BESIII Collaboration

# Probing the internal structure of baryons

Guangshun Huang1,2<EMAIL_ADDRESS>1Department of Modern Physics, University
of Science and Technology of China, Hefei 230026, People’s Republic of China
2State Key Laboratory of Particle Detection and Electronics, Hefei 230026,
People’s Republic of China  Rinaldo Baldini Ferroli3,4<EMAIL_ADDRESS>3INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
4Institute of High Energy Physics, Beijing 100049, People’s Republic of China

###### Abstract

Electromagnetic form factors are fundamental observables that describe the
electric and magnetic structure of hadrons and provide keys to understand the
strong interaction. At the Beijing Spectrometer (BESIII), form factors have
been measured for different baryons in the time-like region for the first time
or with the best precision. The results are presented with examples focus on
but not limited to the proton/neutron, the $\Lambda$, with a strange quark,
and the $\Lambda_{c}$, with a charm quark.

baryon structure, form factor, threshold effect, abnormal production

###### pacs:

##  I Introduction

Baryons and mesons are both hadrons, i.e., bound systems of quarks in naive
quark model quarkmodel  or more accurately also gluons in modern theory.
Baryons are half-integer spin fermions, comprised, in a first approximation,
of 3 quarks held together by the strong interactions. Protons ($p$) and
neutrons ($n$), collectively known as nucleons ($N$), are the lightest
baryons, and are the major components of the observable matter of the
Universe. A nucleon has three valence light quarks ($u$ or $d$); if one or
more of its $u$ or $d$ quarks are replaced by heavier quarks ($s$, $c$, $b$ or
$t$), it becomes a hyperon. The most known baryons are the spin 1/2 SU(3)
octet, including isospin doublet $p/n$, singly-stranged isospin singlet
$\Lambda$, singly-stranged isospin triplet $\Sigma^{-}/\Sigma^{0}/\Sigma^{+}$
and doubly-stranged isospin doublet $\Xi^{-}/\Xi^{0}$ PDG2020 . The lightest
charmed baryon is the $\Lambda_{c}^{+}$ PDG2020 . Hadrons are not point-like
particles, and their internal electric and magnetic structure is characterized
by their electromagnetic form factors (FF).

The particles are so tiny (at the order of 10-15 m, or fm) that they cannot be
observed directly by the human eye (ability of 10-4 m, or 0.1 mm), an optical
microscope (resolution of 10-7 m, or 0.1 $\mu$m), or even an electric
microscope (resolution of 10-10 m, or 0.1 nm, size of an atom). Instead, their
properties are studied through collisions. When two particles traverse each
other, they interact by exchanging force carriers called bosons that transfer
some energy and momentum (i.e., four momentum) from one to the other. For
electron-nucleon scattering, the electron is a probe that spies on the secrets
hidden inside the nucleon, and in this case the four-momentum transfer squared
has a negative value ($q^{2}<0$), and is categorized as a space-like process.
When a particle and an anti-particle meet, for example the case of an electron
and a positron, they can annihilate - i.e., disappear into a virtual photon,
and then produce fermion-antifermion pair that eventually materializes as a
system of hadrons, of which a baryon-antibaryon pair is one possibility. In
this case the four-momentum transfer squared has a positive value ($q^{2}>0$),
and is classified as a time-like process. The Feynman diagrams for these two
processes are shown in Fig. 1 (a) and (b), respectively. For the latter, the
form factors of the participating baryon can be deduced from the behavior of
the outgoing baryon-antibaryon pair, which is subject of the study covered in
this paper.

(a)(b)

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

Hadronic production data from electron-positron annihilations at low energies
(around GeV order) is important input to the understanding of the structure of
hadrons and the strong interactions of their constituent quarks. Moreover,
since hyperons are not stable, they can be studied only in the time-like
domain. The Beijing Spectrometer (BESIII) BES3  at the Beijng Electron
Positron Collider (BEPCII) BEPC2  operates in the center-of-mass energy range
from 2.0 and 4.6 GeV, which is a transition region between perturbative and
non-perturbative Quantum Chromodynamics (QCD). Using the initial state
radiation (ISR) technique, BESIII can also access energies below 2.0 GeV. The
$e^{+}e^{-}$ collision data that is used for QCD studies at BESIII include an
integrated luminosity of 12 pb-1 at four energies (2.23, 2.4, 2.8 and 3.4 GeV)
in the continuum taken in 2012, about 800 pb-1 at 104 energies between 3.85
and 4.6 GeV taken in the 2013-2014 run, and about 650 pb-1 at 22 energies from
2.0 to 3.08 GeV taken in 2015. These are the so-called scan data, with
moderate luminosity at each energy point, nonetheless, for these energies they
are the largest data samples in the world. There are also much larger samples
for charm physics or XYZ particle search, some as large as a few fb-1’s at a
single energy, which are suitable for ISR-type analyses. With these huge data
samples, BESIII is uniquely well suited to make baryon form factor
measurements with unprecedentedly high precision.

##  II Baryon Mysteries

The standard wisdom is that baryons are bound states of three quarks, but this
description is incomplete. For example, though nucleons are the basic building
blocks of observable matter in the Universe, not all of their basic properties
such as their size, spin, magnetic moment and mass are fully understood, even
after 100 years of study Rutherford ; Chadwick .

The charge radius of proton measured by muonic Lamb shift once differed from
that determined by electron-proton scattering or electronic Lamb shift as
large as five standard deviations Pohl , but recent measurements from electron
scattering PRad  and hydrogen spectroscopy elLamb  eliminated the
discrepancies, and this so-called proton-radius puzzle has been essentially
solved SciBull ; NatureRevPhys .

The proton spin has also been in a crisis in the era of the constituent quark
model. The European Muon Collaboration (EMC) experiment found that the baryon
spin is not only due to the spins of the valence quark SpinAsym . It has been
commonly assumed that the proton’s spin=1/2 was formed by two quarks with
parallel spins and the third quark with opposite spin. In the EMC experiment,
a quark of a polarized proton target was struck by a polarized muon beam, and
the quark’s instantaneous spin was measured. It was expected that the spin of
two of the three quarks would cancel out and the spin of the third quark would
be polarized in the direction of the proton’s spin. Thus, the sum of the
quarks’ spin was expected to be equal to the proton’s spin. Surprisingly, it
was found that the number of quarks with spin in the proton’s spin direction
was almost the same as the number of quarks whose spin was in the opposite
direction. Similar results have been obtained in many experiments afterwards,
demonstrating clearly that both generalized parton distributions and
transverse momentum distributions are important in the nucleon spin structure
NucleonSpin . Our modern understanding is that the nucleon spin comes not only
from quarks but also from gluons, and various contributions can be calculated
using e.g. Ji’s sum rule Ji . The abnormal magnetic moment of proton (much
larger than that for a Dirac point-like particle) is generally considered as
an indication of a more complicated internal structure than simply three
spin=1/2 quarks in a relative $S$-wave.

Moreover, the mass of proton also cannot be explained by Higgs mechanism,
since the sum of mass of quarks inside a proton is too small, which means
there are considerable contributions to its mass from the strong interactions
among quarks and gluons. Nowadays these contributions can be calculated
precisely in the lattice QCD, so the proton mass is largely understood
sigmaterm ; npmass .

##  III Baryon form factor measurements at BESIII

The differential cross section of electron-positron annihilation to a baryon-
antibaryon pair can be written as a function of the center-of-mass (c.m.)
energy squared $s$ as Zichichi ,

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

where $\theta$ is the polar angle of the baryon in the $e^{+}e^{-}$ c.m. frame
and $\beta=\sqrt{1-4m_{B}^{2}/s}$ is the speed of baryon. The Gamov-Sommerfeld
factor $C$ Sommerfeld ; Brodsky ; Sakharov  describes the Coulomb enhancement
effect: for charged baryon pair $C=y/(1-\mathrm{e}^{-y})$ with
$y=\pi\alpha\sqrt{1-\beta^{2}}/\beta$, accounts for the electromagnetic
interaction between the outgoing baryons; while for neutral baryon pair $C=1$.
The form factors, $G_{E}$ and $G_{M}$, essentially describe the electric and
magnetic distributions inside the baryon and basically provide a measure of
its boundary or size. These are functions of the four-momentum transfer
$s=q^{2}$, so more accurately should be written as $G_{E}(q^{2})$ and
$G_{M}(q^{2})$. In the time-like domain, the form factors are complex with a
nonzero imaginary part, and the translation into the internal structure is not
straightforward, in contrary to the case in the space-like region. It is
noteworthy that final-state interactions become prevailing close to threshold
and thus should be properly dealt with. By definition the electric and the
magnetic form factors are equal at the baryon-antibaryon pair’s mass threshold
where only $s$-wave production contributes Griffiths , i.e.,
$G_{E}(4m_{B}^{2})=G_{M}(4m_{B}^{2})$, but generally they are not. In analyses
of data with limited statistics it is often assumed that they are equal and
the two form factors are replaced by an effective form factor,
$G_{eff}=G_{E}=G_{M}$.

In principle, the Coulomb interaction between the outgoing charged baryon-pair
$B^{+}B^{-}$ should play an important role, in particular by producing an
abrupt jump in the cross section at threshold, since the phase space factor
$\beta$ is cancelled by a $1/\beta$ factor in the Coulomb correction (however
there is no full consensus on that), which is a non-perturbative correction to
the Born approximation to account for the Coulomb interaction between the
outgoing charged baryons. In fact, the cross section for the $e^{+}e^{-}\to
p\bar{p}$ at threshold has been measured to be very close to the pointlike
value, which is consistent with the prediction, but then it is followed by a
flat behavior, which is unexpected. While for a neutral-baryon-pair
$B^{0}\bar{B}^{0}$, the cross section at threshold should be zero according to
Eqn. 1. The minimum c.m. energy for BESIII data is 2.0 GeV, which is about 122
MeV above the nucleon-antinucleon threshold, so no solid conclusion can be
drawn for the proton-pair and neutron-pair cases, but BESIII can test these
effects for charged baryons by seeing if there is a step with a value close to
the pointlike one for $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ production, and
for neutral baryons by seeing if the cross section is vanishing at the
$\Lambda\bar{\Lambda}$ at threshold. Present BESIII results seem to indicate
that at both the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and
$\Lambda\bar{\Lambda}$ thresholds there is a step that is close to the
pointlike value for charged particles, although maybe not exactly the same.

###  III.1 Proton

Space-like proton form factors have been measured with very high precision by
many experiments JLab1 ; JLab2 . In the time-like region, there have been a
few measurements of $G_{eff}$, by DM2 DM2pp ; DM2BB , E760 E760 , PS170 PS170
, FENICE FENICE , E835 E835a ; E835b , BaBar ppbarBaBar1 ; ppbarBaBar2  and
CMD-3 ppCMD3 ; CMD3 , but these have relatively poor precision and mutual
agreement. For $|G_{E}/G_{M}|$ ratio, the measurements were rare and there is
a long-time tension between PS170 and BaBar. The BESII experiment also
measured the proton effective form factor, but with poor statistical precision
lihh . BESIII continued this effort using the 2012 and 2015 scan data, and
produced the most accurate $|G_{E}/G_{M}|$ ratio measurements at 16 c.m.
energies between 2.0 and 3.08 GeV ppbar2015 ; ppbar2020  that favor BaBar over
PS170 and helped clarifying the puzzle. BESIII also performed the measurements
using the ISR technique ppbaruntag ; ppbartag , with results that are
consistent with BaBar’s. The BESIII measurements are shown in Fig. 2 (a) for
$p\bar{p}$ production cross section in $2.0-3.08$ GeV, (b) the effective
proton time-like form factor, (c) the form factor ratio $R=|G_{E}/G_{M}|$, and
(d) the effective form factor residual, together with results from other
experiments. The best precision in the time-like region was reported by
BESIII, and the electric form factor was extracted for the first time. The
unprecedented 3.5% uncertainty that was achieved at 2.125 GeV by BESIII is
close to that of the best measurements in the space-like region, which have
been at per cent level for a long time. The CMD-3 experiment measured the
production cross section of proton pair and observed an abrupt rise at the
nucleon-antinucleon threshold CMD3 , as expected for point-like charged
particles according to Eqn. 1. BESIII did not extend down to the threshold
energy, but the results around 2 GeV agree with CMD-3. This information
improves our understanding of the proton inner structure from a different
dimension and helps to test theoretical models that depend on non-perturbative
QCD, e.g. charge distribution within the proton can be deduced unitary1 ;
unitary2 . The near threshold behavior of the electromagnetic form factor of a
hadron is mostly determined by the interaction of the hadron-antihadron in the
final state, and therefore the measurements of the form factor properties can
also serve as a fruitful source of information about hadron-antihadron
interaction Dalkarov .

Interestingly there are oscillations in the effective proton form factor,
first seen by BaBar and later confirmed by BESIII ppbaruntag . These
oscillations were subsequently studied with more precise data by BESIII
ppbar2020 . Ref. PeriodicFF  speculated that possible origins of this curious
behavior are rescattering processes at relative distances of 0.7 - 1.5 fm
between the centers of the forming hadrons, leading to a large fraction of
inelastic processes in $p-\bar{p}$ interactions, and a large imaginary
component to the rescattering processes.

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

Figure 2:  (a) the cross sections for $e^{+}e^{-}\to p\bar{p}$ cross section.
(b) the effective proton time-like form factor. The blue curve is the results
of an attempt to fit the measurements with smooth dipole-like function. (c)
the ratio $R=|G_{E}/G_{M}|$. (d) effective form factor residual $F(p)$ after
subtracting the one calculated by QCD theory (the blue curve shown in (b)), as
a function of the relative motion $p$ of the final proton and antiproton.
Plots are from Ref. ppbar2020 .

###  III.2 Neutron

Prior to the BESIII experiment, there was a long standing puzzle related to
differences between the neutron and proton production rates. QCD-motivated
models predict that the cross section for the proton should be 4 times larger
than for neutron QCDcalc , or they should be same Pacetti . In contrast the
FENICE experiment found that the neutron cross section was twice as large as
the the proton’s, albeit with statistics that were very limited, only 74
$n\bar{n}$ events in total for five energy bins FENICE . More recent
measurements in the vicinity of the nucleon-antinucleon threshold are from the
SND experiment SND2014 ; SNDnn . The cross sections of $e^{+}e^{-}\to
n\bar{n}$ and the neutron form factors between 2 GeV and 3.08 GeV have been
measured by BESIII with a good deal more data, over 2000 $n\bar{n}$ events at
18 energies nnbar . Because the final state neutron and anti-neutron are both
neutral, with no tracks recorded in the drift chamber, the event selection is
a challenge. The information in the calorimeter and the time of flight
counters has to be used to identify the signal; as such the selection
efficiency is much lower and the number of observed neutron events is
significantly less than that for protons. Neutron measurements from SND
SND2014 ; SNDnn  and BESIII nnbar  overlap and roughly agree at 2 GeV, where a
cross-section behavior that is close to the $e^{+}e^{-}\to p\bar{p}$ case is
observed, in particular a flat behavior above threshold up to 2 GeV as seen by
CMD-3 CMD3 , but this challenges the expected behavior from Eqn. 1. For
energies above 2 GeV, the BESIII measurements of the ratio of the proton to
neutron cross sections is more compatible with the QCD-motivated model
predictions: as shown in Fig. 3, the cross section for $e^{+}e^{-}\to
p\bar{p}$ is larger than for $e^{+}e^{-}\to n\bar{n}$ in general.

Figure 3: Ratio of Born cross section of $e^{+}e^{-}\to p\bar{p}$ to that of
$e^{+}e^{-}\to n\bar{n}$.

From BESIII measurements of the angular distributions for $e^{+}e^{-}\to
N\bar{N}$ events, the $S$-wave and $D$-wave contributions are disentangled for
the first time, that is currently under further investigation in the
Collaboration. Moreover from comparisons of the $e^{+}e^{-}\to n\bar{n}$ and
$e^{+}e^{-}\to p\bar{p}$ cross sections, the isoscalar and isovector
components of $e^{+}e^{-}\to N\bar{N}$ can, in principle, be separated IsoDomi
. One of the components dominates and is nearly constant up to 2 GeV, similar
to $e^{+}e^{-}\to p\bar{p}$, but at present it is difficult to identify
whether the isoscalar (very likely the largest) or the isovector one. With
more data in the future, this identification could be achieved by BESIII.

###  III.3 The $\Lambda$ hyperon

The $\Lambda$, which is the lightest hyperon that contains an $s$ quark, is
more difficult to study than the nucleon because of its smaller production
cross section. It was measured previously by the DM2 DM2BB  and BaBar
LambdaBaBar  experiments, but with results were not conclusive. BESIII has
studied the channel $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ Lambdapair
with an analysis that used a $40.5~{}\mathrm{pb}^{-1}$ data sample that was
collected at four different energy scan points during 2011 and 2012. The
lowest energy point is 2.2324 GeV, only 1 MeV above the
$\Lambda\bar{\Lambda}$-threshold. These data made it possible to measure the
Born cross section very near threshold. To use the data as efficiently as
possible, both events where $\Lambda$ and $\bar{\Lambda}$ decayed to the
charged mode ($\mathrm{Br}(\Lambda\rightarrow p\pi^{-})=64\%$) and events
where the $\bar{\Lambda}$ decayed to the neutral mode
($\mathrm{Br}(\bar{\Lambda}\rightarrow\bar{n}\pi^{0})=36\%$) were selected. In
the first case, the identification relied on finding two mono-energetic
charged pions with evidence for a $\bar{p}$-annihilation in the material of
the beam pipe or the inner wall of the tracking chamber. In the second case,
the $\bar{n}$-annihilation was identified with a multi-variate analysis of
variables provided by the electromagnetic calorimeter. Additonally, a mono-
energetic $\pi^{0}$ was reconstructed to fully identify this decay channel.
For the higher energy points, only the charged decay modes of $\Lambda$ and
$\bar{\Lambda}$ were reconstructed by identifying all the charged tracks and
using the event kinematics. The resulting measurement Lambdapair  of the Born
cross section are shown in Fig. 4(a) together with previous measurements DM2BB
; LambdaBaBar . The Born cross section near threshold is found to be $312\pm
51(\rm stat.)^{+72}_{-45}$(sys.) pb. This result confirms BaBar’s measurement
LambdaBaBar  but with much higher momentum transfer squared accuracy. Since
the Coulomb factor is equal to 1 for neutral baryon pairs, the cross section
is expected to go to zero at threshold. Therefore the observed threshold
enhancement implies the existence of a complicated underlying physics
scenario. The unexpected features of baryon pair production near threshold
have driven a lot of theoretical studies, including scenarios that invoke
bound states or unobserved meson resonances Dalkarov ; TheoRes ; Xiao:2019qhl
. It was also interpreted as an attractive Coulomb interaction on the
constituent quark level Resumfct1 ; Resumfct2 . Another possible explanation
is the final-state interactions which play an important role near the
threshold FSinter1 ; FSinter2 ; FSinter3 . The BESIII measurement improves
previous results at low invariant masses at least by 10$\%$ and even more
above 2.4 GeV/c. The $\Lambda$ effective form factor extracted from the cross
section measurement is shown in Fig. 4(b).

(a)

(b)

Figure 4: (a) Measurements of $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$
cross section. (b) $\Lambda$ effective form factor. Plots are from Ref.
Lambdapair .

According to the optical theorem, there is a nonzero relative phase between
$G_{E}$ and $G_{M}$. At $M_{\Lambda\bar{\Lambda}}=2.396$ GeV, where we have
the largest $\Lambda\bar{\Lambda}$ sample of 555 events from 66.9 pb-1 data, a
multidimensional analysis was used to make a full determination of the
$\Lambda$ electromagnetic form factors for the first time for any baryon; the
relative phase difference is $\Delta\Phi=37^{\circ}\pm 12^{\circ}\pm
6^{\circ}$ LambdaFF  with the input parameter $\alpha_{\Lambda}=0.750\pm
0.010$ measured from $J/\psi$ decays LambdaPolar . The improved determination
of $\alpha_{\Lambda}$ also has profound implications for the baryon spectrum,
since fits to such observables by theoretical models are a crucial element in
determining the light baryon resonance spectrum, which provides a point of
comparison for theoretical approaches alphaLambda . The $|G_{E}/G_{M}|$ ratio
was determined to be $R=0.96\pm 0.14(\rm stat.)\pm 0.02(\rm sys.)$ and the
effective form factor at $M_{\Lambda\bar{\Lambda}}=2.396$ GeV was determined
to be $|G_{eff}|=0.123\pm 0.003(\rm stat.)\pm 0.003(\rm sys.)$. The $\Lambda$
angular distribution and the polarization as a function of the scattering
angle are shown in Fig. 5(a) and (b), respectively. This first complete
measurement of the hyperon electromagnetic form factor is a milestone in the
study of hyperon structure, while the long-term goal is to describe charge and
magnetization densities of the hyperons.

(a)(b)

Figure 5: (a) The acceptance corrected $\Lambda$ scattering angle distribution
for $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ at
$M_{\Lambda\bar{\Lambda}}=2.396$ GeV. (b) The product of the $\Lambda$ decay
parameter $\alpha_{\Lambda}$ and $\Lambda$ polarization $P_{y}$ as a function
of the scattering angle. Plots are from Ref. LambdaFF .

###  III.4 The $\Lambda_{c}$ charmed baryon

Experimental studies on the charmed baryons have been rather sparse. The only
previous study of process $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
is from the Belle experiment, which measured the cross section using ISR
technique LambdacBelle , and reported a lineshape that implied the existence
of a likely resonance, called the Y(4660). Based on $631.3~{}\mathrm{pb}^{-1}$
data collected in 2014 at four energy points $\sqrt{s}=$4.5745, 4.5809, 4.5900
and 4.5995 GeV, BESIII measured the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
cross section with unprecedented precision Lambdacpair . The lowest energy
point is only 1.6 MeV above the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
threshold. At each of the energy points, ten Cabibbo-favored hadronic decay
modes, $\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$, $pK_{S}^{0}$,
$\Lambda\pi^{+}$, $pK^{-}\pi^{+}\pi^{0}$, $pK^{0}\pi^{0}$,
$\Lambda\pi^{+}\pi^{0}$, $pK_{S}\pi^{+}\pi^{-}$,
$\Lambda\pi^{+}\pi^{+}\pi^{-}$, $\Sigma^{0}\pi^{+}$, and
$\Sigma^{+}\pi^{+}\pi^{-}$, as well as the corresponding charge-conjugate
modes were studied. The total Born cross section is obtained from the weighted
average of the 20 individual measurements, and the results are shown in Fig.
6(a). Similar to the case for $e^{+}e^{-}\to p\bar{p}$, an abrupt rise in the
cross-section just above threshold that is much steeper than phase-space
expectations is discerned, which was not seen by Belle due to limitations of
the ISR method. BESIII’s measured cross section lineshape is different from
Belle’s, disfavoring a resonance like Y(4660) in the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ channel. The BESIII results have driven
discussions in the theoretical literature Dai:2017fwx .

High statistic data samples at $\sqrt{s}=$ 4.5745 and 4.5995 GeV enabled
studies of the polar angular distribution of $\Lambda_{c}$ in the $e^{+}e^{-}$
center-of-mass system. The shape function
$f(\theta)\propto(1+\alpha_{\Lambda_{c}}\cos^{2}\theta)$ is fitted to the
combined data contaning the yields of $\Lambda_{c}^{+}$ and
$\bar{\Lambda}_{c}^{-}$ for all ten decay modes as shown in Fig. 6(b). The
ratio between the electric and magnetic form factors $|G_{E}/G_{M}|$ can be
extracted using
$|G_{E}/G_{M}|^{2}(1-\beta^{2})=(1-\alpha_{\Lambda_{c}})/(1+\alpha_{\Lambda_{c}})$.
From these distributions, the ratios $|G_{E}/G_{M}|$ of $\Lambda_{c}^{+}$ have
been extracted for the first time: they are $1.14\pm 0.14~{}(\rm stat.)\pm
0.07~{}(\rm sys.)$ and $1.23\pm 0.05~{}(\rm stat.)\pm 0.03~{}(\rm sys.)$ at
$\sqrt{s}=$ 4.5745 and 4.5995 GeV, respectively.

(a)

(b)

Figure 6: The Born cross section of
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ obtained by BESIII and
Belle (a). The angular distribution and corresponding fit results in data at
$\sqrt{s}=4.5995$ GeV (b). Plots are from Ref. Lambdacpair .

##  IV Baryon challenges at BESIII

The energy thresholds for pair production of all of the ground state spin 1/2
SU(3) octet and spin 3/2 decuplet are accessible to BESIII. Baryon form factor
measurements are among the most important reasons why BESIII has collected an
unprecedented amount of off-resonance data. From the analysis of existing
data, it is expected that the ratio of the absolute values of the $\Lambda$
electromagnetic form factors, $|G_{E}/G_{M}|$, can be measured at five energy
points. The most interesting findings are the abrupt cross section jumps at
threshold followed by a nearly flat that has been observed for
$\Lambda\bar{\Lambda}$, $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$, $p\bar{p}$,
$n\bar{n}$, etc. If the BEPCII energy could be lowered to the vicinity of
nucleon-antinucleon threshold, BESIII will be able to confirm the $p\bar{p}$
and $n\bar{n}$ cases with much better precision. Figure 7 shows the cross
section lineshapes for a variety of baryon-antibaryon pairs, including those
that were recently measured for singly-stranged
$\Sigma^{+}\bar{\Sigma}^{-}$/$\Sigma^{-}\bar{\Sigma}^{+}$ chargedSigma ,
doubly-stranged $\Xi^{-}\bar{\Xi}^{+}$ chargedXi  and $\Xi^{0}\bar{\Xi}^{0}$
neutralXi .

Figure 7: A compilation of cross sections, revealing similar pattern for all
$B\bar{B}$ pairs measured so far: $p\bar{p}$ by BaBar ppbarBaBar1 ;
ppbarBaBar2  and BESIII ppbar2020 , $n\bar{n}$ by SND SND2014 ; SNDnn  and
BESIII nnbar , $\Lambda\bar{\Lambda}$ by BESIII Lambdapair ,
$\Sigma^{+}\bar{\Sigma}^{-}$/$\Sigma^{-}\bar{\Sigma}^{+}$ by BESIII
chargedSigma , $\Xi^{-}\bar{\Xi}^{+}$/$\Xi^{0}\bar{\Xi}^{0}$ by BESIII
chargedXi ; neutralXi , $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ by Belle
LambdacBelle  and BESIII Lambdacpair .

They all seem to share the common feature with a plateau starting from the
baryon-pair production threshold, though for some channels ideally more
statistics are needed. The behavior of $\Sigma^{0}\bar{\Sigma}^{0}$ (the last
member to be covered for the spin 1/2 SU(3) octet baryons) and other baryon-
pairs will be reported in the near future.

##  V Summary and Prospects

The measurements of baryon form factors have been an important ongoing
activity at BESIII. Form factors of proton with the best precision were
obtained in the time-like region, and the electric form factor of proton was
measured for the first time. Measurements of the neurtron time-like form
factor with unprecedented precision have also been reported. The $\Lambda$ and
$\Lambda_{c}$ were studied and in both cases abnormal cross section
enhancements were observed near the production thresholds. The form factors of
the $\Lambda_{c}$ were extracted for the first time.

In addition $\Sigma^{+}/\Sigma^{-}$ chargedSigma , $\Xi^{-}$ chargedXi  and
$\Xi^{0}$ neutralXi  form factor measurements were recently reported, and
results for $\Sigma^{0}$ will soon be released. BESIII also has a plan to
explore the nucleon production threshold by taking data in $1.8-2.0$ GeV for
$\sim 100$ pb-1 at 23 enengy points WhitePaper , in order to study the
anomalous threshold cross section behavior in more detail. With numerous first
measurements and interesting discoveries, these studies shed new light for
understanding the interactions and fundamental structure of particles.

It will take long time to ultimately unravel the fundamental structure of
baryons. Further improvements in form factor measurement of baryons will
continue to be the focus of future powerful electron-ion colliders in America
(EiC) EiC  and China (EicC) EicC , super electron-positron colliders in China
STCF  and Russia SCTF  for the space-like and time-like regions, respectively.

###### Acknowledgements.

The authors thank Prof. S. L. Olsen for helpful suggestions and proofreading.
This work is supported in part by National Natural Science Foundation of China
(NSFC) under Contract Nos. 12035013, 12061131003, 11911530140 and 11335008;
Joint Large-Scale Scientific Facility Funds of the NSFC and the Chinese
Academy of Sciences under Contract No. U1832103; National Key Research and
Development Program of China under Contract No. 2020YFA0406403; National Key
Basic Research Program of China under Contract No. 2015CB856705. Conflict of
interest statement: none declared.

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