aainstitutetext: Maryland Center for Fundamental Physics, Department of
Physics, University of Maryland,
4296 Stadium Dr., College Park, MD 20742, USAbbinstitutetext: Center for
Nuclear Femtography, SURA,
1201 New York Ave. NW, Washington, DC 20005, USA

# Generalized parton distributions through universal moment parameterization:
zero skewness case

Yuxun Guo a,b , Xiangdong Ji b and Kyle Shiells<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a global analysis program for the generalized parton distributions
(GPDs) based on conformal moment expansion. We apply the strategy of universal
moment parameterization to fit both the collinear parton distribution
functions (PDFs) from phenomenology and generalized form factors from lattice
calculations, and show that the parameterization is flexible enough to
accommodate these constraints. In addition, we can also fit direct lattice
calculations of GPDs from large-momentum effective theory. In this work we
focus on the analysis of $t$-dependent PDFs which correspond to GPDs in the
$\xi\to 0$ limit. The strategy also applies to the $\xi\not=0$ region with
extra parameters, and therefore can be fitted to experimental observables in
the future. With a demonstrative example of fitted GPDs, we exhibit the quark
transverse angular momentum densities of the proton as well as the impact
parameter space distributions of quarks in both unpolarized and transversely
polarized protons.

###### Keywords:

Generalized parton distributions; Generalized form factors; GUMP;

††preprint: CNF-UMD-2022

##  1 Introduction

Generalized parton distributions (GPDs) Muller:1994ses ; Ji:1996ek ; Ji:1998pc
which describe the parton distributions in the nucleon with transverse
displacement contain important information about the nucleon such as the mass,
angular momentum and mechanical properties Ji:1994av ; Ji:1996ek ;
Polyakov:2002yz  as well as the 3-dimensional (3D) structure of the nucleon
Burkardt:2000za ; Burkardt:2002hr ; Ji:2003ak ; Belitsky:2003nz . There have
been experimental measurements of deeply virtual Compton scattering (DVCS)
Ji:1996nm  cross-sections aiming to explore GPDs at HERA (H1 H1:2001nez ;
H1:2005gdw ; H1:2007vrx ; H1:2009wnw , ZEUS ZEUS:2003pwh ; ZEUS:2008hcd ,
HERMES HERMES:2012gbh ; HERMES:2012idp ) and Jefferson Lab (CLAS CLAS:2007clm
; CLAS:2008ahu ; Niccolai:2012sq ; CLAS:2015bqi ; CLAS:2015uuo ; CLAS:2018bgk
; CLAS:2018ddh ; CLAS:2021gwi  and Hall A JeffersonLabHallA:2006prd ;
JeffersonLabHallA:2007jdm ; JeffersonLabHallA:2012zwt ; Georges:2017xjy ;
JeffersonLabHallA:2022pnx ) spanning decades. In addition, lattice quantum
chromodynamics (QCD) provides us another opportunity to study the parton
physics in the nucleon from first-principles calculations. The moments of GPDs
are calculated on the lattice for the quark LHPC:2007blg ; Hagler:2009ni  and
the gluon Shanahan:2018nnv . In recent years, the large momentum effective
theory (LaMET) Ji:2013dva ; Ji:2014gla ; Ji:2020ect  was proposed, which
expands lattice technology further to allow the direct calculations of the
$x$-dependence of the parton distributions including GPDs Alexandrou:2020zbe ;
Constantinou:2020hdm ; Lin:2020rxa ; Lin:2021brq .

Despite the efforts put into investigating GPDs, limited knowledge of them has
been obtained at this point due to their higher-dimension nature. More
specifically, GPDs, collectively denoted as $F$, are functions of $(x,\xi,t)$
with $x$ the parton momentum fraction, $\xi$ the skewness parameter describing
the momentum transfer in the longitudinal direction and $t$ the total momentum
transfer squared. Typically, we have constraints from: parton distribution
functions (PDFs) in the forward limit $\xi=t=0$, generalized form factors
(GFFs) with $t$-dependence and Compton form factors (CFFs) with $\xi$\- and
$t$-dependence. None of these place 3D constraints, causing a huge gap in the
determination of GPDs.

Due to the lack of knowledge, we are faced with the so-called inverse problem
when reconstructing GPDs, which states that GPDs can not be uniquely
determined with, for instance, the combination of PDFs and the CFFs
Bertone:2021yyz . Therefore, we are in need of a global analysis program that
can combine different constraints on GPDs to solve the inverse problem. One
possible way is to parameterize the GPDs with finite (and extendable) numbers
of free parameters, such that one can match the number of free parameters to
the number of constraints on the GPDs till they are completely determined with
controlled uncertainties. This is in the same spirit of determining proton
collinear PDFs from various experimental data Hou:2019efy ; Xie:2021equ .

With the above motivation, lots of GPD parameterization methods are proposed
in the literature Radyushkin:1998bz ; Radyushkin:1998es ; Polyakov:2002wz ;
Guidal:2004nd ; Goloskokov:2005sd ; Mueller:2005ed ; Kumericki:2007sa ;
Kumericki:2009uq ; Goldstein:2010gu ; Gonzalez-Hernandez:2012xap ;
Kriesten:2021sqc . They generally fall into three categories: the double
distribution representations, the moment space representations and other
dynamic GPD models. The Radyushkin double distribution ansatz (DDA) is one of
the earliest GPD parameterization method Radyushkin:1998bz ; Radyushkin:1998es
. Under the double distribution framework, the Vanderhaeghen-Guichon-Guidal
(VGG) model Vanderhaeghen:1999xj ; Goeke:2001tz ; Guidal:2004nd  and the
Goloskokov-Kroll (GK) model Goloskokov:2005sd ; Goloskokov:2007nt  were
developed for phenomenological applications. In general, the DDA expresses the
GPD in terms of an integral expression of the double distribution. Then by
constructing proper double distributions, the GPD will be reduced to the
corresponding PDF in the forward limit and satisfies the polynomiality
conditions. However, since the original double distributions have only one
free parameter to control the overall $\xi$ dependence, such GPDs
parameterization methods lack the flexibility when fitting to measurements
with different $\xi$ Freund:2002qf .

The limitation of DDA leads to explorations of other possibilities. Another
approach to constructing GPDs is to consider the polynomial expansion of GPDs.
Two major GPD parameterization methods based on the polynomial expansion are
the dual parameterization Shuvaev:1999fm ; Noritzsch:2000pr ; Polyakov:2002wz
and the Kumerički-Müller (KM) model Mueller:2005ed ; Kumericki:2009uq
commonly used in DVCS analysis, which are shown to be mathematically
equivalent in ref. Muller:2014wxa . The basic idea of the polynomial expansion
or the moment expansion is to expand the GPD in terms of a complete set of
polynomials where the expansion coefficients are the corresponding moments.
Then the GPD can be expressed as the sum of these polynomials without loss of
generality. The polynomial expansion requires the polynomials to properly
imitate the behavior of the GPD and extra care will be needed for practical
uses.

In addition, there are various dynamic models of GPDs Goldstein:2010gu ;
Gonzalez-Hernandez:2012xap ; Kriesten:2021sqc ; Kumar:2017dbf ; Liu:2022fvl .
These models generally approximate the hadron states as expansions in the Fock
space of quarks and gluons, usually with the help of light front wave
functions. Since the parton distributions are the matrix elements of certain
light-cone operators between hadrons, they are calculable when the hadron
states are replaced with the bound states of quarks and gluons with some
effective interactions. In this sense, the parameterization of GPDs is turned
into the task of determining the free parameters in the wave functions and the
effective interactions.

However, a general parameterization strategy that combines various physical
constraints on GPDs with the lattice calculations still seems to be lacking in
the literature. Therefore, in this work we present a program to parameterize
GPDs through universal moment parameterization (GUMP), which can be fitted to
lattice calculations, PDFs and experimental measurements of CFFs altogether
and help solve the inverse problem. We will continue exploring the moment
expansion, especially the conformal moment expansion, for reasons to be
explained. Our work here is closely related to the KM model, but the goal of
our program is quite different. The KM model aims to just fit to the
experiment data, and thus only the sea distributions are parameterized by the
general moment expansion, whereas the valence distributions are given by some
functional forms on the crossover line only. More efforts are put into the sea
rather than the valence distributions in the KM model because experiments such
as DVCS can only measure the CFFs, which are mostly from the sea distributions
that dominate at small $x$ or $\xi$. As for this program aiming to combine all
the possible constraints on GPDs including both experiments and lattice
calculations, both sea and quark distributions are important. Therefore, a
general extension of the previous parameterization methods to all GPDs is
necessary in order to combine the different constraints.

The organization of the paper is as follows. In section 2, we discuss the
physical properties of GPDs and introduce the well-established conformal
moment expansion of GPDs. In section 3, we make our assumptions based on
phenomenological consideration. In section 4 we apply them to build the
$t$-dependent PDFs and study the nucleon 3D structures with the fitting
results. In the end, we conclude in section 5.

##  2 Review on conformal moment expansion of GPDs

In this section, we will introduce the general technique and review salient
features of conformal moment expansion for GPDs Mueller:2005ed , which will be
our basis for the GUMP program. We start with the physical constraints on GPDs
which are important guides to finding proper parameterization strategies.

###  2.1 Physical constraints on GPDs

GPDs are simply functions $F$ with 3 arguments $(x,\xi,t)$, which can be
defined as the off-forward matrix element of light-cone operators as,

$F(x,\xi,t)\equiv\int\frac{\text{d}\lambda}{2\pi}e^{i\lambda
x}\left<P^{\prime}\left|O(\lambda n)\right|P\right>\ ,$ (1)

with $n$ the light-cone vector $(n^{2}=0)$, $O(\lambda n)$ the light-cone
operators of the quark/gluon and $P$/$P^{\prime}$ the initial/final nucleon
momenta. We define the average nucleon momentum
$\bar{P}\equiv(P+P^{\prime})/2$, the momentum transfer $\Delta\equiv
P^{\prime}-P$ and the momentum transfer square $t\equiv\Delta^{2}$. Besides,
the skewness parameter $\xi$ is defined as
$\xi=-n\cdot\Delta/(2n\cdot\bar{P})$. In the forward limit, GPDs are reduced
to the corresponding PDF $f(x)$, so we have

$\lim_{\xi,t\to 0}F(x,\xi,t)=f(x)\ ,$ (2)

which is the most important and stringent constraint on GPDs.

Figure 1:  A plot showing the kinematically allowed region on the $(\xi,t)$
plane for proton scattering with proton mass $M=0.938\text{ GeV}$. Typically,
larger momentum transfer $|t|$ is needed to reach the region with larger
skewness $\xi$.

The two arguments $x$ and $\xi$ of GPDs have support $x,\xi\in[-1,1]$, whereas
the momentum transfer squared $t$ could take any negative value $t<0$. Since
GPDs are typically defined symmetric in $\xi$, we will consider only positive
$\xi>0$ without loss of generality. GPDs do not exist everywhere in the above
3D subspace. For instance, since the skewness parameter $\xi$ corresponds to
the longitudinal projection of the momentum transfer, there exists a minimum
$|t|_{\rm{min}}$ for each $\xi$ as

$|t|\geq|t|_{\rm{min}}=\frac{4\xi^{2}M^{2}}{1-\xi^{2}}\ ,$ (3)

with $M$ the nucleon mass such that it takes infinite momentum transfer
$|t|\to\infty$ to reach $|\xi|\to 1$, as shown in figure 1. Accordingly, we
restrain ourselves to the kinematically allowed region.

Furthermore, there exists non-analyticity on the $(x,\xi)$ plane, specifically
on the crossover line $|x|=\xi$. Since GPDs are essentially parton
distributions with non-zero momentum transfer $\Delta$, their physical
interpretation varies accordingly Ji:1998pc : with small momentum transfer in
the near forward region $\xi<|x|$, GPDs can still be interpreted as the
amplitudes of emitting and reabsorbing a parton in the nucleon and behave like
the PDFs, whereas with large momentum transfer in the region $\xi>|x|$, they
are interpreted as the amplitudes of emitting/absorbing a pair of partons and
behave like the distribution amplitudes (DAs). The two regions will be
referred to as the PDF region ($\xi<|x|$) and the DA region ($\xi>|x|$)
respectively, and they are joined together at the crossover line $\xi=|x|$
where one of the partons has zero momentum fraction. The collinear
factorization requires the GPDs to be continuous at the crossover lines, but
their derivatives might not be. Consequently, these crossover lines can be
extremely subtle for the parameterization of GPDs on the $(x,\xi)$ plane.

Figure 2:  Graphical representation of the kinematical regions of GPDs on the
$(x,\xi)$ plane. The GPDs need to be continuous at the crossover line
$|x|=\xi$ in order for factorization to work, while their derivatives might
not exist there.

In addition, GPDs are also constrained by the polynomiality condition, which
states that the Mellin moments of GPDs

$F_{n}(\xi,t)\equiv\int_{-1}^{1}\text{d}xx^{n-1}F(x,\xi,t)\ ,$ (4)

must be finite order polynomials of $\xi$ such that Ji:1998pc ,

$F_{n}(\xi,t)=\sum_{k=0,{\rm{even}}}^{n}\xi^{k}F_{n,k}(t)\ .$ (5)

The polynomiality condition is the result of Lorentz invariance, and it holds
even though the integral in eq. (4) involves both the PDF region ($|x|>\xi$)
and the DA region ($|x|<\xi$) in spite of the non-analyticity at the crossover
lines.

While other physical constraints exist for GPDs, the above constraints will be
the main focus of this work.

###  2.2 Conformal moment expansion and scale evolution of GPDs

With the physical constraints discussed above, we now introduce the conformal
moment expansion and show how the constraints can be imposed with such
expansion. Consider if we expand the GPD in terms of a complete set of
polynomials, denoted as $C_{i}(x)$, such that

$F(x,\xi,t)=\sum_{i}\rho_{C}(x)C_{i}(x)F_{i}^{C}(\xi,t)\ ,$ (6)

with $F_{i}^{C}(\xi,t)$ the corresponding coefficient for fixed $\xi$ and $t$.
Since $C_{i}(x)$ form a complete set of basis, we have the completeness
condition

$\sum_{i}C_{i}(x)C_{i}(y)=\rho^{-1}_{C}(x)\delta(x-y)\ ,$ (7)

and the orthonormal condition

$\int_{-1}^{1}\text{d}x\rho_{C}(x)C_{i}(x)C_{j}(x)=\delta_{ij}\ ,$ (8)

with $\rho_{C}(x)$ the weight function of the polynomials $C_{i}(x)$. Then the
$F_{i}^{C}(\xi,t)$ is simply the moment of $F(x,\xi,t)$:

$F_{i}^{C}(\xi,t)=\int_{-1}^{1}\text{d}xC_{i}(x)F(x,\xi,t)\ .$ (9)

The above is generally true for any complete set of polynomials $C_{i}(x)$ and
there are infinite possible choices. Therefore, one naturally would ask: which
choice is the most suitable one for GPDs? An answer is motivated by the
renormalization group equations of GPDs. But before getting into that, we
start with a simpler case — the scale evolution of PDFs, or the
Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations Altarelli:1977zs
; Dokshitzer:1977sg ; Gribov:1972ri .

The leading-order DGLAP equations for non-singlet quark distributions reads,

$\frac{\text{d}}{\text{d}\ln
Q^{2}}q(x,Q)=\frac{\alpha_{s}(Q)}{2\pi}\int_{x}^{1}\text{d}y\frac{q(y,Q)}{y}\cdot
P\left(\frac{x}{y}\right)~{}+~{}\mathcal{O}(\alpha_{s}(Q)^{2})\ ,$ (10)

with $P(z)$ the splitting function. Generally, one needs to solve the integro-
differential equation in order to perform the scale evolution of PDFs.
However, if one performs the Mellin transform of the DGLAP equation according
to

$f_{n}=\int_{-1}^{1}\text{d}x~{}x^{n-1}f(x)\ ,$ (11)

the DGLAP equation will read,

$\frac{\text{d}}{\text{d}\ln
Q^{2}}q_{n}(Q)=\frac{\alpha_{s}(Q)}{2\pi}q_{n}(Q)\cdot
P_{n}~{}+~{}\mathcal{O}(\alpha_{s}(Q)^{2})\ ,$ (12)

in Mellin space, where $q_{n}(Q)$ and $P_{n}$ are the Mellin transformations
of $q(x,Q)$ and $P(x)$ according to eq. (11) and the DGLAP equation becomes
multiplicative in the Mellin moment space which holds true to all order in
perturbation theory. Therefore, if one parameterizes the PDFs in the Mellin
moment space, the scale evolution equations can be easily solved.

The above moment representation also applies to the construction of GPDs.
Consider the scale evolution equations of non-singlet quark GPDs in the form
of Muller:1994ses ; Ji:1996nm ; Radyushkin:1996nd ; Muller:2014wxa ;
Belitsky:1997pc ; Geyer:1982fk ; Balitsky:1983sw ; Braunschweig:1987dr ;
Balitsky:1987bk ,

$\frac{\text{d}}{\text{d}\ln
Q^{2}}F\left(x,\xi,t,Q^{2}\right)=\frac{\alpha_{s}(Q)}{2\pi}\int_{-1}^{1}\frac{\text{d}x^{\prime}}{|\xi|}\left[V\left(\frac{x}{\xi},\frac{x^{\prime}}{\xi}\right)\right]_{+}F\left(x^{\prime},\xi,t,Q^{2}\right)$
(13)

with $\left[V\left(\frac{x}{\xi},\frac{x^{\prime}}{\xi}\right)\right]_{+}$ the
evolution kernel. It can be shown that in the forward limit, the evolution
kernel is reduced to the splitting function of DGLAP equation:

$\lim_{\xi\to
0}\frac{1}{|\xi|}\left[V\left(\frac{x}{\xi},\frac{1}{\xi}\right)\right]_{+}=P(x)\
,$ (14)

while for $|x|,|x^{\prime}|<1$ it is related to the Efremov-Radyushkin-
Brodsky-Lepage (ERBL) kernel Lepage:1979zb ; Lepage:1980fj ; Efremov:1978rn ;
Efremov:1979qk  for the DA scale evolution:

$V(2x-1,2y-1)_{0<x,y<1}=V_{\rm{ERBL}}(x,y)\ .$ (15)

For this reason, the PDF and DA regions of GPDs have also been called DGLAP
and ERBL regions respectively.

Analogous to the analysis of PDFs and their Mellin moments, the GPD moments
that diagonalize the right-hand side (RHS) of eq. (13) will renormalize
multiplicatively. To the leading order, such a solution is found to be the
Gegenbauer polynomials $C_{j}^{\left(\lambda\right)}(x)$, which satisfy
Belitsky:1997pc

$\int_{-1}^{1}\frac{\text{d}x^{\prime}}{|\xi|}\left[V^{(0)}\left(\frac{x}{\xi},\frac{x^{\prime}}{\xi}\right)\right]_{+}C_{j}^{\frac{3}{2}}\left(\frac{x}{\xi}\right)=\gamma_{j}C_{j}^{\frac{3}{2}}\left(\frac{x^{\prime}}{\xi}\right)\
,$ (16)

with $V^{(0)}$ the leading-order evolution kernel and $\gamma_{j}$ the
corresponding anomalous dimension (eigenvalue) of each Gegenbauer polynomial.
For the quark we have $\lambda=3/2$, and for the gluon $\lambda=5/2$ should be
used. Accordingly, the conformal moments defined as

$\mathcal{F}_{j}(\xi,t)\equiv\xi^{j}\frac{\Gamma{\left(\frac{3}{2}\right)}\Gamma(j+1)}{2^{j}\Gamma\left(\frac{3}{2}+j\right)}\int_{-1}^{1}\text{d}xC_{j}^{\frac{3}{2}}\left(\frac{x}{\xi}\right)F(x,\xi,t)\
,$ (17)

are multiplicatively renormalizable, which will not hold if the higher order
evolution effects are taken into account and more care will be needed then
Kumericki:2007sa . Here an extra normalization constant is multiplied to
ensure that in the forward limit, the conformal moments are reduced to the
Mellin moments,

$\lim_{\xi\to
0}\mathcal{F}_{j}(\xi,t)=\int_{-1}^{1}\text{d}x~{}x^{j}F(x,\xi,t)\ ,$ (18)

which shall be expected as the evolution equation itself will be reduced to
the DGLAP equation. Also note that there exists a mismatch of conventions in
the literature that the $j$th conformal moment $F_{j}(\xi,t)$ actually
corresponds to the $n$th Mellin moment with $n=j+1$ instead of $j$, according
to eq. (11). In this work, the two will be distinguished by having $j$ for the
conformal moments and $n$ or $s$ for the Mellin moments respectively, and they
are related by $n=s=j+1$.

Inspired by the leading-order evolution and with eq. (6), in general GPDs can
be written in terms of a formal summation as Muller:2014wxa ,

$F(x,\xi,t)=\sum_{j=0}^{\infty}\xi^{-j-1}\frac{2^{j}\Gamma\left(\frac{5}{2}+j\right)}{\Gamma{\left(\frac{3}{2}\right)}\Gamma(j+3)}\left[1-\left(\frac{x}{\xi}\right)^{2}\right]C_{j}^{\frac{3}{2}}\left(\frac{x}{\xi}\right)\mathcal{F}_{j}(\xi,t)\quad\text{for
}|x|<\xi\ ,$ (19)

where we have the weight function of Gegenbauer polynomials
$\rho_{C}^{\frac{3}{2}}(x)=1-x^{2}$ and some extra normalization constants to
match normalization of the conformal moments $\mathcal{F}_{j}(\xi,t)$ in eq.
(17). The whole prefactors of the conformal moments can be collected to define

$(-1)^{j}p_{j}(x,\xi)\equiv\xi^{-j-1}\frac{2^{j}\Gamma\left(\frac{5}{2}+j\right)}{\Gamma{\left(\frac{3}{2}\right)}\Gamma(j+3)}\left[1-\left(\frac{x}{\xi}\right)^{2}\right]C_{j}^{\frac{3}{2}}\left(\frac{x}{\xi}\right)\quad\text{for
}|x|<\xi\ ,$ (20)

where an extra $(-1)^{j}$ factor is extracted for future convenience.
Analogous to the partial wave expansion, the $p_{j}(x,\xi)$s are called the
conformal wave functions and $\mathcal{F}_{j}(\xi,t)$ the conformal partial
wave amplitude respectively Mueller:2005ed .

Also note that with the above conformal partial wave expansion, the non-
analyticity at $|x|=\xi$ can be taken care of — GPDs might not have well-
defined derivatives there, but the moments still exist such that the
polynomiality condition can be imposed.

###  2.3 Resummation of conformal partial wave expansion

In moment expansion, the reconstruction of PDFs or GPDs requires summing over
the contributions of all moments. Such summation does not always converge, and
analytic methods shall be developed to perform the formal summation.

Consider the Mellin moment representation of the PDF $f(x)$ for example. In
order to express the PDF in terms of its Mellin moments, one can write $f(x)$
in terms of a formal summation as

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\delta^{(n)}(x)f_{n+1}\ ,$ (21)

which is obviously not summable, since the $\delta(x)$ function and its $n$th
derivatives $\delta^{(n)}(x)$ are singular. Even though the RHS of eq. (21)
can be shown to be equivalent to the PDF $f(x)$ formally, the expression is of
no practical use. Therefore, the concept of formal summation has to be
introduced, which implies that we consider the RHS of eq. (21) as an abstract
mathematical entity regardless of the convergence of the series.

To actually reconstruct the PDF $f(x)$, the inverse Mellin transformation can
be employed:

$f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}f_{s}\text{d}s\ ,$
(22)

where the Mellin moments $f_{s}$ are analytically continued in the sense that
$s$ can take complex values rather than just positive integers $n$. This
method has been adopted for the numerical analysis of PDFs in the literatures
Graudenz:1995sk ; Stratmann:2001pb . Note that the two equations (21) and (22)
define the same PDF $f(x)$, though they look totally different.

To resum eq. (19) for GPDs, one obstacle is that the Gegenbauer polynomials
are only complete on the interval $[-1,1]$, while the argument of the
Gegenbauer polynomials $x/\xi$ in the above expansion can take any real value.
For this reason, we restrict the expression eq. (19) to the region $|x|<\xi$,
while extension to the PDF region $|x|>\xi$ has to be defined through analytic
continuation. This can be done with the Schläfli integral, as discussed in
ref. Mueller:2005ed ; Muller:2014wxa ,

$p_{j}(x,\xi)=-\frac{\Gamma(5/2+j)}{\Gamma(1/2)\Gamma(2+j)}\frac{1}{2i\pi}\oint_{-1}^{1}du\frac{\left(u^{2}-1\right)^{j+1}}{(x+u\xi)^{j+1}}\
,$ (23)

where the integration contour encircles the line segment $u\in[-1,1]$. The
analytic continued wave functions can then be derived with this integral
representation and one gets Muller:2014wxa ,

$p_{j}(|x|\leq\xi,\xi)=\frac{2^{j+1}\Gamma(5/2+j)\xi^{-j-1}}{\Gamma(1/2)\Gamma(1+j)}(1+x/\xi)_{2}F_{1}\left(-1-j,j+2,2\mid\frac{\xi+x}{2\xi}\right)\
,$ (24)

and

$p_{j}(x>\xi,\xi)=\frac{\sin(\pi[j+1])}{\pi}x^{-j-1}{}_{2}F_{1}\left(\begin{array}[]{c|}(j+1)/2,(j+2)/2\\\
5/2+j\end{array}~{}\frac{\xi^{2}}{x^{2}}\right)\ .$ (25)

These wave functions are equivalent to the Gegenbauer polynomials definition
of conformal wave functions when $j$ is integer, and they can be used for more
general propose, for instance, the resummation of moments. Note that the above
conformal wave functions are continuous at $|x|=\xi$ but their derivatives are
not, which is consistent with the behavior of GPDs at the crossover lines.

Two techniques, the Mellin-Barnes integral Kumericki:2007sa ; Kumericki:2009uq
and the Shuvaev-Noritzsch transform Shuvaev:1999fm ; Noritzsch:2000pr  have
been used in the literature to reconstruct the GPDs with moments. Both methods
are shown equivalent mathematically Muller:2014wxa , and we will focus on the
former one here. The Mellin-Barnes integral is based on the Sommerfeld-Watson
transformation,

$\sum_{n=-\infty}^{\infty}(-1)^{n}f_{n}=\frac{1}{2i}\oint_{C}\frac{f(z)}{\sin\pi
z}\text{d}z\ ,$ (26)

assuming $f(z)$ is analytic. The proof is straightforward, as the $\sin(\pi
z)$ in the denominator of RHS leads to poles at all integers $z$ with the
residue of $\pi(-1)^{n}f(z)$ and the equation is simply the residual theorem
applied to the RHS. Then, the formal summation in eq. (19) can be written in
terms of the following Mellin-Barnes integral in the same manner
Kumericki:2007sa ; Kumericki:2009uq

$F(x,\xi,t)=\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}dj\frac{p_{j}(x,\xi)}{\sin(\pi[j+1])}\mathcal{F}_{j}(\xi,t)\
,$ (27)

where the constant $c$ should be carefully chosen to have all the non-negative
poles while the poles of conformal wave functions $p_{j}(x,\xi)$ and conformal
moments $\mathcal{F}_{j}(\xi,t)$ should be outside the contour, see more
discussions in ref. Muller:2014wxa .

Figure 3:  A comparison of the inverse Mellin transformation expression of the
PDFs and the Mellin-Barnes integral expression of GPDs together with the
corresponding formal summation expressions.

The comparison of the different methods used for PDFs and GPDs are shown in
figure 3. In principle, the Mellin-Barnes integral should reproduce the PDFs
in the forward limit $\xi=0$. Indeed, since the conformal wave functions
$p_{j}(x,\xi)$s have the limit behavior

$\lim_{\xi\to 0}p_{j}(x,\xi)=\frac{1}{j!}\delta^{(j)}(x)\ ,$ (28)

the formal summation in eq. (19) will be reduced to eq. (21). However, due to
the singular behavior of $\delta$ functions, the Mellin-Barnes integral will
not converge in the $\xi\to 0$ limit. Instead, one can use a finite but small
$\xi$ as the regulator for the $\delta$ function, and the Mellin-Barnes
integral will reproduce the PDF when $x\gg\xi$, as shown in figure 4.

Figure 4:  A plot showing that the Mellin-Barnes (MB) integral can indeed
reproduce the reference PDF $f(x)=18.6x^{-0.2}(1-x)^{3.5}$ when $x\gg\xi$,
where $\xi$ serves as the regulator. On the other hand, in the $x\lesssim\xi$
region, the PDFs calculated this way are strong affected by the none-zero
skewness $\xi$ and deviate from the reference value.

Following the above framework, for any (well-behaved) given set of moments
$\mathcal{F}_{j}(\xi,t)$, the corresponding GPDs can be calculated and then
used for all kinds of phenomenological applications. The GUMP program for
global analysis follows this approach by parametrizing the general forms of
the moments or universal moment parametrization. More details concerning the
parameterization of moments will be discussed in the next section.

Note that while the GPDs calculated above are ready for all kinds of
applications, there exist some shortcuts, such as the evaluation of CFFs and
scale evolution, as these calculations are easier to perform in moment space.
The detailed discussion of this topic is in ref. Kumericki:2007sa .

##  3 Universal parameterization of GPD moments

In the last section, we show that the GPDs $F(x,\xi,t)$ can be equivalently
expressed in terms of their moments $\mathcal{F}_{j}(\xi,t)$ without model
assumption. In this section, we will discuss how to find out the universal
parameterization of moments.

Generally, we are interested in GPDs of (anti)quarks with different flavors as
well as the gluons, and conventionally the quarks and antiquark GPDs are
collected together by defining the $F_{(q)}(x,\xi,t)$ such that Muller:2014wxa

$\displaystyle F_{q}(x>-\xi,\xi,t)$ $\displaystyle=F_{(q)}(x,\xi,t)\ ,$ (29)
$\displaystyle F_{\bar{q}}(x>-\xi,\xi,t)$ $\displaystyle=-F_{(q)}(-x,\xi,t)\
,$ (30)

where $F_{q}(x,\xi,t)$ is the quark GPD and $F_{\bar{q}}(x,\xi,t)$ the
antiquark GPD with support $1>x>-\xi$. It is common to reexpress them in terms
of the valence and sea distributions analogous to the forward PDF as,

$\displaystyle F_{q,\rm{val}}(x,\xi,t)$ $\displaystyle\equiv
F_{q}(x,\xi,t)-F_{\bar{q}}(x,\xi,t)\ ,$ (31) $\displaystyle F_{q,\rm{sea}}$
$\displaystyle\equiv 2F_{\bar{q}}(x,\xi,t)\ ,$ (32)

such that for each quark flavor, the different distributions can be expressed
in terms of the corresponding valence and sea distributions.

###  3.1 Polynomiality condition and small $\xi$ expansion of GPD moments

To impose the physical constraints on GPDs, we start by noting that the
polynomiality condition, originally proved for the Mellin moments of the GPDs,
can be equivalently applied to the conformal moments. Since each Gegenbauer
polynomial $C_{j}^{(\lambda)}(x)$ can be expressed in terms of linear
combination of $x^{k}$,

$C_{j}^{(\lambda)}(x)=\sum_{k=0}^{j}c^{(\lambda)}_{j,k}x^{k}\ ,$ (33)

and vice versa:

$x^{j}=\sum_{k=0}^{j}c^{-1,(\lambda)}_{j,k}C_{k}^{(\lambda)}(x)\ ,$ (34)

where $c^{(\lambda)}_{j,k}$ and $c^{-1,(\lambda)}_{j,k}$ are some constant
coefficients, according to eq. (17) we have

$\displaystyle\begin{split}\mathcal{F}_{j}(\xi,t)&\propto\int_{-1}^{1}\text{d}x~{}\xi^{j}C_{j}^{\frac{3}{2}}\left(\frac{x}{\xi}\right)F(x,\xi,t)\
,\\\
&=\int_{-1}^{1}\text{d}x\sum_{k=0}^{j}c^{(\lambda)}_{j,k}\xi^{j-k}x^{k}F(x,\xi,t)\
,\\\
&=\sum_{k=0}^{j}c^{(\lambda)}_{j,k}\xi^{j-k}\int_{-1}^{1}\text{d}xx^{k}F(x,\xi,t)\
.\end{split}$ (35)

Thus the polynomiality condition of Mellin moments immediately leads to the
polynomiality condition of conformal moments. The reverse can be proved in the
same way using eq (34), so the two polynomiality conditions are equivalent.
Therefore, we can write the conformal moments of GPDs as

$\mathcal{F}_{j}(\xi,t)=\sum_{k=0,\rm{even}}^{j+1}\xi^{k}\mathcal{F}_{j,k}(t)\
,$ (36)

where the form factors $\mathcal{F}_{j,k}(t)$ with double subscripts do not
depend on $\xi$. While the GPDs $F(x,\xi,t)$ are completely determined by
$\mathcal{F}_{j,k}(t)$, limited information on them is known. Two limit
behaviors, the PDF limit $\xi=0$ and the DA limit $\xi=1$, can be considered
separately. In the semi-forward limit $\xi=0$, the conformal moments
$\mathcal{F}_{j}(\xi,t)$ are given by $\mathcal{F}_{j,k}(t)$ with $k=0$:

$\mathcal{F}_{j}(\xi=0,t)=\mathcal{F}_{j,0}(t)\ ,$ (37)

where we know that the GPD will be reduced to the PDF and the conformal
moments correspond to the Mellin moments of the PDF if we further take $t=0$.
Thus, the $\mathcal{F}_{j,0}(t)$ at zero momentum transfer $t=0$ is entirely
determined by the forward PDF $f(x)$, which can always be determined well from
global fitting, and we have the following requirements

$\mathcal{F}_{j,0}(t=0)=\int\text{d}x~{}x^{j}f(x)\ .$ (38)

Commonly in the PDF analysis, for instance in the CTEQ global analysis
Hou:2019efy , the ansatz of the PDF $f(x)$

$f(x)=\sum_{i=1}^{i_{\rm{max}}}N_{i}x^{-\alpha_{i}}(1-x)^{\beta_{i}}\ ,$ (39)

is chosen, whereas further improvements can be done based on this. This ansatz
can be immediately transformed to the moment space, and we have the following
ansatz for the Mellin moments of the PDF

$\mathcal{F}_{j,0}(t=0)=\int\text{d}x~{}x^{j}f(x)=\sum_{i=1}^{i_{\rm{max}}}N_{i}B(j+1-\alpha_{i},1+\beta_{i})\
,$ (40)

which will be our building blocks for the GPD at $\xi=0$ or small $\xi$.

On the other hand, in the DA limit $\xi=1$, we have

$\mathcal{F}_{j}(\xi=1,t)=\sum_{k=0,\rm{even}}^{j+1}\mathcal{F}_{j,k}(t)\ ,$
(41)

which could have completely different behavior than the forward case.
Actually, they must behave differently. The reason is that the formal
summation in eq. (19) with the moments in eq. (40) does not converge, and thus
it can only be resummed by the Mellin-Barnes integral method or other
equivalent methods. Consequently, GPDs corresponding to such moments do not
generally vanish at $|x|=\xi$. While it is expected as GPDs do not generally
vanish at the crossover line, for large $\xi$, especially for $\xi$ close to
1, such moments lead to GPDs that do not vanish at the end point $|x|=1$.
Therefore, the moments at large $\xi$ must converge faster than the moments in
eq. (40) in order to have proper end-point behavior.

Due to the lack of constraints from physically observable to the large $\xi$
behavior of GPDs, we will restrain ourselves to the region with small $\xi$ to
avoid such complexity, whereas the large $\xi$ behavior (which also
corresponds to the large $t$ behavior as shown in figure 1) of GPDs requires
separate studies. Then, though the moments of the GPDs are generally unknown,
in the small $\xi$ region, the first few terms will dominate:

$\mathcal{F}_{j}(\xi,t)=\sum_{k=0,\rm{even}}^{k_{\rm{cut}}}\xi^{k}\mathcal{F}_{j,k}(t)\
,$ (42)

with $k_{\rm{cut}}$ the cut-off parameter which controls the $\xi$ dependence,
and generally larger $k_{\rm{cut}}$ is needed for more flexible $\xi$
dependence to contain the larger $\xi$ region. In the region with small $\xi$,
it is assumed that the ansatz used for the PDF in eq. (40) applies not only to
the leading moment with $k=0$, but also the first few moments with $k=2,4...$
which can certainly break down as $k$ gets large.

Note that the small $\xi$ expansion does not apply to the GPDs themselves,
since GPDs are not analytic at the crossover line $|x|=\xi$. However, the
moments of GPDs are always polynomials of $\xi$, and they are always regular
and expandable in the small $\xi$ region, which, on the other hand, need
proper resummation in order to recover the GPDs.

###  3.2 Regge trajectory and $t$-dependence of GPDs

The last piece to complete the universal moment parameterization is the
$t$-dependence of GPDs. In the framework of moment expansion, the
$t$-dependence is encoded in all the form factors $\mathcal{F}_{j,k}(t)$ and
thus the $t$-dependence of the form factors $\mathcal{F}_{j,k}(t)$ are
crucial. While each term can be in principle fitted to the experimental
measurements or lattice calculations, it is not practical to do so, because
the resummation procedure requires a set of $\mathcal{F}_{j,k}(t)$ with $j$
goes to infinity. Neither experimental measurements nor lattice calculations
allow us to access all the moments, especially the higher order ones, and thus
some extrapolation methods that take the known moments into account and allow
us to extrapolate the higher moments will be necessary. Motivated by that, we
introduce the Regge trajectory in this subsection and discuss how it can help
build the $t$-dependence of GPDs.

The starting point is the crossing symmetry, with which processes of different
channels but with the same Feynman diagrams can be connected, as shown in
figure. 5. The scattering amplitude $\mathcal{T}$ can be written with these
Feynman diagrams regardless of which channel one chooses as a function of the
Mandelstam variables $s,t,u$, ignoring the spin degrees of freedom for
simplicity. The Mandelstam variables $s,t,u$ for $2\to 2$ processes are
defined as $s\equiv(P+q)^{2},t\equiv(P^{\prime}-P)^{2}$ and
$u\equiv(P-q^{\prime})^{2}$. The 3 Mandelstam variables $s,t,u$ satisfy the
condition $s+t+u=P^{2}+P^{\prime 2}+q^{2}+q^{\prime 2}$, with which we can
eliminate the variable $u$ for simplicity.

Figure 5:  Crossing symmetry shown in terms of Feynman diagrams. The two
processes are related to the same set of Feynman diagrams by simply exchanging
the initial and final particles. The right plot corresponds to the $t$-channel
process of the left one.

One key hypothetical property in Regge theory is the duality Veneziano:1968yb
, which states that the scattering amplitude is given by the sum of all
$t$-channel resonances and also equals to the sum of all $s$-channel
resonances. Therefore, the $t$-dependence of the scattering amplitude can be
tentatively written as

$\mathcal{T}(s,t,u)=\sum_{\rm{res.}}\frac{g_{t}(s,u)}{t-m_{\rm{res}}^{2}}\ ,$
(43)

with $g_{t}(s,u)$ the residuals of the resonance poles and the RHS sums over
all the $t$-channel resonances. In principle, the scattering amplitude can
then be written directly with all the resonances. However, there are
potentially infinite resonances and summing over their contributions are
highly non-trivial. Thus, one will need to collect the poles together in order
to sum over them, which leads to the Regge theory Regge:1959mz ; Regge:1960zc
.

Regge theory is, in short, a theory about complex angular momentum $J$
(different from the conformal spin $j$). The key observation is that the mass
of the resonances increases with increasing angular momentum $J$ on the hadron
spectrum. Thus, a linear Regge trajectory usually written as
$\alpha(t)\equiv\alpha+\alpha^{\prime}t$ with $\alpha(t)$ the angular momentum
and $t$ the mass squared, also known as the Reggeon. The major assumption in
Regge theory is that the resonances and their analytic continuation to complex
$J$ are the only singularities in the scattering amplitudes. Then the
scattering amplitude can be expressed in terms of all the Regge trajectories
or Reggeons.

Note that the scattering amplitude $\mathcal{T}(s,t,u)$ is not explicitly a
function of $J$. To separate the contributions of different $J$, it can be
written in terms of the partial wave expansion of a $t$-channel scattering as

$\mathcal{T}(s,t,u)=16\pi\sum_{J=0}^{\infty}(2J+1)\mathcal{T}_{J}(t,u)P_{J}(\cos\theta_{t})\
.$ (44)

Then the partial wave amplitudes $\mathcal{T}_{J}(t,u)$ will have poles at
$J=\alpha(t)$ for each Regge trajectory according to the assumption made for
Regge theory, and they can be written with these poles as

$\mathcal{T}_{J}(t,u)=\sum_{\rm{Reg.}}\frac{r(t,u)}{J-\alpha(t)}\ ,$ (45)

which sums over all the Regge trajectory and $r(t,u)$ is the corresponding
residual. Note that eq. (43) and eq. (45) are equivalent descriptions of those
resonance poles. Their relation is intuitive — eq. (43) approaches those poles
with a fixed integer $J$ in the $M^{2}$ direction, whereas eq. (45) approaches
those poles with fixed $M^{2}$ in the $J$ direction.

Then, we can discuss the $t$-dependence of the GPDs or equivalently the form
factors $\mathcal{F}_{j,k}(t)$. Since they are essentially the nucleon matrix
elements of certain quark (gluon) operators which are closely related to the
quark (gluon)-nucleon scattering amplitude, the above arguments also apply to
them, at least qualitatively. Consider the partial wave expansion in eq. (44).
In the higher energy limit $s\to\infty$ and $\xi\to 0$, the $\cos(\theta_{t})$
can be expressed in terms of $\xi$ as

$\cos(\theta_{t})\propto-\frac{1}{\xi}\to\infty\ .$ (46)

Then with the asymptotic behavior of the Legendre polynomials,

$\lim_{z\to\infty}P_{J}(z)\propto z^{J}\ ,$ (47)

one finds that the partial wave amplitude with angular momentum $J$ has the
asymptotic behavior of $\xi^{-J}$ in the high energy limit. Then it can be
quickly identified that the term $\mathcal{F}_{j,k}(t)$ corresponds to the
angular momentum $J=j+1-k$, which leads to

$\mathcal{F}_{j,k}(t)=\sum_{\rm{Reg.}}\frac{r_{j,k}(t)}{j+1-k-\alpha(t)}\ ,$
(48)

with $r(t)$ the corresponding residuals of the Regge poles. Again, if we stay
in the region where momentum transfer $|t|$ is small, the leading lightest
Regge trajectory is sufficiently good to describe the $t$-dependence of the
GPDs, while more terms can always be added to make it more flexible.
Therefore, we reach the finial proposed ansatz for the moments of GPDs as

$\mathcal{F}_{j,k}(t)=\sum_{i=1}^{i_{\rm{max}}}N_{i,k}B(j+1-\alpha_{i,k},1+\beta_{i,k})\frac{r^{\prime}_{i,j,k}(t)}{j+1-k-\alpha_{i,k}(t)}\
,$ (49)

with $k=0,2,4$ which breaks down at large $k$. Note that the residuals
$r^{\prime}_{i,j,k}(t)$ are smooth functions of $t$. In order to restore the
corresponding forward limit in eq. (40), it should satisfy
$r^{\prime}_{i,j,k}(t=0)=j+1-k-\alpha_{i,k}$. However, given the present
amount of data, it should be safe to set it to be simply constant:

$r^{\prime}_{i,j,k}(t)=j+1-k-\alpha_{i,k}\ ,$ (50)

which can be further improved once more data are available. Then there are
essentially $4$ parameters for each term in the summation on the RHS of eq
(49).

In the end, note that the proposed form in eq. (49) is equivalent to the
ansatz in KM model Kumericki:2009uq  and the similar proposed forward-like
function in the dual parameterization Muller:2014wxa  except for the $SO(3)$
partial wave expansion that we dropped for simplicity. However, the main goal
of this work is to focus on both sea and valence distributions and the
corresponding constraints from lattice calculation in for instance
LHPC:2007blg ; Hagler:2009ni ; Shanahan:2018nnv ; Alexandrou:2020zbe ;
Constantinou:2020hdm ; Lin:2020rxa ; Lin:2021brq  besides the experimental
measurements which only partially constrain the GPDs.

##  4 A demonstrative example: $t$-dependent PDF and proton tomography

Following the above global analysis strategy for GPDs, in this section we show
an example to apply it to the $\xi=0$, $t$-dependent PDF and proton tomography
from various constraints. Note that once the parameterization is specified, in
principle one just needs to put all the constraints together, and try fitting
to them with for instance, $\chi^{2}$ minimization. However, the amount of
parameters needed for a global fitting is huge — we needed at least 4
parameters for each different GPDs ($H$, $E$, $\tilde{H}$ and $\tilde{E}$) and
each flavors ($u$, $d$, $g$ and even more) at $\xi=0$ and even more for non-
zero skewness $\xi$. Therefore, as a proof of principle we start by
considering the $H$ and $E$ GPDs of $u$ and $d$ quarks at $\xi=0$, which are
the so-called $t$-dependent PDF $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$. These two
quantities are necessary to understand the angular momentum densities of $u$
and $d$ quarks Ji:1996ek

$J_{u/d}(x,t)=\frac{1}{2}x\left(H_{u/d}(x,t)+E_{u/d}(x,t)\right)\ ,$ (51)

which will be our main focus in this section.

###  4.1 Fitting the valence $t$-dependent PDF

Figure 6:  A simultaneous fit to the forward PDF $H_{u-d}(x)$ (corresponding
to the PDF $q_{u-d}(x)$) Hou:2019efy ; Xie:2021equ , lattice calculated PDF
$H_{u-d}(x,t=-0.39\text{ GeV}^{2})$ Lin:2020rxa  and generalized form factors
$A^{u-d}_{10}(t)$, $A^{u-d}_{20}(t)$ and $A^{u-d}_{30}(t)$ LHPC:2007blg . The
solid lines are the central value of $H_{u-d}(x,t)$, the gray bands are the
regions within 1 $\sigma$ uncertainty and the dots are measurements that are
fitted to. The $\chi^{2}$ per d.o.f. is 3.0 for our 4 parameters fitting. The
parameters from the fitting read $N_{H\rm{vec}}=4.518~{}(327)$,
$\alpha_{H\rm{vec}}=-0.027~{}(26)$, $\beta_{H\rm{vec}}=3.269~{}(76)$ and
$\alpha^{\prime}_{H\rm{vec}}=1.286~{}(51)$.

In this subsection, we use the proposed form in eq. (49) for the $t$-dependent
PDF $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$. We take $k=0$ in the semi-forward
$\xi=0$ limit, and set $i_{\rm{max}}=1$ for the simplest ansatz. Since in
lattice QCD the flavor singlet $u+d$ and non-singlet $u-d$ combinations are
commonly separately calculated, we apply such ansatz for the $H/E_{u+d}(x,t)$
and $H/E_{u-d}(x,t)$ respectively.

We will neglect the sea contributions in the fitting to lattice results here
since the sea contributions to GFFs are suppressed compared to the valence
contributions. According to the CTEQ18 PDFs Hou:2019efy , the moments of sea
distributions are about one order of magnitude smaller than the valence ones,
where similar suppression is expected for the GFFs. Therefore, one should not
expect strong constraints to the sea distributions from lattice results, and
the experimental measurements are crucial to get the sea distributions.

Figure 7:  A simultaneous fit to the forward PDF $H_{u+d}(x)$ (corresponding
to the PDF $q_{u+d}(x)$) Hou:2019efy ; Xie:2021equ  and generalized form
factors $A^{u+d}_{10}(t)$, $A^{u+d}_{20}(t)$ and $A^{u+d}_{30}(t)$
LHPC:2007blg . The solid lines are the central value of $H_{u+d}(x,t)$, the
gray bands are the regions within 1 $\sigma$ uncertainty and the dots are
measurements that are fitted to. The $\chi^{2}$ per d.o.f. is 8.47 for our 4
parameters fitting. The parameters from the fitting read
$N_{H\rm{scal}}=8.340~{}(263)$, $\alpha_{H\rm{scal}}=0.210~{}(10)$,
$\beta_{H\rm{scal}}=3.584~{}(43)$ and
$\alpha^{\prime}_{H\rm{scal}}=1.338~{}(27)$.

Figure 8:  A simultaneous fit to the $t$-dependent PDF $E_{u-d}(x)$
Lin:2020rxa  and generalized form factors $B^{u-d}_{10}(t)$, $B^{u-d}_{20}(t)$
and $B^{u-d}_{30}(t)$ LHPC:2007blg . The solid lines are the central value of
$E_{u-d}(x,t)$, the gray bands are the regions within 1 $\sigma$ uncertainty
and the dots are measurements that are fitted to. The $\chi^{2}$ per d.o.f. is
0.8 for our 4 parameters fitting. The parameters from the fitting read
$N_{E\rm{vec}}=6.770~{}(1.354)$, $\alpha_{E\rm{vec}}=0.437~{}(62)$,
$\beta_{E\rm{vec}}=4.527~{}(240)$ and
$\alpha^{\prime}_{E\rm{vec}}=1.864~{}(222)$.

For the $t$-dependent PDF $H^{\rm{val}}_{u/d}(x,t)$, we can simultaneously fit
to the forward PDF $q(x)$, for which we choose the recommended CTEQ18qed
results Hou:2019efy ; Xie:2021equ , the quark GFFs LHPC:2007blg  and lattice
calculated isovector $t$-dependent PDF $H^{\rm{val}}_{u-d}(x,t)$ Lin:2020rxa .
We used the ManeParse Clark:2016jgm  Mathematica package to read the CTEQ18qed
PDF and the MINUIT2 minimizer Hatlo:2005cj  for $\chi^{2}$ analysis, and the
results are given in figure 6, where the generalized form factors are defined
by Ji:1996ek ; Ji:1998pc

$\displaystyle\int_{-1}^{+1}dxx^{n-1}H\left(x,\xi,t\right)$
$\displaystyle=\sum^{n-1}_{i=0,\rm{even}}(-2\xi)^{i}A_{ni}\left(t\right)+\left.(-2\xi)^{n}C_{n0}\left(Q^{2}\right)\right|_{n\text{
even }}\ ,$ (52) $\displaystyle\int_{-1}^{+1}dxx^{n-1}E\left(x,\xi,t\right)$
$\displaystyle=\sum^{n-1}_{i=0,\rm{even}}(-2\xi)^{i}B_{ni}\left(t\right)-\left.(-2\xi)^{n}C_{n0}\left(Q^{2}\right)\right|_{n\text{
even }}\ ,$

thus only $A_{n0}(t)$ and $B_{n0}(t)$ survive in the semi-forward limit
$\xi=0$. Note that the $\chi^{2}$ per degree of freedom (d.o.f.) is 3.0 for
our 4 parameters fitting. The major source of $\chi^{2}$ is the discrepancy
between lattice calculated form factors and the global fitted PDFs — the
moments of PDF from global fitting are significantly lower than the lattice
calculations. Therefore, the $\chi^{2}$ per d.o.f. indicates the systematic
uncertainties including the effects of unphysical pion mass of
$m_{\pi}=496\text{ MeV}$, omitted disconnected diagrams and other effects in
lattice calculation that are not taken into account.

Similar fitting can be done for the isoscalar valence distributions
$H^{\rm{val}}_{u+d}(x,t)$ as shown in figure 7. However, since the isoscalar
$t$-dependent PDF $H^{\rm{val}}_{u+d}(x,t)$ is not calculated in ref.
Lin:2020rxa , we fit to the forward PDF $H^{\rm{val}}_{u+d}(x,t=0)$ and the
lattice isoscalar form factors $A^{u+d}_{10}(t)$, $A^{u+d}_{20}(t)$ and
$A^{u+d}_{30}(t)$ LHPC:2007blg . Again, the major source of $\chi^{2}$ is the
discrepancy between lattice calculated form factors and the global fitted PDFs
which do not indicate the failure of the parameterization form used and can be
mitigated if the systematical uncertainties are under control.

For the $t$-dependent PDF $E^{\rm{val}}_{u/d}(x,t)$, the forward constraints
from PDFs no longer exist, and we have to rely on the lattice calculated
$t$-dependent GPDs $E^{\rm{val}}_{u/d}(x,t)$ Lin:2020rxa . Though the
isoscalar GPD $E_{u+d}(x,t)$ is not calculated there, there is strong evidence
showing that the isoscalar form factors $B_{u+d}(t)$s are consistent with zero
LHPC:2007blg ; Shanahan:2018nnv . Therefore, we will assume
$E^{\rm{val}}_{u+d}(x,t)=0$ with which the flavor separation becomes trivial.
The results for the $E^{\rm{val}}_{u-d}(x,t)$ are shown in figure 8. The
$\chi^{2}$ per d.o.f. is about 0.8, which is really due to the lack of
precision in the lattice results, since there are no constraints from better
determined forward PDFs. Therefore, the uncertainties of the GPD
$E^{\rm{val}}_{u-d}(x,t)$ are significantly larger.

###  4.2 The angular momentum sum rule and proton tomography

Figure 9:  A plot of the quark transverse angular momentum densities
$J_{u/d}(x)$ with the fitted GPDs $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$. Most of
the uncertainties are from the GPDs $E_{u/d}(x,t)$ which are less constrained
without corresponding forward PDFs.

With the fitted $t$-dependent PDFs $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$,
interesting information of the nucleon 3D structure can be revealed. On the
one hand, as mentioned at the beginning of the section, these GPDs carry
information about the transverse angular momentum densities of quarks. On the
other hand, as first pointed out by M. Burkardt in refs. Burkardt:2000za ;
Burkardt:2002hr , the GPDs in the $\xi\to 0$ limit can be Fourier transformed
in the transverse space to get the impact parameter space parton distribution
which describe the 3D structure of the nucleon. In this section, we will
explore the proton 3D structure with the above fitted GPDs.

In figure 9, we plot the transverse angular momentum densities of quark with
eq. (51). Note again that, this plot needs to be further improved with the sea
distributions, whose contributions to the angular momentum are, however,
mostly in the small $x$ region and one order of magnitude smaller than the
valence ones. The contributions from the $d$ quark are always negative
according to the fitted results, which are essentially the consequence of the
assumption $E_{u+d}(0)=0$ supported by the lattice calculations of quark and
gluon generalized form factors.

Figure 10:  Plots of quark densities $\rho_{q}(x,\bm{b})$ for both $u$ (the
upper row) and $d$ (the lower row) quarks in unpolarized proton with
increasing momentum fraction $x$. The densities typically get more
concentrated but suppressed for increasing $x$.

In addition to the parton transverse angular momentum densities which only
make use of the GPD in the forward limit $\xi=t=0$, we can also take advantage
of the $t$-dependent of the GPDs and study the impact parameter space
distribution of the quarks. For instance, the quark densities in unpolarized
protons are given by the Fourier transform of the GPDs $H_{u/d}(x,t)$:

$\begin{split}\rho_{q}(x,\bm{b})=\int\frac{\text{d}^{2}\bm{\Delta}}{(2\pi)^{2}}e^{-i\bm{\Delta}\cdot\bm{b}}H_{q}(x,-\bm{\Delta}^{2})=\mathscr{H}_{q}(x,\bm{b})\
,\end{split}$ (53)

where $\mathscr{H}_{q}(x,\bm{b})$ is defined as the 2-dimensional Fourier
transformation of $H_{q}(x,-\bm{\Delta}^{2})$. Then we plot the quark
distribution with the above expression and the fitted GPDs in figure 10. The
quark densities typically get more concentrated but suppressed for increasing
$x$, since the number densities should decrease for quarks with larger
momentum due to momentum conservation, and the less deflected they will be.

Figure 11:  Plots of the intrinsic quark densities
$\rho_{q,\rm{In}}^{X}(x,\bm{b})$ for both $u$ (the upper row) and $d$ (the
lower row) quarks in transversely polarized proton (in the $X$ direction) with
increasing momentum fraction $x$. Both the $u$ and $d$ quark densities are
shifted in the $y$ direction which contribute to the transverse angular
momentum $J^{X}$, while the $u$ contributions are positive ($+y$ direction)
and the $d$ contributions are negative ($-y$ direction).

Another set of densities that is of interest is the quark densities in a
polarized proton. It is worth noting that the transverse angular momentum
densities defined in eq. (51) are interpreted as the parton angular momentum
densities in a transversely polarized nucleon only as pointed out in ref.
Burkardt:2002hr , see also Burkardt:2005hp ; Ji:2012sj ; Ji:2012ba . The
original quark density in a transversely polarized nucleon defined in ref.
Burkardt:2002hr  reads

$\begin{split}\rho_{q}^{X}(x,\bm{b})=&\int\frac{\text{d}^{2}\bm{\Delta}}{(2\pi)^{2}}e^{-i\bm{\Delta}\cdot\bm{b}}\left(H_{q}(x,-\bm{\Delta}^{2})+\frac{i\Delta_{y}}{2M}E_{q}(x,-\bm{\Delta}^{2})\right)\
,\\\ =&\mathscr{H}_{q}(x,\bm{b})-\frac{1}{2M}\frac{\partial}{\partial
b^{y}}\mathscr{E}_{q}(x,\bm{b})\ ,\end{split}$ (54)

with $\mathscr{E}_{q}(x,\bm{b})$ the 2-dimensional Fourier transformation of
$E_{q}(x,-\bm{\Delta}^{2})$ and the nucleon polarized in the $X$
(distinguished from the parton momentum $x$) direction. However, with such
quark densities, one find

$\begin{split}J_{q}^{X}(x)\not=\int\text{d}^{2}\bm{b}(b^{y}\times
xP^{+})\rho_{q}^{X}(x,\bm{b})\ ,\end{split}$ (55)

so the transverse angular momentum densities can not be recovered this way. It
is later found that this is due to the nucleon’s center of mass motion — the
center of the nucleon has transverse displacement when transversely polarized
and thus the longitudinal motion of the whole nucleon will contribute to the
transverse angular momentum which does NOT correspond to the intrinsic angular
momentum (spin) of the nucleon. This has been discussed in ref. Ji:2020hii
and carefully studied in ref. Guo:2021aik  where the intrinsic quark densities
are defined as

$\begin{split}\rho_{q,\rm{In}}^{X}(x,\bm{b})=&\int\frac{\text{d}^{2}\bm{\Delta}}{(2\pi)^{2}}e^{-i\bm{\Delta}\cdot\bm{b}}\left[H_{q}(x,-\bm{\Delta}^{2})+\frac{i\Delta_{y}}{2M}\left(H_{q}(x,-\bm{\Delta}^{2})+E_{q}(x,-\bm{\Delta}^{2})\right)\right]\
,\\\ =&\mathscr{H}_{q}(x,\bm{b})-\frac{1}{2M}\frac{\partial}{\partial
b^{y}}\left(\mathscr{H}_{q}(x,\bm{b})+\mathscr{E}_{q}(x,\bm{b})\right)\ ,\\\
=&\mathscr{H}_{q}(x,\bm{b})-\frac{1}{2M}\frac{b_{y}}{|\bm{b}|}\frac{\partial}{\partial|\bm{b}|}\left(\mathscr{H}_{q}(x,\bm{b})+\mathscr{E}_{q}(x,\bm{b})\right)\
,\end{split}$ (56)

with which one has

$\begin{split}J_{q}^{X}(x)=\int\text{d}^{2}\bm{b}(b^{y}\times
xP^{+})\rho_{q,\rm{In}}^{X}(x,\bm{b})\ .\end{split}$ (57)

With the above fitted GPDs $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$, the intrinsic
quark densities in transversely polarized nucleon and be studied as well as
shown in figure 11. While the quark densities in a polarized proton get more
concentrated but suppressed for increasing $x$ just like in the unpolarized
case, we find that clearly the quarks have non-zero transverse displacement,
which contributes to the transverse angular momentum of the nucleon along with
their large longitudinal momenta.

###  4.3 Sea distributions and experimental observables

Here we discuss the effects of the sea distributions, which are important
sources of systematic uncertainties in the above analysis, since in principle
both sea and valence distributions contribute to the generalized form factors.
As mentioned above, the sea distributions can be dropped because they are
mostly in the small $x$ regions and their contributions to the form factors
are suppressed compared to that of the valence distributions. On the other
hand, this indicates that even if we consider the sea distributions in the
above analysis, they will not be constrained as much. Therefore, it is crucial
to also include the experimental observables, especially the ones with small
$\xi$ (since it can not be zero) in order to constrain the sea distributions.
Then, we are inevitably faced with the effects of both none-zero skewness
$\xi$ and the sea distributions simultaneously.

Fortunately, in the KM model Kumericki:2009uq , they show that the sea
distributions with such parameterization method can successfully fit to the
experimental data which proves the flexibility of this parameterization when
extending to non-zero $\xi$. In this sense, our results here with the valence
distributions are complementary to the sea distributions obtained in the KM
models. On the other hand, GPDs at none-zero $\xi$ are faced with the inverse
problem Bertone:2021yyz , as the experimental measurements of CFFs alone
cannot determine the $x$-dependence at non-zero $\xi$. The solution
(compromise) in the KM model is to model the $x$-dependence of the GPDs at
non-zero $\xi$ with the same parameters as the PDFs, except for a different
overall normalization constant, which might be too constraining. However,
there have been results of lattice calculated $x$-dependence of GPDs with
LaMET Alexandrou:2020zbe  at none-zero $\xi$. These results, even just
qualitatively, can be very helpful in mitigating or resolving the inverse
problem at non-zero $\xi$.

Therefore, to obtain the GPDs at non-zero $\xi$ reliably, we need to extend
our present results with sea distributions and fit to both experimental
observables and lattice results at none-zero $\xi$ as well. A global fitting
program is then necessary for this purpose, which will be studied in the
future coming paper.

##  5 Conclusion

We present a flexible parameterization program for GPDs employing the
universal moment expansion which can be fitted to the forward PDFs, the
lattice calculation and the experimental measurements when the skewness
parameter $\xi$ is small. Under such parameterization, the polynomiality
conditions of GPDs can be easily imposed and the non-analyticity at $x=\xi$
will be taken care of. Since the leading order QCD evolution is multiplicative
in the conformal moment space, the QCD scale evolution can be performed
easily.

We briefly discuss the major parameterization methods of GPDs in the
literatures and introduce the necessary techniques for building the GPDs in
moment space. Following the previous works on parameterizing GPDs with
moments, the $t$-dependence of GPDs is parameterized with the Regge theory
which shows reasonable agreement with lattice calculated generalized form
factors. Then by expanding the moments as powers of $\xi$, the
$\xi$-dependence of GPDs can be gradually extended and fitted to observables.
The $x$-dependence of GPDs is the least known except in the forward limit,
which needs further inputs, most likely from lattice QCD.

As a proof of principle, we apply our fitting program to construct the quark
$t$-dependent PDFs $H_{u/d}(x,t)$ and $E_{u/d}(x,t)$ by fitting to the forward
PDFs, lattice calculated $t$-dependent PDFs and lattice calculated generalized
form factors. With these fitted results, we study the quark transverse angular
momentum densities as well as the impact parameter space distributions of
quarks in both unpolarized and transversely polarized protons, and show that
the quark densities in transversely polarized nucleons are shifted in the
transverse direction of impact parameter space, such that the transverse
angular momentum can be generated. We also comment on the future development
of this program to include sea distributions, non-zero skewness $\xi$ and
experimental observables.

## Acknowledgments

We thank B. Kriesten and M. Gabriel Santiago for discussions related to the
subject of this paper. This research is supported by the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics, under contract number
DE-SC0020682, and the Center for Nuclear Femtography, Southeastern
Universities Research Association, Washington D.C. Y. Guo is partially
supported by the JSA/JLab Graduate Fellowship.

## References

  * (1) D. Müller, D. Robaschik, B. Geyer, F.M. Dittes and J. Hořejši, _Wave functions, evolution equations and evolution kernels from light ray operators of QCD_ , _Fortsch. Phys._ 42 (1994) 101 [hep-ph/9812448].
  * (2) X.-D. Ji, _Gauge-Invariant Decomposition of Nucleon Spin_ , _Phys. Rev. Lett._ 78 (1997) 610 [hep-ph/9603249].
  * (3) X.-D. Ji, _Off forward parton distributions_ , _J. Phys. G_ 24 (1998) 1181 [hep-ph/9807358].
  * (4) X.-D. Ji, _A QCD analysis of the mass structure of the nucleon_ , _Phys. Rev. Lett._ 74 (1995) 1071 [hep-ph/9410274].
  * (5) M.V. Polyakov, _Generalized parton distributions and strong forces inside nucleons and nuclei_ , _Phys. Lett. B_ 555 (2003) 57 [hep-ph/0210165].
  * (6) M. Burkardt, _Impact parameter dependent parton distributions and off forward parton distributions for zeta — $>$ 0_, _Phys. Rev. D_ 62 (2000) 071503 [hep-ph/0005108].
  * (7) M. Burkardt, _Impact parameter space interpretation for generalized parton distributions_ , _Int. J. Mod. Phys. A_ 18 (2003) 173 [hep-ph/0207047].
  * (8) X.-d. Ji, _Viewing the proton through ’color’ filters_ , _Phys. Rev. Lett._ 91 (2003) 062001 [hep-ph/0304037].
  * (9) A.V. Belitsky, X.-d. Ji and F. Yuan, _Quark imaging in the proton via quantum phase space distributions_ , _Phys. Rev. D_ 69 (2004) 074014 [hep-ph/0307383].
  * (10) X.-D. Ji, _Deeply virtual Compton scattering_ , _Phys. Rev. D_ 55 (1997) 7114 [hep-ph/9609381].
  * (11) H1 collaboration, _Measurement of deeply virtual Compton scattering at HERA_ , _Phys. Lett. B_ 517 (2001) 47 [hep-ex/0107005].
  * (12) H1 collaboration, _Measurement of deeply virtual compton scattering at HERA_ , _Eur. Phys. J. C_ 44 (2005) 1 [hep-ex/0505061].
  * (13) H1 collaboration, _Measurement of deeply virtual Compton scattering and its t-dependence at HERA_ , _Phys. Lett. B_ 659 (2008) 796 [0709.4114].
  * (14) H1 collaboration, _Deeply Virtual Compton Scattering and its Beam Charge Asymmetry in e+- Collisions at HERA_ , _Phys. Lett. B_ 681 (2009) 391 [0907.5289].
  * (15) ZEUS collaboration, _Measurement of deeply virtual Compton scattering at HERA_ , _Phys. Lett. B_ 573 (2003) 46 [hep-ex/0305028].
  * (16) ZEUS collaboration, _A Measurement of the Q**2, W and t dependences of deeply virtual Compton scattering at HERA_ , _JHEP_ 05 (2009) 108 [0812.2517].
  * (17) HERMES collaboration, _Beam-helicity and beam-charge asymmetries associated with deeply virtual Compton scattering on the unpolarised proton_ , _JHEP_ 07 (2012) 032 [1203.6287].
  * (18) HERMES collaboration, _Beam-helicity asymmetry arising from deeply virtual Compton scattering measured with kinematically complete event reconstruction_ , _JHEP_ 10 (2012) 042 [1206.5683].
  * (19) CLAS collaboration, _Measurement of Deeply virtual Compton scattering beam-spin asymmetries_ , _Phys. Rev. Lett._ 100 (2008) 162002 [0711.4805].
  * (20) CLAS collaboration, _Beam spin asymmetries in deeply virtual Compton scattering (DVCS) with CLAS at 4.8 GeV_ , _Phys. Rev. C_ 80 (2009) 035206 [0812.2950].
  * (21) CLAS collaboration, _Deeply virtual Compton scattering on longitudinally polarized protons and neutrons at CLAS_ , _PoS_ QNP2012 (2012) 053 [1207.3191].
  * (22) CLAS collaboration, _Single and double spin asymmetries for deeply virtual Compton scattering measured with CLAS and a longitudinally polarized proton target_ , _Phys. Rev. D_ 91 (2015) 052014 [1501.07052].
  * (23) CLAS collaboration, _Cross sections for the exclusive photon electroproduction on the proton and Generalized Parton Distributions_ , _Phys. Rev. Lett._ 115 (2015) 212003 [1504.02009].
  * (24) CLAS collaboration, _Measurement of Unpolarized and Polarized Cross Sections for Deeply Virtual Compton Scattering on the Proton at Jefferson Laboratory with CLAS_ , _Phys. Rev. C_ 98 (2018) 045203 [1810.02110].
  * (25) CLAS collaboration, _Exploring the Structure of the Bound Proton with Deeply Virtual Compton Scattering_ , _Phys. Rev. Lett._ 123 (2019) 032502 [1812.07628].
  * (26) CLAS collaboration, _Beam charge asymmetries for deeply virtual Compton scattering off the proton_ , _Eur. Phys. J. A_ 57 (2021) 186 [2103.12651].
  * (27) Jefferson Lab Hall A, Hall A DVCS collaboration, _Scaling tests of the cross-section for deeply virtual compton scattering_ , _Phys. Rev. Lett._ 97 (2006) 262002 [nucl-ex/0607029].
  * (28) Jefferson Lab Hall A collaboration, _Deeply virtual compton scattering off the neutron_ , _Phys. Rev. Lett._ 99 (2007) 242501 [0709.0450].
  * (29) Jefferson Lab Hall A collaboration, _Virtual Compton Scattering and the Generalized Polarizabilities of the Proton at $Q^{2}=0.92$ and 1.76 GeV2_, _Phys. Rev. C_ 86 (2012) 015210 [1205.3387].
  * (30) Jefferson Lab Hall A collaboration, _Deeply Virtual Compton Scattering at 11GeV in Jefferson Lab Hall A_ , _PoS_ Hadron2017 (2018) 170.
  * (31) Jefferson Lab Hall A collaboration, _Deeply Virtual Compton Scattering Cross Section at High Bjorken xB_ , _Phys. Rev. Lett._ 128 (2022) 252002 [2201.03714].
  * (32) LHPC collaboration, _Nucleon Generalized Parton Distributions from Full Lattice QCD_ , _Phys. Rev. D_ 77 (2008) 094502 [0705.4295].
  * (33) P. Hagler, _Hadron structure from lattice quantum chromodynamics_ , _Phys. Rept._ 490 (2010) 49 [0912.5483].
  * (34) P.E. Shanahan and W. Detmold, _Pressure Distribution and Shear Forces inside the Proton_ , _Phys. Rev. Lett._ 122 (2019) 072003 [1810.07589].
  * (35) X. Ji, _Parton Physics on a Euclidean Lattice_ , _Phys. Rev. Lett._ 110 (2013) 262002 [1305.1539].
  * (36) X. Ji, _Parton Physics from Large-Momentum Effective Field Theory_ , _Sci. China Phys. Mech. Astron._ 57 (2014) 1407 [1404.6680].
  * (37) X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang and Y. Zhao, _Large-momentum effective theory_ , _Rev. Mod. Phys._ 93 (2021) 035005 [2004.03543].
  * (38) C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato et al., _Unpolarized and helicity generalized parton distributions of the proton within lattice QCD_ , _Phys. Rev. Lett._ 125 (2020) 262001 [2008.10573].
  * (39) M. Constantinou et al., _Parton distributions and lattice-QCD calculations: Toward 3D structure_ , _Prog. Part. Nucl. Phys._ 121 (2021) 103908 [2006.08636].
  * (40) H.-W. Lin, _Nucleon Tomography and Generalized Parton Distribution at Physical Pion Mass from Lattice QCD_ , _Phys. Rev. Lett._ 127 (2021) 182001 [2008.12474].
  * (41) H.-W. Lin, _Nucleon helicity generalized parton distribution at physical pion mass from lattice QCD_ , _Phys. Lett. B_ 824 (2022) 136821 [2112.07519].
  * (42) V. Bertone, H. Dutrieux, C. Mezrag, H. Moutarde and P. Sznajder, _Deconvolution problem of deeply virtual Compton scattering_ , _Phys. Rev. D_ 103 (2021) 114019 [2104.03836].
  * (43) T.-J. Hou et al., _New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC_ , _Phys. Rev. D_ 103 (2021) 014013 [1912.10053].
  * (44) CTEQ-TEA collaboration, _Photon PDF within the CT18 global analysis_ , _Phys. Rev. D_ 105 (2022) 054006 [2106.10299].
  * (45) A.V. Radyushkin, _Symmetries and structure of skewed and double distributions_ , _Phys. Lett. B_ 449 (1999) 81 [hep-ph/9810466].
  * (46) A.V. Radyushkin, _Double distributions and evolution equations_ , _Phys. Rev. D_ 59 (1999) 014030 [hep-ph/9805342].
  * (47) M.V. Polyakov and A.G. Shuvaev, _On’dual’ parametrizations of generalized parton distributions_ , hep-ph/0207153.
  * (48) M. Guidal, M.V. Polyakov, A.V. Radyushkin and M. Vanderhaeghen, _Nucleon form-factors from generalized parton distributions_ , _Phys. Rev. D_ 72 (2005) 054013 [hep-ph/0410251].
  * (49) S.V. Goloskokov and P. Kroll, _Vector meson electroproduction at small Bjorken-x and generalized parton distributions_ , _Eur. Phys. J. C_ 42 (2005) 281 [hep-ph/0501242].
  * (50) D. Mueller and A. Schafer, _Complex conformal spin partial wave expansion of generalized parton distributions and distribution amplitudes_ , _Nucl. Phys. B_ 739 (2006) 1 [hep-ph/0509204].
  * (51) K. Kumericki, D. Mueller and K. Passek-Kumericki, _Towards a fitting procedure for deeply virtual Compton scattering at next-to-leading order and beyond_ , _Nucl. Phys. B_ 794 (2008) 244 [hep-ph/0703179].
  * (52) K. Kumerički and D. Mueller, _Deeply virtual Compton scattering at small $x_{B}$ and the access to the GPD H_, _Nucl. Phys. B_ 841 (2010) 1 [0904.0458].
  * (53) G.R. Goldstein, J.O. Hernandez and S. Liuti, _Flexible Parametrization of Generalized Parton Distributions from Deeply Virtual Compton Scattering Observables_ , _Phys. Rev. D_ 84 (2011) 034007 [1012.3776].
  * (54) J.O. Gonzalez-Hernandez, S. Liuti, G.R. Goldstein and K. Kathuria, _Interpretation of the Flavor Dependence of Nucleon Form Factors in a Generalized Parton Distribution Model_ , _Phys. Rev. C_ 88 (2013) 065206 [1206.1876].
  * (55) B. Kriesten, P. Velie, E. Yeats, F.Y. Lopez and S. Liuti, _Parametrization of Quark and Gluon Generalized Parton Distributions in a Dynamical Framework_ , 2101.01826.
  * (56) M. Vanderhaeghen, P.A.M. Guichon and M. Guidal, _Deeply virtual electroproduction of photons and mesons on the nucleon: Leading order amplitudes and power corrections_ , _Phys. Rev. D_ 60 (1999) 094017 [hep-ph/9905372].
  * (57) K. Goeke, M.V. Polyakov and M. Vanderhaeghen, _Hard exclusive reactions and the structure of hadrons_ , _Prog. Part. Nucl. Phys._ 47 (2001) 401 [hep-ph/0106012].
  * (58) S.V. Goloskokov and P. Kroll, _The Role of the quark and gluon GPDs in hard vector-meson electroproduction_ , _Eur. Phys. J. C_ 53 (2008) 367 [0708.3569].
  * (59) A. Freund, M. McDermott and M. Strikman, _Modeling generalized parton distributions to describe deeply virtual Compton scattering data_ , _Phys. Rev. D_ 67 (2003) 036001 [hep-ph/0208160].
  * (60) A. Shuvaev, _Solution of the off forward leading logarithmic evolution equation based on the Gegenbauer moments inversion_ , _Phys. Rev. D_ 60 (1999) 116005 [hep-ph/9902318].
  * (61) J.D. Noritzsch, _Off forward parton distributions and Shuvaev’s transformations_ , _Phys. Rev. D_ 62 (2000) 054015 [hep-ph/0004012].
  * (62) D. Müller, M.V. Polyakov and K.M. Semenov-Tian-Shansky, _Dual parametrization of generalized parton distributions in two equivalent representations_ , _JHEP_ 03 (2015) 052 [1412.4165].
  * (63) N. Kumar, C. Mondal and N. Sharma, _Gravitational form factors and angular momentum densities in light-front quark-diquark model_ , _Eur. Phys. J. A_ 53 (2017) 237 [1712.02110].
  * (64) BLFQ collaboration, _Angular momentum and generalized parton distributions for the proton with basis light-front quantization_ , 2202.00985.
  * (65) G. Altarelli and G. Parisi, _Asymptotic Freedom in Parton Language_ , _Nucl. Phys. B_ 126 (1977) 298.
  * (66) Y.L. Dokshitzer, _Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics._ , _Sov. Phys. JETP_ 46 (1977) 641.
  * (67) V.N. Gribov and L.N. Lipatov, _Deep inelastic e p scattering in perturbation theory_ , _Sov. J. Nucl. Phys._ 15 (1972) 438\.
  * (68) A.V. Radyushkin, _Scaling limit of deeply virtual Compton scattering_ , _Phys. Lett. B_ 380 (1996) 417 [hep-ph/9604317].
  * (69) A.V. Belitsky, B. Geyer, D. Mueller and A. Schafer, _On the leading logarithmic evolution of the off forward distributions_ , _Phys. Lett. B_ 421 (1998) 312 [hep-ph/9710427].
  * (70) B. Geyer, _ANOMALOUS DIMENSIONS IN LOCAL AND NONLOCAL LIGHT CONE EXPANSION. (TALK)_ , _Czech. J. Phys. B_ 32 (1982) 645.
  * (71) I.I. Balitsky, _STRING OPERATOR EXPANSION OF THE T PRODUCT OF TWO CURRENTS NEAR THE LIGHT CONE_ , _Phys. Lett. B_ 124 (1983) 230.
  * (72) T. Braunschweig, B. Geyer and D. Robaschik, _Anomalous Dimensions of Flavor Singlet Light Cone Operators_ , _Annalen Phys._ 44 (1987) 403.
  * (73) I.I. Balitsky and V.M. Braun, _Evolution Equations for QCD String Operators_ , _Nucl. Phys. B_ 311 (1989) 541.
  * (74) G.P. Lepage and S.J. Brodsky, _Exclusive Processes in Quantum Chromodynamics: Evolution Equations for Hadronic Wave Functions and the Form-Factors of Mesons_ , _Phys. Lett. B_ 87 (1979) 359.
  * (75) G.P. Lepage and S.J. Brodsky, _Exclusive Processes in Perturbative Quantum Chromodynamics_ , _Phys. Rev. D_ 22 (1980) 2157.
  * (76) A.V. Efremov and A.V. Radyushkin, _Asymptotical Behavior of Pion Electromagnetic Form-Factor in QCD_ , _Theor. Math. Phys._ 42 (1980) 97.
  * (77) A.V. Efremov and A.V. Radyushkin, _Factorization and Asymptotical Behavior of Pion Form-Factor in QCD_ , _Phys. Lett. B_ 94 (1980) 245.
  * (78) D. Graudenz, M. Hampel, A. Vogt and C. Berger, _The Mellin transform technique for the extraction of the gluon density_ , _Z. Phys. C_ 70 (1996) 77 [hep-ph/9506333].
  * (79) M. Stratmann and W. Vogelsang, _Towards a global analysis of polarized parton distributions_ , _Phys. Rev. D_ 64 (2001) 114007 [hep-ph/0107064].
  * (80) G. Veneziano, _Construction of a crossing - symmetric, Regge behaved amplitude for linearly rising trajectories_ , _Nuovo Cim. A_ 57 (1968) 190.
  * (81) T. Regge, _Introduction to complex orbital momenta_ , _Nuovo Cim._ 14 (1959) 951.
  * (82) T. Regge, _Bound states, shadow states and Mandelstam representation_ , _Nuovo Cim._ 18 (1960) 947.
  * (83) D.B. Clark, E. Godat and F.I. Olness, _ManeParse : A Mathematica reader for Parton Distribution Functions_ , _Comput. Phys. Commun._ 216 (2017) 126 [1605.08012].
  * (84) M. Hatlo, F. James, P. Mato, L. Moneta, M. Winkler and A. Zsenei, _Developments of mathematical software libraries for the LHC experiments_ , _IEEE Trans. Nucl. Sci._ 52 (2005) 2818.
  * (85) M. Burkardt, _Transverse deformation of parton distributions and transversity decomposition of angular momentum_ , _Phys. Rev. D_ 72 (2005) 094020 [hep-ph/0505189].
  * (86) X. Ji, X. Xiong and F. Yuan, _Proton Spin Structure from Measurable Parton Distributions_ , _Phys. Rev. Lett._ 109 (2012) 152005 [1202.2843].
  * (87) X. Ji, X. Xiong and F. Yuan, _Probing Parton Orbital Angular Momentum in Longitudinally Polarized Nucleon_ , _Phys. Rev. D_ 88 (2013) 014041 [1207.5221].
  * (88) X. Ji and F. Yuan, _Transverse spin sum rule of the proton_ , _Phys. Lett. B_ 810 (2020) 135786 [2008.04349].
  * (89) Y. Guo, X. Ji and K. Shiells, _Novel twist-three transverse-spin sum rule for the proton and related generalized parton distributions_ , _Nucl. Phys. B_ 969 (2021) 115440 [2101.05243].