###### Abstract

For over 100 years, scientists have investigated the properties of the proton,
which is one of the most abundant components of visible matter in the
universe. Nevertheless, researchers do not fully understand many details about
its internal structure and dynamics. Time-like electromagnetic form factors
are one of the observable quantities that can help us achieve a deeper
understanding. In this review article, we present an overview of the current
experimental status in this field, consisting of measurements of the time-like
reactions $e^{+}e^{-}\to p\bar{p}$, $p\bar{p}\to e^{+}e^{-}$, and future
measurements of $p\bar{p}\to\mu^{+}\mu^{-}$. A focus is put on recent high
precision results of the reaction $e^{+}e^{-}\to p\bar{p}$ that have been
obtained after analyzing 688.5 pb-1 of data taken at the BESIII experiment.
They are compared and put into perspective to results from previous
measurements in this channel. We discuss the channels $p\bar{p}\to e^{+}e^{-}$
and $p\bar{p}\to\mu^{+}\mu^{-}$ in terms of the few existing as well as future
measurements, which the PANDA experiment will perform. Finally, we review
several new theoretical models and phenomenological approaches inspired by the
BESIII high precision results and then discuss their implications for a deeper
understanding of the proton’s structure and inner dynamics.

###### keywords:

Proton; Form Factors; Time-like Region; Scan Technique

1 1 0 https://doi.org/ Proton Electromagnetic Form Factors in the Time-like
Region through the Scan Technique Proton Electromagnetic Form Factors in the
Time-like Region through the Scan Technique L. Xia$\orcidA{}^{1,2*}$, Ch.
Rosner$\orcidB{}^{3*}$, Y. D. Wang$\orcidC{}^{4*}$, X. R.
Zhou$\orcidD{}^{1,2*}$, F. E. Maas$\orcidE{}^{3*}$, R. Baldini
Ferroli$\orcidF{}^{5*}$, H. M. Hu$\orcidG{}^{6,7*}$, G. S.
Huang$\orcidH{}^{1,2*}$ L. Xia, Ch. Rosner, Y. D. Wang, X. R. Zhou, F. E.
Maas, R. Baldini Ferroli, H. M. Hu and G. S. Huang Xia, L.; Rosner, Ch.; Wang,
Y. D.; Maas, F. E.; Ferroli, R. Baldini; Hu, H. M; Huang, G. S.
Correspondence<EMAIL_ADDRESS>(Xia, L<EMAIL_ADDRESS>(Rosner,
Ch<EMAIL_ADDRESS>(Y. D. Wang<EMAIL_ADDRESS>(X. R. Zhou),
<EMAIL_ADDRESS>(F. E. Maas<EMAIL_ADDRESS>(R. Baldini
Ferroli<EMAIL_ADDRESS>(H. M. Hu<EMAIL_ADDRESS>(G. S. Huang)

##  1 Introduction

The proton, along with its partner nucleon, the neutron, makes up more than
99.9% of the visible matter in the universe. The internal structure and
dynamics of the proton, which originate from the strong interaction, has
therefore been the target of intense investigations over the past 100 years
since the discovery of the proton. Despite that, many questions are still not
answered satisfactorily, recent examples being the proton radius puzzle, the
proton spin crisis, or the origin of the proton mass. A detailed theoretical
description of the internal proton structure and its constituents’ dynamics is
made difficult by the non-perturbative nature of the underlying theory,
quantum chromodynamics (QCD), within the energy regime of the nucleon.
Therefore, precise knowledge of one of the most simple observables that
parametrize the proton’s structure and dynamics, the electromagnetic (EM) form
factors (FFs), is crucial for understanding the proton structure. In turn,
these quantities also provide a perfect testing ground for our understanding
of QCD.

The discovery of the proton dates back to the earliest investigations of the
atomic structure. W. Wien and J. J. Thomson identified a positively charged
particle with a mass equal to the hydrogen atom when studying streams of
ionized gaseous atoms and molecules whose electrons had been stripped. E.
Rutherford indicated that the nitrogen under $\alpha$ particle bombardment
ejects what appear to be hydrogen nuclei in 1919. Subsequently, in 1920, E.
Rutherford had accepted the hydrogen nucleus as an elementary particle,
denominating its proton. In 1937, O Stern observed an anomalous magnetic
moment of nucleons ($\mu_{p}=2.79\mu_{N}$, $\mu_{n}=-1.91\mu_{N}$), where
$\mu_{N}=\frac{e\hbar}{2m_{N}c}$ is the nuclear magneton, while theoretically,
the magnetic moment of point-like proton and neutron is $\mu_{N}$ and 0,
respectively. Furthermore, it is the most direct evidence that nucleons are
not point-like particles I. Estermann, O. C. Simpson and O. Stern . Afterward,
R. Hofstadter established an apparatus for discovering nuclei’s internal
structure by elastic scattering of electrons and protons (see Figure 1 (a)).
In the 1950s, R. Hofstadter investigated the charge distribution in atomic
nuclei and the charge and magnetic moment distributions in the proton and
neutron. Nuclei were thereby proven not to be homogeneous but to have internal
structures R. Hofstadter, R. W. McAllister . After that, J. I. Friedman, H. W.
Kendall, and R. E. Taylor investigated deep inelastic scattering of electrons
on protons and bound neutrons (also see Figure 1 (a)). This research has been
of essential significance for the development of the quark model in particle
physics E. D. Bloom, D. H. Coward, H. DeStaebler, J. Drees, G. Miller, L. W.
Mo, R. E. Taylor, M. Breidenbach, J. I. Friedman, G. C. Hartmann and H. W.
Kendall .

Until today, the research of the internal structures of protons is one of the
hottest topics in nuclear physics. The proton-spin crisis and proton-radius
puzzle are the most critical issues of the research. In 2017, $\chi$QCD
Collaboration reported that the gluon helicity contributes to half of the
total proton spin by Lattice QCD Y. B. Yang, R. S. Sufian . Afterwards,
Jefferson Laboratory (PRad) reported a proton radius of $r_{p}=0.831\pm
0.007({\text{stat}})\pm 0.012({\text{syst}})$ W. Xiong, A. Gasparian , which
supports the value found by two previous muonic hydrogen experiments R. Pohl,
A. Antognini ; radiusMuon2 .

Nucleon EM FFs parametrize the difference between a point-like photon-nucleon
vertex to one that considers the internal structure of the nucleon. For spin
1/2 particles, such as proton and neutron, this structure is encoded in two
FFs, the Dirac FF ($F_{1}$) and the Pauli FF ($F_{2}$), where the first
describes the difference to a point-like charge distribution and the second to
a point-like magnetization distribution. More commonly used than these Dirac-
and Pauli FFs are the so-called Sachs FFs $G_{E}$ and $G_{M}$, which are
simple linear combinations of the former. Proton EM FFs can be explored in two
kinematic regions, the spacelike (SL, momentum transfer $q^{2}<0$) and
timelike (TL, momentum transfer $q^{2}>0$) region. The former region has been
extensively investigated through electron-proton scattering experiments (see
Figure 1 (a)) since the 1950s, and consequently, proton TL EM FFs are known
with low uncertainties of the order of a few percent. Experimental data on the
TL region, commonly obtained by investigating annihilation reactions (Figure 1
(b)), has been much more scarce for a long time. Only within the last two
decades, more experiments have provided data within this region, and only very
recent high luminosity measurements have brought the precision up to par with
the SL region. FFs enter explicitly in the coupling of a virtual photon with
the hadron electromagnetic current. Their measurements can be directly
compared to hadron models and thereby provide constraints to the description
of the internal structure of hadrons R. Pohl, A. Antognini . The extension of
the proton EM FFs to the TL region opens further possibilities to investigate
peculiarities in the structure of the proton. While the information contained
in the TL FFs has a less intuitive interpretation compared to the electric and
magnetic distribution densities of the nucleon deduced from SL FFs, they play
a crucial role in understanding the long-range behavior of strong
interactions. Moreover, proton EM FFs can be used to make theoretical
predictions for the behavior of other baryons’ FFs, e.g. the neutrons  A. Z.
Dubničková, S. Dubnička .

\begin{overpic}[width=156.10345pt]{./feyngraph_SL.pdf} \put(45.0,0.0){(a)}
\end{overpic}\begin{overpic}[width=156.10345pt]{./feyngraph_TL_scan.pdf}
\put(45.0,0.0){(b)} \end{overpic}

Figure 1: Lowest-order Feynman diagrams for elastic electron-proton scattering
$e^{-}p\rightarrow e^{-}p$ (a), and for the annihilation process
$e^{+}e^{-}\rightarrow p\bar{p}$ (b).

In terms of $G_{E}$ and $G_{M}$, the cross section of the process
$e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ reads:

$\displaystyle\sigma_{e^{+}e^{-}\xleftrightarrow{}p\bar{p}}(s)=\frac{4\pi\alpha^{2}\beta
C}{3s}\left[|G_{M}(s)^{2}|+\frac{2m_{p}^{2}}{s}|G_{E}(s)^{2}|\right],$ (1)

where $\alpha$ is the fine structure constant, $\beta=\sqrt{1-4m_{p}^{2}/s}$
is the proton velocity, $m_{p}$ is the proton mass, and C is the coulomb
enhancement factor R. Baldini Ferroli, S. Pacetti, A. Zallo, A. Zichichi .
From this integrated cross section, a so called effective FF
($G_{\text{eff}}$) can be deduced under the assumption of
$|G_{E}(s)|=|G_{M}(s)|$:

$\displaystyle|G_{\text{eff}}(s)|=\sqrt{\frac{\sigma_{e^{+}e^{-}\xleftrightarrow{}p\bar{p}}(s)}{\frac{4\pi\alpha^{2}\beta
C}{3s}(1+\frac{2m_{p}^{2}}{s})}}=\sqrt{\frac{|G_{M}(s)^{2}|+\frac{2m_{p}^{2}}{s}|G_{E}(s)^{2}|}{1+\frac{2m_{p}^{2}}{s}}}.$
(2)

This $|G_{\text{eff}}|$ was mainly used by older experiments with limited
statistics and was calculated from the measured cross section using the middle
part of the above equation. More recent experiments have been able to perform
a measurement of the differential cross section through an angular analysis in
one-photon exchange approximation in the $e^{+}e^{-}$ center-of-mass (c.m.)
system, which allows to determine the individual FFs $|G_{E}(s)|$ and
$|G_{M}(s)|$:

$\displaystyle\frac{d\sigma_{p\bar{p}}(s)}{d\Omega}=\frac{\alpha^{2}\beta
C}{4s}\left[|G_{M}(s)|^{2}(1+\cos^{2}\theta)+\frac{4m_{p}^{2}}{s}|G_{E}(s)|^{2}\sin^{2}\theta\right],$
(3)

where $\theta$ is the polar angle of the outgoing particles.

Various approaches to providing a theoretical description of the proton EM FFs
exist, among them approaches based on modified versions of the old, yet still
relevant, vector meson dominance (VMD) model E. L. Lomon, S. Pacetti , models
based on the unitarity and analyticity of the proton FFs using dispersion
theoretical approaches Y. H. Lin, H. W. Hammer, Ulf-G. Meißner , descriptions
based on a relativistic constituent quark model J. P. B. C. de Melo, T.
Frederico, E. Pace, S. Pisano, and G. Salmè , and more. A comprehensive review
of all these approaches is outside the scope of this article. Instead, a focus
is put on a review of the experimental results for the proton TL FFs,
especially from the most recent high luminosity BESIII energy scan, and the
current, primarily phenomenological theoretical approaches to describe the
newly observed features in the effective TL proton FF as well as the
individual electric and magnetic FF.

##  2 Scan method

Measuring nucleon EM FFs over a wide kinematical range is possible through two
different methods, the so-called initial state radiation (ISR) and scan
methods. The ISR method uses collider data at a fixed c.m. energy and analyses
events where a photon is emitted from the initial state (see Figure 2 (a)),
thus reducing the momentum transfer $q^{2}$ of the process. This allows
measuring FFs at $q^{2}$ values from threshold up to the fixed c.m. energy of
the collider. For the scan method, this is achieved by taking energy scan data
sets, where the c.m. energy of the collider is systematically varied. Within
this paper, only former results, as well as future experiments employing the
scan method, are reviewed, whereas an overview of results with the ISR method
can be found in Ref D. X. Lin, A. Dbeyssi, F. E. Maas . Further reviews about
nucleon EM FFs in the TL region in general and other baryon FFs can be found
in Ref. A. Denig, G. Salmè and Ref. G. S. Huang, R. Baldini Ferroli ,
respectively.

\begin{overpic}[width=99.73074pt]{./feyngraph_TL_ISR.pdf} \put(45.0,0.0){(a)}
\end{overpic}\begin{overpic}[width=99.73074pt]{./feyngraph_TL_scan.pdf}
\put(45.0,0.0){(b)} \end{overpic}

\begin{overpic}[width=99.73074pt]{./feyngraph_TL_scanP.pdf}
\put(45.0,0.0){(c)} \end{overpic}

Figure 2: Feynman diagrams for the ISR (a) and born (b) process of
$e^{+}e^{-}\rightarrow p\bar{p}$, as well as the time-reverse process
$p\bar{p}\rightarrow e^{+}e^{-}$ (c).

##  3 Overview of Timelike Electromagnetic Form Factor measurements for the
Proton

###  3.1 Previous measurements of the proton EM FFs

Previous measurements of the proton EM FFs using the scan method have been
performed with the processes $e^{+}e^{-}\rightarrow p\bar{p}$ as well as the
time-reverse process, $p\bar{p}\rightarrow e^{+}e^{-}$ (see Figure 2 (b) and
(c)). While processes where muons replace the electron and positron, are
possible, no FF measurements have been performed employing these channels yet.
So far, such measurements were infeasible either due to the lack of
$\mu^{+}\mu^{-}$ colliders, or in the time-reversed case due to a combination
of low statistics and high background contamination, and the same is true for
tauons. Proposals for measurements in the muon channel will be discussed in
the next section.

The earliest attempts to measure proton EM FFs in the time-like region have
been performed in the 1970s, in case of electron-positron annihilation at the
ADONE collider in Frascati M. Castellano, G. Di Giugno and in case of
$p\bar{p}$ annihilation reactions with the Proton Synchrotron (PS) at CERN G.
Bassompierre, G. Binder . In the 1980s and 90s, measurements in electron-
positron annihilation were continued at these facilities with the FENICE
experiment A. Antonelli, R. Baldini Ferroli at ADONE as well as the DM1 B.
Delcourt, I. Derado and DM2 D. Bisello, S. Limentani, G. Busetto experiments
at the Orsay colliding beam facility (DCI) in Orsay. Fenice measured at 5
energy scan points between 1.90 and 2.44 GeV with a total integrated
luminosity ($\mathcal{L}_{\text{int}}$) of 0.36 pb-1, whereas DM1 and its
successor DM2 both did scans with an integrated luminosity of about 0.4 pb-1
between 1.925 and 2.226 GeV in 4 and 6 energy steps, respectively. The time-
reversed channel was further explored with the PS170 G. Bardin, G. Burgun
experiment, using the $\bar{p}$ beam available at the Low Energy Antiproton
Ring (LEAR) at CERN, in a series of scan measurements performed between 1991
and 1994 with momentum transfer from the proton-antiproton production
threshold up to 2.05 GeV. Additional measurements using this reaction at
higher momentum transfer were also performed at the E760 T. A. Armstrong, D.
Bettoni and E835 M. Ambrogiani, M. Andreotti experiments at Fermilab, in case
of E760 in 3 steps between 2.98 and 3.6 GeV ($\mathcal{L}_{\text{int}}=26.5$
pb-1) and in case of E835 in a total of 10 steps between 2.97 and 4.29 GeV
($\mathcal{L}_{\text{int}}=194.5$ pb-1). In the 2000s, the first high
precision, high luminosity measurements started in the electron-positron
channel in the form of an extended energy scan between 2.00 and 3.07 GeV
performed with the Beijing Spectrometer (BES) at the Beijing electron positron
collider (BEPC) M. Ablikim, J. Z. Bai , as well as a single high luminosity
scan point at 3.671 GeV obtained with the CLEO-c detector at the Cornell
Electron Storage Ring (CESR) T. K. Pedlar, D. Cronin-Hennessy . The CMD-3
experiment at the VEPP-2000 $e^{+}e^{-}$ collider performed a fine energy scan
of the proton TL EM FFs between threshold and 2 GeV, divided into 10 energy
points in 2016, and an additional 11 energy points in 2019 R. R. Akhmetshin,
A. N. Amirkhanov . Finally, the most recent and by far most precise results
for the proton time-like EM FFs obtained with the energy scan technique stem
from a new high luminosity scan measurement performed in 2015 X. R. Zhou, G.
S. Huang and 2020 L. Xia, R. Christoph with the BESIII detector at BEPCII.
This new data comprises 22 energy scan points between 2.00 and 3.08 GeV, with
a total integrated luminosity of 688.5 pb-1. The high luminosity of this new
data allows a precise determination of the individual FFs $|G_{E}|$ and
$|G_{M}|$ as well as their ratio R, whereas most previous experiments have
been limited to the determination of $|G_{\text{eff}}|$.

###  3.2 Future experimental prospects

Further measurements of the proton TL EM FFs are necessary to confirm the
results obtained by the newest BESIII measurement for the individual FF
$|G_{E}|$ and $|G_{M}|$ as well as their ratio. In addition, more data is
necessary to investigate newly found structures that appear both in the
effective and the individual FFs. An extension of the measured kinematical
range for the separate FFs, both towards higher c.m. energies and the
threshold region, is also desirable.

For proton TL EM FF measurements with the scan technique, the most promising
upcoming experiment is the PANDA experiment currently under construction at
the Facility for Antiproton and Ion Research (FAIR) in Darmstadt. Located at
the High Energy Storage Ring (HESR), which will provide an $\bar{p}$ beam with
luminosities up to $10^{32}$ cm-2s-1 and momenta between 1.5 and 15 GeV/$c$,
the PANDA detector will be divided into a target spectrometer and a forward
spectrometer. A detailed description of the detector can be found in Ref
PandaDetector .

Proposals for proton TL EM FF measurements exist both for the channel
$p\bar{p}\rightarrow e^{+}e^{-}$ as well as
$p\bar{p}\rightarrow\mu^{+}\mu^{-}$. Measurements, especially of the
individual proton TL EM FFs employing the first channel, will be of high
interest since here the only existing. Rather an old measurement by PS170 is
in disagreement with newer $e^{+}e^{-}\rightarrow p\bar{p}$ results both from
the scan and ISR method. The second channel will be the first access to proton
TL EM FFs using leptons other than electrons, which will also allow for tests
of lepton universality. Feasibility studies exist for both channels: for the
electron channel PandaFeasibility1 , expected relative uncertainties range
between 1% to 5% for the FF ratio ($|G_{E}/G_{M}|$) and 1% to 4% for $|G_{E}|$
in the already explored $q^{2}$ range up to 3 GeV/$c$ . The expected
uncertainty for both R and $|G_{E}|$ grows to more than 50% at 3.73 GeV/$c$.
However, this would still be an essential extension of the kinematical range
of the measurement of the individual FFs. Uncertainties for $|G_{M}|$ would be
well below 1% for all momentum transfers. In case of the muon channel
PandaFeasibility2 , uncertainties would range between 3% to 27% for $|G_{E}|$,
2% to 10% for $|G_{M}|$, and 5% to 37% for the $|G_{E}/G_{M}|$ in the energy
range between 2.25 and 2.86 GeV.

##  4 Discussion of proton TL EM FF results

Within this section, the different TL EM FF measurements of the experiments
introduced in the last section are compared and put into perspective. The
comparison starts with the cross section of the process
$e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ in Section 4.1, which was determined by
most experiments performing scan measurements of proton FFs. While it is not
directly a measurement of proton FFs, the cross section is closely related to
the effective FF $|G_{\text{eff}}|$ (see Eq. (2), which is discussed in
Section 4.2. Therefore, the discussion in the latter section applies to both
the cross section and $|G_{\text{eff}}|$. Finally, the measurements of the FF
ratio as well as the individual electric ($|G_{E}|$) and magnetic ($|G_{E}|$)
proton FF are compared in Section 4.3 and 4.4, respectively.

###  4.1 Measurements of the cross section of
$e^{+}e^{-}\xleftrightarrow{}p\bar{p}$

Measurements of the cross section of $e^{+}e^{-}\xleftrightarrow{}p\bar{p}$
from $e^{+}e^{-}$ colliders employing the scan technique are summarized in
Figure 3 (a) for c.m. energies from threshold up to 2.35 GeV, and in Figure 3
(b) for c.m. energies between 2.35 and 4.00 GeV.

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Xsection1.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Xsection2.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 3:  Comparison of the results for the
$e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ cross section using the energy scan
strategy in (a) $2m_{p}<\sqrt{s}<2.35$ GeV, (b) $2.35<\sqrt{s}<4.00$ GeV.
Shown are the published measurements from BESIII X. R. Zhou, G. S. Huang ; L.
Xia, R. Christoph , CMD3 R. R. Akhmetshin, A. N. Amirkhanov , CLEO T. K.
Pedlar, D. Cronin-Hennessy , BES M. Ablikim, J. Z. Bai , FENICE A. Antonelli,
R. Baldini Ferroli , E835 M. Ambrogiani, M. Andreotti , E760 T. A. Armstrong,
D. Bettoni , DM2 D. Bisello, S. Limentani, G. Busetto , DM1 B. Delcourt, I.
Derado , and ADONE M. Castellano, G. Di Giugno .

###  4.2 Measurements of the effective FF $|G_{\text{eff}}|$ of the proton

Measurements of the proton EM FFs of most previous experiments in the TL
region were restricted to measuring $|G_{\text{eff}}|$, as shown in Eq. (2),
due to limited statistics. Instead of the assumption of $|G_{E}|=|G_{M}|$,
some experiments at high values of $q^{2}$ (e. g. E760 and E835) also measured
the magnetic FF $|G_{M}|$ under the assumption that the $|G_{E}|$ contribution
is negligible at high energies due to the suppression by a factor of $1/\tau$
(see Eq. (3)), with $\tau=\frac{s}{4m_{p}^{2}}$.

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_FF1.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_FF2.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 4:  Comparison of the results for $|G_{\text{eff}}|$ using the energy
scan strategy in (a) $2m_{p}<\sqrt{s}<2.35$ GeV, (b) $2.35<\sqrt{s}<4.00$ GeV.
Shown are all the published measurements from BESIII X. R. Zhou, G. S. Huang ;
L. Xia, R. Christoph , CMD3 R. R. Akhmetshin, A. N. Amirkhanov , CLEO T. K.
Pedlar, D. Cronin-Hennessy , BES M. Ablikim, J. Z. Bai , FENICE A. Antonelli,
R. Baldini Ferroli , E835 M. Ambrogiani, M. Andreotti , E760 T. A. Armstrong,
D. Bettoni , PS170 G. Bardin, G. Burgun , DM2 D. Bisello, S. Limentani, G.
Busetto , DM1 B. Delcourt, I. Derado , M. S. T G. Bassompierre, G. Binder ,
and ADONE M. Castellano, G. Di Giugno .

A comparison of $|G_{\text{eff}}|$ of the proton determined by the different
experiments is shown in Figure 4 (a) for c.m. energies from threshold up to
2.35 GeV, and in Figure 4 (b) for c.m. energies between 2.35 and 4.00 GeV.
Within their respective uncertainties, the measurements agree well with each
other. Measurements close to the threshold, especially by the PS170
experiment, show a steep rise of $|G_{\text{eff}}|$ towards the threshold,
while data at higher momentum transfer mostly follows a dipole behavior. A
deviation from this behavior can only be seen in the most precise measurement
recently performed at BESIII, where a periodic behavior was found to be
superimposed over the monotonous decrease of $|G_{\text{eff}}|$. The origin of
this structure is still under debate, with possible sources being rescattering
of the forming final state particles or intermediate resonance states before
the formation of the proton-antiproton state which enhances the cross section
of the process within a certain momentum transfer region. A more detailed
discussion of this phenomenon can be found in Section 5.3.

###  4.3 Measurements of the FF ratio $|G_{E}/G_{M}|$ of the proton

Measurements with the scan technique of the individual proton TL EM FFs over a
wider momentum transfer range are limited to three published measurements, two
of which were performed with the BESIII detector. A determination of the ratio
of the electric and magnetic FF requires an angular analysis of the data,
while a separate measurement of $|G_{E}|$ and $|G_{M}|$ additionally requires
precise knowledge of the luminosity of the acquired data. For most proton TL
EM FF measurements, either this knowledge was missing or the amount of
detected events did not allow for a measurement of the differential cross
section.

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./mod1_03.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./mod1_10.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 5:  Fit to the $|\cos\theta|$ distributions at 2.125 GeV and 2.396 GeV
at BESIII L. Xia, R. Christoph , after the application of angular-dependent
$\epsilon(1+\delta)$ factors.

In one-photon approximation, the FFs $|G_{E}|$ and $|G_{M}|$, or equivalently
their ratio $|G_{E}/G_{M}|$ and $|G_{M}|$ can be determined from a fit to the
proton angular distribution for energy points with a sufficiently high number
of selected candidates. The range of the angular analysis in most experiments
is limited to $\cos\theta$ from $-0.8$ to $0.8$. The formula used to fit the
proton angular distribution, deduced from Eq. (3), can be expressed as

$\displaystyle\begin{split}\frac{dN}{\epsilon\left(1+\delta\right)\times
d\cos\theta}=\frac{\mathcal{L}_{\text{int}}\pi\alpha^{2}\beta
C}{2s}|G_{M}|^{2}\Bigl{[}\left(1+\cos^{2}\theta\right)+\frac{4m_{p}^{2}}{s}\left|\frac{G_{E}}{G_{M}}\right|^{2}\left(1-\cos^{2}\theta\right)\Bigr{]},\end{split}$
(4)

where $\epsilon(\cos\theta)$ is the angular-dependent efficiency, including
both the detector efficiency and the efficiency from the event selection, and
$(1+\delta)(\cos\theta)$ is the correction factor for both radiative
corrections, such as ISR and final state radiation (FSR), as well as vacuum
polarisation effects. Both correction factors are obtained from Monte Carlo
(MC) simulation. After applying the corrections, the $|\cos\theta|$
distribution is fitted with Eq. (4). An example of the angular distribution of
outgoing protons including such a fit is shown in Figure 5 at 2.125 GeV and
2.396 GeV from the most recent BESIII measurement L. Xia, R. Christoph .

\begin{overpic}[width=284.52756pt,height=213.39566pt,angle={0}]{./com_GeGm1.pdf}
\end{overpic}

Figure 6:  Comparison of the results for $|G_{E}/G_{M}|$ using the energy scan
strategy. Shown are all the published measurements from BESIII X. R. Zhou, G.
S. Huang ; L. Xia, R. Christoph , CMD3 R. R. Akhmetshin, A. N. Amirkhanov ,
PS170 G. Bardin, G. Burgun , FENICE+DM2 A. Antonelli, R. Baldini Ferroli ; D.
Bisello, S. Limentani, G. Busetto ; R. Baldini Ferroli, C. Bini, P. Gauzzi ,
E835 M. Ambrogiani, M. Andreotti ; R. Baldini Ferroli, C. Bini, P. Gauzzi .

The published results for the existing measurements of the $|G_{E}/G_{M}|$
using the scan technique are shown in Figure 6. The most recent BESIII results
are the most extensive and precise ones. They agree well with the previous
BESIII results obtained with smaller statistics. The single measurement point
from CMD3 is also in good agreement within its uncertainties. In contrast, the
PS170 results show a systematic trend towards smaller values of
$|G_{E}/G_{M}|$, which is not confirmed by the BESIII results.

###  4.4 Measurements of the $|G_{E}|$ and $|G_{M}|$ of proton

In the case of $|G_{E}|$, the only published results are from the recent
BESIII high luminosity scan measurement. However, both the smaller, previous
BESIII scan as well as PS170 have measured $|G_{E}/G_{M}|$ and $|G_{M}|$,
which would allow for a calculation of $|G_{E}|$. The sole direct measurement
of $|G_{E}|$ is shown on the left side of Figure 7 (a), while the right side
(b) shows a comparison of the results for $|G_{M}|$.

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Ge1.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Gm1.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 7:  Comparison of the results for (a) $|G_{M}|$ and (b) $|G_{E}|$ using
the energy scan strategy. Shown are all the published measurements from BESIII
X. R. Zhou, G. S. Huang ; L. Xia, R. Christoph , CLEO T. K. Pedlar, D. Cronin-
Hennessy , E835 M. Ambrogiani, M. Andreotti , E760 T. A. Armstrong, D. Bettoni
, PS170 G. Bardin, G. Burgun .

For $|G_{M}|$, the available measurements are all in agreement, albeit the
BESIII measurement from 2020 is by far the most precise. It should be noted
that the three measurements at high momentum transfer by CLEO, E780, and E835
were performed under the assumption of $|G_{E}|=0$ instead of extraction
through angular analysis. Therefore, they are more of an $|G_{\text{eff}}|$,
albeit the assumption may be justified due to the strong suppression of
$|G_{E}|$ at higher energies.

Both the $|G_{E}/G_{M}|$ as well as the individual FF measurement, especially
that of $|G_{E}|$, seem to show a periodic structure on top of a dipole like,
monotonously falling behavior. This structure is similar to the one already
observed in $|G_{\text{eff}}|$, however, in the case of the individual FFs it
is only visible for the high precision fine scan data from 2020. First
approaches to describe these features have been made from the theoretical side
with mostly phenomenological approaches. However, more investigations
especially concerning the origin of these structures, are necessary. From the
experimental side, independent confirmation of the BESIII measurement of
$|G_{E}/G_{M}|$, $|G_{E}|$ and $|G_{M}|$ is needed to confirm this behavior.
An attempt at a phenomenological description of these structures can be found
in Section 5.4.

##  5 Theoretical approaches to the description of Timelike Electromagnetic
Proton Form Factors

###  5.1 Test of perturbative QCD and low energy effective theory for QCD

The recently obtained high precision results in the TL region provide
information to improve our understanding of the proton inner structure and
test theoretical models that depend on perturbative QCD (pQCD) and low energy
effective theory for QCD (low-enegy-eff-QCD). The low energy region is the
regime of low-enegy-eff-QCD due to the growing of the running QCD coupling
constant and the associated confinement of quarks and gluons. The Energy
region between 2.00 and 3.08 GeV connects the low-enegy-eff-QCD and the pQCD
regime. Therefore, Ref. L. Xia, R. Christoph suggests a fit function for the
energy dependence of $\sigma_{p\bar{p}}$ within this energy region, consisting
of a low-enegy-eff-QCD and a pQCD part:

$\displaystyle\sigma_{p\bar{p}}(s)=\left\\{\begin{aligned}
\hfil\displaystyle\begin{split}\sigma_{\text{eff-
QCD}}(s)&=\frac{e^{a_{c}}\pi^{2}\alpha^{3}}{s\left[1-e^{-\frac{\pi\alpha_{s}(s)}{\beta(s)}}\right]\left[1+\left(\frac{\sqrt{s}-2m_{p}c^{2}}{a_{0}}\right)^{a_{1}}\right]},~{}\sqrt{s}\leq
2.3094~{}{\text{GeV}},\\\ \end{split}\\\
\hfil\displaystyle\begin{split}\sigma_{\text{pQCD}}(s)&=\frac{2\pi\alpha^{2}\beta(s)C\left[2+\left(\frac{2m_{p}c^{2}}{\sqrt{s}}\right)^{2}\right]e^{2a_{2}}}{3s^{5}\left[4\ln^{2}\left(\frac{\sqrt{s}}{a_{0}}\right)+\pi^{2}\right]^{2}},~{}\sqrt{s}>2.3094~{}{\text{GeV}},\end{split}\end{aligned}\right.$
(5)

with the strong coupling constant $\alpha_{s}(s)$ and the fine-structure
constant $\alpha$. For the strong coupling constant, the following
parametrization is used:

$\displaystyle\alpha_{s}(s)$
$\displaystyle=\left[\frac{1}{\alpha_{s}\left(m_{Z}^{2}c^{4}\right)}+\frac{25}{12\pi}\ln\left(\frac{s}{m_{Z}^{2}c^{4}}\right)\right]^{-1},$
(6)

with the mass of the $Z$ boson $m_{Z}=91.1876$ GeV/$c^{2}$ and the strong
coupling constant at the $Z$ pole $\alpha_{s}(m_{Z}^{2}c^{4})=0.11856$.
Instead of introducing the Coulomb enhancement factor, the suggested fit
function in Ref. L. Xia, R. Christoph (Eq. 5) uses an alternative approach
close to the $p\bar{p}$ threshold by considering gluon exchange between the
$B\bar{B}$ pair. In addition, the first part of Eq. (5) describes the behavior
near threshold by considering strong interaction effects from the low-enegy-
eff-QCD for low the energy region near the $p\bar{p}$ threshold O. P.
Solovtsova, Y. D. Chernichenko ; R. Baldini Ferroli, S. Pacetti, A. Gauzzi .
The second part of the formula describes the pQCD behavior of the cross
section, computed in leading order for the continuum region at high momentum
transfer A. Bianconi, E. Tomasi-Gustafsson .

\begin{overpic}[width=284.52756pt,height=213.39566pt,angle={0}]{./com_Xsectionfit.pdf}
\end{overpic}

Figure 8: Summary for the $e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ cross section
and a fit to the data (blue solid line and band) according to Eq. (5). The
shown data are the published measurements from BESIII X. R. Zhou, G. S. Huang
; A. Dbeyssi, F. E. Maas ; L. Xia, R. Christoph ; D. X. Lin, F. E. Maas ,
BABAR B. Aubert, J. P. Lees , CMD3 R. R. Akhmetshin, A. N. Amirkhanov , CLEO
T. K. Pedlar, D. Cronin-Hennessy , BES M. Ablikim, J. Z. Bai , FENICE A.
Antonelli, R. Baldini Ferroli , E835 M. Ambrogiani, M. Andreotti , E760 T. A.
Armstrong, D. Bettoni , DM2 D. Bisello, S. Limentani, G. Busetto , DM1 B.
Delcourt, I. Derado , and ADONE M. Castellano, G. Di Giugno . In the fit,
$\chi^{2}$ is defined as
$\chi^{2}=\sum_{i}[f(x_{i})-y_{i}]^{2}/{\text{err}}_{i}^{2}$, where
$\text{err}_{i}$ is the error of the measured results including statistical
and correlated systematic uncertainties, $f$ is the fit function, and n.d.f.
is the number of degrees of freedom.

In conclusion, the low-enegy-eff-QCD based description is suitable for the
lower energy behavior below 2.3094 GeV, while the pQCD based approach provides
a good description for the cross section data at higher energy, which is
illustrated in Figure 8. The results and meaning of the fit parameters are as
follows: $a_{0}=332\pm 3$ MeV is the overall QCD parameter, $a_{1}=4.78\pm
0.13$ is the $\sigma_{p\bar{p}}(s)$ power-law dependence, which is related to
the number of valence quarks, and $a_{2}=4.39\pm 0.01$ is a normalization
constant. Finally, $a_{c}$ makes $\sigma_{\text{low-enegy-eff-
QCD}}({\text{2.3094~{}GeV}})=\sigma_{\text{pQCD}}({\text{2.3094~{}GeV}})$, and
does not attend the global fit.

###  5.2 Possible Resonance Structures in the $p\bar{p}$ system

A description of the cross section of $e^{+}e^{-}\xleftrightarrow{}p\bar{p}$
according to a pQCD and low-enegy-eff-QCD model has been introduced in Section
5.1. The slight ridges and bump structures occurring here imply that the model
in Section 5.1 and its description of the low-enegy-eff-QCD and pQCD regime
are not perfect. Various approaches have been proposed to describe these
deviations, including resonance structures I. T. Lorenz, H. W. Hammer and
Ulf-G. Meißner and periodic interference structures A. Bianconi, E. Tomasi-
Gustafsson . In the first instance, possible resonance structures are
considered as convex modifications to the overall concave function describing
$|G_{\text{eff}}|$ at invariant masses of around 2.00–2.25 GeV. The approach
will be introduced in the following, and the fit parameters are extracted with
improved precision by including the new high luminosity data from BESIII. The
line shape of $|G_{\text{eff}}|$ is fitted using a coherent sum of a Breit-
Wigner function and a nonresonant term I. T. Lorenz, H. W. Hammer and Ulf-G.
Meißner :

$\displaystyle\sigma_{e^{+}e^{-}\xleftrightarrow{}p\bar{p}}(s)=\frac{4\pi\alpha^{2}\beta
C}{3s}\left(1+\frac{2m_{p}^{2}}{s}\right)|G_{\text{eff}}^{\text{n.r.}}+A_{p\bar{p}}^{\text{Res(2150)}}|^{2}.$
(7)

The first term $G_{\text{eff}}^{\text{n.r.}}$ is the nonresonant contribution,

$\displaystyle
G_{\text{eff}}^{\text{n.r.}}=\sqrt{\frac{3s\sigma_{p\bar{p}}(s)}{4\pi\alpha^{2}\beta
C(1+\frac{2m_{p}^{2}}{s})}},$ (8)

where $\sigma_{p\bar{p}}(s)$ is from Eq. 5. The second term
$A_{p\bar{p}}^{\text{Res(2150)}}$ is the Breit-Wigner amplitude:

$\displaystyle
A_{p\bar{p}}^{\text{Res(2150)}}=\frac{e^{a_{3}+i\Psi_{\text{Res(2150)}}}}{m_{\text{Res}(2150)}^{2}-s-i\Gamma_{\text{Res}(2150)}m_{\text{Res}(2150)}},$
(9)

with normalization factor $a_{3}$, mass $m_{\text{Res}(2150)}$, width
$\Gamma_{\text{Res}(2150)}$ and relative phase angle to the nonresonant
component $\Psi_{\text{Res(2150)}}$. The results of the fit parameters are as
follows: $a_{0}=387_{-22}^{+21}$ MeV, $a_{1}=7.43_{-1.12}^{+1.53}$,
$a_{2}=4.33_{-0.08}^{+0.06}$, $a_{3}=7.73_{-0.18}^{+0.12}$,
$m_{\text{Res}(2150)}=2150_{-12}^{+10}$ MeV/$c^{2}$,
$\Gamma_{\text{Res}(2150)}=359_{-23}^{+30}$ MeV and
$\Psi_{\text{Res(2150)}}=-1.17_{-0.13}^{+0.11}$. This structure at around 2.15
GeV can be attributed, e.g., to the $\rho(2150)$ resonance M. E. Biagini, S.
Dubnicka, E. Etim, P. Kolàr .

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Xsectionfit1BB.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Xsectionfit2B.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 9: Summary for the $e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ cross section
and a fit to the data (blue solid line and band) according to (a) Eq. (7) and
(b) Eq. (10), with the low-enegy-eff-QCD and pQCD model (green dash-dotted
curve and band), and contributions from possible resonances at around 2.15 GeV
(magenta dotted curve and band) and around 2.3 GeV (teal dashed curve and
band). The data shown are the published measurements from BESIII X. R. Zhou,
G. S. Huang ; A. Dbeyssi, F. E. Maas ; L. Xia, R. Christoph ; D. X. Lin, F. E.
Maas , BABAR B. Aubert, J. P. Lees , CMD3 R. R. Akhmetshin, A. N. Amirkhanov ,
CLEO T. K. Pedlar, D. Cronin-Hennessy , BES M. Ablikim, J. Z. Bai , FENICE A.
Antonelli, R. Baldini Ferroli , E835 M. Ambrogiani, M. Andreotti , E760 T. A.
Armstrong, D. Bettoni , DM2 D. Bisello, S. Limentani, G. Busetto , DM1 B.
Delcourt, I. Derado , and ADONE M. Castellano, G. Di Giugno . The definition
of $\chi^{2}$ is the same as in Figure 8.

An additional resonance at around 2.3 GeV can also be considered in the fit I.
T. Lorenz, H. W. Hammer and Ulf-G. Meißner :

$\displaystyle\sigma_{e^{+}e^{-}\xleftrightarrow{}p\bar{p}}(s)=\frac{4\pi\alpha^{2}\beta
C}{3s}\left(1+\frac{2m_{p}^{2}}{s}\right)|G_{\text{eff}}^{\text{n.r.}}+A_{p\bar{p}}^{\text{Res(2150)}}+A_{p\bar{p}}^{\text{Res(2300)}}|^{2},$
(10)

where $A_{p\bar{p}}^{\text{Res(2300)}}$ is the Breit-Wigner amplitude:

$\displaystyle
A_{p\bar{p}}^{\text{Res(2300)}}=\frac{e^{a_{4}+i\Psi_{\text{Res(2300)}}}}{m_{\text{Res}(2300)}^{2}-s-i\Gamma_{\text{Res}(2300)}m_{\text{Res}(2300)}},$
(11)

with normalization factor $a_{4}$, mass $m_{\text{Res}(2300)}$, width
$\Gamma_{\text{Res}(2300)}$ and relative phase angle to the nonresonant
component $\Psi_{\text{Res(2300)}}$. The results of the fit parameters are as
follows: $a_{0}=334_{-24}^{+42}$ MeV, $a_{1}=2.66_{-0.65}^{+0.77}$,
$a_{2}=4.52_{-0.06}^{+0.04}$, $a_{3}=6.03_{-0.91}^{+0.86}$,
$m_{\text{Res}(2150)}=2169_{-16}^{+26}$ MeV/$c^{2}$,
$\Gamma_{\text{Res}(2150)}=187_{-85}^{+60}$ MeV,
$\Psi_{\text{Res(2150)}}=0.46_{-0.59}^{+0.51}$, $a_{4}=7.00_{-0.38}^{+0.23}$,
$m_{\text{Res}(2300)}=2302_{-14}^{+16}$ MeV/$c^{2}$,
$\Gamma_{\text{Res}(2300)}=188_{-51}^{+34}$ MeV and
$\Psi_{\text{Res(2300)}}=-1.36_{-0.27}^{+0.13}$. The significance of the two
possible resonance structures is determined to be 2.8$\sigma$.

The corresponding work states that this is not a rigorous analysis since these
resonance structures overlap and are not separable from the background.
Nevertheless, the addition of two resonances peaking at around 2.15 and 2.30
GeV significantly improves the description of the data I. T. Lorenz, H. W.
Hammer and Ulf-G. Meißner .

###  5.3 Periodic Interference Structures in the TL Proton FF

The second mentioned approach to describe the periodic structures in
$|G_{\text{eff}}|$ is presented in this section. The corresponding variables
of the fit are determined with improved precision with the new high luminosity
data from BESIII.

The data of the timelike $|G_{\text{eff}}|$ are best reproduced by the 3-pole
model function proposed in Ref. E. Tomasi-Gustafsson, M. P. Rekalo ,

$\displaystyle
F_{\text{3p}}(s)=\frac{\mathcal{A}}{(1+\frac{s}{m_{a}^{2}c^{4}})[1-\frac{s}{0.71~{}\text{GeV}^{2}}]^{2}}.$
(12)

A global fit to all available results for $|G_{\text{eff}}|$ using the above
equation has been performed and is shown as the green dash-dotted curve and
band in Figure 10 (a). The residuals between the fit function and the
experimental data indicate a periodic structure when they are represented as a
function of the relative momentum $p(s)=\sqrt{s(\frac{s}{4m_{p}^{2}}-1)}$ of
the $p\bar{p}$ pair A. Bianconi, E. Tomasi-Gustafsson . Phenomenologically,
the structure can be described by an oscillating function suggested in Ref A.
Bianconi, E. Tomasi-Gustafsson :

$\displaystyle
F_{\text{osc}}[p(s)]=b^{{\text{osc}}}_{0}e^{-b^{{\text{osc}}}_{1}p(s)}\cos[b^{{\text{osc}}}_{2}p(s)+b^{{\text{osc}}}_{3}].$
(13)

The corresponding fit to the residuals is shown as the magenta dotted curve
and band in Figure 10 (a) and the magenta solid curve and band in Figure 10
(b). Additionally, a global fit to $|G_{\text{eff}}|$ using the sum of two
contributions A. Bianconi, E. Tomasi-Gustafsson ; A. Bianconi, E. Tomasi-
Gustafsson has been performed:

$\displaystyle|G_{\text{eff}}(s)|=F_{\text{3p}}(s)+F_{\text{osc}}[p(s)].$ (14)

Here, $\mathcal{A}=8.79_{-0.21}^{+0.22}$, $m_{a}^{2}=9.04_{-0.58}^{+0.62}$
GeV${}^{2}/c^{4}$, $b^{{\text{osc}}}_{0}=0.06\pm 0.01$,
$b^{{\text{osc}}}_{1}=0.93\pm 0.07$ GeV/$c^{-1}$,
$b^{{\text{osc}}}_{2}=5.02_{-0.09}^{+0.10}$ GeV/$c^{-1}$ and
$b^{{\text{osc}}}_{3}=0.61_{-0.16}^{+0.15}$ are obtained from our fit, which
is illustrated as the blue solid line and band in Figure 10 (a).

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_FFfit2.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_FFfitos.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 10: Summary for (a) the effective FF of the proton $|G_{\text{eff}}|$
and a fit to the data (blue solid line and band) according to Eq. (14)
suggested in Ref. A. Bianconi, E. Tomasi-Gustafsson ; A. Bianconi, E. Tomasi-
Gustafsson with the 3-pole (green dash-dotted curve and band) and oscillation
(magenta dotted curve and band) contributions; (b) residuals of the proton
$|G_{\text{eff}}|$, after subtraction of the smooth function described by Eq.
(12), as a function of the relative momentum
$p(s)=\sqrt{s(\frac{s}{4m_{p}^{2}}-1)}$ with a fit of the oscillation
according to Eq. (13) (magenta solid curve and band). The shown data are the
published measurements from BESIII X. R. Zhou, G. S. Huang ; A. Dbeyssi, F. E.
Maas ; L. Xia, R. Christoph ; D. X. Lin, F. E. Maas , BABAR B. Aubert, J. P.
Lees , CMD3 R. R. Akhmetshin, A. N. Amirkhanov , CLEO T. K. Pedlar, D. Cronin-
Hennessy , BES M. Ablikim, J. Z. Bai , FENICE A. Antonelli, R. Baldini Ferroli
, E835 M. Ambrogiani, M. Andreotti , E760 T. A. Armstrong, D. Bettoni , PS170
G. Bardin, G. Burgun , DM2 D. Bisello, S. Limentani, G. Busetto , DM1 B.
Delcourt, I. Derado , M. S. T G. Bassompierre, G. Binder , and ADONE M.
Castellano, G. Di Giugno . The definition of $\chi^{2}$ is the same as in
Figure 8.

Periodic interference structures of $|G_{\text{eff}}|$ for the proton manifest
as a deviation from a modified dipole behavior. A similar oscillation with a
comparable frequency is observed for the neutron, albeit with a significant
phase difference, as illustrated in Figure 11. The fit yields
$b^{{\text{osc}}}_{2}=5.05_{-0.09}^{+0.10}$ GeV/$c^{-1}$, the phase of proton,
$b^{{\text{osc}}}_{3(p)}=0.57_{-0.16}^{+0.14}$,
$b^{{\text{osc}}}_{3(n)}=2.82_{-0.21}^{+0.22}$, and a phase difference of
$\Delta
b^{{\text{osc}}}_{3}=|b^{{\text{osc}}}_{3(p)}-b^{{\text{osc}}}_{3(n)}|=(129_{-10}^{+11})^{\circ}$
The result indicates unexplored intrinsic dynamics that lead to almost
orthogonal oscillations.

\begin{overpic}[width=284.52756pt,height=213.39566pt,angle={0}]{./com_FFfit_nucos.pdf}
\end{overpic}

Figure 11: Summary for residuals of the proton and neutron $|G_{\text{eff}}|$,
after subtraction of the smooth function described by Eq. (12), as a function
of the relative momentum $p(s)=\sqrt{s(\frac{s}{4m_{p}^{2}}-1)}$ with a fit of
the oscillation according to Eq. (13) (magenta and cyan solid curve and band
for proton and neutron, respectively). The shown data are the published
measurements from BESIII (proton X. R. Zhou, G. S. Huang ; A. Dbeyssi, F. E.
Maas ; L. Xia, R. Christoph ; D. X. Lin, F. E. Maas and neutron X. R. Zhou, J.
F. Hu, and P. Larin ), BABAR B. Aubert, J. P. Lees and CMD3 R. R. Akhmetshin,
A. N. Amirkhanov . The definition of $\chi^{2}$ is the same as in Figure 8.

###  5.4 Phenomenological analysis of $|G_{E}/G_{M}|$

Following a similar approach as for $|G_{\text{eff}}|$ in section 5.3, a fit
is performed on the ratio of $|G_{E}|$ and $|G_{M}|$ in the TL region, using a
function that consists of a damped oscillation on top of a decreasing monopole
part A. Bianconi, E. Tomasi-Gustafsson :

$\displaystyle|\frac{G_{E}[\omega(s)]}{G_{M}[\omega(s)]}|=\frac{1}{1+\frac{\omega(s)^{2}}{c_{0}}}\\{1+c_{1}e^{-c_{2}\omega(s)}\sin[c_{3}\omega(s)]\\},$
(15)

where $\omega(s)=\sqrt{s}-2m_{p}c^{2}$, and the unitary normalization at the
production threshold
$|\frac{G_{E}[\omega(4m_{p}^{2})]}{G_{M}[\omega(4m_{p}^{2})]}|=1$ is imposed.
A fit to the available $|G_{E}/G_{M}|$ results returns the parameters
$c_{0}=3.56_{-1.47}^{+4.14}$ GeV2, $c_{1}=0.58_{-0.13}^{+0.21}$,
$c_{2}=2.17_{-1.05}^{+1.50}$ GeV-1 and $c_{3}=9.48_{-0.57}^{+0.46}$ GeV-1. The
results of the fit are shown in Figure 12 as the blue solid curve and band.
The green dash-dotted curve and its band in the same Figure represent the
monopole component, the magenta dotted curve, and its band represent the
oscillation component (shifted up by 0.5). The red dash-dotted curve in Figure
12 represents the Kuraev model, another form of monopole model E. A. Kuraev,
A. Dbeyssi, E. Tomasi-Gustafsson :

$\displaystyle|\frac{G_{E}(s)}{G_{M}(s)}|=\frac{\mu_{p}}{1+\frac{s-4m_{p}^{2}c^{4}}{c_{4}}},$
(16)

where $\mu_{p}$ is the proton anomalous magnetic moment ($\mu_{p}=2.79$ in
units of Bohr magnetons), $c_{4}=0.74\pm 0.03$ GeV from fit. The unitary
normalization at the production threshold
$|\frac{G_{E}(2m_{p})}{G_{M}(2m_{p})}|=\mu_{p}$ is imposed.

\begin{overpic}[width=284.52756pt,height=213.39566pt,angle={0}]{./com_GeGm.pdf}
\end{overpic}

Figure 12: Summary for the ratio $|G_{E}/G_{M}|$ of the proton and a fit to
the data (blue solid line and band) according to Eq. (15) with the monopole
(green dash-dotted curve and band) and oscillation (magenta dotted curve and
band) component (shifted up by 0.5), and the Kuraev model E. A. Kuraev, A.
Dbeyssi, E. Tomasi-Gustafsson (red dash-dotted curve and band) according to
Eq. (16). The shown data are the published measurements from BESIII X. R.
Zhou, G. S. Huang ; A. Dbeyssi, F. E. Maas ; L. Xia, R. Christoph ; D. X. Lin,
F. E. Maas , BABAR B. Aubert, J. P. Lees , CMD3 R. R. Akhmetshin, A. N.
Amirkhanov , FENICE A. Antonelli, R. Baldini Ferroli ; R. Baldini Ferroli, C.
Bini, P. Gauzzi , E835 M. Ambrogiani, M. Andreotti ; R. Baldini Ferroli, C.
Bini, P. Gauzzi , PS170 G. Bardin, G. Burgun , DM2 D. Bisello, S. Limentani,
G. Busetto ; R. Baldini Ferroli, C. Bini, P. Gauzzi . The definition of
$\chi^{2}$ is the same as in Figure 8.

###  5.5 Individual FFs $|G_{E}|$ and $|G_{M}|$

Using the definition of the effective FF $|G_{\text{eff}}|$ in Eq. (2), the
electric and magnetic FF can be expressed in terms of $|G_{\text{eff}}|$ and
their ratio $|G_{E}/G_{M}|$ A. Bianconi, E. Tomasi-Gustafsson :

$\displaystyle|G_{E}(s)|=|G_{\text{eff}}(s)|\sqrt{\frac{1+\frac{s}{2m_{p}^{2}}}{1+\frac{s}{2m_{p}^{2}|\frac{G_{E}(s)}{G_{M}(s)}|^{2}}}},$
(17)
$\displaystyle|G_{M}(s)|=|G_{\text{eff}}(s)|\sqrt{\frac{1+\frac{s}{2m_{p}^{2}}}{|\frac{G_{E}(s)}{G_{M}(s)}|^{2}+\frac{s}{2m_{p}^{2}}}}.$
(18)

These relations allow to calculate $|G_{E}|$ and $|G_{M}|$ from the fit
results obtained for $|G_{E}/G_{M}|$ (Eq. (15)) and $|G_{\text{eff}}|$ (Eq.
(14)) in the previous section. The resulting curves are shown in Figure 13 (a)
and (b). Here, the blue solid curves and bands represent the fit functions
according to Eq. (17) and Eq. (18), deduced from the fit results of Figure 10
and Figure 12, and the red dash-dotted curves represent the the Kuraev model
E. A. Kuraev, A. Dbeyssi, E. Tomasi-Gustafsson :

$\displaystyle|G_{E}(s)|=\frac{1}{1+\frac{s-4m_{p}^{2}c^{4}}{d_{0}}}\frac{1}{(1+\frac{s-4m_{p}^{2}c^{4}}{d_{1}})^{2}},$
(19)
$\displaystyle|G_{M}(s)|=\frac{1}{\mu_{p}(1+\frac{s-4m_{p}^{2}c^{4}}{d_{2}})^{2}},$
(20)

where $d_{0}=0.41_{-0.02}^{+0.03}$ GeV2, $d_{1}=9.61_{-1.87}^{+2.60}$ GeV2 and
$d_{2}=2.79\pm 0.03$ GeV2 are obtained from a fit to the available data.

\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Ge.pdf}
\put(0.0,69.0){$(a)$} \end{overpic}
\begin{overpic}[width=187.78836pt,height=142.26378pt,angle={0}]{./com_Gm.pdf}
\put(0.0,69.0){$(b)$} \end{overpic}

Figure 13: Summary for (a) the electric FF of the proton $|G_{E}|$ and a fit
to the data (blue solid line and band) according to Eq. (17), and the Kuraev
model E. A. Kuraev, A. Dbeyssi, E. Tomasi-Gustafsson Eq. (19) (red dash-dotted
line and band); (b) the magnetic FF of the proton $|G_{M}|$ and a fit to the
data (blue solid line and band) according to Eq. (18), and the Kuraev model E.
A. Kuraev, A. Dbeyssi, E. Tomasi-Gustafsson Eq. (20) (red dash-dotted line and
band). The shown data are the published measurements from BESIII X. R. Zhou,
G. S. Huang ; A. Dbeyssi, F. E. Maas ; L. Xia, R. Christoph ; D. X. Lin, F. E.
Maas , BABAR B. Aubert, J. P. Lees , PS170 G. Bardin, G. Burgun . The
definition of $\chi^{2}$ is the same as in Figure 8.

###  5.6 Theoretical estimates of the proton radius

Precise measurements of FFs in the TL region can also be used to improve the
theoretical estimates of the proton radius A. Bianconi, E. Tomasi-Gustafsson ;
I. T. Lorenz, H. W. Hammer, and U. G. Meißner . A direct comparison of the
measured TL and space-like (SL) results of the proton EM FF ratio is shown in
Figure 14, together with a simple fit according to the model proposed in Ref.
E. A. Kuraev, A. Dbeyssi, E. Tomasi-Gustafsson :

$\displaystyle|\frac{G_{E}(s)}{G_{M}(s)}|=\frac{1}{\mu_{p}(1+\frac{s}{c_{0}})},$
(21)

where $c_{0}$ is the same parameter as in Eq. (15), and is determined to be
$c_{0}=5.61_{-1.33}^{+1.37}$ GeV2 from the TL-SL joint fit.

\begin{overpic}[width=284.52756pt,height=213.39566pt,angle={0}]{./com_GeGmcom1.pdf}
\end{overpic}

Figure 14: Measured $|G_{E}/G_{M}|$ in the timelike region from BESIII X. R.
Zhou, G. S. Huang ; A. Dbeyssi, F. E. Maas ; L. Xia, R. Christoph ; D. X. Lin,
F. E. Maas , BABAR B. Aubert, J. P. Lees , CMD3 R. R. Akhmetshin, A. N.
Amirkhanov , BES M. Ablikim, J. Z. Bai , FENICE+DM2 A. Antonelli, R. Baldini
Ferroli ; D. Bisello, S. Limentani, G. Busetto ; R. Baldini Ferroli, C. Bini,
P. Gauzzi , E835 M. Ambrogiani, M. Andreotti , PS170 G. Bardin, G. Burgun and
in the spacelike region from the GEp Collaboration A. J. R. Puckett, E. J.
Brash, M. K. Jones . The red dash-dotted line and band are monopole-like fits.
The definition of $\chi^{2}$ is the same as in Figure 8.

The model of Eq. (21) allows for a prediction of proton EM FFs by connecting
annihilation reactions ($l^{+}l^{-}\xleftrightarrow{}p\bar{p}$) in the TL
region to $ep$ elastic scattering in the SL region. Since there is more data
available for SL FFs, they have a larger weight compared to TL FFs. Therefore
more TL experimental data over a wider energy range is highly desirable.

##  6 Conclusions and Prospect

Within this review, the progress in the determination and understanding of the
proton EM FF in the TL region, both from the experimental and the theoretical
point of view, has been highlighted, on the experimental side focusing on
results obtained with the energy scan technique. The development of the field,
from early pioneering experimental works, for example, at Frascati M.
Castellano, G. Di Giugno ; A. Antonelli, R. Baldini Ferroli up to the
unprecedented accuracy results from the BESIII experiment X. R. Zhou, G. S.
Huang ; L. Xia, R. Christoph , has been outlined, and exciting future
prospects such as the PANDA experiment at HESR PandaDetector show that the
investigation of proton EM FFs is still a hot topic even 100 years after the
discovery of the proton.

The first successful investigations of the proton TL EM FFs with the scan
technique have been conducted by measurements of the total cross section of
$e^{+}e^{-}\xleftrightarrow{}p\bar{p}$ at $e^{+}e^{-}$ or $p\bar{p}$ colliders
(illustrated in Figure 3), which resulted in a determination of an effective
FF $|G_{\text{eff}}|$, assuming $|G_{E}|=|G_{M}|$ (illustrated in Figure 4).
Today, most available measurements only provide this auxiliary quantity, which
is extracted under an assumption that does not hold over the whole momentum
transfer range. Only recently, experiments have been successful in
disentangling the proton EM FFs in the TL region through angular analysis
(illustrated in Figure 5), measuring the ratio of the two FFs $|G_{E}/G_{M}|$
(illustrated in Figure 6). The high luminosity of the most recent BESIII scan
data L. Xia, R. Christoph in addition to a precise knowledge of its integrated
luminosity, allowed for a breakthrough in the comprehension of the proton
through the individual determination of $|G_{E}|$ and $|G_{M}|$ (illustrated
in Figure 7).

As shown in the theoretical interpretation of the experimental data for the
cross section of the reaction, $e^{+}e^{-}\xleftrightarrow{}p\bar{p}$, pQCD
and low-enegy-eff-QCD have been authenticated. Interesting phenomenology, in
particular the superposition of small periodic structures on an otherwise
smooth dipole parametrization of $|G_{\text{eff}}|$, has been discovered. Two
different approaches currently under discussion to explain these structures
have been presented: firstly, the possibility of resonant structures around
2.15 and 2.30 GeV, represented as Breit-Wigner functions on top of the non-
resonant pQCD and low-enegy-eff-QCD description (illustrated in Figure 9 (a)
and (b)), and secondly, a description based on an oscillating function
superimposed on the smooth dipole parametrization (illustrated in Figure 10
(a) and (b)). Furthermore, a similar periodic behavior for the neutron has
been observed (illustrated in Figure 11). Using the oscillation approach, a
the conjoint frequency with the proton oscillation of
$b^{{\text{osc}}}_{2}=5.05_{-0.09}^{+0.10}$ GeV/$c^{-1}$ has been extracted,
with a phase difference to the proton case of $\Delta
b^{{\text{osc}}}_{3}=|b^{{\text{osc}}}_{3(p)}-b^{{\text{osc}}}_{3(n)}|=(129_{-10}^{+11})^{\circ}$.
Similar to the oscillations of $|G_{\text{eff}}|$, a periodic behavior of the
ratio $|G_{E}/G_{M}|$ has also been observed, which can be extended to
$|G_{E}|$ and $|G_{M}|$ (illustrated in Figure 12 and Figure 13 (a) and (b)).
The period of these oscillations can be related to subhadronic scale processes
A. Bianconi, E. Tomasi-Gustafsson ; A. Bianconi, E. Tomasi-Gustafsson . We
have discussed a faster average decrease in $|G_{E}|$ and $|G_{M}|$, following
a similar behavior as in the SL region, which is in agreement with the
predictions of Ref E. A. Kuraev, A. Dbeyssi, E. Tomasi-Gustafsson (illustrated
in Figure 12 and Figure 13 (a) and (b)). Finally, we have examined the
connection of the TL and SL region by a TL-SL joint fit to improve our
comprehension of the radius of the proton (illustrated in Figure (14)).

Though a lot of progress has been made to deepen our understanding of the
proton structure, most experimental results were reported under the scenario
of one-photon approximation. In recent years, the scenario of two-photon
exchange (TPE) has been re-discussed, starting from a precise measurement of
$G_{E}/G_{M}$ ratio by polarized $ep$ scattering experiments E. Tomasi-
Gustafsson, M. Osipenko ; V. Punjabi, C. F. Perdrisat ; V. Punjabi, K.
Wijesooriya ; V. Punjabi, K. A. Aniol . TPE would manifest in a forward-
backward asymmetry in the angular distribution in the TL region. Precise
measurements of such an asymmetry at BESIII are ongoing, bringing valuable new
insights into the contribution of TPE to the process. The ultimate goal in the
field of TL nucleon FF’s would be the investigation of the phases between the
two complex valued FF’s. Theoretical efforts in this direction have already
been carried out and will be further pursued in the near future.
Experimentally, a measurement of the phase would require a measurement of the
proton polarization perpendicular to the plane of the incoming particles,
which could be achieved by including a polarimeter in the experimental setup
of a collider experiment.

In summary, the study of the proton EM FFs in the TL region has come a long
way. Measurements with the scan strategy have been the working horse of
previous experiments, as well as future prospects such as PANDA. Independent
confirmation of recent high precision results, as well as further improvements
both on both the precision as well as the measured $q^{2}$ range, will further
improve our understanding of the proton’s inner structure and dynamics.

All authors have read and agreed to the published version of the manuscript.
This work is supported in part by National Key Basic Research Program of China
under Contract No. 2015CB856700, No. 2020YFA0406403; National Natural Science
Foundation of China (NSFC) under Contracts No. 11335008, No. 11375170, No.
11425524, No. 11475164, No. 11475169, No. 11605196, No. 11605198, No.
11625523, No. 11635010, No. 11705192, No. 11735014, No. 11911530140, No.
11950410506, No. 12005219, No. 12035013, No. 12061131003; Joint LargeScale
Scientific Facility Funds of the NSFC and CAS under Contracts No. U1532102,
No. U1732263, No. U1832103, No. U2032111; China Postdoctoral Science
Foundation No. 2021M693097; ERC under contract no. 758462; European Union
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement no. 645664, 872901, and 894790; German Research
Foundation DFG under contract no. 443159800; Collaborative Research Center CRC
1044,

###### Acknowledgements.

Authors would like to thank the editors of the special issue of the journal,
Dr. Monica Bertani, Prof. Simone Pacetti and Dr. Alessio Mangoni for the
organization and helps. Authors would like to thank Dr. Weiping Wang for their
careful reading of this paper. References

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