Dressed Quark Model in QCD

• Dressed quark models are QCD-motivated frameworks where quarks are effectively modified by gluon exchange, meson-cloud effects, and dynamical chiral symmetry breaking.
• These models employ techniques like Dyson–Schwinger equations, light-front Hamiltonian methods, and constituent approaches to derive observables such as tensor charges and form factors.
• They preserve key QCD symmetries and offer practical insights into hadron structure by embedding nonperturbative dynamics into the effective quark degrees of freedom.

Searching arXiv for recent and foundational papers on dressed quark models and related frameworks. In the literature on hadron structure, a dressed quark is not a free or bare quark, and it is not merely a static constituent-quark surrogate either. It is a quark fully embedded in QCD dynamics: its propagator and interactions are modified by repeated gluon exchange, dynamical chiral symmetry breaking, meson-cloud effects, or explicit quark–gluon Fock components, depending on the framework. Taken together, these works suggest that the “dressed quark model” is not a single model but a family of QCD-motivated constructions in which the effective quark degree of freedom already contains nontrivial gluonic and, in some approaches, mesonic substructure. Representative realizations range from symmetry-preserving Dyson–Schwinger/Bethe–Salpeter equation treatments of the dressed propagator and quark vertices (Liu et al., 2019), to light-front quark-plus-gluon composite states (More et al., 2016), to chiral constituent and meson-cloud descriptions (Watanabe et al., 2017), and to effective light-front mass operators built from a running quark mass extracted from a dressed Minkowski-space propagator (Marinho et al., 24 Apr 2026).

1. Core definition and scope

A common starting point is the dressed quark propagator

$S^{-1}(p)=i\gamma\cdot p\,A(p^2)+B(p^2),$

or, equivalently,

$S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$

In this notation, $A$ is the wave-function dressing, $B$ is the scalar mass function, and $\mathcal M$ is the momentum-dependent dressed mass. In the chiral limit, any nonzero $B(p)$ is dynamically generated by chiral symmetry breaking (Liu et al., 2019, Ferreira et al., 4 Dec 2025).

This common propagator language does not imply a unique ontology. In some approaches, the dressed quark is a nonperturbative solution of a gap equation; in others, it is a light-front composite state with explicit quark and gluon Fock sectors; in still others, it is a constituent quasiparticle dressed by a Goldstone-boson cloud or by pion and vector-meson corrections to the quark–photon vertex. This suggests a taxonomy of dressed-quark models by the mechanism of dressing rather than by a single formal definition.

Realization	Dressing content	Representative use
DSE/BSE frameworks (Liu et al., 2019, Ferreira et al., 4 Dec 2025)	Momentum-dependent quark propagator, dressed quark-gluon vertex, symmetry-preserving kernels	Tensor charge, pion Goldstone dynamics
Light-front dressed-quark state [(More et al., 2016); (Mukherjee et al., 2014); (Mukherjee et al., 2015); (More et al., 2023); (Kasemets et al., 2016)]	Bare quark sector plus quark–gluon sector	Wigner distributions, GTMDs, GFFs, DPDs
Chiral constituent and meson-cloud models [(Watanabe et al., 2017); (Traini, 2013)]	Constituent quark plus pseudoscalar or meson–baryon cloud	Kaon and nucleon PDFs
NJL, Euclidean, and effective LF-mass constructions [(Ninomiya et al., 2014); (Hobbs et al., 2016); (Marinho et al., 24 Apr 2026)]	Dynamical mass generation, confinement-inspired regularization, pion cloud, Euclidean dressing functions, running LF mass operator	Form factors, axial charge, pion TMDs/PDFs/DAs

2. Symmetry-preserving nonperturbative frameworks

In Dyson–Schwinger/Bethe–Salpeter equation treatments, the dressed quark model is defined by a coupled system for the quark propagator, the relevant quark vertex, and a kernel constructed so that these equations remain compatible. A central construction derives the two-particle irreducible Bethe–Salpeter kernel directly from the quark self-energy by

$$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$

which enforces consistency between the quark DSE and the vertex BSE. In the tensor-vertex application, this framework is used with both the rainbow-ladder truncation, $\Gamma_\nu\to\gamma_\nu$, and a Munczek ansatz for the longitudinal quark–gluon vertex, and then inserted into a constituent quark plus scalar diquark nucleon model to compute the nucleon tensor charge (Liu et al., 2019).

Beyond rainbow-ladder, the central issue is preserving the Ward–Takahashi identities while incorporating full quark–gluon vertex structure. A symmetry-preserving truncation for pion physics treats the quark propagator, the quark–gluon vertex, and the axial-vector vertex self-consistently, simplifies the quark–gluon Schwinger–Dyson equation, and yields a renormalized pion Bethe–Salpeter equation composed of a pair of one-loop diagrams containing the full quark–gluon vertex plus a single two-loop crossed diagram. The crossed two-loop contribution is numerically only about $1\%$, but it is instrumental for the eigenvalue condition $\lambda(P^2=0)=1$ and therefore for the masslessness of the pion in the chiral limit. In the same calculation, the Goldstone relation

$S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$0

is satisfied to within about $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$1 or better (Ferreira et al., 4 Dec 2025).

A related line of work uses a recursively dressed quark–gluon vertex in a Munczek–Nemirovsky interaction model. There the vertex obeys

$S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$2

with $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$3 corresponding to rainbow-ladder and $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$4 to the fully resummed vertex. The same dressing is propagated into the meson BSE kernel so that the axial-vector Ward–Takahashi identity remains satisfied. In heavy-light pseudoscalar and vector mesons, this construction provides a controlled measure of corrections beyond rainbow-ladder (Gomez-Rocha et al., 2015, Gomez-Rocha et al., 2016).

3. Light-front dressed-quark states

In light-front Hamiltonian perturbation theory, the dressed quark model is usually the minimal composite spin-$S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$5 system consisting of a one-particle quark sector plus a two-particle quark–gluon sector. The target is not a proton but a quark dressed with a gluon, and the essential dynamical input is the perturbative two-body light-front wave function. This setup retains explicit gluonic degrees of freedom while remaining analytically tractable, and it is therefore well suited to phase-space and correlation observables (More et al., 2016).

Within this framework, quark Wigner distributions are constructed from generalized transverse-momentum dependent distributions. For different polarization configurations, the model finds that only $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$6 of the $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$7 leading-twist Wigner distributions are independent, with $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$8, $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$9 for $A$0, and $A$1. The distributions $A$2, $A$3, and $A$4 have similar overall behavior, whereas $A$5 exhibits a dipole structure in momentum and impact-parameter space and a quadrupole-like pattern in mixed space, reflecting spin–orbit correlations (More et al., 2016).

The same light-front dressed-quark state has been used to compare kinetic and canonical orbital angular momentum. In that calculation, the quark Wigner distributions are parameterized by GTMDs such as $A$6 and $A$7, which are pure GTMDs and do not reduce to GPDs or TMDs in any limit. The model yields both kinetic and canonical quark OAM negative in the studied parameter range, but with different magnitudes, and the quark spin–orbit correlation is also negative, implying anti-alignment between quark spin and quark OAM. The difference between kinetic and canonical OAM is attributed to the presence of explicit gluonic degrees of freedom (Mukherjee et al., 2014).

An analogous calculation for gluons defines gluon Wigner distributions and relates canonical gluon OAM to $A$8 and gluon spin–orbit correlation to $A$9. In the same light-front setting, gravitational form factors satisfy the expected sum rules $B$0, $B$1, and $B$2, with the quark and gluon $B$3 canceling in the total energy-momentum tensor. The D-term then controls pressure and shear distributions in impact-parameter space (Mukherjee et al., 2015, More et al., 2023).

For two-parton observables, quark–gluon double parton distributions can be computed explicitly from the same two-body wave function. The model finds sizable spin-spin and spin-kinematic correlations, nearly saturated positivity bounds, and, crucially, non-factorization of the transverse dependence: $B$4 is not supported because the dependence occurs through $B$5 inside Bessel functions. This conclusion is specific to the dressed-quark two-body state, whose kinematics also enforces $B$6 (Kasemets et al., 2016).

4. Constituent, meson-cloud, and Euclidean dressed-quark pictures

A different usage of the dressed quark model appears in the chiral constituent quark model. There a dressed constituent quark is a superposition of a bare quark state plus meson–quark Fock components generated by spontaneous chiral symmetry breaking. For the kaon, the valence $B$7 and $B$8 quarks are dressed by virtual $B$9, $\mathcal M$0, and $\mathcal M$1 fluctuations, with wave-function renormalization constants $\mathcal M$2 and $\mathcal M$3 for the quoted parameter set. After NLO DGLAP evolution from $\mathcal M$4, the resulting dressed valence distributions reproduce the empirical ratio $\mathcal M$5, while the final $\mathcal M$6 flavor symmetry breaking is found to be smaller than in several earlier approaches (Watanabe et al., 2017).

An analogous meson-cloud logic is applied to the nucleon in a light-front three-quark model dressed by virtual meson–baryon fluctuations. The physical nucleon is written as a bare core plus one-meson Fock components, and the Sullivan-process convolution

$\mathcal M$7

generates nonperturbative sea structure. In this picture, $\mathcal M$8- and $\mathcal M$9-cloud effects produce the $B(p)$0 asymmetry, while $B(p)$1 and $B(p)$2 fluctuations generate $B(p)$3. The work emphasizes that NNLO evolution is necessary because perturbative evolution itself produces a nonzero strange asymmetry at NNLO if it is absent at the starting scale (Traini, 2013).

The three-flavor NJL model provides another dressed-quark framework. There the dressed light-quark mass $B(p)$4 is generated by the gap equation, confinement effects are mimicked by proper-time infrared regularization with $B(p)$5, and pion-cloud plus vector-meson dressing modifies the quark–photon vertex. Scanning $B(p)$6, the analysis finds that $B(p)$7 gives the best simultaneous description of $B(p)$8, $B(p)$9, condensate ratios, and pion and kaon electromagnetic form factors (Ninomiya et al., 2014).

A Euclidean constituent quark model provides an explicit bridge to Euclidean DSE/BSE analyses. In a quark plus scalar diquark picture of the nucleon, the model introduces Euclidean density functions $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$0 and $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$1, which recover $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$2 and the axial-singlet charge by radial integration and can absorb momentum-dependent dressing functions from Bethe–Salpeter analyses. After fitting electromagnetic form factors, the gluonic dressing of the quark axial-vector vertex changes the proton axial-singlet charge by $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$3 in one truncation and $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$4 in another, leading to the conclusion that the effect is small and consistent with zero at the few-percent level (Hobbs et al., 2016).

5. Phenomenology and representative results

The dressed quark model is primarily an observable-producing framework. Depending on the realization, it has been used to compute tensor charges, meson masses, form factors, pion Bethe–Salpeter amplitudes, parton distributions, Wigner distributions, gravitational form factors, and effective light-front wave-function observables.

Observable	Framework	Representative result
Proton $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$5-quark tensor charge	Consistent DSE/BSE tensor vertex (Liu et al., 2019)	$$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$6, $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$7, $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$8; suppression about $$[K(k,q;0)]^{GH}_{EF} = -\frac{\delta [\Sigma(k)]_{EF}}{\delta [S(q)]_{GH}},$$9 in RL and about $\Gamma_\nu\to\gamma_\nu$0 in the Munczek-consistent framework
Chiral-limit pion BSE	Symmetry-preserving dressed quark-gluon vertex (Ferreira et al., 4 Dec 2025)	Dressed RL $\Gamma_\nu\to\gamma_\nu$1, quantum $\Gamma_\nu\to\gamma_\nu$2, crossed two-loop $\Gamma_\nu\to\gamma_\nu$3; $\Gamma_\nu\to\gamma_\nu$4 to within about $\Gamma_\nu\to\gamma_\nu$5 or better
Gluon momentum fraction	DSE/RL quark-dressing PDFs (Freese et al., 2021)	$\Gamma_\nu\to\gamma_\nu$6 for pion and nucleon at the model scale; at $\Gamma_\nu\to\gamma_\nu$7 GeV, $\Gamma_\nu\to\gamma_\nu$8
Intrinsic-glue quark target	QTM with Wilson line (Costa et al., 2021)	For $\Gamma_\nu\to\gamma_\nu$9 and $1\%$0 GeV: $1\%$1, $1\%$2, Wilson-line contribution $1\%$3
Heavy-light vector spectroscopy	DSBSE with dressed QGV (Gomez-Rocha et al., 2016)	$1\%$4 GeV; theory average $1\%$5 GeV

These results illustrate a recurring pattern: dressing typically suppresses simple bare-current estimates, redistributes momentum and spin between quark and gluon sectors, and improves agreement with lattice QCD or phenomenology when the kernel and vertex dressing are implemented consistently.

A recent light-front Hamiltonian formulation pushes this logic further by constructing an effective dressed mass-squared operator directly from a Minkowski-space running quark mass. Using

$1\%$6

with $1\%$7 GeV, $1\%$8 GeV, and $1\%$9 GeV, the resulting operator-level effective self-energy is strongly enhanced in the infrared, reaching values around $\lambda(P^2=0)=1$0–$\lambda(P^2=0)=1$1 GeV at small relative momentum, and tends to the bare/current quark mass $\lambda(P^2=0)=1$2 in the ultraviolet. In pion applications, the dressed Gaussian model becomes more sharply peaked around $\lambda(P^2=0)=1$3, while power-law models are less sensitive because the vertex structure dominates over the modified resolvent (Marinho et al., 24 Apr 2026).

6. Model dependence, limitations, and nonstandard directions

Despite the shared language of quark dressing, the frameworks differ sharply in truncation, degrees of freedom, and domain of validity. In the tensor-charge calculation, the nucleon is not obtained from the full Faddeev equation; the model retains only a scalar diquark component, neglects axial-vector structures, and employs a simplified quark–diquark form factor. The calculation is therefore useful for isolating vertex-dressing effects, but it is not a complete first-principles nucleon description (Liu et al., 2019).

The symmetry-preserving pion framework is exact enough to maintain the axial Ward–Takahashi identity in the chiral limit, but extension beyond the chiral limit requires nonzero current masses and Green’s functions at complex momenta, because bound states satisfy $\lambda(P^2=0)=1$4 and the internal ingredients must be known off the Euclidean real axis (Ferreira et al., 4 Dec 2025). Similarly, the light-front dressed-quark DPD analysis makes clear that the two-body truncation enforces $\lambda(P^2=0)=1$5; richer $\lambda(P^2=0)=1$6 dependence would require higher Fock states (Kasemets et al., 2016).

Other limitations are tied to the chosen effective interaction. In the Munczek–Nemirovsky DSBSE models for heavy-light mesons, observables acquire an artificial dependence on the momentum-partitioning parameter $\lambda(P^2=0)=1$7, explicitly identified as a model artifact. The same studies therefore use the $\lambda(P^2=0)=1$8-dependence to estimate systematic uncertainties rather than interpret it physically (Gomez-Rocha et al., 2016). In the intrinsic-glue quark target model, combining the dressing kernel with a pion body PDF can produce negative support near $\lambda(P^2=0)=1$9, which is interpreted as an artifact of using a DGLAP-like kernel inside an impulse-approximation convolution (Costa et al., 2021).

A separate, highly nonstandard branch of the literature treats a bare massless quark as the seed of a self-generated, absolutely confined spherical object obtained from a self-consistent solution of the QCD field equations. After an exact operator reduction, the model finds three distinct solitonlike solutions, interpreted as the three quark generations, with radii $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$00 and binding energies linked to the SU(3) structure constants by $S^{-1}(p)=A(p)\slashed{p}-B(p), \qquad \mathcal{M}(p)=\frac{B(p)}{A(p)}.$01. The same construction then invokes general relativity and a cosmological vacuum term to stabilize the Planck-scale system and generate MeV-scale dressed quark masses. Because the paper itself characterizes the framework as highly nonstandard, it is best regarded as a distinct speculative realization of quark dressing rather than part of the standard DSE/BSE, light-front, or constituent-model lineages (Greben, 2019).

Across all of these realizations, the durable content of the dressed quark model is the replacement of a bare quark by an effective degree of freedom whose mass, vertices, and multiparton structure are modified by QCD interactions. The specific implementation varies—from symmetry-preserving kernels, to one-gluon light-front Fock sectors, to meson clouds and Euclidean density functions—but the common objective is the same: to encode nonperturbative QCD dynamics into the effective quark that enters hadron structure calculations.