Generalized parton distributions and transverse densities in a light-front quark-diquark model for the nucleons

We present a study of the generalized parton distributions for the quarks in a proton in both momentum and position spaces using the light-front wave functions of a quark-diquark model for the nucleon predicted by the soft-wall model of AdS/QCD. The results are compared with the soft-wall AdS/QCD model of proton GPDs for zero skewness. We also calculate the GPDs for nonzero skewness. We observe that the GPDs have a diffraction pattern in longitudinal position space, as seen before in other models. Then we present a comparitive study of the nucleon charge and anomalous magnetization densities in the transverse plane. Flavor decompositions of the form factors and transverse densities are also discussed.


I. INTRODUCTION
Hadronic structure and their properties being nonperturbative in nature are always very difficult to evaluate from QCD first principle and there have been numerous attempts to gain insight into hadrons by studying QCD inspired models. The quark-diquark model, where a nucleon is considered to be a bound state of a single quark and a scalar or vector diquark state, is proven to reproduce many interesting properties of nucleons and has been extensively used to investigate the proton structure. Recently, a light front quark-diquark model for the nucleons has been proposed in Ref. [1], where the light front wavefuctions are modeled by the In this paper, we study the proton structure and evaluate the Generalized Parton Distributions(GPDs), transverse charge and magnetization densities in the light front quark-diquark model. Contrary to ordinary parton distribution function, GPDs are functions of three variables, namely, longitudinal momentum faction x of the quark or gluon, square of the total momentum transferred (t) and the skewness ζ, which represents the longitudinal momentum transferred in the process and provide interesting information about the spin and orbital angular momentum of the constituents, as well as the spatial structure, of the nucleons(see [2] for reviews on GPDs). The GPDs appear in the exclusive processes like deeply virtual Compton scattering (DVCS) or vector meson productions and they reduce to the ordinary parton distributions in the forward limit. Their first moments are related to the form factors and the second moment of the of the sum of the GPDs are related to the angular momentum by a sum rule proposed by Ji [3]. Being off-forward matrix elements, the GPDs have no probabilistic interpretation. But for zero skewness, the Fourier transforms of the GPDs with respect to the transverse momentum transfer (∆ ⊥ ) give the impact parameter dependent GPDs which satisfy the positivity condition and can be interpreted as distribution functions [4]. The transverse impact parameter dependent GPDs provide us with the information about partonic distributions in the impact parameter or the transverse position space for a given longitudinal momentum (x). The impact parameter b ⊥ gives the separation of the struck quark from the center of momentum. In parallel to the efforts to understand the GPDs by theoretical modeling, different experiments are also measuring deeply virtual Compton scattering and deeply virtual meson production to gain insight and experimentally constrain the GPDs [5].
We evaluate the proton GPDs for both zero and nonzero skewness and compare with the results in a soft-wall AdS/QCD model [6](for hard-wall and soft-wall AdS/QCD models of hadrons, see [7,8]) . For zero skewness, the GPDs are investigated in the impact parameter or transverse position space. The LF diquark results for GPD H(x, b ⊥ ) for u-quark is almost the same as AdS/QCD results whereas there is a little difference for d-quark. But the LF diquark model results for E(x, b ⊥ ) for both u and d quarks are different from AdS/QCD results. For nonzero skewness, the GPDs in longitudinal impact parameter space show a diffraction pattern.
It is interesting to note that similar diffraction patterns were observed in simple QED model for DVCS amplitude [9] and GPDs [10] and in a phenomenological model of proton GPDs [11].
Electric charge and magnetization densities in the transverse plane also provide insights into the structure of nucleons. The charge and magnetization densities in the transverse plane are defined as the Fourier transform of the electromagnetic form factors. The form factors involve initial and final states with different momenta and the three dimensional Fourier transforms cannot be interpreted as densities whereas the transverse densities(i.e., Fourier transformed only for transverse momenta) defined at fixed light front time are free from this difficulty and have proper density interpretation [12,13]. We calculate the transverse charge and anomalous magnetization densities for both proton and neutron in the light-front diquark model and compare with the two different global parametrizations proposed by Kelly [14] and Bradford et al [15]. We present results for both unpolarized and transversely polarized nucleons. We also present a comparison with the AdS/QCD results citeCM3 for the transverse charge and magnetization densities.
The paper is organized as follows. In Section II, we give a brief introductions about the nucleon LFWFs of quark-diquark model as well as the electromagnetic flavor form factors. We show the results for proton GPDs of u and d quarks in momentum space in Section III. Then we discuss the GPDs in the transverse as well as the longitudinal impact parameter space in Sections III A and III B. We present the results of the charge and anomalous magnetization densities in the transverse plane in section IV. Finally we provide a brief summary and conclusions in Section V. For GPDs with nonzero skewness, we present a comparison of the quark-diquark model results with a Double Distribution(DD) model in the appendix.

II. LIGHT-FRONT QUARK-DIQUARK MODEL FOR THE NUCLEON
In quark-scalar diquark model, the three valence quarks of nucleon are considered as an effectively composite system composed of a fermion and a neutral scalar bound state of diquark based on one loop quantum fluctuations. In the light-cone formalism for a spin 1 2 composite system the Dirac and Pauli form factors F 1 (q 2 ) and F 2 (q 2 ) are indentified to the helicityconserving and helicity-flip matrix elements of the J + current [17] here M n is the nucleon mass. Writing proton as a two particle bound state of a quark and a scalar diquark in the light front quark-diquark model, the Dirac and Pauli form factors for the quarks can be written in the light-front representation [17,18] as are the LFWFs with specific nucleon helicities λ N = ± and for the struck quark λ q = ±, where plus and minus correspond to + 1 2 and − 1 2 respectively. We consider the frame where q = (0, 0, q ⊥ ), thus Q 2 = −q 2 = q 2 ⊥ .
We adopt the generic ansatz for the quark-diquark model of the valence Fock state of the nucleon LFWFs at an initial scale µ 0 = 313 MeV as proposed in [1] : where ϕ (1) q (x, k ⊥ ) and ϕ (2) q (x, k ⊥ ) are the wave functions predicted by soft-wall AdS/QCD [19] Following the convention of [1], we fix the normalizations of the Dirac and Pauli form factors so that F q 1 (0) = n q and F q 2 (0) = κ q where n u = 2, n d = 1 and the anomalous magnetic moments for the u and d quarks are κ u = 1.673 and κ d = −2.033. The advantage of the modified formulae in Eq.(7) is that, irrespective of the values of the parameters, the normalization conditions for the form factors are automatically satisfied. The structure integrals, I q i (Q 2 ) have the form as with It is straightforward to write down the flavor decompositions of the Dirac and Pauli form factors of nucleon as q (with i = 1, 2) for each quark. In Ref. [1], κ is taken to be 0.35 GeV and the parameters are evaluated to fit the electromagnetic properties of the nucleon. But the results for the form factors presented in that paper are not converged with respect to the lower limit of x integrations in Eqs. (8 and 9). The comparison with experimental data presented in several plots in Ref. [1] are true only for an unrealistically large value of lower limit for the x integrations and are not stable under lowering that limit toward x → 0. In this work, we use a different scale parameter κ = 0.4066 GeV which was obtained by fitting the nucleon form factors in AdS/QCD soft-wall model [6,20]. Here, we show that we can reproduce the nucleon form factors with the new parameters a  Table I are in excellent agreement with the data. For F d 1 , we can see a clear improvement in the quark-diquark model over the AdS/QCD model. It is important to note that other models fail to reproduce the form factors data for d quark [22]. In Fig.2, we have shown the fit of light-front quark-diquark model results with experimental data of proton form factors. We get excellent agreement with the  [23,24]. The red dashed lines represent the soft-wall AdS/QCD model [20]. The plots show the ratio of Pauli and Dirac form factors for the proton, (a) the ratio is multiplied by Q 2 = −q 2 = −t, (b) the ratio is divided by κ p . The experimental data are taken from Refs. [25][26][27][28][29].
The red dashed lines represent the soft-wall AdS/QCD model [20]. data. In the same plots, we also show comparisons of the light-front quark-diquark model and the soft-wall AdS/QCD model with the same value of κ [20]. The results of the light-front quark-diquark model agree with the data better than AdS/QCD, specially at large Q 2 values we achieve substantial improvement. The Sach form factor G E (Q 2 ) for the neutron is shown in Fig.3. Again, our results agree with the experimental data much better than the AdS/QCD results. The fitted results for the electromagnetic radii of the nucleons are listed in Table II. The standard formulae for the electromagnetic radii of nucleon used here are given below: where N stands for nucleon(N = p/n) and the Sachs form factors are defined as This domain corresponds to the situation where one removes a quark from the initial proton with light-front momentum fraction x = k + P + and the transverse momentum k ⊥ and re-insert it into the final state of the proton with longitudinal momentum fraction x − ζ and transverse momentum k ⊥ −q ⊥ . The GPDs H and E are defined through the matrix element of the bilocal vector current on the light-front: The proton state | P, λ is written in two particle Fock states with one femion and a scalar boson in the light front quark-diquark model. Using the relations whereP = (P + P ′ )/2 and λ(λ ′ ) = ± 1 2 is the initinal(final) proton spin, we have the following expressions for the GPDs in terms of the LFWFs in the quark-diquark model where Substituting the LFWFs (Eq.(5)) in Eqs. (18) and (19) and integrating over k ⊥ , we get the following expressions for GPDs where the functions F q i (x, ζ, t) are given by where A is a function of x and x ′ , I q 1 (0) and I q 2 (0) are the integrals defined in Eqs. (8) and (9) for Q 2 = 0. The GPDs are normalized as where n q denotes the number of u or d valence quarks in the proton and the quark anomalous magnetic moment is denoted by κ q . The GPDs for zero skewness(ζ = 0) in light-front quark-   A. GPDs in transverse impact parameter space GPDs in transverse impact parameter space are defined as [4,39]: Here, b ⊥ is the transverse impact parameter. For zero skewness, b ⊥ gives a measure of the transverse distance between the struck parton and the center of momentum of the hadron. b ⊥ satisfies the condition i x i b ⊥i = 0, where the sum is over the number of partons. The relative distance between the struck parton and the center of momentum of the spectator system is given by |b ⊥ | 1−x , which provides us an estimate of the size of the bound state [40]. However, the exact estimation of the nuclear size is not possible as the spatial extension of the spectator system is not available from the GPDs. In the DGLAP domain x > ζ, the impact parameter b ⊥ implies the location where the quark is pulled out and pushed back to the nucleon. In the ERBL region x < ζ, b ⊥ gives the transverse location of the quark-antiquark pair inside the nucleon. that the AdS/QCD model is unable to reproduce F d 1 to match with experimental data whereas the form factor in the diquark model agrees well with the data (see Fig.1(b)). In Fig.9 we The similar behavior of the GPDs of a phenomenological model was observed in [41]. Another interesting behavior of all the GPDs is that the width of all the distributions in transverse b. It is interesting to note that the peaks of all the distributions also become broader as ζ increases for a fixed value of x. This means that the probability of hitting the active quark at a larger transverse impact parameter b increases as the momentum transfer in the longitudinal direction increases.

B. GPDs in longitudinal impact parameter space
The boost invariant longitudinal impact parameter is defined as σ = 1 2 b − P + which is conjugate to the skewness ζ, the measure of longitudinal momentum transfer. The parameter σ was first Introduced in [9] and it was shown that the DVCS amplitude in a QED model of a dressed electron shows an interesting diffraction pattern in the longitudinal impact parameter space. Since Lorentz boosts are kinematical in the light front, the correlation defined in the three dimensional position space b ⊥ and σ is frame independent. It was shown in the same simple relativistic spin half system of an electron dressed with a photon that the GPDs also exhibit the similar diffraction pattern in the longitudinal impact parameter space [10]. Similar diffraction pattern was also observed in a phenomenological model for proton GPDs [11]. So, it is very interesting to investigate if the similar pattern is also observed in this light front quark model. The GPDs in longitudinal position space are defined as: E(x, σ, t) = 1 2π Since we are considering the region ζ < x < 1, the upper limit of ζ integration ζ f is given by ζ max if x is larger than ζ max , otherwise by x if x is smaller than ζ max where the maximum value of ζ for a fixed −t is given by In Fig.12, we show the GPDs in longitudinal position space σ considering the DGLAP region.
We observe that the GPDs show diffraction pattern in longitudinal impact parameter space, similar to the nature of a dressed electron in QED or in a holographic model for the meson [9].
This effect has also been observed for the GPDs of a phenomenological model [11] as well as for the chiral odd GPDs of light-front QED model [10].

IV. TRANSVERSE CHARGE AND MAGNETIZATION DENSITIES
The two dimensional Fourier transform of the Dirac form factor gives the transverse charge density in the transverse plane for the unpolarized nucleons, where b represents the impact parameter and J 0 is the cylindrical Bessel function of order zero.
We can write a similar formula for charge density for flavor ρ q f ch (b) with F 1 is replaced by F q 1 . In a similar fashion, one defines the magnetization density in the transverse plane by the Fourier transform of the Pauli form factor, whereas, has the interpretation of anomalous magnetization density [42]. Since these quantities are not directly measured in experiments, actual experimental data are not available. In Ref. [13], an estimation of the proton charge and magnetization densities has been done from experimental data of electromagnetic form factors. To get an insight into the contributions of the different flavors, we evaluate the charge and anomalous magnetization densities for the u and d quarks.
We can define the decompositions of the transverse charge and magnetization densities for nucleons in the similar way as electromagnetic form factors. The charge densities decompositions in terms of two flavors can be written as where e u and e d are charge of u and d quarks respectively. Due to the charge and isospin symmetry the u and d quark densities in the proton are the same as the d and u densities in the neutron [16,45]. Under the charge and isospin symmetry, we can write where ρ q ch (b) is the charge density of each quark and ρ q f ch is the charge density for each flavor. We can similarly decompose ρ m into magnetization densities for each flavor.     [14], and the lines with circles represent the parametrization of Bradford at el [15], dot-dashed lines are for soft-wall model [16]. The solid lines represent the light-front scalar diquark model.
In Fig. 13, we show the charge and anomalous magnetization densities for proton and neutron. The plots suggest that the light-front diquark model's results for the charge and magnetization density of proton and the magnetization density of neutron are in excellent agreement with the two different global parametrizations of Kelly [14] and Bradford at el [15]. Though the diquark model is unable to reproduce the data for the neutron charge density at small b, still it is better than the AdS/QCD Model-I predictions presented in Ref. [16].
In Fig.13(c), one can notice a negatively charged core surrounded by a ring of positive charge    [14], and the lines with circles represent the parametrization of Bradford at el [15], dot-dashed lines are soft-wall model [16]. The solid lines represent the light-front scalar diquark model.   [14], and line with circles represents the parametrization of Bradford at el [15], dot-dashed line is soft-wall Model-I in [16]. The solid line represents this work for light-front scalar diquark model.
For transversely polarized nucleon, the charge density in the transverse plane is given by [43] where M is the mass of nucleon and the transverse polarization of the nucleon is given by S ⊥ = (cos φ sx + sin φ sŷ ) and the transverse impact parameter b ⊥ = b(cos φ bx + sin φ bŷ ). Without loss of generality, the polarization of the nucleon is taken along x-axis ie., φ s = 0. The second term in Eq. (35), provides the deviation from circular symmetry of the unpolarized charge Polarization is along x-direction.
pattern in the case of neutron [43]. The behaviors are in agreement with the results reported in Refs. [42][43][44].
We compare the up quark charge densities for the transversely polarized and unpolarized nucleon in Fig.17 (a) and (b) and the similar plots for d quark are shown in Fig. 17 (c) and (d).
The deviation or distortion from the symmetric unpolarized density is more for down quark than the up quark. For the nucleons polarized in x direction, the charge density shifts towards positive b y direction for d quark but in opposite direction for the u quark.

V. SUMMARY AND CONCLUSIONS
The parameters in a light front quark-diquark model of nucleons [1] are found to be inconsistent with the experimental data. We have re-evaluated the parameters in this model for the AdS/QCD scale parameter κ = 0.4066 GeV which was previously obtained by fitting the nucleon form factors in sofft wall QdS/QCD [6,20]. The GPDs for ζ = 0 admit a density interpretation when one takes the Fourier transform to the impact parameter space but in experiments, ζ is always nonzero. In recent past, there have been a lot of works to model GPDs with nonzero skewness by modeling relevant DDs [46,47].
In this section, we compare our results for nonzero skewness with the GPDs modeled from the Double Distributions(DD) [48][49][50]. The GPDs have an integral representation in terms of the double distributions f (β, α, t). For the valance quarks, the GPDs can be written as where F q v = H q v , E q v . Here, we use the factorized DD ansatz for the GPDs as suggested by Musatov and Radyushkin [51] f q v (β, α, t) = F q v (β, 0, t)h(β, α), where the weight function h(β, α) generates the skewness dependence of the GPDs and satisfies the nornalization condition h(β, α)dα = 1.
The similar profile function for N = 2 has been used in many phenomenological model of DVCS and exclusive meson production [2,[52][53][54]. Inserting Eq. A2 in the Eq. A1, with the help of delta-function one can perform the integral over α and obtains for x > ζ, the integration boundaries are In Fig. 18 and Fig. 19, we show the skewness dependent GPDs calculated using double distribution parametrization and compare with the results directly calculated in the quarkdiquark model. Fig.18 suggests that for small ζ and large −t, the results of double distribution are more or less in agreement with the diquark model results, while Fig.19 shows that at moderate or high values of skewness ζ, the agreement is completely lost.