Chiral Soliton Model in QCD

• Chiral soliton model is a framework of low-energy QCD where baryons emerge as stable solitonic configurations of chiral fields.
• In the mesonic Skyrme model, baryons are generated from hedgehog field configurations and refined through collective-coordinate quantization.
• Quark-based realizations, such as the Chiral Quark Soliton Model, incorporate explicit valence quarks and a polarized Dirac sea to bridge quark dynamics and effective chiral fields.

Searching arXiv for the cited papers to ground the article in current records. The chiral soliton model is a class of low-energy QCD effective theories in which baryons are described as stable, localized classical field configurations of chiral degrees of freedom. In its mesonic realization, exemplified by the Skyrme model, the baryon is a topological soliton of the chiral field $U(\mathbf{x})$; in quark-based realizations, such as the Chiral Quark Soliton Model (CQSM) or bosonized Nambu–Jona-Lasinio (NJL) constructions, the baryon is a self-consistent chiral mean field with explicit valence quarks and a polarized Dirac sea (Weigel, 26 Mar 2025, Wakamatsu, 2 Jun 2026). Large-$N_c$ QCD provides the principal theoretical motivation: baryons scale as heavy collective objects while meson interactions become weak, which suggests a solitonic description in an effective meson theory (Weigel, 26 Mar 2025).

1. Chiral symmetry, topology, and the soliton concept

The underlying symmetry structure is the approximate chiral symmetry of massless light-quark QCD,

$$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$

together with spontaneous chiral symmetry breaking and the appearance of Nambu–Goldstone bosons. For $N_f=2$, the pion field is collected into

$$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$

and for $N_f=3$ into the analogous $SU(3)$ matrix built with the Gell-Mann matrices (Wakamatsu, 2 Jun 2026).

A soliton in these theories is a stable, localized classical field configuration. In the simplest $SU(2)$ case, the standard hedgehog ansatz is

$$U(\mathbf{x})=\exp\!\big(i\,\vec{\tau}\cdot\hat{\mathbf{r}}\,F(r)\big),$$

with $\hat{\mathbf{r}}=\mathbf{r}/|\mathbf{r}|$ and a profile $N_c$0 satisfying

$N_c$1

For $N_c$2, the winding number is one and the configuration is interpreted as a single baryon (Wakamatsu, 2 Jun 2026). In the Skyrme language, baryon number is identified with the topological charge

$N_c$3

so that the baryon is not inserted as a fundamental field but emerges from the topology of the meson configuration (Pavlovsky et al., 2010).

Large-$N_c$4 arguments supply the conceptual bridge between QCD and this soliton picture. In the low-energy meson theory, the effective interaction strength decreases with the number of colors, while baryon masses scale as $N_c$5, which is the characteristic scaling of a classical soliton (Weigel, 26 Mar 2025). This large-$N_c$6 logic underlies both purely mesonic and quark-based chiral soliton models.

2. The Skyrme model and mesonic chiral solitons

The canonical mesonic realization is the Skyrme model. Its basic Lagrangian is

$N_c$7

where the four-derivative Skyrme term stabilizes the finite-energy soliton (Pavlovsky et al., 2010, Weigel, 26 Mar 2025). In this framework, baryons are purely mesonic topological solitons of $N_c$8; quarks and gluons are not explicit degrees of freedom (Wakamatsu, 2 Jun 2026).

The static hedgehog does not carry good spin or isospin quantum numbers. Physical nucleon and $N_c$9 states are obtained by quantizing the collective rotations of the soliton in coordinate and isospin space. This collective-coordinate procedure is one of the defining structural features of the model and remains central in its $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$0 extensions to hyperons and multibaryon systems (Wakamatsu, 2 Jun 2026, Kopeliovich, 2012).

The mesonic realization has several systematic extensions. One important direction is the inclusion of vector mesons, which improves agreement with empirical baryon properties (Weigel, 26 Mar 2025). Another is the gauged Skyrme model in a magnetic field and at finite baryon chemical potential, where a suitable domain-wall ansatz reduces the theory to an effective one-dimensional sine-Gordon system describing a chiral soliton lattice; in that setting the Callan–Witten term allows nonvanishing baryon number even when the usual Skyrme topological density vanishes (Canfora et al., 13 Oct 2025).

3. Quark-based realizations

A second major branch replaces the purely mesonic picture by an effective quark theory in which quarks move in a self-consistent chiral background. The CQSM is the paradigmatic case. Its basic $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$1 Lagrangian is

$$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$2

with $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$3 the constituent quark mass generated by spontaneous chiral symmetry breaking (Wakamatsu, 2 Jun 2026). For a static hedgehog pion field, the Dirac Hamiltonian develops a deep bound state, the valence level, while the Dirac sea is deformed by the background. Baryon number is obtained by filling the discrete bound level with $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$4 quarks and allowing the negative-energy sea to be vacuum polarized (Wakamatsu, 2 Jun 2026).

The bosonized NJL realization gives a more explicit construction. Starting from the two-flavor NJL model,

$$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$5

one bosonizes to scalar and pseudoscalar meson fields and regularizes the resulting action with Pauli–Villars subtraction (Takyi et al., 2019). The chiral soliton is then built with the hedgehog ansatz

$$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$6

and the profile $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$7 is obtained by minimizing the full static energy functional, which includes the valence level and the regularized sea contribution (Takyi et al., 2019). In this realization, the baryon is stabilized dynamically by quark vacuum polarization rather than by an explicit Skyrme term (Takyi et al., 2019).

Quark–meson models provide another non-topological soliton realization. In the two-flavor linear sigma model,

$$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$8

the baryon is a bound state of three constituent quarks in a hedgehog background $$SU(N_f)_L \times SU(N_f)_R \longrightarrow SU(N_f)_V,$$9, $N_f=2$0 (Shu et al., 2014, Mao et al., 2013).

A further extension incorporates broken scale invariance through a logarithmic potential. In the hybrid-chiral soliton model and its vector-meson generalization, the scalar sector is modified to reflect the QCD trace anomaly, either with a frozen dilaton $N_f=2$1 or with vector mesons $N_f=2$2, $N_f=2$3, and $N_f=2$4 introduced as massive gauge fields (Drago et al., 2011, Drago et al., 2011). In these models the logarithmic term softens the restrictions of the Mexican-hat potential and changes the in-medium scalar dynamics.

Realization	Degrees of freedom	Characteristic mechanism
Skyrme model	Chiral meson field $N_f=2$5	Topological baryon number
CQSM / bosonized NJL	Quarks plus self-consistent chiral field	Valence level plus polarized Dirac sea
Linear sigma / quark–meson model	Quarks, $N_f=2$6, $N_f=2$7	Non-topological quark–meson soliton
Scale-breaking/vector-meson variants	Quarks, chiral fields, logarithmic potential, vector mesons	Broken scale invariance and short-range vector dynamics

4. Collective quantization and nucleon structure observables

Collective-coordinate quantization is required in both mesonic and quark-based formulations. In the CQSM and NJL soliton, the static hedgehog carries grand spin rather than good spin and isospin. One therefore introduces time-dependent flavor rotations $N_f=2$8,

$N_f=2$9

and quantizes the associated zero mode to obtain physical nucleon states (Wakamatsu, 2 Jun 2026). The same procedure, generalized to $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$0, underlies hypernuclear and singly heavy baryon spectroscopy in soliton approaches (Kopeliovich, 2012, Won et al., 2021).

A principal advantage of quark-based chiral soliton models is access to nonlocal quark observables. Pure meson Skyrme-type models have a severe problem for deep inelastic scattering: they do not reproduce the correct partonic Callan–Gross relation, because quark bilocal operators cannot be faithfully represented in a purely mesonic theory (Weigel et al., 2021). By contrast, the CQSM can handle non-local quark–quark correlators and bilocal light-cone operators, which is decisive for parton distribution functions (Wakamatsu, 2 Jun 2026).

In the bosonized NJL chiral soliton model, structure functions are extracted from quark wave functions in the soliton background, with particular emphasis on a consistent regularization of the Dirac-sea contribution. Numerical simulations show that the comparison with experiment is convincing for the polarized structure functions but exhibits some discrepancies in the unpolarized case, and the vacuum contribution to the polarized structure functions is particularly small (Takyi et al., 2019). Weigel and Takyi stressed that the bosonized NJL soliton provides a bridge between the quark-level description of QCD and an effective hadronic description in terms of meson fields, precisely because the quark bilinears remain accessible (Weigel et al., 2021).

5. Finite temperature, density, and nuclear matter

Chiral soliton models have been widely used to study hot and dense matter. In the linear sigma model at finite temperature and chemical potential, the soliton is embedded in a thermal constituent-quark medium through an effective potential

$$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$1

with asymptotic boundary condition $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$2, $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$3 (Shu et al., 2014, Mao et al., 2013). In one formulation, the stable soliton satisfies $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$4 for $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$5, while for $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$6 there is a sharp delocalization phase transition from hadron matter to quark matter coincident with the restoration of chiral symmetry (Mao et al., 2013).

The finite-$$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$7 literature contains a genuine definitional issue concerning the baryon mass. In one study based on the thermal linear sigma model, the baryon mass was defined to include the meson interaction energy $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$8, and all of $$U(x)=\exp\!\left(i\,\frac{\pi^a(x)\tau^a}{f_\pi}\right)\in SU(2),$$9, $N_f=3$0, $N_f=3$1, and $N_f=3$2 were found to increase with $N_f=3$3 and $N_f=3$4 (Shu et al., 2014). In another treatment, the effective soliton energy was defined relative to the thermal background and decreased with temperature and density, approaching the free three-quark threshold at the delocalization transition (Mao et al., 2013). A subsequent analysis emphasized that the thermal effective potential must be properly treated so as to derive a finite and well-defined baryon mass out of the thermal background, and with that subtraction the baryon mass again decreases with $N_f=3$5 and $N_f=3$6 (Zhang et al., 2015). This divergence of trends reflects different energy definitions rather than a single settled result.

At finite density, the Wigner–Seitz approximation is a standard tool for approximating nuclear matter as a lattice of spherical cells each containing one soliton (Drago et al., 2011, Drago et al., 2011). In the hybrid-chiral model with a logarithmic scale-breaking potential, the soliton can be followed to higher densities than in the linear sigma model, up to approximately $N_f=3$7 for $N_f=3$8 (Drago et al., 2011). In the version with vector mesons introduced as massive gauge fields, the model reaches higher densities than the linear-$N_f=3$9 model and the introduction of vector mesons allows one to obtain saturation; the $SU(3)$0 field supplies repulsive isoscalar vector self-energy, while $SU(3)$1 and $SU(3)$2 contribute in isovector and axial channels (Drago et al., 2011).

6. $SU(3)$3, heavy flavors, and other extensions

The $SU(3)$4 generalization allows the treatment of hyperons, hypernuclei, and heavy baryons. In the bound-state rigid-oscillator approach to neutron-rich hypernuclei, a classical $SU(3)$5 skyrmion of baryon number $SU(3)$6 is dressed by a bound antikaon mode, and the resulting $SU(3)$7 excitation lowers the isospin from $SU(3)$8 to $SU(3)$9. The model predicts additional binding of strange hypernuclei in comparison with nonstrange neutron-rich nuclei at not large atomic numbers $SU(2)$0; total binding energies of $SU(2)$1 and the recently discovered $SU(2)$2 satisfactorily agree with experiment in the nuclear variant with rescaled Skyrme constant $SU(2)$3 (Kopeliovich, 2012).

In a medium-modified $SU(2)$4 chiral soliton model with heavy meson bound states, singly heavy baryons are treated as a soliton of the light meson sector coupled to a heavy meson multiplet. The resulting masses in nuclear matter are rather sensitive to the medium modifications of the heavy meson mass (Won et al., 2021). In a related direction, the Chiral Quark-Soliton Model has been applied to heavy tetraquarks by replacing the heavy quark with a heavy anti-diquark while keeping the same soliton background; within that construction only $SU(2)$5 tetraquarks are found to be bound (Kucab et al., 2024).

The term also appears in adjacent literatures with a different physical referent. In monoaxial chiral magnets and related sine-Gordon systems, a chiral soliton model describes topological domain walls whose chirality is fixed by Dzyaloshinskii–Moriya interactions; the corresponding chiral soliton lattice controls magnetic structure and, in lattice-electron models, negative magnetoresistance proportional to the number of solitons (Okumura et al., 2017, Chinellato et al., 9 Dec 2025). In supersymmetric field theory, the neutral pion sector of a supersymmetric chiral Lagrangian with a Wess–Zumino–Witten term reduces to a chiral sine-Gordon model whose ground state can become a chiral soliton lattice in a strong magnetic field or at large baryon chemical potential (Nitta et al., 2024). These usages share the localized chiral domain-wall structure, but they are distinct from the baryonic chiral soliton models of low-energy QCD.

Taken together, the chiral soliton model denotes not a single Lagrangian but a coherent family of chiral effective theories in which localized nonperturbative configurations organize baryon structure, nuclear matter, and, in some extensions, inhomogeneous phases of matter. Its mesonic and quark-based realizations differ sharply in microscopic degrees of freedom, yet both are built around the same low-energy QCD ingredients: spontaneous chiral symmetry breaking, nonlinear pion dynamics, and collective quantization (Weigel, 26 Mar 2025, Wakamatsu, 2 Jun 2026).