# Measurement of the proton form factor by studying $e^{+}e^{-}\rightarrow
p\bar{p}$

M. Ablikim1, M. N. Achasov9,a, X. C. Ai1, O. Albayrak5, M. Albrecht4, D. J.
Ambrose44, A. Amoroso48A,48C, F. F. An1, Q. An45, J. Z. Bai1, R. Baldini
Ferroli20A, Y. Ban31, D. W. Bennett19, J. V. Bennett5, M. Bertani20A, D.
Bettoni21A, J. M. Bian43, F. Bianchi48A,48C, E. Boger23,h, O. Bondarenko25, I.
Boyko23, R. A. Briere5, H. Cai50, X. Cai1, O. Cakir40A,b, A. Calcaterra20A, G.
F. Cao1, S. A. Cetin40B, J. F. Chang1, G. Chelkov23,c, G. Chen1, H. S. Chen1,
H. Y. Chen2, J. C. Chen1, M. L. Chen1, S. J. Chen29, X. Chen1, X. R. Chen26,
Y. B. Chen1, H. P. Cheng17, X. K. Chu31, G. Cibinetto21A, D. Cronin-
Hennessy43, H. L. Dai1, J. P. Dai34, A. Dbeyssi14, D. Dedovich23, Z. Y. Deng1,
A. Denig22, I. Denysenko23, M. Destefanis48A,48C, F. De Mori48A,48C, Y.
Ding27, C. Dong30, J. Dong1, L. Y. Dong1, M. Y. Dong1, S. X. Du52, P. F.
Duan1, J. Z. Fan39, J. Fang1, S. S. Fang1, X. Fang45, Y. Fang1, L.
Fava48B,48C, F. Feldbauer22, G. Felici20A, C. Q. Feng45, E. Fioravanti21A, M.
Fritsch14,22, C. D. Fu1, Q. Gao1, X. Y. Gao2, Y. Gao39, Z. Gao45, I.
Garzia21A, C. Geng45, K. Goetzen10, W. X. Gong1, W. Gradl22, M. Greco48A,48C,
M. H. Gu1, Y. T. Gu12, Y. H. Guan1, A. Q. Guo1, L. B. Guo28, Y. Guo1, Y. P.
Guo22, Z. Haddadi25, A. Hafner22, S. Han50, Y. L. Han1, X. Q. Hao15, F. A.
Harris42, K. L. He1, Z. Y. He30, T. Held4, Y. K. Heng1, Z. L. Hou1, C. Hu28,
H. M. Hu1, J. F. Hu48A,48C, T. Hu1, Y. Hu1, G. M. Huang6, G. S. Huang45, H. P.
Huang50, J. S. Huang15, X. T. Huang33, Y. Huang29, T. Hussain47, Q. Ji1, Q. P.
Ji30, X. B. Ji1, X. L. Ji1, L. L. Jiang1, L. W. Jiang50, X. S. Jiang1, J. B.
Jiao33, Z. Jiao17, D. P. Jin1, S. Jin1, T. Johansson49, A. Julin43, N.
Kalantar-Nayestanaki25, X. L. Kang1, X. S. Kang30, M. Kavatsyuk25, B. C. Ke5,
R. Kliemt14, B. Kloss22, O. B. Kolcu40B,d, B. Kopf4, M. Kornicer42, W. Kühn24,
A. Kupsc49, W. Lai1, J. S. Lange24, M. Lara19, P. Larin14, C. Leng48C, C. H.
Li1, Cheng Li45, D. M. Li52, F. Li1, G. Li1, H. B. Li1, J. C. Li1, Jin Li32,
K. Li13, K. Li33, Lei Li3, P. R. Li41, T. Li33, W. D. Li1, W. G. Li1, X. L.
Li33, X. M. Li12, X. N. Li1, X. Q. Li30, Z. B. Li38, H. Liang45, Y. F.
Liang36, Y. T. Liang24, G. R. Liao11, D. X. Lin14, B. J. Liu1, C. X. Liu1, F.
H. Liu35, Fang Liu1, Feng Liu6, H. B. Liu12, H. H. Liu1, H. H. Liu16, H. M.
Liu1, J. Liu1, J. P. Liu50, J. Y. Liu1, K. Liu39, K. Y. Liu27, L. D. Liu31, P.
L. Liu1, Q. Liu41, S. B. Liu45, X. Liu26, X. X. Liu41, Y. B. Liu30, Z. A.
Liu1, Zhiqiang Liu1, Zhiqing Liu22, H. Loehner25, X. C. Lou1,e, H. J. Lu17, J.
G. Lu1, R. Q. Lu18, Y. Lu1, Y. P. Lu1, C. L. Luo28, M. X. Luo51, T. Luo42, X.
L. Luo1, M. Lv1, X. R. Lyu41, F. C. Ma27, H. L. Ma1, L. L. Ma33, Q. M. Ma1, S.
Ma1, T. Ma1, X. N. Ma30, X. Y. Ma1, F. E. Maas14, M. Maggiora48A,48C, Q. A.
Malik47, Y. J. Mao31, Z. P. Mao1, S. Marcello48A,48C, J. G. Messchendorp25, J.
Min1, T. J. Min1, R. E. Mitchell19, X. H. Mo1, Y. J. Mo6, C. Morales
Morales14, K. Moriya19, N. Yu. Muchnoi9,a, H. Muramatsu43, Y. Nefedov23, F.
Nerling14, I. B. Nikolaev9,a, Z. Ning1, S. Nisar8, S. L. Niu1, X. Y. Niu1, S.
L. Olsen32, Q. Ouyang1, S. Pacetti20B, P. Patteri20A, M. Pelizaeus4, H. P.
Peng45, K. Peters10, J. Pettersson49, J. L. Ping28, R. G. Ping1, R. Poling43,
Y. N. Pu18, M. Qi29, S. Qian1, C. F. Qiao41, L. Q. Qin33, N. Qin50, X. S.
Qin1, Y. Qin31, Z. H. Qin1, J. F. Qiu1, K. H. Rashid47, C. F. Redmer22, H. L.
Ren18, M. Ripka22, G. Rong1, X. D. Ruan12, V. Santoro21A, A. Sarantsev23,f, M.
Savrié21B, K. Schoenning49, S. Schumann22, W. Shan31, M. Shao45, C. P. Shen2,
P. X. Shen30, X. Y. Shen1, H. Y. Sheng1, W. M. Song1, X. Y. Song1, S.
Sosio48A,48C, S. Spataro48A,48C, G. X. Sun1, J. F. Sun15, S. S. Sun1, Y. J.
Sun45, Y. Z. Sun1, Z. J. Sun1, Z. T. Sun19, C. J. Tang36, X. Tang1, I.
Tapan40C, E. H. Thorndike44, M. Tiemens25, D. Toth43, M. Ullrich24, I.
Uman40B, G. S. Varner42, B. Wang30, B. L. Wang41, D. Wang31, D. Y. Wang31, K.
Wang1, L. L. Wang1, L. S. Wang1, M. Wang33, P. Wang1, P. L. Wang1, Q. J.
Wang1, S. G. Wang31, W. Wang1, X. F. Wang39, Y. D. Wang20A, Y. F. Wang1, Y. Q.
Wang22, Z. Wang1, Z. G. Wang1, Z. H. Wang45, Z. Y. Wang1, T. Weber22, D. H.
Wei11, J. B. Wei31, P. Weidenkaff22, S. P. Wen1, U. Wiedner4, M. Wolke49, L.
H. Wu1, Z. Wu1, L. G. Xia39, Y. Xia18, D. Xiao1, Z. J. Xiao28, Y. G. Xie1, Q.
L. Xiu1, G. F. Xu1, L. Xu1, Q. J. Xu13, Q. N. Xu41, X. P. Xu37, L. Yan45, W.
B. Yan45, W. C. Yan45, Y. H. Yan18, H. X. Yang1, L. Yang50, Y. Yang6, Y. X.
Yang11, H. Ye1, M. Ye1, M. H. Ye7, J. H. Yin1, B. X. Yu1, C. X. Yu30, H. W.
Yu31, J. S. Yu26, C. Z. Yuan1, W. L. Yuan29, Y. Yuan1, A. Yuncu40B,g, A. A.
Zafar47, A. Zallo20A, Y. Zeng18, B. X. Zhang1, B. Y. Zhang1, C. Zhang29, C. C.
Zhang1, D. H. Zhang1, H. H. Zhang38, H. Y. Zhang1, J. J. Zhang1, J. L. Zhang1,
J. Q. Zhang1, J. W. Zhang1, J. Y. Zhang1, J. Z. Zhang1, K. Zhang1, L. Zhang1,
S. H. Zhang1, X. Y. Zhang33, Y. Zhang1, Y. H. Zhang1, Y. T. Zhang45, Z. H.
Zhang6, Z. P. Zhang45, Z. Y. Zhang50, G. Zhao1, J. W. Zhao1, J. Y. Zhao1, J.
Z. Zhao1, Lei Zhao45, Ling Zhao1, M. G. Zhao30, Q. Zhao1, Q. W. Zhao1, S. J.
Zhao52, T. C. Zhao1, Y. B. Zhao1, Z. G. Zhao45, A. Zhemchugov23,h, B. Zheng46,
J. P. Zheng1, W. J. Zheng33, Y. H. Zheng41, B. Zhong28, L. Zhou1, Li Zhou30,
X. Zhou50, X. K. Zhou45, X. R. Zhou45, X. Y. Zhou1, K. Zhu1, K. J. Zhu1, S.
Zhu1, X. L. Zhu39, Y. C. Zhu45, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, L.
Zotti48A,48C, B. S. Zou1, J. H. Zou1 (BESIII Collaboration) 1 Institute of
High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS Institute of Information Technology, Lahore, Defence Road, Off
Raiwind Road, 54000 Lahore, Pakistan
9 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
10 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
11 Guangxi Normal University, Guilin 541004, People’s Republic of China
12 GuangXi University, Nanning 530004, People’s Republic of China
13 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
14 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
15 Henan Normal University, Xinxiang 453007, People’s Republic of China
16 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
17 Huangshan College, Huangshan 245000, People’s Republic of China
18 Hunan University, Changsha 410082, People’s Republic of China
19 Indiana University, Bloomington, Indiana 47405, USA
20 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
21 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
22 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
23 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
24 Justus Liebig University Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
25 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
26 Lanzhou University, Lanzhou 730000, People’s Republic of China
27 Liaoning University, Shenyang 110036, People’s Republic of China
28 Nanjing Normal University, Nanjing 210023, People’s Republic of China
29 Nanjing University, Nanjing 210093, People’s Republic of China
30 Nankai University, Tianjin 300071, People’s Republic of China
31 Peking University, Beijing 100871, People’s Republic of China
32 Seoul National University, Seoul, 151-747 Korea
33 Shandong University, Jinan 250100, People’s Republic of China
34 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
35 Shanxi University, Taiyuan 030006, People’s Republic of China
36 Sichuan University, Chengdu 610064, People’s Republic of China
37 Soochow University, Suzhou 215006, People’s Republic of China
38 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
39 Tsinghua University, Beijing 100084, People’s Republic of China
40 (A)Istanbul Aydin University, 34295 Sefakoy, Istanbul, Turkey; (B)Dogus
University, 34722 Istanbul, Turkey; (C)Uludag University, 16059 Bursa, Turkey
41 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
42 University of Hawaii, Honolulu, Hawaii 96822, USA
43 University of Minnesota, Minneapolis, Minnesota 55455, USA
44 University of Rochester, Rochester, New York 14627, USA
45 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
46 University of South China, Hengyang 421001, People’s Republic of China
47 University of the Punjab, Lahore-54590, Pakistan
48 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
49 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
50 Wuhan University, Wuhan 430072, People’s Republic of China
51 Zhejiang University, Hangzhou 310027, People’s Republic of China
52 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
b Also at Ankara University, 06100 Tandogan, Ankara, Turkey
c Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia and at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
d Currently at Istanbul Arel University, 34295 Istanbul, Turkey
e Also at University of Texas at Dallas, Richardson, Texas 75083, USA
f Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
g Also at Bogazici University, 34342 Istanbul, Turkey
h Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia

###### Abstract

Using data samples collected with the BESIII detector at the BEPCII collider,
we measure the Born cross section of $e^{+}e^{-}\rightarrow p\bar{p}$ at 12
center-of-mass energies from 2232.4 to 3671.0 MeV. The corresponding effective
electromagnetic form factor of the proton is deduced under the assumption that
the electric and magnetic form factors are equal $(|G_{E}|=|G_{M}|)$. In
addition, the ratio of electric to magnetic form factors, $|G_{E}/G_{M}|$, and
$|G_{M}|$ are extracted by fitting the polar angle distribution of the proton
for the data samples with larger statistics, namely at $\sqrt{s}=$ 2232.4 and
2400.0 MeV and a combined sample at $\sqrt{s}$ = 3050.0, 3060.0 and 3080.0
MeV, respectively. The measured cross sections are in agreement with recent
results from BaBar, improving the overall uncertainty by about 30%. The
$|G_{E}/G_{M}|$ ratios are close to unity and consistent with BaBar results in
the same $q^{2}$ region, which indicates the data are consistent with the
assumption that $|G_{E}|=|G_{M}|$ within uncertainties.

###### pacs:

13.66.Bc, 14.20.Dh, 13.40.Gp

##  I Introduction

Electromagnetic form factors (FFs) of the nucleon provide fundamental
information about its internal structure and dynamics. They constitute a
rigorous test of non-perturbative QCD as well as of phenomenological models.

Proton FFs can be measured in different kinematic regions by i) lepton-proton
elastic scattering (space-like, labeled SL) ii) electron-positron annihilation
into a proton-antiproton pair or proton-antiproton annihilation into an
electron-positron (time-like, labeled TL). The lowest order Feynman diagram of
lepton-proton scattering is shown in Fig. 1(a). The momentum transfer squared,
$q^{2}$, is negative and the FFs are real functions of $q^{2}$. The lowest
order $e^{+}e^{-}$ annihilation process is shown in Fig. 1(b). Here, $q^{2}$
is positive and the FFs are complex functions of $q^{2}$. The basic kinematic
variables are also shown in Fig. 1, where $k$, $k^{\prime}$ are the initial
and final electron momenta and $p$, $p^{\prime}$ are the initial and final
proton momenta. Since the electromagnetic vertex of the lepton is well known,
one can reliably extract the proton electromagnetic vertex $\Gamma^{\mu}$ by
measuring the cross section and the polarization. Assuming one-photon
exchange, i.e. in Born approximation, and under the basic requirements of
Lorentz invariance, the hadronic vertex can be parameterized in terms of two
FFs, $F_{1}$ and $F_{2}$ ff ,

$\Gamma_{\mu}(p^{\prime},p)=\gamma_{\mu}F_{1}(q^{2})+\frac{i\sigma_{\mu\nu}q^{\nu}}{2m_{p}}\kappa_{p}F_{2}(q^{2}),$
(1)

where the element
$\sigma_{\mu\nu}=\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}$ is a
representation of the Lorentz group, $m_{p}$ is the mass of the proton,
$\kappa_{p}=\frac{g_{p}-2}{2}$ is the anomalous magnetic moment of the proton,
$g_{p}=\frac{\mu_{p}}{J}$, $\mu_{p}=2.79$ is the magnetic moment of the proton
and $J=\frac{1}{2}$ is its spin. The functions $F_{1}$ and $F_{2}$ are the so
called Dirac and Pauli FFs, respectively. The optical theorem, applied to
lepton-nucleon scattering, implies that at the lowest order the FFs are real
in the SL region optical  optical2 , i.e. the complex conjugate of the
amplitude in Fig. 1(a), $\mathcal{M}^{\mathcal{\dagger}}$, is identical to
$\mathcal{M}$. In the TL region, as in in Fig. 1(b), the FFs can be complex
above the first hadronic threshold, that is, above twice the pion mass.

\begin{overpic}[width=208.13574pt]{SL-3.eps} \put(45.0,45.0){ (a)}
\end{overpic}\begin{overpic}[width=208.13574pt]{TL-3.eps} \put(45.0,45.0){
(b)} \end{overpic}

Figure 1: (a) Feynman diagram of $ep\rightarrow ep$ elastic scattering at the
lowest order. (b) Feynman diagram of $e^{+}e^{-}\rightarrow p\bar{p}$
annihilation at the lowest order (identical to that of the reverse reaction
$p\bar{p}\rightarrow e^{+}e^{-}$ with $e\leftrightarrow p$ exchange.)

The Sachs FFs, electric $G_{E}$ and magnetic $G_{M}$, are introduced as linear
combinations of the Dirac and Pauli FFs sachs . Concerning the SL region in
the Breit frame, $G_{E}$ and $G_{M}$ are the Fourier transforms of the charge
and magnetization distribution of the nucleon, respectively. $G_{M}$ and
$G_{E}$ are proportional to spin-flip and non spin-flip amplitudes,
respectively. They are expressed as

$G_{E}(q^{2})=F_{1}(q^{2})+\frac{q^{2}}{4m_{p}^{2}}\kappa_{p}F_{2}(q^{2}),$
(2) $G_{M}(q^{2})=F_{1}(q^{2})+\kappa_{p}F_{2}(q^{2}).$ (3)

In the TL region, the center-of-mass (c.m.) system is equivalent to the Breit
frame since the helicities of baryons are opposite for the spinors aligned in
$G_{M}$ and are the same for the spinors aligned in $G_{E}$.

In the SL region, FFs have been extracted by the Rosenbluth separation method
Rosenbluth , as well as, more recently, by the recoil proton polarization
transfer method Polarization . The latter has been applied to obtain the
$\mu_{p}G_{E}/G_{M}$ ratio. Results from the GEp-II experiment at JLab’s Hall
A jlab1 ; jlab2  for $\mu_{p}G_{E}/G_{M}$ show that this ratio decreases
rather quickly with increasing $Q^{2}$, where $Q^{2}=-q^{2}\geq 0$, while
results achieved by the Rosenbluth method show an almost constant ratio And .
The discrepancy between the Rosenbluth and the polarization transfer method
may be resolved by including higher order corrections like two-photon
exchange. A small correction to the Rosenbluth separation could imply a large
correction for the extraction of $G_{E}$, since $G_{E}$ is the slope of the
Rosenbluth plot. However, the correction of including two-photon exchange is
small and cannot significantly influence the results of the polarization
transfer experiment.

In the TL region, measurements have been performed in the direct production
channel $e^{+}e^{-}\rightarrow p\bar{p}$ DM1 ; DM2 ; FENICE ; bes2 ; cleoform
, in the radiative return channel $e^{+}e^{-}\rightarrow
p\bar{p}(\gamma_{ISR})$ babar2 ; babar3  where $\gamma_{ISR}$ refers to a
photon emitted by initial state radiation (ISR), and in $\bar{p}p\rightarrow
e^{+}e^{-}$ ps170 ; E760 ; E835  experiments. In cases where the data sample
is too small to extract angular distributions and disentangle $|G_{E}|$ and
$|G_{M}|$, the effective proton FF $|G|$ can be calculated from the total
cross section, assuming $|G_{E}|=|G_{M}|$. This assumption is valid at the
$p\bar{p}$ mass threshold, if analyticity of the FFs holds, implying that at
threshold the angular distribution should be isotropic. In the PS170
experiment at LEAR ps170 , the effective proton FF was obtained, as well as
the $|G_{E}/G_{M}|$ ratio, from $p\bar{p}$ threshold up to $\sqrt{s}=2.05$
GeV. In the BaBar experiment at PEP-II babar2 ; babar3 , the cross section was
measured using the ISR method from the $p\bar{p}$ production threshold up to
$\sqrt{s}=6.5$ GeV. The $|G_{E}/G_{M}|$ ratio was measured from threshold up
to $\sqrt{s}=3.0$ GeV and the result shows an inconsistency with respect to
the PS170 results.

The presence of vector resonances, like $\rho$, $\omega$ and $\phi$ in the
_unphysical region_ , below the $p\bar{p}$ threshold, can influence the
functional form of the FFs in the physical region. Hence the FFs, in
particular the ratio $|G_{E}/G_{M}|$, in the TL region cannot be simply
extrapolated from the SL ones. Until now it has been assumed that all FFs
respect analyticity, which should allow to calculate their behavior in the
unphysical region thanks to dispersion relations dispersion  using the
available data in both the TL and SL regions. In the SL region, the ratio
$\mu_{p}G_{E}/G_{M}$ has been measured at 16 $Q^{2}$ values in (0.5, 8.5) GeV2
with the best precision to 1.7$\%$ jlab1 ; jlab2 , while the present precision
of $|G_{E}/G_{M}|$ in the TL region exceeds $10\%$ by far. Therefore, it is
necessary to improve the measurement of $|G_{E}/G_{M}|$ ratio in the TL
region.

The experimental determinations of proton FFs are important input for various
QCD-based theoretical models. There are plenty of theoretical approaches
applied to explain TL FFs: Chiral Perturbation Theory chiral , Lattice QCD
lattice  lattice2 , Vector Meson Dominance (VMD) VMD , the Relativistic
Constituent Quark Model (CQM) CQM , and, at high energies, perturbative QCD
predictions pQCD .

In this paper, we present an investigation of the process
$e^{+}e^{-}\rightarrow p\bar{p}$ based on data samples collected with the
Beijing Spectrometer III (BESIII) BESIII  at the Beijing Electron Positron
Collider II (BEPCII) at 12 c.m. energies ($\sqrt{s}$). The Born cross section
at these energy points are measured and the corresponding effective FFs are
determined. The ratio of electric to magnetic FFs, $|G_{E}/G_{M}|$, and
$|G_{M}|$ are measured at those c.m. energies where the statistics are large
enough. The results are consistent with those from BaBar in the same $q^{2}$
region.

##  II The BESIII Experiment And Data Sets

BEPCII is a double-ring $e^{+}e^{-}$ collider running at c.m. energies between
2.0-4.6 GeV and reached a peak luminosity of $0.85\times 10^{33}$cm-2s-1 at a
c.m. energy of 3770 MeV. The cylindrical BESIII detector has an effective
geometrical acceptance of $93\%$ of 4$\pi$ and is divided into a barrel
section and two endcaps. It contains a small cell, helium-based (40$\%$ He,
60$\%$ C3H8) main drift chamber (MDC) which provides momentum measurement for
charged particles with a resolution of $0.5\%$ at a momentum of 1 GeV/c in a
magnetic field of 1 Tesla. The energy loss measurement ($dE/dx$) provided by
the MDC has a resolution better than $6\%$. A time-of-flight system (TOF)
consisting of 5-cm-thick plastic scintillators can measure the flight time of
charged particles with a time resolution of 80 ps in the barrel and 110 ps in
the end-caps. An electromagnetic calorimeter (EMC) consisting of 6240 CsI (Tl)
in a cylindrical structure and two end-caps is used to measure the energies of
photons and electrons. The energy resolution of the EMC is $2.5\%$ in the
barrel and $5.0\%$ in the end-caps for photon/electron of 1 GeV energy. The
position resolution of the EMC is 6 mm in the barrel and 9 mm in the end caps.
A muon system (MUC) consisting of about 1000 m2 of Resistive Plate Chambers
(RPC) is used to identify muons and provides a spatial resolution better than
2 cm.

Monte Carlo (MC) simulated signal and background samples are used to optimize
the event selection criteria, estimate the background contamination and
evaluate the selection efficiencies. The MC samples are generated using a
Geant4-based geant4  simulation software package BESIII Object Oriented
Simulation Tool (BOOST) Deng , which includes the description of geometry and
material, the detector response and the digitization model, as well as a
database of the detector running conditions and performances. In this
analysis, the generator software package Conexc ping  is used to simulate the
signal MC samples $e^{+}e^{-}\rightarrow p\bar{p}$, and calculate the
corresponding correction factors for higher order process with one radiative
photon in the final states. Another generator Phokhara phokhara  serves as a
cross check of the radiative correction factors. At each c.m. energy, a large
signal MC sample with more than 10 times of the produced events in data for
the process $e^{+}e^{-}\rightarrow p\bar{p}$, contributing $0.15\%$
statistical uncertainty on the detection efficiency, is generated. Simulated
samples of the QED background processes $e^{+}e^{-}\rightarrow l^{+}l^{-}$ (l
= e, $\mu$) and $e^{+}e^{-}\rightarrow\gamma\gamma$ are generated with the
generator Babayaga babayaga . The other background MC samples for the
processes with the hadronic final states $e^{+}e^{-}\rightarrow h^{+}h^{-}$
($h=\pi$, $K$), $e^{+}e^{-}\rightarrow p\bar{p}\pi^{0}$,
$e^{+}e^{-}\rightarrow p\bar{p}\pi^{0}\pi^{0}$ and
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ are generated with uniform phase
space distributions. The background samples are generated with equivalent
luminosities at least as large as the data samples.

##  III Analysis Strategy

###  III.1 Event selection

Charged tracks are reconstructed with the hit information from the MDC. A good
charged track must be within the MDC coverage, $|\cos\theta|<0.93$, and is
required to pass within 1 cm of the $e^{+}e^{-}$ interaction point (IP) in the
plane perpendicular to the beam and within $\pm 10$ cm in the direction along
the beam. The combined information of $dE/dx$ and TOF is used to calculate
particle identification (PID) probabilities for the pion, kaon and proton
hypothesis, respectively, and the particle type with the highest probability
is assigned to the track. In this analysis, exactly two good charged tracks,
one proton and one antiproton, are required. To suppress Bhabha background
events, the ratio $E/p$ of each proton candidate is required to be smaller
than 0.5, where $E$ and $p$ are the energy deposited in the EMC and the
momentum measured in the MDC, respectively. The cosmic ray background is
rejected by requiring $|T_{\text{trk1}}-T_{\text{trk2}}|<$4 ns, where
$T_{\text{trk1}}$ and $T_{\text{trk2}}$ are the measured time of flight in the
TOF detector for the two tracks. For the samples with c.m. energy
$\sqrt{s}>2400.0$ MeV, the proton is further required to satisfy
$\cos\theta<0.8$ to suppress Bhabha background.

After applying the above selection criteria, the distributions of the opening
angle between proton and antiproton, $\theta_{p\bar{p}}$, at c.m. energies
$\sqrt{s}=$ 2232.4 and 3080.0 MeV are shown in Fig. 2. Good agreement between
data and MC samples is observed, and a better resolution is achieved with
increasing c.m. energy due to the smaller effects on the small angle multiple
scattering. A c.m. energy dependent requirement, $i.e.$,
$\theta_{p\bar{p}}>178^{\circ}$ at $\sqrt{s}\leq 2400.0$ MeV, and
$\theta_{p\bar{p}}>179^{\circ}$ at $\sqrt{s}>2400.0$ MeV, is further applied.
Figure 3 shows the distribution of the momentum of proton or antiproton at
c.m. energies $\sqrt{s}=2232.4$ and 3080.0 MeV. A momentum window of 5 times
the momentum resolution, $|p_{\text{mea}}-p_{\text{exp}}|<5\sigma_{p}$, is
applied to extract the signals, where $p_{\text{mea}}$ and $p_{\text{exp}}$
are the measured and expected momentum of the proton or antiproton in the c.m.
system, respectively, and $\sigma_{p}$ is the corresponding resolution.

\begin{overpic}[width=216.81pt]{6-anglemcdata-223-9.eps} \put(70.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=216.81pt]{6-anglemcdata-308-9.eps}
\put(70.0,60.0){(b)} \end{overpic}

Figure 2: Opening angle distributions between proton and antiproton at the
c.m. energies of (a) 2232.4 MeV, and (b) 3080.0 MeV. The dots with error bars
are data, the histograms represent the distributions of signal MC samples. The
arrows show the selection applied.

\begin{overpic}[width=216.81pt]{p223datamc-6.eps} \put(70.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=216.81pt]{p308datamc-6.eps}
\put(70.0,60.0){(b)} \end{overpic}

Figure 3: Momentum distribution of the proton or antiproton at the c.m.
energies (a) 2232.4 MeV, and (b) 3080.0 MeV, two entries per event. The dots
with error bars are data, the histograms represent the distributions of signal
MC samples. The arrows show the momentum window requirements.

###  III.2 Background study

The potential background contamination can be classified into two categories,
the beam associated background and the physical background.

The beam associated background includes interactions between the beam and the
beam pipe, beam and residual gas, and the Touschek effect touschek . Dedicated
data samples with separated beams were collected with the BESIII detector at
$\sqrt{s}=$ 2400.0 and 3400.0 MeV; these are used to study the beam associated
background. Since the two beams do not interact with each other, all of the
observed events are beam associated background, and can be used to evaluate
the beam associated background at different c.m. energies by normalizing the
data-taking time and efficiencies. No events from the separated beam data
samples survive the signal selection criteria. Considering that the
normalization factor is less than 5 for most of energy points (other than 3.08
and 3.65 GeV), the beam associated background at all c.m. energy points is
negligible.

The physical background may come from the $e^{+}e^{-}$ annihilation processes
with two-body final states, e.g. Bhabha or di-muon events, where leptons are
misidentified as protons or antiprotons, or processes with multi-body final
states including $p\bar{p}$, e.g. $e^{+}e^{-}\to p\bar{p}\pi^{0}(\pi^{0})$.
The contamination from physical background is evaluated by MC samples, and are
listed in Table 1 for $\sqrt{s}=2232.4$ and $3080.0$ MeV, respectively.

The number of the surviving background events after normalization,
$N^{\text{MC}}_{\text{nor}}$, is very small at the low c.m. energies and can
therefore be safely neglected. However, at higher c.m. energies ($\sqrt{s}\geq
3.40$ GeV), due to the rapid decrease of the cross section of
$e^{+}e^{-}\rightarrow p\bar{p}$, the background level which is mainly from
Bhabha events is higher, and $N^{\text{MC}}_{\text{nor}}$ needs to be
corrected for.

Table 1: Physical background processes estimated from the MC samples at $\sqrt{s}=2232.4$ and $3080.0$ MeV. $N^{\text{MC}}_{\text{gen}}$ is the number of generated MC events, $N^{\text{MC}}_{\text{sur}}$ is the number of events remaining after the selection criteria, $\sigma$ is the production cross section in the $e^{+}e^{-}$ annihilation process, which is obtained using the Babayaga generator for Bhabha, di-muon, and di-photon processes, and from the previous experimental results for others processes xsection1 ; xsection2 . $N^{\text{MC}}_{\text{uplimit}}$ and $N^{\text{MC}}_{\text{nor}}$ are the estimated upper limit at the 90$\%$ confidence level (C.L.) and the normalized number of background events.  |  $\sqrt{s}=2232.4$ MeV ($2.63$ pb-1) |  $\sqrt{s}=3080.0$ MeV ($30.73$ pb-1)
---|---|---
Bkg. |  $N^{\text{MC}}_{\text{gen}}$ ($\times 10^{6}$) | $N^{\text{MC}}_{\text{sur}}$ |  $\sigma$ (nb) | $N^{\text{MC}}_{\text{uplimit}}$ | $N^{\text{MC}}_{\text{nor}}$ |  $N^{\text{MC}}_{\text{gen}}$ ($\times 10^{6}$) | $N^{\text{MC}}_{\text{sur}}$ |  $\sigma$ (nb) | $N^{\text{MC}}_{\text{uplimit}}$ | $N^{\text{MC}}_{\text{nor}}$
$e^{+}e^{-}$ | 9.6 | 0 | 1435.01 | $<0.96$ | 0 | 39.9 | 1 | 756.86 | $<2.54$ | 1
$\mu^{+}\mu^{-}$ | 0.7 | 0 | 17.41 | $<0.16$ | 0 | 1.5 | 0 | 8.45 | $<0.42$ | 0
$\gamma\gamma$ | 1.9 | 0 | 70.44 | $<0.24$ | 0 | 4.5 | 0 | 37.05 | $<0.62$ | 0
$\pi^{+}\pi^{-}$ | 0.1 | 0 | 0.17 | $<0.01$ | 0 | 0.1 | 0 | $<0.11$ | $<0.02$ | 0
$K^{+}K^{-}$ | 0.1 | 0 | 0.14 | $<0.008$ | 0 | 0.1 | 0 | 0.093 | $<0.02$ | 0
$p\bar{p}\pi^{0}$ | 0.1 | 0 | $<0.1$ | $<0.006$ | 0 | 0.1 | 0 | $<0.1$ | $<0.07$ | 0
$p\bar{p}\pi^{0}\pi^{0}$ | 0.1 | 0 | $<0.1$ | $<0.006$ | 0 | 0.1 | 0 | $<0.1$ | $<0.07$ | 0
$\Lambda\overline{\Lambda}$ | 0.1 | 0 | $<0.4$ | $<0.02$ | 0 | 0.1 | 0 | 0.002 | $<0.001$ | 0

The ratio of $p\bar{p}$ invariant mass and the c.m. energy,
$M_{p\bar{p}}/\sqrt{s}$, from data and MC has been compared and is shown in
Fig. 4 at different c.m. energies. The integral luminosity of the data set at
each c.m. energy is listed in Table 2. There is good agreement between data
and MC simulations. The signal yields are extracted by counting the number of
events and are listed in Table 2, where the quoted uncertainties are
statistical only. The data sample at $3550.7$ MeV is a combination of three
data sub-samples with very close c.m. energies, $\sqrt{s}=$ 3542.4, 3553.8,
3561.1 MeV, and the value of 3550.7 MeV is the average c.m. energy weighted
with their luminosity values.

\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-223-datamc.eps}
\put(70.0,60.0){\footnotesize(a)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-240-datamc.eps}
\put(70.0,60.0){\footnotesize(b)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-280-datamc.eps}
\put(70.0,60.0){\footnotesize(c)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-305-datamc.eps}
\put(70.0,60.0){\footnotesize(d)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-306-datamc.eps}
\put(70.0,60.0){\footnotesize(e)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-308-datamc.eps}
\put(70.0,60.0){\footnotesize(f)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-340-datamc.eps}
\put(70.0,60.0){\footnotesize(g)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-350-datamc.eps}
\put(70.0,60.0){\footnotesize(h)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-355-datamc.eps}
\put(70.0,60.0){\footnotesize(i)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-360-datamc.eps}
\put(70.0,60.0){\footnotesize(j)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-365-datamc.eps}
\put(70.0,60.0){\footnotesize(k)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{4-mpps-3671-datamc.eps}
\put(70.0,60.0){\footnotesize(l)} \end{overpic}

Figure 4: Comparison of $M_{p\bar{p}}/\sqrt{s}$ distributions at different
c.m. energies for data (dots with error bars) and MC (histograms): (a) 2232.4,
(b) 2400.0, (c) 2800.0, (d) 3050.0, (e) 3060.0, (f) 3080.0, (g) 3400.0, (h)
3500.0, (i) 3550.7, (j) 3600.2, (k) 3650.0, (l) 3671.0 MeV. The sample (i) is
a combination of three data sub-samples with very close c.m. energies,
$\sqrt{s}=$ 3542.4, 3553.8, 3561.1 MeV, and the value of 3550.7 MeV is the
average c.m. energy weighted with their luminosity values.

###  III.3  Extraction of the Born cross section of $e^{+}e^{-}\rightarrow
p\bar{p}$ and the effective FF

The differential Born cross section of $e^{+}e^{-}\rightarrow p\bar{p}$ can be
written as a function of FFs, $|G_{E}|$ and $|G_{M}|$ formula ,

$\displaystyle\begin{split}\frac{d\sigma_{\text{Born}}(s)}{d\Omega}=\frac{\alpha^{2}\beta
C}{4s}[&|G_{M}(s)|^{2}(1+\cos^{2}\theta_{p})+\\\
&\frac{4m_{p}^{2}}{s}|G_{E}(s)|^{2}\sin^{2}\theta_{p}],\end{split}$ (4)

where $\alpha\approx\frac{1}{137}$ is the fine structure constant,
$\beta=\sqrt{1-\frac{4m_{p}^{2}}{s}}$ is the velocity of the proton in the
$e^{+}e^{-}$ c.m. system,
$C=\frac{\pi\alpha}{\beta}\frac{1}{1-\exp(-\pi\alpha/\beta)}$ is the Coulomb
correction factor for a point-like proton, $s$ is the square of the c.m.
energy, and $\theta_{p}$ is the polar angle of the proton in the $e^{+}e^{-}$
c.m. system. We assume that the proton is point-like above the $p\bar{p}$
production threshold, meaning that the Coulomb force acts only on the already
formed hadrons. At the energies we are considering here, the Coulomb
correction factor can be safely assumed to be 1. Furthermore, under the
assumption of the effective FF $|G|=|G_{E}|=|G_{M}|$ and by integrating over
$\theta_{p}$, it can be deduced:

$|G|=\sqrt{\frac{\sigma_{\text{Born}}}{86.83\cdot\frac{\beta}{s}(1+\frac{2m_{p}^{2}}{s})}},$
(5)

where $\sigma_{\text{Born}}$ is in nb and $m_{p}$, $s$ in GeV.

Experimentally, the Born cross section of $e^{+}e^{-}\rightarrow p\bar{p}$ is
calculated by

$\sigma_{\text{Born}}=\frac{N_{\text{obs}}-N_{\text{bkg}}}{L\cdot\varepsilon\cdot(1+\delta)},$
(6)

where $N_{\text{obs}}$ is the observed number of candidate events, extracted
by counting the number of signal events, $N_{\text{bkg}}$ is the expected
number of background events estimated by MC simulations, $L$ is the integrated
luminosity estimated with large-angle Bhabha events, $\varepsilon$ is the
detection efficiency determined from a MC sample generated using the Conexc
generator ping , which includes radiative corrections (which will be discussed
in detail in next paragraph), and $(1+\delta)$ is the radiative correction
factor which has also been determined using the Conexc generator. The derived
Born cross section $\sigma_{\text{Born}}$, the effective FF $|G|$, as well as
the related variables used to calculate $\sigma_{\text{Born}}$ are shown in
Table 2 at different c.m. energies. In the table, the product value
$\varepsilon^{\prime}=\varepsilon\times(1+\delta)$ is presented to account for
the effective efficiency. Comparisons of $\sigma_{\text{Born}}$ and $|G|$ to
the previous experimental measurements are shown in Fig. 5. Compared to the
BaBar results babar2 , the precision of the Born cross section is improved by
30% for data sets with $\sqrt{s}\leq 3080.0$ MeV, and the corresponding
precision of effective FF is improved, too.

From Eq. 4, it is obvious that the detection efficiency depends on the ratio
of the electric and magnetic FFs, $|G_{E}/G_{M}|$, due to the different polar
angle $\theta_{p}$ distributions. In this analysis, the detection efficiency
is evaluated with the MC samples. The ratio of $|G_{E}/G_{M}|$ is measured for
data samples at c.m. energies $\sqrt{s}$ = 2232.4 and 2400.0 MeV, and for a
combined data with sub-data samples at $\sqrt{s}$ = 3050.0, 3060.0, and 3080.0
MeV, which have close c.m. energy. The corresponding measured $|G_{E}/G_{M}|$
ratios are used as the inputs for MC generation. Details of the
$|G_{E}/G_{M}|$ ratio measurement can be found in Sec. III.4. For other c.m.
energy points, where the $|G_{E}/G_{M}|$ ratios are not measured due to
limited statistics, the detection efficiencies are obtained by averaging the
efficiencies with setting $|G_{E}|=0$ and $|G_{M}|=0$, respectively. The
corresponding product values of detection efficiencies and the radiative
correction factors at different c.m. energies are listed in Table 2. The
interference of $p\bar{p}$ final states between $e^{+}e^{-}$ annihilation and
$J/\psi$ decay in the lower tail is assumed to be negligible bian .

Table 2: Summary of the Born cross section $\sigma_{\text{Born}}$, the effective FF $|G|$, and the related variables used to calculate the Born cross sections at the different c.m. energies $\sqrt{s}$, where $N_{\text{obs}}$ is the number of candidate events, $N_{\text{bkg}}$ is the estimated background yield, $\varepsilon^{\prime}=\varepsilon\times(1+\delta)$ is the product of detection efficiency $\varepsilon$ and the radiative correction factor $(1+\delta)$, and $L$ is the integrated luminosity. The first errors are statistical, and the second systematic. $\sqrt{s}$ (MeV) | $N_{\text{obs}}$ | $N_{\text{bkg}}$ |  $\varepsilon^{\prime}$ $(\%)$ |  $L$ (pb-1) |  $\sigma_{\text{Born}}$ (pb) |  $|G|$ $(\times 10^{-2})$
---|---|---|---|---|---|---
2232.4 | $614\pm 25$ | 1 | 66.00 | 2.63 | $353.0\pm 14.3\pm 15.5$ | $16.10\pm 0.32\pm 0.35$
2400.0 | $297\pm 17$ | 1 | 65.79 | 3.42 | $132.7\pm 7.7\pm 8.1$ | $10.07\pm 0.29\pm 0.31$
2800.0 | $53\pm 7$ | 1 | 65.08 | 3.75 | $21.3\pm 3.0\pm 2.8$ | $4.45\pm 0.31\pm 0.29$
3050.0 | $91\pm 10$ | 2 | 59.11 | 14.90 | $10.1\pm 1.1\pm 0.6$ | $3.29\pm 0.17\pm 0.09$
3060.0 | $78\pm 9$ | 2 | 59.21 | 15.06 | $8.5\pm 1.0\pm 0.6$ | $3.03\pm 0.17\pm 0.10$
3080.0 | $162\pm 13$ | 1 | 58.97 | 30.73 | $8.9\pm 0.7\pm 0.5$ | $3.11\pm 0.12\pm 0.08$
3400.0 | $2\pm 1$ | 0 | 63.34 | 1.73 | $1.8\pm 1.3\pm 0.4$ | $1.54\pm 0.55\pm 0.18$
3500.0 | $5\pm 2$ | 0 | 63.70 | 3.61 | $2.2\pm 1.0\pm 0.6$ | $1.73\pm 0.39\pm 0.22$
3550.7 | $24\pm 5$ | 1 | 62.23 | 18.15 | $2.0\pm 0.4\pm 0.6$ | $1.67\pm 0.17\pm 0.23$
3600.2 | $14\pm 4$ | 1 | 62.24 | 9.55 | $2.2\pm 0.6\pm 0.9$ | $1.78\pm 0.25\pm 0.35$
3650.0 | $36\pm 6$ | 4 | 61.20 | 48.82 | $1.1\pm 0.2\pm 0.1$ | $1.26\pm 0.11\pm 0.07$
3671.0 | $6\pm 2$ | 0 | 51.17 | 4.59 | $2.2\pm 0.9\pm 0.8$ | $1.84\pm 0.37\pm 0.33$

\begin{overpic}[width=216.81pt]{com_Xsection_log_3.eps} \put(60.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=216.81pt]{com_FF_log_3.eps}
\put(60.0,60.0){(b)} \end{overpic}

Figure 5: Comparison of (a) the Born cross section and (b) the effective FF
$|G|$ between this measurement and previous experiments, shown on a
logarithmic scale for invariant $p\bar{p}$ masses from 2.20 to 3.70
GeV/$c^{2}$.

Several sources of systematic uncertainties are considered in the measurement
of the Born cross sections and the corresponding effective FFs, including
those of tracking, PID, $E/p$ requirement, background estimation, theory
uncertainty from radiative corrections, FF model dependence and integrated
luminosity.

(a) Tracking and PID : The uncertainties of tracking and PID efficiencies for
proton/antiproton are investigated using almost background-free control
samples $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$ and
$\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi\rightarrow\pi^{+}\pi^{-}p\bar{p}$.
The differences of tracking and PID efficiencies between data and MC
simulation is $1.0\%$ per track, respectively, and they are taken as
systematic uncertainties. (b) $E/p$ requirement : The uncertainty of the $E/p$
requirement is also estimated using the $J/\psi\rightarrow
p\bar{p}\pi^{+}\pi^{-}$ control sample. The difference between data and MC in
efficiency is found to be 1.0% applying the same $E/p$ criteria on the proton
sample, and is taken as a systematic uncertainty. (c) Background estimation :
In the analysis, the background contamination is estimated by the MC samples.
An alternative method, 2-dimensional sidebands in the proton momentum versus
antiproton momentum space, is applied to estimate the background
contamination, and the difference is taken as the systematic uncertainty. The
proton/antiproton momentum sideband region is defined by
$6~{}\sigma_{p}<|p_{\text{mea}}-p_{\text{exp}}|<11~{}\sigma_{p}$, where
$p_{\text{exp}}$ and $\sigma_{p}$ are the expected momentum and resolution of
proton/antiproton at a given c.m. energy. (d) Radiative correction : In the
nominal results, the radiative correction factors are estimated with the
Conexc generator. An alternative generator, Phokhara, is used to evaluate the
theoretical calculation of the radiative correction factors, and the
differences in the resulting products $\varepsilon^{\prime}$ of detection
efficiency and radiative correction factor are taken as the systematic
uncertainty. (e) FFs model dependence : For those c.m. energies with measured
$|G_{E}/G_{M}|$ ratios, the uncertainties on the detection efficiencies are
estimated by varying the $|G_{E}/G_{M}|$ ratios with 1 standard deviation
measured in this analysis. These systematic uncertainties are found to be less
than 5.0%. For other c.m. energy points, whose $|G_{E}/G_{M}|$ ratios are
unknown, the uncertainties on the detection efficiencies are evaluated to be
half of the differences between the detection efficiencies with setting
$|G_{E}|=0$ or $|G_{M}|=0$, respectively, which give larger uncertainties
exceeding 10.0%. (f) Integrated luminosity : The integrated luminosity is
measured by analyzing large-angle Bhabha scattering process, and achieves 1.0%
in precision.

All systematic uncertainties are summarized in Table 3. The total systematic
uncertainty of the Born cross section is obtained by summing the individual
contributions in quadrature. The effective FF $|G|$ is proportional to the
square root of the Born cross section, and its systematic uncertainty is half
of that of the Born cross section.

Table 3: Summary of systematic uncertainties (in %) for the Born cross sections $\sigma_{\text{Born}}$ and the effective form factor $|G|$ measurements. $\sqrt{s}$ (MeV) |  Trk. | PID | $E/p$ | Bkg. | MC gen. | Model | Lum. |  Total ($\sigma_{\text{Born}}$) |  Total ($|G|$)
---|---|---|---|---|---|---|---|---|---
2232.4 | $2.0$ | $2.0$ | $1.0$ | $2.6$ | $0.4$ | $1.5$ | $1.0$ | $4.4$ | $2.2$
2400.0 | $2.0$ | $2.0$ | $1.0$ | $2.0$ | $1.8$ | $4.5$ | $1.0$ | $6.1$ | $3.1$
2800.0 | $2.0$ | $2.0$ | $1.0$ | $1.9$ | $7.5$ | $10.2$ | $1.0$ | $13.2$ | $6.6$
3050.0 | $2.0$ | $2.0$ | $1.0$ | $2.2$ | $0.9$ | $4.0$ | $1.0$ | $5.6$ | $2.8$
3060.0 | $2.0$ | $2.0$ | $1.0$ | $3.8$ | $0.1$ | $4.1$ | $1.0$ | $6.4$ | $3.2$
3080.0 | $2.0$ | $2.0$ | $1.0$ | $0.0$ | $0.1$ | $4.3$ | $1.0$ | $5.3$ | $2.7$
3400.0 | $2.0$ | $2.0$ | $1.0$ | $0.0$ | $7.8$ | $21.9$ | $1.0$ | $23.5$ | $11.8$
3500.0 | $2.0$ | $2.0$ | $1.0$ | $20.0$ | $7.0$ | $12.9$ | $1.0$ | $25.0$ | $12.5$
3550.7 | $2.0$ | $2.0$ | $1.0$ | $20.8$ | $9.0$ | $14.3$ | $1.0$ | $27.0$ | $13.5$
3600.2 | $2.0$ | $2.0$ | $1.0$ | $35.7$ | $4.3$ | $11.6$ | $1.0$ | $37.9$ | $18.9$
3650.0 | $2.0$ | $2.0$ | $1.0$ | $3.3$ | $0.9$ | $9.7$ | $1.0$ | $10.8$ | $5.4$
3671.0 | $2.0$ | $2.0$ | $1.0$ | $33.3$ | $0.7$ | $13.3$ | $1.0$ | $36.0$ | $18.0$

###  III.4 Extraction of the electromagnetic $|G_{E}/G_{M}|$ ratio

The distribution of the proton polar angle $\theta_{p}$ depends on the
electric and magnetic FFs. The Eq. 4 can be rewritten as:

$\displaystyle\begin{split}F(\cos\theta_{p})=&N_{\text{norm}}[1+\cos^{2}\theta_{p}+\\\
&\frac{4m_{p}^{2}}{s}R^{2}(1-\cos^{2}\theta_{p})],\end{split}$ (7)

where $R=|G_{E}/G_{M}|$ is the ratio of electric to magnetic FFs, and
$N_{\text{norm}}=\frac{2\pi\alpha^{2}\beta
L}{4s}[1.94+5.04\frac{m_{p}^{2}}{s}R^{2}]G_{M}(s)^{2}$ is the overall
normalization factor. Both $R$ and $N_{norm}$ ($G_{M}(s)$) can be extracted
directly by fitting the $\cos\theta_{p}$ distributions with Eq. 7. The polar
angular distributions $\cos\theta_{p}$ are shown in Fig. 6 for $\sqrt{s}$ =
2232.4 and 2400.0 MeV, as well as for a combined data sample with sub-data
samples at $\sqrt{s}$ = 3050.0, 3060.0 and 3080.0 MeV. The distributions are
corrected with the detection efficiencies in different $\cos\theta_{p}$ bins
which are evaluated by MC simulation samples. The distributions are fitted
with Eq. 7, and the fit results are also shown in Fig. 6. The fit results as
well as the corresponding qualities of fit, $\chi^{2}/ndf$, are summarized in
Table 4. The corresponding ratios $R=|G_{E}/G_{M}|$ are shown in Fig. 7, and
the results from the previous experiments are also presented on the same plot
for comparison.

\begin{overpic}[width=138.76157pt,angle={0}]{3-gegm223.eps}
\put(78.0,60.0){\footnotesize(a)}
\end{overpic}\begin{overpic}[width=138.76157pt,angle={0}]{3-gegm240.eps}
\put(70.0,58.0){(b)} \end{overpic}

\begin{overpic}[width=138.76157pt,angle={0}]{3-gegm308.eps}
\put(70.0,58.0){(c)} \end{overpic}

Figure 6:  Efficiency corrected distributions of $\cos\theta_{p}$ and fit results for data at c.m. energies (a) 2232.4, (b) 2400.0 MeV and (c) a combined sample with c.m. energy at 3050.0, 3060.0 and 3080.0 MeV. The dots with error bars represent data. The solid line (black) represents the overall fit result. The dot-dashed line (in red) shows the contribution of the magnetic FF and the dashed line (in blue) of the electric FF. Table 4: Summary of the ratio of electric to magnetic FFs $|G_{E}/G_{M}|$, magnetic FF $|G_{M}|$ by fitting on the distribution of $\cos{\theta_{p}}$ and _method of moments_ at different c.m. energies. For the method of fitting on $\cos\theta_{p}$, the statistical and systematic uncertainties are quoted for $|G_{E}/G_{M}|$ and $|G_{M}|$, and the fitting quality $\chi^{2}/n.d.o.f.$ is presented. Only statistical uncertainty is shown for the _method of moments_. $\sqrt{s}$ (MeV) | $~{}~{}~{}|G_{E}/G_{M}|~{}~{}~{}~{}~{}~{}~{}~{}~{}$ |  $~{}~{}~{}|G_{M}|$ $(\times 10^{-2})~{}~{}~{}~{}~{}~{}~{}~{}$ | $\chi^{2}/ndf$
---|---|---|---
| Fit on $\cos\theta_{p}$
2232.4 | $0.87\pm 0.24\pm 0.05$ | $18.42\pm 5.09\pm 0.98$ | 1.04
2400.0 | $0.91\pm 0.38\pm 0.12$ | $11.30\pm 4.73\pm 1.53$ | 0.74
(3050.0, 3080.0) | $0.95\pm 0.45\pm 0.21$ | $3.61\pm 1.71\pm 0.82$ | 0.61
| _method of moments_
2232.4 | $0.83\pm 0.24$ | $18.60\pm 5.38$ | -
2400.0 | $0.85\pm 0.37$ | $11.52\pm 5.01$ | -
(3050.0, 3080.0) | $0.88\pm 0.46$ | $3.34\pm 1.72$ | -
\begin{overpic}[width=289.07999pt]{com-GeGm-3.eps} \end{overpic} Figure 7: The
measured ratio of electric to magnetic FFs $|G_{E}/G_{M}|$ at different c.m.
energy from BESIII (filled circles), BaBar at SLAC (open crosses) and PS170 at
LEAR/CERN (open circles).

The systematic uncertainties of the $|G_{E}/G_{M}|$ ratio and $|G_{M}|$
measurements are mainly from background contamination, the difference of
detection efficiency between data and MC, and the different fit range of
$\cos\theta_{p}$. The small background contamination as listed in Table 2 is
not considered in the nominal fit. An alternative fit with background
subtraction is performed, where the background contamination is estimated by
the two-dimension sideband method, and the differences are considered as the
systematic uncertainties related to background contamination. In the fit, the
detection efficiency is evaluated with the MC simulation. An alternative fit
with corrected detection efficiency which takes into account the differences
in tracking, PID and $E/p$ selection efficiency between data and MC is
performed, and the results in differences are taken as the systematic
uncertainties. Fits with ranges $[-0.8,0.6]$ and $[-0.7,0.7]$ in
$\cos\theta_{p}$ are performed, and the largest differences to the nominal
values are taken as the uncertainties. Table 5 summarizes the related
systematic uncertainties for the $|G_{E}/G_{M}|$ and $|G_{M}|$ measurements.
The overall systematic uncertainties are obtained by summing all the three
systematic uncertainties in quadrature.

Table 5: Summary of systematic uncertainties (in %) in the $|G_{E}/G_{M}|$ ratio and $|G_{M}|$ measurement. Source | $|G_{E}/G_{M}|$ | $|G_{M}|$
---|---|---
$\sqrt{s}$ (MeV) | $~{}2232.4~{}$ | $~{}2400.0~{}$ | $~{}(3050.0,3080.0)~{}$ | $~{}2232.4~{}$ | $~{}2400.0~{}$ | $~{}(3050.0,3080.0)~{}$
Background contamination | 1.1 | 7.7 | 3.2 | 1.4 | 7.7 | 3.2
Detection efficiency | 2.3 | 1.1 | 4.2 | 2.3 | 1.1 | 4.2
Fit range | 4.6 | 11.0 | 22.1 | 4.6 | 11.0 | 22.1
Total | 5.3 | 13.5 | 22.7 | 5.3 | 13.5 | 22.7

As a crosscheck, a different method, named _method of moments_ (MM) mm , is
applied to extract the $|G_{E}/G_{M}|$ ratio, where the weighted factors in
front of $G_{E}$ and $G_{M}$ may be used to evaluate the electric or magnetic
FF from moments of the angular distribution directly. The expectation value,
or moment, of $\cos^{2}\theta_{p}$, for a distribution following Eq. 7 is
given by:

$\displaystyle\begin{split}\left<\cos^{2}\theta_{p}\right>&=\frac{1}{N_{\text{norm}}}\int\frac{2\pi\alpha^{2}\beta
C}{4s}\cos^{2}\theta_{p}[(1+\cos^{2}\theta_{p})|G_{M}|^{2}\\\
&+\frac{4m_{p}^{2}}{s}(1-\cos^{2}\theta_{p})|R^{2}|G_{M}|^{2}]d\cos\theta_{p}.\end{split}$
(8)

Calculating this within the interval $[-0.8,0.8]$ where the acceptance is non-
zero and smooth, gives for the acceptance correction:

$R=\sqrt{\frac{s}{4m^{2}_{p}}\frac{\left<\cos^{2}\theta_{p}\right>-0.243}{0.108-0.648\left<\cos^{2}\theta_{p}\right>}},$
(9)

and the corresponding uncertainty:

$\sigma_{R}=\frac{0.0741}{R(0.167-\left<\cos^{2}\theta\right>)^{2}}\frac{s}{4m^{2}_{p}}\sigma_{\left<\cos^{2}\theta_{p}\right>},$
(10)

where $\sigma_{\left<\cos^{2}\theta_{p}\right>}$ is given by

$\displaystyle\begin{split}&\sigma_{\left<\cos^{2}\theta_{p}\right>}=\sqrt{\frac{1}{N-1}\left[\left<\cos^{4}\theta_{p}\right>-\left<\cos^{2}\theta_{p}\right>^{2}\right]}.\end{split}$
(11)

In the analysis of experimental data, $\left<\cos^{2}\theta_{p}\right>$ and
$\left<\cos^{4}\theta_{p}\right>$ are the average of $\cos^{2}\theta_{p}$ and
$\cos^{4}\theta_{p}$ which are calculated taking the detection efficiency
event-by-event into account:

$\left<\cos^{2,4}\theta_{p}\right>=\overline{\cos^{2,4}\theta_{p}}=\frac{1}{N}\sum_{i=1}^{N}\cos^{2,4}\theta_{pi}/\varepsilon_{i},$
(12)

where $\varepsilon_{i}$ is the detection efficiency with the $i$th event’s
kinematics as estimated by the MC simulation.

The extracted $|G_{E}/G_{M}|$ ratios and $|G_{M}|$ by MM at different c.m.
energies are also shown in Table 4, where $|G_{M}|$ is calculated by
$N_{norm}$ in Eq. 7 using the measured $|G_{E}/G_{M}|$ ratio. The results are
well consistent with those extracted by fitting the distribution of polar
angle $\cos\theta_{p}$, and the statistical uncertainty is found to be
comparable between the two different methods due to the same number of events.

##  IV SUMMARY

Using data at 12 c.m. energies between 2232.4 MeV and 3671.0 MeV collected
with the BESIII detector, we measure the Born cross sections of
$e^{+}e^{-}\rightarrow p\bar{p}$ and extract the corresponding effective FF
$|G|$ under the assumption $|G_{E}|=|G_{M}|$. The results are in good
agreement with previous experiments. The precision of the Born cross section
with $\sqrt{s}\leq 3.08$ GeV is between $6.0\%$ and $18.9\%$ which is much
improved comparing with the best precision of previous results (between
$9.4\%$ and $26.9\%$) from BaBar experiment babar2 ; and the precision is
comparable with those of previous results at $\sqrt{s}>$ 3.08 GeV. The
$|G_{E}/G_{M}|$ ratios and $|G_{M}|$ are extracted at the c.m. energies
$\sqrt{s}=2232.4$ and 2400.0 MeV and a combined data sample with c.m. energy
of 3050.0, 3060.0 and 3080.0 MeV, with comparable uncertainties to previous
experiments. The measured $|G_{E}/G_{M}|$ ratios are close to unity which are
consistent with those of the BaBar experiment in the same $q^{2}$ region. At
present, the precision of the $|G_{E}/G_{M}|$ ratio is dominated by
statistics. A MC simulation study shows that the precision can achieve 10% or
3.0% if we have a factor of 5 or 50 times higher integrated luminosity. In the
near future, a new scan at BEPCII with c.m. energy ranging between 2.0 GeV and
3.1 GeV is foreseen to improve the precision of the measurement on
$|G_{E}/G_{M}|$ ratio in a wide range.

##  V ACKNOWLEDGMENTS

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. We are grateful to Henryk Czyz for providing
us the new Phokhara generator with the scan mode. This work is supported in
part by National Key Basic Research Program of China under Contract No.
2015CB856700; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 10935007, 11121092, 11125525, 11235011, 11322544, 11335008,
11375170, 11275189, 11078030, 11475164, 11005109, 11475169, 11425524; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under
Contracts Nos. 11079008, 11179007, U1232201, U1332201; CAS under Contracts
Nos. KJCX2-YW-N29, KJCX2-YW-N45; 100 Talents Program of CAS; INPAC and
Shanghai Key Laboratory for Particle Physics and Cosmology; German Research
Foundation DFG under Contract No. Collaborative Research Center CRC-1044;
Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of
Turkey under Contract No. DPT2006K-120470; Russian Foundation for Basic
Research under Contract No. 14-07-91152; U. S. Department of Energy under
Contracts Nos. DE-FG02-04ER41291, DE-FG02-05ER41374, DE-FG02-94ER40823,
DESC0010118; U.S. National Science Foundation; University of Groningen (RuG)
and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU
Program of National Research Foundation of Korea under Contract No.
R32-2008-000-10155-0.

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