Baryonic Extended Linear Sigma Model

• The baryonic eLSM is a chiral effective theory that integrates mesons and baryons via a mirror assignment to reproduce parity doublets and empirical mass hierarchies.
• It incorporates symmetry breaking and empirical constraints to achieve accurate predictions for vacuum hadron masses, decay widths, and nuclear matter saturation properties.
• The model extends to dense and hot matter, enabling predictions for neutron star structure and the QCD phase diagram through in-medium modifications of hadronic properties.

The baryonic extended Linear Sigma Model (eLSM) is a chiral effective theory that systematically incorporates the full spectrum of low-lying hadrons—(pseudo)scalar and (axial-)vector mesons, glueballs, and baryons—into a Lagrangian framework respecting the symmetry structure of Quantum Chromodynamics (QCD) at low energies. By extending the original linear sigma model to three flavors and coupling in baryonic degrees of freedom, particularly via the mirror assignment, the eLSM enables unified descriptions of vacuum hadron phenomenology, nuclear matter, and strongly interacting matter at high density and temperature. The approach is characterized by a global $SU(3)_L \times SU(3)_R$ chiral symmetry, extended mass and interaction terms, explicit symmetry breaking through the inclusion of current quark masses, and constraints from empirical hadronic data and nuclear saturation properties (Kovacs et al., 2013, Kovacs et al., 2013, Ma et al., 29 Dec 2025, Olbrich et al., 2016).

1. Model Structure and Symmetry Content

The baryonic eLSM is constructed to realize chiral $SU(3)_L \times SU(3)_R$ symmetry linearly, embedding both mesonic and baryonic multiplets in a manner consistent with QCD. The field content encompasses:

• A nonet of scalar and pseudoscalar mesons ($\Phi = S + iP$)
• Nonets of vector and axial-vector mesons ($L_\mu = V_\mu + A_\mu$, $R_\mu = V_\mu - A_\mu$)
• Baryon octet and decuplet fields, most effectively treated using the quark–diquark formalism and mirror assignment for chiral transformation properties

The generic Lagrangian form is

$$\mathcal{L} = \mathcal{L}_{\rm meson} + \mathcal{L}_{\rm baryon} + \mathcal{L}_{\rm vec} + \mathcal{L}_{\rm SB}$$

where $\mathcal{L}_{\rm meson}$ and $\mathcal{L}_{\rm vec}$ contain mesonic kinetic, mass, and (self-)interaction terms, while $\mathcal{L}_{\rm baryon}$ encodes baryon kinetic terms, chirally invariant mass generation (both through spontaneous symmetry breaking and explicit mirror assignment), and Yukawa-type couplings to the meson fields (Kovacs et al., 2013, Olbrich et al., 2016).

Spontaneous chiral symmetry breaking is realized via vacuum expectation values (vevs) of nonstrange and strange scalar fields ($\langle\sigma_N\rangle = \phi_N$, $\langle\sigma_S\rangle = \phi_S$), which generate tree-level hadron masses and reproduce the empirical mass hierarchy.

2. Mirror Assignment and Baryon Parity Doubling

Crucially, the baryonic sector employs the mirror assignment, introducing two (three-flavor) Dirac multiplets with distinct chiral transformation properties:

• "Standard" multiplets ($N_1$, $N_2$) with conventional chiral transformation
• "Mirror" multiplets ($M_1$, $M_2$) transforming oppositely under $SU(3)_L \times SU(3)_R$

This construction allows for chirally invariant mass terms, enabling parity doublets for baryons. After symmetry breaking, physical nucleons and their negative-parity partners mix as dictated by the full mass matrix. In the two-flavor sector, diagonalization yields four doublets corresponding to $N(939)$, $N(1440)$, $N(1535)$, and $N(1650)$, reproducing both masses and decay widths to empirical accuracy (Olbrich et al., 2016). In the chiral-restoration limit, states organize into parity doublets consistent with the predicted structure:

• $N(939)$, $N(1535)$ (first doublet)
• $N(1440)$, $N(1650)$ (second doublet)

3. Parameter Determination and Empirical Constraints

Model parameters, including couplings, chiral-invariant bare masses, and explicit-breaking coefficients, are fitted using multiparametric minimization procedures to a wide array of observables:

• Vacuum masses of octet and decuplet baryons and low-lying mesons
• Nuclear matter saturation properties ($n_0$, $E/A$, $K$, $E_{\rm sym}$)
• Hyperon single-particle potentials
• Meson and baryon decay widths
• Axial couplings and empirical sigma-term constraints

Typical parameter sets reproduce all light baryon and meson masses within $1-2\%$ of experimental values, with baryon octet and decuplet decay widths agreeing to within $20-40\%$ depending on the sector (Kovacs et al., 2013, Ma et al., 29 Dec 2025). The fit requires attention to the pion–nucleon sigma term $\sigma_{\pi N}$, which is shown to require significant density dependence at higher baryon densities to ensure physical solutions in compact star environments (Ma et al., 29 Dec 2025).

Parameter	Example Value	Sector/Fit Role
$c_2$, $c_4$	$2.38\times10^5$ MeV$^2$	Meson mass/gap terms
$g_{\sigma NN}$	$-9.35$	Scalar Yukawa (nuclear matter saturation)
$M_0$ (chiral-inv.)	$0.8$ GeV	Parity-doublet mass splitting/fit
$\sigma_{\pi N}$	$-600$ MeV (high density)	EOS, neutron-star support

4. Nuclear Matter, Neutron Stars, and the Equation of State

The bELSM framework is extended to dense and/or hot matter via relativistic mean-field (RMF) and mean-field thermodynamic treatment. The mesonic mean fields $\sigma, \omega, \rho$ are treated as classical backgrounds, coupling to baryon scalar and vector densities, and field equations are solved self-consistently for arbitrary proton, neutron, and hyperon densities in beta-equilibrium (Ma et al., 29 Dec 2025, Giacosa, 2016).

Key results include:

• Realistic reproduction of nuclear matter saturation properties and the emergence of a stiff EOS when the $\sigma_{\pi N}$ is made strongly negative at high density, avoiding pathologies in scalar gap equations at $n \gtrsim 3n_0$
• Delayed hyperon onset in neutron-star matter due to vector coupling structure, addressing the "hyperon softening" problem
• Equation of state predictions supporting maximum neutron star masses $M_{\rm max} \sim 2.0-2.1\,M_\odot$ with $R(1.4\,M_\odot)\sim 11-12$ km, consistent with astrophysical constraints (Ma et al., 29 Dec 2025)

5. Extended Phenomenology: Meson Mass Shifts and Phase Structure

The eLSM framework incorporates in-medium modifications of hadron properties, with one-loop nucleon corrections yielding decreasing meson masses at finite density (except for the pion, kaon, and lightest scalar-isoscalar) and density-dependent shifts in vector meson masses, sensitive to the parity-doublet chiral invariant mass $M_0$ (Suenaga et al., 2019). Empirically preferred values $M_0\sim 0.8$ GeV reproduce small downward shifts for the $\omega$ in medium and correct nuclear compressibility.

At finite temperature and chemical potential, the Polyakov-extended eLSM predicts a critical end point (CEP) in the QCD phase diagram with baryon number kurtosis $\kappa\sigma^2$ diverging at the CEP, quantitatively matching lattice QCD baryon susceptibilities at $\mu_B = 0$ (Kovacs et al., 2020).

6. Role of Exotic Components and Inhomogeneous Phases

The model is adapted to account for exotic low-energy degrees of freedom—e.g., including the light four-quark scalar $f_0(500)$ as a chiral singlet to accurately capture the intermediate range nucleon–nucleon attraction and nuclear matter saturation (Giacosa, 2016). The field $\chi$ replaces the explicit $m_0$ mass term, leading to a dynamically generated mirror mass.

Inhomogeneous chiral symmetry breaking phases, modeled by chiral–spiral ansatz for mesonic fields, are energetically favored at moderate baryon densities ($\rho\sim2-3\rho_0$), potentially delaying or precluding the complete restoration of chiral symmetry with increasing density (Giacosa, 2016).

7. Significance and Outlook

The baryonic extended Linear Sigma Model provides a chiral-symmetry-based, QCD-constrained approach to describing the spectrum and interactions of low-energy hadrons, the structure of nuclear and neutron-star matter, and the phase structure of QCD. Its simultaneous constraints on empirical masses, decays, nuclear matter properties, and compact star observables underline its utility for nuclear and hadron phenomenology. The observed requirement for a density-dependent explicit chiral symmetry breaking term suggests an important, previously underappreciated, interplay between in-medium QCD symmetry structure and astrophysical observables, motivating further investigations of the medium dependence of chiral order parameters and hadronic effective interactions (Ma et al., 29 Dec 2025, Kovacs et al., 2013, Olbrich et al., 2016).