Light-Front Quark-Scalar-Diquark Model

• The light-front quark-scalar-diquark model is a two-body valence framework where a proton is described by an active quark and a spectator scalar diquark with dynamics encoded in LFWFs.
• It utilizes both soft-wall AdS/QCD-inspired profiles and spectator formulations to compute key distributions such as PDFs, TMDs, GPDs, and GTMDs, ensuring consistency with experimental data.
• The model enables analytic derivations of overlap representations, evolution equations, and spatial imaging of nucleon structure, while highlighting its validity as a low-scale, valence-focused framework.

Searching arXiv for recent and foundational papers on the light-front quark–scalar-diquark model. arXiv search query: "light-front quark scalar diquark model proton TMD GPD GTMD" The light-front quark-scalar-diquark model is a two-body valence description of the proton in which an active quark is accompanied by a spectator scalar diquark, and the nonperturbative dynamics are encoded in light-front wave functions (LFWFs). In the cited literature, this framework appears both in soft-wall AdS/QCD-inspired form and in spectator formulations with a point-like quark-diquark vertex, and it is used to compute PDFs, TMDs, GPDs, GTMDs, Wigner distributions, gravitational form factors, and higher-twist correlators (Maji et al., 2016, Maji et al., 2017, Kaur et al., 2018).

1. Effective degrees of freedom and model definition

In its minimal form, the model truncates the proton Fock expansion to a single active quark plus a scalar diquark spectator. For the scalar channel, the diquark has spin $0$ and helicity $\lambda_S=0$, while the proton retains helicity $J^z=\pm \tfrac12$ and the quark helicity takes $\lambda_q=\pm \tfrac12$ (Maji et al., 2017, Kaur et al., 2018). The basic state content is therefore

$$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$

with $x$ the quark light-cone momentum fraction and $p_\perp$ its transverse momentum (Sharma et al., 2023).

A broader spin-flavor decomposition often embeds the scalar channel into a quark-diquark basis containing both scalar and axial-vector spectators. In the SU(4) basis used for nucleon modeling,

$$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$

with the scalar sector isolated by retaining only $|u\,S^0\rangle^\pm$ or, in a pure scalar-diquark treatment, by setting the scalar weight $C_S=1$ (Maji et al., 2016, Sharma et al., 2023).

The model is also used in a spectator realization in which the proton is treated as a quark of mass $\lambda_S=0$0 coupled to a scalar diquark of mass $\lambda_S=0$1 through a point-like vertex of strength $\lambda_S=0$2. In that formulation, no explicit Lagrangian is required for practical calculations because the dynamics are carried by the LFWFs themselves, although an effective Yukawa-type interaction $\lambda_S=0$3 is a standard underlying picture (Kaur et al., 2018, Sharma et al., 2023).

2. Light-front state expansion and helicity structure

For the scalar-diquark sector, the proton state is expanded as

$\lambda_S=0$4

with

$\lambda_S=0$5

In the soft-wall AdS/QCD-based construction, the four nonzero helicity amplitudes are (Maji et al., 2017, Maji et al., 2016)

$\lambda_S=0$6

$\lambda_S=0$7

A different but closely related spectator parameterization writes the scalar-channel spinor components as (Kaur et al., 2018, Sharma et al., 2023)

$\lambda_S=0$8

with $\lambda_S=0$9. In both implementations, the spin dependence is carried by explicit transverse-momentum prefactors and the radial dependence is isolated in one or two scalar profile functions.

Normalization is imposed sector by sector. A typical condition is

$J^z=\pm \tfrac12$0

or its equivalent with the light-front measure written explicitly (Maji et al., 2017, Maji et al., 2016). In the AdS/QCD-based nucleon model, the free two-body invariant mass is

$J^z=\pm \tfrac12$1

with $J^z=\pm \tfrac12$2 in the massless-quark version of the model (Maji et al., 2016).

3. Radial wave-function ansätze and parameterizations

The best-known implementation uses soft-wall AdS/QCD-inspired profile functions. In momentum space,

$J^z=\pm \tfrac12$3

with $J^z=\pm \tfrac12$4; the strict AdS/QCD limit corresponds to $J^z=\pm \tfrac12$5 and $J^z=\pm \tfrac12$6 (Maji et al., 2017, Maji et al., 2016). The parameters $J^z=\pm \tfrac12$7 are fitted either to unpolarized PDFs or to nucleon form-factor data, depending on the analysis (Maji et al., 2017, Maji et al., 2016).

A spectator alternative employs a single scalar profile

$J^z=\pm \tfrac12$8

together with constituent masses and couplings fitted to form factors (Kaur et al., 2018). Higher-twist analyses also use an AdS/QCD-motivated Gaussian form,

$J^z=\pm \tfrac12$9

with $\lambda_q=\pm \tfrac12$0 fixed by $\lambda_q=\pm \tfrac12$1 (Sharma et al., 2023).

Implementation	Radial input	Representative parameters
Soft-wall AdS/QCD nucleon model	Two profiles $\lambda_q=\pm \tfrac12$2 with Gaussian $\lambda_q=\pm \tfrac12$3-dependence and powers $\lambda_q=\pm \tfrac12$4	$\lambda_q=\pm \tfrac12$5; at $\lambda_q=\pm \tfrac12$6: for $\lambda_q=\pm \tfrac12$7, $\lambda_q=\pm \tfrac12$8, $\lambda_q=\pm \tfrac12$9, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$0, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$1, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$2, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$3 (Maji et al., 2016, Sharma et al., 2023)
TMD-focused AdS/QCD scalar sector	Same functional form, fitted to PDFs at a different initial scale	$$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$4; at $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$5: $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$6, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$7, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$8, $$|q^{\lambda_q}(xP^+,p_\perp),\,S((1-x)P^+,-p_\perp)\rangle,$$9, $x$0 (Maji et al., 2017)
Spectator scalar-diquark model	Single propagator-like profile $x$1	$x$2, $x$3, $x$4 (Kaur et al., 2018)

A more dynamical realization arises in the light-front Faddeev-Bethe-Salpeter approach, where a contact two-quark interaction produces a scalar-diquark subamplitude $x$5, the proton valence wave function is obtained by symmetrizing a bachelor-quark vertex $x$6, and model parameters are tuned to reproduce the proton mass and Dirac form factor (Ydrefors et al., 2022). This suggests a route from phenomenological quark-diquark LFWFs to a bound-state construction in which the scalar diquark is not merely an external spectator parameter.

4. Overlap representations for PDFs and TMDs

The scalar-diquark model is particularly suited to overlap calculations. For collinear PDFs, the unpolarized, helicity, and transversity distributions are obtained by integrating LFWF overlaps over transverse momentum. In the scalar channel these take the form (Maji et al., 2018, Maji et al., 2016)

$x$7

$x$8

and $x$9 from the corresponding chiral-odd overlap. In the nucleon model based on soft-wall AdS/QCD LFWFs, the helicity and transversity distributions were reported to agree with phenomenological fits, axial and tensor charges were reported to agree with experimental or phenomenological extractions, and the Soffer bound was verified numerically at all scales studied (Maji et al., 2016, Maji et al., 2018).

At leading twist and for T-even TMDs, the scalar sector yields explicit overlap expressions. Representative examples are (Maji et al., 2017)

$p_\perp$0

$p_\perp$1

$p_\perp$2

The remaining leading-twist T-even functions $p_\perp$3, $p_\perp$4, $p_\perp$5, and $p_\perp$6 follow from off-diagonal helicity overlaps (Maji et al., 2017).

A notable numerical result is the approximate $p_\perp$7-$p_\perp$8 factorization,

$p_\perp$9

with

$$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$0

The AdS/QCD scalar-diquark model does not factorize analytically into a product of pure $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$1- and $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$2-functions, but the Gaussian ansatz was found to be realized to very good approximation (Maji et al., 2017).

5. Generalized distributions, spatial imaging, and orbital structure

The same overlap machinery extends to GPDs, GTMDs, and Wigner distributions. For chiral-odd GPDs, the scalar-diquark model provides overlap formulas for $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$3, $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$4, $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$5, and $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$6 at both zero and nonzero skewness, with active-quark momentum arguments shifted as

$$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$7

and similarly for $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$8 and $$|P;\pm\rangle = C_S\,|u\,S^0\rangle^\pm + C_V\,|u\,A^0\rangle^\pm + C_{VV}\,|d\,A^1\rangle^\pm,$$9 (Chakrabarti et al., 2015). Their two-dimensional Fourier transforms define impact-parameter-space GPDs, and a one-dimensional Fourier transform in $|u\,S^0\rangle^\pm$0 introduces a longitudinal impact coordinate $|u\,S^0\rangle^\pm$1. In the scalar model, $|u\,S^0\rangle^\pm$2, $|u\,S^0\rangle^\pm$3, and $|u\,S^0\rangle^\pm$4 show diffraction-like patterns in $|u\,S^0\rangle^\pm$5-space, whereas $|u\,S^0\rangle^\pm$6 does not display a pronounced diffraction pattern because it is odd in $|u\,S^0\rangle^\pm$7 (Chakrabarti et al., 2015).

Phase-space imaging is developed further through Wigner distributions and GTMDs. The twist-2 Wigner operator is

$|u\,S^0\rangle^\pm$8

and its Fourier transform in $|u\,S^0\rangle^\pm$9 yields

$C_S=1$0

Combining proton and quark polarizations generates 16 independent twist-2 Wigner distributions, while the GTMD decomposition contains 16 complex functions of

$C_S=1$1

of which only 10 survive at $C_S=1$2 (Kaur et al., 2018). In the scalar-diquark sector, $C_S=1$3 and $C_S=1$4 are circularly symmetric in both $C_S=1$5 and $C_S=1$6; $C_S=1$7 and $C_S=1$8 display dipolar patterns and a quadrupole in mixed space; and $C_S=1$9 at $\lambda_S=0$00 is directly related to quark orbital angular momentum (Kaur et al., 2018).

Momentum-space densities also reveal transverse-shape information. In the scalar model,

$\lambda_S=0$01

with $\lambda_S=0$02. The transverse-shape relation

$\lambda_S=0$03

shows non-spherical, “pretzel” distortions driven by $\lambda_S=0$04 (Maji et al., 2017).

For energy-momentum structure, the scalar-diquark LFWFs generate gravitational form factors through overlaps with $\lambda_S=0$05: $\lambda_S=0$06 Their two-dimensional Fourier transforms define the longitudinal momentum density $\lambda_S=0$07 and the angular momentum density $\lambda_S=0$08 in the transverse plane (Kumar et al., 2017).

T-odd extensions require an explicit final-state interaction. In the holographic scalar-diquark model with one-gluon-exchange phase,

$\lambda_S=0$09

the T-odd GTMDs $\lambda_S=0$10 and $\lambda_S=0$11 emerge from the imaginary part of $\lambda_S=0$12, and in the forward limit $\lambda_S=0$13 they reduce to the Sivers and Boer-Mulders functions (Chakrabarti et al., 2019).

6. Evolution, higher-twist sectors, and limitations

The scalar-diquark model is defined at a low hadronic scale and then evolved. For integrated PDFs, ordinary DGLAP evolution is applied, either directly or through a simulated DGLAP procedure in which the profile parameters $\lambda_S=0$14, $\lambda_S=0$15, and $\lambda_S=0$16 acquire scale dependence (Maji et al., 2016, Maji et al., 2018). For TMDs, a Collins-Soper-Sterman-type evolution is written in $\lambda_S=0$17-space: $\lambda_S=0$18 with

$\lambda_S=0$19

followed by the Fourier transform back to momentum space (Maji et al., 2017).

Beyond leading twist, the model remains analytically tractable. For T-even twist-3 TMDs, the scalar sector yields explicit formulas for $\lambda_S=0$20, $\lambda_S=0$21, $\lambda_S=0$22, $\lambda_S=0$23, $\lambda_S=0$24, $\lambda_S=0$25, $\lambda_S=0$26, and $\lambda_S=0$27. Because explicit quark-gluon interaction terms are absent and the gauge link is set to unity for T-even observables, Wandzura-Wilczek-type relations become exact in the model, including

$\lambda_S=0$28

for the corresponding twist-3 functions (Sharma et al., 2023).

At twist 4, the scalar-diquark sector gives closed expressions for $\lambda_S=0$29, $\lambda_S=0$30, $\lambda_S=0$31, $\lambda_S=0$32, $\lambda_S=0$33, and $\lambda_S=0$34, with model relations tying them to leading-twist TMDs. The simplest is

$\lambda_S=0$35

and analogous formulas relate $\lambda_S=0$36, $\lambda_S=0$37, $\lambda_S=0$38, $\lambda_S=0$39, and $\lambda_S=0$40 to $\lambda_S=0$41, $\lambda_S=0$42, $\lambda_S=0$43, $\lambda_S=0$44, $\lambda_S=0$45, and $\lambda_S=0$46 (Sharma et al., 2023). For the twist-4 T-even function $\lambda_S=0$47, the scalar channel gives

$\lambda_S=0$48

with reported numerical averages

$\lambda_S=0$49

(Sharma et al., 2023). Twist-4 GTMDs have also been parameterized in the scalar sector, and their TMD limit reproduces the previously derived twist-4 T-even TMDs (Sharma et al., 2023).

The main limitations are tied to the same simplifications that make the model analytic. Many calculations are valence-only and retain only the minimal quark-diquark Fock sector (Maji et al., 2016, Ydrefors et al., 2022). In the chiral-odd GPD construction, the analysis is restricted to the DGLAP region $\lambda_S=0$50, and all valence GPDs vanish at $\lambda_S=0$51 because no explicit ERBL contribution is included (Chakrabarti et al., 2015). T-odd observables are absent unless an explicit final-state interaction phase is introduced (Chakrabarti et al., 2019). In the light-front Faddeev-Bethe-Salpeter approach, the evolved non-polarized valence PDFs suggest that higher Fock components beyond the valence one are needed for comparison with NNPDF4.0 (Ydrefors et al., 2022). A plausible implication is that the light-front quark-scalar-diquark model is most reliable as a controlled low-scale valence framework and as a laboratory for analytic relations among light-front distributions, rather than as a complete description of proton structure at all $\lambda_S=0$52 and all twists.