# Quark and gluon GPDs at finite skewness from strings in holographic QCD:
evolved and compared to experiment

Kiminad A. Mamo<EMAIL_ADDRESS>Physics Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA  Ismail Zahed<EMAIL_ADDRESS>Center
for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794-3800, USA

###### Abstract

The generalized parton distributions (GPDs) provide a versatile tool for
studying hadronic form factors, parton distribution amplitudes and parton
densities, all in one consolidated framework. At low-x and finite skewness,
they are dominated by Reggeon and Pomeron exchanges. In the large number of
colors limit, we suggest that their leading twist Gegenbauer (conformal)
moments at low resolution scale, can be extracted from the leading Witten
diagrams in holographic QCD. We specifically derive the corresponding moments
for few exclusive processes, mostly the electroproduction of photons and
mesons. The Gegenbauer moments are then evolved to a higher resolution scale,
using established QCD evolution equations in leading order. We use the
explicit results for the evolved singlet and non-singlet quark GPDs, as well
as the gluon GPDs, to analyze the electroproduction of longitudinal
$\rho^{0,+},\phi$ mesons. The results are compared to the available data. Our
explicit holographic results should prove useful for the extraction of GPDs
from the electroproduction of photons and mesons, in current and future global
GPDs data analyses.

##  I Introduction

General parton distributions (GPDs) provide a unique tool for addressing the
longitudinal momentum and spatial distributions of partons in hadrons. They
consolidate in a single framework, the physical content of form factors,
parton distribution amplitudes (DA) and parton distribution functions (PDFs).
A number of current and future electron machines (EIC, EIcC) Abdul Khalek et
al. (2021); Anderle et al. (2021) will be dedicated to their measurement.

The GDPs capture several invariants of the off-forward and non-local quark or
gluon bilinears. We will be mostly interested in the unpolarized quark and
gluon GPDs $H(x,\eta,t)$, and quark axial GPDs $\widetilde{H}(x,\eta,t)$, in
the large $N_{c}$ limit. For the quarks, they can be combined into valence
(isovector) and singlet GPDs. GPDs are a function of the parton longitudinal
momentum fraction $x$, the skewness $\eta\sim\xi=-\Delta^{+}/2P^{+}$, and the
momentum transfer $\Delta^{2}=-t$. $\Delta^{+}$ and $P^{+}$ are the light
front momentum transfer and averaged momentum of the in-out protons,
respectively.

For fixed Mandelstam $t$ and positive skewness, the GDPs are characterized by
two distinct kinematical regions: the DGLAP regions for $|x|>\xi$ and the ERBL
region for $|x|<\xi$. In the DGLAP region with positive (negative) $x$, the
GDPs correspond to removing a quark (antiquark) with momentum $k$ and re-
inserting it with momentum $k+\Delta$. In the ERBL region, the GDPs correspond
to emitting a meson-like quark-antiquark pair with momentum $-\Delta$. As a
result, the forward limit with zero skewness of the GDPs, coincide with the
quark and gluon PDFs, while the x-integrated GPDs with finite skewness,
coincide with form factors. We will be mostly interested by the ERBL region,
in the double limit of large $N_{c}$ and strong gauge coupling.

GPDs play an important role in both inclusive and exclusive processes, and are
at the corner stone of hadronic tomography. Deeply virtual Compton scattering
(DVCS) is one of the process suggested for the empirical extraction of the
GPDs Ji (1997); Radyushkin (1997). Assuming factorization, the invariant
Compton amplitudes are usually expressed as integral transforms of the leading
twist-2 quark operators, in the form of generalized sum rules. The extraction
of the quark GPDs require their inversion, usually using a perturbative
analysis of the pertinent moments. This is a non-trivial deconvolution problem
Bertone et al. (2021).

Holographic QCD provides a non-perturbative approach to a variety of
scattering processes in QCD, in the double limit of a large number of colors
and strong ′t Hooft gauge coupling. It is a proposal following on the AdS/CFT
or gauge/gravity duality established in string theory Nastase (2015) (and
references therein). In short, a strongly coupled gauge theory in 4-dimensions
is dual to weakly coupled string theory in higher dimensions. The original
correspondence holds for type IIB superstring theory in $AdS_{5}\times S_{5}$,
but is commonly assumed to hold for a string theory in a general background.

We propose to use holographic QCD with a soft wall, to evaluate
electroproduction of photons and mesons, at low-x and low resolution, all at
finite skewness. A key outcome from this construction, are exclusive
amplitudes with well defined dependence on all kinematical variables. This
yields well defined expressions for the initial quark and gluon GPDs in the
ERBL region, to evolve at higher resolution with the help of the perturbative
renormalization group equations. The interface between the perturbative and
non-perturbative and stringy aspects of QCD, will be shown to set in for a
gauge coupling $\alpha_{s}(\mu_{0})\sim\frac{1}{3}$ at a resolution
$\mu_{0}\sim 1.8$ GeV.

In our analysis, the deconvolution problem is altogether by-passed, through
the use of Gegenbauer moments for pertinent GPDs. The holographic construction
allows their identification for any value of the conformal spin-j, in the
large number of colors limit. Their spin-j resummation in the GPDs, can be
carried explicitly in the ERBL region in the Regge limit. The extension to the
DGLAP region can be sought using a re-organized Gegenbauer expansion. Once
evolved, the results can be used in any semi-inclusive process. They also
should prove useful, for the extraction of the GPDs in global data analyses.
For completeness, we note the holographic GPDs analysis in Nishio and Watari
(2014a).

To our knowledge, the holographic framework provides the most economical way
to encode the stringy properties of QCD at low resolution, in line with the
dual Veneziano amplitudes, and empirical Regge phenomenology. In the double
limit of large $N_{c}$ and strong ′t Hooft coupling, the holographic
amplitudes capture the key aspects used in QCD dispersive analyses, such as
crossing symmetry, unitarity and spectral densities, all while enforcing the
essential symmetries of QCD, with very few stringy parameters.

The outline of the paper is as follows: In section II, we recall how the
leading twist-2 quark and gluon off-diagonal matrix elements QCD, are tied to
the Gegenbauer (conformal) moments of pertinent GPDs. In section III, we use
large $N_{c}$ arguments to match these moments to their holographic exclusive
counterparts, at low-x and low resolution $\mu_{0}$. This matching is done
explicitly for the electroproduction of mesons, photons (DVCS), a pair of
pions, and a neutral pion. In section IV, we summarise how these moments are
evolved to higher resolution $\mu>\mu_{0}$, using DGLAP type equations at one-
loop. In section V, we show how to match the fixed ′t Hooft coupling at
$\mu_{0}$ to the running pQCD coupling, to extract the initial value of
$\mu_{0}\sim 1.8$ GeV. In section VI, we use the evolved singlet quark GPDs,
to analyse in detail the electroproduction of neutral $\rho^{0}$. The total
and differential cross sections are derived, and compared to the available
data for a wide range of energies $\sqrt{s}$, and momentum transfer. In
section VII, we extend the analysis to the electroproduction of $\phi$, using
the evolved gluon GPDs. The results are compared to some existing data. In
section VIII, we make use of the evolved quark non-singlet GPDs, to derive the
total and differential cross sections for the electroproduction of charged
$\rho^{+}$. The results are also compared to existing data, mostly at low
$\sqrt{s}$. Our conclusions are in section IX. We have included a number of
Appendices to detail and support some of the derivations.

##  II Quark and Gluon GPDs from their Gegenbauer (Conformal) Moments at
$\mu=\mu_{0}$

The framework for analysing the evolution equations for the GPDs, consists of
defining the operators in the OPE in the representations of the conformal
group. The conformal operators transform multiplicatively under
renormalization. As a result, the anomalous dimensions of conformal operators
are only labeled by spin-j, in reference to the conformal spin of the
composite operator Belitsky and Radyushkin (2005) (and references therein). In
this section, we will briefly review the relevant quark and gluon spin-j
operators, which enter the leading GPDs at twist-2. The spin-j operators are
the Gegenbauer moments of the various quark and gluon GPDs.

Classically, massless 4D QCD is invariant under the conformal group $SO(4,2)$
with 15 generators (1-dilatation, 4-conformal, 4-translation, 6-Lorentz). The
OPE expansion on the light cone involves Wilsonian operators which transform
covariantly under the Lorentz group $SO(3,1)$, but not under the conformal
group $SO(4,2)$. Since $SL(2)$ on the light front, is a subgroup of the
conformal group, a way to re-organize the Wilsonian expansion is in terms of
conformal tensors of spin $j=\frac{1}{2}(d+s)$ (the eigenvalue of the
collinear subgroup of the conformal operator) where the twist is $\tau=d-s$
Belitsky and Radyushkin (2005). More specifically, the twist-2 vector quark
and gluon operators are (see Eq.4.138 in Belitsky and Radyushkin (2005))

$\mathbb{O}^{qq}_{j}=\bar{\psi}(i\partial^{+})^{j-1}\gamma^{+}C_{j-1}^{3/2}\left(\stackrel{{\scriptstyle{}_{\leftrightarrow}}}{{\cal
D}}{\\!}^{+}\\!/\partial^{+}\right)\psi\,,\qquad\mathbb{O}^{gg}_{j}=F^{+}{}_{\mu}(i\partial^{+})^{j-2}C_{j-2}^{5/2}\left(\stackrel{{\scriptstyle{}_{\leftrightarrow}}}{{\cal
D}}{\\!}^{+}\\!/\partial^{+}\right)F^{\mu+}\,,$ (II.1)

and the twist-2 axial-vector quark operator is

$\tilde{\mathbb{O}}^{qq}_{j}=\bar{\psi}(i\partial^{+})^{j-1}\gamma^{+}\gamma_{5}C_{j-1}^{3/2}\left(\stackrel{{\scriptstyle{}_{\leftrightarrow}}}{{\cal
D}}{\\!}^{+}\\!/\partial^{+}\right)\psi\,,$ (II.2)

Here $\stackrel{{\scriptstyle{}_{\leftrightarrow}}}{{\cal D}}{\\!}^{+}$ is the
symmetrized long derivative on LF+. $C_{n}^{\nu}(x)$ are the Gegenbauer
polynomials, a generalization of the Legendre polynomials to $(2\nu+2)$-D
space. Their explicit form in terms of hypergeometric functions is

$\displaystyle C_{n}^{\nu}(x)=\begin{pmatrix}n+2\nu+1\\\
n\end{pmatrix}\bigg{(}\frac{x+1}{2}\bigg{)}^{2}{}_{2}F_{1}\bigg{(}-n,-n-\nu+\frac{1}{2};\nu+\frac{1}{2};\frac{x-1}{x+1}\bigg{)}$
(II.3)

with support in $[-1,+1]$, and the ortho-normalization with weight factor
$w(x|\nu)=(1-x^{2})^{\nu-1/2}$,

$\displaystyle\int_{-1}^{1}dx\,(1-x^{2})^{\nu-\frac{1}{2}}\,C_{n}^{\nu}(x)C_{m}^{\nu}(x)=\frac{\pi
2^{1-2\nu}\Gamma(n+2\nu}{(n+\nu)\gamma^{2}(\nu)\Gamma(n+1)}\,\delta_{mn}$
(II.4)

###  II.1 Gegenbauer Moments of Quark and Gluon GPDs at $\mu=\mu_{0}$

Valence vector quark:
The Gegenbauer moments of the valence vector quark GPDs, are directly related
to the matrix elements of the spin-j conformal operators in QCD (see Eq.4.243
in Belitsky and Radyushkin (2005))

$\displaystyle\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})=\eta^{j-1}\,\int_{0}^{1}dx\
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)F_{valence}^{q}(x,\eta,t;\mu_{0}^{2})=(p^{+})^{-(j-1)-1}\langle
p_{2}|\mathbb{O}^{qq}_{j}(0)|p_{1}\rangle\,,$

for odd $j=1,3,...$. The valence and x-even combination

$F_{valence}^{q}(x,\eta;\mu^{2})=F^{q}(x,\eta;\mu^{2})+F^{q}(-x,\eta;\mu^{2})$

stands for either $H_{valence}^{q}(x,\eta;\mu^{2})$ or
$E_{valence}^{q}(x,\eta;\mu^{2})$.

Singlet vector quark:
The Gegenbauer moments for the vector-singlet combinations, are also related
to the spin-j conformal operators by (see Eq.4.243 in Belitsky and Radyushkin
(2005))

$\displaystyle\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2})=\eta^{j-1}\,\int_{0}^{1}dx\
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)F_{singlet}^{q}(x,\eta,t;\mu_{0}^{2})=(p^{+})^{-(j-1)-1}\langle
p_{2}|\mathbb{O}^{qq}_{j}(0)|p_{1}\rangle\,,$

for even $j=2,4,...$. The valence and x-odd combination

$F_{singlet}^{q}(x,\eta;\mu^{2})=F^{q}(x,\eta;\mu^{2})-F^{q}(-x,\eta;\mu^{2})$

is either $H_{singlet}^{q}(x,\eta;\mu^{2})$ or
$E_{singlet}^{q}(x,\eta;\mu^{2})$.

Singlet axial quark:
The Gegenbauer moments of the singlet axial quark GPDs are related to the
conformal operators in QCD via (straightforward generalization of Eq.4.243 in
Belitsky and Radyushkin (2005) for the axial GPDs)

$\displaystyle\widetilde{\mathbb{F}}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2})=\eta^{j-1}\,\int_{0}^{1}dx\
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\widetilde{F}_{singlet}^{q}(x,\eta,t;\mu_{0}^{2})=(p^{+})^{-(j-1)-1}\langle
p_{2}|\widetilde{\mathbb{O}}^{qq}_{j}(0)|p_{1}\rangle\,,$

for odd $j=1,3,...$. Again, the x-even singlet combination

$\widetilde{F}_{singlet}^{q}(x,\eta,t;\mu^{2})=\widetilde{F}^{q}(x,\eta,t;\mu^{2})+\widetilde{F}^{q}(-x,\eta,t;\mu^{2})$

will refer to either $\widetilde{H}_{singlet}^{q}(x,\eta,t;\mu^{2})$ or
$\widetilde{E}_{singlet}^{q}(x,\eta,t;\mu^{2})$.

Valence axial quark:
The Gegenbauer moments for the axial-valence combinations, are tied to the
spin-j twist-2 axial operator (straightforward generalization of Eq.4.243 in
Belitsky and Radyushkin (2005) for the axial GPDs)

$\displaystyle\tilde{\mathbb{F}}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})=\eta^{j-1}\,\int_{0}^{1}dx\
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\widetilde{F}_{valence}^{q}(x,\eta,t;\mu_{0}^{2})=(p^{+})^{-(j-1)-1}\langle
p_{2}|\widetilde{\mathbb{O}}^{qq}_{j}(0)|p_{1}\rangle\,,$

for even $j=2,4,...$. The x-odd valence axial combination

$\widetilde{F}_{valence}^{q}(x,\eta,t;\mu^{2})=\widetilde{F}^{q}(x,\eta,t;\mu^{2})-\widetilde{F}^{q}(-x,\eta,t;\mu^{2})$

will refer to either $\widetilde{H}_{valence}^{q}(x,\eta,t;\mu^{2})$ or
$\widetilde{E}_{valence}^{q}(x,\eta,t;\mu^{2})$.

Gluons:
For gluon operators, we have (see Eq.4.284 in Belitsky and Radyushkin (2005))

$\displaystyle\mathbb{F}^{g}_{j}(\eta,t;\mu_{0}^{2})=2\eta^{j-2}\,\int_{0}^{1}dx\
C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)F^{g}(x,\eta,t;\mu_{0}^{2})=4(p^{+})^{-(j-1)-1}\langle
p_{2}|\mathbb{O}^{gg}_{j}(0)|p_{1}\rangle\,,$

for even $j=2,4,...$.

###  II.2 Reconstruction of Quark and Gluon GPDs from their Gegenbauer Moments
at $\mu=\mu_{0}$

The conformal (Gegenbauer) moments can be inverted using their orthonormality
(II.4), to give the quark and gluon GPDs at a given resolution in explicit
form. We now give the inversion with explicit support in $|x|\leq\eta$, and
briefly comment on the inversion in the full support $|x|\leq 1$.

Valence vector quark:
The valence vector quark GPD, follows by inversion(see Eq.4.242 in Belitsky
and Radyushkin (2005))

$\displaystyle H_{valence}^{q}(x,\eta,t;\mu_{0}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=1}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{3}{2}\big{)}}{\eta^{j-1}\,N_{j-1}(\frac{3}{2})}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})\,,$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\sum_{j=1}^{\infty}\frac{1}{\eta^{j-1}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})\,,$

for odd $j=1,3,...$, with

$\displaystyle
N_{j}(\nu)=2^{1-2\nu}\frac{{\mit\Gamma}^{2}(1/2){\mit\Gamma}(2\nu+j)}{{\mit\Gamma}^{2}(\nu){\mit\Gamma}(j+1)(\nu+j)}$
(II.11)

Singlet vector quark:
Similarly, the singlet vector quark GPD, is (see Eq.4.264 in Belitsky and
Radyushkin (2005))

$\displaystyle H_{singlet}^{q}(x,\eta,t;\mu_{0}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{3}{2}\big{)}}{\eta^{j-1}\,N_{j-1}(\frac{3}{2})}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2})$
(II.12) $\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-1}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2})$

for even $j=2,4,...$.

Valence axial quark:
The valence axial quark GPD, is

$\displaystyle\widetilde{H}_{valence}^{q}(x,\eta,t;\mu_{0}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{3}{2}\big{)}}{\eta^{j-1}\,N_{j-1}(\frac{3}{2})}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\widetilde{\mathbb{F}}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})$
(II.13) $\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-1}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\widetilde{\mathbb{F}}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})$

for even $j=2,4,...$.

Singlet axial quark:
The singlet axial quark GPD, is

$\displaystyle\tilde{H}_{singlet}^{q}(x,\eta,t;\mu_{0}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{3}{2}\big{)}}{\eta^{j-1}\,N_{j-1}(\frac{3}{2})}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\tilde{\mathbb{F}}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2}$
(II.14) $\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\sum_{j=1}^{\infty}\frac{1}{N_{j-1}(\frac{3}{2})}\times\frac{1}{\eta^{j-1}}\times
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\tilde{\mathbb{F}}^{q}_{j(singlet)}(\eta,t;\mu_{0}^{2})\,,$

for odd $j=1,3,...$.

Gluons:
The gluon GPD is (see Eq.4.264 in Belitsky and Radyushkin (2005))

$\displaystyle H^{g}(x,\eta,t;\mu_{0}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{5}{2}\big{)}}{\eta^{j-2}\,N_{j-2}(\frac{5}{2})}C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)\mathbb{F}^{g}_{j}(\eta,t;\mu_{0}^{2})\,,$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)^{2}\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-2}}\times\frac{1}{N_{j-2}(\frac{5}{2})}\times
C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)\times\mathbb{F}^{g}_{j}(\eta,t;\mu_{0}^{2})\,,$

for even $j=2,4,...$.

These representations of the GPDs make manifest their asymptotic forms, with
total support in the ERBL region (see section IV.3 below). Indeed, since the
support of $C_{j}^{\nu}(x/\eta)$ is $|x|\leq\eta$, it follows that at
asymptotic resolution, the contribution to the GPDs is dominated by the
leading conformal spin, with vanishing contribution for $|x|>\eta$. The
reorganized GPDs with the Gegenbauer polynomials in the region $|x|\leq\eta$,
will prove very useful in extracting them from the holographic dual
construction in the Regge limit at large $N_{c}$. The restoration of the full
support including the DGLAP region, is achieved by using the Gegenbauer
polynomials with their natural support. For instance, we may equally expand
(II.2) using instead

$\displaystyle
H_{valence}^{q}(x,\eta,t;\mu_{0}^{2})=\sum_{j=1}^{\infty}\frac{w\big{(}x|\frac{3}{2}\big{)}}{N_{j}(\frac{3}{2})}C_{j}^{3/2}(x)\bigg{[}\sum_{k=0}^{j}E_{jk}\big{(}\eta|\frac{3}{2}\big{)}\,\mathbb{F}^{q}_{k(valence)}(\eta,t;\mu_{0}^{2})\bigg{]}$
(II.16)

with the Gegenbauer overlaps

$\displaystyle
E_{jk}\big{(}\eta|\frac{3}{2}\big{)}=\int_{-1}^{1}dx\,\frac{w(x|\nu)}{\eta^{k}N_{k}(\nu)}\,C_{k}^{\nu}(x)C_{j}^{\nu}(\eta
x)$ (II.17)

and similarly for the other GPDs, which are more pertinent at lower
resolution(see Fig. 34 in Belitsky and Radyushkin (2005)).

##  III Gegenbauer (conformal) moments of GPDs at $\mu=\mu_{0}$ from t-channel
string exchange in AdS/QCD

Throughout, we will assume that factorization works, for most of the exclusive
processes (leptoproduction of photons and hadrons), with the GPDs capturing
the off-forward partonic content of a hadron. These matrix elements are
defined at some initial soft renormalization scale $\mu=\mu_{0}$, which
carries information on confinement and chiral symmetry breaking, and then
evolved using the RGE to higher $\mu$. We propose to determine these matrix
elements using holographic QCD, a dual string approach in AdS, in the double
limit of large $N_{c}$ and strong ′t Hooft coupling.

The proposal consits in identifying the twist-2 quark and gluon GPD conformal
moments listed above, at finite skewness $\eta\neq 0$ and an initial
renormalization scale $\mu_{0}\sim Q_{0}$, from various exclusive t-channel
open and closed string exchange electroproduction amplitudes, in soft-wall
holographic QCD, see Appendix J for their detailed derivation. The closed
strings (dual to spin-j glueballs in QCD) are Type IIB closed strings in soft-
wall AdS. The open strings (dual to spin-j vector and axial mesons) are
attached to bulk filling $N_{f}$ D9-branes in soft-wall AdS.

We match various exclusive electroproduction amplitudes in holographic QCD, to
various electroproduction amplitudes in QCD (written in terms of quark and
gluon GPDs based on the leading-order factorization theorems), see Figs. 1 \-
2. This is done by matching their respective large-$N_{c}$ dependence at fixed
′t Hooft coupling coupling $\lambda_{s}=g_{s}^{2}N_{c}$. In the holographic
side, we determine the large-$N_{c}$ dependence of the exclusive processes, by
using the bulk gravitational constant $2\kappa^{2}\sim{1}/{N_{c}^{2}}$, and
the bulk D9-brane coupling constant $g_{5}^{2}\sim{1}/{N_{c}}$.

We consider only tree-level Witten diagrams (Feynman diagrams in AdS), as
these are the only diagrams that contribute in the large-$N_{c}$ limit, see
Fig. 1 \- 2 right. Similarly, we consider only the leading-order exclusive
elecroproduction ampitudes in QCD, as these are the only QCD Feynman diagrams
that contribute in the large-$N_{c}$ limit, see Fig. 1 \- 2 left. This section
summarises the major results of this paper.

Electroproduction of mesons :
The leading order electroproduction amplitude for vector mesons (with quark DA
and gluon GPD) is of order $\frac{g_{s}^{2}}{N_{c}}$ as illustrated in Fig. 1
left. In the large-$N_{c}$ limit, it is of order
$\frac{g_{s}^{2}N_{c}}{N_{c}^{2}}=\frac{\lambda_{s}}{N_{c}^{2}}\,.$ This
matches the $\frac{1}{N_{c}^{2}}$ dependence for the holographic vector meson
electroproduction amplitude, using the corresponding tree-level Witten diagram
illustrated in Fig. 1 right. More specifically, the $N_{c}$ dependence of the
electroproduction of vector mesons written in terms of Gegenbauer moments,

(a)  (b)

Figure 1: Electro-production of a vector meson probing the
$\mathbf{gluon~{}GPD}$: (a) leading QCD contribution in the Regge limit; (b)
leading Witten diagram in the large-$N_{c}$ limit.

$\displaystyle{\cal A}^{LL}_{\gamma^{*}p\rightarrow
Vp}(s,t,Q_{0},\epsilon_{L},\epsilon^{\prime}_{L})$ $\displaystyle\approx$
$\displaystyle
e\times\frac{1}{4}\times\frac{1}{N_{c}^{2}}\times\frac{1}{Q_{0}}\times\lambda_{s}(\mu^{2})\times\left[\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\right]\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{j=2}^{\infty}\frac{1}{\xi^{j}}\times\frac{1}{N_{j-2}(\frac{5}{2})}\times\left[\int_{0}^{\xi}\frac{dx}{\xi}\,\left(1-\frac{x^{2}}{\xi^{2}}\right)\times
C_{j-2}^{5/2}\left(\frac{x}{\xi}\right)\right]\times\mathbb{F}^{g}_{j}(\xi,t;\mu_{0}^{2})\,,$

for even $j=2,4,...$, matches the holographic result

$\displaystyle{\cal A}^{LL}_{\gamma^{*}p\rightarrow
Vp}(s,t,Q_{0},\epsilon_{L},\epsilon^{\prime}_{L})$ $\displaystyle=$
$\displaystyle e\times\frac{1}{g_{5}}\times g_{5}\times
2\kappa^{2}\times\frac{1}{2}\times\frac{1}{Q_{0}}\times\left[f_{V}\right]\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
(III.19) $\displaystyle\times$
$\displaystyle\sum_{j=2}^{\infty}\,\frac{1}{\xi^{j}}\times\Bigg{[}\frac{1}{4^{j/2}}\frac{8}{(2+\Delta_{g}(j))\Delta_{g}(j)}\,\left(\frac{1}{2}\right)\,\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{V}^{2}}\right)^{j+2}\widetilde{\mathcal{I}}(Q_{0},\tilde{\kappa}_{V},j)$
$\displaystyle\times$
$\displaystyle\frac{1}{\Gamma(\Delta_{g}(j)-2)}\times\left[\mathcal{A}(j,\tau,\Delta_{g}(j),t)+\mathcal{D}_{\eta}(j,\tau,\Delta_{g}(j),t)\right]\Bigg{]}\,,$

for even $j=2,4,...$, with the details of the derivation given in Appendix J.
The 5-dimensional bulk gravitational coupling is
$2\kappa^{2}={8\pi^{2}}/{N_{c}^{2}}$, and $g_{5}\sim 1/\sqrt{N_{c}}$.

Electroproduction of pion pair :
The electroproduction amplitude for pair of mesons (with gluon DA and quark
GPDs) at leading-order is order of $\frac{g_{s}^{2}}{N_{c}}$. Therefore, in
the large-$N_{c}$ limit, it is of order
$\frac{g_{s}^{2}N_{c}}{N_{c}^{2}}=\frac{\lambda_{s}}{N_{c}^{2}}\,,$ (see Fig.
2 left). We find the same $\frac{1}{N_{c}^{2}}$ dependence for the holographic
meson pair electroproduction amplitude, using the corresponding tree-level
Witten diagram (see Fig. 2 right, and compare the $N_{c}$ dependence of the
electroproduction of pair mesons written in terms of Gegenbauer momnents
(III), and the holographic one (III) below, which are derived in detail in
Appendix J,

(a)  (b)

Figure 2: Electro-production of double pions probing the $\bf{non-
singlet~{}(valence)~{}vector~{}quark~{}GPDs}$ : (a) leading QCD contribution
in the Regge limit; (b) leading Witten diagram in the large-$N_{c}$ limit.

$\displaystyle{\cal A}^{LL}_{\gamma^{*}p\rightarrow\pi\pi
p}(s,t,Q_{0},\epsilon_{L};m_{\pi\pi})$ $\displaystyle\approx$ $\displaystyle
e\times\lambda_{s}(\mu_{0})\times\frac{4}{N_{c}^{2}}\times\frac{1}{Q_{0}}\times\left[\int_{0}^{1}dz\,\frac{\Phi^{g}(z)}{z(1-z)}\right]\times\frac{1}{2m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{j=1}^{\infty}\frac{1}{\xi^{j}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times\left[\int_{0}^{\xi}\frac{dx}{\xi}\,C_{j-1}^{3/2}\left(\frac{x}{\xi}\right)\right]\times\sum_{q}Q_{q}\,\mathbb{F}^{q}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})\,,$

for odd $j=1,3,..$, and

$\displaystyle{\cal A}^{LL}_{\gamma^{*}p\rightarrow\pi\pi
p}(s,t,Q_{0},\epsilon_{L};m_{\pi\pi})=e\times\frac{1}{g_{5}}\times g_{5}\times
2\kappa^{2}\times\frac{1}{4}\times\frac{1}{Q_{0}}\times\left[A_{gluon}^{\pi}(m_{\pi\pi})\right]\times\frac{1}{2m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{j=1}^{\infty}\,\frac{1}{\xi^{j}}\times\left[\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j}\times(-1)\times\mathcal{I}_{valence}^{vector}(j,Q_{0},\tilde{\kappa}_{N})\times\frac{1}{\Gamma(\Delta_{vv}(j)-2)}\times\mathcal{F}_{1}(j,\tau,\Delta_{vv}(j),t)\right]\,,$

for odd $j=1,3,....$, with the 5-dimensional bulk gravitational coupling
$2\kappa^{2}={8\pi^{2}}/{N_{c}^{2}}$, and $g_{5}\sim 1/\sqrt{N_{c}}$.

DVCS amplitude :
The DVCS amplitude in QCD at leading-order is order of $N_{c}^{0}$ (see Fig. 3
left), the NLO correction (due to the gluon GPD contribution) is order of
$g_{s}^{2}=\frac{g_{s}^{2}N_{c}}{N_{c}}=\frac{\lambda_{s}}{N_{c}}\,$, and can
be ignored in the large-$N_{c}$ and small ’t Hooft coupling limit. Using the
corresponding tree-level Witten diagrams (see Fig. 3 right), we find the same
$N_{c}^{0}$ dependence for the holographic DVCS amplitudes in the
large-$N_{c}$ limit (compare the $N_{c}$ dependence of the axial part of DVCS
written in terms of Gegenbauer moments (III) and the holographic DVCS (III)
below, the details of which are given in Appendix J,

(a)  (b)

Figure 3: Deeply virtual Compton scattering (DVCS) probing the
$\bf{singlet~{}axial~{}quark~{}GPDs}$ : (a) leading QCD contribution in the
Regge limit; (b) leading Witten diagram in the large-$N_{c}$ limit.

$\displaystyle{\cal A}^{LT}_{\gamma^{*}p\rightarrow\gamma
p}(s,t,Q_{0},\epsilon_{L},\epsilon^{\prime}_{T})$ $\displaystyle\approx$
$\displaystyle\\!\\!\\!\epsilon_{L\mu}\epsilon_{T\nu}^{\prime*}\times
N_{c}^{0}\times
i\varepsilon^{\mu\nu\rho\sigma}\tilde{q}_{\sigma}p_{\rho}\times\frac{1}{p^{+}}\times\bar{u}(p_{2})\gamma^{+}\gamma_{5}u(p_{1})$
$\displaystyle\times$
$\displaystyle\frac{4}{Q_{0}^{2}}\times\sum_{j=1}^{\infty}\,\frac{1}{\xi^{j-1}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times\left[\int_{0}^{\xi}\frac{dx}{\xi}\,C_{j-1}^{3/2}\left(\frac{x}{\xi}\right)\right]\times\sum_{q}Q_{q}^{2}\,\tilde{\mathbb{F}}^{q}_{j(singlet)}(\eta\sim\xi,t;\mu_{0}^{2})\,,$

for odd $j=1,3,...$, and

$\displaystyle{\cal A}^{LT}_{\gamma^{*}p\rightarrow\gamma
p}(s,t,Q_{0},\epsilon_{L},\epsilon^{\prime}_{T})$ $\displaystyle=$
$\displaystyle\frac{1}{g_{5}}\times\frac{1}{g_{5}}\times g_{5}\times
g_{5}^{3}\kappa_{cs}\times\frac{1}{\tilde{\kappa}_{N}^{2}}\epsilon_{L\mu}\epsilon_{T\nu}^{\prime*}\times
i\varepsilon^{\mu\nu\rho\sigma}\tilde{q}_{\sigma}\times\bar{u}(p_{2})\gamma_{\rho}\gamma_{5}u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{j=1}^{\infty}\,\frac{1}{\xi^{j-1}}\times\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j-1}\times\mathcal{I}_{singlet}^{axial}(j,Q_{0},\tilde{\kappa}_{N})\times\frac{1}{\Gamma(\Delta_{sa}(j)-2)}\times\mathcal{F}_{A}(j,\tau,\Delta_{sa}(j),t)\,,$

for odd $j=1,3,...$ with $g_{5}\sim 1/\sqrt{N_{c}}$ and the Chern-Simons
coupling $\kappa_{cs}\sim N_{c}$.

Electroproduction of neutral pion :
Finally, the electroproduction amplitude for mesons (with quark DAs and quark
GPDs) at leading-order is order of $\frac{C_{F}g_{s}^{2}}{N_{c}}$ with
$C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$ (see Fig. 4 left). Therefore, the
electroproduction amplitude for mesons (with quark DAs and quark GPDs) at
leading-order is of order
$\frac{g_{s}^{2}N_{c}}{2N_{c}}=\frac{\lambda_{s}}{2N_{c}}\,,$ using
$C_{F}\sim\frac{N_{c}}{2}$ in the large-$N_{c}$ limit. We find the same
$\frac{1}{N_{c}}$ dependence for the holographic meson electroproduction
amplitude using the corresponding tree-level Witten diagram (see Fig. 4 right,
and compare the $N_{c}$ dependence of the electroproduction of $\pi^{0}$
written in terms of Gegenbauer momnents (III) and the holographic one (III)
below, with more details in Appendix J,

(a)  (b)

Figure 4: Electro-production of neutral pion probing the $\bf{non-
singlet~{}(valence)~{}axial~{}quark~{}GPDs}$ : (a) leading QCD contribution in
the Regge limit; (b) leading Witten diagram in the large-$N_{c}$ limit.

$\displaystyle{\cal
A}^{LL}_{\gamma^{*}p\rightarrow\pi^{0}p}(s,t,Q_{0},\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle\,\approx\,$ $\displaystyle e\times{1\over
2}\times\frac{1}{N_{c}}\times\lambda_{s}(\mu_{0})\times{1\over{Q_{0}}}\times\left[\,\int_{0}^{1}dz{{\Phi_{\pi}(z)}\over
z}\right]\times\frac{1}{p^{+}}\times
u(p_{2})\gamma^{+}\,\gamma_{5}u(p_{1})\times{-1\over{\sqrt{2}}}$
$\displaystyle\times$
$\displaystyle\sum_{j=2}^{\infty}\frac{1}{\xi^{j}}\times\frac{1}{N_{j-1}(\frac{3}{2})}\times\left[\int_{0}^{\xi}\frac{dx}{\xi}\,\frac{x}{\xi}\,C_{j-1}^{3/2}\left(\frac{x}{\xi}\right)\right]$
$\displaystyle\times$ $\displaystyle\left(Q_{u}\
\widetilde{\mathbb{F}}^{u}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})\,-\,Q_{d}\
\widetilde{\mathbb{F}}^{d}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})\right)\,,$

for even $j=2,4,...$, and

$\displaystyle{\cal
A}^{LL}_{\gamma^{*}p\rightarrow\pi^{0}p}(s,t,Q_{0},\epsilon_{L},\epsilon_{L}^{\prime})=e\times\frac{1}{g_{5}}\times
g_{5}^{2}\times
g_{5}\times\left[f_{\pi}\right]\times\frac{1}{2}\times\frac{1}{Q_{0}}\times\frac{1}{p^{+}}\times\bar{u}(p_{2})\gamma^{+}\,\gamma_{5}u(p_{1})\times\frac{-1}{\sqrt{2}}$
$\displaystyle\times\sum_{j=2}^{\infty}\,\frac{1}{\xi^{j}}\times\left[\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j}\,\left(\frac{\tilde{\kappa}_{N}}{m_{\pi}}\right)^{2}\,(-1)\,\mathcal{I}_{valence}^{axial}(j,Q_{0},\tilde{\kappa}_{N})\,\frac{1}{\Gamma(\Delta_{va}(j)-2)}\,\mathcal{F}_{A}(j,\tau,\Delta_{va}(j),t)\right]\times\mathcal{O}\left(1/N_{c}\right)\,,$

for even $j=2,4,...$, with $g_{5}\sim 1/\sqrt{N_{c}}$. Note that the
holographic amplitude (III) is $1/N_{c}$ suppressed since there is no direct
coupling between axial-axial-vector mesons at tree level.

The leading twist-2 electroproduction amplitudes for vector mesons (III),
photons (III), neutral pions (III) and double neutral pions (III), are to be
identified with their holographic counterparts (III.19), (III), (III) and
(III) respectively. This leads to the identification of the pertinent
Gegenbauer or conformal moments of the GPDs at strong coupling and large
$N_{c}$, as we now detail.

###  III.1 Gegenbauer (conformal) moments of gluon GPDs at $\mu=\mu_{0}$ from
t-channel closed string exchange, in soft-wall AdS/QCD

A comparison of (III) and (III.19), allows the extraction of the Gegenbauer
(conformal) moments of the gluon GPDs, in terms of the t-channel closed string
exchange in the soft-wall AdS background geometry (see Fig. 1). More
specifically in the double limit of large $N_{c}$ and strong ′t Hooft
coupling, the moments are given by (J.225)

$\displaystyle\mathbb{F}^{g}_{j}(\eta\sim\xi,t;\mu_{0}^{2})=\frac{\mathcal{N}_{GgbrGluon}(j,\mu_{0})}{\mathcal{N}_{GgbrGluon}(2,\mu_{0})}\times\frac{1}{\Gamma(\Delta_{g}(j)-2)}\times\left[\mathcal{A}(j,\tau,\Delta_{g}(j),t)+\mathcal{D}_{\eta}(j,\tau,\Delta_{g}(j),t)\right]\,,$

with $\eta\sim\xi$ following from $\sqrt{s}\gg-t$ (see Appendix A). The
anomalous dimension of the spin-j conformal gluon operator at $\mu=\mu_{0}\sim
Q_{0}$, and even $j=2,4,...$ is

$\Delta_{g}(j)=2+\sqrt{2\sqrt{\lambda}(j-j_{0g})}\,,$

with $j_{0g}=2-2/\sqrt{\lambda}$, and the ’t Hooft coupling of QCD at
$\mu=\mu_{0}$ given by

$\lambda\equiv\lambda_{s}(\mu_{0})=4\pi\alpha_{s}(\mu_{0})N_{c}\rightarrow
11.243\,.$

Below we suggest that the matching scale $\mu_{0}=1.808$ GeV (see section V).
We have also defined the spin-$j$ and scale $\mu_{0}$ dependent normalization
factor $\mathcal{N}_{GgbrGluon}(j,\mu_{0})$, of the Gegenbauer (conformal)
moments of the gluon GPD at $\mu=\mu_{0}$ as

$\displaystyle\mathcal{N}_{GgbrGluon}(j,\mu_{0})\equiv\frac{N_{j-2}(\frac{5}{2})}{I_{j-2}^{g}}\times\frac{1}{\lambda_{s}(\mu_{0})}\times\Bigg{[}\frac{1}{4^{j/2}}\frac{8}{(2+\Delta_{g}(j))\Delta_{g}(j)}\left(\frac{1}{2}\right)\,\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{V}^{2}}\right)^{j+2}\widetilde{\mathcal{I}}(Q_{0},\tilde{\kappa}_{V},j)\Bigg{]}_{Q_{0}\sim\mu_{0}}\,.$

In addition, we have set the Mandelstam variable $t=-K^{2}$, and defined the
$\eta$-independent gluonic spin-j (with even $j=2,4,...$) form factors of the
proton (with twist $\tau$) as

$\displaystyle\mathcal{A}(j,\tau,\Delta_{g}(j),-t=K^{2})=$
$\displaystyle\frac{2^{1-\Delta_{g}(j)}}{\Gamma(\tau)}\Bigg{(}(\tau-1)\Gamma\left(\frac{j-2}{2}+\tau-\frac{\Delta_{g}(j)}{2}+1\right)\Gamma\left(\frac{j-2}{2}+\frac{\Delta_{g}(j)}{2}+\tau-1\right)$
$\displaystyle\times$
$\,{}_{2}\tilde{F}_{1}\left(\frac{a_{K}}{2}-\frac{\Delta_{g}(j)}{2}+2,\frac{j-2}{2}+\tau-\frac{\Delta_{g}(j)}{2}+1;\frac{a_{K}}{2}+\frac{j-2}{2}+\tau+1;-1\right)$
$\displaystyle+\Gamma\left(\frac{j-2}{2}+\tau-\frac{\Delta_{g}(j)}{2}+2\right)\Gamma\left(\frac{j-2}{2}+\frac{\Delta_{g}(j)}{2}+\tau\right)$
$\displaystyle\times$
${}_{2}\tilde{F}_{1}\left(\frac{a_{K}}{2}-\frac{\Delta_{g}(j)}{2}+2,\frac{j-2}{2}+\tau-\frac{\Delta_{g}(j)}{2}+2;\frac{a_{K}}{2}+\frac{j-2}{2}+\tau+2;-1\right)\Bigg{)}\,,$

where ${}_{2}\tilde{F}_{1}$ is the regularized hypergeometric function, with
$a_{K}=K^{2}/4\tilde{\kappa}_{T}^{2}$. The skewness or $\eta$-dependent spin-j
$\mathcal{D}_{\eta}$-terms are given by

$\displaystyle\mathcal{D}_{\eta}(j,\tau,\Delta_{g}(j),-t=K^{2})$
$\displaystyle=$
$\displaystyle\left(\hat{d}_{j}(\eta,-K^{2})-1\right)\times\left[\mathcal{A}(j,\tau,\Delta_{g}(j),K,\tilde{\kappa}_{T})-\mathcal{A}_{S}(j,\tau,\Delta_{g}(j),K,\tilde{\kappa}_{S})\right]\,,$

where

$\displaystyle\mathcal{A}_{S}(j,\tau,\Delta_{g}(j),K,\tilde{\kappa}_{S})$
$\displaystyle\equiv$
$\displaystyle\mathcal{A}(j,\tau,\Delta_{g}(j),K,\tilde{\kappa}_{T}\rightarrow\tilde{\kappa}_{S})\,.$
(III.30)

and

$\displaystyle\hat{d}_{j}(\eta,-K^{2})=\,_{2}F_{1}\left(-\frac{j}{2},\frac{1-j}{2};\frac{1}{2}-j;\frac{4m_{N}^{2}}{K^{2}}\times\eta^{2}\right)\,.$
(III.31)

Note that the A-form factor for the spin-2 exchange is given by

$A(t)=\mathcal{A}(j=2,\tau,\Delta_{g}(j=2)=4,-t=K^{2})\,,$

and the D-form factor for the same spin-2 exchange (the spin-2 D-term) is
given by

$\eta^{2}D(t)=\mathcal{D}_{\eta}(j=2,\tau,\Delta_{g}(j=2)=4,-t=K^{2})\,.$

Both form factors play an important role in the holographic electroproduction
of heavy mesons near treshold Mamo and Zahed (2020, 2021a, 2022) (and
references therein).

###  III.2 Gegenbauer (conformal) moments of singlet quark GPDs at
$\mu=\mu_{0}$ from t-channel open string exchange, in soft-wall AdS/QCD

Since at tree-level, there is no direct holographic coupling between three
spin-1 (or two spin-1 and a spin-j) vector bulk gauge fields in soft-wall AdS,
we can set the Gegenbauer (conformal) moments of the singlet vector quark GPDs
to zero,

$\displaystyle\sum_{q}Q_{q}^{2}\times\mathbb{F}^{q}_{j(singlet)}(\eta\sim\xi,t;\mu_{0}^{2})$
$\displaystyle\approx$
$\displaystyle\frac{N_{j-1}(\frac{3}{2})}{I^{singlet}_{j-1}}\times\Bigg{[}\left(\frac{Q_{0}^{2}}{4\tilde{\kappa}_{N}^{2}}\right)^{j}\times\frac{1}{4}\mathcal{I}_{singlet}^{vector}(j,Q_{0},\tilde{\kappa}_{N})\times\frac{1}{\Gamma(\Delta_{sv}(j)-2)}$
$\displaystyle\times$
$\displaystyle\left[\mathcal{A}_{q}(j,\tau,\Delta_{sv}(j),K)+\mathcal{D}_{q\eta}(j,\tau,\Delta_{sv}(j),K)\right]\Bigg{]}_{Q_{0}\sim\mu_{0}}\times\mathcal{O}(1/N_{c})\longrightarrow
0\,,$

at the initial scale $\mu=\mu_{0}$, and in leading order in $1/N_{c}$. As a
result, our singlet vector quark GPDs at $\mu\neq\mu_{0}$ will be generated by
the initial gluon GPD at $\mu=\mu_{0}$, through evolution.

###  III.3 Gegenbauer (conformal) moments of non-singlet (valence) quark GPDs
at $\mu=\mu_{0}$ from t-channel open string exchange, in soft-wall AdS/QCD

To identify the Gegenbauer (conformal) moments of the non-singlet (valence)
quark GPDs, we match (III) to (III). In leading order in $1/N_{c}$, the non-
singlet moments at the scale $\mu=\mu_{0}$, are given by the t-channel open
string exchange in the soft-wall AdS background geometry, as illustrated in
Fig. 2. More specifically, we have

$\displaystyle\mathbb{F}^{u}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})$
$\displaystyle=$ $\displaystyle
2\times\frac{\mathcal{N}_{GgbrValence}(j,\mu_{0})}{\mathcal{N}_{GgbrValence}(1,\mu_{0})}\times\frac{1}{\Gamma(\Delta_{vv}(j)-2)}\times\mathcal{F}_{1}(j,\tau,\Delta_{vv}(j),t)\,,$
(III.33) $\displaystyle\mathbb{F}^{d}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})$
$\displaystyle=$
$\displaystyle\frac{\mathcal{N}_{GgbrValence}(j,\mu_{0})}{\mathcal{N}_{GgbrValence}(1,\mu_{0})}\times\frac{1}{\Gamma(\Delta_{vv}(j)-2)}\times\mathcal{F}_{1}(j,\tau,\Delta_{vv}(j),t)\,,$
(III.34)

following from (J.3.3). The anomalous dimension of the spin-j conformal
valence quark operator at $\mu=\mu_{0}\sim Q_{0}$, and odd $j=1,3,...$ is
given by

$\Delta_{vv}(j)=2+\sqrt{\sqrt{\lambda}(j-j_{0q})}\,,$

with $j_{0q}=1-1/\sqrt{\lambda}$, and the ’t Hooft coupling at $\mu=\mu_{0}$
fixed as

$\lambda\equiv\lambda_{s}(\mu_{0})=4\pi\alpha_{s}(\mu_{0})N_{c}=11.243\,.$

We have also defined the spin-$j$ and $\mu_{0}$ dependent normalization factor
$\mathcal{N}_{GgbrValence}(j,\mu_{0})$ of the Gegenbauer (conformal) moments
of the valence quark GPD at $\mu=\mu_{0}$ as

$\displaystyle\mathcal{N}_{GgbrValence}(j,\mu_{0})\equiv\frac{1}{\lambda_{s}(\mu_{0})}\times\frac{N_{j-1}(\frac{3}{2})}{I_{j-1}^{valence}}\left[\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j}\times\frac{-\pi^{2}}{2}\mathcal{I}_{valence}^{vector}(j,Q_{0},\tilde{\kappa}_{N})\right]_{Q_{0}\sim\mu_{0}}\,.$

The $t$-dependent valence quark spin-j (with odd $j=1,3,...$) form factors of
the proton with twist $\tau=d-s$, is

$\displaystyle\mathcal{F}_{1}(j,\tau,\Delta_{vv}(j),-t=K^{2})$
$\displaystyle=$
$\displaystyle\frac{(4a_{K}(\tau-1)+(\Delta_{vv}(j)+j+2\tau-4)(-\Delta_{vv}(j)+j+4\tau-2))}{4(\Delta_{vv}(j)-1)\Gamma(\tau)}$
$\displaystyle\times$
$\displaystyle\frac{\Gamma(a_{K}+\Delta_{vv}(j)-2)\Gamma\left(\frac{j-\Delta_{vv}(j)}{2}+\tau\right)\Gamma\left(\frac{1}{2}(j+\Delta_{vv}(j)-4)+\tau\right)}{\Gamma\left(a_{K}+\frac{1}{2}(\Delta_{vv}(j)+j-2)+\tau\right)}$

with $a_{K}=K^{2}/4\tilde{\kappa}_{N}^{2}$. Note that the Dirac
electromagnetic form factor (ignoring the Pauli contribution) for the spin-1
exchange is given by

$F_{1}(t)=\mathcal{F}_{1}(j=1,\tau,\Delta_{vv}(j=1)=3,-t=K^{2})\,.$

###  III.4 Gegenbauer (conformal) moments of singlet axial quark GPDs at
$\mu=\mu_{0}$ from t-channel open string exchange in soft-wall AdS/QCD

The Gegenbauer (conformal) moments of the singlet axial quark GPDs at
$\mu=\mu_{0}$, are identified with the t-channel open string exchange, in the
soft-wall AdS background geometry as illustrated in Fig. 3. The matching of
(III) to (III) in leading order in $1/N_{c}$, yields

$\displaystyle\sum_{q}Q_{q}^{2}\,\tilde{\mathbb{F}}^{q}_{j(singlet)}(\eta\sim\xi,t;\mu_{0}^{2})$
$\displaystyle=$
$\displaystyle\frac{\mathcal{N}_{GgbrSingletAxial}(j,\mu_{0})}{\mathcal{N}_{GgbrSingletAxial}(1,\mu_{0})}\times\frac{1}{\Gamma(\Delta_{sa}(j)-2)}\times\mathcal{F}_{A}(j,\tau,\Delta_{sa}(j),t)\Bigg{]}\,,$

using the result (J.2.3). The anomalous dimension of the spin-j conformal
singlet axial quark operator at $\mu=\mu_{0}\sim Q_{0}$, and odd $j=1,3,...$
is given by

$\Delta_{sa}(j)=2+\sqrt{\sqrt{\lambda}(j-j_{0})}\,,$

with $j_{0}=1-1/\sqrt{\lambda}$. We have also defined the spin-$j$ and
$\mu_{0}$ dependent normalization factor
$\mathcal{N}_{GgbrSingletAxial}(j,\mu_{0})$ of the Gegenbauer (conformal)
moments of the singlet axial quark GPD at $\mu=\mu_{0}$ as

$\displaystyle\mathcal{N}_{GgbrSingletAxial}(j,\mu_{0})\equiv\frac{N_{j-1}(\frac{3}{2})}{\tilde{I}^{singlet}_{j-1}}\times\Bigg{[}\frac{g_{5}^{2}\kappa_{cs}}{2}\,\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j}\,\mathcal{I}_{singlet}^{axial}(j,Q_{0},\tilde{\kappa}_{N})\Bigg{]}_{Q_{0}\sim\mu_{0}}\,.$

The $t$-dependent singlet axial quark spin-j (with odd $j=1,3,...$) form
factors of the proton is

$\displaystyle\mathcal{F}_{A}(j,\tau,\Delta_{sa}(j),-t=K^{2})$
$\displaystyle=$
$\displaystyle\frac{(4a_{K}(\tau-1)-(\Delta_{sa}(j)+j+2\tau-4)(-\Delta_{sa}(j)+j+4\tau-2))}{4(\Delta_{sa}(j)-1)\Gamma(\tau)}$
$\displaystyle\times$
$\displaystyle\frac{\Gamma(a_{K}+\Delta_{sa}(j)-2)\Gamma\left(\frac{j-\Delta_{sa}(j)}{2}+\tau\right)\Gamma\left(\frac{1}{2}(j+\Delta_{sa}(j)-4)+\tau\right)}{\Gamma\left(a_{K}+\frac{1}{2}(\Delta_{sa}(j)+j-2)+\tau\right)}$

with $a_{K}=K^{2}/4\tilde{\kappa}_{N}^{2}$. Note that the axial
electromagnetic form factor for the spin-1 exchange is given by

$F_{A}(t)=\mathcal{F}_{A}(j=1,\tau=3,\Delta_{sa}(j=1)=3,-t=K^{2})\,.$

###  III.5 Gegenbauer (conformal) moments of non-singlet (valence) axial quark
GPDs at $\mu=\mu_{0}$ from t-channel open string exchange in soft-wall AdS/QCD

Similarly, the Gegenbauer (conformal) moments of the valence axial GPDs at
$\mu=\mu_{0}$, follow by matching (III) to (III) in leading order in
$1/N_{c}$, as illustrated in Fig. 4. The holographic t-channel open string
exchange in the soft-wall AdS background geometry gives

$\displaystyle\left(Q_{u}\
\widetilde{\mathbb{F}}^{u}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})\,-\,Q_{d}\
\widetilde{\mathbb{F}}^{d}_{j(valence)}(\eta\sim\xi,t;\mu_{0}^{2})\right)=$
$\displaystyle\frac{\mathcal{N}_{GgbrValenceAxial}(j,\mu_{0})}{\mathcal{N}_{GgbrValenceAxial}(2,\mu_{0})}\times\frac{1}{\Gamma(\Delta_{va}(j)-2)}\left[\mathcal{F}_{A}(j,\tau,\Delta_{va}(j),t)+\mathcal{D}_{A\eta}(j,\tau,\Delta_{va}(j),t)\right]\times\mathcal{O}(1/N_{c})\longrightarrow
0\,,$

following from (J.4.2). Therefore, the non-singlet (valence) axial quark GPDs
vanish in the large-$N_{c}$ limit at $\mu=\mu_{0}$.

Here, we have defined the anomalous dimension of the spin-j conformal valence
axial operator at $\mu=\mu_{0}\sim Q_{0}$, and even $j=2,4,...$ as

$\Delta_{va}(j)=\Delta_{vv}(j)=2+\sqrt{\sqrt{\lambda}(j-j_{0q})}\,.$

We have also defined the spin-$j$ and $\mu_{0}$ dependent normalization factor
$\mathcal{N}_{GgbrValenceAxial}(j,\mu_{0})$ of the Gegenbauer (conformal)
moments of the valence axial GPDs at $\mu=\mu_{0}$ as

$\displaystyle\mathcal{N}_{GgbrValenceAxial}(j,\mu_{0})\equiv\frac{N_{j-1}(\frac{3}{2})}{\tilde{I}_{j-1}}\times\frac{1}{\lambda_{s}(\mu_{0})}\times\Bigg{[}\left(\frac{Q_{0}^{2}}{2\tilde{\kappa}_{N}^{2}}\right)^{j}\times\left(\frac{\tilde{\kappa}_{N}}{m_{\pi}}\right)^{2}\times
g_{5}^{2}N_{c}\times(-1)\mathcal{I}_{valence}^{axial}(j,Q_{0},\tilde{\kappa}_{N})\Bigg{]}\,.$

The valence axial spin-j form factors of the proton with even $j=2,4,...$, are

$\displaystyle\mathcal{F}_{A}(j,\tau,\Delta_{va}(j),-t=K^{2})$
$\displaystyle=$
$\displaystyle\frac{(4a_{K}(\tau-1)-(\Delta_{sa}(j)+j+2\tau-4)(-\Delta_{sa}(j)+j+4\tau-2))}{4(\Delta_{sa}(j)-1)\Gamma(\tau)}$
$\displaystyle\times$
$\displaystyle\frac{\Gamma(a_{K}+\Delta_{sa}(j)-2)\Gamma\left(\frac{j-\Delta_{sa}(j)}{2}+\tau\right)\Gamma\left(\frac{1}{2}(j+\Delta_{sa}(j)-4)+\tau\right)}{\Gamma\left(a_{K}+\frac{1}{2}(\Delta_{sa}(j)+j-2)+\tau\right)}\,,$

with $a_{K}=K^{2}/4\tilde{\kappa}_{N}^{2}$. The $\eta$-dependent spin-j
$\mathcal{D}_{A\eta}$-terms are also defined by the integral (J.274)

$\displaystyle\mathcal{D}_{A\eta}(j,\tau,\Delta_{sv}(j),-t=K^{2})=$
$\displaystyle\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}\,z^{1+2(j-1)}\,\big{(}\psi_{R}^{2}(z)-\psi_{L}^{2}(z)\big{)}\,z^{-(j-1)}$
(III.43)
$\displaystyle\times\sum_{n=0}^{\infty}\,\left(\hat{d}_{j}(\eta,m_{n}^{2})-1\right)\times\frac{g_{5}\mathcal{F}_{n}(j)\phi_{n}(j,z)}{K^{2}+m_{n}^{2}(j)}$
$\displaystyle\times\frac{1}{\Delta_{sv}(j)-1}\Gamma\left(a_{K}+\Delta_{sv}(j)-2\right)\times\tilde{\kappa}_{V}^{j-1+\Delta_{sv}(j)+1}\,.$

Note that the valence axial form factor for the spin-2 axial meson exchange is
given by

$\mathcal{F}_{2A}(t)=\mathcal{F}_{A}(j=2,\tau,\Delta_{va}(j=2),-t=K^{2})\,,$

and the valence axial D-form factor for the same spin-2 axial exchange (the
valence axial spin-2 D-term) is given by

$\eta^{2}D_{A}(t)=\mathcal{D}_{A\eta}(j=2,\tau,\Delta_{sv}(j=2),-t=K^{2})\,.$

Here $\hat{d}_{j}(\eta,m_{n}^{2})$, for even $j=2,4,...$ is a polynomial of
the skewness parameter $\eta$ of degree $j$ which can be written explicitly in
terms of the hypergeometric function ${}_{2}F_{1}(a,b,c,d)$ as

$\displaystyle\hat{d}_{j}(\eta,m_{n}^{2})=\,_{2}F_{1}\left(-\frac{j}{2},\frac{1-j}{2};\frac{1}{2}-j;-\frac{4m_{N}^{2}}{m_{n}^{2}(j)}\times\eta^{2}\right)\,,$
(III.44)

with the mass of spin-j (even $j=2,4,..$) axial meson resonances
$m_{n}^{2}(j)=4\tilde{\kappa}_{N}^{2}(n+1+(\Delta_{sv}(j)-3))$.

##  IV Renormalization Group evolution of quark and gluon GPDs

In an inclusive DIS process, the partonic content of a hadron changes with the
$Q^{2}$ of the probing virtual photon. In a scale free theory such as massless
QCD, this translates to scaling violations, logarithmic deviations to free
partons. DIS structure functions are convolution of perturbatively determined
coefficient functions with fixed $Q^{2}/\mu^{2}$ times pertinent PDFs at a
resolution $\mu$, thanks to factorization. The evolution and mixing of the
PDFs is fixed by DGLAP, which reflects on the RG independence of the original
structure functions.

Unlike the PDFs which are diagonal matrix elements of pertinent gauge
invariant bilocals (single parton distributions), GPDs are off-diagonal
(double parton distributions). Their evolution follows that of exclusive
kernels as described by ERBL. The ERBL evolution is amenable to a DGLAP
evolution, when recast in terms of moments of hadronic DAs for instance. In
this spirit, the evolution of the GPDs can be turned to a DGLAP type
evolution, when expressed in terms of Gegenbauer (conformal) moments, as
detailed in Belitsky and Radyushkin (2005). We now briefly recall their
central RG evolution results, as they apply to the Gegenbauer moments
extracted from the holographic construction at the initial resolution
$\mu_{0}$. They will be used in the analysis of the electroproduction of
vector mesons next.

###  IV.1 RG evolution of non-singlet (valence) quark GPDs

The leading-order RG evolution of Gegenbauer moments of valence (non-singlet)
vector quark GPDs is given by (see Eq.4.244 in Belitsky and Radyushkin (2005))

$\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu^{2})=\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu_{0}^{2})\left(\frac{\alpha_{s}(\mu_{0}^{2})}{\alpha_{s}(\mu^{2})}\right)^{\gamma_{(0)j-1}^{qq;\text{V};\text{NS}}/\beta_{0}}$
(IV.45)

for odd $j=1,3,...$, where the vector non-singlet anomalous dimensions are

$\displaystyle\gamma^{qq;\text{V}}_{(0)j}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-C_{F}\left(-4\psi(j+2)+4\psi(1)+\frac{2}{(j+1)(j+2)}+3\right)\,,$
(IV.46)

The Euler $\psi$-function is given in terms of the harmonic numbers

$\psi(n)=\frac{d}{dn}\ln{\mit\Gamma}(n)=\psi(1)+\sum_{k=1}^{n-1}\frac{1}{k}\,.$
(IV.47)

with the Euler constant $\gamma_{\scriptscriptstyle\rm E}=-\psi(1)\approx
0.577216$.

The leading and one-loop RG evolution equation for the valence (non-singlet)
vector quark GPDs is given by (see Eq.4.240 in Belitsky and Radyushkin (2005))

$\frac{d}{d\ln\mu}H_{valence}^{q}(x,\eta,t;\mu^{2})=-\frac{\alpha_{s}(\mu^{2})}{2\pi}\int_{-1}^{1}\frac{dy}{\eta}k^{qq}_{(0)}\left(\frac{x}{\eta},\frac{y}{\eta}\right)H_{valence}^{q}(y,\eta,t;\mu^{2})+{\cal
O}(\alpha_{s}^{2})\,,$ (IV.48)

$k^{qq}_{(0)}\left({x}/{\eta},{y}/{\eta}\right)$ is the leading-order quark-
quark generalized exclusive kernel (see Eq.4.38 in Belitsky and Radyushkin
(2005) for its explicit form), with Gegenbauer polynomials $C_{j-1}^{3/2}(x)$
as eigenfunctions,

$\int_{-1}^{1}\frac{dx}{\eta}\,C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)k^{qq}_{(0)}\left(\frac{x}{\eta},\frac{y}{\eta}\right)=\gamma_{(0)j-1}^{qq;\text{V}}C_{j-1}^{3/2}\left(\frac{y}{\eta}\right)\,.$
(IV.49)

Conversely, since the exclusive kernel is diagonal in the Gegenbauer basis
which is complete in $L^{2}[-1,1]$, the explicit solution to (IV.48) is (see
Eq.4.242 in Belitsky and Radyushkin (2005))

$H_{valence}^{q}(x,\eta,t;\mu^{2})=\frac{1}{\eta}\sum_{j=1}^{\infty}\frac{w\big{(}\frac{x}{\eta}|\frac{3}{2}\big{)}}{\eta^{j-1}\,N_{j-1}(\frac{3}{2})}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\mathbb{F}^{q}_{j(valence)}(\eta,t;\mu^{2})\,,$
(IV.50)

for odd $j=1,3,...$. Here the weight and normalization factors are

$w(x|\nu)=(1-x^{2})^{\nu-1/2}\,,\qquad
N_{j-1}(\nu)=2^{1-2\nu}\frac{{\mit\Gamma}^{2}(1/2){\mit\Gamma}(2\nu+(j-1))}{{\mit\Gamma}^{2}(\nu){\mit\Gamma}((j-1)+1)(\nu+(j-1))}\,.$

###  IV.2 RG evolution of singlet quark and gluon GPDs

The singlet quark and gluon GPDs mix under evolution. Also as we noted
earlier, the initial holographic input for the singlet quark evolution
vanishes in leading order in $1/N_{c}$, simplifying the initial data. With
this in mind, the combined singlet vector quark and gluon GPDs

$\displaystyle\mathbf{F}(x,\eta,t;\mu^{2})\equiv\left({\sum_{q}F_{singlet}^{q}(x,\eta,t;\mu^{2})\atop
F^{g}(x,\eta,t;\mu^{2})}\right)\,,$ (IV.51)

evolve in leading order as (see Eq.4.264 in Belitsky and Radyushkin (2005))

$\mathbf{F}(x,\eta,t;\mu^{2})=\frac{1}{\eta}\sum_{j=2}^{\infty}{\bf
E}_{j}\left(\frac{x}{\eta}\right)\mathbf{F}_{j}(\eta,t;\mu^{2})\,,$ (IV.52)

for even $j=2,4,...$, with the matrix valued coeffcients

${\mbox{\bf E}}_{j}(x)=\left(\begin{array}[]{cc}E_{j-1}(x|\frac{3}{2})&0\\\
0&E_{j-2}(x|\frac{5}{2})\end{array}\right)\,,\qquad\qquad
E_{j}(x|\nu)=\frac{w\left(x|\nu\right)}{\eta^{j}N_{j}(\nu)}C_{j}^{\nu}(x)\,.$
(IV.53)

Note that we do not need to re-expand the GPDs using additional orthogonal
polynomials, since our input GPDs are non-zero only for $-\eta<x<\eta$.
(IV.52) is invertible

$\mathbf{F}_{j}(\eta,t;\mu^{2})=\int_{0}^{1}dx\,\left(\begin{array}[]{cc}\eta^{j-1}\,C_{j-1}^{3/2}\left(x/\eta\right)&0\\\
0&\eta^{j-2}\,C_{j-2}^{5/2}\left(x/\eta\right)\end{array}\right)\,\mathbf{F}(x,\eta,t;\mu^{2})\,,$
(IV.54)

thanks to the ortogonality and completeness of the Gegenbauer polynomials.

The details of the evolution of (IV.54) are given in Appendix K.3, with the
result

$\displaystyle\mathbf{F}_{j}(\eta,t;\mu^{2})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\sum_{q}\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu^{2})\\\
\mathbb{F}^{g}_{j}(\eta,t;\mu^{2})\end{array}\right)\,,$ (IV.57)

where

$\displaystyle\sum_{q}\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu^{2})$ (IV.58)
$\displaystyle=$
$\displaystyle\frac{2\left(\gamma^{qq}_{(0)j-1}-\gamma^{-}_{j-1}\right)}{\gamma^{+}_{j-1}-\gamma^{-}_{j-1}}\times
F^{+}_{j}(\eta,t;\mu_{0}^{2})\times\left(\frac{\alpha_{s}(\mu^{2}_{0})}{\alpha_{s}(\mu^{2})}\right)^{\gamma^{+}_{j-1}/\beta_{0}}$
$\displaystyle+$
$\displaystyle\frac{-2\left(\gamma^{qq}_{(0)j-1}-\gamma^{+}_{j-1}\right)}{\gamma^{+}_{j-1}-\gamma^{-}_{j-1}}\times
F^{-}_{j}(\eta,t;\mu_{0}^{2})\times\left(\frac{\alpha_{s}(\mu^{2}_{0})}{\alpha_{s}(\mu^{2})}\right)^{\gamma^{-}_{j-1}/\beta_{0}}\,,$
$\displaystyle\mathbb{F}^{g}_{j}(\eta,t;\mu^{2})$ (IV.59) $\displaystyle=$
$\displaystyle\frac{2\left(\gamma^{gq}_{(0)j-1}\right)}{\gamma^{+}_{j-1}-\gamma^{-}_{j-1}}\times
F^{+}_{j}(\eta,t;\mu_{0}^{2})\times\left(\frac{\alpha_{s}(\mu^{2}_{0})}{\alpha_{s}(\mu^{2})}\right)^{\gamma^{+}_{j-1}/\beta_{0}}$
$\displaystyle+$
$\displaystyle\frac{-2\left(\gamma^{gq}_{(0)j-1}\right)}{\gamma^{+}_{j-1}-\gamma^{-}_{j-1}}\times
F^{-}_{j}(\eta,t;\mu_{0}^{2})\times\left(\frac{\alpha_{s}(\mu^{2}_{0})}{\alpha_{s}(\mu^{2})}\right)^{\gamma^{-}_{j-1}/\beta_{0}}\,,$

for even $j=2,4,...$. More explicitly, in terms of the evolved conformal
(Gegenbauer) moments (IV.58-IV.59), the evolved singlet quark and gluon GPDs
are

$\displaystyle\sum_{q}F_{singlet}^{q}(x,\eta,t;\mu^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\left(\frac{x}{\eta}|\frac{3}{2}\right)}{\eta^{j-1}N_{j-1}\left(\frac{3}{2}\right)}C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\sum_{q}\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu^{2})\,,$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-1}}\times\frac{1}{N_{j-1}\left(\frac{3}{2}\right)}\times
C_{j-1}^{3/2}\left(\frac{x}{\eta}\right)\times\sum_{q}\mathbb{F}^{q}_{j(singlet)}(\eta,t;\mu^{2})\,,$

and

$\displaystyle F^{g}(x,\eta,t;\mu^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta}\sum_{j=2}^{\infty}\frac{w\left(\frac{x}{\eta}|\frac{5}{2}\right)}{\eta^{j-2}N_{j-2}\left(\frac{5}{2}\right)}C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)\times\mathbb{F}^{g}_{j}(\eta,t;\mu^{2})\,,$
(IV.61) $\displaystyle=$
$\displaystyle\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)^{2}\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-2}}\times\frac{1}{N_{j-2}\left(\frac{5}{2}\right)}\times
C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)\times\mathbb{F}^{g}_{j}(\eta,t;\mu^{2})\,.$

###  IV.3 Evolution of quark and gluon GPDs to the asymptotic regime
$\mu\rightarrow\infty$

For completeness, we summarise the asymptotic forms of the GPDs at infinite
resolution, which are fixed by the free parton model, and are dominated by the
leading spin-1 or spin-2 Gegenbauer moments. More explicitly, the asymptotic
GPDs for non-singlet (valence) quark GPDs, are (see Eq.4.245 in Belitsky and
Radyushkin (2005))

$\displaystyle H_{valence}^{q}(x,\eta,t;\mu^{2}\to\infty)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times
F^{q}_{1}(t)\times\theta(\eta-|x|)\,,$ $\displaystyle
E_{valence}^{q}(x,\eta,t;\mu^{2}\to\infty)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times
F^{q}_{2}(t)\times\theta(\eta-|x|)\,.$

For singlet quark GPDs, they are (see Eq.4.289 and Eq.4.290 in Belitsky and
Radyushkin (2005))

$\displaystyle\sum_{q}H_{singlet}^{q}(x,\eta,t;\mu^{2}\to\infty)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{15}{4}\frac{N_{f}}{4C_{F}+N_{f}}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\frac{1}{\eta}\times\frac{x}{\eta}\times\sum_{q}\bigg{\\{}A_{q}(t)+\eta^{2}D_{q}(t)\bigg{\\}}\times\theta(\eta-|x|)\,,$
$\displaystyle\sum_{q}E_{singlet}^{q}(x,\eta,t;\mu^{2}\to\infty)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{15}{4}\frac{N_{f}}{4C_{F}+N_{f}}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)\times\frac{1}{\eta}\times\frac{x}{\eta}\times\sum_{q}\bigg{\\{}E_{q}(t)-\eta^{2}D_{q}(t)\bigg{\\}}\times\theta(\eta-|x|)\,.$

And, for gluon GPDs, they are (see Eq.4.291 and Eq.4.292 in Belitsky and
Radyushkin (2005))

$\displaystyle H^{g}(x,\eta,t;\mu^{2}\to\infty)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{15}{8}\frac{4C_{F}}{4C_{F}+N_{f}}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)^{2}\times\bigg{\\{}A_{g}(t)+\eta^{2}D_{g}(t)\bigg{\\}}\times\theta(\eta-|x|)\,,$
$\displaystyle E^{g}(x,\eta,t;\mu^{2}\to\infty)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{15}{8}\frac{4C_{F}}{4C_{F}+N_{f}}\times\frac{1}{\eta}\times\left(1-\frac{x^{2}}{\eta^{2}}\right)^{2}\times\bigg{\\{}E_{g}(t)-\eta^{2}D_{g}(t)\bigg{\\}}\times\theta(\eta-|x|)\,.$

##  V Matching $\lambda$ in soft-wall holographic QCD to
$\lambda_{s}(\mu_{0}^{2})$ in pQCD

In our previous work Mamo and Zahed (2020), we have fixed the ′t Hooft
coupling $\lambda=11.243$ in the soft-wall holographic QCD, by using the
photoproduction data for $J/\Psi$. Now, we suggest to map this fixed and large
′t Hooft coupling in holographic QCD, to the running ′t Hooft coupling in
pQCD, which will allow us to determine the value of the initial scale
$\mu_{0}$. More specifically, the running ′t Hooft coupling
$\lambda_{s}(\mu_{0}^{2})=g_{s}^{2}(\mu_{0}^{2})N_{c}=\alpha_{s}(\mu_{0}^{2})\times
4\pi\times N_{c}$ at $\mu=\mu_{0}$, maps onto

$\displaystyle\lambda_{s}(\mu_{0}^{2})=\lambda=11.243\,.$ (V.65)

which translates to

$\displaystyle\alpha_{s}(\mu_{0}^{2})=\frac{\lambda_{s}(\mu_{0}^{2})}{4\pi
N_{c}}=\frac{\lambda}{4\pi N_{c}}=\frac{11.243}{4\pi N_{c}}=0.298\,.$ (V.66)

This result is remarkably close to the value of $\frac{1}{3}$ commonly used to
initialize pQCD evolutions. This is welcome, since our holographic description
is set to describe the non-perturbative physics at $\mu_{0}$. Therefore, using
$\alpha_{s}(\mu_{0}^{2})=0.298$ in the two-loop exact transcendental equation,
for the QCD running coupling constant $\alpha_{s}(\mu^{2})$, Belitsky and
Radyushkin (2005)

$-\beta_{0}\ln\frac{\mu^{2}}{\Lambda^{2}_{\overline{\rm
MS}}}=\frac{4\pi}{\alpha_{s}(\mu^{2})}-\frac{\beta_{1}}{\beta_{0}}\ln\left(-\frac{4\pi}{\beta_{0}\alpha_{s}(\mu^{2})}-\frac{\beta_{1}}{\beta_{0}^{2}}\right)\,,$
(V.67)

with

$\displaystyle\beta_{0}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{4}{3}T_{F}N_{f}-\frac{11}{3}C_{A}\,,$ (V.68)
$\displaystyle\beta_{1}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{10}{3}C_{A}N_{f}+2C_{F}N_{f}-\frac{34}{3}C_{A}^{2}\,,$
(V.69)

and

$T_{F}=\frac{1}{2}\,,\qquad C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}\,,\qquad
C_{A}=N_{c}\ ,$ (V.70)

we find the input scale $\mu_{0}=1.808~{}\text{GeV}$ (or
$\mu_{0}^{2}=3.269~{}\text{GeV}^{2}$ ), using $N_{c}=N_{f}=3$, and
$\Lambda_{\overline{\rm MS}}=0.34~{}\text{GeV}$.

##  VI Electroproduction of longitudinal $\rho^{0}$ meson with evolved singlet
quark and gluon GPDs

We now apply the preceding results to the electroproduction of vector mesons
as illustrated in Fig. 1. In this section, we will focus mostly on the
electroproduction of longitudinal neutral $\rho^{0}$, and the comparison to
the available detailed data for this process. The extension to the charged
$\rho^{+}$ production, and the $\phi$ mesons, with comparison to data will
also be discussed. Clearly, most of our analysis applies to heavier meson
production such as $J/\Psi$ and $\Upsilon$ with minor kinematical changes.

With this in mind, the electroproduction of neutral rho meson ($\rho^{0}$) in
terms of quark DAs and quark GPDs is given by (see Eqs. 219-221 in Goeke et
al. (2001))

$\displaystyle{\cal
A}^{LL(quark)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})=$
$\displaystyle-e\times{C_{F}\over N_{c}}\times{1\over
2}\times(4\pi\alpha_{s}(\mu^{2}))\times{1\over{Q}}\times\left[\,\int_{0}^{1}dz{{\Phi_{\rho}(z)}\over
z}\right]\times\frac{1}{p^{+}}\left\\{A_{\rho^{0}\,p}\,h^{+}\,+\,B_{\rho^{0}\,p}\,e^{+}\right\\}\,,$
(VI.71)

where $\Phi_{\rho}(z)$ is the neutral rho meson DA, and

$\displaystyle A_{\rho^{0}\,p}\,$ $\displaystyle=$
$\displaystyle\,\int_{-1}^{1}dx\;{-1\over{\sqrt{2}}}\,{\left(Q_{u}\
H^{u}(x,\eta,t;\mu^{2})\,-\,Q_{d}\
H^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{{{1}\over{\xi-
x-i\epsilon}}-{{1}\over{\xi+x-i\epsilon}}\right\\}$ $\displaystyle
B_{\rho^{0}\,p}\,$ $\displaystyle=$
$\displaystyle\,\int_{-1}^{1}dx\;{-1\over{\sqrt{2}}}\,{\left(Q_{u}\
E^{u}(x,\eta,t;\mu^{2})\,-\,Q_{d}\
E^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{{{1}\over{\xi-
x-i\epsilon}}-{{1}\over{\xi+x-i\epsilon}}\right\\}\,.$ (VI.72)

Using $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$, we can rewrite (VI) as

$\displaystyle{\cal
A}^{LL(quark)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})=$
$\displaystyle e\times{1\over 2}\times{1\over
2}\times\left(1-\frac{1}{N_{c}^{2}}\right)\times
4\pi\times\alpha_{s}(\mu^{2})\times{1\over{Q}}\times\left[\,\int_{0}^{1}dz{{\Phi_{\rho}(z)}\over
z}\right]\times\frac{1}{p^{+}}\times
u(p_{2})\gamma^{+}u(p_{1})\times{1\over{\sqrt{2}}}$
$\displaystyle\times\int_{0}^{1}dx\;{\left(Q_{u}\
H_{singlet}^{u}(x,\eta,t;\mu^{2})\,-\,Q_{d}\
H_{singlet}^{d}(x,\eta,t;\mu^{2})\right)}\times\left\\{{{1}\over{\xi-
x-i\epsilon}}-{{1}\over{\xi+x-i\epsilon}}\right\\}\,,$

where we have defined
$H_{singlet}^{q}(x,\eta,t;\mu^{2})=H^{q}(x,\eta,t;\mu^{2})-H^{q}(-x,\eta,t;\mu^{2})$,
and we have ignored the Pauli contribution for $-t\ll 4m_{N}^{2}$.

In addition, the electroproduction of neutral rho meson ($\rho^{0}$) in terms
of quark DAs and gluon GPDs is given by (see Eq.282 in Diehl (2003))

$\displaystyle\mathcal{A}^{LL(gluon)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})=$
$\displaystyle e\times\frac{1}{8}\times\frac{1}{N_{c}}\times
4\pi\times\alpha_{s}(\mu^{2})\times\frac{1}{Q}\times\left[\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\right]\times\int_{-1}^{1}dx\,\frac{F^{g}(x,\eta,t;\mu^{2})}{x}\left\\{\frac{1}{\xi-
x-i\epsilon}-\frac{1}{\xi+x-i\epsilon}\right\\}\,.$

On the light front,

$h^{+}=\bar{u}(p_{2})\gamma^{+}u(p_{1})\approx\frac{p^{+}}{2m_{N}}\times\bar{u}(p_{2})u(p_{1}),$

and ignoring the Pauli contribution $E^{g}(x,\eta,t;\mu^{2})$ to
$F^{g}(x,\eta,t;\mu^{2})$ for $-t\ll 4m_{N}^{2}$, we have

$\displaystyle
F^{g}(x,\eta,t;\mu^{2})\approx\frac{h^{+}}{p^{+}}H^{g}(x,\eta,t;\mu^{2})=H^{g}(x,\eta,t;\mu^{2})\times\frac{1}{2m_{N}}\times\bar{u}(p_{2})u(p_{1})\,,$
(VI.75)

which simplifies (VI)

$\displaystyle\mathcal{A}^{LL(gluon)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle=$ $\displaystyle
e\times\frac{1}{8}\times\frac{1}{N_{c}}\times\frac{1}{Q}\times
4\pi\times\alpha_{s}(\mu^{2})\times\frac{1}{2m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\left[\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\right]\times\int_{0}^{1}dx\,\frac{H^{g}(x,\eta,t;\mu^{2})}{x}\times\left\\{\frac{1}{\xi-
x-i\epsilon}-\frac{1}{\xi+x-i\epsilon}\right\\}\,.$

###  VI.1 Final electro-production amplitude of longitudinal $\rho^{0}$ mesons
after quark and gluon evolution

The details of the fully evolved quark and gluon contributions to the
electroproduction of $\rho^{0}$ are given in Appendix K. The result for the
singlet quark contribution follows from (K.357) as

$\displaystyle{\cal
A}^{LL(quark)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle=$
$\displaystyle\Bigg{(}Q_{u}\widehat{\mathbb{F}}^{u}_{j_{0g}(singlet)}(\xi,t;\mu^{2})-Q_{d}\widehat{\mathbb{F}}^{d}_{j_{0g}(singlet)}(\xi,t;\mu^{2})\Bigg{)}$
(VI.77) $\displaystyle\times$ $\displaystyle\mathcal{N}_{q}(j_{0g})\times
e\,f_{V}\,\alpha_{s}(\mu)\,{1\over{Q}}\,\frac{1}{2\sqrt{2}m_{N}}\,\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\left[\frac{(\sqrt{\lambda}/\pi)}{\xi^{2-2/\sqrt{\lambda}}}+\frac{i}{\xi^{2-2/\sqrt{\lambda}}}\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\times\frac{e^{-\sqrt{\lambda}/2\tilde{\tau}}}{\tilde{\tau}^{3/2}}\,.$

and the one for the fully evolved gluon GPD follows from (K.373) as

$\displaystyle{\cal
A}^{LL(gluon)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle=$
$\displaystyle\mathcal{N}_{g}(j_{0g})\,\widehat{\mathbb{F}}^{g}_{j_{0g}}(\xi,t;\mu^{2})\times
e\,f_{V}^{\prime}\,\alpha_{s}(\mu)\,{1\over{Q}}\,\frac{1}{2\sqrt{2}m_{N}}\,\bar{u}(p_{2})u(p_{1})$
(VI.78) $\displaystyle\times$
$\displaystyle\left[\frac{(\sqrt{\lambda}/\pi)}{\xi^{2-2/\sqrt{\lambda}}}+\frac{i}{\xi^{2-2/\sqrt{\lambda}}}\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\times\frac{e^{-\sqrt{\lambda}/2\tilde{\tau}}}{\tilde{\tau}^{3/2}}\,,$

with $j_{0g}=2-2/\sqrt{\lambda}$, $\tilde{\tau}=\text{log}(1/\xi)$, and
$f_{V}^{\prime}\approx f_{V}$.

We also have the real s-channel background amplitude for the electroproduction
of $\rho^{0}$ meson (which can be determined by slightly modifying the
s-channel amplitude for DVCS (K.2 \- K.2) assuming the bulk wave function of
rho meson is independent of $z$, i.e., $\phi_{V}(z)\approx
g_{5}\times\frac{f_{V}}{M_{V}}$)

$\displaystyle{\cal
A}^{LL(s-channel)}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle=$ $\displaystyle e\times
g_{5}\times\left[\epsilon_{L\mu}\epsilon_{L\nu}^{\prime*}T^{\mu\nu}(s,t;Q^{2})\right]\times
g_{5}\times\frac{f_{V}}{M_{V}}$ $\displaystyle=$ $\displaystyle
e\times\sqrt{2}\times g_{5}^{2}\times\frac{Q}{M_{V}}\times
F_{1}^{P}(Q)\times\frac{f_{V}}{M_{V}}\times\frac{1}{2\sqrt{2}m_{N}}\,\bar{u}(p_{2})u(p_{1})\times\frac{1}{1-\xi^{2}}\,.$

Finally, combining the singlet quark (VI.77) and gluon (VI.78) GPD
contributions, along with the s-channel contribution, we have the total
longitudinal differential cross section (including the real s-channel
background amplitude (VI.1)) for electroproduction of $\rho^{0}$ meson

$\displaystyle\frac{d\sigma_{L}(\rho^{0})}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi s^{2}}\times\frac{1}{2}\sum_{\rm
spin}\Big{|}\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{0}p}^{LL(total)}\Big{|}^{2}\,,$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi s^{2}}\times\frac{1}{2}\sum_{\rm
spin}\Big{|}\frac{\mathcal{N}_{s}}{e\times\sqrt{2}\times
g_{5}^{2}}\times\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{0}p}^{LL(s-channel)}+\mathcal{N}_{t}\times\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{0}p}^{LL(quark)}+\mathcal{N}_{t}\times\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{0}p}^{LL(gluon)}\Big{|}^{2}\,.$

Also note that we sum over the spin as

$\displaystyle\sum_{s,s^{\prime}}\bar{u}_{s^{\prime}}(p_{2})u_{s}(p_{1})\bar{u}_{s}(p_{1})u_{s^{\prime}}(p_{2})$
$\displaystyle=$ $\displaystyle{\rm
Tr}\Big{(}\sum_{s.s^{\prime}}u_{s^{\prime}}(p_{2})\bar{u}_{s^{\prime}}(p_{2})u_{s}(p_{1})\bar{u}_{s}(p_{1})\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{4}{\rm
Tr}\Big{(}\big{(}\gamma_{\mu}p_{2}^{\mu}+m_{N}\big{)}\big{(}\gamma_{\mu}p_{1}^{\mu}+m_{N}\big{)}\Big{)}=8m_{N}^{2}\times\left(1+\frac{K^{2}}{4m_{N}^{2}}\right)\approx
8m_{N}^{2}\,,$

for $\frac{K^{2}}{4m_{N}^{2}}\ll 1$ or $p_{1}\sim p_{2}$ which we are assuming
in the high energy limit $s\gg-t$.

###  VI.2 Cross sections

For $\mathcal{N}_{t}\neq 0$ in (VI.1), we can compute the total cross section
using the optical theorem as

$\displaystyle\sigma_{L}(\rho^{0})=\left(\frac{16\pi\,\frac{d\sigma_{L}(\rho^{0})}{dt}}{1+\rho}\right)^{1/2}_{t=0}\,,$
(VI.82)

where $\rho$ is the ratio of the real and imaginary part of our total complex
amplitude $\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{0}p}^{LL(total)}$ defined
in (VI.1). And, for $\mathcal{N}_{t}=0$ in (VI.1), we can compute the total
cross section near-threshold as

$\displaystyle\sigma_{L}(\rho^{0})$ $\displaystyle=$
$\displaystyle\int_{t_{min}}^{t_{max}}dt\,\frac{d\sigma_{L}(\rho^{0})}{dt}\approx(t_{max}-t_{min})\times\left(\frac{d\sigma_{L}(\rho^{0})}{dt}\right)_{t\approx
0}\,,$ (VI.83)

where $t_{min}\equiv|t|_{\cos\theta=+1}|$ and
$t_{max}\equiv|t|_{\cos\theta=-1}|$ with

$\displaystyle
q_{z}=|\mathbf{q}_{\gamma}|=\frac{1}{2\sqrt{s}}\sqrt{s^{2}-2(-Q^{2}+m_{N}^{2})s+(-Q^{2}-m_{N}^{2})^{2}}\,,$
$\displaystyle|\mathbf{q}_{V}|=\frac{1}{2\sqrt{s}}\sqrt{s^{2}-2(M_{V}^{2}+m_{N}^{2})s+(M_{V}^{2}-m_{N}^{2})^{2}}\,,$
$\displaystyle
t=-Q^{2}+M_{V}^{2}-2E_{\gamma}E_{V}+2|\mathbf{q}_{\gamma}||\mathbf{q}_{V}|\cos\theta\,,$

and $E_{\gamma}=\sqrt{-Q^{2}+q_{z}^{2}}$,
$E_{V}=\sqrt{M_{V}^{2}+|\mathbf{q}_{V}|^{2}}$.

We have compared our total cross sections (VI.82) and (VI.83) to experimental
data in Fig. 5 \- 10. Note that throughout this paper, we use the ’t Hooft
coupling constant $\lambda=11.243$, mass scales
$\tilde{\kappa}_{N}=\tilde{\kappa}_{V}=0.402$ GeV, and
$\tilde{\kappa}_{T}=0.388$ GeV (and also ignore the effect of the D-term for
small-$\xi$ where the contribution from the GPDs is dominant).

AdS/QCD $(This~{}Work)$ |  $\lambda$ |  $\tau$ |  $\tilde{\kappa}_{T,S}(\text{GeV})$ |  $\tilde{\kappa}_{N,V}(\text{GeV})$ |  $\mathcal{N}_{t}(\mu b\times\text{GeV}^{2})$ |  $\mathcal{N}_{s}(\mu b\times\text{GeV}^{2})$
---|---|---|---|---|---|---
Fig. 5,8,9,6 $(blue)$ |  11.243 |  3 |  0.388 |  0.402 |  151.654 or 292.834 |  0
Fig. 5,8,9.6 $(green)$ |  11.243 |  3 |  0.388 |  0.402 |  0 |  -535.996
Fig. 5,8,9,6 $(red)$ |  11.243 |  3 |  0.388 |  0.402 |  151.654 or 292.834 |  -126.078 or -142.114
Table 1: Summary of the values we used for the pre-determined parameters
($\lambda=11.243$, $\tau=3$, $\tilde{\kappa}_{T,S}=0.388~{}\text{GeV}$,
$\tilde{\kappa}_{N,V}=0.402~{}\text{GeV}$) of AdS/QCD with a soft wall. The
overall normalization constants $\mathcal{N}_{t,s}$, are fitted to the
pertinent cross section data (mostly from 5.754 GeV CLAS Morrow et al.
(2009)), as per the corresponding (colored) figures for the neutral rho meson
production.

(a)  (b)  (c)  (d)  (e)

Figure 5: Longitudinal cross section for the electro-production of neutral rho
mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=1.6-1.9$ GeV2 (a),
$Q^{2}=1.9-2.2$ GeV2 (b), $Q^{2}=2.2-2.5$ GeV2 (c), $Q^{2}=2.5-2.8$ GeV2 (d),
$Q^{2}=2.8-3.1$ GeV2 (e). The solid-blue curves are the evolved singlet (up
and down) quark, and gluon GPDs (i.e., (VI.82). The solid-green curves are the
holographic real s-channel contribution (i.e., (VI.83). The dashed-red curves
are the sum of the s-channel background, and the evolved singlet quark plus
gluon GPDs. The brown data points are from 4.2 GeV CLAS Hadjidakis et al.
(2005). The yellow data points are from CORNELL Cassel et al. (1981). The cyan
data points are from HERMES Airapetian et al. (2000). The purple data points
are from E665 Adams et al. (1997). The black data points are from 5.754 GeV
CLAS Morrow et al. (2009).

(a)  (b)  (c)  (d)  (e)

Figure 6: Longitudinal cross section for the electro-production of neutral rho
mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=3.1-3.6$ GeV2 (a),
$Q^{2}=3.6-4.1$ GeV2 (b), $Q^{2}=4.1-4.6$ GeV2 (c), $Q^{2}=4.6-5.1$ GeV2 (d),
$Q^{2}=5.1-5.6$ GeV2 (e). The solid-blue curves are with the evolved up and
down singlet quark, and gluon GPDs (i.e., (VI.82). The solid-green curves are
the holographic real s-channel contribution (i.e., (VI.83). The dashed-red
curves are the sum of the s-channel, and the evolved up and down singlet
quark, and gluon GPD contributions at amplitude level (i.e., (VI.82). The cyan
data points are from HERMES Airapetian et al. (2000). The purple data points
are from E665 Adams et al. (1997). The black data points are from 5.754 GeV
CLAS Morrow et al. (2009). (a)  Figure 7: Longitudinal cross section for the
electro-production of neutral rho mesons versus $W=\sqrt{s}$ in GeV, for
$Q^{2}=4$ GeV2. The solid-blue curves are with the evolved up and down singlet
quark, and gluon GPDs (i.e., (VI.82). The solid-green curves are the
holographic real s-channel background contribution (i.e., (VI.83). The dashed-
red curves are the sum of the s-channel background, and the evolved up and
down singlet quark, and gluon GPD contribution at amplitude level (i.e.,
(VI.82). Note that we have also re-scaled our total cross section by an
overall factor of $10^{3}$. The brown data points are from 4.2 GeV CLAS
Hadjidakis et al. (2005). The black data points are from 5.754 GeV CLAS Morrow
et al. (2009). The cyan data points are from HERMES Airapetian et al. (2000).
The purple data points are from E665 Adams et al. (1997). The grey data points
are from ZEUS Breitweg et al. (1999). The pink data points are from H1 Adloff
et al. (2000a).

(a)  (b)  (c)  (d)

Figure 8: Longitudinal cross section for the electro-production of neutral rho
mesons versus $Q^{2}$ in GeV2, for Bjorken $x_{B}=0.16-0.22$ (a),
$x_{B}=0.22-0.28$ (b), $x_{B}=0.28-0.34$ (c), $x_{B}=0.34-0.4$ (d). The solid-
blue curves are with the evolved up and down singlet quark, and gluon GPD
contributions (i.e., (VI.82). The solid-green curves are the holographic real
s-channel background contribution (i.e., (VI.83). The dashed-red curves are
the sum of the s-channel background, and the evolved up and down singlet
quark, and gluon GPD contributions at amplitude level (i.e., (VI.82) with
$\mathcal{N}_{t}=151.654$ and $\mathcal{N}_{s}=-126.078$). The black data
points are from 5.754 GeV CLAS Morrow et al. (2009).

(a)  (b)  (c)

Figure 9: Longitudinal cross section for the electro-production of neutral rho
mesons versus $Q^{2}$ in GeV2, for Bjorken $x_{B}=0.4-0.46$ (a),
$x_{B}=0.46-0.52$ (b), $x_{B}=0.52-0.58$ (c). The solid-blue curves are with
the evolved up and down singlet quark, and gluon GPDs (i.e., (VI.82). The
solid-green curves are the holographic real s-channel background contribution
(i.e., (VI.83) with $\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-535.996$). The
dashed-red curves are the sum of the s-channel background, and the evolved up
and down singlet quark, and gluon GPD contribution at amplitude level (i.e.,
(VI.82). The black data points are from 5.754 GeV CLAS Morrow et al. (2009).
(a)  Figure 10: Longitudinal cross section for the electro-production of
neutral rho mesons versus $Q^{2}$ in GeV2, for $W=\sqrt{s}=75$ GeV. The solid-
blue curves are with the evolved up and down singlet quark, and gluon GPD
contributions (i.e., (VI.82). The solid-green curves are the holographic real
s-channel contribution (i.e., (VI.83) with $\mathcal{N}_{t}=0$ and
$\mathcal{N}_{s}=-14,576.3$). The dashed-red curves are the sum of the
s-channel, and the evolved up and down singlet quark, and gluon GPDs at
amplitude level (i.e., (VI.82. The black data points are from H1, ZEUS Aaron
et al. (2010).

(a)  (b)  (c)  (d)  (e)

Figure 11: Longitudinal differential cross section for the electro-production
of neutral rho mesons versus $-t$ in GeV2, for $Q^{2}=0-1.9$ GeV2, and
$x_{B}=0.16-0.22$ (a), $Q^{2}=1.9-2.2$ GeV2, and $x_{B}=0.34-0.4$ (b),
$Q^{2}=2.2-2.5$ GeV2, and $x_{B}=0.22-0.28$ (c), $Q^{2}=3.1-3.6$ GeV2, and
$x_{B}=0.4-0.46$ (d), $Q^{2}=4.1-4.6$ GeV2, and $x_{B}=0.58-0.64$ (e). The
solid-blue curves are with the evolved up and down singlet quark, and gluon
GPD contributions, i.e. (VI.1) with $\mathcal{N}_{t}=386.637$ and
$\mathcal{N}_{s}=0$ for (a), (VI.1) with $\mathcal{N}_{t}=6,963.08$ and
$\mathcal{N}_{s}=0$ for (b), (VI.1) with $\mathcal{N}_{t}=9,568.23$ and
$\mathcal{N}_{s}=0$ for (c), and (VI.1) with $\mathcal{N}_{t}=63,344.4$ and
$\mathcal{N}_{s}=0$ for (d)). The solid-green curve in (e) is using (VI.1)
with $\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-577.532$. The black data points
are from 5.754 GeV CLAS Morrow et al. (2009).

###  VI.3 Results for exclusive $\rho^{0}$ production

In Table 1, we list the parameters $\lambda=11.243$, $\tau=3$,
$\tilde{\kappa}_{T,S}=0.388~{}\text{GeV}$,
$\tilde{\kappa}_{N,V}=0.402~{}\text{GeV}$, in the AdS construction with a soft
wall. We recall that $\lambda$ is the ′t Hooft coupling, $\tau$ the nucleon
twist parameter, $\tilde{\kappa}_{T,S}$ the tensor and scalar glueball Regge
slope parameters, and $\tilde{\kappa}_{N,V}$ the nucleon and vector meson
Regge slope parameters, all of which have been fixed by constraining the
electromagnetic and gravitational form factors Mamo and Zahed (2021b). So for
the GPDs discussed here, they are totally pre-determined. The normalization
constants $\mathcal{N}_{t,s}$ lump the remaining parameters $g_{5},1/N_{c}$.
They are fixed by fitting to the analyzed cross sections in Fig. 5,6,8,9, as
summarized in Table 1. We have also used $e=0.3$, $m_{N}=0.94~{}\text{GeV}$,
$m_{\rho^{0,+}}=0.775~{}\text{GeV}$, and $f_{\rho^{0,+}}=0.216~{}\text{GeV}$.

In Fig. 5 we show the longitudinal cross section for the electro-production of
neutral rho mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=1.6-1.9$ GeV2 (a),
$Q^{2}=1.9-2.2$ GeV2 (b), $Q^{2}=2.2-2.5$ GeV2 (c), $Q^{2}=2.5-2.8$ GeV2 (d),
$Q^{2}=2.8-3.1$ GeV2 (e). The solid-blue curves are with the evolved up and
down singlet quark, and gluon GPDs (i.e., (VI.82) with
$\mathcal{N}_{t}=151.654$ and $\mathcal{N}_{s}=0$). The solid-green curves are
the holographic real s-channel contribution (i.e., (VI.83) with
$\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-535.996$). The dashed-red curves are
the sum of the s-channel background, and the evolved up and down singlet
quark, and gluon GPD contributions at amplitude level (i.e., (VI.82) with
$\mathcal{N}_{t}=151.654$ and $\mathcal{N}_{s}=-126.078$). The brown data
points are from 4.2 GeV CLAS Hadjidakis et al. (2005), see also Fig.25 of
Morrow et al. (2009). The yellow data points are from CORNELL Cassel et al.
(1981), see also Fig.25 of Morrow et al. (2009). The cyan data points are from
HERMES Airapetian et al. (2000), see also Fig.25 of Morrow et al. (2009). The
purple data points are from E665 Adams et al. (1997), see also Fig.25 of
Morrow et al. (2009). The black data points are from 5.754 GeV CLAS Morrow et
al. (2009), see Fig. 25 in Morrow et al. (2009). Overall, the cross section is
dominated by the s-channel exchange at low $\sqrt{s}$, and the evolved GPDs
initialized by the holographic t-channel exchanges at large $\sqrt{s}$. More
data will be welcome, for a detailed comparison.

In Fig. 6 we show the longitudinal cross section for the electro-production of
neutral rho mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=3.1-3.6$ GeV2 (a),
$Q^{2}=3.6-4.1$ GeV2 (b), $Q^{2}=4.1-4.6$ GeV2 (c), $Q^{2}=4.6-5.1$ GeV2 (d),
$Q^{2}=5.1-5.6$ GeV2 (e). The solid-blue curves are with the evolved up and
down singlet quark, and gluon GPDs (i.e., (VI.82) with
$\mathcal{N}_{t}=292.834$ and $\mathcal{N}_{s}=0$). The solid-green curves are
the holographic real s-channel contribution (i.e., (VI.83) with
$\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-535.996$). The dashed-red curves are
the sum of the s-channel, and the evolved up and down singlet quark, and gluon
GPD contributions at amplitude level (i.e., (VI.82) with
$\mathcal{N}_{t}=292.834$ and $\mathcal{N}_{s}=-142.114$). The cyan data
points are from HERMES Airapetian et al. (2000), see also Fig.25 of Morrow et
al. (2009). The purple data points are from E665 Adams et al. (1997), see also
Fig.25 of Morrow et al. (2009). The black data points are from 5.754 GeV CLAS
Morrow et al. (2009), see also Fig. 25 of Morrow et al. (2009). The same
observations made earlier regarding the s-channel dominance at low
$W=\sqrt{s}$, and the t-channel dominance at large $\sqrt{s}$. Our results are
in good agreement with the reported data.

In Fig. 7 we show the longitudinal cross section for the electro-production of
neutral rho mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=4$ GeV2. The solid-
blue curves are with the evolved up and down singlet quark, and gluon GPDs
(i.e., (VI.82) with $\mathcal{N}_{t}=255.766$ and $\mathcal{N}_{s}=0$). The
solid-green curves are the holographic real s-channel background contribution
(i.e., (VI.83) with $\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-628.784$). The
dashed-red curves are the sum of the s-channel background, and the evolved up
and down singlet quark, and gluon GPD contribution at amplitude level (i.e.,
(VI.82) with $\mathcal{N}_{t}=255.766$ and $\mathcal{N}_{s}=-211.633$). Note
that we have also re-scaled our total cross section by an overall factor of
$10^{3}$. The dip is the result of a destructive interference between the two
contributions, which becomes manifest for the present choice of kinematics.
The brown data points are from 4.2 GeV CLAS Hadjidakis et al. (2005), see also
Fig.5 in Kroll (2015). The black data points are from 5.754 GeV CLAS Morrow et
al. (2009), see also Fig.5 in Kroll (2015). The cyan data points are from
HERMES Airapetian et al. (2000), see also Fig.5 in Kroll (2015). The purple
data points are from E665 Adams et al. (1997), see also Fig.5 in Kroll (2015).
The grey data points are from ZEUS Breitweg et al. (1999), see also Fig.5 in
Kroll (2015). The pink data points are from H1 Adloff et al. (2000a), see also
Fig.5 in Kroll (2015).

In Fig. 8 we show the longitudinal cross section for the electro-production of
neutral rho mesons versus $Q^{2}$ in GeV2, for Bjorken $x_{B}=0.16-0.22$ (a),
$x_{B}=0.22-0.28$ (b), $x_{B}=0.28-0.34$ (c), $x_{B}=0.34-0.4$ (d). The solid-
blue curves are with the evolved up and down singlet quark, and gluon GPD
contributions (i.e., (VI.82) with $\mathcal{N}_{t}=151.654$ and
$\mathcal{N}_{s}=0$). The solid-green curves are the holographic real
s-channel background contribution (i.e., (VI.83) with $\mathcal{N}_{t}=0$ and
$\mathcal{N}_{s}=-535.996$). The dashed-red curves are the sum of the
s-channel background, and the evolved up and down singlet quark, and gluon GPD
contributions at amplitude level (i.e., (VI.82) with $\mathcal{N}_{t}=151.654$
and $\mathcal{N}_{s}=-126.078$). The black data points are from 5.754 GeV CLAS
Morrow et al. (2009), see Fig.27 of Morrow et al. (2009).

In Fig. 9 we show the longitudinal cross section for the electro-production of
neutral rho mesons versus $Q^{2}$ in GeV2, for Bjorken $x_{B}=0.4-0.46$ (a),
$x_{B}=0.46-0.52$ (b), $x_{B}=0.52-0.58$ (c). The solid-blue curves are with
the evolved up and down singlet quark, and gluon GPDs (i.e., (VI.82) with
$\mathcal{N}_{t}=151.654$ and $\mathcal{N}_{s}=0$). The solid-green curves are
the holographic real s-channel background contribution (i.e., (VI.83) with
$\mathcal{N}_{t}=0$ and $\mathcal{N}_{s}=-535.996$). The dashed-red curves are
the sum of the s-channel background, and the evolved up and down singlet
quark, and gluon GPD contribution at amplitude level (i.e., (VI.82) with
$\mathcal{N}_{t}=151.654$ and $\mathcal{N}_{s}=-126.078$). The black data
points are from 5.754 GeV CLAS Morrow et al. (2009), see Fig. 27 in Morrow et
al. (2009).

In Fig. 10 we show the longitudinal cross section for the electro-production
of neutral rho mesons versus $Q^{2}$ in GeV2, for $W=\sqrt{s}=75$ GeV. The
solid-blue curves are with the evolved up and down singlet quark, and gluon
GPD contributions (i.e., (VI.82) with $\mathcal{N}_{t}=169.407$ and
$\mathcal{N}_{s}=0$). The solid-green curves are the holographic real
s-channel contribution (i.e., (VI.83) with $\mathcal{N}_{t}=0$ and
$\mathcal{N}_{s}=-14,576.3$). The dashed-red curves are the sum of the
s-channel, and the evolved up and down singlet quark, and gluon GPDs at
amplitude level (i.e., (VI.82) with $\mathcal{N}_{t}=169.407$ and
$\mathcal{N}_{s}=-99,721.3$). The black data points are from H1, ZEUS Aaron et
al. (2010), see also Fig.7 of Favart et al. (2016).

In Fig. 11 we show the longitudinal differential cross section for the
electro-production of neutral rho mesons versus $-t$ in GeV2, for
$Q^{2}=0-1.9$ GeV2, and $x_{B}=0.16-0.22$ (a), $Q^{2}=1.9-2.2$ GeV2, and
$x_{B}=0.34-0.4$ (b), $Q^{2}=2.2-2.5$ GeV2, and $x_{B}=0.22-0.28$ (c),
$Q^{2}=3.1-3.6$ GeV2, and $x_{B}=0.4-0.46$ (d), $Q^{2}=4.1-4.6$ GeV2, and
$x_{B}=0.58-0.64$ (e). The solid-blue curves are with the evolved up and down
singlet quark, and gluon GPD contributions (i.e., (VI.1) with
$\mathcal{N}_{t}=386.637$ and $\mathcal{N}_{s}=0$ for (a), (VI.1) with
$\mathcal{N}_{t}=6,963.08$ and $\mathcal{N}_{s}=0$ for (b), (VI.1) with
$\mathcal{N}_{t}=9,568.23$ and $\mathcal{N}_{s}=0$ for (c), and (VI.1) with
$\mathcal{N}_{t}=63,344.4$ and $\mathcal{N}_{s}=0$ for (d)). The solid-green
curve in (e) is using (VI.1) with $\mathcal{N}_{t}=0$ and
$\mathcal{N}_{s}=-577.532$. The black data points are from 5.754 GeV CLAS
Morrow et al. (2009), see Fig.26 of Morrow et al. (2009).

##  VII Electroproduction of longitudinal $\phi$ meson with evolved gluon GPD

###  VII.1 Far-from-threshold electroproduction of $\phi$ meson with evolved
gluon GPD

The electroproduction of $\phi$ meson with the evolved gluon GPD is similar to
the gluon contribution to $\rho^{0}$ meson (VI.78), and is given by

$\displaystyle{\cal A}^{LL(gluon)}_{\gamma^{*}p\rightarrow\phi
p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{N}_{g}(j_{0g})\,\widehat{\mathbb{F}}^{g}_{j_{0g}}(\xi,t;\mu^{2})\times
e\,f_{V}^{\prime}\,\alpha_{s}(\mu)\,{1\over{Q}}\,\frac{1}{2\sqrt{2}m_{N}}\,\bar{u}(p_{2})u(p_{1})$
(VII.87) $\displaystyle\times$
$\displaystyle\left[\frac{(\sqrt{\lambda}/\pi)}{\xi^{2-2/\sqrt{\lambda}}}+\frac{i}{\xi^{2-2/\sqrt{\lambda}}}\right]\times(\sqrt{\lambda}/2\pi)^{1/2}\times\frac{e^{-\sqrt{\lambda}/2\tilde{\tau}}}{\tilde{\tau}^{3/2}}\,,$

with $j_{0g}=2-2/\sqrt{\lambda}$, and $\tilde{\tau}=\text{log}(1/\xi)$. In
terms of (VII.87), the total longitudinal differential cross section for
electroproduction of $\phi$ meson reads

$\displaystyle\frac{d\sigma_{L}(\phi)}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi s^{2}}\times\frac{1}{2}\sum_{\rm
spin}\Big{|}\mathcal{N}_{t}\times\mathcal{A}_{\gamma^{*}p\rightarrow\phi
p}^{LL(gluon)}\Big{|}^{2}\,,$

where the far-from-threshold total amplitude
$\mathcal{A}_{\gamma^{*}p\rightarrow\phi p}^{LL(gluon)}$ is given by (VII.87).
We also compute the far-from-threshold total cross section using the optical
theorem as

$\displaystyle\sigma_{L}(\phi)=\left(\frac{16\pi\,\frac{d\sigma_{L}(\phi)}{dt}}{1+\rho}\right)^{1/2}_{t=0}\,,$
(VII.89)

where $\rho$ is the ratio of the real and imaginary part of the total
amplitude $\mathcal{A}_{\gamma^{*}p\rightarrow\phi p}^{LL(gluon)}$ given in
(VII.87). We have compared our total cross section (VII.89) to experimental
data in Fig. 12 and 13. Note that these relations carry to heavier mesons
$J/\Psi$ and $\Upsilon$ with minor changes.

###  VII.2 Near-threshold electroproduction of $\phi$ meson with evolved gluon
GPD

The near-threshold electroproduction amplitude for longitudinal $\phi$ meson
(or heavy vector meson in general) could be described by keeping only the
leading gluon conformal (Gegenbauer) moment in vector meson electroproduction
amplitude (K.5) or more compactly (K.5). Here we will compute the result in
this regime more explicitly for $\phi$ meson, but the formalism is applicable
for any other heavy vector mesons such as $J/\psi$ or $\Upsilon$.

The electroproduction of $\phi$ meson with the evolved gluon GPD is given by

$\displaystyle\mathcal{A}^{LL(gluon)}_{\gamma^{*}p\rightarrow\phi
p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$ $\displaystyle=$ $\displaystyle
e\times\frac{1}{4}\times\frac{1}{N_{c}}\times\frac{1}{Q}\times\alpha_{s}(\mu^{2})\times\left[\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\right]\times\Bigg{[}\frac{1}{\xi^{2}}\times\int_{0}^{1}dx\,\frac{H^{g}(x,\eta,t;\mu^{2})}{(1-\frac{x^{2}}{\xi^{2}})}$
(VII.90) $\displaystyle+$ $\displaystyle i\pi H^{g}(\xi,\eta,t;\mu^{2})+i\pi
H^{g}(-\xi,\eta,t;\mu^{2})\Bigg{]}\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})\,,$
$\displaystyle=$ $\displaystyle
e\times\frac{1}{4}\times\frac{1}{N_{c}}\times\frac{1}{Q}\times\alpha_{s}(\mu^{2})\times\left[\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\right]\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\Bigg{[}\frac{1}{\xi^{2}}\times\sum_{j=2}^{\infty}\frac{1}{\eta^{j-2}}\times\frac{1}{N_{j-2}(\frac{5}{2})}\times\left[\int_{0}^{\eta}\frac{dx}{\eta}\,\frac{\left(1-\frac{x^{2}}{\eta^{2}}\right)^{2}}{(1-\frac{x^{2}}{\xi^{2}})}\times
C_{j-2}^{5/2}\left(\frac{x}{\eta}\right)\right]\times\mathbb{F}^{g}_{j}(\eta,t;\mu^{2})$
$\displaystyle+$ $\displaystyle i\pi H^{g}(\xi,\eta,t;\mu^{2})+i\pi
H^{g}(-\xi,\eta,t;\mu^{2})\Bigg{]}\,.$

For the near-threshold regime, we can approximate (VII.90) as

$\displaystyle\mathcal{A}^{LL(gluon)-\text{near-
threshold}}_{\gamma^{*}p\rightarrow\rho^{0}p}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle\approx$ $\displaystyle e\times\mathcal{N}_{g}\times
f_{V}^{\prime}\times\alpha_{s}(\mu^{2})\times\frac{1}{Q}\times\frac{1}{\xi^{2}}\times\mathbb{F}^{g}_{2}(\xi,t;\mu^{2})\times\frac{1}{2\sqrt{2}m_{N}}\times\bar{u}(p_{2})u(p_{1})\,,$

where we have defined

$\displaystyle\mathcal{N}_{g}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4}\times\frac{1}{N_{c}}\times
2\sqrt{2}\times\frac{1}{N_{0}(\frac{5}{2})}\times I_{0}\,,$ (VII.92)

and the integrals

$\displaystyle f_{V}\approx
f_{V}^{\prime}=\int_{0}^{1}dz\,\sum_{q}e_{q}\frac{\Phi^{q}(z)}{z(1-z)}\,,\qquad
I_{0}=\int_{0}^{1}d\tilde{x}\,\left(1-\tilde{x}^{2}\right)\,,$ (VII.93)

with $\tilde{x}=x/\xi$. Note that in (VII.90), we have used $\eta\sim\xi$, and
kept only the first term in the series, i.e., $j=2$, in order to find the
near-threshold amplitude (VII.2).

Finally, using (VII.2), we have the total longitudinal differential cross
section for near-threshold electroproduction of $\phi$ meson

$\displaystyle\frac{d\sigma_{L}(\phi)}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi s^{2}}\times\frac{1}{2}\sum_{\rm
spin}\Big{|}\mathcal{A}_{\gamma^{*}p\rightarrow\phi p}^{LL(gluon))-\text{near-
threshold}}\Big{|}^{2}\,.$ (VII.94)

And, using (VII.94), we can compute the total cross section near-threshold as

$\displaystyle\sigma_{L}(\phi)=\int_{t_{min}}^{t_{max}}dt\,\frac{d\sigma_{L}(\phi)}{dt}\approx(t_{max}-t_{min})\times\left(\frac{d\sigma_{L}(\phi)}{dt}\right)_{t\approx
0}\,.$ (VII.95)

We have plotted the near-threshold cross-section (VII.95) (including the far-
from-threshold one (VII.89)) in Fig. 12 and 13. We have also used $e=0.3$,
$m_{N}=0.94~{}\text{GeV}$, $m_{\phi}=1.019~{}\text{GeV}$, and
$f_{\phi}=0.233~{}\text{GeV}$, in addition to the same parameters as neutral
rho meson $\lambda=11.243$, $\tau=3$,
$\tilde{\kappa}_{T,S}=0.388~{}\text{GeV}$,
$\tilde{\kappa}_{N,V}=0.402~{}\text{GeV}$ with the values of the normalization
parameter $\mathcal{N}_{t}$ fixed by the phi meson data as detailed in the
next subsection below.

(a)  Figure 12: Longitudinal cross section for the electro-production of phi
mesons versus $W=\sqrt{s}$ in GeV, for $Q^{2}=2.21$ GeV2 (dashed-purple
curves) and $Q^{2}=3.8$ GeV2 (solid-blue curves) in (VII.89). The dashed-
purple curves and solid-blue curves are only with the evolved gluon GPD. The
orange data point is from CLAS 2008 Santoro et al. (2008). The brown data
point is from CLAS 2001 Lukashin et al. (2001). The cyan data point is from
HERMES 2000. The grey data points are from ZEUS Chekanov et al. (2005). The
pink data points are from H1 Adloff et al. (2000b).  (a)  Figure 13:
Longitudinal cross section (from near-threshold to far-from-threshold) for the
electro-production of phi mesons versus $Q^{2}$ in GeV2, for $W=\sqrt{s}=75$
GeV (solid-blue curve) in (VII.89). The solid-blue curve is only with the
evolved gluon GPD. The black data points are from H1, ZEUS Aaron et al.
(2010).

###  VII.3 Results for exclusive $\phi$ production

In Fig. 12 we show the longitudinal cross section (from near-threshold to far-
from-threshold) for the electro-production of phi mesons versus $W=\sqrt{s}$
in GeV, for $Q^{2}=2.21$ GeV2 (dashed-purple curves) and $Q^{2}=3.8$ GeV2
(solid-blue curves) in (VII.89) with $\mathcal{N}_{t}=162.358\times 10^{3}$
for the far-from-threshold regime (and in (VII.95) with
$\mathcal{N}_{t}=583.248$ for the near-threshold regime). Note that the
dashed-purple curves and solid-blue curves are only with the evolved gluon GPD
(i.e., we have set the strange singlet quark GPD to zero). We have also set
the s-channel background contribution to zero assuming the coupling of phi
meson to proton ($g_{\phi pp}=0$) to be zero. The orange data point is from
CLAS 2008 Santoro et al. (2008). The brown data point is from CLAS 2001
Lukashin et al. (2001), see also Fig.5 in Kroll (2015). The cyan data point is
from HERMES 2000, see also Fig.5 in Kroll (2015). The grey data points are
from ZEUS Chekanov et al. (2005), see also Fig.5 in Kroll (2015). The pink
data points are from H1 Adloff et al. (2000b), see also Fig.5 in Kroll (2015).

In Fig. 13 we show the longitudinal cross section (from near-threshold to far-
from-threshold) for the electro-production of phi mesons versus $Q^{2}$ in
GeV2, for $W=\sqrt{s}=75$ GeV (solid-blue curve) in (VII.89) with
$\mathcal{N}_{t}=154.351$ for the far-from-threshold regime. Note that the
solid-blue curve is only with the evolved gluon GPD (i.e., we have set the
strange singlet quark GPD to zero). We have also set the s-channel
contribution to zero assuming the coupling of phi meson to proton ($g_{\phi
pp}=0$) to be zero. The black data points are from H1, ZEUS Aaron et al.
(2010), see also Fig.7 of Favart et al. (2016).

##  VIII Electroproduction of longitudinal $\rho^{+}$ meson with evolved
valence quark GPDs

The electroproduction of longitudinal charged rho meson ($\rho^{+}$) in terms
of quark DAs and quark GPDs is given by (see Eq.219 with Eq.222 and Eq.223 in
Goeke et al. (2001))

$\displaystyle{\cal
A}^{LL}_{\gamma^{*}p\rightarrow\rho^{+}n}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})=$
$\displaystyle-e\times{C_{F}\over N_{c}}\times{1\over
2}\times(4\pi\alpha_{s}(\mu^{2}))\times{1\over{Q}}\times\left[\,\int_{0}^{1}dz{{\Phi_{\rho^{+}}(z)}\over
z}\right]\times\frac{1}{p^{+}}\left\\{A_{\rho^{+}\,n}\,h^{+}\,+\,B_{\rho^{+}\,n}\,e^{+}\right\\}\,,$
(VIII.96)

where $\Phi_{\rho^{+}}(z)$ is the charged rho meson ($\rho^{+}$) distribution
amplitude (DA), and

$\displaystyle A_{\rho^{+}\,n}\,$ $\displaystyle=$
$\displaystyle\,\int_{-1}^{1}dx\;{\left(H^{u}(x,\eta,t;\mu^{2})\,-\,H^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{{{Q_{u}}\over{\xi-
x-i\epsilon}}-{{Q_{d}}\over{\xi+x-i\epsilon}}\right\\}$ $\displaystyle
B_{\rho^{+}\,n}\,$ $\displaystyle=$
$\displaystyle\,\int_{-1}^{1}dx\;{\left(E^{u}(x,\eta,t;\mu^{2})\,-\,E^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{{{Q_{u}}\over{\xi-
x-i\epsilon}}-{{Q_{d}}\over{\xi+x-i\epsilon}}\right\\}\,.$ (VIII.97)

Using $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$, and $Q_{u}=+2/3\,(Q_{d}=-1/3)$, we
can rewrite (VIII-VIII) as

$\displaystyle{\cal
A}^{LL}_{\gamma^{*}p\rightarrow\rho^{+}n}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle\,=\,$ $\displaystyle-e\times{1\over
2}\times\left(1-\frac{1}{N_{c}^{2}}\right)\times
4\pi\times\alpha_{s}(\mu^{2})$ (VIII.98) $\displaystyle\times$
$\displaystyle{1\over{Q}}\times\left[\,\int_{0}^{1}dz{{\Phi_{\rho^{+}}(z)}\over
z}\right]\times\frac{1}{p^{+}}\times u(p_{2})\gamma^{+}u(p_{1})$
$\displaystyle\times$
$\displaystyle\Bigg{[}\int_{0}^{1}dx\;{\left(H_{valence}^{u}(x,\eta,t;\mu^{2})\,-\,H_{valence}^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{\frac{\xi}{2x}\times\frac{2x}{\xi^{2}-x^{2}}\right\\}$
$\displaystyle+$
$\displaystyle\int_{0}^{1}dx\;{\left(H_{singlet}^{u}(x,\eta,t;\mu^{2})\,-\,H_{singlet}^{d}(x,\eta,t;\mu^{2})\right)}\;\left\\{\frac{1}{6}\times\frac{2x}{\xi^{2}-x^{2}}\right\\}$
$\displaystyle+$ $\displaystyle i\pi\times
Q_{u}\left(H^{u}(\xi,\eta,t;\mu^{2})\,-\,H^{d}(\xi,\eta,t;\mu^{2})\right)$
$\displaystyle-i$ $\displaystyle\pi\times
Q_{d}\left(H^{u}(-\xi,\eta,t;\mu^{2})\,-\,H^{d}(-\xi,\eta,t;\mu^{2})\right)\Bigg{]}\,,$

where we have defined
$H_{valence}^{q}(x,\eta,t;\mu^{2})=H^{q}(x,\eta,t;\mu^{2})+H^{q}(-x,\eta,t;\mu^{2})$,
$H_{singlet}^{q}(x,\eta,t;\mu^{2})=H^{q}(x,\eta,t;\mu^{2})-H^{q}(-x,\eta,t;\mu^{2})$,
and we have ignored the Pauli contribution for $-t\ll 4m_{N}^{2}$.

The details of the reduction of (VIII.98) are given in Appendix L. Using the
result for $\text{Re}\left[{\cal A}^{LL(valence)}\right]$ in (L) and for
$\text{Im}\left[{\cal A}^{LL(valence)}\right]$ in (L.383), the full complex
amplitude with the evolved valence quark GPDs is

$\displaystyle{\cal
A}^{LL(valence)}_{\gamma^{*}p\rightarrow\rho^{+}n}(s,t,Q,\epsilon_{L},\epsilon_{L}^{\prime})$
$\displaystyle\simeq\Bigg{(}\widehat{\mathbb{F}}^{u}_{j_{0q}(valence)}(\xi,t;\mu^{2})-\widehat{\mathbb{F}}^{d}_{j_{0q}(valence)}(\xi,t;\mu^{2})\Bigg{)}\times\Bigg{[}(\sqrt{\lambda}/4\pi)^{1/2}\times\frac{1}{\xi}\times(\sqrt{\lambda}/2\pi)\times
e^{-\tilde{\tau}/\sqrt{\lambda}}\times\frac{e^{-\sqrt{\lambda}/4\tilde{\tau}}}{\tilde{\tau}^{3/2}}$
$\displaystyle+i\times\frac{1}{\xi^{j_{0q}}}\times(\sqrt{\lambda}/4\pi)^{1/2}\times\frac{e^{-\sqrt{\lambda}/4\tilde{\tau}}}{\tilde{\tau}^{3/2}}\Bigg{]}\times\mathcal{N}_{q(valence)}(j_{0q})\times(-1)\times
e\,f_{V}^{+}\,\alpha_{s}(\mu)\,{1\over{Q}}\,\frac{1}{2m_{N}}\,\bar{u}(p_{2})u(p_{1})\,,$
$\displaystyle\simeq\Bigg{(}\widehat{\mathbb{F}}^{u}_{j_{0q}(valence)}(\xi,t;\mu^{2})-\widehat{\mathbb{F}}^{d}_{j_{0q}(valence)}(\xi,t;\mu^{2})\Bigg{)}\times\mathcal{N}_{q(valence)}(j_{0q})\times(-1)\times
e\,f_{V}^{+}\,\alpha_{s}(\mu)\,{1\over{Q}}\,\frac{1}{2m_{N}}\,\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times\left[\frac{(\sqrt{\lambda}/2\pi)}{\xi^{1-1/\sqrt{\lambda}}}+\frac{i}{\xi^{1-1/\sqrt{\lambda}}}\right]\times(\sqrt{\lambda}/4\pi)^{1/2}\times\frac{e^{-\sqrt{\lambda}/4\tilde{\tau}}}{\tilde{\tau}^{3/2}}\,.$
(VIII.99)

Finally, using the valence quark GPDs contribution (VIII), we have the total
longitudinal differential cross section for electroproduction of $\rho^{+}$
meson

$\displaystyle\frac{d\sigma_{L}(\rho^{+})}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi s^{2}}\times\frac{1}{2}\sum_{\rm
spin}\Big{|}\mathcal{A}_{\gamma^{*}p\rightarrow\rho^{+}n}^{LL(valence)}\Big{|}^{2}\,.$

We also compute the total cross section using the optical theorem as

$\displaystyle\sigma_{L}(\rho^{+})=\left(\frac{16\pi\,\frac{d\sigma_{L}(\rho^{+})}{dt}}{1+\rho}\right)^{1/2}_{t=0}\,,$
(VIII.101)

where $\rho$ is the ratio of the real and imaginary part of our total complex
amplitude. We have compared our total cross section (VIII.101) to the
experimental data in Fig. 14 \- 15. We have also used $e=0.3$,
$m_{N}=0.94~{}\text{GeV}$, $m_{\rho^{+}}=0.775~{}\text{GeV}$, and
$f_{\rho^{+}}=0.216~{}\text{GeV}$, in addition to the same parameters as
neutral rho meson $\lambda=11.243$, $\tau=3$,
$\tilde{\kappa}_{T,S}=0.388~{}\text{GeV}$,
$\tilde{\kappa}_{N,V}=0.402~{}\text{GeV}$ with the values of the normalization
parameter $\mathcal{N}_{t}$ fixed by the charged rho meson data as detailed in
the next subsection below.

(a)  (b)  (c)  (d)  (e)  (f)

Figure 14: Longitudinal cross section for the electro-production of charged
rho mesons ($\rho^{+}$) versus $W=\sqrt{s}$ in GeV, for $Q^{2}=1-1.5$ GeV2
(a), $Q^{2}=1.5-2$ GeV2 (b), $Q^{2}=2-2.5$ GeV2 (c), $Q^{2}=2.5-3$ GeV2 (d),
$Q^{2}=3-3.5$ GeV2 (e), $Q^{2}=3.5-4$ GeV2 (f). The shaded-blue curve is with
the evolved valence up and down quark GPDs (i.e., (VIII.101). The black data
points are from 5.776 GeV CLAS Fradi (2011). (a)  Figure 15: Longitudinal
cross section for the electro-production of charged rho mesons ($\rho^{+}$)
versus $W=\sqrt{s}$ in GeV, for $Q^{2}=4-4.5$ GeV2 (a). The shaded-blue curve
is with the evolved valence up and down quark GPDs (i.e., (VIII.101). The
black data points are from 5.776 GeV CLAS Fradi (2011).

###  VIII.1 Results for exclusive $\rho^{+}$ production

In Fig. 14 we show the longitudinal cross section for the electro-production
of charged rho mesons ($\rho^{+}$) versus $W=\sqrt{s}$ in GeV, for
$Q^{2}=1-1.5$ GeV2 (a), $Q^{2}=1.5-2$ GeV2 (b), $Q^{2}=2-2.5$ GeV2 (c),
$Q^{2}=2.5-3$ GeV2 (d), $Q^{2}=3-3.5$ GeV2 (e), $Q^{2}=3.5-4$ GeV2 (f). The
shaded-blue curve is with the evolved valence up and down quark GPDs (i.e.,
(VIII.101) with $\mathcal{N}_{t}=0.00192\pm 0.000742$ and $\mathcal{N}_{s}=0$.
We have have also set the s-channel contribution to zero, assuming the
coupling of the $\rho^{+}$ meson to the proton and neutron to be zero
($g_{\rho^{+}pn}=0$). The black data points are from 5.776 GeV CLAS Fradi
(2011), see Fig.4 of Fradi (2011).

In Fig. 15 we show the longitudinal cross section for the electro-production
of charged rho mesons ($\rho^{+}$) versus $W=\sqrt{s}$ in GeV, for
$Q^{2}=4-4.5$ GeV2 (a). The shaded-blue curve is with the evolved valence up
and down quark GPDs (i.e., (VIII.101) with $\mathcal{N}_{t}=0.00192\pm
0.000742$ and $\mathcal{N}_{s}=0$. We have also set the s-channel contribution
to zero assuming the coupling of the $\rho^{+}$ meson to the proton and
neutron to be zero ($g_{\rho^{+}pn}=0$). The black data points are from 5.776
GeV CLAS Fradi (2011), see Fig.4 of Fradi (2011).

##  IX Summary and Conclusion

We have provided a framework for the construction of the GPDs at low-x and
finite skewness, using holographic QCD, a dual description that captures the
stringy aspects of QCD in the double limit of large $N_{c}$ and strong gauge
coupling. In this limit, we have explicitly constructed the holographic
amplitudes for the electroproduction of photons, vector mesons, pair of pions
and neutral pions. At low-x they are dominated by the bulk spin-j gravitons
and vector or axial meson exchanges in AdS, dual to Pomerons and Reggeons in
QCD.

The leading twist-2 quark and gluon off-diagonal matrix elements, are directly
tied to the Gegenbauer or conformal moments of the gluon and quark GPDs, at a
fixed renormalization scale $\mu$. We have suggested that for the initial or
low renormalization point with $\mu\sim\mu_{0}\sim 1.8$ GeV, these moments can
be explicitly obtained from holographic QCD at finite skewness, in the Regge
limit. The matching in $1/N_{c}$, is found to hold for all of the exclusive
electroproduction processes we have considered.

Using the holographically fixed Gegenbauer moments of the flavored and gluonic
GPDs at $\mu_{0}$, we have used the one-loop pQCD evolution equations to
obtain their counterparts at a higher resolution $\mu$. The evolution enhances
the perturbative partonic content at low-x, in addition to the nonperturbative
one captured by the holographic construction.

The holographic QCD results are derived at strong ′t Hooft coupling
$\lambda=g^{2}_{s}N_{c}$, while the evolution equations are based on weak
gauge coupling $\alpha_{s}=g_{s}^{2}/4\pi$. We have shown that for the fixed
coupling $\lambda\sim 11$ used in all of our holographic analyses, this maps
to $\alpha_{s}(\mu_{0})\sim\frac{1}{3}$ with $\mu_{0}\sim 1.8$ GeV using a
running coupling. Remarkably, this scale is usually regarded as the interface
between perturbative and non-perturbative physics.

We have used our evolved results to analyse specifically the exclusive
production of rho and phi mesons. The electroproduction of neutral $\rho^{0}$,
depends explicitly on the flavor singlet and gluon GPDs. The electroproduction
of charged $\rho^{+}$, is sensitive to the flavor non-singlet up and down
quark GPDs. The electroproduction of $\phi$ is sensitive to the gluon GPDs.
The comparison to the existing data from CLAS, HERMES and H1 although sparse,
is overall consistent with our evolved results at large $\sqrt{s}$ in the
Regge kinematics with $|t|\ll s$. The s-channel holographic contribution at
low $\sqrt{s}$ is in a good agreement with the few available data points.

Although our detailed comparison to the data has been limited to the
longitudinal light vector mesons $\rho^{0,+}$ and $\phi$, our analyses and
results can be extended to the longitudinal heavier vector mesons such as
$J/\Psi$ and $\Upsilon$ with minor changes. They also yield detailed
predictions for the exclusive processes, such as the electroproduction of
photons in DVCS, or the longitudinal electroproduction of single and double
axial mesons. The detailed comparison of our results for these processes to
the existing experimental data will be discussed elsewhere.

Finally, the extracted quark and gluon GPDs for fixed skewness $\eta$, as
fixed by holography at the initial scale $\mu_{0}$, or after evolution at a
scale $\mu$, provide very explicit and detailed dependence on the kinematical
variables $t$, and $\eta$ with their moments satisfying all the polynomilality
conditions in $\eta$. As such, they should prove useful for the extration of
the GPDs using global data analyis. Needless to say that they also carry the
detailed tomographic information about the partonic distributions inside a
nucleon after Fourier transforming to coordinate space with respect to $t$. We
will return to some of these issues next.

Acknowledgments

K.M. is supported by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, contract no. DE-AC02-06CH11357, and an LDRD initiative at
Argonne National Laboratory under Project No. 2020-0020. I.Z. is supported by
the Office of Science, U.S. Department of Energy under Contract No. DE-
FG-88ER40388.

##  Appendix A Kinematics

Following Belitsky and Radyushkin (2005), we define

$\tilde{q}=\frac{1}{2}(q+q^{\prime})\,,\qquad
p=\frac{1}{2}(p_{1}+p_{2})\,,\qquad\Delta=p_{1}-p_{2}=q^{\prime}-q\,.$ (A.102)
$\tilde{q}^{2}=-\tilde{Q}^{2}\,,\qquad\xi=\frac{\tilde{Q}^{2}}{2p\cdot\tilde{q}}\,,\qquad\eta=\frac{\Delta\cdot\tilde{q}}{2p\cdot\tilde{q}}\,.$
(A.103)

$Q^{2}=-q^{2}\,,\qquad Q^{\prime 2}=-q^{\prime 2}\,,\qquad x_{\rm
B}=\frac{{Q}^{2}}{2p_{1}\cdot q}\,,$ (A.104)
$\tilde{Q}^{2}=\frac{1}{2}\left(Q^{2}+Q^{\prime
2}+\frac{\Delta^{2}}{2}\right)\,,$ (A.105)

$\xi=\frac{Q^{2}+Q^{\prime 2}+\Delta^{2}/2}{2Q^{2}/x_{\rm B}-Q^{2}+Q^{\prime
2}+\Delta^{2}}\,,\qquad\eta=\frac{Q^{2}-Q^{\prime 2}}{2Q^{2}/x_{\rm
B}-Q^{2}+Q^{\prime 2}+\Delta^{2}}\,,$ (A.106)

$Q^{\prime
2}=\left(1-\frac{\eta}{\xi}\right)\tilde{Q}^{2}-\frac{\Delta^{2}}{4}\,,\qquad
Q^{2}=\left(1+\frac{\eta}{\xi}\right)\tilde{Q}^{2}-\frac{\Delta^{2}}{4}\,,$
(A.107)

and

$x_{\rm
B}=\frac{(\xi+\eta)\tilde{Q}^{2}-\xi\Delta^{2}/4}{(1+\eta)\tilde{Q}^{2}-\xi\Delta^{2}/2}\,.$
(A.108)

We also introduce a pair of light-cone vectors such that $n^{2}=n^{\ast 2}=0$
and $n\cdot n^{\ast}\equiv n^{\mu}n^{\ast}_{\mu}=1$. They can be chosen as

$n^{\mu}\equiv\frac{1}{\sqrt{2}}(1,0,0,-1)\,,\qquad
n^{\ast\mu}\equiv\frac{1}{\sqrt{2}}(1,0,0,1)\,.$ (A.109)

Any four-vector $a^{\mu}$ can be decomposed into its light-cone components as

$a^{\mu}=a^{+}n^{\ast\mu}+a^{-}n^{\mu}+a^{\mu}_{\perp}\,,$ (A.110)

with

$a^{+}\equiv a\cdot n=\frac{1}{\sqrt{2}}(a^{0}+a^{z})\,,\qquad a^{-}\equiv
a\cdot n^{\ast}=\frac{1}{\sqrt{2}}(a^{0}-a^{z})\,.$ (A.111)

Scalar products are written as

$a\cdot y\equiv
a_{\mu}y^{\mu}=a^{+}y^{-}+a^{-}y^{+}-{a}_{\perp}\cdot{y}_{\perp}\,.$ (A.112)

The light-cone derivatives are defined as follows

$\partial^{+}=\frac{\partial}{\partial
a^{-}}\,,\qquad\partial^{-}=\frac{\partial}{\partial
a^{+}}\,,\qquad\partial^{\mu}_{\perp}=\frac{\partial}{\partial
a_{\mu}^{\perp}}$ (A.113)

##  Appendix B Quark operators

We define leading-twist bilocal quark operators as Belitsky and Radyushkin
(2005):

$\displaystyle{\cal O}^{qq}(w^{-}_{1},w^{-}_{2})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\bar{\psi}(w^{-}_{1})[w^{-}_{1},w^{-}_{2}]\gamma^{+}\psi(w^{-}_{2})\,,$
(B.114) $\displaystyle\widetilde{\cal O}^{qq}(w^{-}_{1},w^{-}_{2})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\bar{\psi}(w^{-}_{1})[w^{-}_{1},w^{-}_{2}]\gamma^{+}\gamma^{5}\psi(w^{-}_{2})\,,$
(B.115)

where have added the Wilson line in the adjoint representation

$[w^{-}_{1},w^{-}_{2}]^{ab}=P\exp\left(g\int_{w^{-}_{2}}^{w^{-}_{1}}dw^{\prime-}f^{abc}A_{c}^{+}(w^{\prime-},{\bf
0}_{\perp})\right)\,,$

to respect gauge invariance.

##  Appendix C Gluon operators

We define the two-dimensional tensors Belitsky and Radyushkin (2005)

$\displaystyle g^{\perp}_{\mu\nu}\\!\\!\\!$ $\displaystyle\equiv$
$\displaystyle\\!\\!\\!g_{\mu\nu}-n_{\mu}n^{\ast}_{\nu}-n_{\nu}n^{\ast}_{\mu}\,,$
(C.116) $\displaystyle\varepsilon^{\perp}_{\mu\nu}\\!\\!\\!$
$\displaystyle\equiv$
$\displaystyle\\!\\!\\!\varepsilon^{\alpha\beta\rho\sigma}g^{\perp}_{\alpha\mu}g^{\perp}_{\beta\nu}n^{\ast}_{\rho}n_{\sigma}\,,$
(C.117)

where the totally antisymmetric tensor is normalized as
$\varepsilon^{0123}=1$.

We also define the gauge potential in the light-cone gauge

$A_{\perp}\equiv A^{x}+iA^{y}\,,\qquad\bar{A}_{\perp}\equiv A^{x}-iA^{y}\,,$
(C.118)

and the gluon field-strength tensors

$\displaystyle\partial^{+}A_{\perp}=F^{+x}-i\widetilde{F}^{+x}=i\left(F^{+y}-i\widetilde{F}^{+y}\right)\,,\qquad\partial^{+}\bar{A}_{\perp}=F^{+x}+i\widetilde{F}^{+x}=-i\left(F^{+y}+i\widetilde{F}^{+y}\right)\,,$

$F^{\
+\mu}_{[-]\,\perp}=F^{+\mu}_{\phantom{+}\,\perp}-i\widetilde{F}^{+\mu}_{\phantom{+}\,\perp}\,,\qquad
F^{\
+\mu}_{[+]\,\perp}=F^{+\mu}_{\phantom{+}\,\perp}+i\widetilde{F}^{+\mu}_{\phantom{+}\,\perp}\,,$
(C.119)

with the dual gluon field strength defined as $\widetilde{F}^{\mu\nu}\equiv
12\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$, so that
$\widetilde{F}^{+}{}^{\mu}_{\perp}=\varepsilon^{\mu\nu}_{\perp}F^{+}{}^{\perp}_{\nu}$
for two-dimensional transverse indices.

Using these definitions we introduce the leading-twist gluonic operators
Belitsky and Radyushkin (2005):

$\displaystyle{\cal O}^{gg}(w^{-}_{1},w^{-}_{2})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!F^{+\mu}_{a}(w^{-}_{1})[w^{-}_{1},w^{-}_{2}]^{ab}g_{\mu\nu}F^{\nu+}_{b}(w^{-}_{2})\,,$
(C.120) $\displaystyle\widetilde{\cal O}^{gg}(w^{-}_{1},w^{-}_{2})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!F^{+\mu}_{a}(w^{-}_{1})[w^{-}_{1},w^{-}_{2}]^{ab}i\varepsilon^{\perp}_{\mu\nu}F^{\nu+}_{b}(w^{-}_{2})\,,$
(C.121)

##  Appendix D Operator matrix elements and GPDs

We define the spinor bilinears Belitsky and Radyushkin (2005)

$\begin{array}[]{ll}b=\bar{u}(p_{2})u(p_{1})\,,&\widetilde{b}=\bar{u}(p_{2})\gamma^{5}u(p_{1})\,,\\\
h^{\mu}=\bar{u}(p_{2})\gamma^{\mu}u(p_{1})\,,&\widetilde{h}^{\mu}=\bar{u}(p_{2})\gamma^{\mu}\gamma^{5}u(p_{1})\,,\\\
t^{\mu\nu}=\bar{u}(p_{2})i\sigma^{\mu\nu}u(p_{1})\,,&\widetilde{t}^{\mu\nu}=\bar{u}(p_{2})i\sigma^{\mu\nu}\gamma^{5}u(p_{1})\,.\end{array}$
(D.122)

The matrix elements for quark operators can be decomposed in terms of the
above spinor bilinears as

$\displaystyle\langle p_{2}|{\cal O}^{qq}(-w^{-},w^{-})|p_{1}\rangle\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\int_{-1}^{1}dx\ {\rm
e}^{-ixp^{+}w^{-}}\left\\{h^{+}H^{q}(x,\eta,\Delta^{2})+e^{+}E^{q}(x,\eta,\Delta^{2})\right\\}$
(D.123) $\displaystyle\equiv$
$\displaystyle\\!\\!\\!p^{+}\int_{-1}^{1}dx\,{\rm
e}^{-ixp^{+}w^{-}}F^{q}(x,\eta,\Delta^{2})\,,$ $\displaystyle\langle
p_{2}|\widetilde{\cal O}^{qq}(-w^{-},w^{-})|p_{1}\rangle\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\int_{-1}^{1}dx\ {\rm
e}^{-ixp^{+}w^{-}}\left\\{\tilde{h}^{+}\widetilde{H}^{q}(x,\eta,\Delta^{2})+\tilde{e}^{+}\widetilde{E}^{q}(x,\eta,\Delta^{2})\right\\}$
(D.124) $\displaystyle\equiv$
$\displaystyle\\!\\!\\!p^{+}\int_{-1}^{1}dx\,{\rm
e}^{-ixp^{+}w^{-}}\widetilde{F}^{q}(x,\eta,\Delta^{2})\,.$

Similarly, the matrix elements for gluon operators can be decomposed in terms
of the spinor bilinears as

$\displaystyle\langle p_{2}|{\cal O}^{gg}(-w^{-},w^{-})|p_{1}\rangle\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\frac{1}{4}p^{+}\int_{-1}^{1}dx\ {\rm
e}^{-ixp^{+}w^{-}}\left\\{h^{+}H^{g}(x,\eta,\Delta^{2})+e^{+}E^{g}(x,\eta,\Delta^{2})\right\\}$
(D.125) $\displaystyle\equiv$
$\displaystyle\\!\\!\\!\frac{(p^{+})^{2}}{4}\int_{-1}^{1}dx\,{\rm
e}^{-ixp^{+}w^{-}}F^{g}(x,\eta,\Delta^{2})\,,$ $\displaystyle\langle
p_{2}|\widetilde{\cal O}^{gg}(-w^{-},w^{-})|p_{1}\rangle\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\frac{1}{4}p^{+}\int_{-1}^{1}dx\ {\rm
e}^{-ixp^{+}w^{-}}\left\\{\tilde{h}^{+}\widetilde{H}^{g}(x,\eta,\Delta^{2})+\tilde{e}^{+}\widetilde{E}^{g}(x,\eta,\Delta^{2})\right\\}$
(D.126) $\displaystyle\equiv$
$\displaystyle\\!\\!\\!\frac{(p^{+})^{2}}{4}\int_{-1}^{1}dx\,{\rm
e}^{-ixp^{+}w^{-}}\widetilde{F}^{g}(x,\eta,\Delta^{2})\,.$

##  Appendix E Soft-wall holographic QCD

A simple way to capture AdS/CFT duality in the non-conformal limit is to model
it using a slice of AdS5 with various bulk fields with assigned anomalous
dimensions and pertinent boundary values, in the so-called bottom-up approach
which we will follow here using the conventions in our recent work in DIS
scattering Mamo and Zahed (2020, 2021c). We consider AdS5 with a soft wall
with a background metric $g_{MN}=(\eta_{\mu\nu},-1)R^{2}/z^{2}$ with the flat
metric $\eta_{\mu\nu}=(1,-1,-1,-1)$ at the boundary. Confinement will be
described by a harmonic background dilaton
$\phi={\tilde{\kappa}_{N}}^{2}z^{2}$.

###  E.1 Bulk vector mesons

The vector mesons fields $L,R$ are described by the bulk effective action Hirn
and Sanz (2005); Domokos et al. (2009)

$\displaystyle S_{M}=$ $\displaystyle-\frac{1}{4g_{5}^{2}}\int
d^{5}xe^{-\phi(z)}\sqrt{g}\,g^{MP}g^{NQ}{\rm Tr}\bigg{(}{\cal F}^{L}_{MN}{\cal
F}^{L}_{PQ}+{\cal F}^{R}_{MN}{\cal F}^{R}_{PQ}\bigg{)}+\int
d^{5}x\,\bigg{(}\omega^{L}_{5}({\cal A})-\omega^{R}_{5}({\cal A})\bigg{)}$
(E.127)

with the Chern-Simons contribution

$\displaystyle\omega_{5}({\cal A})=\frac{N_{c}}{24\pi^{2}}\int d^{5}x\,{\rm
Tr}\left({\cal AF}^{2}+\frac{1}{2}{\cal A}^{3}{\cal F}-\frac{1}{10}{\cal
A}^{5}\right)$ (E.128)

Here ${\cal F}=d{\cal A}-i{\cal A}^{2}$ and ${\cal A}={\cal A}^{a}T^{a}$ with
$T^{0}=\frac{3}{4}{\bf 1_{2}}$ and $T^{i}=\frac{1}{4}\tau^{i}$, with the form
notation subsumed. Also the vector fields are given by $V=(R+L)/2$ and the
axial-vector fields are given by $A=(R-L)/2$. The coupling $g_{5}$ in (E.127)
is fixed by the brane embeddings in bulk, or phenomenologically as
${1}/{g_{5}^{2}}\equiv{N_{c}}/(12\pi^{2})$ Cherman et al. (2009).

The flavor gauge fields solve

$\displaystyle\Box
V^{\mu}+ze^{\tilde{\kappa}_{N}^{2}z^{2}}\partial_{z}\Big{(}e^{-\tilde{\kappa}_{N}^{2}z^{2}}\frac{1}{z}\partial_{z}V^{\mu}\Big{)}\,=\,0\qquad\Box
V_{z}-\partial_{z}\Big{(}\partial_{\mu}V^{\mu}\Big{)}\,=\,0\,.$ (E.129)

subject to the gauge condition

$\partial_{\mu}V^{\mu}\,+\,ze^{\kappa^{2}z^{2}}\partial_{z}\Big{(}e^{-\kappa^{2}z^{2}}\frac{1}{z}V_{z}\Big{)}\,=\,0\,,$
(E.130)

with the boundary condition $V_{\mu}(z,y)|_{z\to
0}\,=\,\epsilon_{\mu}(q)\,e^{-iq\cdot y}$. The non-normalizable solutions are

$\displaystyle V_{\mu}(z,y)$ $\displaystyle=$
$\displaystyle\epsilon_{\mu}(q)\,e^{-iq\cdot
y}\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}_{V}^{2}}\bigg{)}\,\,\tilde{\kappa}_{N}^{2}z^{2}\,\,{\cal
U}\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}_{N}^{2}};2;\tilde{\kappa}_{N}^{2}z^{2}\bigg{)}$
$\displaystyle V_{z}(z,y)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\,\epsilon(q)\cdot q\,e^{-iq\cdot
y}\,\,\Gamma\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}_{V}^{2}}\bigg{)}\,\,z\,\,{\cal
U}\bigg{(}1+\frac{Q^{2}}{4\tilde{\kappa}_{V}^{2}};1;\tilde{\kappa}_{V}^{2}z^{2}\bigg{)}\,,$
(E.131)

with $\,{\cal U}(a;b;w)\,$ the confluent hypergeometric functions of the
second kind.

###  E.2 Bulk Dirac fermions

The bulk Dirac fermion action in a sliced of AdS5 is

$\displaystyle S_{F}=\frac{1}{2g_{5}^{2}}\int
d^{5}x\,e^{-\phi(z)}\,\sqrt{g}\,\Big{(}\mathcal{L}_{F1}+\mathcal{L}_{F2}\Big{)}+\frac{1}{2g_{5}^{2}}\int
d^{4}x\sqrt{-g^{(4)}}\,\Big{(}\mathcal{L}_{UV1}+\mathcal{L}_{UV2}\Big{)}\,,$
(E.132)

The Dirac and Pauli contributions to ${\cal L}_{F1,2}$ are respectively

$\displaystyle\mathcal{L}_{\rm Dirac1,2}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{i}{2}\overline{\Psi}_{1,2}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{D}_{N}^{L,R}-\overleftarrow{D}_{N}^{L,R}\big{)}\Psi_{1,2}-(\pm
M+V(z))\bar{\Psi}_{1,2}\Psi_{1,2}\bigg{)}\,,$ $\displaystyle\mathcal{L}_{\rm
Pauli1,2}$ $\displaystyle=$ $\displaystyle\pm
2g_{5}^{2}\times\eta\,\bar{\Psi}_{1,2}e^{M}_{A}e^{N}_{B}\sigma^{AB}{\cal
F}^{L,R}_{MN}\Psi_{1,2}\,,$

with $V(z)={\tilde{\kappa}}^{2}z^{2}$, $e^{N}_{A}=z\delta^{N}_{A}$,
$\sigma^{AB}=\frac{i}{2}[\Gamma^{A},\Gamma^{B}]$, and $\omega_{\mu
z\nu}=-\omega_{\mu\nu z}=\frac{1}{z}\eta_{\mu\nu}$. The Dirac gamma matrices
$\Gamma^{A}=(\gamma^{\mu},-i\gamma^{5})$ are chosen in the chiral
representation. They satisfy the flat anti-commutation relation
$\\{\Gamma^{A},\Gamma^{B}\\}=2\eta^{AB}$. The left and right covariant
derivatives are defined as

$\displaystyle\overrightarrow{D}_{N}^{X=L,R}=$
$\displaystyle\overrightarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]-iX_{N}^{a}T^{a}\equiv\overrightarrow{\mathcal{D}}_{N}-iX_{N}^{a}T^{a}$
$\displaystyle\overleftarrow{D}_{N}^{X=L,R}=$
$\displaystyle\overleftarrow{\partial}_{N}+\frac{1}{8}\omega_{NAB}[\Gamma^{A},\Gamma^{B}]+iX_{N}^{a}T^{a}\equiv\overleftarrow{\mathcal{D}}_{N}+iX_{N}^{a}T^{a}$
(E.134)

The nucleon doublet refers to

$\displaystyle\Psi_{1,2}\equiv\begin{pmatrix}\Psi_{p1,2}\\\
\Psi_{n1,2}\end{pmatrix}\,.$ (E.135)

The nucleon fields in bulk form an iso-doublet $p,n$ with $1,2$ referring to
their boundary chirality $1,2=\pm=R,L$ Hong et al. (2007). They are dual to
the boundary sources $\Psi_{p1,2}\leftrightarrow{\cal O}_{p,\pm}$ and
$\Psi_{n1,2}\leftrightarrow{\cal O}_{n,\pm}$ with anomalous dimensions $\pm
M=\pm(\Delta-2)=\pm(\tau-3/2)$.

The equation of motions for the bulk Dirac chiral doublet is

$\displaystyle\bigg{(}ie^{N}_{A}\Gamma^{A}D_{N}^{L,R}-\frac{i}{2}(\partial_{N}\phi)\,e^{N}_{A}\Gamma^{A}-(\pm
M+V(z))\bigg{)}\Psi_{1,2}=0\,,$ (E.136)

The normalizable solution to (E.136) are

$\displaystyle\Psi_{1}(p,z;n)$ $\displaystyle=$
$\displaystyle\psi_{R}(z;n)\Psi^{0}_{R}(p)+\psi_{L}(z;n)\Psi^{0}_{L}(p)$
$\displaystyle\Psi_{2}(p,z;n)$ $\displaystyle=$
$\displaystyle\psi_{R}(z;n)\Psi^{0}_{L}(p)+\psi_{L}(z;n)\Psi^{0}_{R}(p)$
(E.137)

with the normalized bulk wave functions

$\displaystyle\psi_{R}(z;n)=z^{\Delta}\times\tilde{\psi}_{R}(z;n)=n_{R}{\tilde{\xi}_{N}^{\tau-\frac{3}{2}}}L^{(\tau-2)}_{n}(\tilde{\xi}_{N})\times
z^{\Delta}\,,$
$\displaystyle\psi_{L}(z;n)=z^{\Delta}\times\tilde{\psi}_{L}(z;n)=\tilde{n}_{L}{\tilde{\xi}_{N}^{\tau-1}}L^{(\tau-1)}_{n}(\tilde{\xi}_{N})\times
z^{\Delta}\,,$

Here $\Delta=\tau+\frac{1}{2}$, $\tilde{\xi}_{N}=\tilde{\kappa}_{N}^{2}z^{2}$,
$L_{n}^{(\alpha)}(\tilde{\xi})$ are the generalized Laguerre, and
$n_{R}=n_{L}\sqrt{\tau-1}$ and
$n_{L}=\tilde{\kappa}^{-(\tau-2)}\sqrt{{2}/{\Gamma(\tau)}}$. The free Weyl
spinors $\Psi^{0}_{R/L}(p)=P_{\pm}u(p)$ and
$\bar{\Psi}^{0}_{R/L}(p)=\bar{u}(p)P_{\mp}$, and the free boundary spinors
satisfy

$\displaystyle\bar{u}(p)u(p)=2m_{N}\qquad\qquad
2m_{N}\times\bar{u}(p^{\prime})\gamma^{\mu}u(p)=\bar{u}(p^{\prime})(p^{\prime}+p)^{\mu}u(p)\,.$
(E.139)

The fermionic spectrum Reggeizes
$m_{n}^{2}=4\tilde{\kappa}_{N}^{2}(n+\tau-1)$. The assignments $1=+$ and $2=-$
at the boundary are commensurate with the substitutions
$\psi_{R,L}\leftrightarrow\mp\psi_{L,R}$ by parity.

Using the Dirac 1-form currents

$\displaystyle J^{aN}_{L}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Dirac1}}{\partial
L^{a}_{N}}=\overline{\Psi}_{1}e^{N}_{A}\Gamma^{A}T^{a}\Psi_{1}\,,$
$\displaystyle J^{aN}_{R}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Dirac2}}{\partial
R^{a}_{N}}=\overline{\Psi}_{2}e^{N}_{A}\Gamma^{A}T^{a}\Psi_{2}\,,$ (E.140)

and Pauli 2-form currents

$\displaystyle J^{aMN}_{L}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Pauli1}}{\partial
L^{a}_{MN}}=+2g_{5}^{2}\times\eta^{a}\overline{\Psi}_{1}e^{M}_{A}e^{N}_{B}\sigma^{AB}T^{a}\Psi_{1}\,,$
$\displaystyle J^{aMN}_{R}=$ $\displaystyle\frac{\partial\mathcal{L}_{\rm
Pauli2}}{\partial
R^{a}_{MN}}=-2g_{5}^{2}\times\eta^{a}\overline{\Psi}_{2}e^{M}_{A}e^{N}_{B}\sigma^{AB}T^{a}\Psi_{2}\,.$
(E.141)

we can rewrite (E.2) with the explicit isoscalar ($a=0$) and isovector ($a=3$)
contributions

$\displaystyle\mathcal{L}_{F1}+\mathcal{L}_{F2}\supset$
$\displaystyle\frac{i}{2}\bar{\Psi}_{1}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{\mathcal{D}}_{N}-\overleftarrow{\mathcal{D}}_{N}\big{)}\Psi_{1}-(M+V(z))\bar{\Psi}_{1}\Psi_{1}+\frac{i}{2}\bar{\Psi}_{2}e^{N}_{A}\Gamma^{A}\big{(}\overrightarrow{\mathcal{D}}_{N}-\overleftarrow{\mathcal{D}}_{N}\big{)}\Psi_{2}-(-M+V(z))\bar{\Psi}_{2}\Psi_{2}$
$\displaystyle+V_{N}^{0}J_{V}^{0N}+A_{N}^{0}J_{A}^{0N}+V_{N}^{3}J_{V}^{3N}+A_{N}^{3}J_{A}^{3N}+V^{0}_{MN}J_{V}^{0MN}+A^{0}_{MN}J_{A}^{0MN}+V^{3}_{MN}J_{V}^{3MN}+A^{3}_{MN}J_{A}^{3MN}$

with $J_{V,A}^{aN}=J_{L}^{aN}\pm J_{R}^{aN}$ and $J_{V,A}^{aMN}=J_{L}^{aMN}\mp
J_{R}^{aMN}$.

##  Appendix F t-channel spin-1 meson exchange

In the soft-wall model the normalized wave function for spin-1 (vector) mesons
is given by Grigoryan and Radyushkin (2007)

$\displaystyle\phi_{n}(z)=c_{n}\tilde{\kappa}_{V}^{2}z^{2}L_{n}^{1}(\tilde{\kappa}_{V}^{2}z^{2})\equiv
J_{A}(m_{n},z)\,,$

with $c_{n}=\sqrt{{2}/{(n+1)}}$ which is determined from the normalization
condition (for the soft-wall model with background dilaton
$\phi=\tilde{\kappa}_{V}^{2}z^{2}$)

$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,(g^{xx})^{2}\,\phi_{n}(z)\phi_{m}(z)=\delta_{nm}\,.$

Therefore, we have

$\displaystyle
F_{n}=\frac{1}{g_{5}}\bigg{(}-e^{-\phi}\frac{1}{z^{\prime}}\partial_{z^{\prime}}\phi_{n}(z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}=-\frac{2}{g_{5}}c_{n}(n+1)\tilde{\kappa}_{V}^{2}\,,$

with $\phi_{n}(z\rightarrow 0)\approx c_{n}\tilde{\kappa}_{V}^{2}z^{2}(n+1)$.
If we define the decay constant as $f_{n}=-{F_{n}}/{m_{n}}$, we have

$\displaystyle\phi_{n}(z)=g_{5}\times\frac{f_{n}}{m_{n}}\times
2\tilde{\kappa}_{V}^{2}z^{2}L_{n}^{1}(\tilde{\kappa}_{V}^{2}z^{2})\,.$ (F.146)

Note that for $z\rightarrow 0$, we can write the bulk-to-bulk propagator as

$\displaystyle G(z\rightarrow 0,z^{\prime})$
$\displaystyle\approx\frac{\phi_{n}(z\rightarrow
0)}{-g_{5}F_{n}}\sum_{n}\frac{-g_{5}F_{n}\phi_{n}(z^{\prime})}{k^{2}-m_{n}^{2}}=\frac{z^{2}}{2}\sum_{n}\frac{-g_{5}F_{n}\phi_{n}(z^{\prime})}{k^{2}-m_{n}^{2}}=\frac{z^{2}}{2}V(k,z^{\prime}).$

For space-like momenta ($t=k^{2}=-K^{2}$), we have the bulk-to-bulk propagator
near the boundary

$\displaystyle G(z\rightarrow
0,z^{\prime})\approx\frac{z^{2}}{2}\sum_{n}\frac{g_{5}F_{n}\phi_{n}(z^{\prime})}{K^{2}+m_{n}^{2}}=\frac{z^{2}}{2}\mathcal{V}(K,z^{\prime})\,,$
(F.148)

where Grigoryan and Radyushkin (2007)

$\displaystyle\mathcal{V}(K,z)=g_{5}\sum_{n}\frac{F_{n}\phi_{n}(z)}{K^{2}+m_{n}^{2}}=\tilde{\kappa}_{V}^{2}z^{2}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{a_{K}}{\rm
exp}\Big{[}-\frac{x}{1-x}\tilde{\kappa}_{V}^{2}z^{2}\Big{]}\,,$ (F.149)

with the normalization ${\cal V}(0,z)={\cal V}(K,0)=1$, and we have defined
$a_{K}=\frac{K^{2}}{4\tilde{\kappa}_{V}^{2}}$.

##  Appendix G t-channel spin-j meson exchange

The spin-j normalized meson wavefunctions $J_{V}(m_{n}(j),z)$ are expressed in
terms of the wavefunctions of massive scalar fields $\tilde{\phi}_{n}(j,z)$
which are given, for the soft wall model, in terms of the generalized Laguerre
polynomials as Ballon Bayona et al. (2008)

$\displaystyle
zJ_{V}(m_{n}(j),z)\equiv\tilde{\phi}_{n}(j,z)=c_{n}(j)\,z^{\Delta(j)}L_{n}^{\Delta(j)-2}(\tilde{\xi})\,,$
(G.150)

with $\tilde{\xi}=\tilde{\kappa}_{V}^{2}z^{2}$. The normalization coefficients
are

$\displaystyle
c_{n}(j)=\Big{(}\frac{2\tilde{\kappa}_{V}^{2(\Delta(j)-1)}\Gamma(n+1)}{\Gamma(n+\Delta(j)-1)}\Big{)}^{\frac{1}{2}}\,,$
(G.151)

and the dimension of the massive scalar fields (with an additional mass coming
from the massive open string states attached to the D9 or D7-branes)
$\Delta(j)$ is given by

$\displaystyle\Delta(j)=2+\sqrt{4+m^{2}R^{2}+\frac{R^{2}}{\alpha^{\prime}}(j-1)}=2+\sqrt{\sqrt{\lambda}(j-j_{0})}\,,$
(G.152)

where, in the last line, we have used the fact that $m^{2}R^{2}=-3$. The
spin-1 transverse bulk gauge field defined as $zV_{\mu}(m_{n},z)$ obeys the
same bulk equation of motion as a bulk massive scalar field
$\tilde{\phi}_{n}(j=1,z)$ with $m^{2}R^{2}=-3$. We have also used the open
string quantized mass spectrum
$m_{j}^{2}R^{2}=(j-1)({R^{2}}/{\alpha^{\prime}})=\sqrt{\lambda}(j-1)$ for open
strings attached to the D9-brane in bulk, and we have defined
$j_{0}=1-{1}/{\sqrt{\lambda}}$.

We now recall that the non-normalized bulk-to-boundary propagators of massive
scalar fields are given in terms of Kummer’s (confluent hypergeometric)
function of the second kind, and their integral representations are (for
space-like momenta $k^{2}=-K^{2}$)

$\displaystyle\mathcal{\tilde{V}}(j,K,z)=$ $\displaystyle
z^{\Delta(j)}U\Big{(}a_{K}+\frac{\Delta(j)}{2},\Delta(j)-1;\tilde{\xi}\Big{)}=z^{\Delta(j)}\tilde{\xi}^{2-\Delta(j)}U\Big{(}\hat{a}(j),\hat{b}(j);\tilde{\xi}\Big{)}$
$\displaystyle=$ $\displaystyle
z^{\Delta(j)}\tilde{\xi}^{2-\Delta(j)}\frac{1}{\Gamma(\tilde{a}(j))}\int_{0}^{1}dx\,x^{\hat{a}(j)-1}(1-x)^{-\hat{b}(j)}{\rm
exp}\Big{(}-\frac{x}{1-x}\tilde{\xi}\Big{)}\,,$ (G.153)

with $\tilde{\xi}=\tilde{\kappa}_{V}^{2}z^{2}$

$\displaystyle
a_{K}=\frac{K^{2}}{4\tilde{\kappa}_{V}^{2}}\qquad\hat{a}(j)=a_{K}+2-\frac{\Delta(j)}{2}\qquad\hat{b}(j)=3-\Delta(j)$
(G.154)

after using the identity $U(m,n;y)=y^{1-n}U(1+m-n,2-n,y)$. Therefore, the
bulk-to-bulk propagator of spin-j mesons

$\displaystyle
J_{V}(m_{n}(j),z)\,G(j,z,z^{\prime})\,J_{V}(m_{n}(j),z^{\prime})=z^{-(j-1)}\,G(j,z,z^{\prime})\,z^{\prime-(j-1)}$

can be approximated at the boundary as (for space-like momenta $k^{2}=-K^{2}$)

$\displaystyle G(j,z\rightarrow 0,z^{\prime})\approx$
$\displaystyle-\bigg{[}\frac{\phi_{n}(j,z\rightarrow
0)}{g_{5}\mathcal{F}_{n}(j)}\bigg{]}\times\sum_{n}\frac{g_{5}\mathcal{F}_{n}(j)\phi_{n}(j,z^{\prime})}{K^{2}+m_{n}^{2}(j)}$
$\displaystyle=$
$\displaystyle(\tilde{\kappa}_{V}^{2})^{\Delta(j)-2}\frac{z^{\Delta(j)-1}}{\Delta(j)-1}\frac{\Gamma(\Delta(j)-2+a_{K})}{\Gamma(\Delta(j)-2)}\,\mathcal{V}(j,K,z^{\prime})$

where $\phi_{n}(j,z\rightarrow 0)=J_{V}(m_{n}(j),z\rightarrow
0)=\frac{1}{z}\tilde{\phi}_{n}(j,z\rightarrow 0)$. We have defined the non-
normalized bulk-to-boundary propagator of spin-j mesons

$\displaystyle\mathcal{V}(j,K,z^{\prime})=\sum_{n}\frac{g_{5}\mathcal{F}_{n}(j)\phi_{n}(j,z^{\prime})}{K^{2}+m_{n}^{2}(j)}=\frac{1}{z^{\prime}}\mathcal{\tilde{V}}(j,K,z^{\prime})|_{a_{K}+\frac{\Delta(j)}{2}\rightarrow
a_{K}+1+(\Delta(j)-3)}\,,$

with the shift $a_{K}+\frac{\Delta(j)}{2}\rightarrow a_{K}+1+(\Delta(j)-3)$
defined in such a way that the mass spectrum of massive scalar fields
$m_{n}^{2}=4\tilde{\kappa}_{V}^{2}(n+\frac{\Delta(j=1)}{2})$ and the mass
spectrum of spin-1 gauge fields $m_{n}^{2}=4\tilde{\kappa}_{V}^{2}(n+1)$
match, i.e., we shift $n+\frac{\Delta(j=1)}{2}\rightarrow
n+1+(\Delta(j=1)-3)$, giving the mass spectrum of spin-j mesons

$\displaystyle m_{n}^{2}(j)=4\tilde{\kappa}_{V}^{2}(n+1+(\Delta(j)-3))\,.$
(G.158)

We have also used

$\displaystyle\mathcal{F}_{n}(j)=$
$\displaystyle\frac{\mathcal{C}(j,K,\epsilon)}{g_{5}}\bigg{(}-\sqrt{g}\,e^{-\phi}\,\big{(}g^{xx}\big{)}^{2}\,\partial_{z^{\prime}}\phi_{n}(j,z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}\,,$
$\displaystyle\mathcal{C}(j,K,\epsilon)=$
$\displaystyle\mathcal{V}(j,K,\epsilon)$ (G.159)

with $\phi_{n}(j,z\rightarrow 0)=\frac{1}{z}\tilde{\phi}_{n}(j,z\rightarrow
0)\approx c_{n}(j)\,z^{\Delta(j)-1}L_{n}^{\Delta(j)-2}(0)$ for the soft wall
model.

##  Appendix H t-channel spin-2 glueball exchange

In the soft-wall model the normalized wave function for spin-2 glueballs is
given by Ballon Bayona et al. (2008) (note that the discussion in Ballon
Bayona et al. (2008) is for general massive bulk scalar fluctuation but can be
used for spin-2 glueball which has an effective bulk action similar to
massless bulk scalar fluctuation)

$\displaystyle
J_{h}(m_{n},z)\equiv\psi_{n}(z)=c_{n}\,z^{4}L_{n}^{2}(2\tilde{\xi}_{T})\,,$

with $\tilde{\xi}_{T}=\tilde{\kappa}_{T}^{2}z^{2}$, and

$\displaystyle
c_{n}=\Bigg{(}\frac{2^{4}\tilde{\kappa}_{T}^{6}\Gamma(n+1)}{\Gamma(n+3)}\Bigg{)}^{\frac{1}{2}}=\frac{4\tilde{\kappa}_{T}^{3}}{\sqrt{(n+2)(n+1)}}\,,$
(H.161)

which is determined from the normalization condition (for soft-wall model with
background dilaton $\phi=\tilde{\kappa}_{T}^{2}z^{2}$)

$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,{\left|g^{xx}\right|}\,\psi_{n}(z)\psi_{m}(z)=\delta_{nm}\,.$

Therefore we have

$\displaystyle F_{n}=\frac{1}{\sqrt{2}\kappa}\bigg{(}-\frac{1}{z^{\prime