Double parton correlations in mesons within AdS/QCD soft-wall models: a first comparison with lattice data

Double parton distribution functions (dPDFs), entering the double parton scattering (DPS) cross section, are unknown fundamental quantities encoding new interesting properties of hadrons. Here, the pion dPDFs are investigated within holographic QCD inspired quark models in order to access their basic features. Dynamical information obtained within the AdS/QCD soft-wall approach, also including further improvements, have been compared with predictions of lattice QCD calculations of the pion two-current correlation functions. The present analysis confirms that double parton correlations, affecting dPDFs, are very important and not direct accessible from generalised parton distribution functions and electromagnetic form factors. The comparison between lattice data and quark model calculations unveils the relevance of the contributions of high partonic Fock states in the pion. Nevertheless, by using a complete general procedure, results of lattice QCD have been used, for the first time, to estimate the mean value of the so called $\sigma_{eff}$, an experimental observable, relevant for DPS processes. In addition, the results of a first study of the dPDFs of the $\rho$ meson are discussed in order to make predictions.


Introduction
In the last few years, great attention has been devoted to theoretical and experimental studies of multiple parton interactions (MPI), due to the large demand of detailed description of hadronic final states required at the LHC [1,2]. The inclusion of MPI in experimental analyses is fundamental for the research of New Physics, being MPI a source of background. The simplest case of MPI is the double parton scattering (DPS) [3,4], where two partons of an hadron simultaneously interact with other two partons of the other colliding hadron. As discussed in a recent review [5], the measurements of DPS processes are mandatory to access unknown double parton correlations (DPCs) in the proton. Moreover, the DPS cross section depends on a new quantity called double parton distribution functions (dPFDs) which encode the probability of finding two partons, with given flavors, longitudinal momentum fractions (x 1 , x 2 ) and relative transverse distance d ⊥ [1,6,7]. If measured, dPDFs would therefore represent a novel tool to access the three-dimensional hadron structure. In fact, dPDFs provide new fundamental information, complementary to those obtained by using generalised parton distribution functions (GPDs) [8]. However, for the moment being, no data for the proton dPDFs have been so far collected. Furthermore, dPDFs are non perturbative objects in QCD not directly accessible from the theory. It is therefore useful to estimate them at low momentum scales (∼ Λ QCD ), for example by using quark models [9,10,11,12]. In addition to several general analyses on dPDFs [6,13,14,15,16], a lattice QCD investigation on two-current correlations in the pion has been published very recently [17]. In the present analysis, we take advantage of the lattice data, to test quark model predictions for the pion dPDFs. The calculations of the latter have been shown for the first time in Ref. [18] and then within the Nambu-Jona-Lasinio (NJL) model [19,20]. In particular, following the line of Ref. [18], we consider here the AdS/QCD soft-wall inspired quark models. Let us mention that the mean value of σ ef f , sensitive to dPDFs, has been already calculated within an holographic QCD model for the proton target [21]. The models here used are inspired by the so called AdS/CFT correspondence [22,23], which relates a supersymmetric conformal field theory with a classical graviational one in an antide-Sitter space. In the so called bottom-up approach, one implements fundamental properties of QCD by generating a theory in which conformal symmetry is asymptotically restored [24,25,26,27]. Let us mention that this approach has been successfully applied to access non perturbative features of QCD, for example the description of the spectrum of glueballs, hadrons, form factors (ffs) and different kind of parton distribution functions (PDFs) [28,29,30,31,32,33,34,35,36,37,38,39]. In the present investigation we discuss the calculations of the pion dPDFs and their first moments within the AdS/QCD approach. Comparisons with lattice outcomes will test the predictive power of these models and provide new fundamental constraints for their future improvements. In the last part of the present investigation, predictions for the ρ dPDFs will be shown for the first time. The paper is organised as follows.
In Sect. 2 the formalism to describe dPDFs and related quantities within the Light-Front (LF) approach is shown. In Sect. 3 a brief recapitulation of the main lattice evaluation of moments of dPDFs [17] will be presented. In Sect. 4 details on the adopted AdS/QCD models we adopted will be discussed. In Sect. 5 numerical calculations of dPDFs and related quantities will be shown, also including the comparisons with lattice data. In Sect. 6 the first study of the ρ dPDFs is presented. 2 Meson Double PDF within the Light-Front approach In this formal section, the main strategy to obtain a suitable expression of the mesonic dPDFs, for quark model calculations, will be presented. In particular, the essential steps of this procedure has been previously developed in Refs. [18,40] and it is here summarised. In particular we consider the Light-Front (LF) approach [41,42] together with the LF wave function representation of the hadronic state [43,44]. In this scenario, the meson (M) state |M, P , with momentum P µ , can be decomposed in a coherent sum over partonic Fock states. The relative contribution of a given Fock state to the meson is encoded in the so called LF wave function (w.f.) ψ. The latter contains all non-perturbative information on the meson structure. Of course, the LF w.f. cannot be evaluated from first principles, i.e. the QCD. In this scenario, constituent quark models represent suitable tools to evaluate the w.f. and then to explore basic non perturbative features of different kind of distributions, such as parton distribution functions, form factors and dPDFs. Indeed, all these quantities can be described in terms of the LF wave function. In the present analysis we focus our attention on dPDFs. As already mentioned, these quantities encodes novel information on the hadron structure which cannot be obtained through one-body functions such as generalised parton distributions (GPDs) and transverse momentum dependent PDFs (TMDs).
Since the main purpose of this investigation is the comparison between quark model calculations with those obtained within the lattice framework [17], here we consider the unpolarized dPDFs which depend on the Dirac matrix γ µ . The double PDFs can be formally defined through a light-cone correlator [6]: where, for generic 4-vectors y and z, the operator O q (y, z) for the quark of flavor q reads: and q(z) is the LF quark field operator. In order to find a suitable expression of the dPDF, we consider the Fock decomposition of the mesonic state [43,44] and keep only the |qq contribution [18]. In fact, for the moment being, an explicit expression for the LF wave function of, e.g., the |qqqq , or higher states, is not available. Therefore, the meson state reads: Here, h andh represent the parton helicities, x i = k + i /P + and k i⊥ the quark longitudinal momentum fraction and its transverse momentum, respectively, and P µ is the meson 4-momentum. The light cone components of a generic 4-vector are defined by l ± = l 0 ± l 3 . In Eq.
is the LF meson wave-function, whose normalisation is chosen to be The w.f. ψ π h,h (x 1 , x 2 , k 1⊥ , k 2⊥ ) determines the structure of the state. The direct expression of the dPDF in terms of the above quantity can be obtained by following the procedure developed in Refs. [12,18,40]. In the Appendix A, details on the convention for the quark-antiquark field operator and anticommutation relations, between creation-annihilation operators [28], are shown. Finally, the meson dPDF reads: In the above expression, q 1 andq 2 are the flavors of the constituent quarks. Due to momentum conservation, ) for brevity. Since as already mentioned, the comparison with lattice data is fundamental in the present investigation, we are mainly interested in moments of dPDFs, i.e. the integrals over x 1 and x 2 of Eq. (5). Thus f M 2 (x 1 , k ⊥ ) in Eq. (5), is the quantity that will be calculated within constituent quark models: As shown in Refs. [12,14,15], dPDFs evaluated at k ⊥ = 0 are related to the PDF. Here and in the following, the meson PDFs are specified by the subscript "1" , i.e. f M 1 (x). As one might notice, if only a two-body Fock state is considered in Eq. (3), the dPDFs would be essentially an unintegrated PDF. In the proton case, where the |qqq state is the dominant one, the above feature is not valid. Since in this analysis we make use of different quark models, in order to identify general non perturbative features of mesonic dPDFs, the following ratio is studied [11,12] to emphasise the role of correlations between the x and k ⊥ dependence: in fact, if a factorised ansatz for dPDFs were valid, for example f M 2 (x, k ⊥ ) ∼ f 2,x (x)f 2,k ⊥ (k ⊥ ), then the ratio r k (x, k ⊥ ) would not depend on k ⊥ . For details on the calculations of this quantity in the proton case, see Refs. [11,12,13,45]. Let us remind that this kind of ansatz is often used in experimental analyses.
In closing this section, we note that the dPDFs depend on two momentum scales. However, since for the moment being we are mainly interested in the first moment of dPDFs, we take both scales equal to the hadronic one. For evolution effects in the pion dPDFs see Ref. [18].

Moments of dPDFs
As already mentioned, in the present study we are mainly interested on the first moment of the pion dPDF. Results of the calculations of this quantity will be compared to those obtained within the lattice [17]. Thus, the physical interpretation of the first moment of dPDFs is here discussed. As shown in Refs. [13,46,47], the latter can be interpreted as a double form factor. This quantity, that we usually call effective form factor (eff), can be defined as follows: For unpolarized dPDFs, the eff does not depend on the direction of k ⊥ . The above definition is general and also valid for many-body systems. Moreover, one should notice that the normalisation of the LF wave function relies in the condition: F 2 (0) = 1. Physically, the latter ensures that the Fourier Transform (FT) of the eff can be interpreted as the probability of finding two partons with a given transverse distance d ⊥ . This quantity is indeed the conjugate variable to k ⊥ . Let us stress that a pre-factor in Eq. (8), depending on the kind of hadron, could appear according to the dPDF sum rules [14]. In the meson case, where only a qq state is considered, the eff reads: The above quantity will be calculated in the next sections and compared with that extracted from the lattice QCD [17].

An approximation in terms of one body quantities
In order to phenomenologically estimate the magnitude of the DPS cross section in proton-proton collisions, an approximate relation between GPDs and dPDFs is often assumed in experimental analyses [48,49]. In fact, by introducing a complete set of states in the correlator (1) and keeping only the mesonic contribution, one gets By using then the strategy already discussed in the previous section, one finds: where H q (x, k ⊥ ) = H q (x, ξ = 0, k ⊥ ), is the meson GPD at zero skewness (see Refs. [50,51] for useful reports on GPDs). Let us mention that the above expression has been tested, in the proton case, by using a LF quark model [52]. The integral over x 2 of Eqs. (5) and (11) leads to where F M (k ⊥ ) is the standard e.m. form factor of the meson M. We denote the meson dPDF, evaluated within the above ansatz, as f M 2,A . The difference between the full calculation of the dPDF and its approximation can be interpreted as the sign of the presence of correlations not encoded in one-body quantities, such as GPDs and ffs. A dedicated numerical section about the impact of correlations in dPDFs will follow. Let us mention that an approximated expression of the moments of dPDFs can be also obtained. In this case, the integration over x of the expression (12) leads to: A similar approximation has been tested in the lattice investigation of Ref. [17]. In Eq. (13), the relation between the GPDs and ffs has been used [50]: The above quantity can be described in terms of the LF wave function. For a meson described by the first Fock state, one gets the following expression [28,53,54]: where k ⊥ = |k ⊥ |. The approximation Eq. (13) will be numerically tested by means of holographic quark models.

The effective cross section
In this section, a relevant observable for DPS studies, called effective cross section, σ ef f [7], is introduced. This quantity is defined as the ratio of the product of two single parton scattering process cross sections to the DPS one with the same final states. Usually σ ef f is extracted from data by using model assumptions, such as the factorisation of dPDFs in terms of PDFs. Experimental analyses, for proton-proton collisions, have been already compared with quark model calculations of σ ef f [13,21,46,55,56]. Let us mention that in Ref. [21] an AdS/QCD soft-wall model for the proton has been used to calculate this quantity. The common feature, pointed out in the analyses of Refs. [13,21,46,55], is the dependence of σ ef f on the longitudinal momentum fractions carried by the acting partons. This behaviour is interpreted as the effects of non trivial double parton correlations. Although no experimental analyses for the extraction of σ ef f for meson-meson collisions are available, in the present investigation the above quantity will be evaluated to make predictions for DPS processes involving mesons. Let us mention that for the pion case, the estimate of σ ef f , shown in Ref. [18], has been used in the experimental investigation of Ref. [57]. The general definition of this quantity is [58]: m is a process-dependent combinatorial factor: m = 1 if A and B are identical and m = 2 if they are different. σ pp ′ A(B) is the differential cross section for the inclusive process pp ′ → A(B) + X. As a first approximation for experimental analyses, σ ef f is considered rather independent from the flavors of the partons, the final states of the processes and the experimental kinematic conditions. However, recent studies on quarkonia production suggest that this ansatz might be violated [59]. Due to the lack of experimental data for meson-meson DPS processes, in the present study we calculate the mean value of σ ef f in order to discuss its geometrical interpretation [47]: Let us mention that if the constant conditions of σ ef f would be valid then σ ef f = σ ef f . The above expression shows how σ ef f encodes non perturbative insight on the hadronic structure, such as the geometrical information on the system.

On the geometric interpretation of σ ef f
As already pointed out in the previous section, due to the lack of experimental information on double parton scattering processes, in particular for meson targets, calculations on σ ef f could be relevant to make predictions for experimental analyses, such as the one of Ref. [57]. In this scenario, the interpretation of σ ef f , in terms of geometrical properties of the incoming hadron, is fundamental. To this aim, in this section, we explore an intuitive relation between σ ef f and the mean partonic distance between two partons acting in a DPS process. This study has been discussed in detail in Refs. [13,47]. The procedure is somehow similar to that applied in the case of elastic processes, where the e.m. form factor can be extracted from the relative cross section and then the charge/magnetic ratio can be obtained. However, since σ ef f depends on the integral over k ⊥ of the product of two effs, see Eq. (17), a direct extraction of the eff is precluded. Nevertheless, basic probabilistic properties of the FT of this quantity allow to relate σ ef f to the main partonic transverse distance between two partons d 2 ⊥ . The effective form factor [46], for a generic system, can be indeed defined as follows: being ρ(d ⊥ ) the two-body density of the system for two particles whose distance in the transverse plane is d ⊥ .
Thanks to this relation, one finds: Due to this connection between the effective form factor and the mean distance of two partons, one can relate σ ef f (17) to the above quantity. Here and in the following we refer to σ ef f as the geometrical effective cross section. The latter is indeed a process independent constant depending only on the functional behaviour of the eff. In Ref. [47], the relation between the numerical value of σ ef f and the partonic distance has been properly understood. Here the main outcome of Ref. [47] is shown. By considering the definition ofσ ef f (17) and the probabilistic interpretation of the FT of the eff, one can show that the main partonic distance (19) lies in a range depending onσ ef f as follows:σ Such a result is extremely useful to get some information on the geometrical structure of an hadron once some data on σ ef f are collected. Since in the present analysis the mean partonic distance will be calculated within quark models and compared to that obtained from the lattice QCD, the above inequality (RC) will be tested. Let us remind that in the proton case the RC inequality has been verified by using all quark models and ansatz of dPDFs at our disposal [13,47]. Furthermore, in the pion case, the above relation has been also validated by the NJL model [20].

Lattice analysis of moments of dPDFs
In this section, we briefly recall the main formalism introduced in Ref. [17]. Here, the expectation for the two-current distribution, a quantity related the first moment of the pion dPDF, has been evaluated within the lattice framework. In momentum space, this quantity reads: The main differences, with respect to the light-cone derivation of dPDFs, are: i) the gamma matrix considered in Eq. (21) is γ 0 , instead of γ + in Eq. (2); ii) the distance between the quark field operators y is chosen along the condition y 0 = 0, instead of y + = 0, see Eq. (1). However, as discussed in the Appendix A, kinematic corrections, due to the choices of the gamma matrix and the separation condition, can be neglected in the infinite momentum frame (IMF), i.e. the natural reference system where a partonic description of hadrons can be provided. Thus numerical comparisons, between lattice and quark models calculations, are allowed in this frame. Indeed, one of the main consequences of the conditions i and ii is the frame dependence of numerical evaluations of Eq. (21) within the lattice approach [17]. In other words, moments of dPDFs depend upon the the pion momentum p. This feature will be explicitly relevant in the analysis of the approximation (13). In the next section, the comparisons of the double parton correlations effects, highlighted in the analysis of Ref. [17], with those addressed in constituent quark model calculations, will be presented. To this aim the lattice data, we are interested for, are here shown. For simplicity, all distributions will be evaluated in momentum space.

The pion form factor
The standard electromagnetic (e.m.) ff has been directly fitted from lattice results, in order to test the approximation (13). The expression reads: where the parameters leading to a good fit with lattice data are: M = 0.872 ± 0.016 GeV and n = 1.173 ± 0.069 (configuration A) or M = 0.777 ± 0.012 GeV and n = 1 (configuration B) [17]. As one can observe in the left panel of Fig. 1, differences between the two configurations are minimal. In the present investigation, we consider the ff evaluated in the A configuration as a benchmark for further comparisons.

Effective form factor
As already pointed out, the first moment of a dPDF is the eff (8). However, within the lattice framework, one finds that M (0) = −2m π , thus, following the procedure of Ref. [17], the eff is properly defined as follows here the parameter n is the same of that of F L of Eq. (22). Let us stress that the FT of the above expression has the probabilistic interpretation shown in Eq. (18). In fact, within the above functional form, the mean distance between the two partons is: d 2 = 1.046 fm [17]. As deeply discussed in Ref. [17], the quantity M (q 2 ), has been numerically evaluated in the pion rest frame, i.e. p = 0. However, as previously mentioned, a comparison between lattice data and quark model calculations of dPDFs is possible in the IMF, (see discussion in Ref. [17]). Thus in order to proceed with the present study, it is necessary to realise that the IMF can be approximately mimicked whenever q 2 << m 2 π . We recall that in the lattice QCD analysis [17], the pion mass is fixed to be m π = 0.3 GeV. The above condition will be numerically tested in the next section.

3.3
An approximation for the moment of dPDFs in lattice QCD As mentioned in Sect. 2.2, a direct measure of the impact of unknown DPCs in the effs is the discrepancy between the latter and their approximations in terms of e.m. form factors, see Eq. (13). To this aim, the procedure discussed in Sect. 2.2 has been considered also in the lattice analysis [17]. However, since in this framework frame dependent effects appear, the following result is obtained: As already explained [17], the above result comes from the procedure discussed in Sect. 2.2 but using the lattice conditions described in the first part of Sect. 3. In particular, the above expression has been obtained in the pion rest frame, i.e. p = 0 [17]. As one can see, the approximation (24) is different from that derived within the light-cone formalism, see Eq. (13). However, it is remarkable that in the IMF the standard expression Eq. (13) is recovered from Eq. (24). In fact, by replacing the pion energy at rest with that of a moving target with an extremely large momentum p: m π → E p = m 2 π + p 2 , one gets: for q 2 << p 2 . This is exactly the result found by following the standard strategy discussed in Sect. 2.2, see Eq. (13). Let us remark that such a conclusion can be reached also by imposing q 2 << m 2 π (see discussion on Ref. [17]). A direct consequence of this approximation is the relation between the mean partonic distance and the mean pion radius. In fact by using that: and by considering the relations Eqs. (24)(25) between the eff and the e.m. ff, one finds that d 2 ∼ 2 r 2 . Numerical values, obtained within the lattice techniques [17], for d 2 and r 2 , immediately shows that d 2 = 1.046 = 2 r 2 = 0.85 fm. As one can observe, correlations effects prevent a simple relation between these two quantities. Let us stress that since both d 2 and r 2 depends on the small Q 2 behaviour of effs and ffs, they are rather independent on the chosen frame. Such a feature has been confirmed by numerical calculations of each sides of Eq. (26), see Ref. [17]. In addition, details on the impact of DPCs can be obtained by comparing both sides of Eqs. (24)(25) as a function of Q 2 . As one can see in the right panel of Fig. 1, the presence of correlations prevents a simple description of the eff in terms of standard ff. See the difference between the full line (left hand side of Eq. (24)) and the dashed line (right hand side of Eq. (24)). However, in the lattice framework, DPCs mix with frame dependent effects, thus, in the right panel of Fig. 1 we also plot the right hand side of Eq. (25) i.e. the approximation in the infinite momentum frames (dotted lines). The comparison between dotted and dashed lines, provides a numerical estimate of the region where frame dependent effects are minimal. One can observe that calculations obtained in the pion rest frame are close to those obtained in the IMF up to q 2 < m 2 π GeV 2 . Numerically, from this check one can deduce that a comparisons, between lattice data and predictions of holographic QCD models, are allowed for q 2 < 0.07 GeV 2 . Before closing this section, the explicit expression of the pion eff (23) has been used to evaluate the geometrical effective cross section:σ ef f = 26.3 mb. Since this result correspond to the case of pions in their rest frame, this numerical result is rather useless for experimental analyses. On the contrary, d 2 is almost frame independent. In fact, this quantity depends on the behaviour of the eff at k ⊥ ∼ 0 (see discussion in Ref. [17]). Therefore, by inverting the RC inequality (20), one can estimate a range of possible σ ef f once the value of d 2 is established: From the above expression, an allowed range of σ ef f , valid also in the IMF, can be estimated. Starting from the lattice data d 2 = 1.046 fm 2 , one gets: From this general results of the lattice QCD, we conclude that the mean value of σ ef f for a pion-pion collision is almost bigger then that extracted in proton-proton collisions.

The pion dPDF within the holographic QCD
In this section, details on the constituent quark models adopted to investigate basic feature of pion dPDF will be discussed. In particular, we are interested in the mesonic wave function calculated by using different Light-Front holographic QCD models. The first w.f. described in this section has been introduced in Ref. [28].
The pion dPDF has been evaluated for the first time within this model [18]. However, since the aim of this analysis is to provide a first comparison with lattice data, the pion wave function has been also evaluated by improving that of Ref. [28]. To this aim, we also considered the model where dynamical spin effects have been taken into account [38]. In addition, in order to include other fundamental phenomenological effects, such as the Regge trajectory of the x-dependence of PDFs, the model of Ref. [30] has been also adopted.

Pion in AdS/QCD I: The original version
In this section, we discuss the calculation of the pion dPDF evaluated within the model described in Refs. [28,29]. Since the w.f. obtained in this scenario is the starting point for any further implementations, here and in the following, we refer to it as the "original" model. Indeed, it can reproduce basic properties of the meson spectroscopy and structure functions. In momentum space representation, the pion wave function reads [28]: being κ o = 0.548 GeV fixed to reproduce the Regge behaviour of the mass spectrum of mesons. Moreover, in order to include a dependence on the quark masses, the wave function has been written in terms of the invariant mass [29]: where m = m 1 ∼ m 2 , x = x 1 , x 2 = 1 − x 1 and k 2⊥ = −k 1⊥ . In this scenario the pion wave function now reads: The mass parameter is usually chosen to be m o ∼ 0.33 GeV [60]. The constant A o is fixed by the following normalisation condition: Within this approach the dPDF expression for the pion can be analytically found [18]:

Pion in AdS/QCD II: Dynamical spin effects in holographic QCD
In Ref. [38], dynamical spin effects have been included into the holographic pion wave function in order to predict the mean charge radius of the pion and its ff without including high Fock states in the meson expansion (3). To account these contributions, let us promote the function appearing in Eq. (31) as an helicity dependent quantity, i.e.
Without going into details, let us discuss only the main outcomes of Ref. [38]. The spin operator reads: where k ⊥ = k ⊥ e iθ k ⊥ . The original model, described in the previous section is restored for B = 0 and A = 1/m 2 π , i.e.: Where the last condition ensures the normalisation of the pion wave function. Let us call the dPDF evaluated within the present model, f πAB 2 , where A and B can take different values. By following Ref. [38], we consider two configurations, i.e. A = B = 1 and A = 0, B = 1. In particular, the w.f. entering Eq. (34) is the same of that obtained within the original model discussed in the above section, see Eq. (31). However, in order to recover phenomenological predictions, the parameter entering the w.f. Eq. (31) is κ 0 = 0.523 GeV [38]. Also in this case, analytic expressions for dPDFs can be found: for the case where A = 0 and B = 1. Here and in the following we refer to this model as the "dynamical spin model".

Pion in AdS/QCD III: A universal wave function
In this last part of this section, a new and promising pion wave function, obtained from the holographic correspondence, will be presented [30]. In this case, the basic idea is to consider the most general analytic structure of GPDs, obtained within holographic QCD, and then incorporate the Regge trajectories for small x in PDFs. In this procedure, the mathematical structure preserves the poles of the ff in the physical region.
Here and in the following we indicate this model as the "Universal model" (UM). Let us here just remind the main outcomes of the analysis of Ref. [30]. Within this model, the effects of two Fock states in the hadron expansion (3) are considered: the valence configuration |qq and the |qqqq contribution. These two different states are addressed with the index τ = 2 and τ = 4, respectively. A remarkable result shown in [30] is that nucleon and pion PDFs, GPDs and ffs can be described within the same model. Of course, free parameters are chosen to describe ffs, PDFs and hadron spectroscopy at the same time. The w.f., related to a given τ state can be effectively expressed as follows: where here q τ (x) is the τ contribution to the pion PDF. The analytical structure of this quantity is fixed by the holographic QCD approach: where: Thanks to this choice the Regge trajectory is correctly reproduced. Moreover, the parameters a and λ are phenomenologically fixed, by fitting the mesonic mass spectrum and the e.m. form factor. Results are found for a = 0.531 and λ = 0.548 GeV. In Ref. [30], the authors fixed the weight of these contributions by using the pion moment of PDFs: where γ = 0.125. Let us point out that the wave function of the τ = 4 state is computed only to calculate PDFs. Thus, the dependence of the latter upon the other two particle momenta is integrated out. Thanks to all these ingredients, the pion dPDFs f πU 2 (x, k ⊥ ), can be evaluated. In the next section numerical results will be discussed.

Numerical Results
In this section, numerical results of the calculations of dPDFs within holographic models will be presented. In particular, we will mainly focus on quantities which allows to qualitatively estimate the impact of non perturbative double parton correlations, not directly accessible via one-body distributions.

Calculation of dPDFs
In the left panel of Fig. 2, Fig. 3 and the left panel of Fig. 4, we show the calculations of the pion dPDFs (6). As one can see, we have considered here the x dependence of dPDFs by fixing different values of k ⊥ . In the cases of the original and dynamical spin models, the shape of these quantities are symmetric, reflecting the symmetry between x and 1 − x, see left panel of Fig. 2 and Fig. 3. The different behaviour, observed in the case of the universal model, is related to the implementation of the Regge trajectory at small x. A common feature shared by all these models is the decreasing shape w.r.t. the increasing of k ⊥ . Such a result is directly related to the behaviour of the relative eff. Details on the evaluation of the latter quantity are presented later on this section. As shown for the proton case [12,52], the impact of DPCs effects is enhanced for dPDFs depending on x 1 − x 2 with x 1 and x 2 which are almost independent and bound by x 1 + x 2 ≤ 1. However, for a meson, where only the two body Fock state contribution is considered in Eq. (3), the dPDF only depends on x = x 1 and x 2 = 1 − x 1 due to momentum conservation [18]. For the moment being, an expression for the LF wave function corresponding to, e.g. a |qqqq , is not available. In fact, let us remind that, in the UM, such a contribution is included only to describe PDFs, thus a possible non trivial dependence of the dPDFs on x 1 , x 2 and x 3 can not be obtained. In this scenario, the most relevant sign of DPCs is given by studying the x − k ⊥ dependence of dPDFs. In particular, an unfactorized dependence of dPDFs, w.r.t. the x and k ⊥ , represents a possible signals of double parton correlations. We recall here that in order estimate the impact of these effects, the ratio f π 2 (x, k ⊥ )/f π 2 (x = 0.4, k ⊥ ) is evaluated as a function of x for different values of k ⊥ . One should notice that if correlations were neglected, then the latter quantity would be constant w.r.t. variations of k ⊥ . As one can see in the right panel of Fig. 2 and in Fig. 5, correlations are very strong for the original and dynamical spin models. However, as one can observe in the right panel of Fig. 4, the impact of DPCs, encoded in the UM, is less relevant w.r.t. the other models. This feature is related to the poor general knowledge of these effects. In any case, for all models here considered, the factorisation in the k ⊥ and x dependence is not fully supported.

The pion form factor
Since the main purpose of the present study is to compare lattice data with holographic quark model calculations, here we show results for the pion e.m. form factor. This quantity has been extensively investigated from a theoretical and experimental point of view [17,28,30,38,61]. To this aim, we consider the pion ff evaluated within the lattice techniques in the A configuration. As one can see in the left panel of Fig. 6, the AdS/QCD models are able to reproduce the essential behaviour of the pion ff. In particular, the original model [28] fits the ff in the small Q 2 region, while the dynamical spin and universal ones [30,38] provide an impressive agreement.   However, by comparing the values of the mean pion radius, one can conclude that the model which includes dynamical spin effects reproduce very well experimental data [62], see Table 1.

The effective form factor of the pion
Here we show the first comparison between the calculations of the eff within AdS/QCD inspired models and that from lattice QCD, see Eq. (23). Let us first discuss some differences between the pion ff and eff. As discussed in Refs. [13,46], in the proton case, the two objects are completely different. In particular the eff involves two particle correlations and depends on k ⊥ , i.e. the momentum unbalance between the first and the second parton in the initial and final states. In the e.m. form factor, q ⊥ represents the exchanged momentum between the initial and final state of one of the partons. However, in the mesonic case, if one considers only the |qq contributions, the formal expression of the eff (9) and the e.m. (15) one are extremely similar (see Ref. [18] for details on this topic). In addition, k ⊥ represents the conjugate variable to d ⊥ , i.e. the transverse distance between the two partons, while q ⊥ is the conjugate variable to r ⊥ , i.e. the transverse distance of a parton w.r.t. the centre of the hadron. In the right panel of Fig. 6, results of the calculations of F 2π (Q 2 ) 2 has been shown. We remind that this combination of two effs enters the expression of σ ef f , see Eq. (17), and encodes the hadron geometrical properties which affect this experimental observable. Thus, in the right panel of Fig. 6, we highlighted the main discrepancies, between lattice and model calculations, which also affect the mean value of σ ef f . As one can see, only the original model is able to reproduce the eff in the allowed kinematic region. Let us stress again that the comparison is well motivated only for Q 2 < m 2 π . In the forward region, frame dependent effects are important but not included in the LF formalism. It is fundamental to point out that, while in the lattice framework there are no truncation of the meson Fock state, in all AdS/QCD models, but the UM one, only the first |qq contribution is included. Thus the dPDF is restricted to be considered as an unintegrated PDFs where the momentum conservation unambiguously fixes the relation between x 1 and x 2 = 1 − x 1 . In this scenario, lattice data of the first moment of dPDFs of the pion represent a reach starting point to understand in details the contribution of high Fock states in the meson expansion Eq. (3). Thanks to this analysis, further implementations of holographic models could include two-body effects based on the lattice data [17]. As one can see in Table 2, the lattice calculation of d 2 is comparable to that obtained within the original model. On the contrary, the UM largely underestimates the mean partonic distance.

Calculation of σ ef f
Here we discuss a possible predictions for an ideal DPS process involving two pions. Due to the lack of data and experimental analyses, we only study the mean value σ ef f (17). In this scenario, only geometrical properties, affecting σ ef f , have been taken into account. The evaluation of this quantity within the lattice framework would be extremely valuable in order to guide future experimental analyses. However, as extensively discussed in the previous sections, lattice data have been obtained in the pion rest frame. Due to this restrictive condition, we have evaluated the mean value of σ ef f by changing the higher extreme value of the integral Eq. (17). In fact, for k ⊥ < m π , frame dependent effects are small. For the purpose of the present investigation, in Table 3, we have reported the results of the calculations of σ ef f . As one can observe in the first row, up to m π , the original model predicts a σ ef f very close to that obtained from the lattice. Let us remind that the full value of σ ef f , evaluated within this model, has been used in the experimental analysis of Ref. [57]. This result is completely coherent with the comparison between the eff evaluated within the lattice QCD and the original model. From just a mathematical point of view, we also displayed the calculation of σ ef f in the full range of k ⊥ . As one can observe, in this case the universal model fits very well the lattice data. However, let us stress again that in this case, frame dependent effects, preventing a clear comparison between lattice outcomes and holographic calculations, cannot be neglected. The full evaluation of σ ef f is anyhow relevant to verify the validity of the RC inequality (20). As one can observe in Table 4, the latter perfectly works for all models and lattice calculations. Let us stress again that the relation between the mean value of σ ef f and the mean distance between two partons has been obtained in a complete general manner in Ref. [47]. Therefore, the validation of the RC inequality, in model independent frameworks, such as the lattice QCD, is extremely precious.

Comparison between two-body distributions and the product of one-body functions
In this last part of this section, devoted to the study of the pion dPDFs, we discuss the validity of Eqs. (12) and (13). Let us start with the comparison between f π 2 (x, k ⊥ ) and its approximation f π 2,A (x, k ⊥ ), i.e. the product of the pion GPD and its form factor, see Eq. (12). Since the dPDFs of nucleons and mesons are basically unknown, as discussed in the previous section, in order to estimate the magnitude of DPS cross section, the approximation (10) is often used. In this framework, model calculations allow to verify if this simplification holds. Here we consider the same strategy developed in Ref. [18], i.e. we directly compare f π 2 (x, k ⊥ ) and f π 2,A (x, k ⊥ ) by remarking their difference, see Figs. 7 and 8, where distributions have been evaluated for three different values of k ⊥ as functions of x. As one can see, in all model calculations, but the UM case, the shape of dPDFs is symmetric , at variance of the product of ffs and GPDs. Such a feature can be explained by considering that GPDs and form factors depend on the transverse momentum: k 1,⊥ ± (1 − x)k ⊥ , see Eq. (15). Such a dependence, produces an asymmetry in the x distribution, not present in dPDF, see Eq. (6). Moreover, since in the GPDs the momentum unbalance in the wave function is multiplied by the pre-factor 1 − x < 1, the GPD goes to zero slower then the dPDF [18]. The last feature is partially discussed in Ref. [47] for the proton case. Furthermore, one should notice, that for the original and dynamical spin models the approximation Eq. (12) underestimates the full calculation of the dPDF, at the variance of the universal case. In order to compare the impact of DPCs described within the lattice QCD and holographic models, the following quantity will be also evaluated: In order to minimise frame dependent effects, we will focus on the region where Q 2 < m 2 π . As one can see, if the approximation Eq. (13) holds in some kinematic region, then the above quantity would be small. In Fig. 9 we report the calculations of the above quantity. As one can observe, in the allowed region of Q 2 , the original and dynamical spin models can almost reproduce the behaviour of DPC effects which prevent a description of the eff in terms of the e.m. form factor. In any case, one should notice that there is no model able to reproduce both the effective and the e.m. form factors at the same time with the same precision. In fact, both the dynamical spin and universal models, fit very well data on the e.m. form factor but fail in the description of the eff. On the other hand side, the original model can qualitatively reproduce both the effective and e.m. form factors only in the small Q 2 region. The main outcome of this analysis is the evident need of the inclusion of more Fock states in the pion expansion (3) in order to include all possible DPCs and to describe both the e.m. and effective form factors. In the peculiar case of the universal model, which effectively takes into the |qqqq state, the full dependence of this contribution upon the parton momenta should be investigated.  Figure 9: The quantity ∆(Q 2 ) (45). Full line stands for lattice data [17], dotted line for the calculation performed within the universal model [30], dashed line represent the result of the evaluation within the original model [28] and dot-dashed for the dynamical spin model [38] in the A = B = 1 configuration.

The ρ meson within AdS/QCD
In this section, we introduce and calculate the dPDFs of the ρ meson. This is the first analysis of DPS involving a vector meson. However, in the future, the kind of information for this target could be accessed via lattice techniques. Here we consider possible predictions provided by AdS/QCD based models. In order to evaluate the ρ w.f., the procedure developed in Refs. [31,63,64] has been adopted. In particular, for the three polarisation of the ρ meson, the wave function is built from that of the pion. In this case, the input will be the w.f. Eq. (31). Of course, the parameters entering this quantity, i.e. κ o and m 0 , in principle could be different in the ρ case. In addition, the normalisation constant will depend on the ρ polarisation. In Ref. [64] the parameters have been chosen to describe several observable for both the ρ and the φ mesons. In particular κ o = 0.54 GeV and m o = 0.14 GeV (configuration A). In the present work we propose to use instead the parameters already adopted to calculate the pion w.f. (31), i.e. κ 0 = 0.548 and m o = 0.33 GeV (configuration B). This choice leads to a good comparison with the moments of ρ PDFs evaluated within the lattice framework. Details on this topic will be discussed in the next section. The ρ w.f., built from that of the pion, reads as follows: Where here and in the following, we denote with Ψ λ h,h (x, b ⊥ ) the ρ meson wave function in coordinates space for λ = L, T polarisation and quark-antiquark helicities h andh, respectively. Moreover, ζ is the usual variable introduced in the AdS/QCD framework, i.e.
The gaussian like function, appearing in Eqs. (46,47), describes the scalar part of the meson w.f. in AdS/QCD (31): In Eq. (47), b ⊥ e iθ is the complex form of the vector b ⊥ . The normalisation condition of the ρ w.f. is the following: The normalisation constant, N λ appearing in Eq. (48), depends on the polarisation [31]. From the above expressions, the dPDF, for a longitudinal polarised ρ meson, can be obtained as follows: In the transverse polarisation case, since dPDFs are diagonal distributions in coordinate space, the main quantity we need to evaluate is the following one: The direct expression for dPDF for transverse polarisation reads as follows:

Numerical results
Here we show the numerical predictions for the dPDFs and effs of the ρ mesons. Since this is the first analysis about this topic, we present the evaluation of the ρ PDFs to motivate the choice of the free parameters appearing in Eq. (48). To this aim, we compare the calculations of these quantities obtained within the AdS/QCD approach, with those of the lattice QCD [65]. Furthermore, also for this hadron, the role of DPCs in dPDFs and effs will be investigated. In addition, we have also calculated the mean value of σ ef f in order to provide a first prediction for this experimental quantity. Thus, for the moment being, only the geometrical contributions have been taken into account in the meson σ ef f .

The parameters entering the ρ wave functions
In Refs. [31,63,64] the parameters appearing in Eq. (48) have been properly chosen to reproduce the diffractive cross section for the ρ and φ meson productions. Within the A configuration, also the decay constants are well reproduced. In the present analysis, we propose to use the parameters of the B configuration, in order to have a good agreement with lattice data of the ρ PDF moments. To this aim, we show the different predictions for the decay constant, obtained within the configurations A and B. We consider the following expressions: where Ψ λ (x, b) = h,h Ψ λ h,h (x, b), b = |b ⊥ | and µ = 1 GeV. As one can observe in Table 5, the results of the calculations of f ρ and f ⊥ ρ , within the A and B configurations, are very similar and comparable to the analyses of Refs. [66,67].

Comparison with lattice QCD
Here we discuss the comparison between the moments of the ρ PDFs, evaluated within the holographic model [31,63,64], with those obtained within the lattice QCD [65]. Let us remind that for the moment being their are no analyses of dPDF moments for the ρ meson. Thus, in order to investigate to what extent the adopted model could be compared to lattice predictions, here we only consider moments of the following structure function: where here q is the quark flavor with charge Q q , f ρT 1,qq (x) is the ρ PDF with transverse polarisation and f ρL↑ 1,qq (x) is the PDF of the ρ meson longitudinally polarised and evaluated for a quark with positive helicity. Since, in the holographic model, the latter quantity does not depends on the spin orientation, nor the flavor of the quarks, the above structure functions can be rewritten in terms of PDFs for unpolarised quarks: In Ref. [65], the following quantity has been calculated: As one can observe in Fig. 10, the model calculations, in the B configuration of the parameters of Eq. (48), are close to that of the lattice predictions. In further analyses, implementations of the ρ w.f. to improve the comparison with the lattice outcomes will be available. For example, by using the pion wave function evaluated within the models of Refs. [30,38] as input of the procedure.

Calculations of dPDFs, effs and σ ef f
In the present section, we show and discuss the results of numerical evaluations of ρ dPDFs, effs and σ ef f in the B configurations of the parameters entering Eq. (48). We separately calculated the above quantities for the longitudinal and transverse polarisations. As for the pion case, in Fig. 11, we display the ρ dPDFs for the two possible polarizations: left panel for the longitudinal and right panel for the transverse, respectively. As one might notice, the k ⊥ behaviour of the distributions is similar to that obtained for the pion evaluated within the original models, see left panel of Fig. 2. Such a result is coherent with the choice of the scalar w.f. entering Eqs. (46) and (47). Moreover, in the transverse polarisation case, the distribution has two pronounced peaks. In addition, as one can see in Fig. 12, also in this case a possible factorisation between the x and k ⊥ dependence is violated, thus reflecting the presence of correlations. The amount of these effects is slightly different from those addressed in the right panel of Fig. 2 and studied within the original model. This feature is related to the presence of derivatives w.r.t. b ⊥ in Eqs. (46) and (47). Thus, the overall dependence on k ⊥ and b ⊥ of ρ dPDFs is somehow different from that of the pion. The main interpretation of the present outcome is that  the procedure, used to generate the ρ w.f., introduces additional correlations. Moreover, the mean value of the effective cross section reads: σ ef f = 32.2 mb for the longitudinal case and σ ef f = 54.7 mb for the transverse one. Let us remark that the model used as input of the calculation provides a good description of the pion eff, in comparison with the lattice. Due to this feature, it is reasonable to conclude that the DPS cross section is dominated by the longitudinal component of the ρ meson. In fact, we recall that the most σ ef f is small the most the DPS contribution is big with respect to the SPS case, see Eq. (16). Therefore, the predictions of this holographic model is that a DPS process involving ρ mesons could be dominated by the longitudinal polarization. In Fig. 13 the eff of the ρ meson, obtained by disentangling the two polarisation contributions, is shown. Full line represents the transverse polarisation and dotted line stands for the longitudinal one. Also for the ρ meson, the mean partonic distance has been calculated by following Eq. (19): d 2 = 0.895 fm and d 2 = 1.159 fm, for the longitudinal and transversal polarizations, respectively. One should notice that in the first case the mean distance is lower then that evaluated for the pion target within the same original model [28], see Table 1. On the contrary, for the transversely polarised ρ, the latter quantity is bigger w.r.t. the pion case. Since, the adopted model reproduces quite well the mean value of d 2 , computed within the lattice QCD, a possible interesting conclusion of the present analysis is that valence quarks in the ρ meson are closer to each other then in the pion case if the the ρ is longitudinally polarised. This feature is intimately related to the nature of the interaction, parameterised by the AdS/QCD potential. In addition, we also show that the RC inequality (20) perfectly works also for the ρ meson described within the holographic approach, see Table 6.

Conclusions
Double parton distribution functions are new fundamental quantities encoding information on the three dimensional partonic structure of hadrons. Double PDFs enter the double parton scattering cross section for which theoretical and experimental analyses are ongoing. However, for the moment being, only proton-proton and proton-ion collisions are actually investigated from an experimental and phenomenological point of view. Nevertheless, lattice data of quantities related to the first moment of the pion dPDFs are now available. These quantities encode double parton correlations which cannot be accessed via one-body functions such as standard form factors. This conclusion is qualitatively coherent with the quark model analyses for the proton target. The main purpose of the present study is to compare lattice QCD predictions of the effective form factor with quark model calculations. In particular, here we have considered AdS/QCD soft-wall inspired pion models for which phenomenological implementations are also included. Double PDFs have been calculated by showing their full dependence on the longitudinal momentum fraction and the transverse momentum unbalance k ⊥ . Ratios sensitive to DPCs have been calculated and results show that DPCs are relevant. An important comparison between dPDFs and their approximations in terms of GPDs and form factors have been also investigated. Holographic model predictions shows that even if the pion is described by considering only the first |qq state, dPDFs cannot be described in terms of one-body functions. Such a conclusion is consistent with previous studies of the proton dPDFs. Let us stress here that such an approximation is largely used in phenomenological analyses of DPS processes. In order then to provide useful predictions, an estimate of the experimental observable σ ef f has been provided via quark models and lattice QCD. These results have been properly interpreted in terms of geometrical properties of the pion partonic structure by verifying the RC inequality. Furthermore, moments of dPDFs, i.e. the effective form factors, have been calculated within the adopted quark models and then compared with lattice data for the first time. Despite the limited region in Q 2 , which minimises the impact of frame dependent effects, one can conclude that for the moment being the absence of a complete evaluation of high Fock states, in the pion expansion, prevents a simultaneous description of the electric-magnetic and effective form factors via holographic models. Nevertheless, the original AdS/QCD model almost matches the lattice eff and qualitatively reproduces the impact of double parton correlations. On the contrary, even if the other models provide an impressive description of the e.m. form factor, they fail in the evaluation of the eff. These first comparisons, between lattice and quark model analyses, point to the necessity of an accurate description of the contributions of high Fock states in the pion. Thus, lattice data can be used to add new constraints on future implementations of holographic models. Even if the frame dependence of the lattice eff prevents to get a useful mean value of σ ef f , the RC inequality has been inverted in order to provide a range of frame independent values of σ ef f starting from the d 2 addressed by lattice QCD. Such a procedure leads to a value of σ ef f , for a pion-pion collision, which is bigger then to the proton-proton case. This conclusion is directly obtained from lattice QCD data and could guide future experimental and theoretical analyses. In the final part of this study, predictions for dPDFs and effs of the ρ meson have been discussed for the first time showing how the impact of DPCs change with the meson polarisation.
dy z e y(k1−k1) = e y ⊥ ·(k 1⊥ −k 1⊥ ) dy − e y − (k + 1 −k + 1 ) e y + (k − 1 −k − 1 ) = e y ⊥ ·(k 1⊥ −k 1⊥ ) dy − e y − (k + where we have used that for y 0 = 0 one gets y z = y + = −y − . We recall that k andk are the momentum of partons in the hadron in the initial and final states, respectively. Thus from Eq. (65), we get: where O y represents the corrections due to the choice of y 0 = 0 w.r.t. y + = 0. One can show that this quantity reads: In this case the correction is proportional to 1/(P + ) 2 . In the IMF such a contribution is small. Thus, from only a kinematic point of view, the light-cone expression of dPDFs is similar to that obtained within the lattice framework in the IMF. In closing we stress that also in the analysis of Ref. [17], the authors claim that the standard expression (13), which relates the eff to the product of one-body quantities, is restored for q 2 << m 2 π .