Chiral Quark Model: Effective QCD Theories

• The chiral quark model is a family of effective low-energy QCD theories that implements spontaneous chiral symmetry breaking directly in quark dynamics through mechanisms like Goldstone boson dressing and soliton fields.
• It encompasses diverse formulations—including constituent quark, soliton, NJL-type, quark–meson, and quark–diquark models—that address nucleon flavor asymmetries, meson PDFs, and dense matter properties.
• These models offer practical insights into QCD phenomena, contributing to our understanding of sea-quark asymmetries, hadronic reactions, neutron-star matter, and the interplay between confinement and chiral symmetry.

Searching arXiv for the cited papers to ground the article in the current record. arXiv search: chiral quark model CQSM NJL parity doublet neutron stars The expression chiral quark model is used in the cited literature for a family of effective low-energy QCD descriptions in which spontaneous chiral symmetry breaking is encoded directly in quark dynamics. Depending on the realization, the active degrees of freedom are constituent quarks dressed by Goldstone bosons, quarks moving in a self-consistent chiral soliton, quarks coupled to $\sigma$ and $\pi$ fields, quark–diquark baryonic multiplets with mirror assignment, or NJL-type quark matter embedded in hadron–quark crossover constructions (Wakamatsu, 2 Jun 2026, Dahiya, 2015, Rabhi et al., 2011, Minamikawa et al., 2022). In all of these cases, the defining theme is that chiral symmetry and its breaking are not imposed only at the hadronic level: they enter the quark sector itself through nonlinear pion fields, scalar condensates, chirally invariant mass terms, or quark–antiquark condensates.

Representative uses in the cited literature can be organized as follows.

Realization	Characteristic degrees of freedom	Representative use
Chiral constituent quark model	Constituent quarks + Goldstone bosons	Sea asymmetries, octet flavor observables
Chiral quark soliton model	Quarks in a rotating hedgehog pion field	Nucleon PDFs and polarized sea
NJL-type chiral quark model	Three-flavor quarks with scalar, vector, diquark channels	Dense matter and neutron stars
Quark–meson / sigma model	Quarks + $\sigma,\pi$ mean fields	Magnetized quark matter
Quark–diquark mirror model	Baryonic multiplets built from quarks and diquarks	Parity partners in the nucleon sector

1. Conceptual scope and chiral structure

In the constituent-quark versions, the low-energy region between the confinement scale and the chiral-symmetry-breaking scale is described in terms of constituent quarks, Goldstone bosons, and weakly interacting gluons, with the chiral fields encoding the Nambu–Goldstone sector generated by spontaneous symmetry breaking (Dahiya, 2015, Song et al., 2010). A minimal interaction can be written as

$\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$

or, in derivative form,

$${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$

depending on the formulation (Dahiya, 2015, Watanabe et al., 2017).

A more microscopic route starts from QCD with confinement. In the vacuum correlator formalism, the confining kernel generates a scalar mass operator dressed by a nonlinear chiral field, leading to

$$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$

so that the effective chiral Lagrangian and, in a local limit, the chiral quark model emerge from the same confining structure (Simonov, 2015). This formulation makes the relation between confinement and chiral symmetry breaking explicit: the scalar confining kernel produces the mass operator, while $U$ supplies the nonlinear Goldstone sector.

The surveyed papers also show that the label is not restricted to one degree-of-freedom content. In some cases baryons are explicit quark systems; in others they are solitons, or effective hadronic fields whose transformation properties are motivated by quark and diquark substructure (Wakamatsu, 2 Jun 2026, Olbrich et al., 2015). Taken together, these works indicate that chiral quark model functions less as a single Lagrangian than as a class of quark-based chiral effective theories.

2. Constituent-quark and Goldstone-boson formulations

The standard chiral constituent-quark picture represents a constituent quark as a dressed object that fluctuates through Goldstone-boson emission,

$q \to q' + {\rm GB} \to q' + (q\bar q'),$

thereby generating a nonperturbative sea (Song et al., 2010). In the octet-baryon application, this mechanism yields explicit sea-quark flavor distributions and predicts, for the nucleon, $\bar u^N-\bar d^N=-0.118$ and $\bar u^N/\bar d^N=0.652$, while for $\pi$0 it gives $\pi$1 (Dahiya, 2015). The same framework is used to compute flavor fractions, generalized Gottfried integrals, and sigma terms such as $\pi$2, $\pi$3, and $\pi$4 across the octet (Dahiya, 2015).

When proton–neutron isospin symmetry breaking is introduced through mass splittings within isospin multiplets, the same Goldstone-boson dressing generates nonzero $\pi$5, $\pi$6, $\pi$7, and $\pi$8. Within this framework, the violation of the Gottfried sum rule is still dominated by flavor asymmetry rather than by isospin breaking, and the correction to the NuTeV anomaly has the opposite sign from what would be needed to remove the anomaly (Song et al., 2010).

The chiral constituent-quark model has also been extended to meson PDFs. For the kaon, the dressed valence distributions are obtained from bare inputs plus meson-cloud convolutions and then evolved from $\pi$9 with NLO DGLAP evolution. The fitted bare exponents are $\sigma,\pi$0 and $\sigma,\pi$1, while the renormalization constants are $\sigma,\pi$2 and $\sigma,\pi$3 (Watanabe et al., 2017). A central conclusion is that the resulting $\sigma,\pi$4-flavor symmetry breaking in kaon valence PDFs is smaller than in several preceding approaches (Watanabe et al., 2017).

A distinct nonlocal variant extends the same logic into heavy-light systems by combining HQEFT kinematics with nonlocal constituent masses,

$\sigma,\pi$5

with $\sigma,\pi$6 MeV (Nam, 2012). In that model, the computed heavy-meson decay constants are

$\sigma,\pi$7

showing how chiral dressing can be merged with heavy-quark effective kinematics in a single effective framework (Nam, 2012).

3. Solitonic, confining, and mirror-assignment realizations

In the chiral quark soliton model, baryons are not treated as fixed three-quark bound states but as quarks moving in a self-consistent hedgehog pion field. The basic Lagrangian is

$\sigma,\pi$8

and the nucleon emerges after collective quantization of the rotating soliton (Wakamatsu, 2 Jun 2026). Because the model retains explicit quark fields, it can evaluate nonlocal bilinears and hence parton distributions. At the model scale $\sigma,\pi$9, it predicts $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$0, a negative small-$\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$1 isoscalar helicity distribution, and $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$2; for the polarized sea it finds $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$3 and $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$4 (Wakamatsu, 2 Jun 2026).

A bag-based realization makes the bag surface itself dynamical by replacing the sharp MIT/chiral-bag boundary with a smooth bag function built from the nonlinear pion field. For the hedgehog profile, the bag functions become

$\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$5

and, after the quark-field redefinition $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$6, the model turns into a chiral quark theory with an effective position-dependent mass

$\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$7

for a confined quark moving in a nonlinear pion background (Jia et al., 2013). This construction gives the estimate

$\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$8

and with $\mathcal{L}_{\rm int}= c_8 \bar \psi\, \Phi' \psi,$9 and $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$0 yields a proton charge radius $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$1 fm and magnetic moment $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$2 (Jia et al., 2013).

A different issue is whether scalar confinement can be made compatible with chiral symmetry. In the Covariant Spectator Theory, this is achieved with an interaction kernel whose confining part contains equal-weight scalar and pseudoscalar structures, $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$3, and satisfies the decoupling condition

$${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$4

Under these conditions the dressed propagator and axial vertex satisfy the axial-vector Ward–Takahashi identity, and the model obeys the Adler-zero constraint in $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$5-$${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$6 scattering (Biernat et al., 2015). This shows that scalar confinement is not automatically incompatible with chiral low-energy theorems.

In quark–diquark-inspired effective theories, chiral symmetry can instead be realized through multiplet structure. In the three-flavor mirror-assignment model embedded in the extended Linear Sigma Model, scalar and pseudoscalar diquarks transform like antiquarks, permitting four spin-$${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$7 baryonic multiplets and chirally invariant mass terms $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$8 and $${\mathcal L}_{\rm int}=-\frac{g_A}{f}\,\bar\psi\gamma^\mu\gamma_5(\partial_\mu \Pi)\psi,$$9 (Olbrich et al., 2015). After reduction to two flavors and fitting the nucleonic sector, the model identifies the chiral partner pairs as

$$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$0

with the mirror assignment providing the mass contributions that survive when the quark condensate vanishes (Olbrich et al., 2015).

4. Spectroscopy, reactions, and multiquark applications

In hadron spectroscopy and hadron–hadron dynamics, the chiral $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$1 quark model typically uses the Hamiltonian

$$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$2

where $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$3 consists of scalar and pseudoscalar nonet exchanges, and in the extended version also vector meson exchange (Yang et al., 2015). When applied to $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$4, $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$5, $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$6, and $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$7 systems, the dominant attraction is traced to $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$8-exchange between light quarks, while OGE and confinement do not contribute directly between the color-singlet hadron clusters (Yang et al., 2015). The extended model predicts, for example, $$L_{\rm eff}(M_s,\phi) = - N_c\, \mathrm{tr}\,\log\left[i\hat\partial + \hat m + M_s \hat U\right], \qquad \hat U=e^{i\gamma_5\hat\phi},$$9 and $U$0 bound in channels that can be identified with $U$1 and $U$2, whereas the standard model is less attractive (Yang et al., 2015).

The same constituent-chiral machinery has been applied to heavy-light tetraquarks. In the $U$3 problem, the light antiquark pair feels confinement, OGE, and chiral interaction, while heavy-heavy and heavy-light pairs are treated with confinement plus OGE only (0711.1029). The clearest bound-state result is

$U$4

whereas the corresponding $U$5 state is slightly above the $U$6 threshold (0711.1029). The binding mechanism is a combination of attractive color-magnetic and pseudoscalar exchange in the light antidiquark sector, together with suppression of repulsive heavy-pair color magnetism by the larger bottom mass (0711.1029).

In meson–baryon reactions, chiral quark models use quark-level transition operators together with $U$7 wave functions. For $U$8 and $U$9, the approach explains near-threshold $q \to q' + {\rm GB} \to q' + (q\bar q'),$0-wave dominance and fixes the relative signs of interfering resonance contributions through spin-isospin matrix elements (Zhao, 2010). In particular, $q \to q' + {\rm GB} \to q' + (q\bar q'),$1 and $q \to q' + {\rm GB} \to q' + (q\bar q'),$2 interfere destructively in $q \to q' + {\rm GB} \to q' + (q\bar q'),$3, while $q \to q' + {\rm GB} \to q' + (q\bar q'),$4 and $q \to q' + {\rm GB} \to q' + (q\bar q'),$5 play the analogous role in $q \to q' + {\rm GB} \to q' + (q\bar q'),$6 (Zhao, 2010).

A more elaborate coupled-channel realization uses Cloudy Bag Model quark wave functions and meson sources in the $q \to q' + {\rm GB} \to q' + (q\bar q'),$7 partial wave. There, the $q \to q' + {\rm GB} \to q' + (q\bar q'),$8 and $q \to q' + {\rm GB} \to q' + (q\bar q'),$9 are treated as bare quark-model states dressed by meson loops and coupled-channel rescattering, and the resulting $\bar u^N-\bar d^N=-0.118$0-matrix poles lie at 1535 MeV and 1690 MeV (Golli et al., 2011). The model gives a good overall description of the $\bar u^N-\bar d^N=-0.118$1 scattering and electroproduction observables and argues that the $\bar u^N-\bar d^N=-0.118$2 is dominantly a genuine three-quark state rather than a purely meson-baryon quasi-bound configuration (Golli et al., 2011).

5. Dense matter, magnetized quark matter, and hadron–quark hybridizations

In dense-matter applications, the phrase chiral quark model often refers to quark phases embedded in broader hadronic frameworks. One example is the two-flavor linear sigma model with quarks in a magnetic background,

$\bar u^N-\bar d^N=-0.118$3

used to study quark matter under strong magnetic fields (Rabhi et al., 2011). The decisive issue there is the treatment of magnetic vacuum corrections. If these corrections are not included explicitly through $\bar u^N-\bar d^N=-0.118$4, the model parameters must be refitted to low-density meson properties in the presence of the magnetic field; the simplified prescription remains reasonable only up to about $\bar u^N-\bar d^N=-0.118$5 G (Rabhi et al., 2011).

A neutron-star realization uses a three-window construction in which hadronic matter is described by a parity doublet model for $\bar u^N-\bar d^N=-0.118$6, quark matter by a three-flavor NJL-type model for $\bar u^N-\bar d^N=-0.118$7, and the intermediate region by an interpolated pressure

$\bar u^N-\bar d^N=-0.118$8

(Minamikawa et al., 2022). In this setting the quark sector is the high-density chiral quark model proper: a standard three-flavor NJL model supplemented by a diquark interaction $\bar u^N-\bar d^N=-0.118$9 and a vector interaction $\bar u^N/\bar d^N=0.652$0 (Minamikawa et al., 2022). The favored chirally invariant mass in the hadronic sector is

$\bar u^N/\bar d^N=0.652$1

and the interpolated condensates remain positive and approach zero gradually; in the NJL high-density regime the in-medium condensates stay at about $\bar u^N/\bar d^N=0.652$2 of the vacuum value rather than changing sign (Minamikawa et al., 2022).

A different dense-matter strategy is the chiral quark–meson coupling model, in which baryons are composite quark systems whose internal structure responds to scalar mean fields. In the version applied to neutron stars, quark–quark hyperfine interactions due to gluon and pion exchange are included, and extending vector-meson couplings from SU(6) to SU(3) allows maximum masses up to the observed

$\bar u^N/\bar d^N=0.652$3

(Miyatsu et al., 2012). The resulting composition contains $\bar u^N/\bar d^N=0.652$4 and $\bar u^N/\bar d^N=0.652$5 hyperons, while $\bar u^N/\bar d^N=0.652$6 hyperons and $\bar u^N/\bar d^N=0.652$7-isobars do not appear (Miyatsu et al., 2012).

A broader hadron–quark hybrid model augments a nonlinear hadronic $\bar u^N/\bar d^N=0.652$8 $\bar u^N/\bar d^N=0.652$9-$\pi$00 mean-field theory by quarks and a Polyakov-loop potential,

$\pi$01

and suppresses hadrons at high $\pi$02 and $\pi$03 by excluded-volume effects (Rau et al., 2013). In this construction, both chiral restoration and deconfinement are smooth crossovers, the nuclear liquid-gas endpoint is around $\pi$04 MeV, and the transition region at $\pi$05 lies around $\pi$06–$\pi$07 MeV depending on the order parameter discussed (Rau et al., 2013).

6. Conceptual achievements, ambiguities, and open problems

A recurring achievement of chiral quark models is that they tie observables directly to chiral dynamics in the quark sector. Across the cited literature this includes sea-quark asymmetries in nucleon and hyperon flavor structure, nonlocal PDFs in the nucleon, low-energy theorems such as the Adler zero, realistic heavy-meson decay constants, hadronic molecules generated by light-quark chiral forces, and dense-matter equations of state constrained by neutron-star phenomenology (Dahiya, 2015, Wakamatsu, 2 Jun 2026, Biernat et al., 2015, Nam, 2012, Minamikawa et al., 2022).

At the same time, the term is intrinsically ambiguous. Some formulations are explicitly confining, such as the dynamical bag model or the CST construction; some are quark soliton theories without explicit gluons; some are NJL-type models without confinement; and some are effectively hadronic theories whose baryonic multiplets are organized by quark and diquark transformation properties (Jia et al., 2013, Wakamatsu, 2 Jun 2026, Minamikawa et al., 2022, Olbrich et al., 2015). This suggests that “chiral quark model” should be read as a structural descriptor—quark-level effective dynamics constrained by chiral symmetry—rather than as the name of a unique Hamiltonian or Lagrangian.

Several limitations recur. NJL-based dense-matter models lack confinement and therefore are restricted to high density; the CQSM has no explicit gluons and usually assumes $\pi$08 and $\pi$09 at the starting scale; the heavy-sector nonlocal masses in the extended nonlocal model are phenomenological rather than derived; and tree-level reaction models built from $\pi$10 wave functions do not enforce unitarity (Minamikawa et al., 2022, Wakamatsu, 2 Jun 2026, Nam, 2012, Zhao, 2010). In the mirror-assignment baryon model, the decay $\pi$11 remains underpredicted by about an order of magnitude, indicating missing dynamics such as additional flavor structure or anomaly-related couplings (Olbrich et al., 2015). In strong magnetic fields, the reliability of the quark–meson description depends crucially on including magnetic vacuum corrections explicitly (Rabhi et al., 2011).

Open issues remain correspondingly diverse. The cited neutron-star crossover study leaves the low-density hadronic sector and the high-density three-flavor quark sector with a flavor mismatch and only a modeled interpolation between them (Minamikawa et al., 2022). In hadron spectroscopy, the status of states such as $\pi$12 remains formulation-dependent: one coupled-channel chiral quark model supports a predominantly three-quark interpretation, whereas other reaction frameworks emphasize strong meson-baryon dressing (Golli et al., 2011). In kaon structure, the size of $\pi$13-breaking effects is still model-sensitive, with the chiral constituent-quark treatment finding much milder asymmetry than several competing approaches (Watanabe et al., 2017).

The common thread is that all of these models attempt to encode the same QCD fact—spontaneous chiral symmetry breaking—at the quark level, but they do so with markedly different realizations of confinement, baryon structure, and many-body dynamics. That plurality is not incidental; it is a defining feature of the chiral quark model literature itself.