The effective cross section for double parton scattering within a holographic AdS/QCD approach

A first attempt to apply the AdS/QCD framework for a bottom-up approach to the evaluation of the effective cross section for double parton scattering in proton-proton collisions is presented. The main goal is the analytic evaluation of the dependence of the effective cross section on the longitudinal momenta of the involved partons, obtained within the holographic Soft-Wall model. If measured in high-energy processes at hadron colliders, this momentum dependence could open a new window on 2-parton correlations in a proton.


INTRODUCTION
The effects of multiple parton interactions (MPI) in proton-proton scattering have been the object of several studies which have a long history (see, e.g. Ref. [1]) and, at the same time, continue to be an active field of interest. From an experimental point of view, the Large Hadron Collider (LHC) has opened the possibility to observe specific signatures of these effects (see [2][3][4][5][6] for recent reports), useful to constrain the background for the search of New Physics; from a theoretical point of view, the investigation of two-parton correlations will become possible, opening a new field in the description of the non-perturbative three dimensional (3D) proton structure (see, e.g., Ref. [7]). The simplest MPI process is double parton scattering (DPS), whose description is based on specific non-perturbative elements: the double Parton Distribution Functions (dPDFs). These quantities describe the number densities of two partons, located at a given transverse distance (b ⊥ ) in coordinate space, which carry given longitudinal momentum fractions (x i = x 1 , x 2 ) of the parent proton. The calculation of dPDFs, non-perturbative quantities, is particularly cumbersome and therefore one can perform model calculations able to focus on the relevant features [8][9][10][11]. Usually, in the literature, the Fourier transform of the dPDFs w.r.t. b ⊥ , depending therefore on k ⊥ , the relative transverse momentum between the two acting partons, sometimes called 2 GPDs, have been studied. At present, it has not yet been possible to extract dPDFs from experimental data, but a specific observable, related to DPS, has received much attention in the past: the so called effective cross section, σ ef f . It is defined through the ratio of the product of two single parton scattering cross sections to the DPS cross section with the same final states and can be parameterized in terms of dPDFs and parton distribution functions (PDFs). The effective cross sec-tion has been extracted, although in a model dependent way, in several experiments [12][13][14][15][16][17]. The apparent conclusion, within the present scenario and despite the large error bars, is that σ ef f remains constant as a function of the center-of-mass energy of the collision.
In Ref. [18] we have recently investigated σ ef f , using the dPDFs calculated within the Light-Front (LF) approach developed in Ref. [10]. A clear dependence on the fractions of proton longitudinal momentum carried by the four partons involved in the DPS process has been predicted. This feature could represent a first access to the experimental observation of two-parton correlations in the proton.
The aim of the present work is to provide confirmation on the x i dependence in σ ef f by using an AdS/QCD framework, a completely different approach to hadron structure than the LF formalism used in Ref. [18]. In the AdS/QCD framework the evaluation of the required (double) parton distributions is straightforward, leading in some cases to analytic expressions which can be very helpful to guide experiments. The leitmotif of this approach is the duality between conformal field theories and gravitation in an anti de Sitter space [19]. Since QCD is not a conformal theory, people have not been able to develop yet the fundamental top-down approach. We shall proceed therefore by a bottom-up approach where important features of QCD are implemented generating a theory in which conformal symmetry is only asymptotically restored [20,21]. In this scheme we make use of the well established Soft-Wall model [22] of the AdS/CFT framework. Within this approach it has been proven that the gauge/gravity duality provides a (holographic) mapping of the string model Φ(z), z being the fifth dimension, to the hadron Light-Font wave functions (LFWFs) in four dimensional space-time. The approach has been successfully applied to the description of the mass spectrum of mesons and baryons and it reproduces the Regge trajec-tories (e.g. Ref. [21] and references therein).
The structure of the paper is the following. In section II we propose a formulation of dPDFs within an AdS/QCD holographic approach. In section III we investigate explicitly the effective cross section calculating its x i -dependence in a simple and analytic way and, eventually, conclusions are drawn in section IV.

II. DPDFS FROM GPDS IN ADS/QCD
The approach we are using, a semiclassical approximation to QCD, is often called Light Front Holography (LFH) [23]. It is based on the realization of a mapping relating AdS modes to LFWFs; it is obtained by matching specific matrix elements (e.g. the electromagnetic form factors) in the two approaches -string theory in AdS and Light-Front QCD in Minkowski space-time [24]. An interesting straightforward application of the gauge/gravity correspondence to hadronic properties in the strong coupling regime, where QCD cannot be used in a direct and simple way, is the calculation of the Generalized Parton Distributions (GPDs) of the nucleon, described in Ref. [25][26][27][28]. Recently, GPDs have been evaluated within generalized approaches breaking conformal symmetry in the hadronic sector [29].
We refer to Ref. [25] for the detailed aspects of the calculation of GPDs and of the holographic mapping; in the next sections, we will recall only basic results to be used in the study of dPDFs.

A. Factorization
In actual analyses, dPDFs are usually approximated by factorized forms. In particular, as firstly proposed in Ref. [30] and widely used, the dPDF in momentum space, F uV uV (x 1 , x 2 , k ⊥ , µ 2 0 ), can be written as a product of two spin independent, quark helicity conserving GPDs We recall that GPDs depend on the longitudinal momentum fraction of the active quark (x), on the momentum transferred in the longitudinal direction (ξ, the so called skewdness) and the invariant momentum transfer, t = −k 2 ⊥ . As indicated, GPDs depend also on the momentum scale µ 0 1 . To be more precise, let us concentrate first on the chiral even (helicity conserving) distribution H q (x, ξ, t, Q 2 ) for partons of q-flavor, and taking deeply virtual Compton scattering (DVCS) as a typical process. A virtual photon of momentum q µ is exchanged by a lepton to a nucleon of momentum P µ and a real photon of momentum q ′ µ is produced, together with a recoiling nucleon with momentum P ′ µ . The space-like virtuality is therefore Q 2 = −q µ q µ and it identifies the scale of the process (in the expression (1), Q 2 = µ 2 0 ). The invariant momentum transfer is t = −k 2 ⊥ = (P ′ µ − P µ )/2 and the skewedness ξ encodes the change of the longitudinal nucleon momentum (2ξ = k + /P + , with 2P µ = (P µ + P ′ µ )). The factorized form (1) contains only the GPDs at ξ = 0; it is remarkable that, when Fourier transformed to coordinate space, these quantities become densities, the so called impact paramater dependent parton distributions (the reader can find in Ref. [31] a recent update on GPDs physics). It is also interesting to note that the dPDF, Eq. (1), Fourier transformed to coordinate space, is given by a convolution of impact parameter dependent parton distributions. In this approximation, the longitudinal momenta of the quarks described by the dPDF are not correlated, while these momenta and k ⊥ are correlated (see Ref. [3] for a discussion on this issue).
The H u V are normalized in the natural way and the factorization (1) is valid in the region x 1 +x 2 < 1, i.e. in the region kinematically accessible to the two partons whose total momentum cannot exceed the nucleon momentum. In Ref. [3] also a first order correction to Eq. (1) has been evaluated and the total expression reads which includes a correction containing E q , the nucleon spin independent, helicity flip GPD, and M p is the proton mass.

B. GPDs in SW-model
Expressions for the GPDs in terms of the AdS modes are obtained making use of the holographic mapping suggested by Brodsky and de Teramond [24]. In the present case, as in the calculation of the nucleon form factors, the procedure is based on the use of the integral representation for the bulk-to-boundary propagator introduced by  Grigoryan and Radyushkin [33]: where α is the parameter entering the Soft-Wall potential in the holographic coordinate z to be identified with the Light-Front coordinate ζ in the mapping procedure: where α = 0.41 GeV [27,29]. As a conclusion, the helicity independent GPDs assume the exact form [25]: Analogous expressions can be written for the helicity dependent GPDs E q . One should notice that, in the obtained GPDs, the dependence on the longitudinal momentum and that on the momentum transfer are not factorized, as it happens, to our knowledge, in all the microscopic model calculations of GPDs (see, e.g., Refs. [34] and [35]).

III. xi-DEPENDENCE OF THE PROTON EFFECTIVE CROSS SECTION
The effective cross section, σ ef f , is a relevant quantity in the experimental analysis of DPS (for a recent update, see, e.g., Ref. [18] and references therein).
An expression for σ ef f , suitable for theoretical evaluations, has been developed in Ref. [18] and can be written as follows: where F i , F k , F j , F l are the PDFs entering the process in study (globally, i, k, j, l = q,q, g), F ij (x 1 , x 2 , k ⊥ ) are the related dPDFs (in Eq. (6) the explicit dependence on the scale µ 2 0 has been suppressed for simplicity) and C ik are color factors. In principle, σ ef f depends on four momentum fractions. In order to discuss the main features of σ ef f , one can restrict the analysis to the zero rapidity region (y = 0), and therefore to x i = x ′ i , and to valence u V which remains the dominant component of the Fock space in the AdS approach and it is identified with valence quarks [29,32]: where the explicit dependence (5) has been used. Eq. (7) shows that an analytic dependence on x 1 x 2 is predicted by the holographic AdS approach.
In particular in the valence region, the behavior is qualitatively similar to the one found previously within a LF approach [18]. Quantitatively, taking for example x 1 = x 2 = 0.4, one finds, from (7) σ ef f ≃ 2π α 2 [ ln(1/x 1 ) + ln(1/x 2 ) ] ≃ 26.6 mbarn , a value which is not far from the result of Ref. [18] and from those extracted by the experimental collaborations. As shown in Ref. [18], at least in the valence region, QCD evolution does not change substantially the x-dependence and the absolute values of σ ef f . However, in the present analysis, we are especially interested in the x i dependence of σ ef f , which is found to be a largely model independent feature. To that aim we normalize the cross section at some low-x 2 value (x 2 = x 0 2 ) obtaining (for x 1 + x 2 < 0, and which represents the essential result of the present work, illustrated in Fig. 1, where the ratios (8), is shown as a function of x 1 and at different values of x 2 . A relevant x i -dependence of the cross section is found. It turns out to be rather strong in the valence region, as already indicated in Ref. [18]. It is important to notice that this dependence is sizable also at lower values of x i , manifesting a suppression of 20 − 30% at x 1 = 0.01 (depending on the value of x 2 ). At x 1 = 0.1, the suppression is around 50%. Before concluding the section let us discuss the further correction due to the k 2 ⊥ /M 2 p contribution in Eq. (2), as proposed in Ref. [3]. An explicit calculation shows that Eq. (8) still holds with the simple replacement where and κ u = 2κ p + κ n ≈ 1.673 is related to the anomalous magnetic moments of proton and neutron, κ p and κ n , respectively.
The correction is very small as it borns out from Fig.1, where its effects are shown for x 2 = 0.2 (the other cases being quite similar).

IV. SUMMARY AND CONCLUSIONS
The present work addresses a topic which has a specific relevance in extracting double parton correlations from high-energy proton-proton scattering data: the x idependence of the (so called) effective cross section, a de-pendence put in numerical evidence in Ref. [18]. The relevance of such a dependence deserves some further study and we have investigated it within an AdS/QCD holographic approach. In fact it is largely recognized that such a technique is a good analytic tool to investigate physical systems, and their electromagnetic interactions, within non-perturbative QCD (see Ref. [21] for a recent report). The approach here proposed applies, for the first time, AdS/QCD to the evaluation of dPDFs and parton correlations. The result is rather direct, showing a clear x i dependence of the effective cross section. Experimentally, such a dependence is not evident, most likely because of the large errorbars. A better identification of the behavior of the cross section as a function of the center-of-mass energy of the collision would open interesting windows on the parton-parton correlations and, consequently, on a novel way to look at specific features of the 3-D structure of the nucleon.