Modified Godfrey-Isgur Quark Model

• The modified Godfrey-Isgur quark model is a family of relativistic frameworks that replace the linear confining term with a screened potential to better capture the dynamics of excited hadrons.
• It retains the original GI model's relativistic kinetic terms, Gaussian smearing, and spin-dependent interactions, while addressing mass overestimation in higher excitations.
• The model is extended to include coupled-channel dynamics and diquark-antidiquark systems, enhancing its application to meson, baryon, and tetraquark spectroscopy.

The modified Godfrey-Isgur quark model denotes a set of relativized constituent-quark frameworks derived from the Godfrey-Isgur (GI) model and used for hadron spectroscopy beyond the domain where the original linear-confinement formulation is most accurate. In the literature summarized here, the most common modification is the replacement of the linear confining interaction by a screened potential, while retaining the GI relativistic kinetic term, Gaussian smearing, and spin-dependent interactions. Closely related usages extend the GI framework by explicit coupled-channel dynamics or by adapting it to diquark-antidiquark systems for tetraquarks. Taken together, these constructions form a phenomenological program for describing higher radial and orbital excitations, threshold-sensitive states, heavy baryons, and selected multiquark candidates within a unified relativized-potential language (Li et al., 2021, Feng et al., 2022, Weng et al., 2024, Yang et al., 2023).

1. Conceptual origin and scope

The original GI model is a relativized quark model in which the Hamiltonian combines relativistic kinetic energy with a quark-antiquark potential containing spin-independent confinement, Coulomb-like one-gluon exchange, hyperfine terms, tensor forces, and spin-orbit interactions. In meson applications it is typically written as

$$\tilde{H}=\sqrt{m_1^2+\mathbf{p}^2}+\sqrt{m_2^2+\mathbf{p}^2}+\tilde{V}_{\mathrm{eff}}(\mathbf{p},\mathbf{r}),$$

or equivalently as a kinetic operator plus relativized interaction terms. The model has been widely used over light, strange, charm, and bottom sectors, and related baryon constructions follow the Capstick-Isgur extension with three relativistic quarks and QCD-motivated interactions (Sultan et al., 13 Mar 2025, Weng et al., 2024).

The impetus for modification is recurrent across the literature. For higher excitations, the original linear confining term can overestimate masses; one summary explicitly states that the original GI model produces significant discrepancies, “hundreds of MeV,” for higher excited states in light mesons, while another shows that for highly excited $\rho$ states the screened version lowers masses by as much as $300$–$500$ MeV relative to the GI model. In heavy sectors, a distinct but related issue is that continuum coupling and higher Fock components can shift bare GI masses substantially, especially near open-flavor thresholds (Li et al., 2021, Feng et al., 2022, Ferretti et al., 2013, Sultan et al., 13 Mar 2025).

Accordingly, “modified GI” is not a single universal prescription. In much of recent meson and baryon spectroscopy it specifically means screened confinement. In other works it refers to explicit unquenching or to a diquark-antidiquark reduction that preserves the GI interaction structure but changes color factors or constituent interpretation. This suggests that the modified GI model is best understood as a family of GI-based phenomenological extensions rather than a uniquely fixed Hamiltonian.

2. Hamiltonian structure and screened confinement

The canonical screened version preserves the relativized GI architecture while modifying the long-range confining piece. The basic substitution is

$$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$

or, in equivalent notation used in several papers,

$S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$

For $\mu r \ll 1$, the screened form reduces to $br$; for $\mu r \gg 1$, it saturates. The intended physics is the flattening of the confining interaction at large distance, interpreted as a phenomenological representation of string breaking, vacuum polarization, or coupled-channel effects (Li et al., 2021, Feng et al., 2022, Pang et al., 2018, Weng et al., 2024).

This replacement is not made in isolation. The screened interaction is subjected to the same relativizing procedures characteristic of GI: Gaussian smearing in coordinate space, momentum-dependent factors, and modified spin-dependent operators. One summary gives the smeared potential as

$\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$

with $\rho$0 Gaussian, while another writes the screened confinement after smearing as $\rho$1. In light-meson applications, representative parameter choices include $\rho$2, $\rho$3, $\rho$4, and $\rho$5; related studies use $\rho$6, $\rho$7, $\rho$8, and $\rho$9 (Li et al., 2021, Feng et al., 2022, Pang et al., 16 Mar 2025, Chen et al., 19 Oct 2025).

In baryons, the original confinement is formulated through a QCD flux-tube picture,

$300$0

and the modified GI version replaces the linear behavior by a screened one. The heavy-baryon study reports that the original GI and modified GI models give similar results for currently observed heavy baryons, while advocating the screened form as conceptually more appropriate for higher excitations and threshold-sensitive states (Weng et al., 2024).

3. Alternative GI-based modifications: unquenching and diquark reductions

A second major line of modification introduces explicit continuum coupling. In this formulation the GI Hamiltonian supplies the bare spectrum, and physical masses are obtained from

$300$1

with

$300$2

The interaction $300$3 is modeled by a $300$4 quark-pair-creation operator, often with a Gaussian quark form factor and an effective pair-creation strength $300$5 to suppress heavy-flavor pair creation. This approach was applied to charmed-strange mesons and bottomonium, where it yields large but often nearly universal downward mass shifts that can be partly or largely absorbed into refitted quenched parameters (Yang et al., 2023, Sultan et al., 13 Mar 2025, Ferretti et al., 2013).

A third line of modification appears in tetraquark studies. There the GI model is first applied to a $300$6 diquark, usually in the color $300$7 representation, with the quark-quark potential taken as

$300$8

The resulting diquark and antidiquark masses are then used in a second GI calculation for the diquark-antidiquark bound state. In open-charm and open-bottom tetraquarks, the Coulomb-like one-gluon-exchange term may be modified by a diquark-size form factor,

$300$9

although one study sets $500$0, treating the diquark as pointlike and thereby maximizing attraction (Lü et al., 2016).

The $500$1 tetraquark study combines this diquark-antidiquark reduction with screened confinement. It solves the $500$2 diquark and then the $500$3 system, taking screening parameters in the range $500$4 to $500$5, with $500$6 preferred by charmed-strange mesons (Lü et al., 2016).

4. Spectroscopy of light and strange mesons

The most extensive use of the modified GI model has been in high-lying light and strange mesons, where screening is introduced precisely because the unscreened GI potential tends to overestimate excited-state masses. In the $500$7 sector, the model supports the assignments $500$8, $500$9, and $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$0. The quoted masses are $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$1, $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$2, and $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$3 MeV in the modified model, compared with $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$4, $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$5, and $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$6 MeV in the original GI calculation; the corresponding predicted total widths, $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$7, $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$8, and $$br \rightarrow V^{\rm scr}(r)=\frac{b(1-e^{-\mu r})}{\mu},$$9 MeV, are reported to agree with experiment (Li et al., 2021).

The same framework has been pushed to still higher $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$0 excitations. For $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$1, $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$2, and $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$3, the modified GI masses are $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$4, $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$5, and $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$6 MeV, versus GI values $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$7, $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$8, and $S(r)=\frac{b(1-e^{-\mu r})}{\mu}+c.$9 MeV. For $\mu r \ll 1$0, $\mu r \ll 1$1, and $\mu r \ll 1$2, the modified values are $\mu r \ll 1$3, $\mu r \ll 1$4, and $\mu r \ll 1$5 MeV, again substantially below the unscreened predictions. The paper explicitly attributes this downward shift to screening (Feng et al., 2022).

Related analyses extend to other light-meson families. The $\mu r \ll 1$6 study predicts $\mu r \ll 1$7 and $\mu r \ll 1$8 at $\mu r \ll 1$9 GeV and $br$0 at $br$1 GeV, all lower than the corresponding GI values. The $br$2 family study gives $br$3 as $br$4, $br$5 as $br$6, and finds that $br$7 is more consistent with $br$8 than $br$9. The $\mu r \gg 1$0 analysis assigns $\mu r \gg 1$1 and $\mu r \gg 1$2 as the same $\mu r \gg 1$3 state, $\mu r \gg 1$4 as $\mu r \gg 1$5, $\mu r \gg 1$6 as $\mu r \gg 1$7, and $\mu r \gg 1$8 as $\mu r \gg 1$9 (Pang et al., 2018, Pang et al., 29 Aug 2025, Chen et al., 19 Oct 2025).

In the strange scalar sector, the newly observed $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$0 is assigned as the $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$1 state. The modified GI model predicts $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$2 at $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$3 MeV, compared with the experimental $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$4 MeV, and the calculated total width of $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$5 MeV is stated to agree with the reported $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$6 MeV. The same study also lists $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$7 at $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$8 MeV and $\tilde{f}(r)=\int f(r)\rho(\vec r-\vec r\,')\,d^3r',$9 at $\rho$00 MeV (Pang et al., 16 Mar 2025).

5. Heavy-quark systems, baryons, and exotic candidates

In heavy hadrons, the modified GI framework serves both as a spectroscopy tool and as a diagnostic for threshold effects. For charmed-strange mesons, the coupled-channel GI study reports that $\rho$01 and $\rho$02 can be interpreted as the $\rho$03 and $\rho$04 states with larger $\rho$05 and $\rho$06 components, respectively. It quotes mass shifts large enough to move $\rho$07 from a bare $\rho$08 MeV to a physical $\rho$09 MeV and $\rho$10 from $\rho$11 MeV to $\rho$12 MeV; the same work gives continuum admixtures of about $\rho$13 and $\rho$14 for these two states (Yang et al., 2023).

In bottomonium, a comparison between a quenched GI model and an unquenched coupled-channel model shows that both can describe the spectrum well. The quoted average relative and absolute errors are $\rho$15 and $\rho$16 MeV for the quenched fit, versus $\rho$17 and $\rho$18 MeV for the unquenched one. The paper further states that continuum mixing typically contributes $\rho$19–$\rho$20 admixtures, leaving valence $\rho$21 probabilities of $\rho$22–$\rho$23, and argues that much of the continuum effect can be absorbed by parameter renormalization (Sultan et al., 13 Mar 2025).

For heavy baryons, the screened and unscreened relativized quark models are both reported to account for all heavy baryons observed so far as three-quark states. Specific assignments include $\rho$24, $\rho$25, $\rho$26, and $\rho$27 as $\rho$28 excitations with $\rho$29, $\rho$30, $\rho$31, and $\rho$32, while $\rho$33 is treated as a $\rho$34 $\rho$35 state and its bottom partner is predicted near $\rho$36 (Weng et al., 2024).

The diquark-antidiquark applications yield mixed results for exotics. In open-bottom tetraquarks, the calculated $\rho$37 masses are found to be much higher than that of $\rho$38, disfavoring a tetraquark interpretation within that scenario. In contrast, the $\rho$39 study assigns $\rho$40 to the ground $\rho$41 tetraquark, $\rho$42 to a $\rho$43 $\rho$44 tetraquark, and $\rho$45 to a configuration built from one $\rho$46 scalar diquark and one scalar antidiquark; it simultaneously concludes that $\rho$47 cannot be explained as a tetraquark in that model and may instead be the conventional $\rho$48 state (Lü et al., 2016, Lü et al., 2016).

6. Interpretation, limitations, and open issues

The modified GI program is phenomenologically successful precisely because it is flexible, but that flexibility also defines its limitations. Screening is introduced as an effective description of long-distance dynamics rather than as a derivation from first-principles QCD, and its quantitative implementation depends on fitted parameters such as $\rho$49. This is evident in the tetraquark literature, where one study uses screened confinement with $\rho$50 varied between $\rho$51 and $\rho$52, whereas another retains the original GI parameters unchanged and instead modifies only color factors and constituent structure (Lü et al., 2016, Lü et al., 2016).

Near thresholds, the distinction between implicit screening and explicit continuum coupling becomes especially important. One bottomonium study argues that the influence of coupled-channel effects can be largely absorbed into quenched parameters, whereas the charmed-strange analysis finds that explicit hadron-loop shifts are crucial for resolving the $\rho$53 and $\rho$54 mass puzzles. A plausible implication is that the screened-potential and explicitly unquenched approaches are complementary effective descriptions rather than strictly interchangeable ones, with the balance depending on how universal the continuum-induced shifts are in a given sector (Sultan et al., 13 Mar 2025, Yang et al., 2023).

Model dependence is also visible in multiquark calculations. In open-bottom tetraquarks, the pointlike approximation $\rho$55 was chosen because it maximally strengthens one-gluon-exchange attraction; the same work notes that using $\rho$56 would only further increase the tetraquark masses. That result strengthens the conclusion that $\rho$57 is disfavored in the diquark-antidiquark GI framework, but it also shows how sensitive exotic assignments can be to assumptions about constituent size and reduction to effective two-body dynamics (Lü et al., 2016).

A further limitation appears in form-factor studies using GI wave functions. In the Bakamjian-Thomas relativistic quark model, GI wave functions give a satisfactory description of elastic heavy-quark transitions and their $\rho$58 corrections, but finite-mass inelastic transitions violate HQET zero-recoil constraints for $\rho$59 channels. The paper concludes that these HQET constraints are crucial for constructing a sensible relativistic quark model of inelastic form factors (Yaouanc et al., 2014).

Within hadron spectroscopy, therefore, the modified Godfrey-Isgur quark model is best regarded as a versatile phenomenological framework: powerful for organizing spectra and decay systematics, especially for higher excitations and threshold-adjacent states, but intrinsically dependent on how screening, unquenching, or effective constituent reduction is implemented in a given application.