Transverse charge and magnetization densities in holographic QCD

We present a study of flavor structures of transverse charge and anomalous magnetization densities for both unpolarized and transversely polarized nucleons. We consider two different models for the electromagnetic form factors in holographic QCD. The flavor form factors are obtained by decomposing the Dirac and Pauli form factors for nucleons using the charge and isospin symmetry. The results are compared with two standard phenomenological parametrizations.


I. INTRODUCTION
or impact parameter space [1]. The form factor involves initial and final states with different momentum and three dimensional Fourier transforms cannot be interpreted as densities whereas the transverse densities defined in the infinite momentum frame are free from this difficulty and have proper density interpretation [2][3][4].
Recently, AdS/QCD has emerged as one of the most promising techniques to unravel the structure of mesons and nucleons. The AdS/CFT conjecture [5] relates a gravity theory in AdS d+1 to a conformal theory at the d dimensional boundary. There are many applications of AdS/CFT to investigate the QCD phenomena [6,7]. A boundary condition in the fifth dimension z in AdS 5 breaks the conformal invariance and allows QCD mass scale and confinement. In the hard-wall model, an IR cutoff is set at z 0 = 1/Λ QCD while in soft-wall model, a confining potential in z is introduced. There is an exact correspondence between the holographic variable z and the light front transverse variable ζ which measures the separation of the quark and gluonic constituents in the hadron [8,9]. The AdS/QCD for the baryon has been developed by several groups [8][9][10][11][12][13][14]. Though it gives only the semiclassical approximation of QCD, so far this method has been successfully applied to describe many hadron properties e.g., hadron mass spectrum, parton distribution functions, GPDs, meson and nucleon form factors, structure functions etc [12,[15][16][17][18][19][20][21][22]. AdS/QCD wave functions are used to predict the experimental data for ρ meson electroproduction [23]. AdS/QCD has also been successfully applied in the meson sector to predict the branching ratio for decays ofB 0 andB 0 s into ρ mesons [24], isospin asymmetry and branching ratio for the B → K * γ decays [25], transition form factors [26,27], etc. There are many other applications in the baryon sector e.g., semi-empirical hadronic momentum density distributions in the transverse plane have been calculated in [28], in [29], the form facfor of spin 3/2 baryons (∆ resonance) and also the transition form factor between ∆ and nucleon have been studied, an AdS/QCD model has been proposed to study the baryon spectrum at finite temperature [30] etc.
The flavor decompositions of the nucleon form factors in a light-front quark model with SU(6) spin-flavor symmetry have been studied in detail in [21] and shown to agree with experimental data. It is interesting and instructive to study the transverse densities and their flavor decomposition in holographic QCD. There are two different holographic QCD models for nucleon form factors developed by Abidin and Carlson [12] and Brodsky and Teramond [18]. Here, we present a detailed analysis of the transverse densities in both the models.
A model-independent transverse charge densities for nucleons in infinite-momentumframe have been studied in [31] whereas the charge densities in the transverse plane for a transversely polarized nucleon are shown in [32,33]. In [34], the long range behaviors of the unpolarized quark transverse charge density of the nucleons have been studied. Transverse charge and magnetization densities in the nucleon's chiral periphery(i.e., at a distance b = O(1/m π )) using methods of dispersion analysis and chiral effective field theory has been analysed in [35]. The transverse densities for the quarks are studied in a chiral quark-soliton model in [36]. Using Laguerre-Gaussian expansion, Kelly [37] proposed a parametrization of the nucleon Sachs form factors in terms of charge and magnetization densities. A study of flavor dependence of the transverse densities in a GPD model has been reported in [38] In [19], the nucleon transverse charge and magnetization densities have been evaluated in the model developed in [12]. In this work, we show that the flavor decompositions of the transverse densities of the nucleons in two different models in the framework of AdS/QCD and compare with the two global parametrizations of Kelly [39] and Bradford at el [40]. By decomposing the nucleon form factors F 1 and F 2 using the charge and isospin symmetry, we obtain the flavor form factors F q 1 and F q 2 for the quarks. The Fourier transforms of these electromagnetic form factors give the charge and magnetization densities in the transverse plane.
The paper is organized as follows. A brief description of the the form factors in AdS/QCD has been has given in Sec.II. In Sec.III, the charge and magnetization densities for both unpolarized and transversely polarized nucleons have been studied. The individual flavor contributions are also studied in this section. Then we provide a brief summary in Sec.IV.

II. NUCLEON AND FLAVOR FORM FACTORS IN ADS/QCD
Here we consider the soft wall model of AdS/QCD, where in place of a sharp cutoff z, one introduces a potential. The action in soft model is written as [18] where e M A = (z/R)δ M A is the inverse vielbein and V (z) is the confining potential which breaks the conformal invariance and R is the AdS radius.
The Dirac equation in AdS derived from the above action is given by With z identified as the light front transverse impact variable ζ which gives the separation of the quark and gluonic constituents in the hadron, it is possible to extract the light-front wave functions for the hadron.
To map with the light front wave equation, we identify z → ζ, where ζ is the light front transverse variable, (2) and set | µR |= ν + 1/2 where ν is related with the orbital angular momentum by ν = L + 1 . For linear confining potential U(ζ) = (R/ζ)V (ζ) = κ 2 ζ, we get the light front wave equation for the baryon in 2 × 2 spinor representation as In case of mesons, the similar potential κ 4 ζ 2 appears in the Klein-Gordon equation which can be generated by introducing a dilaton background φ = e ±κ 2 z 2 in the AdS space which breaks the conformal invariance. But in case of baryon, the dilaton can be scaled out by a field redefinition [18]. So, the confining potential for baryons cannot be produced by dilaton and is put in by hand in the soft wall model. The form of the confining potential (κ 4 ζ 2 ) is unique for both the meson and baryon sectors [41]. The twist-3 nucleon wave functions in the soft wall model are obtained as

Model-I
The SU (6) The form factors are normalized to F p 1 (0) = 1, F n 1 (0) = 0 and F p/n 2 (0) = κ p/n , the anomalous magnetic moments for the nucleons. The bulk-to-boundary propagator for soft wall model is given by where U(a, b, z) is the Tricomi confluent hypergeometric function. The bulk-to-boundary propagator, Eq. (10), can be written in a simple integral form [18,42] We refer the formulae for the form factors given in Eqs. (7,8 and 9) as Model-I. It has been shown [20,21] that the form factors for the nucleons agree with experimental data for

Model-II
The other model of the form factors was formulated by Abidin and Carlson [12]. Since the action defined in Eq.(1) cannot produce the spin flip (Pauli) form factors, they introduced an additional gauge invariant non-minimal coupling. This additional term also gives an anomalous contribution to the Dirac form factor. In this model the form factors are given by [12] F p where C 1 (Q 2 ) = a + 6 (a + 1)(a + 2)(a + 3) , where a = Q 2 /(4κ 2 ). The value of κ is fixed by simultaneous fit to proton and rho meson mass and the best fit gives the value κ = 0.350GeV . The other parameters are determined from the normalization conditions of the Pauli form factor at Q 2 = 0 and are given by η p = 0.224 and η n = −0.239 [12]. We refer the form factors given by Eqs.
with the normalizations F u 1 (0) = 2, F u 2 (0) = κ u and F d 1 (0) = 1, F d 2 (0) = κ d where the anomalous magnetic moments for the up and down quarks are κ u = 2κ p + κ n = 1.673 and κ d = κ p + 2κ n = −2.033. It was shown in [44] that though the ratio of Pauli and Dirac form factors for the proton F p 2 /F p 1 ∝ 1/Q 2 , the Q 2 dependence is almost constant for the ratio of the quark form factors F 2 /F 1 for both u and d.
The transverse charge density inside the nucleons is given by where b represents the impact parameter and J 0 is the cylindrical Bessel function of order zero. Similar formula for charge density for flavor ρ q f ch (b) can be written with F 1 is replaced by F q 1 . One can define the magnetization density in the similar fashion to have the formula Whereas, has the interpretation of anomalous magnetization density [3]. Since these quantities are not directly measured in experiments, actual experimental data are not available. In [4], an approximate estimation of the proton charge and magnetization densities has been done from experimental form factor data. To get an insight into the contributions of the different quark flavors, we evaluate the charge and anomalous magnetization densities for the up and down quarks.
We can define the decompositions of the transverse charge and magnetization densities for nucleons in the similar way as electromagnetic form factors [44]. 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. We should remember that due to the charge and isospin symmetry the u, d quark densities in the proton are the same as the d, u densities in the neutron as shown in [31] ρ u ch (b) = ρ p ch + ρ n 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 also do the similar decompositions as Eq.(23) and Eq.(24) for ρ m .
In Fig.1(a)   fails to agree as shown in Fig.2(b). Model-I results for the u quark contributions to the charge density for both proton and neutron are in excellent agreement with the two different global parametrizations Kelly [39] and Bradford at el [40]. The d quark contributions deviate form these two fits. It is not very surprising as it has been already shown [21] for Model-I that the Dirac form factor for d quark itself does not agree well with the experiment results. In case of anomalous magnetization both the quarks contributions in proton and neutron agree quite well with the fits. The charge density for neutron ( Fig.2(a)) shows a negatively charged core surrounded by a ring of positive charge density (note that b = 0  For transversely polarized nucleon, the charge density is given by [33] 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. (25), provides the deviation from circular symmetry of the unpolarized charge density [33]. We show the charge densities for the transversely polarized proton and neutron in Fig.4(a) and 4(b). The u and d quark charge densities for the transversely polarized nucleon are shown in Fig.4(c) and 4(d). Again, in Model-I, the densities for proton and u quark are in good agreement with the global parametrizations but deviate for neutron and d quark. Only for d-quark charge density as shown in Fig.4(d), Model-II agrees with the phenomenological parametrizations better than Model-I. The comparison of charge densities for the transversely polarized and unpolarized proton is shown in Fig.5(a) and the similar plot for neutron is shown in Fig.5(b). For the nucleons polarized along the +x direction,, the charge densities are shifted towards negative b y direction. The deviation is much larger for the neutron compared to the proton. The behaviors are in agreement with the results reported in [3,33,36]. We compare the up and down quark charge densities for the trans- versely polarized and unpolarized nucleon in Fig.5(c) and 5(d). The deviation or distortion from the symmetric unpolarized density is more for down quark than the up quark. The shifting of charge density for the nucleons polarized in +x direction, is towards positive b y direction for down quark but opposite for up quark.

IV. SUMMARY
In this paper, we have presented a detailed study and comparison of the charge and anomalous magnetization densities for nucleons in the transverse plane in two models in AdS/QCD. We have also compared our results with the two standard phenomenological parametrizations of the form factors. Both the unpolarized and transversely polarized nucle-ons have been considered in this work. The unpolarized densities are symmetric in transverse plane while for the transversely polarized nucleons they become distorted. If the nucleon is polarized along x direction, the densities get shifted towards negative y-direction. We have also studied the flavor decompositions of the transverse densities i.e., the charge and anomalous magnetization densities for individual u and d quark flavors. Our analysis shows that Model-I reproduces the data much better than the Model-II. The agreement is not so good for d quark which is consistent with the findings in [21], where the form factors for the d -quark were shown to deviate from the experimental results. For transversely polarized nucleon, the distortion in d quark charge density is found to be stronger than that for u quark and shifted in opposite direction to each other.