Quark–Diquark Model Overview

• The quark–diquark model is an effective description of baryons that reorganizes the three-quark problem into a two-body system of a constituent quark and a correlated diquark.
• It simplifies baryon spectroscopy and nucleon structure calculations by reducing spatial degrees of freedom and emphasizing dominant spin–flavor correlations, consistent with observed spectra.
• Various formulations, including light-front and Faddeev approaches, leverage this reduction to reproduce form factors, excitation spectra, and lattice-QCD susceptibilities despite inherent model limitations.

Searching arXiv for recent and foundational papers on the quark–diquark model to ground the article in the literature. The quark–diquark model is an effective description of baryons in which the three-quark problem is reorganized into a two-body system composed of a constituent quark and a correlated quark pair, the diquark. In the formulations surveyed in the literature, the diquark is typically taken in the color $\bar{\mathbf 3}$ channel and may be scalar or axial-vector, while the baryon is treated either as a relativistic quark–diquark bound state, a light-front quark–spectator system, or a quark–diquark correlation embedded in a more complete Faddeev framework. Across spectroscopy, nucleon structure, finite-temperature QCD, and lattice-based constructions, the model functions less as a fundamental rewriting of QCD than as a controlled reduction of degrees of freedom that preserves dominant spin–flavor correlations and often reproduces observed spectra and form-factor systematics with fewer states than three-quark constituent models (Santopinto et al., 2011).

1. Effective two-body picture and diquark degrees of freedom

In QCD, a baryon is a color-singlet state of three quarks, schematically $qqq$. The quark–diquark model replaces this by an effective two-body system $q + D$, where $D\equiv (qq)$ denotes a correlated quark pair. In the formulations considered for light, strange, heavy, and doubly heavy baryons, the diquark is taken in the attractive color antitriplet channel, so that $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$ can form a color singlet (Kinutani et al., 2023). A diquark is not an isolated hadron; it carries color and therefore cannot exist freely, but it can act as an effective constituent inside a baryon (Mutuk, 2021).

The basic spin assignment is between scalar and axial-vector diquarks. In SU(6)$_{sf}$ language, the color-antitriplet diquark belongs to the symmetric $\mathbf{21}_{sf}$, which decomposes into a scalar diquark with spin $0$ and flavor $\bar{\mathbf 3}$, and an axial-vector diquark with spin $1$ and flavor $qqq$0 (Sanctis et al., 2016). In nucleon light-front models this appears explicitly as a superposition of scalar isoscalar and axial-vector isoscalar/isovector components, e.g.

$qqq$1

with the coefficients fixed by normalization and form-factor constraints (Maji et al., 2016). For heavy $qqq$2 baryons, the relevant diquark is the scalar, isospin-zero $qqq$3 configuration, which makes $qqq$4 and $qqq$5 natural heavy-quark plus scalar-diquark systems (Kinutani et al., 2023).

This two-body reduction is motivated by several recurring observations. First, the attractive color-$qqq$6 channel favors quark–quark clustering. Second, a quark–diquark organization reduces the number of effective spatial degrees of freedom relative to a genuine three-body problem. Third, in several phenomenological contexts the baryon spectrum behaves more like an effective two-body system than a generic three-body one: a semiclassical estimate gives $qqq$7 for three-body dynamics, whereas the observed baryon growth is closer to $qqq$8–$qqq$9, which suggests effective quark–diquark behavior in excited baryons (Megias et al., 2019). A closely related analysis argues that the cumulative number of baryon states behaves approximately as $q + D$0, again consistent with two-body rather than three-body dynamics (1812.2111).

2. State classification and relativistic mass operators

The quark–diquark reduction sharply constrains the allowed baryon multiplets. Because the quark transforms as $q + D$1 in SU(6)$q + D$2 and the diquark as $q + D$3, the spin–flavor product is

$q + D$4

so the $q + D$5 multiplet of three-quark SU(6)$q + D$6 models is absent (Santopinto et al., 2011). This reduction is central to the model’s role in addressing the missing-resonances problem: many states allowed in three-quark SU(6)$q + D$7$q + D$8O(3) schemes are simply forbidden once the baryon is treated as quark plus diquark (Santopinto et al., 2011). A related relativistic interacting quark–diquark model likewise emphasizes that the number of predicted excited states below $q + D$9 GeV is smaller than in three-quark models, while still accommodating the established strange and nonstrange resonances (Sanctis et al., 2016).

In relativistic spectroscopy, the dynamics is encoded in a mass operator rather than a nonrelativistic Hamiltonian. In the interacting quark–diquark model developed in point-form dynamics, the mass operator is

$D\equiv (qq)$0

where $D\equiv (qq)$1 and $D\equiv (qq)$2 are diquark and quark masses, $D\equiv (qq)$3 is a Coulomb-plus-linear direct interaction, $D\equiv (qq)$4 is a spin–flavor exchange term, $D\equiv (qq)$5 is a short-range contact term used in the nonstrange sector, and $D\equiv (qq)$6 is a spin–isospin transition term mixing scalar and axial-vector diquark configurations (Sanctis et al., 2016). In the nonstrange sector the direct interaction is

$D\equiv (qq)$7

while the exchange interaction contains spin, isospin, and spin–isospin operators, multiplied by $D\equiv (qq)$8 (Sanctis et al., 2016).

A simpler algebraic implementation constructs a mass formula directly for $D\equiv (qq)$9 in order to reproduce rotational and vibrational Regge trajectories. There the mass spectrum is built from terms linear in the orbital angular momentum $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$0, the radial quantum number $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$1, spin $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$2, total angular momentum $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$3, isospin $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$4, SU(3) and SU(6) Casimirs, and a squared constituent-mass term containing the scalar and axial-vector diquark mass splittings (Santopinto et al., 2011). The fitted rotational and vibrational slopes are reported as

$q(\mathbf 3)\otimes D(\bar{\mathbf 3})$5

which are interpreted as effective Regge slopes for the quark–diquark string (Santopinto et al., 2011).

The same general logic underlies the relativistic point-form extension of the interacting model, where Lorentz transformations are kinematic and the baryon mass operator carries the interaction. In that setting the model is used not only for spectroscopy but also for nucleon electromagnetic form factors, while maintaining the scalar and axial-vector diquark content established in the spectral analysis (Sanctis et al., 2015).

3. Light-front quark–diquark models of nucleon structure

A major line of development formulates the quark–diquark model on the light front. In these constructions the nucleon is expanded in two-body light-front Fock components, with one active quark carrying momentum fraction $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$6 and transverse momentum $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$7, and the spectator diquark carrying the remainder (Maji et al., 2016). For the scalar diquark channel, the proton component is written as

$q(\mathbf 3)\otimes D(\bar{\mathbf 3})$8

while the axial-vector sector contains the six helicity amplitudes $q(\mathbf 3)\otimes D(\bar{\mathbf 3})$9 with $_{sf}$0 (Maji et al., 2016).

One influential model constructs the light-front wave functions from soft-wall AdS/QCD. Its radial wave functions are taken as

$_{sf}$1

with $_{sf}$2 GeV and fitted parameters $_{sf}$3, $_{sf}$4, and $_{sf}$5 (Maji et al., 2016). With this input, the model reproduces the scale evolution of unpolarized proton PDFs over a wide range of $_{sf}$6, yields helicity and transversity distributions consistent with phenomenological fits, gives axial and tensor charges in agreement with experiment, and satisfies the Soffer bound (Maji et al., 2016). It is also explicitly constructed to be consistent with the quark counting rule and the Drell–Yan–West relation (Maji et al., 2016).

A different light-front quark–diquark realization is used to analyze full phase-space spin structure through quark Wigner distributions. There the spin-$_{sf}$7 composite state is expanded as

$_{sf}$8

and both scalar and axial-vector diquark contributions are treated via overlap representations of light-front wave functions (Kaur et al., 2019). The Wigner distributions

$_{sf}$9

encode transverse position, transverse momentum, and longitudinal momentum fraction, and interpolate between GPDs, TMDs, PDFs, and GTMDs under the appropriate projections (Kaur et al., 2019). A technical extension of that work is the explicit inclusion of the longitudinal polarization vector of the axial-vector diquark, allowing a comparison between results with only transverse diquark polarizations and results including the $\mathbf{21}_{sf}$0 component (Kaur et al., 2019).

The same light-front architecture has also been used for twist-3 quark–gluon–quark correlation functions. In that setting the transverse-momentum-dependent quark–gluon–quark correlator is evaluated with an axial-vector spectator, and the time-reversal-odd interaction-dependent twist-3 functions $\mathbf{21}_{sf}$1 and $\mathbf{21}_{sf}$2 are extracted from helicity-interference terms in the light-front wave functions (Sharma et al., 2021). Even in the lowest-order approximation where gauge links are neglected, the model yields nonzero T-odd interaction-dependent twist-3 quark TMDs, although point-like nucleon–quark–diquark couplings generate divergences that motivate dipole regularization of the vertex (Sharma et al., 2021).

A more covariant variant embeds the quark–diquark core in a pion cloud while maintaining Poincaré invariance. There the bare nucleon consists of scalar and axial-vector diquark configurations on the light front, and a one-pion loop dresses the electromagnetic current and the quark spin content. With suitably chosen parameters, the model reproduces the proton and neutron electromagnetic form factors and reduces the quark-spin fraction from the bare-core value $\mathbf{21}_{sf}$3 to $\mathbf{21}_{sf}$4, close to polarized-DIS phenomenology (Cloet et al., 2012).

4. Heavy baryons, excitation spectra, and relativistic corrections

Heavy baryons furnish a particularly clean testing ground for the quark–diquark approximation. In $\mathbf{21}_{sf}$5 and $\mathbf{21}_{sf}$6, the light $\mathbf{21}_{sf}$7 pair is naturally treated as a scalar, isospin-zero, color-$\mathbf{21}_{sf}$8 diquark, and the baryon becomes a heavy-quark plus scalar-diquark bound state (Kinutani et al., 2023). In a Cornell-like potential model,

$\mathbf{21}_{sf}$9

the central issue is whether the effective string tension $0$0 should coincide with the quarkonium value (Kinutani et al., 2023).

A nonrelativistic study of $0$1 and $0$2 found that the potential determined from quarkonium spectra, with $0$3 and $0$4 GeV/fm, fails to reproduce the $0$5 and $0$6 excitation spectra in the point-like quark–diquark picture: the 1P excitations come out too high (Sanctis et al., 2016). The same analysis reported that the heavy-baryon spectra are reproduced with a string tension approximately half as large,

$0$7

and interpreted this as a puzzle for the naive universality of confinement in a point-like quark–diquark description (Sanctis et al., 2016). The paper then examined finite-size effects of the diquark and found that introducing a diquark of size $0$8 fm softens the effective potential sufficiently to recover the observed $0$9 1P excitation while retaining the quarkonium string tension, which suggests that the reduced effective confinement may arise from diquark compositeness rather than a genuinely different color string tension (Sanctis et al., 2016).

A later relativistic treatment of $\bar{\mathbf 3}$0 and $\bar{\mathbf 3}$1 analyzed the same systems with two distinct types of relativistic corrections to the quark–diquark potential: a quark-level treatment in which the heavy quark exchanges a gluon with one of the light quarks inside the diquark, and an effective scalar treatment in which the diquark is treated as a point-like scalar color-$\bar{\mathbf 3}$2 particle (Kinutani et al., 2023). The central finding is that these two prescriptions differ strongly because the qq-type treatment contains a large light-quark Darwin term proportional to $\bar{\mathbf 3}$3, whereas the scalar-diquark treatment lacks that enhancement (Kinutani et al., 2023). The qq-type relativistic corrections, especially the Darwin term, reduce the level spacings and allow a universal quarkonium-like string tension to remain compatible with the observed $\bar{\mathbf 3}$4 and $\bar{\mathbf 3}$5 excitation spectra, thereby addressing the string-tension puzzle without changing the confining long-distance interaction (Kinutani et al., 2023).

The quark–diquark picture has also been applied to doubly charmed baryons. For $\bar{\mathbf 3}$6, a nonrelativistic Cornell-potential analysis compared three clusterings,

$\bar{\mathbf 3}$7

and found that the $\bar{\mathbf 3}$8 ground-state masses lie in the expected $\bar{\mathbf 3}$9–$1$0 GeV range, with the $1$1 configuration matching the measured $1$2 mass particularly closely (Mutuk, 2021). However, the same study reported that the $1$3-diquark clustering has much larger $1$4, more reasonable hyperfine splittings, and better consistency with Faddeev-based analyses, which led it to conclude that $1$5 diquark clustering is more favorable dynamically than the widely assumed $1$6 clustering (Mutuk, 2021).

5. Thermodynamics, lattice-QCD input, and relation to other QCD frameworks

The quark–diquark model has been used not only for spectroscopy and partonic structure but also for QCD thermodynamics. In the hadron resonance gas description of the confined phase, one can generate a baryon spectrum from a relativistic quark–diquark Hamiltonian

$1$7

with

$1$8

and then insert the resulting baryon masses into the HRG susceptibilities of conserved charges (Megias et al., 2019). In that program the derivative of the quark–diquark potential is assumed equal to the derivative of the quark–antiquark potential,

$1$9

which is motivated by Polyakov-loop correlators and by lattice results for the static quark–diquark potential (1812.2111). The resulting quark–diquark baryon spectrum gives an overall good agreement both with the relativized quark-model spectrum and with lattice-QCD data for $qqq$00, $qqq$01, and $qqq$02, supporting the claim that baryonic fluctuations below the crossover can be saturated by a dense spectrum of quark–diquark states (Megias et al., 2019).

A distinct finite-temperature application arises in the NJL model, where baryons are treated as quark–diquark bound states in quark matter. There the diquark channel undergoes a Mott transition as temperature increases, but the baryon can remain bound even after the diquark has dissociated because Pauli blocking for quarks and Bose enhancement for diquarks in the quark–diquark Bethe–Salpeter kernel approximately cancel (Blaschke et al., 2015). In that regime the baryon is described as a “Borromean” three-quark state in medium: the two-particle subsystem is unbound while the three-particle state remains bound (Blaschke et al., 2015).

The model has also been anchored more directly in lattice QCD. A lattice method analogous to HAL QCD was proposed to extract the quark–diquark interaction potential and diquark mass from equal-time Nambu–Bethe–Salpeter wave functions in the $qqq$03 system (Watanabe et al., 2021). Using $qqq$04-flavor gauge configurations with $qqq$05 MeV, the analysis obtained a charm-quark mass $qqq$06 GeV, a diquark mass $qqq$07 GeV, and a Cornell-like $qqq$08–$qqq$09 potential, and emphasized that the extracted diquark mass is roughly consistent with naive constituent estimates such as $qqq$10 GeV and $qqq$11 GeV (Watanabe et al., 2021). This provides a first-principles route for building quark–diquark models from lattice data rather than from purely phenomenological fits.

At a more formal level, the quark–diquark approximation has been examined within the Poincaré-covariant Faddeev framework. There the baryon remains fundamentally a three-quark bound state, but strong color-$qqq$12 quark–quark correlations allow a quark–diquark truncation of the full Faddeev equation (Eichmann et al., 2010). In the rainbow–ladder kernel used in that work, replacing the full three-body equation by the quark–diquark approximation changes the nucleon mass by only about $qqq$13, which suggests that diquark correlations capture the dominant quark-core dynamics while remaining embedded in a genuinely three-quark description (Eichmann et al., 2010).

6. Scope, successes, and limitations

The quark–diquark model is successful precisely where a dominant two-body clustering captures most of the relevant dynamics. In baryon spectroscopy it reduces the number of states and thereby mitigates the missing-resonances problem (Santopinto et al., 2011). In nucleon structure it yields practical overlap formulae for PDFs, GPDs, TMDs, GTMDs, Wigner distributions, form factors, and spin-dependent observables (Maji et al., 2016). In heavy baryons it can reproduce excitation spectra with a scalar-diquark light cloud, provided relativistic corrections or finite-size effects are handled carefully (Kinutani et al., 2023). In thermodynamics it generates baryon spectra dense enough to saturate lattice-QCD susceptibilities in the confined phase (Megias et al., 2019).

At the same time, the literature emphasizes that the model is an effective description rather than a literal decomposition of QCD. Common limitations include the neglect of explicit gluon degrees of freedom and higher Fock components, model dependence of diquark masses and couplings, and the absence of full QCD evolution in many implementations (Kaur et al., 2019). In HRG and spectral models diquarks are often treated as point-like effective constituents with fixed masses, while internal structure and form factors are not resolved (1812.2111). In simple potential models spin-dependent interactions, tensor forces, and three-body correlations beyond the quark–diquark approximation are often omitted (Sanctis et al., 2016). In twist-3 light-front applications, point-like nucleon–quark–diquark couplings can generate divergences that require phenomenological vertex form factors (Sharma et al., 2021).

A further limitation is channel dependence. The same quark–diquark picture that works rather well for $qqq$14 and $qqq$15 encounters difficulties in the $qqq$16 excitation spectrum, where the calculated ordering of $qqq$17 and $qqq$18 excitations is opposite to experiment (Sanctis et al., 2016). This suggests that strange baryons may require more elaborate mixing or a less rigid clustering assumption. Similarly, while scalar and axial-vector diquarks are usually sufficient below about $qqq$19 GeV, this suggests that higher excitations or channels with strong continuum coupling may need dynamics beyond a frozen diquark core (Sanctis et al., 2016).

Taken together, these results suggest that the quark–diquark model is best understood as a hierarchy of effective descriptions. At one end are phenomenological two-body models with constituent diquark masses and Cornell-like interactions; at the other are covariant Faddeev and lattice-QCD constructions in which the diquark emerges as a strong but confined correlation. The model’s enduring utility lies in the fact that, across these formulations, the dominant scalar and axial-vector quark–quark correlations repeatedly organize a large amount of baryon phenomenology into a tractable and often quantitatively successful two-body framework (Eichmann et al., 2010).