Light-Front Dressed Quark Model

• The light-front dressed quark model is a framework that represents quarks as dressed states incorporating one-gluon interactions through a truncated Fock space.
• It employs light-front Hamiltonian perturbation theory and two-body wave functions to derive partonic observables like Wigner distributions, GTMDs, and gravitational form factors.
• The approach bridges perturbative and effective constituent models, capturing spin–orbit correlations and nonperturbative effects via quark dressing and zero-mode clouds.

The light-front dressed quark model denotes a family of light-front constructions in which quark degrees of freedom are treated as dressed rather than bare. In one widely used realization, the target is a spin-$\tfrac12$ quark dressed by one gluon in light-front Hamiltonian perturbation theory, truncated to the one-body and two-body Fock sectors, and used to compute Wigner distributions, GTMDs, DPDs, gravitational form factors, and angular-momentum densities. In another realization, constituent quarks in a light-front quark model for mesons are interpreted as already incorporating an effective zero-mode cloud of the QCD vacuum, so that explicit zero-mode terms disappear from final observables while chiral-symmetry constraints remain satisfied. Recent extensions formulate dressed light-front dynamics directly from a Minkowski-space dressed propagator and an effective light-front mass-squared operator (Mukherjee et al., 2014, Choi et al., 2014, Marinho et al., 24 Apr 2026).

1. Light-front construction and dressed-quark degrees of freedom

The perturbative construction is formulated in light-front coordinates, typically with $v^\pm=(v^0\pm v^3)/\sqrt2$ and transverse components $v_\perp$, in light-front gauge $A^+=0$. The dressed-quark state is expanded in Fock space as a bare quark plus a quark–gluon component, with boost-invariant Jacobi variables $x_i$ and $q_{i\perp}$ satisfying $\sum_i x_i=1$ and $\sum_i q_{i\perp}=0$. One-particle states obey the standard light-front normalization

$$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$

In the two-component formalism, only the dynamical quark field and transverse gluon field appear explicitly; constrained components are eliminated through the light-front equations of motion (Mukherjee et al., 2014, More et al., 2017).

The central dynamical object is the two-body light-front wave function. In the one-gluon truncation it takes the form

$$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$

Equivalent expressions appear in spinor and two-component forms. The model is usually evaluated in symmetric frames with purely transverse momentum transfer and zero skewness, or in the Drell–Yan frame $v^\pm=(v^0\pm v^3)/\sqrt2$0, where many overlap formulae simplify (Mukherjee et al., 2014, More et al., 2017).

A defining simplification is the straight gauge link in $v^\pm=(v^0\pm v^3)/\sqrt2$1, for which the Wilson line reduces to unity. This removes final-state interactions and implies that naively $v^\pm=(v^0\pm v^3)/\sqrt2$2-odd limits vanish in the corresponding TMD sector. The same choice makes the model analytically tractable, while retaining explicit gluonic degrees of freedom absent in scalar-spectator or purely constituent constructions (More et al., 2017).

2. Phase-space imaging: Wigner distributions and GTMD structure

In the dressed-quark realization, quark and gluon Wigner distributions are obtained as transverse Fourier transforms of GTMD correlators. For quarks, the leading-twist Dirac structures are $v^\pm=(v^0\pm v^3)/\sqrt2$3; for gluons, the correlators are built from $v^\pm=(v^0\pm v^3)/\sqrt2$4 with projectors such as $v^\pm=(v^0\pm v^3)/\sqrt2$5 and $v^\pm=(v^0\pm v^3)/\sqrt2$6. In both cases the Wigner functions are overlap integrals of light-front wave functions with shifted intrinsic transverse momenta,

$v^\pm=(v^0\pm v^3)/\sqrt2$7

This makes the relation between five-dimensional phase-space densities and GTMDs explicit (More et al., 2017, Mukherjee et al., 2015).

At twist-2, sixteen quark Wigner distributions exist in general, but the one-gluon dressed-quark model yields eight independent nonzero functions: $v^\pm=(v^0\pm v^3)/\sqrt2$8, $v^\pm=(v^0\pm v^3)/\sqrt2$9, $v_\perp$0, $v_\perp$1, and $v_\perp$2, $v_\perp$3, $v_\perp$4, $v_\perp$5. The pretzelous distribution $v_\perp$6 vanishes, and $v_\perp$7-odd limits vanish because the gauge link is trivial. Analytical expressions are available for each nonzero distribution, with denominators $v_\perp$8 inherited from the light-front energy denominators of the $v_\perp$9 intermediate state (More et al., 2017).

The resulting spatial patterns are highly structured. The unpolarized, helicity, and parallel-transverse combinations display central peaks in impact-parameter space. The mixed polarization channels exhibit dipole or quadrupole distortions that encode spin–orbit correlations. For quarks, $A^+=0$0 and $A^+=0$1 show dipole structures in both $A^+=0$2- and $A^+=0$3-space and quadrupole structure in mixed space. For gluons, $A^+=0$4 and $A^+=0$5 are even under $A^+=0$6, whereas $A^+=0$7 and $A^+=0$8 are odd and exhibit dipole behavior; mixed-space quadrupoles again trace angular correlations between position and momentum (Mukherjee et al., 2015, More et al., 2018).

Numerically, the transverse Fourier integrals are oscillatory. The Levin method was introduced to stabilize the $A^+=0$9 integration, with convergence observed for $x_i$0 GeV. In this setup, typical illustrations use $x_i$1 GeV, fixed $x_i$2 GeV $x_i$3 or $x_i$4 GeV$x_i$5, and results displayed after factoring out the overall normalization constant (More et al., 2017, More et al., 2018).

3. Double parton distributions and transverse non-factorization

The same dressed-quark state provides a calculable model for quark–gluon double parton distributions. Here the proton is replaced by a perturbatively dressed quark, and DPDs are written as overlaps of the same two-body light-front wave functions. Because the Fock-space truncation contains only a quark and a gluon, the support is constrained by

$x_i$6

so nontrivial longitudinal two-parton correlations beyond this kinematic relation are absent in the minimal model (Kasemets et al., 2016).

The resulting scalar DPDs are obtained in closed form. The unpolarized, longitudinally polarized, linearly polarized gluon, transversely polarized quark, and mixed transverse–linear structures are expressed through modified Bessel functions $x_i$7 and $x_i$8, with argument $x_i$9. Representative results include

$q_{i\perp}$0

and

$q_{i\perp}$1

The tensor structure $q_{i\perp}$2 vanishes in this truncation (Kasemets et al., 2016).

A central outcome is that the transverse dependence does not factorize. The dependence on $q_{i\perp}$3 always appears together with $q_{i\perp}$4 inside the Bessel-function argument, and distinct $q_{i\perp}$5-dependent prefactors multiply different transverse structures. As a consequence, neither $q_{i\perp}$6–$q_{i\perp}$7 factorization nor a universal profile $q_{i\perp}$8 is supported in the model. The calculation also finds sizable spin–spin and spin–kinematic correlations, strong longitudinal polarization effects over broad regions of $q_{i\perp}$9, and suppression of linear gluon polarization at small $\sum_i x_i=1$0 (Kasemets et al., 2016).

The color structure is equally constrained. For quark–gluon DPDs, the ratios of color-interference to singlet distributions are

$\sum_i x_i=1$1

which saturate the most stringent color positivity bound discussed in the cited analysis. Positivity constraints on the full helicity density matrix are satisfied, and the strongest bounds are approximately saturated within numerical accuracy (Kasemets et al., 2016).

4. Orbital angular momentum, gravitational form factors, and mechanical structure

The dressed-quark model has been used extensively to compare canonical and kinetic angular momentum. For quarks, the kinetic OAM is extracted from GPDs through Ji’s sum rule,

$\sum_i x_i=1$2

whereas the canonical OAM is obtained from the GTMD $\sum_i x_i=1$3,

$\sum_i x_i=1$4

In this model $\sum_i x_i=1$5, so the quark spin–orbit correlation satisfies

$\sum_i x_i=1$6

Both $\sum_i x_i=1$7 and $\sum_i x_i=1$8 are negative, but they are not equal, and the spin–orbit correlation is negative, indicating anti-alignment of quark spin and quark OAM (Mukherjee et al., 2014).

An analogous program has been carried out for gluons. The canonical gluon OAM is extracted from $\sum_i x_i=1$9,

$\sum_i q_{i\perp}=0$0

and the kinetic gluon OAM from the gluon GPD combination entering Ji’s sum rule. In the one-loop dressed-quark model both $\sum_i q_{i\perp}=0$1 and $\sum_i q_{i\perp}=0$2 are negative for typical ultraviolet cutoffs, and the gluon spin–orbit correlation is also negative. Their distinct $\sum_i q_{i\perp}=0$3-weights show that the canonical and kinetic decompositions remain inequivalent at the density level even when the Wilson line is trivial (Mukherjee et al., 2015).

The same dressed-quark state supports a full calculation of quark and gluon gravitational form factors. Using the symmetric QCD energy-momentum tensor in $\sum_i q_{i\perp}=0$4, one introduces the spin-$\sum_i q_{i\perp}=0$5 decomposition

$\sum_i q_{i\perp}=0$6

with $\sum_i q_{i\perp}=0$7 in the convention used. In the dressed-quark calculation, $\sum_i q_{i\perp}=0$8, $\sum_i q_{i\perp}=0$9, and $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$0. The gluon contribution $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$1 is negative, while the quark contribution $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$2 is positive; the total anomalous gravitomagnetic moment therefore vanishes at $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$3, as required by the sum rule (More et al., 2021, More et al., 2023, More et al., 2023).

The $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$4-term governs the mechanical densities. In two-dimensional impact-parameter space,

$$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$5

For the total dressed-quark system, the pressure exhibits a positive core and a negative tail, satisfying the two-dimensional von Laue condition, while the total normal force is positive. In the gluon sector, $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$6 diverges as $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$7, a behavior compared in the source to the photon $$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$8-term in QED (More et al., 2023).

A further development concerns angular-momentum densities in impact-parameter space. In the light-front gauge, canonical and kinetic densities differ by superpotentials, although their integrated values agree. For the dressed quark state the analysis verifies

$$\langle p',s'|p,s\rangle =2p^+(2\pi)^3\delta(p'^+-p^+)\delta^{(2)}(p'_\perp-p_\perp)\delta_{s's}.$$9

and analogous equalities in the canonical and Belinfante decompositions. Locally in $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$0, however, the canonical and kinetic gluon angular-momentum densities differ by the gluon superpotential density, which integrates to zero only after integration over all transverse space (Mukherjee et al., 2024).

5. Dressed constituents in light-front quark models of mesons

A different usage of the term appears in light-front quark-model studies of pseudoscalar mesons. There the starting point is not a perturbatively dressed quark target but a meson described by constituent quark and antiquark degrees of freedom with a Gaussian radial wave function. The key claim is that, after matching an exactly solvable covariant Bethe–Salpeter model to the phenomenological LFQM, explicit zero-mode and instantaneous terms present in the Bethe–Salpeter analysis disappear from the LFQM expressions. This is interpreted as evidence that the constituent quark and antiquark effectively include a “zero-mode cloud” and can therefore be regarded as dressed constituents (Choi et al., 2014).

The analysis uses light-front coordinates $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$1, internal variables $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$2, and the Drell–Yan frame $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$3. In the pseudoscalar channel, the twist-2 and twist-3 distribution amplitudes are defined through the light-cone correlators

$$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$4

$$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$5

with $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$6 and $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$7 by GMOR. Under the LFQM correspondence, the Bethe–Salpeter vertex is replaced by the Gaussian radial wave function $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$8, and $$\Psi^{\sigma a}_{\sigma_1 \sigma_2}(x,q_\perp) = \frac{gT^a}{\sqrt{2(2\pi)^3}} \frac{1}{\sqrt{1-x}} \frac{ \chi^\dagger_{\sigma_1} \left[ -\frac{2q_\perp}{1-x} -\frac{(\sigma_\perp\!\cdot\! q_\perp)\sigma_\perp}{x} +i\,m\,\frac{1-x}{x}\sigma_\perp \right] \chi_\sigma\cdot \epsilon_{\perp,\sigma_2}^* }{ m^2-\frac{m^2+q_\perp^2}{x}-\frac{q_\perp^2}{1-x} }.$$9 in the integrands is replaced by $v^\pm=(v^0\pm v^3)/\sqrt2$00, so that the observables become purely on-shell valence expressions (Choi et al., 2014).

In the pion case, the model reproduces the asymptotic chiral-limit distribution amplitudes

$v^\pm=(v^0\pm v^3)/\sqrt2$01

At finite constituent masses, $v^\pm=(v^0\pm v^3)/\sqrt2$02 is symmetric and endpoint-suppressed relative to unity, whereas for the kaon the strange quark carries larger longitudinal momentum and induces an asymmetric twist-3 DA. For the pion elastic form factor, the $v^\pm=(v^0\pm v^3)/\sqrt2$03 frame yields a standard LFQM convolution free of explicit zero-mode contributions for both $v^\pm=(v^0\pm v^3)/\sqrt2$04 and $v^\pm=(v^0\pm v^3)/\sqrt2$05, with only on-shell valence dynamics contributing. The computed $v^\pm=(v^0\pm v^3)/\sqrt2$06 agrees with data up to $v^\pm=(v^0\pm v^3)/\sqrt2$07 GeV$v^\pm=(v^0\pm v^3)/\sqrt2$08 and approaches the expected power-law behavior for small constituent masses (Choi et al., 2014).

The same work reports numerical values consistent with standard low-energy phenomenology: $v^\pm=(v^0\pm v^3)/\sqrt2$09 MeV, $v^\pm=(v^0\pm v^3)/\sqrt2$10 MeV, and a quark condensate inferred from $v^\pm=(v^0\pm v^3)/\sqrt2$11 normalization of approximately $v^\pm=(v^0\pm v^3)/\sqrt2$12 in the linear potential parameter set and $v^\pm=(v^0\pm v^3)/\sqrt2$13 in the harmonic-oscillator set. These results are presented as evidence that the LFQM, despite the absence of explicit vacuum terms in final observables, remains consistent with the chiral symmetry constraints of QCD (Choi et al., 2014).

6. Spectral-resolvent generalizations and scope

Recent work reformulates the dressed-quark problem at the operator level by starting from a Minkowski-space propagator with a running mass,

$v^\pm=(v^0\pm v^3)/\sqrt2$14

with the parameter set

$v^\pm=(v^0\pm v^3)/\sqrt2$15

implying an infrared mass $v^\pm=(v^0\pm v^3)/\sqrt2$16 GeV. The associated generalized spectral representation separates the instantaneous light-front term proportional to $v^\pm=(v^0\pm v^3)/\sqrt2$17 from the propagating part. From the dressed two-body resolvent one constructs an effective dressed light-front mass-squared operator $v^\pm=(v^0\pm v^3)/\sqrt2$18 and an effective light-front self-energy $v^\pm=(v^0\pm v^3)/\sqrt2$19, with ultraviolet renormalization fixed so that

$v^\pm=(v^0\pm v^3)/\sqrt2$20

In this formulation, dressing induces a large infrared enhancement of the effective quark mass, up to about $v^\pm=(v^0\pm v^3)/\sqrt2$21 GeV at low $v^\pm=(v^0\pm v^3)/\sqrt2$22, while preserving controlled ultraviolet behavior (Marinho et al., 24 Apr 2026).

Applied to the pion, the effective operator is inserted into representative valence light-front wave functions. The model sets $v^\pm=(v^0\pm v^3)/\sqrt2$23 MeV and $v^\pm=(v^0\pm v^3)/\sqrt2$24 to fix parameters. In the Gaussian ansatz with running self-energy,

$v^\pm=(v^0\pm v^3)/\sqrt2$25

with $v^\pm=(v^0\pm v^3)/\sqrt2$26 GeV, the unpolarized TMD develops a narrow peak around $v^\pm=(v^0\pm v^3)/\sqrt2$27 and a slower $v^\pm=(v^0\pm v^3)/\sqrt2$28 falloff up to $v^\pm=(v^0\pm v^3)/\sqrt2$29 GeV than the corresponding fixed-mass Gaussian model. By contrast, the power-law-like symmetric-vertex models with and without $v^\pm=(v^0\pm v^3)/\sqrt2$30 give similar TMDs, PDFs, and DAs, and their distribution amplitudes remain close to the asymptotic form $v^\pm=(v^0\pm v^3)/\sqrt2$31 (Marinho et al., 24 Apr 2026).

A complementary formal development connects these dressed-particle descriptions to the four-dimensional Bethe–Salpeter equation through quasi-potential reduction and the Nakanishi integral representation. In that framework, the light-front valence wave function is obtained by projecting a Minkowski-space bound-state amplitude, and the hierarchy of light-front Green’s functions shows explicitly that, once dressed particles are used, the valence sector couples to higher Fock components. The same program extends to three-body systems through light-front Faddeev–Bethe–Salpeter equations with dressed constituents (Paula et al., 16 Jan 2026).

Across these variants, the limitations are explicit. The perturbative dressed-quark state is nonconfining, truncated to one gluon, and lacks final-state interactions because the gauge link is trivial; consequently, $v^\pm=(v^0\pm v^3)/\sqrt2$32-odd functions vanish. The mesonic LFQM depends on the Gaussian ansatz, hadronic-scale inputs, and perturbative evolution for comparison at higher scales. The GFF calculations note that omission of light-front zero modes can affect $v^\pm=(v^0\pm v^3)/\sqrt2$33 and may contribute to the behavior of $v^\pm=(v^0\pm v^3)/\sqrt2$34 near $v^\pm=(v^0\pm v^3)/\sqrt2$35. The spectral-resolvent approach, while embedding nonperturbative dressing, still derives its kinetic operator from the disconnected resolvent and does not yet include the full interaction kernel required for an ab initio bound-state treatment (Kasemets et al., 2016, Choi et al., 2014, More et al., 2023, Paula et al., 16 Jan 2026).

In this combined sense, the light-front dressed quark model is not a single model but a coherent light-front program: perturbative quark–gluon dressing for partonic imaging and spin structure, constituent dressing through effective zero-mode clouds in phenomenological meson models, and operator-level dressing through spectral light-front mass operators. What unifies these constructions is the attempt to encode nontrivial QCD dynamics into light-front degrees of freedom while preserving the calculational advantages of the null-plane framework (Mukherjee et al., 2014, Choi et al., 2014, Marinho et al., 24 Apr 2026).