Three-Flavor NJL Mean-Field Model

• The paper presents a self-consistent mean-field framework using the three-flavor NJL model to capture chiral symmetry breaking and hadron-quark transitions with dynamical gap equations.
• It utilizes a combination of four-fermion and six-fermion (KMT) interactions, enhanced by the Polyakov loop, to unify hadronic and deconfined quark-gluon phases.
• Numerical parameter calibrations and extensions—including eight-quark terms and magnetic field effects—demonstrate the model’s efficacy in describing QCD thermodynamics and mesonic excitations.

A three-flavor Nambu–Jona-Lasinio (NJL) mean-field calculation is a self-consistent field-theoretical approach used to model the dynamical symmetry breaking, hadron-quark transitions, mesonic correlations, and bulk thermodynamics of strong interaction matter with three quark flavors (up, down, strange). Often combined with the Polyakov loop (yielding the PNJL model), this framework incorporates both four-fermion scalar-pseudoscalar interactions and the $U_A(1)$-breaking Kobayashi-Maskawa-'t Hooft (KMT) six-fermion determinant, allowing for a unified description of hadronic and deconfined quark-gluon phases (Yamazaki et al., 2013, Yamazaki et al., 2013, Hiller et al., 2012, Contrera et al., 2016). Variants include extensions to include eight-quark, tensor, isospin, or vector interactions, nonlocality, and background fields (Moreira et al., 2017, Braghin, 13 Jan 2025, Hell et al., 2011, Morimoto et al., 2020, Liu et al., 2021).

1. Core Lagrangian Structure and Interactions

The three-flavor NJL model is based on the fermionic Lagrangian: $$\mathcal{L}_{\rm NJL} = \bar\psi(i\gamma^\mu D_\mu - \hat m)\psi + G\sum_{a=0}^8 \Big[(\bar\psi\lambda^a\psi)^2 + (\bar\psi i\gamma_5 \lambda^a\psi)^2\Big] - K \Big[\det \bar\psi(1+\gamma_5)\psi + \det \bar\psi(1-\gamma_5)\psi\Big]$$ where $\psi=(u, d, s)^T$ is the quark field, $\hat m=\mathrm{diag}(m_u, m_d, m_s)$ the current quark masses, $D_\mu$ may include static gluonic backgrounds (as for PNJL), and $\lambda^a$ are the Gell-Mann matrices. The four-fermion term $G$ drives spontaneous chiral symmetry breaking; $K$ is the strength of the KMT term, which induces $U_A(1)$ anomaly effects including mixing between different flavors and lifting the $\eta'$ mass (Yamazaki et al., 2013).

Extensions introduce flavor-mixing four-fermion couplings $G_{fg}$, explicit symmetry breaking and eight-fermion terms, or additional tensor, isovector, or vector channels (Braghin, 13 Jan 2025, Moreira et al., 2017). In the PNJL model, the ruling degrees of confinement are incorporated via a Polyakov-loop background; the effective potential $\mathcal{U}(\Phi, \bar\Phi; T)$ models thermally driven gluon dynamics: $$\frac{\mathcal{U}(\Phi, \bar\Phi; T)}{T^4} = -\frac{b_2(T)}{2}\,\bar\Phi\Phi -\frac{b_3}{6}(\Phi^3+\bar\Phi^3) +\frac{b_4}{4}(\bar\Phi\Phi)^2$$ with $b_2(T)$ fitted to lattice QCD pure gauge results (Yamazaki et al., 2013, Yamazaki et al., 2013).

2. Mean-Field Factorization and Gap Equations

The multi-fermion operators are replaced in mean field by linear combinations of scalar condensates: $$\sigma_f = \langle \bar\psi_f \psi_f \rangle, \quad f\in\{u,d,s\}$$ Variations with respect to these condensates yield coupled "gap" equations for the constituent masses: $$M_u = m_u - 4G \sigma_u + 2K \sigma_d \sigma_s, \quad M_d = m_d - 4G \sigma_d + 2K \sigma_s \sigma_u, \quad M_s = m_s - 4G \sigma_s + 2K \sigma_u \sigma_d$$ These equations generalize in the presence of flavor mixing, eight-quark, or explicit breaking interactions (Braghin, 13 Jan 2025, Moreira et al., 2017, Hiller et al., 2012).

For vector or tensor-extended models, additional mean fields and coupled equations must be added (e.g., vector densities $$\rho_f = \langle \bar\psi_f \gamma^0 \psi_f \rangle$$, tensor condensates $F_3, F_8$) with corresponding stationarity conditions (Morimoto et al., 2020, Contrera et al., 2016).

In the presence of a Polyakov loop, one must also extremize $\Omega$ with respect to $\Phi, \bar\Phi$.

3. Thermodynamic Potential and Equation of State

The grand canonical thermodynamic potential per volume is, at $T > 0$, for PNJL: $$\Omega(T; \{M_f\}, \Phi, \bar\Phi) = \mathcal{U}(\Phi, \bar\Phi; T) + 2G \sum_f \sigma_f^2 - 4K \sigma_u \sigma_d \sigma_s + \Omega_{\mathrm{vac}} + \Omega_{\mathrm{th}}$$ with

$$\Omega_{\mathrm{vac}} = -2N_c \sum_f \int_{|\mathbf{p}|<\Lambda} \frac{d^3p}{(2\pi)^3} E_f(p)$$

$$\Omega_{\mathrm{th}} = -2T \!\!\sum_{f=u,d,s} \int \!\frac{d^3p}{(2\pi)^3} \ln\left[1 + 3\Phi e^{-\beta E_f} + 3\bar\Phi e^{-2\beta E_f} + e^{-3\beta E_f}\right]$$

where $E_f(p) = \sqrt{p^2 + M_f^2}$ (Yamazaki et al., 2013, Yamazaki et al., 2013). Regularization is typically implemented via a sharp three-momentum cutoff for vacuum (zero-point) parts; thermal quasiparticle pieces are finite.

At $T=0$, the EoS for cold dense quark matter relevant to stars is governed by the Fermi seas, with the pressure $P = -\Omega$ and energy density $\epsilon = \Omega + \sum_f \mu_f n_f$, where Fermi momenta are set by the (possibly shifted) chemical potentials. The bulk equation of state is then constructed numerically as $p(\epsilon)$ (Contrera et al., 2016, Ranea-Sandoval et al., 2015, Wang et al., 2019).

4. Numerical Computation and Parameter Calibration

The self-consistent solution proceeds by initializing a guess for the set of order parameters (condensates, Polyakov loops, possibly vector/tensor fields), solving simultaneously the system of coupled nonlinear equations for a grid in $T$ and (if relevant) quark chemical potentials $\mu_f$ (Yamazaki et al., 2013, Morimoto et al., 2020). Standard techniques such as Newton-Raphson or fixed-point iteration are employed. Model parameters $\{m_u, m_d, m_s, G, K, \Lambda, T_0, \cdots\}$ are fixed by requiring reproduction of vacuum hadronic observables such as pion and kaon masses, $f_\pi$, and $\eta'$ mass (Yamazaki et al., 2013, Hiller et al., 2012).

For PNJL, the Polyakov-loop potential parameters are fitted to lattice glue pressure. In advanced variants, Pauli-Villars or proper-time regularizations are used for improved UV behavior (Wang et al., 2019, Moreira et al., 2017).

5. Mesonic and Collective Excitations Beyond Mean Field

Mesonic correlations as collective excitations are included by bosonizing the theory via Hubbard–Stratonovich transformation, introducing auxiliary fields $\phi^a,\pi^a$. The quadratic expansion around the mean-field saddle yields mesonic propagators: $$D_\alpha^{-1}(\omega_n, \mathbf{q}) = \frac{1}{2G_\alpha'} - \Pi_\alpha(\omega_n, \mathbf{q})$$ where $G'_\alpha$ are (possibly KMT-shifted) channel-dependent couplings, and $\Pi_\alpha$ are polarization functions evaluated through quark loops with Polyakov-averaged quasiparticle distributions. The locations of the poles in the complex-$\omega$ plane define in-medium meson masses and widths. The mesonic pressure is computed via

$$p_M = -\frac{T}{2V} \sum_{\alpha,a} \mathrm{Tr} \ln D_\alpha^{-1}$$

This mechanism unifies the description of hadronic (meson-dominated) and quark-gluon plasma regimes within a single framework (Yamazaki et al., 2013, Yamazaki et al., 2013).

6. Physical Results: Phase Transitions and Observables

In three-flavor PNJL mean field, below $T_c$ ($\sim 220$ MeV, with typical $T_0=270$ MeV), the system is a hadronic gas dominated by low-mass mesonic collective modes (pions, kaons)—quark thermal excitations are suppressed by the Polyakov loop. As $T$ increases, the condensates $\sigma_{u,d}$ drop sharply near $T_c$, while $\sigma_s$ persists to higher $T$; the Polyakov loop rises from near zero to unity, signaling deconfinement. The sequential "Mott" dissociation of kaons then pions is observed above $T_c$ (melting at $\sim 1.15\,T_c$ and $1.2\,T_c$, respectively) (Yamazaki et al., 2013).

At high $T$, quark and gluon quasiparticles dominate the pressure; the phase change is continuous (crossover) at zero density. The unified PNJL calculation thus captures both chiral restoration and deconfinement, consistently with lattice QCD, and provides the full $p(T)$ from hadron-dominated to quark-gluon-dominated regimes (Yamazaki et al., 2013, Hiller et al., 2012).

7. Extensions, Limitations, and Physical Implications

Three-flavor NJL/PNJL mean-field schemes admit substantial extensions:

• Eight-quark and explicit chiral-breaking interactions are required for vacuum stability and quantitative meson mass fits; their variation tunes the critical endpoint (CEP) location in the $(T, \mu)$ plane (Moreira et al., 2017, Hiller et al., 2012).
• In background magnetic fields, the mean-field solution exhibits magnetic catalysis in vacuum (enhancement of chiral symmetry breaking), but at finite $T, \mu$ the position of critical points is shifted to higher $T$ and lower $\mu$ (Moreira et al., 2017).
• Finite-volume effects suppress the chiral broken phase, leading to a reduction in constituent masses as $L$ decreases, with a smooth crossover to current quark masses around $L \sim 1.5$ fm (Abreu et al., 2019).
• The inclusion of diquark pairing (as in color-flavor-locking) or tensor channels leads to superconducting or spin-polarized phases, important in astrophysical contexts (Paulucci et al., 2013, Morimoto et al., 2020).
• Flavor mixing couplings $G_{fg}$ introduce corrections to the standard SU(3)-symmetric case; even modest off-diagonal $G_{fg}$ significantly couple the $u$, $d$, and $s$ sectors (Braghin, 13 Jan 2025).

The model is nonrenormalizable, requiring regularization; all physical results depend on the chosen cutoff, and absolute predictions (e.g., for absolute binding of strange quark matter) are sensitive to parameter choices (Wang et al., 2019). Empirically, the PNJL and its extensions offer a semi-quantitative but uncontrolled effective field theory approximation to low- and intermediate-energy QCD, with relevance for heavy-ion physics, compact stars, and the QCD phase diagram.