to unlock the doors that nature keeps hidden. It can enable us to seek hidden order within chaos and infer completeness from fragmentation. The quark structure of hadrons is a typical example. Based on the dynamical symmetry and with the help of representation theory pertaining to the underlying group, Gell-Mann et al. found order in the spectra of hadrons [1]. Furthermore, to complete the symmetry-based description, a new particle was predicted and confirmed by subsequent experiments. Therefore, symmetry can be a useful tool guiding us into the unknown. The rotational symmetry of snowflakes, the mirror symmetry of butterflies, the spacial symmetry of crystals, etc., explicitly present the geometric symmetry of objects. The ordering of the spectra of atoms, nuclei, and hadrons present another kind of symmetry—dynamical symmetry. A notable application of dynamical symmetry is the interacting boson model of nuclear collective motion [2]. In the model, various phases of nuclear collective motion, vibration, rotation, and the transition between vibration and rotation, respectively, $U_6\supset U_5$, $U_6\supset S{U}_3$, and $U_6\supset O_6$, describe dynamical symmetry.

This Special Issue brings together diverse applications of dynamical symmetry in hadron physics, spanning from single hadrons to mutliquark states and hadronic matters. It comprises 13 papers—9 original research articles, 3 reviews, and 1 pedagogical review article. These works offer a broad and insightful survey of dynamical symmetry’s role in advancing hadron physics.

1. SelectedReviews on Symmetry and Hadron Physics

This Special Issue contains four review articles, two focusing on experimental progresses and two on theoretical advances.

BESIII at the BEPCII accelerator is a leading facility for high-precision studies in hadron physics and $\tau$-charm physics that has made significant contributions to the field. In “Status and Prospects of the ${\chi }_{c1}\left(3872\right)$ Study at BESIII”, H. Zhou et al. present a concise review and outlook on the particle $X\left(3872\right)$ [3]. The $X\left(3872\right)$ plays a pivotal role in our understanding of hadron structures and remains the most extensively studied exotic state. Its properties are critical to distinguishing between molecular, tetraquark, hybrid, and conventional charmonium interpretations. High-precision measurements of its productions and decays at BESIII continue to shed the light on the nature of the particle.

Another experimental review by Y. Zhao et al., “Experimental Review of the Quarkonium Physics at the LHC”, highlights recent results from LHCb, CMS, and ATLAS collaborations at the LHC [4]. Notably, analyses by CMS and ATLAS collaborations of the $t\overline{t}$ invariant mass spectrum reveal a significant enhancement near the production threshold, interpreted as evidence for the top quarkonium state ${\eta }_t$. This is surprising given the top quark’s short lifetime, which was thought to preclude hadronization into bound states. This discovery not only validates the NRQCD prediction of bound state formation via the Coulomb potential but also opens a new platform for studying the top quark–Yukawa coupling and quantum entanglement.

Collaborative efforts by ATLAS and LHCb have further revealed the suppression of heavy quarkonia in nuclear environments, offering key insights into the properties of Quark–Gluon Plasma (QGP). Additionally, all-charm tetraquark observations by LHCb, CMS, and ATLAS collaborations have stimulated extensive studies on multiquark states, with spin-parity measurements by the CMS collaboration helping to clarify their internal structures.

In the theoretical review “Exotic Heavy Hadrons”, Garcilazo and Valcarce summarize their recent work on exotic heavy hadrons, including hidden heavy-flavor pentaquarks $P_c$ and $P_{cs}$, as well as the bound states of three B mesons [5]. They emphasize the importance of dynamical correlations induced by the Coulomb-like short-range color interaction between heavy quarks in multiquark systems. Through symmetry analysis, they argue that heavy quark pairs in such states should form color singlets. Based on the result of Jaffe [6], they note that the binding energy of a $Qq$ diquark scales with the light quark mass $m_q$, while that of the color singlet $Q\overline{Q}$ pair scales as 2$M_Q$, favoring an internal structure dominated by a color singlet $Q\overline{Q}$ pair rather than a color octet. Extending this approach to three-hadron systems, they find that the $T_{bbb}$ three-meson bound state is robustly stable.

The pedagogical article “Dynamical Symmetry and Hadron Spectrum” by M. Y. Pan et al. explores the dynamical symmetries of hadrons as quark systems [7]. It provides a step-by-step construction of color, spin, and flavor wave functions consistent with phase conventions for two- to six-quark systems, following a general discussion of dynamical symmetries and Casimir operators of $S{U}_n$. Based on these symmetries, the authors construct mass formulas, generalizing the Gürsey-Radicati form that effectively describes hadron spectra. This article serves as a valuable guide for newcomers to the systematic study of multiquark systems.

2. Selected Advances on Symmetry and Hadron Physics

Applying group theory to physical problems requires consistent phase conventions for coefficients such as Clebsh–Gordan coefficients and isoscalar factors. While the Condon–Shortley phase convention is standard for $S{U}_2$, larger groups often suffer from inconsistent usage. Y. Lu et al., in “Phase Conventions in Hadron Physics from the Perspective of the Quark Model”, systematically analyze and compare flavor $S{U}_3$ phase conventions [8]. They identify all sources of discrepancy and propose a consistent framework for combining different conventions, an important contribution in terms of avoiding errors in multiquark calculations.

Multiquark states represents a frontier in hadron physics. Symmetry analysis, especially through flavor $S{U}_3$ and spin $S{U}_2$ representations, enables systematic classification of their masses and decays. In “Triply Heavy Tetraquark States in a Mass-Splitting Model”, S.Y. Li et al. developed a modified chromomagnetic interaction (CMI) model to study triply heavy tetraquarks [9]. They point out that the choice of reference state used to fix the parameter of the model is important, since the inner structures between the meson–meson and compact states are different. Using $X\left(4140\right)$ as a reference state—interpreted as the lowest $1^{++}$ $cs\overline{c}\overline{s}$ tetraquark—they obtain more reliable mass estimates.

In “Exploring the Interpretations of Charmonia and $cc\overline{c}\overline{c}$ Tetraquarks in the Relativistic Flux Tube Model”, W.C. Dong et al. use heavy antiquark–diquark symmetry to describe both $c\overline{c}$ charmonia and $cc\overline{c}\overline{c}$ fully charmed tetraquarks within a unified relativistic flux tube model with spin-dependent interaction [10]; the systematics of hadron states can be embodied as the Regge trajectories in the (L,$M^2$) plane. A good agreement with experimental data is obtained for the entire mass spectra of the charmonium and fully charmed tetraquark.

Since mass agreement alone is insufficient to identify structures, decay properties provide further constraints. Y. W. Jiang et al., in “Strong Decays of the $\varphi$(2170) as a Fully Strange Tetraquark State”, compute decay widths to final states such as $\varphi \eta$, $\varphi {\eta }^{\prime }$, and $\varphi f_0\left(980\right)$ using the QCD sum rule and the Fierz rearrangement technique [11]. Their results support a tetraquark interpretation for $\varphi \left(2170\right)$.

Although non-relativistic quark models successfully describe of the properties of $q\overline{q}$ mesons and $qqq$ baryons, a relativistic field-theoretical treatment is ultimately desirable. In “Quantum Chromodynamics of the Nucleon in Terms of Complex Probabilistic Processes”, Gevorkyan and Bogdanov develop a non-perturbative approach for nucleons based on a relativistic quantum oscillator model, incorporating the hidden symmetries of three-quark dynamics within a quark–antiquark sea [12]. Their formalism reveals topological singularities and, upon averaging, leads to spontaneous chiral symmetry-breaking and effective interquark interactions.

M. Chizhov et al., in “Explanation of the Mass Pattern of the Low-Lying Scalar Nonet”, use the Nambu–Jona-Lasinio (NJL) model to describe scalar mesons as quark–antiquark pairs whose condensates break chiral and flavor symmetry [13]. They find that the $\sigma$ meson is nearly pure $s\overline{s}$, while the $f_0\left(980\right)$ meson is close to a $(u\overline{u}+d\overline{d})/\sqrt{2}$, which is contrary to conventional assignments. In the spontaneous symmetry-breaking of the $U_3$ flavor symmetry, $K_0^{\ast }\left(700\right)$ mesons play the role of massless Goldstone bosons.

Under extreme high temperatures and densities, such as those in the early universe, relativistic heavy-ion collisions at RHIC, LHC, or compact stellar cores, hadronic matter is expected to undergo a transition to deconfined quark matter. The phase diagram of strongly interacting matter is another frontier of hadron physics. In “A Bridge between Trace Anomalies and Deconfinement Phase Transitions”, Sheng and Ma conjecture a connection between the vanishing of the QCD trace anomaly (described by a dilaton potential) and color deconfinement (measured by the Polyakov loop) [14]. This conjecture is inspired by the fact that both the dilaton potential encoding the trace anomalies of QCD and the Polyakov loop potential measuring the deconfinement phase transition can be expressed in their logarithmic forms, as well as the fact that the scale symmetry is expected to be restored and colors deconfined under extreme conditions. In a pure gluon system, a bridge between the restoration of scale symmetry and deconfined phase transition was setup phenomenologically. The conjucture may deepen our understanding of the evolution of the universe, the mechanism of electroweak symmetry-breaking, the phase diagram of QCD matter, and the properties of neutron stars.

Nowadays, it is believed that the QCD phase transition is a first-order phase transition from a chiral symmetry broken hadron phase to a chiral symmetric quark phase at high chemical potentials and low temperatures. For a typical first-order phase transition, a system will go from a metastable high-energy state to a relatively stable lower energy state through bubble nucleation. J. R. Wang, J. S. Jin and H. Mao’s paper “Bubble Dynamics in the Polyakov Quark-Meson Model” investigates the dynamics of a first-order phase transition via homogeneous thermal nucleation within the Polyakov quark–meson model at a finite temperature and density [15]. In order to incorporate the physical aspect of the confinement–deconfinement phase transition in the quark–meson model, the Polyakov loop operator is introduced. They found that, at a low density, the chiral phase transition is a crossover; however, it will terminate and change into a first-order phase transition around the critical endpoint (CEP) at a high chemical potential. By using a geometric method for the effective potential, the position of the CEP is precisely located at $(T_E,{\mu }_E)=$ (301.4 MeV, 62.1 MeV).

Dark matter is a hot topic in physics and astronomy. Yu Zhen, et al. proposed a singular way to search for dark matter by studying the structure and radial oscillations of strange quark stars admixed with fermionic dark matter [16]. In their work, the color–flavor-locked quark model for strange quark matter and ideal Fermi gas model at a temperature of zero for dark matter were used to investigate the equilibrium structure and radial oscillations of dark matter admixed strange stars. Based on the assumption that the dark matter and strange quark matter couple through gravity and oscillate with the same frequency, they found that the stellar maximum mass and radius are reduced by the inclusion of dark matter, as was the oscillation frequency $f_0$, and that there is a discontinuity of $f_0$ as functions of the stellar mass.

3. Summary

Symmetry principles continue to be a cornerstone of hadron physics, providing profound insights into the structure and dynamics of strongly interacting matter. This Special Issue has showcased a wealth of applications of dynamical symmetry across diverse fields of research, from the classification of exotic multiquark states, the internal structure of nucleons, and the precise study of light and heavy quarkonia to explorations of the QCD phase diagram. The collated works, experimental reviews, theoretical advances, and pedagogical article illustrate that symmetry-based approaches enable a systematic understanding of complex hadronic phenomena.

With the data accumulation of high-energy experiments, hadron physics research will enter a new stage—it is expected that more multiquark states and heavy quarkonia will be discovered, and their properties will be measured with precision. Furthermore, new facilities like the Electron-Ion Collider (EIC) will provide complementary research platforms, especially for studies of the light hadron spectrum and nucleon structure. On the theoretical side, further development of non-perturbative QCD methods, such as lattice QCD, QCD sum rules, the phenomenological quark model, etc., are needed to provide more reliable predictions and, in these, symmetry analysis will continue to be an essential tool and indispensable guide for hadron classification and property studies. The fruitful interplay between theoretical innovation and experimental discovery, as highlighted in this Special Issue, promises to deepen our understanding of hadron physics, and it is our hope that this Special Issue will serve as a resource for both novices and active researchers in hadron physics.