Chiral SU(3) Model Overview

• Chiral SU(3) model is a framework that incorporates both spontaneous and explicit chiral symmetry breaking to construct effective theories of baryons and mesons.
• It systematically derives meson-baryon interactions, explains phenomena like the two-pole structure of the Λ(1405), and models phase transitions in dense matter.
• The approach extends to computing thermodynamic properties and transport coefficients in nuclear and astrophysical contexts, bridging quark-meson couplings with hadronic observables.

The chiral SU(3) model is a framework for constructing effective theories of hadronic and nuclear matter that incorporates the spontaneous and explicit breaking of chiral SU(3)$_L\times$SU(3)$_R$ symmetry—fundamental to low-energy QCD—into a quasi-realistic description of baryons, mesons, and their interactions. It provides a systematic basis for deriving interactions among baryons and mesons, describing phase transitions such as chiral symmetry restoration, and connecting with observables ranging from baryon masses to transport properties and astrophysical phenomena. Critical to the chiral SU(3) approach is the interplay between symmetry- and symmetry-breaking terms, their realization at both the quark-meson and hadronic levels, and the nonperturbative treatment of strong coupling and collective phenomena.

1. SU(3) Chiral Lagrangians and Symmetry Breaking

Chiral SU(3) models are built from Lagrangians invariant under SU(3)$_L\times$SU(3)$_R$ symmetry, realized either linearly or non-linearly depending on the physical context. In the meson-baryon sector, the leading-order (LO) Lagrangian in the heavy-baryon formalism for the baryon octet $B$ is

$$\mathcal{L}_{MB}^{(1)} = \langle\bar B(i v\cdot D)B\rangle + D\,\langle\bar B S^\mu\{u_\mu,B\}\rangle + F\,\langle\bar B S^\mu[u_\mu,B]\rangle - M_0\,\langle\bar B B\rangle,$$

where $$u_\mu = i\left(u^\dagger\partial_\mu u - u\,\partial_\mu u^\dagger\right)$$ with $u = \exp(i\Phi/2f)$, and $\Phi$ collects the pseudoscalar octet fields. The NLO ($p^2$) sector introduces contact terms with LECs $b_0, b_D, b_F$ (coupling to the explicit chiral-breaking spurion $$\chi_+ = 2B_0(u^\dagger \mathcal{M} u^\dagger + u \mathcal{M} u)$$ for $\mathcal{M} = \mathrm{diag}(m_u, m_d, m_s)$) and further terms with $d_i$ constants.

The vector meson, scalar meson, and dilaton (to mimic the QCD trace anomaly) sectors are constructed non-linearly or linearly depending on application. For nucleonic and nuclear matter, explicit symmetry breaking through non-zero $m_s$ and $m_{u/d}$ is crucial for splitting the strange and non-strange sectors, as reflected in all modern variants (0712.1613, Mishra et al., 4 Dec 2025, Shivam et al., 2019).

Spontaneous symmetry breaking arises when the scalar meson fields acquire non-zero vevs. In $SU(3)_{L}\times SU(3)_{R}$ linear models, the chiral breaking pattern yields non-strange and strange condensates $\langle\sigma_n\rangle \simeq f_\pi$, $\langle\sigma_s\rangle \simeq 1.4 f_\pi$, breaking $SU(3)_L \times SU(3)_R \to SU(3)_V$ (Zacchi et al., 2016).

2. Implementation in Baryon and Hadronic Sectors

At the quark level, chiral SU(3) models are realized as:

• Chiral quark-meson models for dense and hot matter—where a linear meson nonet field couples via a Yukawa term to $u$, $d$, and $s$ quarks—and the resulting effective potential is minimized to yield mass gap equations and the equation of state (Zacchi et al., 2016, Mishra et al., 4 Dec 2025, 0802.1999).
• Rigidly quantized solitonic approaches, such as the SU(3) chiral quark-soliton model (χQSM), which provides a nonperturbative, semiclassical tool for physical baryons and allows for computation of static properties and form factors, including flavor SU(3) breaking via the explicit $m_s$ corrections (Ledwig et al., 2010, 0806.4072).

Baryonic parity doublet models can implement different chiral representations, allowing for an accurate reproduction of detailed mass hierarchies (notably $\Sigma$-$\Xi$ ordering) by explicitly mixing $(3, \bar 3) + (\bar 3, 3)$ ("good diquark") and $(3,6) + (6,3)$ ("bad diquark") sectors. Both spontaneous and explicit symmetry breaking via bare quark masses are essential to recover observed spectra (Gao et al., 1 Dec 2025).

3. Coupled-Channel Dynamics and the Meson-Baryon Interaction

For the description of $s$-wave meson-baryon scattering (e.g., low-energy $\bar KN$ dynamics), the leading contribution—the Weinberg-Tomozawa term—arises as the LO chiral $s$-wave meson-baryon contact interaction: $$\mathcal{L}_{WT} = -\frac{1}{4f^2} \langle \bar B \gamma^\mu [\Phi, \partial_\mu \Phi] B \rangle$$ with the scattering potential built from SU(3) Clebsch-Gordan coefficients. The full amplitude $T_{ij}(\sqrt s)$ is obtained by solving a coupled-channel unitarized Bethe–Salpeter equation with dimensional regularization and matched to threshold and subthreshold observables using as parameters both decay constants and subtraction constants $a_i(\mu)$ (0712.1613). Higher-order NLO terms can be included as energy-independent local pieces to increase the fit quality.

A key feature in this framework is the emergence of the two-pole structure of the $\Lambda(1405)$, with one pole strongly coupled to $\bar K N$ and another to $\pi\Sigma$, explaining the distinct behaviors of the $\bar K N$ scattering amplitude and the peculiarities of the $\pi\Sigma$ spectrum.

4. Thermodynamic and Transport Properties of Nuclear and Quark Matter

The chiral SU(3) model forms the basis for the construction of equations of state $p(\epsilon)$ for hot, dense matter in astrophysical and heavy-ion contexts. By solving self-consistency conditions for the mean-field expectation values of the scalar ($\sigma$, $\zeta$, $\delta$) and vector ($\omega$, $\rho$) fields, one computes in-medium masses, chemical potentials, pressure, energy density, and entropy (Mishra et al., 4 Dec 2025, 0802.1999, Zacchi et al., 2016).

Transport coefficients (shear viscosity $\eta$, thermal conductivity $\kappa$) are derived within the relaxation-time approximation, with the medium-modified single-particle properties and effective relaxation times entering integrals over thermal distributions. Strong interaction effects, notably the reduction in effective nucleon mass and the presence of strong mean fields, reduce $\eta$, enhance $\kappa$, and lead to a ratio $\eta/s$ approaching the KSS bound at high baryon density and moderate temperature—regimes relevant for the CBM program at FAIR-GSI (Mishra et al., 4 Dec 2025). Isospin asymmetry ($\alpha$) marginally affects $\eta$, but can enhance $\kappa$ significantly.

In neutron star and proto-neutron star applications, the chiral SU(3) EoS provides mass-radius relations consistent with $2M_\odot$ stars, tracks the crossover nature of the chiral phase transition, and describes the effects of finite temperature, lepton fraction, and rotation (0802.1999).

5. Chiral SU(3) Dynamics in Few-Body and Exotic Systems

Chiral SU(3) models have been foundational in the derivation of effective $\bar K N$ interaction potentials relevant to few-body deeply bound systems. The effective potential, derived from elimination of non-$\bar K N$ channels in the full coupled-channel framework, is generally shallower and more energy-dependent than purely phenomenological (Akaishi–Yamazaki) forms—resolving the puzzle of the moderate binding and broad widths of $K^-pp$ and similar states. This substantially reduces the likelihood of narrow, deeply bound kaonic nuclei and has direct implications for ongoing search strategies (0712.1613).

For baryon-baryon and multiquark systems, the chiral SU(3) quark model reproduces the nucleon-nucleon interaction, deuteron binding, and the baryon ground-state spectrum quantitatively, demonstrating the relative roles of OGE, confining, and chiral mesonic forces. Application to heavy-light four-quark states predicts a bound $bb\bar n\bar n$ ($J^P=1^+$, $I=0$) with chiral meson exchange providing non-negligible additional attraction; the analogous $cc\bar n\bar n$ state remains unbound (Huang et al., 2018, 0711.1029).

6. Chiral SU(3) in Many-Body, Finite-Temperature, and Topological Systems

At nonzero temperature, chiral SU(3) models are generalized to include Polyakov loop dynamics (PNJL models), nonlocal interactions, thermal gluon backgrounds, and wave-function renormalization, capturing both chiral restoration and deconfinement transitions. These models reproduce lattice QCD results for order parameters, thermodynamic quantities, and susceptibilities, typically predicting coincident or near-coincident chiral/deconfinement crossovers around $T_c \sim 160$–$170$ MeV (Carlomagno et al., 2013, Pisarski et al., 2016). A two-branch “twin star” structure, with distinct radii but identical masses, can arise from the nonlinearity associated with the chiral phase transition (Zacchi et al., 2016).

In strongly correlated condensed-matter analogs, “chiral SU(3)” phases describe quantized spin liquids and SPT phases in lattice models, featuring topological order, quantized Hall conductivities, and symmetry-breaking transitions—demonstrating the broad applicability of chiral SU(3) symmetry beyond hadronic and nuclear systems (Zhang et al., 11 Sep 2025, Lai, 2013).

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References

• Effective $\bar K N$ interaction and two-pole $\Lambda(1405)$: (0712.1613)
• SU(3) chiral quark-meson model and twin stars: (Zacchi et al., 2016)
• Tensor charges in SU(3) chiral quark-soliton model: (Ledwig et al., 2010)
• Linear SU(3) parity doublet baryon model: (Gao et al., 1 Dec 2025)
• Chiral SU(3) applications to neutron stars and thermodynamics: (0802.1999, Mishra et al., 4 Dec 2025)
• Chiral ring-exchange models and chiral Haldane phases: (Zhang et al., 11 Sep 2025, Lai, 2013)
• Approximating chiral SU(3) amplitudes: (Ecker et al., 2013)
• Four-quark bound states in the chiral SU(3) model: (0711.1029)
• Chiral matrix and nonlocal PNJL models: (Pisarski et al., 2016, Carlomagno et al., 2013)
• Nucleon-nucleon interaction in chiral SU(3) quark models: (Huang et al., 2018)