††thanks: Corresponding author

# Impact of JLab data on the determination of GPDs at zero skewness and new
insights from transition form factors $N\rightarrow\Delta$

The MMGPDs111Modern Multipurpose GPDs Collaboration:
Muhammad Goharipour1,2<EMAIL_ADDRESS>Hadi Hashamipour2
<EMAIL_ADDRESS>Fatemeh Irani1<EMAIL_ADDRESS>K. Azizi1,2,3
<EMAIL_ADDRESS>1Department of Physics, University of Tehran, North
Karegar Avenue, Tehran 14395-547, Iran
2School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
3Department of Physics, Doǧuş University, Dudullu-Ümraniye, 34775 Istanbul,
Turkey

(August 28, 2024)

###### Abstract

It is well established now that the generalized parton distributions (GPDs) at
zero skewness are playing important roles in some physical process such as
elastic electron-nucleon scattering, elastic (anti)neutrino-nucleon
scattering, and wide-angle Compton scattering (WACS) via various types of form
factors (FFs). In this study, we are going to utilize the recent JLab
measurements of the elastic electron-nucleon scattering reduced cross-section,
namely GMp12, as a touchstone to unravel the tension observed between the
measurements of the WACS cross-section and the data of the proton magnetic FF
$G_{M}^{p}$. We also investigate the impact of GMp12 data on valence
unpolarized GPDs $H_{v}^{q}$ and $E_{v}^{q}$ at zero skewness by performing
some $\chi^{2}$ analyses of the related experimental data. By calculating the
electric and scalar quadrupole ratios, $R_{EM}$ and $R_{SM}$, and magnetic
transition FF $G^{*}_{M}/3G_{D}$ related to the nucleon-to-delta
($N\rightarrow\Delta$) transition using the extracted GPDs and comparing the
results with corresponding experimental measurements, we show that our results
are in an excellent consistency with experiments, indicating the universality
property of GPDs. We emphasize that the inclusion of these data in the future
analysis of GPDs can significantly affect the extracted GPDs especially their
uncertainties at smaller values of $Q^{2}$.

##  I Introduction

Parton distribution functions (PDFs) serve as essential inputs for theoretical
calculations of physical observables in high-energy processes involving
hadrons such as deep inelastic scattering (DIS) and hadron-hadron collisions
which provide the precision test of the Standard Model (SM), understanding the
QCD dynamics, and searches for new physics Forte:2013wc ; Ethier:2020way ;
Rojo:2019uip . However, PDFs just describe the longitudinal momentum
distribution of partons within hadrons and hence can not provide any
information in the transverse plane. In this context, one can refer to the
transverse momentum distributions (TMDs) Close:1977mr ; Kimber:2001sc ;
Angeles-Martinez:2015sea ; Signal:2021aum ; Bacchetta:2022awv ;
Boussarie:2023izj  and GPDs Muller:1994ses ; Radyushkin:1996nd ; Ji:1996nm ;
Ji:1996ek ; Burkardt:2000za  to get further information on the 3D structure of
hadrons Diehl:2015uka . The first ones give correlations between the
transverse and longitudinal momentum of partons inside a hadron, and the
latter ones represent correlations between the transverse position and the
longitudinal momentum of partons. The hadron structure can be described even
in more details through the Wigner distribution Wigner:1932eb  which encodes
all the nonperturbative QCD dynamics of parton constituents and hence
represents a comprehensive visualization of the partonic structure of hadron
in five dimensions Belitsky:2003nz ; Lorce:2011ni ; Mukherjee:2015aja ;
Liu:2015eqa ; Chakrabarti:2017teq ; Pasechnik:2023mdd .

Among different objects that describe the nonperturbative QCD dynamics of
partons inside the nucleon ($p$ and $n$ denote the proton and neutron,
respectively, throughout this article), GPDs are of special importance because
of their connections with other objects Diehl:2003ny ; Boffi:2007yc ;
Diehl:2015uka . On one hand, unpolarized (polarized) PDFs $q(x,\mu^{2})$
($\Delta q$), where $x$ is the longitudinal momentum fraction of the nucleon
carried by partons and $\mu$ is the factorization scale at which the partons
are resolved, can be obtained from unpolarized (polarized) GPDs
$\mathcal{F}^{q}(x,\mu^{2},\xi,t)$ ($\mathcal{\tilde{F}}^{q}$) under the so-
called forward limit $t=0$ and $\xi=0$. Here, $\xi$ is the longitudinal
momentum transfer, known as skewness, and $t=-Q^{2}$ is the negative momentum
transfer squared. At zero skewness, a Fourier transform of GPDs with respect
to the transverse part of the momentum transfer yields the impact parameter
distribution $f(x,\bm{b})$, which gives the probability density of finding a
parton with longitudinal momentum fraction $x$ at a transverse distance
$\bm{b}$ from the center of momentum of the proton Burkardt:2000za ;
Burkardt:2002hr . Note that integrating the Wigner distribution
$W(x,\bm{k},\bm{b})$ over the transverse momentum $\bm{k}$ yields also
$f(x,\bm{b})$, that reveals the connection between GPD and Wigner distribution
in turn, and integrating $f(x,\bm{b})$ over $\bm{b}$ leads to the
corresponding PDF Diehl:2015uka . On the other hand, different Mellin moments
of GPDs are associated with different hadron FFs Muller:1994ses ;
Radyushkin:1996nd ; Ji:1996nm ; Ji:1996ek ; Burkardt:2000za ; Bernard:2001rs ;
Pagels:1966zza ; Guidal:2004nd  including the electromagnetic FFs (EMFFs),
axial FFs (AFFs), gravitational FFs (GFFs), and transition FFs (TFFs). These
properties make GPDs an essential ingredient of different types of hard
exclusive and elastic scattering processes (to get more information see Refs.
Goeke:2001tz ; Belitsky:2005qn ; Kumericki:2016ehc ; Hashamipour:2022noy  and
references therein).

In recent years, there have been many efforts to study different kinds of GPDs
and determine them both theoretically and phenomenologically Kumericki:2016ehc
; Hashamipour:2022noy ; Goloskokov:2007nt ; Kumericki:2009uq ; Gonzalez-
Hernandez:2012xap ; Guidal:2013rya ; Diehl:2013xca ; Selyugin:2014sca ;
Berthou:2015oaw ; Kriesten:2019jep ; Hashamipour:2019pgy ; Hashamipour:2020kip
; Hashamipour:2021kes ; Irani:2023lol ; Kriesten:2021sqc ; Xu:2021wwj ;
Ahmady:2021qed ; CLAS:2022iqy ; Guo:2022upw ; Guo:2023ahv ; Qiu:2022pla ;
Fu:2022bpf ; Duplancic:2023kwe ; Constantinou:2020hdm ; Riberdy:2023awf ;
Bhattacharya:2023ays ; Bhattacharya:2023nmv ; Bhattacharya:2023jsc ;
Selyugin:2023hqu ; Luan:2023lmt ; Kaur:2023lun ; Kaur:2023zhn ; Cichy:2023dgk
; Moffat:2023svr ; Duplancic:2022ffo ; Shuryak:2023siq ; Liu:2024umn ;
Pedrak:2017cpp ; Pedrak:2020mfm ; Grocholski:2022rqj . However, there are
still many questions that we should answer and new ideas we can advance.
Focusing on the case of zero-skewness GPDs, in Ref. Hashamipour:2022noy , the
authors were determined three kinds of GPDs, namely $H^{q}(x,t)$,
$E^{q}(x,t)$, and $\tilde{H}^{q}(x,t)$, by performing a simultaneous analysis
of a wide range of the elastic scattering experimental data. They showed that
there is a significant tension between the data of the wide-angle Compton
scattering (WACS) cross section Danagoulian:2007gs  and the proton magnetic FF
($G_{M}^{p}$) at larger values of $-t$, so that it is not possible to find a
set of GPDs that provides a desirable description of them simultaneously.
Consequently, they suggested either reassessing the experimental measurements
of both $G_{M}^{p}$ and WACS or revising the theoretical calculations of the
WACS cross section. In Ref. Irani:2023lol , by considering the new measurement
of the antineutrino-proton scattering cross section from the MINERvA
Collaboration MINERvA:2023avz , the authors showed that such a tension (but
milder) is also observed between the MINERvA and WACS data. They concluded
that the MINERvA data do not recommend the exclusion of the $G_{M}^{p}$ data
from the analysis, while the WACS data do this.

In the present study, considering the new precision measurements of the
elastic electron-proton scattering cross section at high-energy performed in
Hall A of Jefferson Lab Christy:2021snt , referred to as GMp12, we are trying
to unravel the issue raised in Hashamipour:2022noy  concerning the tension
between the WACS and $G_{M}^{p}$ data in Sec. II. Then, we investigate the
impact of GMp12 data on the unpolarized valence GPDs $H_{v}^{q}(x,t)$ and
$E_{v}^{q}(x,t)$ and their uncertainties by performing a standard $\chi^{2}$
analysis of all available elastic electron-nucleon scattering data in Sec.
III. The extracted GPDs are used to calculate the electric and scalar
quadrupole ratios, $R_{EM}$ and $R_{SM}$, and magnetic transition FF
$G^{*}_{M}/3G_{D}$ related to the nucleon-to-delta transition in Sec. IV. The
results are also compared to the available experimental data and some
theoretical predictions. Finally, we summarize our results and conclusions in
Sec. V.

##  II WACS VS $G_{M}^{p}$ data

It is well established now that the study of the elastic electron-nucleon
scattering can provide crucial information about the electromagnetic nucleon
FFs and hence the distribution of charge and magnetic moments within the
nucleon Rosenbluth:1950yq ; Yennie:1957skg ; Hand:1963zz ; Chen:2023dxp . The
precise measurements of such scattering processes are essential for testing
theoretical models, particularly the electroweak sector of the SM, and
improving our understanding of the underlying physics. Elastic electron-
nucleon scattering experiments contribute also to the determination of the
root-mean-square radii of the proton and neutron. From past to present, there
have been many measurements of this process from different experiments as well
as many efforts to extract the electromagnetic FFs or their ratios from the
original cross sections Christy:2021snt ; Kelly:2002if ;
JeffersonLaboratoryE93-038:2005ryd ; Arrington:2007ux ; Qattan:2012zf ;
A1:2013fsc ; Ye:2017gyb ; McRae:2023zgu .

Recently, GMp12 experiment which performed at Hall A of Jefferson Lab using
the basic suite of experimental instrumentation Alcorn:2004sb , has presented
new precision measurements of the elastic electron-proton scattering cross
section for momentum transfer squared ($-t=Q^{2}$) up to 15.75 GeV2
Christy:2021snt . Considering the one-photon exchange approximation, the
differential elastic electron-nucleon scattering cross section can be written
as the product of Mott cross section and a structure-dependent term that
depends on the electric and magnetic FFs (called Sachs FFs), $G_{E}$ and
$G_{M}$, as follows Sachs:1962zzc ,

$\displaystyle\frac{d\sigma}{d\Omega}=\left(\frac{d\sigma}{d\Omega}\right)_{\rm
Mott}\frac{\epsilon G_{E}^{2}(Q^{2})+\tau
G_{M}^{2}(Q^{2})}{\epsilon(1+\tau)}\,.$ (1)

In above equation, Sachs FFs are functions of
$Q^{2}=4EE^{\prime}\sin^{2}(\theta/2)$ where $\theta$ is the scattering angle
of the electron while $E$ and $E^{\prime}$ are its incident and scattered
energies in lab frame. Moreover, the dimensionless kinematic variables
$\epsilon$ and $\tau$ are written as

$\begin{split}\tau=\frac{Q^{2}}{4m^{2}}\,,\quad\epsilon=\left[1+2(1+\tau)\tan^{2}\frac{\theta}{2}\right]^{-1}\,,\end{split}$
(2)

where $m$ is the nucleon mass. Note that the fine-structure constant $\alpha$
appears in the Mott cross section which is the cross section for a recoil-
corrected relativistic structureless object,

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\rm
Mott}=\frac{\alpha^{2}}{4E^{2}\sin^{4}{\theta/2}}\frac{E^{\prime}}{E}\cos^{2}{\frac{\theta}{2}}\,.$
(3)

Note that $G_{E}^{p}(0)=1$, $G_{E}^{n}(0)=0$, and $G^{j}_{M}(0)=\mu_{j}$,
where $\mu_{j}$ is the magnetic moment of the nucleon $j=p,n$. It is also
common to express the structure-dependent term in Eq. (1), isolated in the
reduced cross section $\sigma_{R}$, in terms of the longitudinal and
transverse contributions to the cross section, $\sigma_{T}$ and $\sigma_{L}$
Christy:2021snt ,

$\displaystyle\sigma_{R}$ $\displaystyle=$ $\displaystyle\epsilon
G_{E}^{2}(Q^{2})+\tau G_{M}^{2}(Q^{2})=\sigma_{T}+\epsilon\sigma_{L}\,,$ (4)
$\displaystyle=$ $\displaystyle G_{M}^{2}(Q^{2})[\tau+\epsilon{\rm
RS}(Q^{2})/\mu_{j}^{2}]\,,$

where ${\rm RS}(Q^{2})=(\mu_{j}G_{E}/G_{M})^{2}$ is the normalized Rosenbluth
slope.

The Sachs FFs in turn are expressed in terms of the Dirac and Pauli FFs of the
nucleon, $F_{1}$ and $F_{2}$,

$\displaystyle G_{M}(Q^{2})$ $\displaystyle=F_{1}(Q^{2})+F_{2}(Q^{2})\,,$
$\displaystyle G_{E}(Q^{2})$
$\displaystyle=F_{1}(Q^{2})-\frac{Q^{2}}{4m^{2}}F_{2}(Q^{2})\,.$ (5)

On the other hand, considering the proton case, the Dirac and Pauli FFs are
related to the unpolarized valence GPDs $H_{v}^{q}$ and $E_{v}^{q}$ at zero
skewness ($\xi=0$), where $q=u,d$ stands for the up and down quarks (the
strange-quark contribution can be neglected), using the following sum rules
Diehl:2004cx ; Diehl:2013xca

$\displaystyle
F^{p}_{1}(Q^{2})=\sum_{q}e_{q}F^{q}_{1}(Q^{2})=\sum_{q}e_{q}\int_{0}^{1}dx\,H_{v}^{q}(x,\mu^{2},Q^{2})\,,$
$\displaystyle
F^{p}_{2}(Q^{2})=\sum_{q}e_{q}F^{q}_{2}(Q^{2})=\sum_{q}e_{q}\int_{0}^{1}dx\,E_{v}^{q}(x,\mu^{2},Q^{2})\,,$
(6)

where $e_{q}$ refers to the electric charge of the constituent quark $q$. On
can obtain the related expressions for the neutron FFs $F_{1}^{n}$ and
$F_{2}^{n}$ considering the isospin symmetry $u^{p}=d^{n},d^{p}=u^{n}$.

According to the above equations, by measuring the elastic electron-nucleon
scattering cross section or equivalently the electromagnetic FFs, one can get
important information about the unpolarized valence GPDs $H_{v}^{q}$ and
$E_{v}^{q}$ at zero skewness. Actually, such measurements play the main role
in the extraction of the unpolarized GPDs and the measurements of the WACS
cross section or the elastic (anti)neutrino-proton scattering cross-section
play a complementary role Hashamipour:2022noy ; Irani:2023lol . So, it is of
interest to investigate the impact of new GMp12 measurements Christy:2021snt
on $H_{v}^{q}$ and $E_{v}^{q}$ GPDs. We will do this mission in the next
section. The main question we are going to focus on in this section is how do
GMp12 measurements judge about the tension observed between the WACS cross-
section measurements Danagoulian:2007gs  and $G_{M}^{p}$ data explained
before. With a lesser importance, it is also of interest to address the
tension observed in Ref. Hashamipour:2022noy  between the world data of
$G_{M}^{p}$ from AMT07 Arrington:2007ux  analysis and the corresponding ones
from Mainz A1:2013fsc  data. To answer these questions, we make some
predictions of the GMp12 data using different GPD sets determined in Ref.
Hashamipour:2022noy  in order to investigate how the theoretical predictions
match the GMp12 data. It is worth noting in this context that the GMp12 data
have really a merit to unravel these issues because the measurements have been
presented not only for $G_{M}^{p}$ but also for the reduced cross-section
$\sigma_{R}$ in Eq. (4) which includes both $G_{E}$ and $G_{M}$. In fact, if
there is something wrong with the $G_{M}^{p}$ data versus the WACS cross-
section measurements, the GPD sets obtained by including the $G_{M}^{p}$ data
in the analysis would not be able to describe the $\sigma_{R}$ GMp12 data
well.

Now, let us briefly introduce different GPD sets from Ref. Hashamipour:2022noy
that we are going to use them to make theoretical predictions of the GMp12
measurements of $\sigma_{R}$.

  * •

Set 2: this set has been obtained by analyzing data of the ratio of the proton
electric and magnetic FFs $,R^{p}=\mu_{p}G_{E}^{p}/G_{M}^{p}$, and the neutron
electric and magnetic FFs, $G_{E}^{n}$ and $G_{M}^{n}/\mu_{n}G_{D}$, from
YAHL18 analysis Ye:2017gyb , the data of the proton magnetic FF, $G_{M}^{p}$,
from AMT07 analysis Arrington:2007ux , and finally the data of the charge and
magnetic radii of the nucleons ParticleDataGroup:2018ovx .

  * •

Set 3: this set has been obtained by analyzing the same data of Set 2 with the
difference that AMT07 data of $G_{M}^{p}$ have been replaced with the
corresponding Mainz A1:2013fsc  data.

  * •

Set 11: this set has been obtained by analyzing all data that were used for
Set 2 and Set 3 (both AMT07 and Mainz data have been included) in addition to
data of the proton AFF, $F_{A}$, from various experiments as well as the CLAS
Collaboration measurements at large $-t$ CLAS:2012ich , and finally the WACS
cross-section data from the JLab Hall A Collaboration Danagoulian:2007gs .

  * •

Set 12: this set has been obtained by removing the AMT07 and Mainz data of
$G_{M}^{p}$ from the analysis of Set 11.

Figure 1 shows a comparison between the GMp12 measurements Christy:2021snt  of
the reduced cross-section $\sigma_{R}$ and the corresponding theoretical
calculation obtained using the GPD Set 2 and Set 3 from Ref.
Hashamipour:2022noy . Additionally, the ratios of these predictions to the
data with their uncertainties have been plotted in the lower panel to
investigate the differences more precisely as the value of $Q^{2}$ increases.
The results clearly show that Set 2 has the better description of data
compared with Set 3. Note that Set 2 has been obtained by removing the
$G_{M}^{p}$ Mainz data (which covers only small $Q^{2}$ values) from the
analysis and considering only the AMT07 world data (which cover a wide range
of the $Q^{2}$ values). This indicates that the new and precise GMp12
measurements are more consistent with the AMT07 data. Considering the tension
between the AMT07 and Mainz data at small values of $Q^{2}$, we can conclude
that the GMp12 data give more validity to GPDs of Set 2 than Set 3. As a
result, one may decide to ignore the Mainz data in the future global analysis
of GPDs. Considering the uncertainty bands in the bottom panel of Fig. 1, we
can infer that the inclusion of the GMp12 data in the global analysis of GPDs
can decrease the GPD uncertainties especially at larger values of $Q^{2}$ as
we will show in the next section. Note that the large uncertainties of GPDs
Set 3 is a reflection of the fact that there were no any $G_{M}^{p}$ data
points at large $Q^{2}$ due to removing the AMT07 world data from the
analysis.

Figure 1: A comparison between the experimental data from the GMp12
measurements Christy:2021snt  for the reduced cross-section of Eq. (4) and the
corresponding theoretical predictions obtained using Set 2 and Set 3 of GPDs
taken from Ref. Hashamipour:2022noy .

Figure 2 shows the same comparison as Fig. 1 but for the results calculated
using Set 11 and Set 12 of Ref. Hashamipour:2022noy . According to the results
obtained, Set 12 that has been obtained by removing all $G_{M}^{p}$ data from
the analysis (both AMT07 and Mainz) has a inadequate description of GMp12
measurements while Set 11 is completely in agreement of the data. Note that
the analysis of Set 11 included the WACS and $G_{M}^{p}$ data simultaneously
(besides other experimental data) regardless of the tension between them.
Although Set 12 has a remarkably better description of the WACS data compared
with Set 11 (see Fig. 28 of Ref. Hashamipour:2022noy ) it absolutely cannot
describe the GMp12 measurements of the reduced cross-section $\sigma_{R}$.
Considering the fact that here the comparisons have been made with the GMp12
$\sigma_{R}$ data instead of its extracted $G_{M}^{p}$ data, we can conclude
that there is not something wrong with the previous measurements of
$G_{M}^{p}$. Actually, GMp12 data do not recommend the exclusion of the
$G_{M}^{p}$ data from the analysis unlike the measurements of the WACS cross-
section by JLab Danagoulian:2007gs . Therefore, the tension observed between
the WACS and $G_{M}^{p}$ has to have another origin. For instance, it may
suggest revising the theoretical calculations of the WACS cross section or
improving the phenomenological framework, e.g., by considering the strange
quark or gluon contributions or searching the optimum value of the
factorization scale $\mu$ as done in Refs. Hashamipour:2019pgy ;
Hashamipour:2020kip .

Figure 2: Same as Fig. 1, but for GPD Set 11 and Set 12.

##  III Impact of GMp12 data on GPDs

According to the findings presented in the previous section, it is of interest
now to investigate the impact of new GMp12 data Christy:2021snt  on the
determination of GPDs. Actually, these data can benefit us in two ways. On one
hand, because of the high accuracy of these data, they can lead to a
remarkable decrease in the uncertainties of the extracted GPDs though their
central values might also be tuned. One the other hand, since these data
contain also data points with large $Q^{2}$ value, they can really improve our
knowledge in the kinematic where the older world experimental data of
$G_{M}^{p}$ have not a considerable contribution and have large uncertainties
unlike the small $Q^{2}$ region. In this section, we first introduce briefly
the phenomenological framework we adopt for performing our QCD analysis of
GPDs. Then we study the impact of GMp12 data on the unpolarized valence GPDs
$H_{v}^{q}$ and $E_{v}^{q}$ by performing two separate analyses, one with and
the other without considering the GMp12 data in the analysis, and comparing
their results with each other.

###  III.1 Phenomenological framework

As mentioned in Sec. II, the unpolarized valence GPDs $H_{v}^{q}$ and
$E_{v}^{q}$ at zero skewness ($\xi=0$) are related to the Sachs FFs, $G_{E}$
and $G_{M}$, through Eqs. (II) and (6). So, in order to calculate
electromagnetic FFs and hence the reduced cross-section $\sigma_{R}$ in Eq.
(4), one needs a suitable parameterization of GPDs in terms of $x$ and $Q^{2}$
(or equivalently $t$). In this study, we adopt the ansatz proposed in Refs.
Diehl:2004cx ; Diehl:2013xca  that, as it has been shown in Refs.
Hashamipour:2019pgy ; Hashamipour:2020kip ; Hashamipour:2021kes ;
Irani:2023lol ; Hashamipour:2022noy , leads to a good description of data,

$\displaystyle H_{v}^{q}(x,\mu^{2},t)=q_{v}(x,\mu^{2})\exp[tf_{v}^{q}(x)]\,,$
$\displaystyle
E_{v}^{q}(x,\mu^{2},t)=e_{v}^{q}(x,\mu^{2})\exp[tg_{v}^{q}(x)]\,.$ (7)

Here $q_{v}(x,\mu^{2})$ is the unpolarized PDF for the valence quark $q$ which
is taken from the the next-to-leading order (NLO) NNPDF analysis NNPDF:2021njg
at scale $\mu=2$ GeV, using the LHAPDF package Buckley:2014ana  (as mentioned
before, the contribution of the strange quark is neglected in the present
study). For the profile functions $f_{q}(x)$ and $g_{q}(x)$ we use the
following parametrization form as suggested in Diehl:2004cx ; Diehl:2013xca

${\cal F}(x)=\alpha^{\prime}(1-x)^{3}\log\frac{1}{x}+B(1-x)^{3}+Ax(1-x)^{2}.$
(8)

Although the profile function can take other forms, the above form is flexible
enough and leads to a better fit of the data Hashamipour:2019pgy . Note also
that at forward limit ($t=0$ and $\xi=0$), GPDs in Eq. (7) reduce to the
corresponding PDFs. However, in the case of GPDs $E_{v}^{q}$, there is no
parameterization available for the forward limit $e_{v}^{q}(x,\mu^{2})$ from a
previous determination. Therefore, $e_{v}^{q}(x,\mu^{2})$ must also be
obtained from the fit to data. In this regard, we consider a parametrization
form similar to Ref. Diehl:2013xca  that has also been used in the analyses of
Ref. Hashamipour:2022noy ; Hashamipour:2021kes

$e_{v}^{q}(x,\mu^{2})=\kappa_{q}N_{q}x^{-\alpha_{q}}(1-x)^{\beta_{q}}(1+\gamma_{q}\sqrt{x})\,,$
(9)

which is defined at $\mu=2$ GeV. $\kappa_{u}$ and $\kappa_{d}$ are the
magnetic moments of the up and down quarks that can be computed from the
measured magnetic moments of proton and neutron in the units of nuclear
magneton. In the present study, we use $\kappa_{u}=1.67$ and
$\kappa_{d}=-2.03$ which have been calculated using the values of the magnetic
moments of proton and neutron taken from ParticleDataGroup:2018ovx . Then, the
normalization factor $N_{q}$ can be obtained from the following sum rule:

$\int_{0}^{1}dxe_{v}^{q}(x)=\kappa_{q}.$ (10)

According to the above explanations, the unknown free parameters associated
with each quark flavor in this study are $\alpha^{\prime}$, $B$, $A$,
$\alpha$, $\beta_{q}$, and $\gamma$. The optimum values of these parameters
are obtained as usual by performing a standard $\chi^{2}$ analysis of the
experimental data in which the following function is minimized:

$\chi^{2}=\sum_{i}^{n}\Big{(}\frac{{\cal E}_{i}-{\cal T}_{i}}{\delta{\cal
E}_{i}}\Big{)}^{2}\,.$ (11)

Here ${\cal E}_{i}$ is the measured value of the experimental data point $i$
and ${\cal T}_{i}$ denotes the corresponding theoretical prediction. The
summation is performed over all data point included in the analysis taken from
different datasets. One can obtain the experimental errors $\delta{\cal
E}_{i}$ associated with each measured data point by adding the systematic and
statistical errors in quadrature. To perform the minimization procedure in
this study we use the CERN program library MINUIT James:1975dr . Our method
for calculating uncertainties of GPDs and physical observables is as in Ref.
Pumplin:2001ct  that is the standard Hessian approach. Note that we use the
standard value $\Delta\chi^{2}=1$ for constructing the confidence region.

As discussed in Ref. Diehl:2013xca , the the profile functions and forward
limits of GPDs must obey a positivity condition which comes from the fact that
the quark and antiquark distributions in $x$ cannot be same at a nominal
transverse distance $b$ from the proton center. In this study, we adopt the
method proposed in Hashamipour:2021kes  (Scenario 2) in which the positivity
condition is preserved automatically in a wide range of the $x$ values by
incorporating condition $g_{v}^{q}(x)<f_{v}^{q}(x)$ in the main body of the
fit program. It is well established now that a suitable and efficient approach
to determine the optimum values of the unknown parameters in every global
analysis of the experimental data is utilizing the parametrization scan
procedure Hashamipour:2021kes ; Hashamipour:2022noy ; Irani:2023lol  proposed
first in the QCD analysis of PDFs performed by H1 and ZEUS Collaborations
H1:2009pze . An advantage of this procedure is that the final parametrization
of each distribution is obtained systematically. Accordingly, here we follow
the same approach to find the optimum parameterizations and the best values of
their unknown parameters.

###  III.2 Data selection

Just as for the analyses of Refs. Hashamipour:2021kes ; Hashamipour:2022noy ,
the main body of the experimental data that we consider in the present study
comprises of the YAHL18 data Ye:2017gyb  which include a wide and updated
range of the world electron scattering data off both proton and neutron
targets and have been obtained by incorporating the two-photon exchange (TPE)
corrections. These data include 69, 38, and 33 data points of the world
$R^{p}=\mu_{p}G_{E}^{p}/G_{M}^{p}$ polarization, $G_{E}^{n}$, and
$G_{M}^{n}/\mu_{n}G_{D}$ measurements, respectively. Here,
$G_{D}=(1+Q^{2}/\Lambda^{2})^{-2}$ is the dipole form factor with
$\Lambda^{2}=0.71$ GeV2. Overall, the YAHL18 data cover the $-t$ range from
0.00973 to 10 GeV2 for 140 data points. Since GPDs $H_{v}^{q}$ and $E_{v}^{q}$
at zero skewness are related to the mean squared of the charge and magnetic
radii of the nucleons, $r_{jE}$ and $r_{jM}$, through the Sachs FFs

$\displaystyle\left<r_{jE}^{2}\right>=\left.6\derivative{G_{E}^{j}}{t}\right|_{t=0}\,,$
$\displaystyle\left<r_{jM}^{2}\right>=\left.\frac{6}{\mu_{j}}\derivative{G_{M}^{j}}{t}\right|_{t=0},$
(12)

we also consider their related measurements (four data points) from
ParticleDataGroup:2018ovx . Note that these data can provide crucial
information about the small-$t$ behavior of GPDs, especially the profile
functions $f_{q}(x)$ and $g_{q}(x)$. In addition to these data, in order to
put further constraints on GPDs, we include the data of $G_{E}^{p}$ from AMT07
Arrington:2007ux  and Mainz A1:2013fsc  which contain 47 and 77 data points,
respectively. These data cover the $-t$ range from 0.007 to 5.85 GeV2. An
important point in this context that should be mentioned is that we could use
instead the data of $G_{M}^{p}$ from these analyses, but as it has been shown
in Refs. Hashamipour:2022noy ; Ye:2017gyb  the tension between the world AMT07
data and Mainz measurements is worse in this case. Hence, we decided to use
the $G_{E}^{p}$ data in the present study which show less tension between
AMT07 and Mainz data Ye:2017gyb . Finally, as mentioned before, we include the
new data of the reduced cross-section $\sigma_{R}$ from JLab experiment
Christy:2021snt  for the first time in order to study their impact on the
extracted GPDs. These data cover the $-t$ range from 1.577 to 15.76 GeV2. So,
considering also the fact that they are precise compared with the older data,
they provide crucial information on GPDs especially at larger values of $-t$.

###  III.3 Results

As discussed before, one of the main goals of the present study is
investigating the impact of the new and precise measurements of the reduced
cross-section of the elastic electron-proton scattering defined in Eq. (4)
from JLab experiment Christy:2021snt , namely GMp12, on the unpolarized
valence GPDs at zero skewness. In Sec. II, we showed that these data are more
consistent with GPDs Set 2 of Ref. Hashamipour:2022noy  where the Mainz data
have been excluded from the analysis and also GPDs Set 11 where the
$G_{M}^{p}$ data (both AMT07 and Mainz) have not been excluded from the
analysis and have been analyzed besides the AFFs and WACS data. Now it is of
interest to see how the GMp12 data affect the extracted unpolarized valence
GPDs $H_{v}^{q}$ and $E_{v}^{q}$ especially at larger values of $-t$ where we
expect that these data have grater effect. To this aim, here we construct
three analyses of GPDs. The first analysis which is called the “Base Fit”
includes all data sets described in Sec. III.2 except GMp12 data. In this way,
we obtain a set of GPDs which has no influence from the GMp12 data. Note that
it is different from Set 1 of Hashamipour:2022noy  (where both AMT07 and Mainz
data have been included in the analysis) since we are going to use the
$G_{E}^{p}$ data instead of $G_{M}^{p}$ as explained before. Then, we repeat
the Base Fit but by including also the GMp12 data in the analysis to
investigate their impact on the extracted GPDs. The GPD set that is obtained
from this analysis is called “GMp12”. In addition to these analyses, we
perform an analysis by removing the Mainz data considering the notable tension
observed between the AMT07 and Mainz data Ye:2017gyb  (note again that this
tension is absolutely significant for the case of $G_{M}^{p}$ data). This set
is called “GMp12_NoMainz”.

Following the parametrization scan procedure as described in
Hashamipour:2021kes ; Hashamipour:2022noy ; Irani:2023lol , we find a set of
GPDs with $\alpha^{\prime}_{g_{v}^{u}}=\alpha^{\prime}_{f_{v}^{u}}$,
$B_{g_{v}^{u}}=0$, and $B_{g_{v}^{d}}=0$. Actually, releasing these parameters
does not lead to any decrease in the value of $\chi^{2}$ and hence any
improvement in the quality of the fit. All other parameters are obtained from
the fit to data as usual. Table 1 contains the optimum values of the
parameters obtained from three analyses described above. As can be seen, the
inclusion of the GMp12 data in the analysis has led to a significant decrease
in uncertainties of the parameters as expected. Their central values have also
been change significantly in some cases. Note that removing the Mainz data
from the analysis does not changed considerably the final results, especially
for GPDs $H_{v}^{q}$. However, some changes are seen in the forward limits
$e_{v}^{q}(x)$ of GPDs $E_{v}^{q}$. The impact of GMp12 data on GPDs will be
investigated later in more details where we compare the extracted GPDs to each
other.

Table 1: The optimum values of the parameters of the profile functions (8), and distributions $e_{v}^{q}(x)$ of Eq. (9) obtained from three analyses described in Sec. III.3. Distribution | Parameter | Base Fit | GMp12 | GMp12_NoMainz
---|---|---|---|---
$f_{v}^{u}(x)$ | $\alpha^{\prime}$ | $0.721\pm 0.012$ | $0.712\pm 0.008$ | $0.708\pm 0.009$
| $A$ | $0.987\pm 0.123$ | $0.941\pm 0.039$ | $0.936\pm 0.040$
| $B$ | $0.830\pm 0.057$ | $0.864\pm 0.030$ | $0.876\pm 0.034$
$f_{v}^{d}(x)$ | $\alpha^{\prime}$ | $0.412\pm 0.037$ | $0.396\pm 0.028$ | $0.392\pm 0.027$
| $A$ | $1.644\pm 0.539$ | $1.559\pm 0.365$ | $1.557\pm 0.348$
| $B$ | $1.489\pm 0.205$ | $1.556\pm 0.148$ | $1.567\pm 0.141$
$g_{v}^{u}(x)$ | $\alpha^{\prime}$ | $\alpha^{\prime}_{f_{v}^{u}}$ | $\alpha^{\prime}_{f_{v}^{u}}$ | $\alpha^{\prime}_{f_{v}^{u}}$
| $A$ | $0.577\pm 0.295$ | $0.381\pm 0.154$ | $0.302\pm 0.184$
| $B$ | $0.000$ | $0.000$ | $0.000$
$g_{v}^{d}(x)$ | $\alpha^{\prime}$ | $0.735\pm 0.058$ | $0.730\pm 0.057$ | $0.733\pm 0.057$
| $A$ | $1.773\pm 0.512$ | $1.710\pm 0.460$ | $1.709\pm 0.389$
| $B$ | $0.000$ | $0.000$ | $0.000$
$e_{v}^{u}(x)$ | $\alpha$ | $0.648\pm 0.054$ | $0.678\pm 0.041$ | $0.677\pm 0.055$
| $\beta$ | $7.114\pm 1.111$ | $7.537\pm 0.610$ | $7.904\pm 0.812$
| $\gamma$ | $2.232\pm 2.617$ | $3.301\pm 2.278$ | $3.648\pm 3.415$
$e_{v}^{d}(x)$ | $\alpha$ | $0.739\pm 0.035$ | $0.753\pm 0.025$ | $0.748\pm 0.035$
| $\beta$ | $5.778\pm 1.038$ | $6.101\pm 0.985$ | $5.955\pm 0.810$
| $\gamma$ | $6.649\pm 3.992$ | $8.942\pm 2.855$ | $7.880\pm 4.503$

The $\chi^{2}$ values of three analyses described above are presented in Table
2 where we have listed the data sets used in the analysis with their
references in the first column. The data have been separated according to
their related observables. The range of $-t$ associated to each data set has
been reported in the second column. Note that the GMp12 data cover a wide
range of $-t$ compared with other datasets. This clearly indicates the
importance of these data for constraining GPDs. For each dataset, we have
presented the value of $\chi^{2}$ per number of data points,
$\chi^{2}$/$N_{\textrm{pts.}}$ which makes the deduction on the quality of the
fit easier. The values of total $\chi^{2}$ divided by the number of degrees of
freedom, $\chi^{2}/\mathrm{d.o.f.}$, have also been presented in the last row.

Table 2: The results of three analyses described in Sec. III.3. Observable |  -$t$ (GeV2) |  $\chi^{2}$/$N_{\textrm{pts.}}$
---|---|---
|  | Base Fit | GMp12 | GMp12_NoMainz
$G_{E}^{p}$ A1:2013fsc  | $0.0152-0.5524$ | $219.1/77$ | $220.0/77$ | $-$
$G_{E}^{p}/G_{D}$ Arrington:2007ux  | $0.007-5.85$ | $40.1/47$ | $39.9/47$ | $39.9/47$
$R^{p}=\mu_{p}G_{E}^{p}/G_{M}^{p}$ Ye:2017gyb  | $0.162-8.49$ | $102.7/69$ | $104.9/69$ | $104.9/69$
$G_{E}^{n}$ Ye:2017gyb  | $0.00973-3.41$ | $27.3/38$ | $26.8/38$ | $26.0/38$
$G_{M}^{n}/\mu_{n}G_{D}$ Ye:2017gyb  | $0.071-10.0$ | $38.9/33$ | $37.4/33$ | $36.8/33$
GMp12 $\sigma_{R}$ Christy:2021snt  | $1.577-15.76$ | $-$ | $15.6/13$ | $13.7/13$
$\sqrt{\left<r_{pE}^{2}\right>}$ ParticleDataGroup:2018ovx  | $0$ | $1.8/1$ | $1.6/1$ | $0.0/1$
$\sqrt{\left<r_{pM}^{2}\right>}$ ParticleDataGroup:2018ovx  | $0$ | $2.5/1$ | $2.4/1$ | $2.5/1$
$\left<r_{nE}^{2}\right>$ ParticleDataGroup:2018ovx  | $0$ | $0.1/1$ | $0.8/1$ | $0.9/1$
$\sqrt{\left<r_{nM}^{2}\right>}$ ParticleDataGroup:2018ovx  | $0$ | $15.9/1$ | $18.0/1$ | $16.9/1$
Total $\chi^{2}/\mathrm{d.o.f.}$ |  | $448.4/253$ | $467.4/266$ | $241.6/189$

On one hand, considering the results of the Base Fit in Table 2, a remarkable
tension is seen between the $G_{E}^{p}$ data of AMT07
($\chi^{2}$/$N_{\textrm{pts.}}=0.85$) and Mainz
($\chi^{2}$/$N_{\textrm{pts.}}=2.85$), though it is much more moderate
compared with the case of $G_{M}^{p}$ data reported in Ref.
Hashamipour:2022noy  (2.03 for AMT07 versus 6.02 for Mainz). On the other
hand, comparing the results of three analyses in Table 2, one realizes that
the inclusion of the GMp12 data in the analysis does not significantly affect
the $\chi^{2}$ values of the other datasets. Therefore, considering the good
$\chi^{2}$ values of the GMp12 data in the two last analyses, we can conclude
that the GMp12 data give more credit to the AMT07 data than Mainz data (as it
was deduced from Fig. 1). Note again that the precise GMp12 data are as the
reduced cross-section $\sigma_{R}$, i.e., the combination of $G_{E}$ and
$G_{M}$. So, they are a good benchmark to judge about the tension observed
between the AMT07 and Mainz data. However, it should be noted that in Ref.
Hashamipour:2022noy  the authors concluded that the inclusion of the
$G_{M}^{p}$ Mainz data in the analysis could be useful because they lead to a
better description of the nucleon radii data.

Figure 3 indicates the results obtained for the valence unpolarized GPD
$xH_{v}^{u}(x)$ with their uncertainties from three aforementioned analyses.
The distributions have been compared as ratio to Base Fit at four different
values of $t$, namely $t=-3,-6,-9,-12$ GeV2, in order to study the impact of
GMp12 data on $xH_{v}^{u}(x)$ more clearly in a wide range of $-t$. Note that
the error bands include also the uncertainties of the NNPDF PDFs NNPDF:2021njg
all distributions at $t=0$ are reduced to the $u_{v}(x)$ PDF of NNPDF. As
expected, the GMp12 data have more impact on $xH_{v}^{u}(x)$ as the absolute
value of $-t$ increases. They lead to an increase in $xH_{v}^{u}(x)$ at small
values of $x$ (note that the peak of valence GPDs, e.g., occurs at $x=0.4$ for
$t=-3$ GeV2), and a minor decrease at $x\simeq 0.1-0.2$. The results also show
that without considering the Mainz data, the impact of GMp12 on
$xH_{v}^{u}(x)$ at small $x$ region becomes even stronger. An interesting
finding is the significant reduces in uncertainties of $xH_{v}^{u}(x)$ at
$x\gtrsim 0.02$ especially as the absolute value of $-t$ increases. Note that
the large $x$ region is correlated to the large values of $t$. This clearly
indicates the crucial role of the GMp12 data in the future analysis of GPDs.

Figure 3: A comparison between the results of three analyses performed in Sec.
III.3 for the valence unpolarized GPD $xH_{v}^{u}(x)$ as the ratio to the Base
Fit at four $t$ values shown in panels (a) $t=-3$, (b) $t=-6$, (c) $t=-9$, and
(d) $t=-12$ GeV2.

Figure 4 shows the same results as Fig. 3, but for the valence unpolarized GPD
$xH_{v}^{d}(x)$. The results obtained show that the impact of GMp12 data on
the down quark distribution is almost similar to the case of up quark
considering the trends, though it is somewhat stronger in magnitude. Although
the GMp12 data are led to a decrease in uncertainties of $xH_{v}^{d}(x)$ at
$x\gtrsim 0.02$, the severity of this decrease is less compared to the case of
$xH_{v}^{u}(x)$. This was predictable considering the fact that GMp12 data are
pertained to the electron scattering off the proton. While for a remarkable
reduce in uncertainties of the down quark distribution we need precise
measurements of the neutron (or deuteron) cross-section.

Figure 4: Same as Fig. 3, but for valence unpolarized GPD $xH_{v}^{d}(x)$.

The results obtained for GPD $xE_{v}^{u}(x)$ are shown in Fig. 5 and compared
again at $t=-3,-6,-9,-12$ GeV2. Note that, as mentioned before, there is no
equivalent of PDFs for the forward limits of GPDs $E_{v}^{q}$. So, in this
case, both the forward limits $e_{v}^{q}(x)$ and profile functions
$g_{v}^{q}(x)$ are obtained from the fit to data. In this way, $xE_{v}^{u}(x)$
distributions (and also $xE_{v}^{d}(x)$) obtained from three analyses
performed in this study are different even at $t=0$. However, since the
differences increase as the absolute value of $-t$ increases, we have not
shown here the results corresponding to $t=0$. As can be seen, $xE_{v}^{u}(x)$
has been affected more after the inclusion of the GMp12 data in the analysis
compared with $xH_{v}^{u}(x)$ in Fig. 3. It is logical because according to
Eq. (II) as the value of $Q^{2}$ ($-t$) increases, the contribution of $F_{2}$
which contain GPDs $E_{v}^{q}$ becomes grater. Note also that compared with
GPD $xH_{v}^{u}(x)$, $xE_{v}^{u}(x)$ has been impacted almost at all values of
$x$. An egregious difference is that $xE_{v}^{u}(x)$ undergoes a drastic
suppression at large $x$ region. These findings clearly indicate that
considering the GMp12 data in the global analysis of GPDs at zero skewness can
significantly affect both the shape and uncertainty of GPD $xE_{v}^{u}(x)$.

Figure 5: Same as Fig. 3, but for valence unpolarized GPD $xE_{v}^{u}(x)$.

Figure 6 shows the same results as Fig. 3 but for GPD $xE_{v}^{d}(x)$.
According to the results obtained, one can conclude that the GMp12 data do not
provide any constrain on GPD $xE_{v}^{d}(x)$ at small and medium values of
$x$. However, some changes in $xE_{v}^{d}(x)$ are seen at large $x$ region
both in shape and uncertainty. It should be noted again that GMp12 data have
been measured from the scattering off the proton and it is sensible that they
can not significantly affect the down quark distribution especially
considering the smaller contributions of GPDs $E_{v}^{q}$ rather that
$H_{v}^{q}$ in the total cross-section.

Figure 6: Same as Fig. 3 but for valence unpolarized GPD $xE_{v}^{d}(x)$.

##  IV Nucleon to $\Delta$ electromagnetic transition

Alongside the usual FFs of the nucleon, their transition FFs to another
particle can also provide valuable information about the internal structure
and dynamics of the nucleons Stoler:1993yk ; Burkert:2004sk ; Aznauryan:2011qj
; Aznauryan:2012ba ; Hagelstein:2017cbl ; Proceedings:2020fyd ;
Burkert:2022ioj ; Ramalho:2023hqd . Actually, TFFs are crucial in
understanding processes like electromagnetic and weak interactions in particle
physics. By investigating how TFFs are changed during particle interactions or
decays, one can gain insights into fundamental aspects of particle behavior,
such as their charge distribution, momentum transfer, and the strength of
their interactions with other particles. This knowledge is essential for
developing and refining theoretical models, as well as for interpreting
experimental data. Overall, there are several types of TFFs Ramalho:2023hqd ;
Aung:2024qmf ; Chen:2023zhh ; Ozdem:2022zig ; Lu:2019bjs  including the
electromagnetic TFFs (EM TFFs), weak FFs, gravitational TFFs, and axial TFFs.
Each type of form factor provides information about different aspects of
particle interactions and properties. Among them, EM TFFs Burkert:2022ioj ;
Ramalho:2023hqd ; Aung:2024qmf  provides information about how the particle
interacts electromagnetically with other particles or fields. They quantify
how the charge is distributed within the particle and how it responds to
electromagnetic interactions. In the experimental point of view, EM TFFs are
measured by studying the scattering or decay of particles involving
electromagnetic interactions, such as photon emission or absorption.

Measurements of the nucleon-to-delta ($N\rightarrow\Delta$) electromagnetic
TFFs are typically performed by utilizing electron scattering off a nucleon
target ($\Delta$ can also be produced in scattering pions off a nucleon). By
measuring the differential cross-sections and polarization observables of the
scattered electrons, one can extract information about the TFFs. Experimental
facilities such as Jefferson Lab in the US, MAMI in Germany, and others around
the world have conducted experiments to precisely measure these TFFs. It is
well established now that the $\Delta(1232)$ resonance is the first excited
state of the nucleon and has isospin $3/2$, and therefore can be existed in
four different charge states: $\Delta^{++}$, $\Delta^{+}$, $\Delta^{0}$, and
$\Delta^{-}$. These states have approximately the same mass and width. Note
that the spin of the $\Delta(1232)$ is also $3/2$.

The experiments in the region of the $\Delta(1232)$ resonance are often used
to measure the electric and scalar quadrupole ratios, $R_{EM}$ and $R_{SM}$,
which are defined as Burkert:2022ioj

$R_{EM}\equiv\frac{Im(E_{1+})}{Im(M_{1+})}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ R_{SM}\equiv\frac{Im(S_{1+})}{Im(M_{1+})}\,,$ (13)

where $E_{1+}$, $M_{1+}$, and $S_{1+}$ are the electromagnetic transition
multipoles at the mass of the $\Delta(1232)$ resonance. The multipole ratios
can also be written in terms of the Jones-Scadron FFs as Hagelstein:2017cbl

$R_{EM}(Q^{2})=-\frac{G^{*}_{E}(Q^{2})}{G^{*}_{M}(Q^{2})}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
R_{SM}(Q^{2})=-\frac{Q_{+}Q_{-}}{4M^{2}_{\Delta}}\frac{G^{*}_{C}(Q^{2})}{G^{*}_{M}(Q^{2})}\,,$
(14)

where $G^{*}_{E}$, $G^{*}_{M}$, and $G^{*}_{C}$ are the nucleon-to-delta TFFs.
These TFFs can be connected to the electromagnetic nucleon properties via
large number of color ($N_{c}$) relations and modeled at finite momentum
transfer as Pascalutsa:2007wz

$\displaystyle{}G^{*}_{M}(Q^{2})=$
$\displaystyle\frac{1}{\sqrt{2}}\left[F_{2}^{p}(Q^{2})-F_{2}^{n}(Q^{2})\right]\,,$
$\displaystyle G^{*}_{E}(Q^{2})=$
$\displaystyle\left(\frac{M}{M_{\Delta}}\right)^{3/2}\frac{\Delta
M_{+}}{2\sqrt{2}Q^{2}}G_{E}^{n}(Q^{2})\,,$ $\displaystyle G^{*}_{C}(Q^{2})=$
$\displaystyle\frac{4M_{\Delta}^{2}}{\Delta M_{+}}G^{*}_{E}(Q^{2})\,.$ (15)

In above equations, $\Delta M_{+}=M_{\Delta}^{2}-m^{2}$, where $M_{\Delta}$
and $m$ denote the mass of $\Delta(1232)$ and nucleon respectively.
Considering Eqs. (II), (6), and (IV) together, one can conclude that the
nucleon-to-delta TFFs $G^{*}_{E}$, $G^{*}_{M}$, and $G^{*}_{C}$ are also
related to the valence unpolarized GPDs $H_{v}^{q}$ and $E_{v}^{q}$. Hence, by
measuring these TFFs, or equivalently the electric and scalar quadrupole
ratios, $R_{EM}$ and $R_{SM}$, one can get new and important information on
GPDs at zero skewness. Fortunately, there have been various measurements of
$R_{EM}$ and $R_{SM}$ (note that TFFs can also be extracted in such
measurements) up to now. So, it is interesting to see that how GPDs extracted
in the previous section through the analysis of the electromagnetic FFs
describe the available data relating to the nucleon-to-delta transition.

Figure 7 shows a comparison between the experimental measurements of $R_{EM}$
(top panel) and $R_{SM}$ (bottom panel) from various experiments Stave:2006ea
; Sparveris:2006uk ; Sparveris:2013ena ; Pospischil:2000ad ; CLAS:2001cbm ;
CLAS:2006ezq ; Julia-Diaz:2006ios ; CLAS:2009ces ; Frolov:1998pw ;
Villano:2009sn ; JeffersonLabHallA:2005wfu ; Blomberg:2015zma ; A1:2008ocu ;
Elsner:2005cz ; OOPS:2004kai ; Tiator:2000iy  and the corresponding
theoretical prediction calculated using the GPDs set named GMp12 obtained in
the previous section. Note that we have presented the results in percent as it
is common. In this plot, we have separated the data of CLAS (2002)
CLAS:2001cbm , CLAS (2006) CLAS:2006ezq ; Julia-Diaz:2006ios , CLAS (2009)
CLAS:2009ces , Hall C Frolov:1998pw ; Villano:2009sn  and Hall A
JeffersonLabHallA:2005wfu  from JLab as well as MAMI Stave:2006ea ;
Sparveris:2006uk ; Sparveris:2013ena ; Pospischil:2000ad  and Blomberg (2016)
Blomberg:2015zma . The other experimental data denoted as “Other sources” have
been gathered from Refs. A1:2008ocu ; Elsner:2005cz ; OOPS:2004kai ;
Tiator:2000iy . Note also that there are no remarkable differences between the
predictions of three sets of GPDs obtained in the previous section. So, we
decided to include only the results of the set named GMp12 that has been
obtained by considering all data introduced in Sec. III.2. As can be seen from
Fig. 7, our results are in a very good consistency with the experimental data.
Note again that these data have not included in our analyses of GPDs. The good
agreement between the theoretical predictions and experimental measurements
beautifully proves the universality property of GPDs. Another important point
can be deduced from Fig. 7 is that the inclusion of the $R_{EM}$ and $R_{SM}$
data in the analysis of GPDs at zero skewness can significantly impacted the
extracted GPDs especially their uncertainties at smaller values of $Q^{2}$.
Consequently, we strongly suggest renewing the comprehensive analysis
performed in Ref. Hashamipour:2022noy  by considering also the $R_{EM}$ and
$R_{SM}$ data in the analysis.

Figure 7: A comparison between our results for $R_{EM}$ and $R_{SM}$
(expressed in percent) obtained using the set of GPDs named GMp12 and the
corresponding experimental measurements from MAMI Stave:2006ea ;
Sparveris:2006uk ; Sparveris:2013ena ; Pospischil:2000ad , CLAS (2002)
CLAS:2001cbm , CLAS (2006) CLAS:2006ezq ; Julia-Diaz:2006ios , CLAS (2009)
CLAS:2009ces , Hall C Frolov:1998pw ; Villano:2009sn , Hall A
JeffersonLabHallA:2005wfu , and Blomberg (2016) Blomberg:2015zma . The other
experimental data denoted as “Other sources” have been gathered from Ref.
A1:2008ocu ; Elsner:2005cz ; OOPS:2004kai ; Tiator:2000iy .

It is also of interest to compare the results for the magnetic transition form
factor $G^{*}_{M}$ which is a very sensitive quantity to the valence
unpolarized GPDs $E_{v}^{q}$. Note that our understanding of GPDs $E_{v}^{q}$
have always been less robust compared with that of GPDs $H_{v}^{q}$ because of
fewer constraints on $E_{v}^{q}$ from the experimental measurements
Hashamipour:2022noy ; Hashamipour:2021kes . In Fig. 8, we have presented a
comparison between the experimental measurements of $G^{*}_{M}/3G_{D}$ from
CLAS (2006) CLAS:2006ezq , CLAS (2009) CLAS:2009ces , and Hall C Frolov:1998pw
; Villano:2009sn  and the corresponding theoretical prediction obtained using
GPDs set GMp12 as well as the Light-Front Relativistic Quark Model (LFRQM)
Aznauryan:2012ec ; Aznauryan:2016wwm  and the Dyson-Schwinger Equation (DSE)
Segovia:2014aza . It should be noted that the data points from CLAS (2006) and
Hall C have been presented in their related references, but for CLAS (2009) we
have calculated them according to the information presented in CLAS:2009ces .
As can be seen from Fig. 8, our result is again in a sensible consistency with
the experimental data especially compared with the results of LFRQM and DSE.
The results obtained clearly confirm this statement that the inclusion of the
measurements of the TFFs in the analysis of GPDs at zero skewness can play a
crucial role to constrain them and reduce their uncertainties especially at
small $-t$ region.

Figure 8: A comparison between our result for $G^{*}_{M}/3G_{D}$ obtained
using the set of GPDs named GMp12 and the corresponding experimental
measurements from CLAS (2006) CLAS:2006ezq ; Julia-Diaz:2006ios , CLAS (2009)
CLAS:2009ces , and Hall C Frolov:1998pw ; Villano:2009sn  as well as the pure
theoretical calculations from the LFRQM Aznauryan:2012ec ; Aznauryan:2016wwm
and the Dyson-Schwinger equation Segovia:2014aza .

##  V Summary and conclusion

In this study, we showed that the JLab GMp12 data Christy:2021snt  of the
reduced cross-section of the elastic electron-proton scattering play a crucial
role in disentangling the tension observed in Ref. Hashamipour:2022noy
between the world data of $G_{M}^{p}$ from AMT07 analysis Arrington:2007ux
and the corresponding ones from Mainz A1:2013fsc  as well as that observed
between the measurements of the WACS cross-section Danagoulian:2007gs  and
$G_{M}^{p}$ data. As a result, we found that GMp12 data are more consistent
with AMT07 $G_{M}^{p}$ world data rather than Mainz data. They also do not
recommend the exclusion of the $G_{M}^{p}$ data from the analysis, while the
WACS data do this. This may suggest to revising the theoretical calculations
relating to the WACS cross-section or improving the phenomenological
framework.

As the next step, we investigated the impact of GMp12 data on valence
unpolarized GPDs $H_{v}^{q}$ and $E_{v}^{q}$ at zero skewness by performing
two separate analyses, one with and the other without considering the GMp12
data in the analysis, and comparing their results with each other. As
expected, we found that GMp12 data have more impact on GPDs as the absolute
value of $-t$ increases. They lead to an increase in $xH_{v}^{u}(x)$ and
$xH_{v}^{d}(x)$ at small values of $x$ as well as a minor decrease at medium
$x$ region. For the case of GPD $xE_{v}^{u}(x)$, we found that it has been
affected more compared to $xH_{v}^{u}(x)$ so that it undergoes a drastic
suppression at large $x$ region too. However, we indicated that the GMp12 data
do not provide remarkable constrains on GPD $xE_{v}^{d}(x)$ except at large
$x$ region. Overall, the results show that by excluding the Mainz data from
the analysis, the impact of GMp12 on GPDs at small $x$ region becomes even
stronger. A very important finding of the present study is the significant
reduces in uncertainties of GPDs at $x\gtrsim 0.02$, especially as the
absolute value of $-t$ increases, after the inclusion of the GMp12 data in the
analysis. These findings clearly indicate the crucial role of the GMp12 data
in the future analysis of GPDs at zero skewness. We emphasis that considering
these data can significantly affect both the shape and uncertainty of GPDs.

By calculating the electric and scalar quadrupole ratios, $R_{EM}$ and
$R_{SM}$, of the the $\Delta(1232)$ resonance and the nucleon-to-delta
magnetic transition form factor $G^{*}_{M}/3G_{D}$ theoretically using the
extracted GPDs and comparing the results with corresponding experimental
measurements, we showed that there is an excellent consistency between our
results and experiments that beautifully proves the universality property of
GPDs. Our predictions have also much better description of data compared with
the pure theoretical prediction obtained from LFRQM and DSE. We indicated that
the inclusion of these data in the analysis of GPDs at zero skewness can
significantly impacted the extracted GPDs especially their uncertainties at
smaller values of $Q^{2}$. Consequently, we strongly suggest renewing the
comprehensive analysis performed in Ref. Hashamipour:2022noy  by considering
also the experimental data relating to the $N\rightarrow\Delta$ transition.
This aspect will be further explored in our future research.

## ACKNOWLEDGMENTS

M. Goharipour and H. Hashamipour are thankful to the School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM), for
financial support provided for this research. M. Goharipour thanks the
Theoretical Physics Department, CERN, for providing financial support for a
short-term visit during which a crucial part of the present project was done.

## Note Added

We are pleased to provide the GPDs extracted in the present study with their
uncertainties in any desired values of $x$ and $-t$ upon request.

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