# Generalized parton distributions: confining potential effects within AdS/QCD

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## Abstract

Generalized parton distributions are investigated within a holographic approach where the string modes in the fifth dimension describe the nucleon in a bottom–up or AdS/QCD framework. The aim is to bring the AdS/QCD results in the realm of phenomenology in order to extract consequences and previsions. Two main aspects are studied: (i) the role of the confining potential needed for breaking conformal invariance and introducing confinement (both: classic soft-wall and recent infra-red potentials are investigated); (ii) the extension of the predicted GPDs to the entire range of off-forward kinematics by means of double distributions. Higher Fock states are included describing the nucleon as a superposition of three valence quarks and quark–antiquark pairs and gluons.

## 1 Introduction

Generalized parton distribution functions (GPDs) are a source of fundamental information encoding essential aspects of the nucleon structure [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] as basic ingredients in the description of hard exclusive processes [2, 3, 4]. They are generalization of the well known parton distribution functions and, at the same time, as correlation functions they incorporate quite non-trivial aspects of hadrons in the non-perturbative regime like: electromagnetic form factors, spin and angular momentum of the constituents and their spatial distribution [1, 10]. Their functional structure is usually written as a function of the longitudinal momentum fraction of the active quark (*x*), the momentum transferred in the longitudinal direction (\(\xi \) or skewedness) and the invariant momentum (square) \(t = - \Delta ^2\). The Fourier transform of GPDs (at \(\xi =0\)) in the transverse direction encodes information on the partonic distributions in the transverse plane and it translates in a quantitative information (because of the probabilistic interpretation as density functions) on the separation of the struck quark and the center of momentum of the nucleon [5, 6, 7, 8, 9]. The detailed map of quarks and gluons in the nucleon interior is often called “nucleon tomography” since the traditional information from elastic and deep-inelastic scattering provide static coordinates or momentum space pictures, separately, while GPDs provide pictures of dynamical correlations in both coordinate and momentum spaces [15].

Amplitudes of different hard exclusive processes (like deeply virtual Compton scattering (e.g. [16]), and virtual vector meson production (e.g. [17, 18]) in the new generation of CLAS experiments at Jefferson Lab.) contain GPDs as essential components. On the other hand the experimental results already collected have shed a fundamental light on their role in different processes and kinematical regimes (for example the H1 [19] and ZEUS [20] at HERA, HERMES at DESY [21], Hall A and Hall B at Jefferson Lab. [22], COMPASS at CERN [23]).

GPDs are non-perturbative objects and their evaluation lies in the realm of non-perturbative QCD. The successes are, till now, strongly limited [24, 25]. An alternative approach is the holographic light-front technique. Its fundamentals are in the correspondence between string theory developed in a higher dimensional anti-de Sitter (AdS) space and conformal field theory (CFT) in Minkowski physical space-time [26, 27, 28, 29]. Several consequent models have been constructed and they can be divided in top-down and bottom–up approaches. Starting from some brane configuration in string theory, one can, indeed, try to reproduce basic features of QCD following top-down paths (e.g. Ref. [30, 31]). On the way up one starts from low-energy properties of QCD (like chiral symmetry breaking and quark confinement) to infer elements for a gravity frame with asymptotically AdS space, the models are therefore indicated as AdS/QCD (e.g. Ref. [32, 33] and the references therein). In particular within the bottom–up approach two successful models have been constructed: (i) the hard-wall model, which uses a sharp cut-off in the extra dimension to confine the (dual) hadron field [34, 35]. The model is simple, analytic and appealing, but it does not reproduce the linear Regge behavior of the meson masses. (ii) In the soft-wall model [36] a (quadratic) dilation field is added in the meson sector in order to successful reproduce the Regge behavior, however, chiral symmetry breaking cannot be consistently realized. In particular, it has been shown [37] that the spontaneous chiral symmetry breaking in vacuum and its restoration at finite temperature, can be realized only within a careful choice of the dilaton profile (see also Ref. [38]).

Consequently several authors are investigating how to improve the SW description to incorporate the largest number of QCD properties [39, 40, 41, 42, 43, 44, 45].

An example particularly interesting in the present perspective is the infra-red improved soft-wall AdS/QCD model proposed in Ref. [40]: it is constructed for baryons, taking into account a specific baryonic property of the spectrum, namely the parity-doublet pattern of the excited baryons. It shows consistent properties also in the meson sector [41]. This simplified model is taken, in the present paper, as a prototype to investigate GPDs and illustrating, at the same time, a procedure valid to study generalized parton distributions and other observables in a generically modified confining potential.

Within the AdS/QCD approach deep-inelastic scattering (DIS) has been first addressed by Polchinski and Strassler in Refs. [47, 48], and GPDs have been investigated by many authors both within the hard-wall [49, 50] and soft-wall [51, 52, 53, 54] models. Because of the nature of the AdS–QCD analogy in the region of DIS, the results are restricted to the *forward* limit (\(\xi = 0\)) (cf. Sect. 3).

In the present work an attempt for a step forward is investigated and in two directions: (i) generalizing the study of GPDs for confining potentials more complex than the simple soft-wall model; (ii) extending the GPDs results to the off-forward region, \(\xi > 0\), by means of a technique called double distributions [55].

In Sect. 2 the procedure to evaluate the nucleon holographic wave function in the modified confining potential is discussed and the numerical results illustrated. In Sect. 3 the relation between sum rules and the \(\xi =0\) components of the GPDs is investigated and generalized to include, within a unified framework: (i) the effects of the modified confining potential; (ii) the contributions of higher Fock states. Numerical results for both helicity-independent and -dependent GPDs are discussed in Sect. 4 and compared with a light-front approach. Section 5 is devoted to the application of double-distribution techniques [55] to the AdS/QCD predictions for the soft-wall model. It is shown how AdS/QCD can become predictive in the whole kinematical range \((x,\xi >0,t)\). Conclusions and perspectives in Sect. 6.

## 2 From the soft-wall to the infra-red improved model

*d*-dimensions defined at the boundary. The needed breaking of conformal invariance (QCD is not a conformally invariant theory) of that correspondence for the baryonic case is obtained introducing, in addition to the dilaton term \(\varphi (z)\), an effective interaction \(\rho (z)\) in the action of the Dirac field (propagating in AdS\(_{d+1}\)) [32, 33]:

*z*can be identified with the transverse impact variable \(\zeta \) of the \(n-1\) spectator system with respect the active parton in a

*n*-parton bound state (\(z = \zeta \)). In the \(2 \times 2\) chiral spinor representation one obtains two coupled differential equations (cf. e.g. Ref. [32])

### 2.1 Linear soft-wall potential

*D*-harmonic oscillator)

### 2.2 The IR-improved soft-wall model and its solutions

The IR potential has been constructed to reproduce, with good accuracy, both the meson and the baryon masses. In particular it gives consistent predictions for the mass spectra of scalar, pseudoscalar, vector and axial-vector mesons, and both confinement and chiral symmetry breaking are well characterized [41]. In the case of baryons the parameters \(\lambda _A\) and \(\lambda _B\) are fixed by fitting the masses of the first low-lying baryons with even parity (including nucleon). The predicted masses for odd-parity baryons and high excited states of even-parity baryons are consistently reproduced [40] by using the same values of the parameters.

### 2.3 Numerical results

The numerical values of the coefficients \(a_{\nu n}^\pm \) for the variational expansion (14) in the case of maximum h.o. quanta \(n_{\mathrm{max}}=16\) and \(\nu =3\) (\(l_+ = \nu = 3\), \(l_-=\nu +1=4\)). The h.o. constants are fixed by the minimization procedure at \(\alpha ^+ = 2.35\) fm\(^{-1}\) and \(\alpha ^- = 2.65\) fm\(^{-1}\)

| \(a_{(\nu =3), n}^+\) | \(a_{(\nu =3), n}^-\) |
---|---|---|

0 | 0.9811 | 0.8749 |

1 | −0.1872 | −0.4423 |

2 | 0.0486 | 0.1834 |

3 | −0.0071 | −0.0678 |

4 | 0.0014 | 0.0233 |

5 | −0.0002 | −0.0076 |

6 | 3.6e−05 | 0.0024 |

7 | −5.3e−06 | −7.3e−04 |

8 | 8.3e−07 | 2.2e−04 |

9 | −1.2e−07 | −6.5e−05 |

10 | 1.8e−08 | 1.9e−05 |

11 | −2.7e−09 | −5.4e−06 |

12 | 3.9e−10 | 1.5e−06 |

13 | −5.7e−11 | −4.3e−07 |

14 | 7.3e−12 | 1.2e−07 |

15 | −5.7e−12 | −3.2e−08 |

16 | −6.4e−12 | 8.0e−09 |

## 3 GPDs and sum rules at \(\xi =0\)

*q*-flavor at the scale where one is assuming valid the calculation for the related amplitudes. For example, the amplitude for deeply virtual Compton scattering where a virtual photon of momentum \(q^\mu \) is exchanged by a lepton to a nucleon of momentum \(P^\mu \) and a real photon of momentum \(q'^\mu \) is produced (together with a recoiling nucleon \(P'^\mu \)). The space-like virtuality is therefore \(Q^2=-q^\mu q_\mu \) and it identifies the scale of the process. The invariant momentum square is \(t=-\Delta ^2=(P'^\mu - P^\mu )^2\) and the skewedness \(\xi \) encodes the change of the longitudinal nucleon momentum (\(2 \xi = \Delta ^+/\bar{P}^+\), with \(2 \bar{P}^\mu = (P^\mu + P'^\mu )\)). In the following the common notation of simply three variables \((x,\xi ,t)\) instead of \((x,\xi ,Q^2,t)\) is assumed

^{1}. In addition only the limit \(\xi =0\) will be discussed and therefore one can remain in the \(0 \le x \le 1\) region.

### 3.1 Contribution of the valence quarks (\(\nu =3\))

*q*and one has

*q*(\(N_u =2\), \(N_d = 1\)). The integral properties are therefore model independent and strongly constrain the helicity conserving distributions in any model and/or parametrization (the conditions on \(N_q\) are satisfied within all the models presented). The second moment,

*t*-dependent moments of the GPDs are related to the nucleon elastic form factors [10], i.e.

*q*to the Dirac and Pauli form factors. The property (19) does not depend on \(\xi \) and it holds also in the present approach with \(\xi =0\) and therefore \(0 \le x \le 1\), (cf. Refs. [10, 15]), and one has

*AdS*/

*QCD*, the Dirac form factors for the nucleons in the present soft-wall linear model are given by [32, 49, 50, 51, 52, 53, 56]

### 3.2 Higher Fock states (\(\nu =4\), \(\nu =5\))

## 4 GPDs and confining potentials: results and comments

### 4.1 \(H^u(x,\xi =0,t)\) and \(H^d(x,\xi =0,t)\)

Results for \(H^u(x,\xi =0,t)\) are shown in Figs. 3 and 4. In particular in Fig. 3 the results for the valence components \(H_V^u(x,\xi =0,t)\), i.e. the twist-3 contributions (\(\nu =3\)) are shown for both the SW model and the IR-improved model. One could imagine that the change in the confining potential encodes just refinements producing only small effects on the observables. This comment is true from the point of view of the baryon spectra, however, the modifications induced on the wave functions can show up in a more consistent way in appropriate observables. It is the philosophy of the present work and it is well illustrated in Fig. 3: comparing the SW and the IR-improved results one can appreciate the effects produced by the tuning of the confining potential (cf. Fig. 1). Analogous effects emerge in the analysis of the response of *d*-valence quarks (Fig. 5). In that case the effects of the IR-improved potential seem to be even more evident in the low-*x* region and for both \(t=-0.2\) GeV\(^2\) and \(t=-0.5\) GeV\(^2\).

The *t*-dependence of the *H*-GPDs can be appreciated comparing the upper and lower panels of Figs. 3 and 5, where the responses are shown for two different values of the momentum \(t=-0.2\) GeV\(^2\) and \(t=-0.5\) GeV\(^2\). In particular in Fig. 3 the results of the present AdS/QCD approach are compared with an investigation (cf. Ref. [58]) which makes use of a light-front relativistic quark model developed in Ref. [59] and based on a *q*-*q*-potential with a linear plus a Coulomb-like component: \(V = -{\tau \over r} + \kappa _l \, r\). The predictions of the two approaches look rather different. The constraints due to conformal symmetry breaking imposed by the AdS/QCD approach seem to reduce the response considerably (and in the whole *x*-range) changing, at the same time, their *t*-dependence in a relevant way.

Figures 4 and 5 are devoted to the investigation of the higher Fock states effects. Within the IR-improved potential the \(\nu =3\) and \(\nu =3,4,5\) responses are shown and compared. The effects of higher-Fock states is rather weak, but one has to keep in mind the limited validity of the contribution for \(\xi =0\), the only component here discussed. The role of quark–antiquark and gluon components should show up in a more consistent way in the \(\xi \)-dependence of the response [1, 60, 61]. It would be particularly interesting, in view of the next generation of experiments, to add explicitly such components together with the appropriate perturbative QCD evolution. Work in this direction is in progress.

*t*-dependence of the responses. Often such a dependence is taken following the fall off of the nucleon form factors. Modeling GPDs does not confirm that hypothesis and the results of Ref. [58] already questioned such a

*t*-dependence. The results of the AdS/QCD approach show an even stronger

*t*-dependence, a peculiarity which should be explicitly investigated in future experiments.

### 4.2 \(E^u(x,\xi =0,t)\) and \(E^d(x,\xi =0,t)\)

^{2}On the contrary the AdS/QCD wave functions are normalized at the experimental values. To make the comparison more meaningful Figs. 6, 7 and 8 show the ratios \(E^q(x,\xi ,t)/\kappa ^q\). They are in continuity with Figs. 3, 4 and 5 for the \(H^q\) responses.

*x*-dependence is concerning. The comparison with the LF-approach shows also an important difference in

*t*- dependence between the LF and the AdS/QCD approaches. The inclusion of higher Fock states is illustrated in Figs. 7 and 8.

#### 4.2.1 The \(\alpha \) parameter: a critical analysis

^{3}

*x*-dependence of \(H^q\), it influences its

*t*-dependence. Such a conclusion has the relevant consequence that the differences one can see in Fig. 2 are \(\alpha \)-independent. For \(t < 0\) the effects are more complicated correlating in a critical way the

*x*and

*t*-dependence

^{4}and the choice of the \(\alpha \)-parameter appears to be critical. The discussion of the SW spectrum for baryons shows that the masses obey the Regge behavior and \(\alpha \approx 0.5\) GeV is needed to reproduce the nucleon spectrum (cf. Sect. 2.1). In the literature the values \(\alpha = 0.49\) GeV and \(\alpha = 0.51\) are considered the best choices to reproduce, within the holographic AdS/QCD, the nucleon and the \(\Delta \) spectra respectively [32]. The freedom in the choice of \(\alpha \) is related to the nature of the AdS/QCD approach and the actual value is fixed following physical constraints like the nucleon and the \(\Delta \) masses. In the study of the nucleon electromagnetic form factors, \(\alpha \) is fixed in order to reproduce their momentum transfer behavior and, to this end, it has been chosen [54] \(\alpha = 0.4066 \approx 0.41\) GeV. It is physically sensible to remain within this choice in order to study GPDs. However, just to give a flavor of the \(\alpha \) dependence of the present investigation in Fig. 9 the sensitivity of the helicity-independent GPDs to the \(\alpha \)’s values is shown for both the SW and the IR-improved potential. The values chosen are: (i) the choice made in the previous sections and related to the electromagnetic form factors, \(\alpha = 0.41\) GeV; (ii) the value from the best fit of the nucleon masses, \(\alpha = 0.49\) GeV. The variations shown in Fig. 9 could represent an upper bound to the absolute theoretical error. However, one cannot consider the range of the results shown in the figure as genuine theoretical error bars; in fact the value \(\alpha = 0.41\) GeV is well constrained to be associated to the electromagnetic interactions as described within AdS/QCD. One has to keep in mind, indeed, that \(\alpha \) is the parameter that appears in the dilaton definition used to break conformal invariance in AdS and it affects all fields considered in the model, including the vector massless field which allows the calculation of form factors (and GPDs); cf. Eq. (24). The same parameter appears, in the case of the nucleon, in the soft-wall potential, in the holographic coordinate \(V(z) = \alpha ^2 \, z\). Consistency is mandatory and the \(\alpha \) value has to be fixed by physical constraints connected with the vector massless field dual to the electromagnetic field and the form factors appear a natural choice.

In concluding the present Sect. 4, a general comment can be added in order to justify the large differences one can see in the IR versus SW potential predictions as well as in the comparison with the LF model. The quite different behavior of the potentials at intermediate vales of \(z \approx 0.5\) fm (see Fig. 1), introduce relevant differences in the high-momentum components of the wave corresponding functions and, consequently, on the *H*-distributions. In particular, if the behavior of the IR potential is extrapolated to small distances (\(< 0.5\) fm) to match Coulomb tail like in the case the LF model, the enhancement at small- and intermediate-*x* values is emphasized as it emerges, for instance, from Figs. (3) and (6). The responses of the LF model (in the region \(0 \le x \le 0.3\)) are larger than the IR-potential ones; the IR responses are, in turn, larger than the SW model distributions: a coherent behavior. On the contrary, because of the sum rule constraints, the responses in the large-*x* region follow an inverse behavior.

## 5 Modeling the \(\xi \)-dependence with double distributions

*x*,

*t*-independent part can be parametrized by a two component form [55]

The \(D^q\) contribution in Eq. (39) is defined in the region \(|x| \le \xi \) and therefore does not contribute in the forward limit. The D-term contributes to the singlet-quark and gluon distributions and does not contribute to non-singlet components. Its effect under evolution is restricted at the level of few percent [62, 63, 64] and it will be disregarded in the following.

*b*fixes the width of the profile function \(h(\beta ,\alpha )\) and the strength of the \(\xi \)-dependence. In principle it could be used (within the double-distribution approach) as a fit parameter in the extraction of GPDs from hard electro-production observables. The favored choice is \(b_\mathrm{NS} = b_\mathrm{S} = 1\) (producing a maximum skewedness) and \(b_{\mathrm{gluon}} = 2\) [60, 68]. In the limiting case \(b \rightarrow \infty \), \(h(\beta ,\alpha ) \rightarrow \delta (\alpha ) h(\beta )\) and \(H^q(x,\xi ) \rightarrow H^q(x,\xi =0)\). The explicit evaluation of \(H^q(x,\xi )\) in Eq. (39) makes use of the results of the previous sections within the holographic AdS/QCD approach.

### 5.1 Results at low momentum scale: the soft-wall model

*x*-values by the integrals (40) and the combinations (35), (36) and (37) (cf. also Ref. [68]):

*only*valence quarks at the lowest scale, the holographic approach is intrinsically based on the QCD dynamics. The bound system of valence quarks cannot share momentum among a pure three-quark system. The masses of the quarks are unknown and what is reproduced is the spectrum of the system. The interpretation of the nucleon bound system implies the presence of gluons exchanged among the valence quarks. A natural consequence is an additional gluon distribution filling the gap to the total momentum. A gluon distribution proportional to the valence densities at \(Q_0^2\) (à la Glück, Reya, Vogt [70, 71]) can be a sensible choice,

*small*) \(H^g\) component of Fig. 10. In the previous studies of GPDs within AdS/QCD no mention is made of the fact that the momentum sum rule is not satisfied, i.e. property (44). The main reason to introduce here a conserving momentum sum rule, like Eq. (46), is related to the possibility of a detailed investigation of the perturbative QCD evolution properties of the distributions. The simplified assumption made in Eq. (45) is mostly connected to the fact that the perturbative evolution is dominated by the value of the moment carried by the gluon component rather than by the exact form of the distribution.

## 6 Conclusions and perspectives

- (i)
The role of the confining potential in the holographic coordinate as described within the soft-wall and within more general potential models; in particular the possibility of introducing high Fock states in the calculation of GPDs. A method to study effects due to different confining potentials introduced to break conformal symmetry in the AdS/QCD approach to baryons has been proposed in Sect. 2. In several works devoted to the investigation of AdS/QCD wave functions for baryons, often the complementary aspect is stressed: the potentials must manifest iso-spectral properties and therefore their differences have to be adequately mitigated [39]. On the contrary the use of different (almost iso-spectral potentials) in calculating amplitude and responses in deep inelastic scattering, can put in evidence relevant differences that can discriminate among them. The specific observables discussed in relation with generalized parton distributions are a good example. The method implies the use of the Soft Wall solutions as a complete basis to solve more sophisticated potential models. The results are promising: the power of the holographic approach seems to be preserved and observables can be calculated following well established techniques. Higher Fock states can be accommodated showing their relevance in the whole

*x*-region. - (ii)
The extension of the GPDs results from AdS/QCD methods from the forward (\(\xi =0\)) to the off-forward region (\(\xi >0\)). The procedure used (double distributions) enlarge the phenomenological domain of the GPDs predictions opening the concrete evaluation of the single, non-singlet GPDs in the whole (

*x*, \(\xi \),t) domain. The procedure used identifies also the resolution scale of the results. The example developed is restricted to the soft-wall, but it is easily generalized to more complex confining potentials.

The chosen reference frame is symmetric and \(q^\mu \) and the average moment \(\bar{P}^\mu =(P^\mu +P'^\mu )/2\), are collinear (along the *z* axis) and opposite in directions.

More explicitly, Ref. [58] investigates two L-F quark models: (i) a Hypercentral potential which includes linear and Coulombian interactions and which is *SU*(6) symmetric; (ii) a model with Goldstone Boson exchange (GBE) [65] which breaks *SU*(6). Despite the fact that \(\kappa _{p/n}\) are in principle sensitive to *SU*(6) breaking effects, the vales of the two models do not differ that much. For details cf. Ref. [58].

Results for the helicity-independent components will be discussed, they are illustrative also for the results of the helicity-dependent component.

## Acknowledgements

I would like to thank my colleagues J.-P. Blaizot and J.-Y. Ollitrault of the IPhT CEA-Saclay for the active scientific atmosphere that has stimulated also the present study and for their continuous help. A key mail exchange with Zhen Fang is also gratefully acknowledged. I thank S. Scopetta, V. Vento and M. Rinaldi for a critical reading of the manuscript and their fruitful collaboration. The last version of the present manuscript has been written in Perugia during a visiting period and I thank the Sezione INFN and the Department of Physics and Geology for warm hospitality and support.

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