Scaling of PDFs, TMDs, and GPDs in soft-wall AdS/QCD

We explicitly demonstrate how to correctly define the hadronic parton distributions (PDFs, TMDs, and GPDs) in the soft-wall AdS/QCD approach, based on the use of a quadratic dilaton field, providing confinement and spontaneous breaking of conformal and chiral symmetries. The power behavior of parton distributions at large values of the light-cone variable is consistent with quark counting rules and Drell-Yan-West duality. All parton distributions are defined in terms of profile functions, which depend on the light-cone coordinate and are fixed from PDFs and electromagnetic form factors.


I. INTRODUCTION
During the last decade the soft-wall AdS/QCD [1]- [3] formalism achieved significant progress in the description of hadron structure: mass spectrum, parton distributions, form factors, etc. (for overview, see, e.g., Ref. [4]), based on an effective action constructed with the use of a quadratic dilaton field providing confinement and spontaneous breaking of conformal and chiral symmetry. This dilaton field ϕ(z) has quadratic dependence on the holographic variable z, and is multiplied with the dilaton scale parameter κ (of order of a few hundreds of MeV): ϕ(z) = exp(−κ 2 z 2 ). One of the main advantages of this approach is that it explicitly reproduces the power scaling of hadronic form factors at large Q 2 [4]- [12]. In particular, soft-wall AdS/QCD is consistent with the Drell-Yan-West (DYW) relation [13] between the large-Q 2 behavior of nucleon electromagnetic form factors and the large-x behavior of the structure functions (see also Ref. [14] for the extension to inelastic scattering) and quark counting rules [15]. Based on the findings in Refs. [13][14][15] one can, e.g., relate the behavior of the quark distribution function (PDF) in nucleon q v (x) ∼ (1 − x) p at x → 1 with the scaling of the proton Dirac form factor F p 1 (Q 2 ) ∼ 1/(Q 2 ) (p+1)/2 at large Q 2 , where the parameter p is related to the number of constituents in the proton (or twist τ ) as p = 2τ − 3 [13,16]. At large x and finite Q 2 there are also model-independent predictions of perturbative QCD (pQCD) for the generalized parton distributions (GPDs) [17] pion H π q (x, Q 2 ) and nucleon H N q (x, Q 2 ), E N q (x, Q 2 ): Note that the prediction of pQCD for the pion PDF q π (x) ∼ (1 − x) 2 at large x (it trivially follows from the prediction for GPDs [17]) was supported by the updated analysis [18] of the E615 data [19] on the cross section of the Drell-Yan (DY) process π − N → µ + µ − X, including next-to-leading logarithmic (NLL) threshold resummation effects: q π (x) ∼ (1 − x) 2.03 at the initial scale µ 0 = 0.63 GeV [18]. The importance of the scaling laws and their role in the description of nucleon structure has been stressed and studied in detail in the literature. Moreover, they are important for the proper construction of light-front (LF) QCD approaches [4,5], [20]- [27], motivated by soft-wall AdS/QCD and developed in the past decade. The main advantage of these LF QCD approaches was in the construction of effective wave functions for mesons [4,5,7,8,20,23,[26][27][28], baryons [21,22,[24][25][26][27], and for hadrons with arbitrary number of partons (arbitrary twist) [21,22,24,26,27], which were further used in the calculation of fundamental properties of hadrons -parton distributions and form factors. While form factors and parton distributions in LF QCD were consistent with quark counting rules at large Q 2 and large x → 1 (light-cone variable), there was the problem of achieving full consistency in soft-wall AdS/QCD. As we stressed before, hadronic form factors in soft-wall AdS/QCD obey the power scaling 1/Q 2(τ −1) at large Q 2 and for arbitrary twist τ of a hadron. On the other hand, parton distributions (like PDF and GPDs) calculated in softwall AdS/QCD (see, e.g., Refs. [5,6,9]) have different scaling at large x. In particular, the pion PDF scaled as (1 − x) 0 [5,9], the nucleon charged and magnetization PDFs/GPDs are scaled as (1 − x) and (1 − x) 2 , respectively [6]. Such behavior of PDFs was obtained using the following expression for the hadron form factors: where the integrand contains the square of the holographic wave function in fifth dimension z (dual to hadron wave function), multiplied with the vector field V (Q 2 , z) (dual to the electromagnetic field). The expressions for the φ τ (z) and V (Q 2 , z) are given in analytical form as [4,5,9]: and where a = Q 2 /(4κ 2 ), Γ(n) and U (a, b, z) are the gamma and Tricomi functions, respectively. It is convenient to use the integral representation for V (Q, z) [29] V (Q 2 , z) = κ 2 z 2 In Refs. [5,6,9] the identification of the y variable with the light-cone momentum fraction x after integration over the z variable, leads to an integral representation for the form factor, which can also be written in closed form as the beta function B(α, β) from which one can extracted both PDFs q τ (x) and GPDs H τ (x, Q 2 ) [5,6,22]: Such x dependence of PDF and GPD contradicts model-independent results: the DY inclusive counting rule for q τ (x) at x → 1 [13,16,17] and the prediction of pQCD for GPDs -pion H π q (x, Q 2 ) and nucleon H N q (x, Q 2 ), E N q (x, Q 2 ) at large x and finite Q 2 [17].
It was first noticed in Ref. [21] that the interpretation of the variable y in the integral representation (5) as lightcone variable is not truly correct and that one can think about a generalized light-cone variable y(x) depending on x. Then the power behavior of hadronic PDFs and GPDs ar large x is consistent with model-independent results of Refs. [13,16,17] can be obtained, provided that an appropriate choice of the x dependence of the function y(x) is made. In particular, the simplest choice the function y(x) was found as leading to the correct large-x scaling of PDFs and GPDs in mesons at N = 2τ − 2 and in baryons at N = 2τ − 3. The function y τ (x) obeys the following boundary conditions y τ (0) = 0 and y τ (1) = 1. Notice that a similar idea was recently considered in the framework of light-front holographic QCD (LFHQCD) [26,27] (see also Ref. [30] for an extension of Ref. [26]). In particular, a function [named as w(x)] was introduced in the integral representation of the form factor [26,27]: Obviously, both mathematical extensions considered in Refs. [21] and [26,27] are equivalent. The only difference is that in Refs. [26,27] an extra power −1/2 was included in the [w(x)] Q 2 /4λ−1/2 , while in the soft-wall model [5,6,9] the factor is [w(x)] Q 2 /4λ . In other words, the soft-wall model [5,6,9] and LFHQCD [5,6,9] deal with slightly different analytical expressions for the hadronic form factors: F τ (Q 2 ) ∼ B(τ − 1, 1 + Q 2 /4λ) in soft-wall AdS/QCD and F τ (Q 2 ) ∼ B(τ − 1, 1/2 + Q 2 /4λ) in LFHQCD.
The main objective of this paper is to continue the discussion of ideas started in Ref. [21,26,27] and propose a more simple derivation of PDFs, TMDs, and GPDs of hadrons with arbitrary twist in the context of soft-wall AdS/QCD models. In particular, we explicitly demonstrate how to correctly define the hadronic parton distributions (PDFs, TMDs, and GPDs) in the soft-wall AdS/QCD approach, based on the use of a quadratic dilaton field providing confinement and spontaneous breaking of conformal and chiral symmetry. The obtained power behavior of parton distributions at large values of light-cone variable x are then consistent with quark counting rules and DYW duality. All parton distributions are defined in terms of profile functions depending on the light-cone coordinate and are fixed from PDFs and electromagnetic form factors.
The paper is organized as follows. In Sec. II we consider derivation of PDFs. TMDs will be derived in Sec. III. In Sec. IV we discuss derivation of GPDs. Finally, Sec. V contains our summary and conclusions.

A. General consideration
We start with the derivation of PDFs in soft-wall AdS/QCD. In Ref. [21] and [26,27], this quantity has been derived using an integral representation for the hadronic form factor (see discussion in the Introduction). The easiest way is to start with the hadronic wave function normalization condition, which depends on the holographic variable z: where φ τ (z) is defined in Eq. (3). Next we use the integral representation for unity and insert it into Eq. (12). Here x is the light-cone coordinate and f τ (x) is the profile function with boundary condition f τ (0) = 1, which is specific for a particular hadron and fixed from its PDF. The functions f τ (x) and y τ (x) [see Eq. (8)] are related as: or We remind that at x = 0 the functions y τ (x) and f τ (x) obey the boundary conditions y τ (0) = 0 and f τ (0) = 1. At x = 1 function f τ is finite and its value depends on the specific choice of twist τ (see below), while y τ (1) = 1 is independent on twist. After integration over the variable z we get Here and in the following the superscript ( ′ ) means derivative with respect to variable x. Using a general definition for the hadronic PDF q τ (x), in the form of the integral representation (first moment) over x we get: We require that the hadronic PDF q τ (x) must have the correct scaling at large x and this behavior is governed by the profile function f τ (x).

B. Pion PDF
Now let us consider applications. First we look at the pion PDF at leading twist τ = 2: Following the pQCD prediction presented in Ref. [18], we consider the parametrization for the pion PDF at the initial scale µ 0 = 0.63 GeV as: where N π is the normalization constant, α = 0.70, β = 2.03, γ = 13.8, δ = 2. Notice that in Ref. [23] we derived the LF wave function which produces this PDF. Now we are on the position to fix the profile function f π (x), matching Eqs. (19) and (20). Restricting to leading twist, with good accuracy we can use an approximate value of the parameter β ≃ 2 in Eq. (20). With this and the boundary condition f π (0) = 1 we fix f π (x): It is easy to verify that f π obeys the boundary conditions f π (0) = 1 and f π (1) = 0. At large x it scales as , which leads to the correct scaling of the pion PDF: We can also write down the relation of function f π (x) with y π (x) ≡ y 2 (x): which for large x due to f π (x) ∼ (1 − x) simplifies to In Ref. [11] we proposed a formalism for the inclusion of high-Fock states in soft-wall AdS/QCD. In the case of PDF it is given by the sum: where c τ is the set of mixing coefficients defining the partial contributions to the pion PDF, from specific twists τ = 2, 4, . . ., which obey the normalization condition:

C. Nucleon PDFs
Next we consider the u and d quark PDFs in the nucleon. The nucleon PDFs and GPDs in soft-wall model were calculated for the first time in Ref. [6]. They were extracted from nucleon electromagnetic form factors using an integral presentation for the vector field dual to the electromagnetic current (5). As we stressed in the Introduction, in previous papers using the soft-wall model, the variable of integration in Eq. (5) was identified with the light-cone variable. It led to the results for the PDF and GPDs with much harder scaling at large To solve this problem, one can identify the variable of integration in Eq. (5) with arbitrary function of x, i.e. with y τ (x), and fix y τ (x) to guarantee the consistency of power scaling of PDFs and GPDs with model-independent results known from QCD. One of the solutions for y τ (x) consistent with power counting is [21]: leading to the correct large-x scaling of PDFs and GPDs As we pointed out before, we follow this novel idea in order to introduce the profile function in the normalization condition for the z profiles of the AdS field dual to corresponding hadron wave function. Following the pion example considered above, we derive nucleon PDFs starting from the normalization conditions, and consistent with modelindependent counting rules. In the nucleon case there are two holographic functions dual to its right-(φ R τ ) and left-chirality (φ L τ ) wave functions (see Refs. [31] and [6,9,11]): The normalization conditions for the u and d quark wave functions, which are equivalent to the normalization conditions for their valence PDFs [u v (x) and d v (x)] read: u-quark: d-quark: where are the combinations of right and left holographic wave functions, η u = 2η p + η n and η d = 2η n + η p are the linear combinations of the nucleon couplings with vector field related to nucleon anomalous magnetic moments k N and fixed as [6,31]: , where M N is the nucleon mass. Notice that the contribution of "nonminimal" terms vanish in the normalization condition for wave functions and PDFs due gauge invariance, but they contribute to the x-dependence of PDFs. Moreover, as seen from Eqs. (29) and (30), the "nonminimal contributions" to the quark PDFs are sufficient to violate the symmetry condition For arbitrary twist the expressions for the quark PDFs in the nucleon are given in Appendix A. For leading twist τ = 3, the results for u v (x) and d v (x) read: Both PDFs in Eqs. (32) scale at large x as (1 − x) 3 , as dictated by the counting rules [13,16,17], when the f u (x) and f d (x) go to constants independent on x. In other words, the Taylor expansion for f q (x), q = u, d has the generic form with n c n = 1, due to the boundary condition f q (0) = 1. Here the sum over n starts from n = 0.
World data analysis (see, e.g., Ref. [32]) supports the (1 − x) 3 scaling of u v PDF, while extracted d v PDF has softer behavior (1 − x) 5 . In our approach we can resolve this puzzle. The solution is based on a suppression of (1 − x) 3 term in d v [see Eq. (32)], which can occur when the following constraint on the η d coupling holds: From the latter condition it follows that the dilaton scale parameter κ is related to the nucleon mass as κ = 0.348 M N = 326 MeV, which is very close to the value κ = 350 MeV used in Refs. [6,31]. Adopting the condition (34) and restricting to the leading order in the (1 − x) expansion, we get the following expressions for the quark PDFs in the nucleon: Now we fix the u and d profile functions f u (x) and f d (x), using predictions for the valence PDFs u v (x) and d v (x) extracted from world data analysis. As an example, we use the results of the MSTW 2008 LO global analysis [32]: where µ 0 = 1 GeV is the initial scale. The normalization constants A q and the constants α q , β q , ǫ q , γ q were fixed as Solving the differential equations (35) with the boundary condition f q (0) = 1 and using β u = 2, β d = 5 we get: where Here C k m = m! k!(m−k)! are the binomial coefficients. As in the pion case, we derive the relations between sets of nucleon functions y q (x) and f q (x): For large x → 1 the expressions for f q (x), and therefore the relations (43), are simplified: It is clear that in this limit the quark PDFs in the nucleon obey the correct large x scaling: Now we turn to a discussion of the magnetization PDFs in nucleons E u v (x) and E d v (x). The idea of their derivation is similar to the case of the charged PDFs u v (x) and d v (x). We start with expressions for the contribution to the anomalous magnetic moments k q of u and d quarks in soft-wall AdS/QCD model [6,31], given as integrals over leftand right-chirality nucleon wave functions with specific twist τ (28): Next we use the integral representation for unity (13) and after integration of the holographic variable z, we get the magnetization PDFs in the nucleon for leading twist τ = 3 [expressions for arbitrary twist can be found in Appendix A]: In principle, the f q (x) profile functions can be different in charged and magnetization PDFs. In the case when they are the same we derive the following relation: (48)

III. TMD
TMD can arise in soft-wall AdS/QCD by analogy with PDF, using the generalized integral representation for unity, including integration over the longitudinal x and transverse k ⊥ variables: where D τ (x) is the longitudinal factor derived in Ref. [25], which was fixed from data on the nucleon electromagnetic form factors. The purpose of the function D τ (x) is to include a running scale in TMD, i.e. scale parameter, which accompanies the k ⊥ dependence in TMDs. In our case the running scale parameter is Λ τ (x) = κ/ D τ (x). As was shown in Ref. [25], the appearance of the κ or M N in Λ τ (x) is for convenience, because any different choice can be compensated by rescaling the function D τ (x). Such choice of Λ τ (x) is a generalization of the Gaussian ansatz for TMD with constant scale Λ 2 = k 2 ⊥ in the exponential, proposed by Turin group [34]: This Gaussian ansatz (50) is simple and very useful in practical calculations and analysis of data. However, it is known (see e.g., Ref. [33]), that it presents difficulties in the description of data on DY processes in some kinematical regions (e.g. at Q ⊥ ≤ Q). Therefore, the ansatz for the TMD (50) can be crucially checked. In this vein, one can mention results of AdS/QCD and light-front quark models motivated by AdS/QCD (see Refs. [12,22,25]) where it was shown that the hadronic light-front wave functions, PDFs, and TMDs contain scale parameter depending on the light-cone variable x, i.e. they can be considered as x-dependent scale quantities It was found in Refs. [12,22,25] that x-dependent scale is crucial for a successful description of data on electromagnetic form factors of nucleons and electroexcitation of nucleon resonances. Also we can see below that our result for the unpolarazed quark TMD in nucleon will contain two terms multiplied with a Gaussian: constant term and term proportional to k 2 ⊥ . It is consistent with the form of TMD used by the Pavia group [35]. In the next section we will show that function D τ (x) can be fixed from expression for the electromagnetic form factor and related to functions f τ (x) and y τ (x).
Using the same calculation technique as for the case of PDFs, we insert the integral representation (49) into the normalization condition for the holographic wave function (12) and integrate over the z variable. After that we arrive at the normalization condition for the TMD F τ (x, k ⊥ ), from which the latter can be extracted and expressed through PDF as: Also it is important to stress that from the results for generic PDFs and TMDs derived in present paper one can set up LF quark model in analogy with our previous papers [22,25]. In particular, the LF wave function for generic hadron with twist τ reads: Note that generic TMD and PDF are expressed in term of LF wave function (52) as: Now lets consider as example the result for the unpolarized quark TMD in nucleon f qv 1 (x, k ⊥ ). As in case of PDF it is contributed by two wave functions φ R (z) and φ L (z) (28) corresponding to the leading and subleading twist or having orbital moment L = 0 and L = 1. The φ R (z) function generates the contribution to TMD f qv 1,R (x, k ⊥ ) fixed from condition similar to Eq. (51), while the φ L (z) gives the contribution f qv where Here and δq v (x) are the helicity-independent and helicity-dependent valence quark parton distributions. As we mentioned before, the form of our expression for TMD is very similar to the parametrization used by Pavia group [35]: Using expressions for nucleon PDFs and TMDs one can set up the LF wave functions for the nucleon following Refs. [22,25]: where Here c u = 1, c d = −1, ψ λN λq q (x, k ⊥ ) are the LFWFs at the initial scale µ 0 with specific helicities for the nucleon λ N = ± and for the struck quark λ q = ±, where plus and minus correspond to + 1 2 and − 1 2 , respectively. Note, in terms LF wave functions (58) the unpolarized quark TMD in nucleon reads [36]: Note q ± v (x) and E q v (x) PDFs are related as [25]: The full set of the valence T -even TMDs generated by LF wave functions derived above is listed in Appendix B.

IV. GPD
As we mentioned before, the nucleon GPDs were calculated for the first time in soft-wall AdS/QCD in Ref. [6]. These quantities were expressed in terms of generalized light-cone variable y τ (x), which has direct relation to the profile function f τ (x). Function f τ (x) is more convenient for displaying power behavior of hadronic parton distributions (PDFs, TMDs, and GPDs). In particular, for arbitrary twist τ , a generic GPD in hadron reads [22]: It can be written in more convenient form in terms of PDF: where the PDF q τ (x) and light-cone function y τ (x) are expressed through profile function f τ (x) according to Eqs. (18) and (15). Next we constrain function D τ (x) and relate it to functions y τ (x) and f τ (x) matching the expression for the hadronic form factors in two approaches -soft-wall AdS/QCD and LF QCD. The LF QCD result for the hadron form factor is given by the DYW formula [13] where ψ(x, k ⊥ ) ≡ ψ(x, k ⊥ ; µ 0 ) is wave function derived in Eq. (52), k ′ ⊥ = k ⊥ + (1 − x)q ⊥ , and Q 2 = q 2 ⊥ . We get: For large x function D τ (x) behaves as where f π (x) = 1 − x, f u (x) = f d (x) = 1 and therefore D π (x) = 1 − x, D u (x) = D d (x) = 1. It leads to the following scaling of the TMDs at large x: for pion, for nucleon. Now we consider specific cases for GPDs. In the pion case we have τ = 2 and y π (x) = 1 − f π (x) (1 − x) 2 , where the pion profile function f π (x) is fixed from pion PDF by Eq. (21). The pion PDF q π (x) is fixed from data. Therefore, we give the pion GPD prediction at the initial scale µ 0 = 1 GeV in terms of the pion PDF, or more precisely in terms of constants parametrizing PDF (N π , α, β, γ, δ) fixed in Ref. [18]. At large x the profile functions f π (x) → (1 − x) and y π (x) → 1 [see Eq. (23)], and the scaling of our result for the pion GPD (1 − x) 2 is consistent with the pQCD prediction [17]: it coincides with the leading-order result for the pion PDF and is independent on Q 2 : In the nucleon case we have τ = 3, y u (x) = 1 − f u (x) (1 − x) 2 , and y d ( The quark profile functions f u (x) and f d (x) are fixed from the corresponding nucleon PDFs extracted from global data analysis at the initial scale µ 0 = 1 GeV [32]. The four (charged and magnetization) nucleon GPDs at the initial scale µ 0 = 1 GeV are defined as: (71) Finally we consider the limit of large x. In this case the profile functions f q (x) and functions y q (x) approach 1: The scaling of the nucleon charge and magnetization GPDs are also (as in case of pion) consistent with the pQCD predictions [17]: In the case of the d quark charge GPD H d v (x, Q 2 ) we have two possibilities at large x. In general it scales as (1 − x) 3 in agreement with pQCD [17]. On the other hand, if we suppress the leading-order term (1 − x) 3 in the d quark PDF using the constraint (34), then d v (x) has softer (1 − x) 5 behavior consistent with result of world data analysis [32]. In this vein, we also get (1 − x) 5 scaling of the H d v (x, Q 2 ). Note that the large x scaling of the pion and nucleon GPDs is governed by corresponding PDFs.

V. SUMMARY
In the present paper we have explicitly demonstrated how to correctly define the hadronic parton distributions (PDFs, TMDs, and GPDs) in the soft-wall AdS/QCD approach based on the use of quadratic dilaton. The large x behavior of PDFs and GPDs is consistent with model-independent counting rules. All parton distributions are defined in terms of profile functions f τ (x) depending on the light-cone coordinate. The functions f τ (x) are related to the PDFs and obey the boundary condition f τ (0) = 1. We also proposed a solution to the puzzle related with a softer large x behavior of the valence d quark PDF in nucleon in comparison with the one of the u quark. It can be obtained due to the vanishing of the leading-order term (1 − x) 3 when nonminimal couplings of the nucleons with the electromagnetic field obey the condition (34).
Using our expressions of the LF wave functions we express TMDs through the PDFs Performing the k ⊥ -integration over the TMDs with TMD(x) = d 2 k ⊥ TMD(x, k ⊥ ) , gives the identities The integration over x leads to the normalization conditions where n q is the number of u or d valence quarks in the proton, g q A is the axial charge of a quark with flavor q = u or d, and g q T is the tensor charge. Our TMDs satisfy all relations and inequalities found before in theoretical approaches (see detailed discussion in Ref. [25]).