Covariant Confined Quark Model Overview

• The Covariant Confined Quark Model is a family of Lorentz-covariant constituent quark frameworks that implement confinement via nonlocal vertices and covariant interaction kernels.
• Its nonlocal formulation employs Gaussian vertex functions, compositeness conditions, and an infrared cutoff to eliminate unphysical quark thresholds in loop amplitudes.
• The CST-based formulation reduces the Bethe–Salpeter equation in Minkowski space, ensuring chiral symmetry and stable partial-wave solutions through carefully designed confining kernels.

Searching arXiv for core CCQM and CST references relevant to the Covariant Confined Quark Model. The Covariant Confined Quark Model denotes a family of Lorentz-covariant constituent-quark frameworks in which hadrons are treated as composite quark bound states and confinement is implemented directly at the level of quark-loop amplitudes or quark–antiquark interaction kernels. In the literature covered here, the term spans two closely related but technically distinct realizations. One is the covariant confined constituent quark model based on nonlocal hadron–quark interaction vertices, Gaussian correlation functions, a compositeness condition $Z_H=0$, and an infrared cutoff in Schwinger-parameter space that removes quark thresholds (Gutsche et al., 2012). The other is a Covariant Spectator Theory (CST) formulation in Minkowski space, where confinement is encoded through a covariant linear kernel and consistent quark dressing, with charge-conjugation invariance and chiral constraints built into the bound-state equations (Biernat et al., 2013). In both usages, “covariant” refers to manifest Lorentz covariance, while “confined” refers to the absence of unphysical free-quark production in the hadronic amplitudes or to the vanishing of the bound-state vertex when two quarks are simultaneously on shell (Leitão et al., 2014).

1. Conceptual scope and model classes

The expression “Covariant Confined Quark Model” is used in at least two established senses in the cited literature. In the nonlocal-vertex formulation, hadrons are described by effective nonlocal interaction Lagrangians that couple hadron fields to constituent quark currents with translationally invariant vertex functions. The nonlocality encodes hadron size and regulates ultraviolet behavior, while confinement is implemented by cutting off Schwinger-parameter integrals at an upper limit $1/\lambda^2$, thereby eliminating quark thresholds and ensuring that loop amplitudes are analytic in the external kinematics (Gutsche et al., 2012). This formulation has been applied broadly to mesons, baryons, heavy-flavor decays, electromagnetic observables, and charmonium radiative transitions (Ganbold et al., 2014).

In the CST realization, the same broad designation refers to a covariant Minkowski-space quark–antiquark model in which the interaction kernel itself is the central dynamical object. The bound-state problem is formulated through a CST reduction of the Bethe–Salpeter equation, with dressed quark propagators generated by the same interaction kernel that binds the quarks. In this setting, confinement is associated with a covariant generalization of linear confinement and with the property that bound-state vertices vanish when both quarks are simultaneously on shell (Biernat et al., 2013). This suggests that the label “Covariant Confined Quark Model” is best understood as a methodological umbrella rather than a single unique formalism.

A further distinction concerns how confinement is realized. In the nonlocal-vertex approach, quark propagators are rendered effectively entire in loop amplitudes by the infrared cutoff, so no quark cuts occur (Gutsche et al., 2012). In the CST approach, quark propagators may retain mass poles, but the spectator construction and the confining kernel prevent physically observable quark–antiquark thresholds, and the formalism works directly in Minkowski space without analytic continuation from Euclidean space (Biernat et al., 2013). The two realizations therefore differ in mechanism while sharing the goals of covariance, hadronic compositeness, and effective confinement.

2. Nonlocal hadron–quark formulation

In the nonlocal formulation, the starting point is an effective interaction Lagrangian of the form

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$

with a nonlocal current

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$

for mesons, and analogous three-quark currents for baryons (Dubnička et al., 2023). The vertex function is translationally invariant,

$$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$

and in momentum space a Gaussian ansatz is used,

$$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$

which becomes ultraviolet-suppressing after Wick rotation (Dubnička et al., 2023).

For baryons, the nonlocal current generalizes to three constituent quarks. In the light-baryon sector, a generic interaction Lagrangian is

$$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$

with

$$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$

and a Gaussian momentum-space correlation function characterized by a baryon size parameter $\Lambda_B$ (Gutsche et al., 2018). The formalism supports different Dirac structures for scalar, vector, tensor, pseudoscalar, and axial configurations, depending on the hadron quantum numbers (Gutsche et al., 2012).

The nonlocal formulation is explicitly Poincaré covariant, and the quark-level amplitudes are computed from one-loop or two-loop Feynman diagrams with constituent propagators and nonlocal vertices. In meson transitions such as $B_s\to D_s^{(*)}\ell\nu$, the weak current matrix elements are evaluated from a single quark loop with two nonlocal meson–quark vertices, and the resulting form factors are fitted over the full kinematic range by double-pole parameterizations (Pandya et al., 2023). In baryon processes, two-loop diagrams appear naturally because of the three-quark structure of the hadron currents (Gutsche et al., 2018).

3. Compositeness, confinement, and gauge invariance

A defining element of the nonlocal CCQM is the compositeness condition

$1/\lambda^2$0

which fixes the hadron–quark coupling $1/\lambda^2$1 from the derivative of the hadron mass operator and enforces that the physical hadron has no elementary component (Ganbold et al., 2014). In the meson-spectrum discussion, this condition is paired with the relation

$1/\lambda^2$2

which equates the Yukawa-type hadron–quark theory to a Fermi-type four-fermion theory and gives an interpretation of the meson field as a bound state of constituent quarks (Ganbold et al., 2014). This suggests that compositeness is not merely a normalization convention but part of the model’s ontological definition of hadrons.

Confinement in the nonlocal model is implemented through Schwinger-parameter regularization. With the constituent propagator written as

$1/\lambda^2$3

or in Schwinger form,

$1/\lambda^2$4

the model replaces the upper limit of the proper-time integration by $1/\lambda^2$5. In a generic $1/\lambda^2$6-propagator amplitude this yields

$1/\lambda^2$7

with a universal infrared cutoff $1/\lambda^2$8 (Gutsche et al., 2012). The result is that amplitudes contain no physical quark cuts and quark lines cannot go on shell inside the loop integrals (Gutsche et al., 2012).

Gauge invariance requires special care because the hadron–quark interaction is nonlocal. The standard procedure is to gauge both the local kinetic terms and the nonlocal strong vertex by attaching path-ordered phase factors to the quark fields,

$1/\lambda^2$9

and then expanding to first order in the photon field (Gutsche et al., 2012). This generates not only the usual triangle diagrams in which the photon couples to a quark line, but also bubble or contact diagrams in which the photon couples to the nonlocal vertex function. Their sum satisfies the Ward–Takahashi identities, and in the baryon electromagnetic application the model explicitly shows that the non-gauge-invariant contributions cancel between bubble terms (Gutsche et al., 2012). The same logic underlies later charmonium radiative-transition studies, where gauge-invariant amplitudes are written as sums of triangle, bubble, and derivative-contact terms (Issadykov et al., 17 Sep 2025).

4. CST-based covariant confined quark model

The CST realization starts from the Minkowski-space Bethe–Salpeter equation

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$0

with dressed quark propagators

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$1

(Leitão et al., 2014). CST performs the $$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$2 integration by retaining quark-pole contributions, producing a three-dimensional reduction. When both positive- and negative-energy poles are kept and symmetrized, one obtains a charge-conjugation-invariant four-channel spectator equation (Biernat et al., 2013).

For heavy–light systems, the one-channel spectator equation (1CSE) keeps only the positive-energy pole of the heavier quark: $$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$3 Here $$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$4 is the positive-energy projector

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$5

and the Dirac structure of the interaction may be taken as a scalar/vector mixture,

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$6

(Leitão et al., 2014).

The confining interaction is introduced from a linear potential,

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$7

with covariant momentum-space representation

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$8

Because this kernel is singular, CST introduces an exact subtraction,

$$\mathcal{L}_{\text{int}} = g_H\, H(x)\, J_H(x) + \text{H.c.},$$9

which yields the principal-value identity

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$0

(Leitão et al., 2014). This subtraction removes the nonintegrable singularity and is the key numerical improvement that allows stable partial-wave solutions up to $$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$1, whereas the unsubtracted kernel fails to converge for $$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$2 (Leitão et al., 2014).

The broader CST program extends beyond spectroscopy to quark self-energy and chiral dynamics. The one-body CST-Dyson equation,

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$3

uses the same interaction kernel that appears in the bound-state equation, establishing self-consistency between quark dressing and hadron binding (Biernat et al., 2013). This is structurally reminiscent of Dyson–Schwinger/Bethe–Salpeter approaches, but CST remains explicitly in Minkowski space and employs a spectator reduction instead of Euclidean continuation (Biernat et al., 2013).

5. Chiral symmetry and confinement

One of the central theoretical issues for any covariant confined quark model is whether scalar confinement is compatible with chiral symmetry. In the CST framework, this is addressed through the axial-vector Ward–Takahashi identity and the careful design of the confining kernel (Biernat et al., 2015). The dressed axial vertex satisfies

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$4

while the CST-Bethe–Salpeter equation for the same vertex is

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$5

(Biernat et al., 2015).

The confining kernel is taken in the general form

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$6

A special role is played by the equal-weighted scalar–pseudoscalar condition

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$7

together with the covariant constraint

$$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$8

(Biernat et al., 2015). Under these conditions, the linear-confining part decouples from the chiral-limit pion equation and from the scalar part of the Dyson equation that generates the dynamical mass. The framework then preserves the axial-vector Ward–Takahashi identity and satisfies the Adler-zero constraint in $$J_H(x)=\int d^4x_1\,d^4x_2\,F_H(x;x_1,x_2)\,\bar q_2(x_2)\,\Gamma_H\,q_1(x_1)$$9 scattering (Biernat et al., 2015). This suggests that scalar confinement is not excluded, but it must be embedded in a Lorentz structure and subtraction scheme consistent with chiral identities.

A related CST study shows how dynamical chiral symmetry breaking emerges in Minkowski space when the same interaction kernel dresses the quark propagator and binds the quarks into a pseudoscalar state (Biernat et al., 2013). In the chiral limit, the pseudoscalar bound-state equation reduces to the scalar part of the self-energy equation, and the “massless pion condition” becomes identical to the gap equation. The resulting quark mass function in the simplified kernel model is

$$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$0

with chiral-limit form

$$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$1

and fitted parameters $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$2, $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$3, $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$4, $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$5 (Biernat et al., 2013). The model reproduces Euclidean lattice-QCD mass-function data well in the infrared, with $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$6 in the chiral limit over $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$7 (Biernat et al., 2013).

6. Phenomenology and applications

The nonlocal CCQM has been used extensively for hadronic phenomenology. In the light-baryon sector, the extension from mesons to baryons employs the same constituent masses and universal infrared cutoff as in the meson sector, leaving only hadron-specific size parameters and current-mixing weights to be fitted (Gutsche et al., 2012). For nucleon electromagnetic observables, the model reproduces

$$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$8

with $$F_M(x; x_1, x_2) = \delta\!\left(x - w_1 x_1 - w_2 x_2 \right)\,\Phi_M\!\left[(x_1-x_2)^2\right],\quad w_i=\frac{m_i}{m_1+m_2},$$9 and $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$0, compared with experimental $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$1 and $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$2, and gives

$$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$3

close to the experimental $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$4 (Gutsche et al., 2012). The neutron charge radius is especially sensitive to the mixing of vector and tensor nucleon currents (Gutsche et al., 2012).

Heavy-flavor applications are numerous. In semileptonic $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$5, the model computes the form factors $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$6, $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$7, $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$8, and $$\widetilde{\Phi}_M(-k^2)=\exp\!\left(\frac{k^2}{\Lambda_M^2}\right),$$9 over the full kinematic range from one-loop diagrams with two nonlocal meson–quark vertices (Pandya et al., 2023). The $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$0-dependence is represented by a double-pole ansatz, and the resulting integrated observables include

$$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$1

with angular observables such as $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$2 for $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$3 (Pandya et al., 2023). For rare $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$4, the same framework gives

$$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$5

consistent with modern Standard-Model estimates (Issadykov et al., 2022).

In nonleptonic heavy-hadron decays the model is often combined with naive factorization. For selected $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$6 decays, it gives good agreement in semileptonic channels such as

$$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$7

but more mixed performance in nonleptonic modes, for example overshooting

$$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$8

relative to $$\mathcal{L}_{\rm int}^{B}(x)=g_B\,\bar B(x)\,J_B(x)+\mathrm{h.c.},$$9 (Dubnička et al., 17 Dec 2025). The authors argue that such discrepancies likely reflect the breakdown of naive factorization and the importance of long-distance final-state interactions (Dubnička et al., 17 Dec 2025). This marks one of the recurring limitations of the nonlocal CCQM in its simplest weak-decay implementations.

The baryonic heavy-flavor sector provides another example. In the nonleptonic decay $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$0, a CCQM analysis including both short-distance $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$1-exchange topologies and long-distance pole diagrams finds that the short-distance pieces are more than an order of magnitude too small, while the long-distance resonant contributions dominate and bring the branching fraction to

$$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$2

in excellent agreement with LHCb and Belle (Ivanov et al., 2023). This is a concrete case where the model’s nonlocal quark-loop machinery is combined with current algebra and pole saturation to address hadronic long-distance dynamics.

The CCQM has also been applied to charmonium radiative transitions. A recent study of $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$3, $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$4, $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$5, and $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$6 decays demonstrates explicit gauge invariance of the amplitudes and fits the charm-quark mass and size parameter to three benchmark channels, obtaining

$$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$7

with $$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$8 (Issadykov et al., 17 Sep 2025). Predicted branching fractions such as

$$J_B(x)=\int dx_1\,dx_2\,dx_3\,F_B(x;x_1,x_2,x_3)\,\varepsilon^{a b c}\,\Gamma_1\,q_1^a(x_1)\,q_2^b(x_2)\,C\,\Gamma_2\,q_3^c(x_3)$$9

agree well with experiment (Issadykov et al., 17 Sep 2025). An earlier related charmonium analysis used $\Lambda_B$0, $\Lambda_B$1, and a common size parameter $\Lambda_B$2 to describe the dominant S- and P-wave radiative transitions (Ganbold et al., 2021).

7. Spectroscopy, limitations, and relation to other approaches

The CCQM has been used not only for transition amplitudes but also for spectroscopy. In the meson-spectrum study, the compositeness condition is combined with the bound-state relation

$\Lambda_B$3

to reconstruct a smooth effective Fermi coupling $\Lambda_B$4 across pseudoscalar and vector meson families (Ganbold et al., 2014). With constituent masses

$\Lambda_B$5

and fitted hadron size parameters, the resulting masses agree well with experimental values in the light and charm sectors, though deviations become larger in the bottom sector (Ganbold et al., 2014). A plausible implication is that the Gaussian nonlocal ansatz and effective constituent masses capture much of the gross spectral structure but are less accurate when heavy-quark dynamics and open-flavor thresholds become more delicate.

The CST version faces a different set of limitations. The one-channel spectator equation is well justified for heavy–light systems but must be generalized to the full four-channel system for light–light mesons (Leitão et al., 2014). Most benchmark solutions employ the instantaneous approximation $\Lambda_B$6, although tests with retardation show good convergence (Leitão et al., 2014). Interaction kernels often include only the linear confining part and scalar/time-like-vector or scalar–pseudoscalar–tensor Lorentz structures, while short-range one-gluon exchange and spin-dependent forces remain to be incorporated systematically (Leitão et al., 2014).

Relative to Dyson–Schwinger/Bethe–Salpeter approaches, CST works directly in Minkowski space, retains a more explicit potential-model interpretation of confinement, and can connect the quark self-energy and quark–antiquark kernel within the same formalism (Biernat et al., 2013). Relative to the nonlocal-vertex CCQM, CST does not confine by making quark propagators effectively entire in loop amplitudes; instead it uses spectator kinematics and a confining kernel that prevents simultaneous on-shell propagation of the quark pair (Biernat et al., 2013). The nonlocal CCQM, by contrast, is especially flexible phenomenologically and has been applied to a much broader range of exclusive decay channels (Dubnička et al., 2023).

A persistent source of confusion is therefore terminological. Some authors use “covariant confined quark model” to mean specifically the nonlocal-vertex framework of Gutsche, Ivanov, Körner, Lyubovitskij, and collaborators (Gutsche et al., 2012), whereas others use the phrase more broadly for covariant constituent-quark models with explicit confinement in Minkowski space, including CST-based constructions (Biernat et al., 2013). The overlap in nomenclature reflects shared aims but not identity of formalism.