Generalized parton distributions: confining potential effects within AdS/QCD
Eur. Phys. J. C
Generalized parton distributions: confining potential effects within AdS/QCD
Marco Traini 0 1
0 INFNTIFPA Dipartimento di Fisica, Università degli Studi di Trento , Via Sommarive 14, Povo, 38123 Trento , Italy
1 Institut de Physique Théorique, Université Paris Saclay, CEA , 91191 GifsurYvette , France
Generalized parton distributions are investigated within a holographic approach where the string modes in the fifth dimension describe the nucleon in a bottomup 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 softwall and recent infrared potentials are investigated); (ii) the extension of the predicted GPDs to the entire range of offforward kinematics by means of double distributions. Higher Fock states are included describing the nucleon as a superposition of three valence quarks and quarkantiquark pairs and gluons.

Generalized parton distribution functions (GPDs) are a
source of fundamental information encoding essential aspects
of the nucleon structure [1–14] as basic ingredients in the
description of hard exclusive processes [2–4]. They are
generalization of the well known parton distribution
functions and, at the same time, as correlation functions they
incorporate quite nontrivial aspects of hadrons in the
nonperturbative 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 (ξ or skewedness) and the
invariant momentum (square) t = − 2. The Fourier transform of
GPDs (at ξ = 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 [5–9]. The detailed map of quarks and gluons in the
nucleon interior is often called “nucleon tomography” since
the traditional information from elastic and deepinelastic
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 nonperturbative objects and their evaluation
lies in the realm of nonperturbative QCD. The successes are,
till now, strongly limited [24,25]. An alternative approach is
the holographic lightfront technique. Its fundamentals are
in the correspondence between string theory developed in a
higher dimensional antide Sitter (AdS) space and conformal
field theory (CFT) in Minkowski physical spacetime [26–
29]. Several consequent models have been constructed and
they can be divided in topdown and bottom–up approaches.
Starting from some brane configuration in string theory, one
can, indeed, try to reproduce basic features of QCD
following topdown paths (e.g. Ref. [30,31]). On the way up one
starts from lowenergy 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 hardwall model, which uses a sharp cutoff 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 softwall 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 [39–45].
An example particularly interesting in the present
perspective is the infrared improved softwall AdS/QCD model
proposed in Ref. [40]: it is constructed for baryons, taking
into account a specific baryonic property of the spectrum,
namely the paritydoublet 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 deepinelastic 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 hardwall [49,50] and softwall [51–
54] models. Because of the nature of the AdS–QCD analogy
in the region of DIS, the results are restricted to the forward
limit (ξ = 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 softwall model; (ii) extending the GPDs results to the
offforward region, ξ > 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 ξ = 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 helicityindependent and dependent
GPDs are discussed in Sect. 4 and compared with a lightfront
approach. Section 5 is devoted to the application of
doubledistribution techniques [55] to the AdS/QCD predictions for
the softwall model. It is shown how AdS/QCD can become
predictive in the whole kinematical range (x , ξ > 0, t ).
Conclusions and perspectives in Sect. 6.
2 From the softwall to the infrared improved model
The AdS/QCD framework relates a gravitationally
interacting theory in the antide Sitter space AdSd+1 with a
conformal gauge theory in ddimensions 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 ϕ(z), an effective interaction ρ(z) in the action
of the Dirac field (propagating in AdSd+1) [32,33]:
dd x dz √g eϕ(z)
i AeMA DM − μ − ρ(z)
Maximal symmetry is restored for ϕ(z) = ρ(z) = 0. One
has √(g) = Rz d+1, while eMA is the inverse vielbein,
emMAatr=icesRz anδtAMi.coDmMmiusttehe[ coAv,ariBa]nt =deri2vηatAivBe. aAndDt hireacDilrikace
wave equation can be derived from Eq. (1) and the
dynamical effect due to the dilaton field reabsorbed rescaling the
spinor → eϕ(z)/2 . For that reason the term eϕ(z) is
sufficient to break maximal symmetry for mesons but not for
the baryon sector. The additional interaction term ρ(z)
provides the needed breaking (and confining) contributions to
generate the correct baryon spectrum [32,46]. The absence
of dynamical effects of the dilaton background field has a
particular disappointing side effect in the lack of guidance
from gravity to solve the equations.
A solution is given by a lightfront holographic mapping
where the LF wave equation can be identified with the
equation of motion. In the case of d = 4, A = (γμ, i γ5) and
V (z) = Rz ρ(z), the holographic variable z can be
identified with the transverse impact variable ζ of the n − 1
spectator system with respect the active parton in a nparton bound
state (z = ζ ). In the 2 × 2 chiral spinor representation one
obtains two coupled differential equations (cf. e.g. Ref. [32])
ddζ φ+ − ν +ζ1/2 φ+ − V (ζ )φ+ = Mφ−, (2)
− ddζ φ− − ν +ζ1/2 φ− − V (ζ )φ− = Mφ+; (3)
here ν can be identified with the lightfront angular
momentum, i.e. the relative angular momentum between the active
parton and the spectator cluster. Equations (2) and (3) are
easily reduced to the equivalent system of second order
differential equations:
φ+ + V 2(ζ )φ+ = M2φ+,
φ− + V 2(ζ )φ− = M2φ−.
2.1 Linear softwall potential
For a quadratic interaction (and z = ζ within the holographic
model), ρ(ζ ) ∼ ζ 2, V (ζ ) = α2ζ (the so called softwall
linear potential) and Eqs. (4) (5) become:
φ− + α4ζ 2φ− + 2να2φ−
with normalized solutions (equivalent to 2Dharmonic
oscillator)
Lln(x ) are the associated Laguerre polynomials and one
identifies common eigenvalues M2 = 4α2(n + ν + 1). The linear
confining potential generates a mass gap of the order of α. ν is
related to the h.o. angular momentum by l+ = ν, l− = ν + 1.
In the following α2 = (0.41)2 GeV2 will be selected, a value
which interpolates among different choices in the literature
(cf. Ref. [32] and references therein) and it gives a good fit
to the form factors [54]. A critical analysis of the influence
of the α’s value on the results of the present approach will be
given in Sect. 4.2.1.
2.2 The IRimproved softwall model and its solutions The infrared improved softwall AdS/QCD model proposed in Ref. [40] (in the following: IR) exhibits a confining potential of the form
√α (αζ )l++1/2 e−α2ζ 2/2 Lln+ (α2ζ 2),
VI R (ζ ) = λAkgμg ζ 1 − λB μ2g ζ 2 e−μ2gζ 2 ,
√α (αζ )l−+1/2 e−α2ζ 2/2 Lln− (α2ζ 2);
−20
Fig. 1 The confining softwall linear potential V (ζ ) = α2ζ (α =
0.41 GeV) as function of ζ (fm) (dashed line), is compared with the
IRimproved potential introduced in Refs. [40,41] (see Eq. (11) and
Sect. 2.2 for comments). Also a hardwall potential at ζ0 ∼ 1/ QCD is
sketched (dotted line)
Table 1 Values of the parameters for VI R of Eq. (11). μg, λA, λB are
from Ref. [40]. For kg see Eq. (12) and discussion
shown in Fig. 1. The numerical values of the parameters are
as in Table 1.
The potential (11) belongs to the class of potentials
obeying V (ζ → 0) = α2ζ , and V (ζ μ−g1) = α2ζ , i.e. la class
of potentials matching the linear wall both in the IR and UV
regimes [39].
Therefore the potential VI R must reduce to the linear
softwall potential in the limiting case λB = 0, and one has
λAkgμg = α2 = (0.41)2 GeV2
→ kg ≈ 0.089 GeV,
parameters used in Fig. 1 and in the following.
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 axialvector mesons,
and both confinement and chiral symmetry breaking are well
characterized [41]. In the case of baryons the parameters
λA and λB are fixed by fitting the masses of the first
lowlying baryons with even parity (including nucleon). The
predicted masses for oddparity baryons and high excited states
of evenparity baryons are consistently reproduced [40] by
using the same values of the parameters.
Let us introduce the form (11) in Eqs. (4) and (5), one gets
d2 D±
− dζ 2 + ζ 2 + E ± ζ 2 + F ± ζ 2 e−μgζ 2 + G± ζ 4 e−μgζ 2
+H ± ζ 6 e−2μgζ 2 + I ± φν±,I R
= Rˆ ±φν±,I R = M2I R φν±,I R ,
2
where ( A = λAkgμg, B = λB μg)
n=0
D+ = −(1 − 4ν2)/4; D− = −[1 − 4(ν + 1)2]/4
E + = A2; E − = A2
F + = [−3 − (1 + 2ν)] A B; F − = [+3 − (1 + 2ν)] A B
G+ = [−2 A + 2μ2)] A B; G− = [−2 A − 2μ2)] A B
H + = A2 B2; H − = A2 B2
I + = 2(1 + ν) A; I − = 2ν A .
A convenient technique to solve Eqs. (13) is an expansion
aolnl tthhee baalrseiasdoyf eφsn±tla±blished properties of the solutions (8, 9)
of Eqs. (8, 9), in this way one can keep
within a linear combination of them. Consequently
where ν = 3 and l+ = ν and l− = ν + 1 for the lowest
threequark Fock state of the nucleon [56]. The natural parameter
to be chosen to minimize M2I R looking for the ground state
wave function (the nucleon) is the harmonic oscillator
constant which has to be diversifyed in two components α → α±
in order to respect the essential property M+IR = M−IR .
→ (M± I R )2 → (MI R )2.
Of course the restricted Hilbert space used in solving the
minimization will result in an upper bound for M2I R . However,
as will become clearer in the next section, the convergence
is rapid and one has to expect only few percent deviations.
2.3 Numerical results
The minimization procedure is performed in the two
components φν±,I R varying the parameter α− and reaching the
minimum value for (M−IR )2 = 2.61 GeV2 for α− = 2.65 fm−1,
with the corresponding (M+IR )2 = 2.61 GeV2 for α+ =
2.35 fm−1, and involving 17 oscillator quanta (nmax = 16).
The harmonic oscillator angular momentum quantum
numbers l− = ν + 1 and l+ = ν are fixed by the twist operator
ν = 3 for the lowest number of active quarks [56] (of course
nmax (an±,l± )2 = 1). In Table 2 the actual vales of the
coeffin=0
cbiaesnistsisan±i,nl±f.aOctntehecamnaaxpipmreucmiatneutmheerriacpalidbcaosinsvseurgpepnocrtee.dTbhye
Table 2 The numerical values
of the coefficients aν±n for the
variational expansion (14) in the
case of maximum h.o. quanta
nmax = 16 and ν = 3
(l+ = ν = 3, l− = ν + 1 = 4).
The h.o. constants are fixed by
the minimization procedure at
α+ = 2.35 fm−1 and α− = 2.65
fm−1
the Matlab code used for the minimization; however, it is
evident that remaining within nmax = 10 is a quite good
approximation. The numerical calculations of the next
sections will make use of the restricted basis nmax = 10.
3 GPDs and sum rules at ξ = 0
In order to introduce the explicit calculations of the GPDs,
let us concentrate first on the chiral even (helicity
conserving) distribution H q (x , ξ, Q2, t ) for partons of qflavor 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 qμ is exchanged by a lepton to a nucleon of momentum
Pμ and a real photon of momentum q μ is produced (together
with a recoiling nucleon P μ). The spacelike virtuality is
therefore Q2 = −qμqμ and it identifies the scale of the
process. The invariant momentum square is t = − 2 =
( P μ − Pμ)2 and the skewedness ξ encodes the change of
the longitudinal nucleon momentum (2ξ = +/ P¯ +, with
2 P¯ μ = ( Pμ + P μ)). In the following the common notation
of simply three variables (x , ξ, t ) instead of (x , ξ, Q2, t ) is
assumed1. In addition only the limit ξ = 0 will be discussed
and therefore one can remain in the 0 ≤ x ≤ 1 region.
1 The chosen reference frame is symmetric and qμ and the average
moment P¯ μ = (Pμ + P μ)/2, are collinear (along the z axis) and
opposite in directions.
Fig. 2 The distributions xu(x) and xd(x) as a function of x. From
the softwall linear potential (solid lines) and from the IRimproved
potential model (dotdashed), when higherFock states are included
(cf. Sect. 3.2). The total momentum sum rule reads Mu+d = 0.92 for
the SW model, Mu+d = 0.91 for the IR when higher Fock states are
included
3.1 Contribution of the valence quarks (ν = 3)
The helicity conserving H q distributions, in the limit t = 0
and ξ = 0 reduce to ordinary parton distributions
H q (x , 0, 0) = q(x ),
the unpolarized quark distribution of flavor q and one has
dx H q (x , 0, 0) =
dx q(x ) = Nq ,
where Nq fixes the number of valence quarks of flavor q
(Nu = 2, Nd = 1). The integral properties are therefore
model independent and strongly constrain the helicity
conserving distributions in any model and/or parametrization
(the conditions on Nq are satisfied within all the models
presented). The second moment,
dx x H q (x , 0, 0) =
dx x q(x ) = Mu+d ,
q=u,d
q=u,d
−1
is related to the momentum sum rule (cf. Fig. 2) and the
models discussed differ: the LFmodel is based on lightfront
wave functions and obeys Mu+d = 1 since the valence
contribution is the only component at low momentum scale. The
numerical calculations give: Mu+d = 0.92 for the SW, while
Mu+d = 0.91 for the IR when higher Fock states are
considered (cf. Sect. 3.2). In addition the first t dependent moments
of the GPDs are related to the nucleon elastic form factors
[10], i.e.
−1
q q
where F1 ( 2) and F2 ( 2) are the contribution of quark q
to the Dirac and Pauli form factors. The property (19) does
not depend on ξ and it holds also in the present approach
with ξ = 0 and therefore 0 ≤ x ≤ 1, (cf. Refs. [10,15]), and
one has
F1p( 2) =
F2p( 2) =
dx + 23 HVu (x, ξ = 0, t) − 31 HVd (x, ξ = 0, t) ,
dx − 13 HVu (x, ξ = 0, t) + 23 HVd (x, ξ = 0, t) ,
dx + 23 EVu (x, ξ = 0, t) − 31 EVd (x, ξ = 0, t) ,
dx − 13 EVu (x, ξ = 0, t) + 23 HVd (x, ξ = 0, t) ,
where t = − 2 and isospin symmetry has been assumed.
In terms of the holographic wave functions φ± derived from
Ad S/QC D, the Dirac form factors for the nucleons in the
present softwall linear model are given by [32,49–53,56]
F1p( 2) =
dζ V +( 2, ζ ) φν+,I R (ζ )2 (N +)2,
(α+ζ )4
−V −( 2, ζ ) φν−,I R (ζ )2 (N −)2 ,
(α−ζ )4
φν−,I R (ζ )V −( 2, ζ )φν+,I R (ζ )
(N∓)2
φν+,I R (ζ )V +( 2, ζ )φν−,I R (ζ )
here κ p/n are the proton and neutron anomalous
gyromagnetic factors, respectively. The kernels V ± have a simple and
analytic integral form [57]:
(1 − x )2
(α±ζ )2 x 2/[4(α±)2]e−(α±ζ )2x/(1−x).
The specific boundary condition V ±( 2 = 0, ζ ) = 1
imposes the normalizations:
(N∓)2 = 1 ,
(N±)2 = 1 ;
and the results of the SW [51–53] model are recovered in the
limit
and therefore:
an±l± → anl± = 1,
(N +)2 → 2/(2/3!) = 6,
The resulting expressions for the GPDs are
Fx+( 2, ζ ) φν+,I R (ζ )2 (N +)2
(α+ζ )4
2) − 13 E Vd (x , ξ = 0, −
φν−,I R (ζ )Fx−( 2, ζ )φν+,I R (ζ )
(N∓)2
φν+,I R (ζ )Fx+( 2, ζ )φν−,I R (ζ )
2) + 23 E Vd (x , ξ = 0, −
φν−,I R (ζ )Fx−( 2, ζ )φν+,I R (ζ )
(N∓)2
φν+,I R (ζ )Fx+( 2, ζ )φν−,I R (ζ )
3.2 Higher Fock states (ν = 4, ν = 5)
The formalism developed in the previous sections for the
solution of the improved IR potential at the lowest twist
(ν = 3), can easily accommodate also higher Fock states
in the wave functions opening the possibility of studying
their effects on the generalized parton distributions even in
the presence of a modified potential. In particular additional
gluons (ν = 4) or a quark–antiquark pair (ν = 5) as
discussed in Ref. [56]. One obtains
with c3 = 1.25, c4 = 0.16, and c5 = 1 − c3 − c4 = −0.41.
The previous conditions and values are taken from Ref.
[56] where they are established for the linear SW potential.
However, the criteria are rather general and directly related
to experimental observables, their application also for the
IR potential seems quite natural and it represents, in any
case, a first sensible approximation. The minimization has
to be repeated for ν = 4, 5 in analogy with the numerical
analysis of Sect. 2.3. All the expressions derived in Sect. 3 are
generalized in a straightforward way replacing (φν±,I R (ζ ))2
and φν±,I R (ζ ) φν∓,I R (ζ ) with the linear combinations (31). In
order to comment in more detail, the generalization of Eq.
(27) is given as an example:
One has to notice that the normalization factors N 2 will
depend on ν while the harmonic oscillator parameters α±
will not. In fact the baryon masses (fixed by the explicit form
of the confining potential) will get their minimum values for
the same α±, as it has been checked numerically. The
generalization is straightforward. In the appendix are the numerical
details.
4 GPDs and confining potentials: results and comments
4.1 H u (x , ξ = 0, t ) and H d (x , ξ = 0, t )
Results for H u (x , ξ = 0, t ) are shown in Figs. 3 and 4. In
particular in Fig. 3 the results for the valence components
HVu (x , ξ = 0, t ), i.e. the twist3 contributions (ν = 3) are
shown for both the SW model and the IRimproved 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 IRimproved results one can appreciate the effects
produced by the tuning of the confining potential (cf. Fig. 1).
I R
I R
Fig. 3 Upper panel The results for HVu (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model (continuous line) are compared
with the same results for the corresponding SW model (dotted) and the
LF model calculation of Ref. [58] (dashed). Only twist3 contributions
are included (cf. Sect. 3.1) and therefore the analysis is restricted to the
valence sector. Lower panel As in the upper panel for t = −0.5 GeV2
Analogous effects emerge in the analysis of the response of
dvalence quarks (Fig. 5). In that case the effects of the
IRimproved potential seem to be even more evident in the lowx
region and for both t = −0.2 GeV2 and t = −0.5 GeV2.
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 GeV2 and t = −0.5 GeV2. 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 lightfront relativistic quark model
developed in Ref. [59] and based on a qqpotential with a
linτ
ear plus a Coulomblike component: V = − r + κl 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 IRimproved potential
Fig. 4 Upper panel The results for HVu (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model and twist3 contribution only
(continuous line, the same results of Fig. 3) are compared with the results
obtained including higher Fock states (dotdashed), namely ν = 4, 5
(cf. Sect. 3.2). Lower panel As in the upper panel for t = −0.5 GeV2
the ν = 3 and ν = 3, 4, 5 responses are shown and compared.
The effects of higherFock states is rather weak, but one has
to keep in mind the limited validity of the contribution for
ξ = 0, the only component here discussed. The role of quark–
antiquark and gluon components should show up in a more
consistent way in the ξ 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.
The comparison with experiments seems also particularly
interesting from the point of view of the 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.
SWν = 3
I R ν = 3
I R ν = 3, 4, 5 SWν = 3
I R ν = 3
I R ν = 3, 4, 5
0−1.2
Fig. 5 Upper panel The results for H d (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model and twist3 contribution only
(continuous line) are compared with the results obtained including higher
Fock states (dotdashed), namely ν = 3, 4, 5 (cf. Sect. 3.2). The
predictions of the corresponding SW model are also shown (dotted). Lower
panel As in the upper panel for t = −0.5 GeV2
4.2 E u (x , ξ = 0, t ) and E d (x , ξ = 0, t )
The integral properties of the helicity nonconserving
responses E q are more model dependent:
where κq is the anomalous magnetic moment.
Experimentally κu = 2κ p + κn = 1.67 and κd = 2κn + κ p = −2.03.
From a theoretical point of view the calculation of κq is
affected by the dynamical hypothesis of the approach used. In
particular for the LFapproach of Ref. [58] one has κu = 1.02
and κd = −0.74.2 On the contrary the AdS/QCD wave
functions are normalized at the experimental values. To make the
2 More explicitly, Ref. [58] investigates two LF 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 κp/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].
Fig. 6 Upper panel The results for EVu (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model (continuous line) are compared
with the same results for a the corresponding SW model (dotted) and the
LF model calculation of Ref. [58] (dashed). Only twist3 contributions
are included (cf. Sect. 3.1) and therefore the analysis is restricted to the
valence sector. Lower panel As in the upper panel for t = −0.5 GeV2
comparison more meaningful Figs. 6, 7 and 8 show the ratios
E q (x , ξ, t )/κq . They are in continuity with Figs. 3, 4 and 5
for the H q responses.
Also for the E q distributions the SW and the IRimproved
potentials predict significantly different results as far as their
x dependence is concerning. The comparison with the
LFapproach 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 α parameter: a critical analysis
Before discussing some application of the GPDs in AdS/QCD,
a critical analysis of the parameter α characterizing the SW
potential (cf. Sect. 2.1) could help in fixing the precision one
can expect in the present, and analogous, investigations. To
this end it is convenient to write explicitly the GPDs within
the SW approach, as they result from Eqs. (27)–(30) in the
I R
I R
SWν = 3
I R ν = 3
I R ν = 3, 4, 5 SWν = 3
I R ν = 3
I R ν = 3, 4, 5
Fig. 7 Upper panel The results for EVu (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model and twist3 contribution only
(continuous line, the same results of Fig. 6) are compared with the results
obtained including higher Fock states (dotdashed), namely ν = 3, 4, 5;
cf. Sect. 3.2). Lower panel As in the upper panel for t = −0.5 GeV2
SWlimit of Eq. (26):3
HVu (x , ξ = 0, μ02, t ) = uV (x , μ02) x − 4αt 2 ,
HVd (x , ξ = 0, μ02, t ) = dV (x , μ02) x − 4αt 2 .
(with t = − 2). It is evident from Eq. (34), (and analogous
expressions can be written for the helicitydependent
components) that, for t = 0, the parameter α does not affect the
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 αindependent. For t < 0
the effects are more complicated correlating in a critical way
the x and t dependence4 and the choice of the αparameter
appears to be critical. The discussion of the SW spectrum
for baryons shows that the masses obey the Regge behavior
and α ≈ 0.5 GeV is needed to reproduce the nucleon
spec
3 Results for the helicityindependent components will be discussed,
they are illustrative also for the results of the helicitydependent
component.
4 These correlations can have important physical consequences in dou
ble (or multiple) parton scattering (e.g. Ref. [69]).
Fig. 8 Upper panel The results for Ed (x, ξ = 0, t = −0.2 GeV2)
predicted by the improvedIR model and twist3 contribution only
(continuous line) are compared with the results obtained including higher
Fock states (dotdashed), namely ν = 3, 4, 5; cf. Sect. 3.2). The
predictions of the corresponding SW model are also shown (dotted). Lower
panel As in the upper panel for t = −0.5 GeV2
trum (cf. Sect. 2.1). In the literature the values α = 0.49 GeV
and α = 0.51 are considered the best choices to reproduce,
within the holographic AdS/QCD, the nucleon and the
spectra respectively [32]. The freedom in the choice of α is
related to the nature of the AdS/QCD approach and the actual
value is fixed following physical constraints like the nucleon
and the masses. In the study of the nucleon electromagnetic
form factors, α is fixed in order to reproduce their
momentum transfer behavior and, to this end, it has been chosen [54]
α = 0.4066 ≈ 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 α dependence of the present investigation
in Fig. 9 the sensitivity of the helicityindependent GPDs to
the α’s values is shown for both the SW and the IRimproved
potential. The values chosen are: (i) the choice made in the
previous sections and related to the electromagnetic form
factors, α = 0.41 GeV; (ii) the value from the best fit of
the nucleon masses, α = 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;
α = 0.41 GeV
α = 0.49 GeV
α = 0.41 GeV
α = 0.49 GeV
Fig. 9 Upper panel The sensitivity of HVu (x, ξ = 0, t = −0.2 GeV2)
to the variation of the α parameter in the case of SW predictions. Lower
panel As in the upper panel for I R improved potential. See text
in fact the value α = 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 α
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 softwall potential, in the holographic
coordinate V (z) = α2 z. Consistency is mandatory and the α 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 ≈ 0.5 fm (see
Fig. 1), introduce relevant differences in the highmomentum
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 intermediatex values is
emphasized as it emerges, for instance, from Figs. (3) and (6). The
responses of the LF model (in the region 0 ≤ x ≤ 0.3) are
larger than the IRpotential 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 largex region follow an inverse behavior.
5 Modeling the ξ dependence with double distributions
In the present section the results obtained at ξ = 0, are
generalized to the whole ξ domain by means of a
doubledistribution approach developed by Radyushkin in Ref. [55].
The approach involves a given profile function and the
forward parton distribution as evaluated in the previous sections
(or in a generic model). In order to be specific let us
concentrate on the chiral even (helicity conserving) distributions
H q (x , ξ, Q2, t ). One can introduce nonsinglet (valence)
and singletquark distributions:
H NS(x , ξ, t ) ≡
H q (x , ξ, t ) + H q (−x , ξ, t )
H S(x , ξ, t ) ≡
H q (x , ξ, t ) − H q (−x , ξ, t ) ,
The analogous distribution for gluons is symmetric in x ,
H g(x , ξ, t ) = H g(−x , ξ, t ),
H g(x , ξ = 0, t = 0) = xg(x ), x > 0.
Once again, the Q2 dependence has been omitted following
the common simplified notation, it will be discussed in Sect.
5.1 when the hadronic scale Q20 will be introduced. Due to the
polynomiality property [67] the symmetry characters, (35),
(36) and (37), hold also under ξ → −ξ . The singlet and
gluon components mix under evolution, while the nonsinglet
distribution evolve independently.
The t independent part can be parametrized by a two
component form [55]
H q (x , ξ ) = H Dq D(x , ξ ) + θ (ξ − x ) Dq
H Dq D(x, ξ ) =
and H q (x , ξ ) ≡ H q (x , ξ, t = 0).
−1
dα δ(x − β − αξ ) Fq (β, α),
The Dq contribution in Eq. (39) is defined in the region
x  ≤ ξ and therefore does not contribute in the forward
limit. The Dterm contributes to the singletquark and gluon
distributions and does not contribute to nonsinglet
components. Its effect under evolution is restricted at the level of few
percent [62–64] and it will be disregarded in the following.
Following Radyushkin the DD terms entering Eq. (40) are
written as
F q (β, α) = h(β, α) H q (β, 0, 0),
where H q (β, 0, 0) = q(β) (cf. Eq. (16)) and the profile
function is parametrized as [68]
(2b + 2)
h(β, α) = 22b+1 2(b + 1)
The parameter b fixes the width of the profile function h(β, α)
and the strength of the ξ dependence. In principle it could be
used (within the doubledistribution approach) as a fit
parameter in the extraction of GPDs from hard electroproduction
observables. The favored choice is bNS = bS = 1
(producing a maximum skewedness) and bgluon = 2 [60,68].
In the limiting case b → ∞, h(β, α) → δ(α)h(β) and
H q (x , ξ ) → H q (x , ξ = 0). The explicit evaluation of
H q (x , ξ ) 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 softwall model
In the present section the results for the chiral even
distributions of the softwall model valid at ξ = 0 as discussed
in Sect. 4.1, are generalized to ξ > 0 by means of the
double distributions presented in the previous section. They are
defined in the different regions of the (generalized) x values
by the integrals (40) and the combinations (35), (36) and (37)
(cf. also Ref. [68]):
H Dq D(x , ξ, t = 0) ≡ H Dq D(x , ξ ) = θ (+ξ ≤ x ≤ +1)
H Dq D(x , ξ, t = −
2) = H Dq D(x , ξ, t = 0) x 2/(4α2);
−10−1
Fig. 10 The softwall predictions for GPDs. Singletquark (upper
panel), nonsingletquark (middle panel) and gluon (amplified 20 times,
lower panel) GPDs at the low momentum scale Q20 [see Eqs. (35), (36)
and (37)] at t = − 2 = 0, using the DDs (43). Full lines for ξ = 0.1,
dashed lines for ξ = 0.2 and dotdashed lines for ξ = 0.3
The results at low momentum scale, Q20, where the softwall
model is supposed to be valid, are shown in Fig. 10 for three
different values of the skewedness parameter ξ = 0.1, 0.2,
and 0.3 and invariant momentum t = − 2 = 0. The value
of the low momentum scale is identified by means of the
momentum sum rule. In fact the number of particles are well
defined at the initial scale [cf. Eq. (17)], and the momentum
sum rule is not fulfilled by valence quarks only. As a matter
of fact one has
Differently from a quark model (relativistic or nonrelativistic)
based on the presence of 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 threequark 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 Q20 (à la Glück, Reya, Vogt [70,71]) can be a
sensible choice,
dx x (uV + dV + g)Q20 = 1,
with Ag = 0.091 and dx xg(x , Q02) = 0.08. The (small)
H g gluon distribution of Eq. (37) as consequence of the
density (45), is shown in the lowest panel of Fig. 10. The factor 20
is needed to make H g comparable with the results shown in
the other panel of the same figure, H S and H N S. The choice
(45) is only one of the possible choices one can make. One
could assume a different parametrization of the gluon
distribution (45), and deduce a different behavior of the (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.
In order to appreciate the role of the invariant momentum
transfer t = − 2, the results of Fig. 10 valid at t = 0 are
summarized in Fig. 11 and compared with the analogous
predictions for t = −0.5 GeV2. The SW model gives a
nonvanishing contribution to quark GPDs in the region x  < ξ
at the lowest scale Q20 without introducing discontinuities
at x  = ξ , the ξ dependence is rather weak (cf. Fig. 10).
One can check, in particular, that H S = H N S at x > ξ , a
peculiarity due to the absence of a sea contribution at Q2.
0
H−4
−6−1
Fig. 11 Comparison of the SW predictions for H N S,S,g at ξ = 0.3
and t = 0 GeV2, thin lines, results as in Fig. 10, and t = 0.5 GeV2,
thick lines. The gluon distributions are amplified by a factor of 20
6 Conclusions and perspectives
A study of GPDs within a general AdS/QCD framework has
been presented. Two main features have been emphasized
and investigated in detail:
(i) The role of the confining potential in the holographic
coordinate as described within the softwall 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 isospectral properties and therefore their differences
have to be adequately mitigated [39]. On the contrary the
use of different (almost isospectral 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 (ξ = 0) to the offforward region
(ξ > 0). The procedure used (double distributions)
enlarge the phenomenological domain of the GPDs
predictions opening the concrete evaluation of the single,
nonsinglet GPDs in the whole (x , ξ ,t) domain. The
procedure used identifies also the resolution scale of the
results. The example developed is restricted to the
softwall, but it is easily generalized to more complex
confining potentials.
The calculated helicityindependent and dependent GPDs
show differences and properties to be further investigated in
order to compare their predictions with the new generation
of experimental data. In particular the important
contributions due additional degrees of freedom like nonperturbative
gluon and sea components should be further investigated
together with a detailed analysis of the perturbative effects
due to QCD evolution. The elegance and the effectiveness of
the AdS/QCD approaches has to be integrated in a complete
predictive scheme for a large variety of observables in the
perspectives of modeling the nucleon structure [66]; work in
that direction is in progress.
Acknowledgements I would like to thank my colleagues J.P. Blaizot
and J.Y. Ollitrault of the IPhT CEASaclay 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.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Higher Fock states
In this appendix some details of the method proposed in Sect.
2 are illustrated.
The procedures can be generalized in order to
accommodate higher Fock states. As discussed in Sect. 3.2 the values
of α± remain the same also for the wave functions with ν = 4
and ν = 5 (as a numerical check has confirmed). The
minimization produces the values of the coefficients shown in
Table 3.
Table 3 The numerical values of the coefficients aν±n for the variational
expansion (14) in the case of maximum h.o. quanta nmax = 16 and
ν = 4 (l+ = ν = 4, l− = ν + 1 = 5 and ν = 5 (l+ = ν = 5,
l− = ν + 1 = 6). The h.o. constants are fixed by the minimization
procedure at α+ = 2.35 fm−1 and α− = 2.65 fm−1
−0.2753
−0.0125
−4.0e−04
6.8e−05
−1.1e−05
1.8e−06
−2.8e−07
4.3e−08
−6.6e−09
1.0e−09
−1.5e−10
2.9e−11
1.6e−11
−3.3e−11
−0.4960
−0.0914
−0.0117
−0.0013
3.9e−04
−1.2e−04
3.7e−05
−1.1e−05
3.2e−06
−9.2e−07
2.6e−07
−7.2e−08
1.8e−08
−0.3038
−0.0159
−5.7e−04
1.0e−04
−1.7e−05
2.9e−06
−4.7e−07
7.6e−08
−1.2e−08
1.9e−09
−2.9e−10
4.1e−11
1.3e−11
−5.0e−11
−0.5231
−0.1100
−0.0157
−0.0018
6.0e−04
−1.9e−04
5.9e−05
−1.8e−05
5.5e−06
−1.6e−06
4.8e−07
−1.5e−07
4.7e−08
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