Quark models: What can they teach us?

TL;DR

The paper assesses how quark models can illuminate chiral dynamics in QCD by focusing on a chiral, field-theory–based quark model (GNJL) with confinement. It derives a mass-gap equation, demonstrates a BCS-like chirally broken vacuum, and identifies the pion as a Goldstone boson that also appears as the light $q\bar{q}$ bound state, connecting vacuum structure to hadron spectra via a Bethe–Salpeter framework and a Bogoliubov-like meson formalism. At finite temperature, the model predicts chiral restoration driven by Pauli blocking rather than deconfinement, with the spectrum reorganizing into chiral multiplets and indicating emergent symmetries like $SU(2)_{CS}$ for highly excited states. The GNJL approach thus offers a tractable, microscopic picture that links vacuum condensation, bound-state dynamics, and thermal QCD phenomena, providing qualitative—and in places quantitative—insights that complement lattice studies and experiment.

Abstract

Quark models have a more than 60-year history and through this time they served as a powerful investigation and prediction tool in hadronic physics. In recent years, a lot of new experimental information has been arriving on hadrons that do not qualify as simple quark model states. Yet, quark models remain the cornerstone of the classification scheme for hadrons, provide valuable insights into various phenomena inherent in QCD, and facilitate gaining a clear and physically transparent picture of the underlying physics. In the spotlight of this review is a chiral quark model inspired by quantum field theory approach to confined quarks. The model is well suited for studies of spontaneous breaking of chiral symmetry in the vacuum of QCD as well as its implications in the spectrum of hadrons. It can also be employed to investigate chiral restoration at finite temperatures.

Figures (14)

• Figure 1: The Dyson series in Eq. \ref{['Ds']} for the dressed quark propagator $S$ in Eq. \ref{['Feynman']}, with the bare propagator $S_0$ in Eq. \ref{['S0T0']} and the quark mass operator $\Sigma$ in Eq. \ref{['Sigma0']} taken in the rainbow approximation. The thin and thick solid lines are for the bare and dressed quark propagator, respectively, and the curly line is for the confining "gluon propagator" --- see footnote \ref{['gluonprop']}. Adapted from Glozman:2024dzz.
• Figure 2: The profile of the chiral angle --- solution to the mass-gap equation in Eq. \ref{['mge']} (left plot) and the effective quark mass $M_p$ (right plot) for the linear confinement in Eq. \ref{['Vlin2']}.
• Figure 3: The behaviour of the minimally subtracted dressed quark dispersion law $E_p^{\rm fin}$ defined in Eq. \ref{['Epir']} (the blue solid line) and the effective quark energy $\omega_p$ in Eq. \ref{['omegadef']} (the yellow solid line) for the linear confinement in Eq. \ref{['Vlin2']}. The black dashed line shows the approximation $(E_p^{\rm fin})_{\rm app}$ in Eq. \ref{['Epfit']}.
• Figure 4: The temperature dependence of (i) the chiral angle (upper left plot), (ii) the effective quark mass (upper right plot), and (iii) the damping factor $1-n_p-\bar{n}_p$ in Eq. \ref{['tildeAB']} (lower left plot) for a vanishing chemical potential ($\mu=0$) and $T=0$ (the blue curve), $T=0.95T_{\rm ch}$ (the yellow curve), and $T=0.99T_{\rm ch}$ (the green curve). The lower right plot: the temperature dependence of the chiral condensate. The value of $T_{\rm ch}$ is quoted in Table \ref{['tab:res']}. The confining potential is chosen in the linear form in Eq. \ref{['Vlin2']} and all dimensional quantities are provided in the appropriate units of the string tension $\sigma$. The current quark mass $m$ is set to zero. Adapted from Glozman:2024xll.
• Figure 5: The Regge trajectories for the experimentally measured masses (black dots) of several low-lying $\bar{q}q$ (with $q$ for the $u$ or $d$ quark) mesons with isospin $I=1$ and isoscalar $\bar{s}s$ mesons ParticleDataGroup:2024cfk. Linear fits are shown as blue lines.