Measurement of proton electromagnetic form factors in e + e − →p p ¯ in the energy region 2.00-3.08 GeV
M. Ablikim, M. N. Achasov, P. Adlarson, S. Ahmed, M. Albrecht, M. Alekseev, A. Amoroso, F. F. An, Q. An, Y. Bai, O. Bakina, R. Baldini Ferroli, Y. Ban, K. Begzsuren, J. V. Bennett, N. Berger, M. Bertani, D. Bettoni, F. Bianchi, J Biernat, J. Bloms, I. Boyko, R. A. Briere, H. Cai, X. Cai, A. Calcaterra, G. F. Cao, N. Cao, S. A. Cetin, J. Chai, J. F. Chang, W. L. Chang, G. Chelkov, D. Y. Chen, G. Chen, H. S. Chen, J. C. Chen, M. L. Chen, S. J. Chen, Y. B. Chen, W. Cheng, G. Cibinetto, F. Cossio, X. F. Cui, H. L. Dai, J. P. Dai, X. C. Dai, A. Dbeyssi, D. Dedovich, Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis, F. De Mori, Y. Ding, C. Dong, J. Dong, L. Y. Dong, M. Y. Dong, Z. L. Dou, S. X. Du, J. Z. Fan, J. Fang, S. S. Fang, Y. Fang, R. Farinelli, L. Fava, F. Feldbauer, G. Felici, C. Q. Feng, M. Fritsch, C. D. Fu, Y. Fu, Q. Gao, X. L. Gao, Y. Gao, Y. Gao, Y. G. Gao, Z. Gao, B. Garillon, I. Garzia, E. M. Gersabeck, A. Gilman, K. Goetzen, L. Gong, W. X. Gong, W. Gradl, M. Greco, L. M. Gu, M. H. Gu, Y. T. Gu, A. Q. Guo, L. B. Guo, R. P. Guo, Y. P. Guo, A. Guskov, S. Han, X. Q. Hao, F. A. Harris, K. L. He, et al. (379 additional authors not shown)
MMeasurement of Proton Electromagnetic Form Factors in e + e − → p ¯ p in the Energy Region 2.00 – M. Ablikim, M. N. Achasov,
P. Adlarson, S. Ahmed, M. Albrecht, M. Alekseev,
A. Amoroso,
F. F. An, Q. An,
Anita, Y. Bai, O. Bakina, R. Baldini Ferroli,
I. Balossino,
Y. Ban,
K. Begzsuren, J. V. Bennett, N. Berger, M. Bertani,
D. Bettoni,
F. Bianchi,
J. Biernat, J. Bloms, I. Boyko, R. A. Briere, H. Cai, X. Cai,
A. Calcaterra,
G. F. Cao,
N. Cao,
S. A. Cetin,
J. Chai,
J. F. Chang,
W. L. Chang,
G. Chelkov,
D. Y. Chen, G. Chen, H. S. Chen,
J. Chen, M. L. Chen,
S. J. Chen, X. R. Chen, Y. B. Chen,
W. Cheng,
G. Cibinetto,
F. Cossio,
X. F. Cui, H. L. Dai,
J. P. Dai,
X. C. Dai,
A. Dbeyssi, D. Dedovich, Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis,
F. De Mori,
Y. Ding, C. Dong, J. Dong,
L. Y. Dong,
M. Y. Dong,
S. X. Du, J. Fang,
S. S. Fang,
Y. Fang, R. Farinelli,
L. Fava,
F. Feldbauer, G. Felici,
C. Q. Feng,
M. Fritsch, C. D. Fu, Y. Fu, Q. Gao, Y. Gao, Y. Gao, Y. G. Gao, B. Garillon, I. Garzia,
E. M. Gersabeck, A. Gilman, K. Goetzen, L. Gong, W. X. Gong,
W. Gradl, M. Greco,
L. M. Gu, M. H. Gu,
S. Gu, Y. T. Gu, C. Y. Guan,
A. Q. Guo, L. B. Guo, R. P. Guo, Y. P. Guo, A. Guskov, S. Han, T. Z. Han,
X. Q. Hao, F. A. Harris, K. L. He,
F. H. Heinsius, T. Held, Y. K. Heng,
M. Himmelreich,
Y. R. Hou, Z. L. Hou, H. M. Hu,
J. F. Hu,
T. Hu,
Y. Hu, G. S. Huang,
J. S. Huang, L. Q. Huang, X. T. Huang, N. Huesken, T. Hussain, W. Ikegami Andersson, W. Imoehl, M. Irshad,
Q. Ji, Q. P. Ji, X. B. Ji,
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H. L. Jiang, X. S. Jiang,
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S. Jin, Y. Jin, T. Johansson, N. Kalantar-Nayestanaki, X. S. Kang, R. Kappert, M. Kavatsyuk, B. C. Ke,
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G. Li, H. B. Li,
H. J. Li,
J. C. Li, Ke Li, L. K. Li, Lei Li, P. L. Li,
P. R. Li, W. D. Li,
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Huanhuan Liu, Huihui Liu, J. B. Liu,
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K. Liu, K. Y. Liu, Ke Liu, L. Liu,
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T. Liu,
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Zhiqing Liu, Y. F. Long,
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H. Qi,
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T. Weber, D. H. Wei, P. Weidenkaff, F. Weidner, H. W. Wen, S. P. Wen, U. Wiedner, G. Wilkinson, M. Wolke, J. F. Wu,
L. H. Wu, L. J. Wu,
Z. Wu,
L. Xia ,
Y. Xia, S. Y. Xiao, Y. J. Xiao,
Z. J. Xiao, Y. G. Xie,
Y. H. Xie, T. Y. Xing,
X. A. Xiong,
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B. S. Zou, and J. H. Zou (BESIII Collaboration) Institute of High Energy Physics, Beijing 100049, People ’ s Republic of China Beihang University, Beijing 100191, People ’ s Republic of China Beijing Institute of Petrochemical Technology, Beijing 102617, People ’ s Republic of China Bochum Ruhr-University, D-44780 Bochum, Germany Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Central China Normal University, Wuhan 430079, People ’ s Republic of China China Center of Advanced Science and Technology, Beijing 100190, People ’ s Republic of China COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan Fudan University, Shanghai 200443, People ’ s Republic of China G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany Guangxi Normal University, Guilin 541004, People ’ s Republic of China Guangxi University, Nanning 530004, People ’ s Republic of China Hangzhou Normal University, Hangzhou 310036, People ’ s Republic of China Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Henan Normal University, Xinxiang 453007, People ’ s Republic of China Henan University of Science and Technology, Luoyang 471003, People ’ s Republic of China Huangshan College, Huangshan 245000, People ’ s Republic of China Hunan Normal University, Changsha 410081, People ’ s Republic of China Hunan University, Changsha 410082, People ’ s Republic of China Indian Institute of Technology Madras, Chennai 600036, India Indiana University, Bloomington, Indiana 47405, USA
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University of Ferrara, I-44122 Ferrara, Italy Institute of Modern Physics, Lanzhou 730000, People ’ s Republic of China Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330, Mongolia Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany KVI-CART, University of Groningen, NL-9747 AA Groningen, Netherlands Lanzhou University, Lanzhou 730000, People ’ s Republic of China Liaoning Normal University, Dalian 116029, People ’ s Republic of China Liaoning University, Shenyang 110036, People ’ s Republic of China Nanjing Normal University, Nanjing 210023, People ’ s Republic of China Nanjing University, Nanjing 210093, People ’ s Republic of China Nankai University, Tianjin 300071, People ’ s Republic of China Peking University, Beijing 100871, People ’ s Republic of China Qufu Normal University, Qufu 273165, People ’ s Republic of China Shandong Normal University, Jinan 250014, People ’ s Republic of China Shandong University, Jinan 250100, People ’ s Republic of China Shanghai Jiao Tong University, Shanghai 200240, People ’ s Republic of China Shanxi Normal University, Linfen 041004, People ’ s Republic of China PHYSICAL REVIEW LETTERS124,
University of Ferrara, I-44122 Ferrara, Italy Institute of Modern Physics, Lanzhou 730000, People ’ s Republic of China Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330, Mongolia Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany KVI-CART, University of Groningen, NL-9747 AA Groningen, Netherlands Lanzhou University, Lanzhou 730000, People ’ s Republic of China Liaoning Normal University, Dalian 116029, People ’ s Republic of China Liaoning University, Shenyang 110036, People ’ s Republic of China Nanjing Normal University, Nanjing 210023, People ’ s Republic of China Nanjing University, Nanjing 210093, People ’ s Republic of China Nankai University, Tianjin 300071, People ’ s Republic of China Peking University, Beijing 100871, People ’ s Republic of China Qufu Normal University, Qufu 273165, People ’ s Republic of China Shandong Normal University, Jinan 250014, People ’ s Republic of China Shandong University, Jinan 250100, People ’ s Republic of China Shanghai Jiao Tong University, Shanghai 200240, People ’ s Republic of China Shanxi Normal University, Linfen 041004, People ’ s Republic of China PHYSICAL REVIEW LETTERS124, Shanxi University, Taiyuan 030006, People ’ s Republic of China Sichuan University, Chengdu 610064, People ’ s Republic of China Soochow University, Suzhou 215006, People ’ s Republic of China Southeast University, Nanjing 211100, People ’ s Republic of China State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People ’ s Republic of China Sun Yat-Sen University, Guangzhou 510275, People ’ s Republic of China Tsinghua University, Beijing 100084, People ’ s Republic of China Ankara University, 06100 Tandogan, Ankara, Turkey
Istanbul Bilgi University, 34060 Eyup, Istanbul, Turkey
Uludag University, 16059 Bursa, Turkey
Near East University, Nicosia, North Cyprus, Mersin 10, Turkey University of Chinese Academy of Sciences, Beijing 100049, People ’ s Republic of China University of Hawaii, Honolulu, Hawaii 96822, USA University of Jinan, Jinan 250022, People ’ s Republic of China University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom University of Minnesota, Minneapolis, Minnesota 55455, USA University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany University of Oxford, Keble Road, Oxford OX13RH, United Kingdom University of Science and Technology Liaoning, Anshan 114051, People ’ s Republic of China University of Science and Technology of China, Hefei 230026, People ’ s Republic of China University of South China, Hengyang 421001, People ’ s Republic of China University of the Punjab, Lahore-54590, Pakistan
University of Turin, I-10125 Turin, Italy
University of Eastern Piedmont, I-15121 Alessandria, Italy
INFN, I-10125 Turin, Italy Uppsala University, Box 516, SE-75120 Uppsala, Sweden Wuhan University, Wuhan 430072, People ’ s Republic of China Xinyang Normal University, Xinyang 464000, People ’ s Republic of China Zhejiang University, Hangzhou 310027, People ’ s Republic of China Zhengzhou University, Zhengzhou 450001, People ’ s Republic of China (Received 28 May 2019; revised manuscript received 19 September 2019; published 28 January 2020)The process of e þ e − → p ¯ p is studied at 22 center-of-mass energy points ( ffiffiffi s p ) from 2.00 to 3.08 GeV,exploiting . pb − of data collected with the BESIII detector operating at the BEPCII collider. The Borncross section ( σ p ¯ p ) of e þ e − → p ¯ p is measured with the energy-scan technique and it is found to beconsistent with previously published data, but with much improved accuracy. In addition, the electro-magnetic form-factor ratio ( j G E =G M j ) and the value of the effective ( j G eff j ), electric ( j G E j ), and magnetic( j G M j ) form factors are measured by studying the helicity angle of the proton at 16 center-of-mass energypoints. j G E =G M j and j G M j are determined with high accuracy, providing uncertainties comparable to datain the spacelike region, and j G E j is measured for the first time. We reach unprecedented accuracy, andprecision results in the timelike region provide information to improve our understanding of the protoninner structure and to test theoretical models which depend on nonperturbative quantum chromodynamics. DOI: 10.1103/PhysRevLett.124.042001
Despite the proton being one of the fundamental buildingblocks of atomic matter, its internal structure and dynamicsare not well understood. Improving knowledge of theseproperties in terms of the proton ’ s quark and gluonicdegrees of freedom is one of the most challenging problems of modern nuclear physics. In addition, unsolved problemssuch as the proton-radius puzzle have recently attractedmuch attention [1].The electric and magnetic form factors (FFs), G E ð q Þ and G M ð q Þ , are fundamental quantities that can providevaluable insight into both the structure and dynamics ofnucleons. FFs enter explicitly in the coupling of a virtualphoton with the hadron electromagnetic current, andmeasurements can be directly compared to hadron models]1 ] giving, thereby, constraints in the description of theinternal structure of hadrons. In the spacelike (SL) kin-ematic region (momentum transfer q < ), FFs have been Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article ’ s title, journal citation,and DOI. Funded by SCOAP . PHYSICAL REVIEW LETTERS124,
Despite the proton being one of the fundamental buildingblocks of atomic matter, its internal structure and dynamicsare not well understood. Improving knowledge of theseproperties in terms of the proton ’ s quark and gluonicdegrees of freedom is one of the most challenging problems of modern nuclear physics. In addition, unsolved problemssuch as the proton-radius puzzle have recently attractedmuch attention [1].The electric and magnetic form factors (FFs), G E ð q Þ and G M ð q Þ , are fundamental quantities that can providevaluable insight into both the structure and dynamics ofnucleons. FFs enter explicitly in the coupling of a virtualphoton with the hadron electromagnetic current, andmeasurements can be directly compared to hadron models]1 ] giving, thereby, constraints in the description of theinternal structure of hadrons. In the spacelike (SL) kin-ematic region (momentum transfer q < ), FFs have been Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article ’ s title, journal citation,and DOI. Funded by SCOAP . PHYSICAL REVIEW LETTERS124, q > ), measured inannihilation reactions [Fig. 1(b)]. In most cases thesemeasurements only extracted the effective FF ( G eff ) orthe ratio of G E and G M with uncertainties above 10%. Sincethe FFs in the SL and TL regions are connected viaanalyticity, precise knowledge of them in the TL regioncan help to solve problems in the SL region, such as thediscrepancy found between the ratio G E =G M determinedvia Rosenbluth separation and that found by experimentsusing polarized electron beams or targets [2].The moduli of the FFs can be determined from the studyof the angular distribution of the annihilation process [3],while the relative phase between the two FFs can bedetermined by measuring the polarization of the outgoingbaryons.The Born differential cross section as a function of the e þ e − center-of-mass (c.m.) energy squared s reads [3] d σ p ¯ p ð s Þ d Ω ¼ α β C s (cid:3) j G M ð s Þj ð þ cos θ Þþ m p s j G E ð s Þj sin θ (cid:4) ; ð Þ where G E and G M are the Sachs FFs, θ is the polar angle ofthe proton in the e þ e − c.m. frame, m p is the proton massand β ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − m p =s q . The Coulomb enhancement factor, C , accounts for the electromagnetic interaction between theoutgoing baryons. This factor is usually considered as afinal-state interaction and it is C ¼ y= ð − e − y Þ for point-like fermions with y ¼ πα ffiffiffiffiffiffiffiffiffiffiffiffiffi − β p = β . Since the Coulombinteraction is long range, a pointlike correction is assumedwhen the two charged baryons are far apart.In the TL region, the proton FFs can be accessed by threereactions: e þ e − → p ¯ p [4 – p ¯ p → e þ e − [11 – e þ e − → p ¯ p γ ISR [14,15]. Whilethere are many, generally consistent, measurementsconcerning the total σ p ¯ p , there are few and inconsistentdata on the ratio j G E =G M j , mostly from PS170 [11] and BABAR [14]. So far only two experiments [8,11] have beenable to extract the value of j G M j , which together with theknowledge of j G E =G M j allows j G E j to be determined.Precise measurements of FFs in the TL region may alsobe helpful for improved theoretical estimates of the protonradius [16,17]. From threshold energies to 3 GeV, theamplitude for the process is the sum of a leading term dueto a bare formation process taking place on a time scale = ffiffiffiffiffi q p , and a relatively small perturbation associated withrescattering processes taking place on a longer time scale[16]. The combination of these effects is expected to lead tointeresting phenomenology, in particular the superpositionof small oscillations on an otherwise smooth dipoleparameterization of the G eff .In this Letter, we present a study of the process e þ e − → p ¯ p at c.m. energies ffiffiffi s p ¼ . – . GeV, including ameasurement of the Born cross section ( σ p ¯ p ), the electro-magnetic FF ratio ( j G E =G M j ), the absolute value of theeffective FF ( j G eff j ,) the magnetic FF ( j G M j ) as well as, forthe first time, the electric FF ( j G E j ) of the proton using theenergy-scan technique. The precision of our measurementis greatly improved with respect to that of previousexperiments. Our results in the TL region have unprec-edented precision with uncertainties comparable to FFmeasurements in the SL region.The collision data were taken with the BESIII spec-trometer at BEPCII. A detailed description of the detectorand its performance can be found in Ref. [18]. The detectorresponse, including the interaction of secondary particleswith the detector material, is simulated using a GEANT e þ e − → p ¯ p events per energy point generated with CONEXC [20] are used for the efficiency determinationand to calculate the correction factors for radiation up tonext-to-leading order (NLO), as well as those for thevacuum polarization (VP). MC samples of QED back-ground processes generated with
BABAYAGA [21] andinclusive hadronic events generated with
CONEXC [20]are used for background studies.The final state of the process of interest is characterizedby one proton and one antiproton. Hence, selected eventsmust have exactly two charged tracks with opposite charge.A vertex fit is performed on both tracks under thehypothesis that the two particles in the final state are aproton and an antiproton to constrain them to one commonvertex. A fit quality of χ < is required to selectcandidate events. The opening angle between the protonand antiproton in the rest frame of the e þ e − c.m. system isrequired to be > ° at 2.00 GeV and 2.05 GeV, > ° at2.1000 to 2.3094 GeV, and > ° at 2.3864 to3.0800 GeV. This condition ensures a back-to-back sig-nature between the tracks. Cosmic-ray background isrejected by requiring j T trk − T trk j < ns, where T trk and T trk are the measurements from the time-of-flight(TOF) system for each track. For ffiffiffi s p between 2.000 and2.396 GeV, events are selected even if one of the two tracks (a) (b) FIG. 1. Lowest-order Feynman diagrams for elastic electron-baryon scattering e − B → e − B (a), and for the annihilationprocess e − e þ → B ¯ B (b). B is a baryon. PHYSICAL REVIEW LETTERS124,
CONEXC [20]are used for background studies.The final state of the process of interest is characterizedby one proton and one antiproton. Hence, selected eventsmust have exactly two charged tracks with opposite charge.A vertex fit is performed on both tracks under thehypothesis that the two particles in the final state are aproton and an antiproton to constrain them to one commonvertex. A fit quality of χ < is required to selectcandidate events. The opening angle between the protonand antiproton in the rest frame of the e þ e − c.m. system isrequired to be > ° at 2.00 GeV and 2.05 GeV, > ° at2.1000 to 2.3094 GeV, and > ° at 2.3864 to3.0800 GeV. This condition ensures a back-to-back sig-nature between the tracks. Cosmic-ray background isrejected by requiring j T trk − T trk j < ns, where T trk and T trk are the measurements from the time-of-flight(TOF) system for each track. For ffiffiffi s p between 2.000 and2.396 GeV, events are selected even if one of the two tracks (a) (b) FIG. 1. Lowest-order Feynman diagrams for elastic electron-baryon scattering e − B → e − B (a), and for the annihilationprocess e − e þ → B ¯ B (b). B is a baryon. PHYSICAL REVIEW LETTERS124, p mean ,determined from a fit to the momentum distribution afterbeing boosted into the e þ e − c.m. system, namely ð p mean − σ Þ < p < ð p mean þ σ Þ , where the spread σ (standard deviation) is taken from the fit.Particle identification (PID) is performed using the TOFand the dE=dx measurement from the main drift chamber(MDC). At c.m. energies above 2.150 GeV this informationis used to construct a probability for each track to conformto a particular (pion, kaon, electron, or proton) particlehypothesis to select the proton and antiproton candidates.For events at lower c.m. energies, the selection is madebased on the normalized pulse height of the raw dE=dx information. To remove Bhabha events, a requirement on E=p , defined as the ratio between the energy deposited bythe track in the electromagnetic calorimeter (EMC) and itsmomentum measured in the MDC, is imposed for energypoints above 2.150 GeV. Possible contamination from QEDprocesses and hadronic final states are estimated to be lessthan 0.5% from studies performed on appropriate MCsamples, and are neglected in the subsequent analysis.With the number of events N obs selected, the crosssection σ p ¯ p of the process e þ e − → p ¯ p and j G eff j of theproton can be calculated with σ p ¯ p ð s Þ ¼ N obs L · ϵ · ð þ δ Þ ; ð Þ j G eff ð s Þj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ p ¯ p πα β C s ð þ m p s Þ s ; ð Þ where the efficiency ϵ and the correction factor ð þ δ Þ ¼ σ obs = σ Born are determined with MC simulations. Here, σ obs is the cross section including NLO radiation and VPcorrections, and σ Born is the born cross section. Resultsfor the σ p ¯ p and G eff measurement are summarized inTable I.The FFs j G E j and j G M j , or equivalently their ratio j G E =G M j and j G M j , can be determined from a fit to theproton angular distribution for energy points with asufficiently high number of selected candidates. This isthe case for 15 out of 22 energy points, as well as acombined sample of the individual data sets taken at c.m.energy points of 2.950, 2.981, 3.000, and 3.020 GeV with aluminosity weighted average energy of 2.988 GeV. Therange of the angular analysis is limited to cos θ from − . to 0.8, because of the lack of efficiency in the gap betweenthe barrel and end cap regions of the TOF system and EMC.The formula used to fit the proton angular distribution,deduced from Eqs. (1) and (2), can be expressed as dN ϵ ð þ δ Þ × d cos θ ¼ L πα β C s j G M j (cid:3) ð þ cos θ Þþ m p s (cid:5)(cid:5)(cid:5)(cid:5) G E G M (cid:5)(cid:5)(cid:5)(cid:5) ð − cos θ Þ (cid:4) ; ð Þ TABLE I. The integrated luminosity, the number of p ¯ p events, the Born cross section σ p ¯ p , j G E =G M j , j G eff j , j G E j , and j G M j . ffiffiffi s p [GeV] L ½ pb − (cid:2) N obs σ p ¯ p ½ pb (cid:2) j G eff j½ − (cid:2) j G E =G M j j G E j½ − (cid:2) j G M j½ − (cid:2) . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
40 1 . (cid:3) . (cid:3) .
03 33 . (cid:3) . (cid:3) .
31 24 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
39 1 . (cid:3) . (cid:3) .
04 29 . (cid:3) . (cid:3) .
40 23 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
36 1 . (cid:3) . (cid:3) .
02 28 . (cid:3) . (cid:3) .
31 22 . (cid:3) . (cid:3) . (cid:3) . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
34 1 . (cid:3) . (cid:3) .
01 25 . (cid:3) . (cid:3) .
18 21 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
32 1 . (cid:3) . (cid:3) .
06 28 . (cid:3) . (cid:3) .
46 17 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
29 1 . (cid:3) . (cid:3) .
02 22 . (cid:3) . (cid:3) .
28 18 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
27 1 . (cid:3) . (cid:3) .
02 18 . (cid:3) . (cid:3) .
28 17 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
24 0 . (cid:3) . (cid:3) .
03 14 . (cid:3) . (cid:3) .
42 16 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
21 0 . (cid:3) . (cid:3) .
02 6 . (cid:3) . (cid:3) .
25 11 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
15 0 . (cid:3) . (cid:3) .
02 5 . (cid:3) . (cid:3) .
19 10 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
15 0 . (cid:3) . (cid:3) .
02 7 . (cid:3) . (cid:3) .
21 10 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
09 0 . (cid:3) . (cid:3) .
05 5 . (cid:3) . (cid:3) .
24 5 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
10 0 . (cid:3) . (cid:3) .
04 5 . (cid:3) . (cid:3) .
21 5 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:3) . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
06 0 . (cid:3) . (cid:3) .
03 2 . (cid:3) . (cid:3) .
11 4 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
07 0 . (cid:3) . (cid:3) .
06 3 . (cid:3) . (cid:3) .
17 3 . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . . (cid:3) . (cid:3) . . (cid:3) . (cid:3) . (cid:3) . (cid:3) . (cid:3) . . (cid:3) . (cid:3) .
05 0 . (cid:3) . (cid:3) .
04 1 . (cid:3) . (cid:3) .
12 3 . (cid:3) . (cid:3) . PHYSICAL REVIEW LETTERS124,
12 3 . (cid:3) . (cid:3) . PHYSICAL REVIEW LETTERS124, ϵ ð cos θ Þ is the angular-dependent efficiencyobtained from MC simulations. The correction factor, ð þ δ Þð cos θ Þ , is calculated by dividing the cos θ distri-bution of a MC sample generated with radiation up to NLOand VP corrections by the distribution of a samplegenerated with the Born process alone. A control sampleof e þ e − → p ¯ p π þ π − events is studied to determine correc-tion factors for discrepancies between data and MCsimulation in the angular-dependent efficiency.After applying these corrections, the j cos θ j distribution isfitted with Eq. (4). The results at 2.125 GeVand 2.396 GeVareshown in Fig. 2, while the results for all energy points aresummarized in Table I. Fits to the j cos θ j distributions as wellas ϵ ð þ δ Þð cos θ Þ distributions for all energy points can befound in the Supplemental Material [22].The model used in the MC simulation takes as input σ p ¯ p ð s Þ and j G E ð s Þ =G M ð s Þj . Therefore, the correctionfactors, and hence the measurements themselves, have asignificant dependence on these inputs. For this reason, thecomplete analysis is performed in an iterative manner,where the obtained results are fed back into the MCsimulation. After three iterations, the results for σ p ¯ p ð s Þ and j G E ð s Þ =G M ð s Þj are stable to within 1%.Several sources of systematic uncertainties are consideredin the determination of σ p ¯ p . The uncertainty associated withthe knowledge of the reconstruction efficiency of the twocharged tracks, as well as from the PID efficiency and the E=p selection criteria, are studied with the e þ e − → p ¯ p π þ π − control sample. The difference of the efficiencymeasured in data and MC simulation is assigned as theuncertainty, and it is found to be 1.0% for both tracking andPID, and 0.2% for the E=p selection. The uncertainties dueto the selection based on the TOF difference between thetracks, the angle between the tracks, and the momentumwindow are studied by varying the selection criteria. Theuncertainty associated with the residual background con-tamination is estimated by comparing the populations of dataand MC simulation in a momentum window of the same sizeas the signal region, but separated by σ . The uncertaintyfrom the luminosity measurement is found to be on a 1.0%level from Ref. [23]. The uncertainty due to the iterative MC-tuning procedure is assigned to be the difference between thenominal result and the result from the second-iteration step. To assess the size of any bias from the choice of the used FFmodel in the MC simulations, we use the model from PHOKHARA [24] to generate an alternative set of MC events.The difference in the final result obtained with this newmodel and the default one is taken as the uncertainty.Many of the uncertainties in the j G E =G M j measurementare assigned with the same method as used in the cross-section analysis. This is true for all selection requirements,the uncertainties associated with the background, theiterative fit procedure, and the model used in the MCsimulation. To account for any imperfections due toasymmetries between the fit model and the observedangular distributions, we fit the cos θ distributions insteadof the j cos θ j ones and assign the difference as anuncertainty. The uncertainty from the luminosity measure-ment is taken as an independent systematic component for j G M j , again taken from Ref. [23]. The total systematicuncertainties on j G E =G M j range from 0.93% to 7.40%,while the total systematic uncertainties on j G M j range from0.60% to 2.10%.We study the energy dependence of σ p ¯ p by fitting theexpression σ p ¯ p ð s Þ ¼ e a π α s ½ − e − πα s ð s Þ = β ð s Þ (cid:2)½ þð ffiffi s p − mpa Þ a (cid:2) ; ffiffiffi s p ≤ . GeV ; πα β ð s Þ C ½ þð mp ffiffi s p Þ (cid:2) e a s ½ ln ð ffiffi s p a Þþ π (cid:2) ; ffiffiffi s p > . GeV ; ð Þ where α s ð s Þ is the strong coupling constant and α is theelectromagnetic constant. The running coupling constant α s ð s Þ is parameterized as follows: α s ð s Þ ¼ (cid:3) α s ð m Z Þ þ π ln (cid:6) sm Z (cid:7)(cid:4) − ; ð Þ where m Z ¼ . GeV is the mass of the Z boson and α s ð m Z Þ ¼ . is the strong coupling constant at the Z pole. Near the p ¯ p threshold, an alternative approach tothe Coulomb enhancement factor should be considered inthe cross section; concerning B ¯ B , we have proposed gluonexchange. At large momentum transfer, the cross sectionis computed in perturbative QCD to leading order.Equation (5) takes into account strong-interaction effectsnear the threshold in a manner dependent on the perturba-tive-QCD prediction in the continuum region awayfrom the threshold [16]. Correlations between the system-atic uncertainties of the measurements at each energypoint are taken into account. The results and meaning ofthe fit parameters are as follows: a ¼ . (cid:3) . and a ¼ . þ . − . are normalization constants, a ¼ . (cid:3) . GeVis the QCD parameter near the threshold, a ¼ . (cid:3) . is the σ p ¯ p power-law dependence, which is related to thenumber of valence quarks, and a ¼ . þ . − . GeV is theQCD parameter Λ QCD in the continuum region. θ|| cos E v en t s /[ . ] Sig = 75000000 0.030 ± c0 = 1.183 /ndf = 0.904 χ =2.125 GeV s (a) |θ| cos E v en t s /[ . ] Sig = 75000000 0.093 ± c0 = 0.757 /ndf = 0.657 χ =2.396 GeV s (b) FIG. 2. Fit to the j cos θ j distributions at (a) 2.125 GeV and(b) 2.396 GeVafter the application of angular-dependent ϵ ð þ δ Þ factors. PHYSICAL REVIEW LETTERS124,
12 3 . (cid:3) . (cid:3) . PHYSICAL REVIEW LETTERS124, ϵ ð cos θ Þ is the angular-dependent efficiencyobtained from MC simulations. The correction factor, ð þ δ Þð cos θ Þ , is calculated by dividing the cos θ distri-bution of a MC sample generated with radiation up to NLOand VP corrections by the distribution of a samplegenerated with the Born process alone. A control sampleof e þ e − → p ¯ p π þ π − events is studied to determine correc-tion factors for discrepancies between data and MCsimulation in the angular-dependent efficiency.After applying these corrections, the j cos θ j distribution isfitted with Eq. (4). The results at 2.125 GeVand 2.396 GeVareshown in Fig. 2, while the results for all energy points aresummarized in Table I. Fits to the j cos θ j distributions as wellas ϵ ð þ δ Þð cos θ Þ distributions for all energy points can befound in the Supplemental Material [22].The model used in the MC simulation takes as input σ p ¯ p ð s Þ and j G E ð s Þ =G M ð s Þj . Therefore, the correctionfactors, and hence the measurements themselves, have asignificant dependence on these inputs. For this reason, thecomplete analysis is performed in an iterative manner,where the obtained results are fed back into the MCsimulation. After three iterations, the results for σ p ¯ p ð s Þ and j G E ð s Þ =G M ð s Þj are stable to within 1%.Several sources of systematic uncertainties are consideredin the determination of σ p ¯ p . The uncertainty associated withthe knowledge of the reconstruction efficiency of the twocharged tracks, as well as from the PID efficiency and the E=p selection criteria, are studied with the e þ e − → p ¯ p π þ π − control sample. The difference of the efficiencymeasured in data and MC simulation is assigned as theuncertainty, and it is found to be 1.0% for both tracking andPID, and 0.2% for the E=p selection. The uncertainties dueto the selection based on the TOF difference between thetracks, the angle between the tracks, and the momentumwindow are studied by varying the selection criteria. Theuncertainty associated with the residual background con-tamination is estimated by comparing the populations of dataand MC simulation in a momentum window of the same sizeas the signal region, but separated by σ . The uncertaintyfrom the luminosity measurement is found to be on a 1.0%level from Ref. [23]. The uncertainty due to the iterative MC-tuning procedure is assigned to be the difference between thenominal result and the result from the second-iteration step. To assess the size of any bias from the choice of the used FFmodel in the MC simulations, we use the model from PHOKHARA [24] to generate an alternative set of MC events.The difference in the final result obtained with this newmodel and the default one is taken as the uncertainty.Many of the uncertainties in the j G E =G M j measurementare assigned with the same method as used in the cross-section analysis. This is true for all selection requirements,the uncertainties associated with the background, theiterative fit procedure, and the model used in the MCsimulation. To account for any imperfections due toasymmetries between the fit model and the observedangular distributions, we fit the cos θ distributions insteadof the j cos θ j ones and assign the difference as anuncertainty. The uncertainty from the luminosity measure-ment is taken as an independent systematic component for j G M j , again taken from Ref. [23]. The total systematicuncertainties on j G E =G M j range from 0.93% to 7.40%,while the total systematic uncertainties on j G M j range from0.60% to 2.10%.We study the energy dependence of σ p ¯ p by fitting theexpression σ p ¯ p ð s Þ ¼ e a π α s ½ − e − πα s ð s Þ = β ð s Þ (cid:2)½ þð ffiffi s p − mpa Þ a (cid:2) ; ffiffiffi s p ≤ . GeV ; πα β ð s Þ C ½ þð mp ffiffi s p Þ (cid:2) e a s ½ ln ð ffiffi s p a Þþ π (cid:2) ; ffiffiffi s p > . GeV ; ð Þ where α s ð s Þ is the strong coupling constant and α is theelectromagnetic constant. The running coupling constant α s ð s Þ is parameterized as follows: α s ð s Þ ¼ (cid:3) α s ð m Z Þ þ π ln (cid:6) sm Z (cid:7)(cid:4) − ; ð Þ where m Z ¼ . GeV is the mass of the Z boson and α s ð m Z Þ ¼ . is the strong coupling constant at the Z pole. Near the p ¯ p threshold, an alternative approach tothe Coulomb enhancement factor should be considered inthe cross section; concerning B ¯ B , we have proposed gluonexchange. At large momentum transfer, the cross sectionis computed in perturbative QCD to leading order.Equation (5) takes into account strong-interaction effectsnear the threshold in a manner dependent on the perturba-tive-QCD prediction in the continuum region awayfrom the threshold [16]. Correlations between the system-atic uncertainties of the measurements at each energypoint are taken into account. The results and meaning ofthe fit parameters are as follows: a ¼ . (cid:3) . and a ¼ . þ . − . are normalization constants, a ¼ . (cid:3) . GeVis the QCD parameter near the threshold, a ¼ . (cid:3) . is the σ p ¯ p power-law dependence, which is related to thenumber of valence quarks, and a ¼ . þ . − . GeV is theQCD parameter Λ QCD in the continuum region. θ|| cos E v en t s /[ . ] Sig = 75000000 0.030 ± c0 = 1.183 /ndf = 0.904 χ =2.125 GeV s (a) |θ| cos E v en t s /[ . ] Sig = 75000000 0.093 ± c0 = 0.757 /ndf = 0.657 χ =2.396 GeV s (b) FIG. 2. Fit to the j cos θ j distributions at (a) 2.125 GeV and(b) 2.396 GeVafter the application of angular-dependent ϵ ð þ δ Þ factors. PHYSICAL REVIEW LETTERS124, G eff are best reproduced bythe function proposed in Ref. [25], j G eff ð s Þj ¼ A ð þ sm a Þ½ − s . ð GeV =c Þ (cid:2) ; ð Þ where A ¼ . (cid:3) . and m a ¼ . (cid:3) . ð GeV =c Þ are obtained from our fit, illustrated in Fig. 3(e). Theresults indicate some oscillating structures which areclearly seen when the residuals are plotted as a functionof the relative momentum p of the p ¯ p pair [26]. The bluesolid curve in Fig. 3(f) describes the periodic oscillationsand has the form [26] F p ¼ b osc e − b osc p cos ð b osc p þ b osc Þ ; ð Þ where b osc ¼ . (cid:3) . , b osc ¼ . (cid:3) . ð GeV =c Þ − , b osc ¼ . (cid:3) . ð GeV =c Þ − , and b osc ¼ . (cid:3) . are obtained from our fit.The data points and results of these fits are shown inFig. 3 together with the data points for j G E =G M j , j G E j ,and j G M j .This Letter presents the most accurate measurement ofthe Born cross section of the process e þ e − → p ¯ p , σ p ¯ p , forc.m. energies in the interval from 2.00 – [GeV]s ( s ) | e ff | G /ndf=4.5104 χ FitBESIII 2020BESIII 2015BESIII(unTagged)BaBar(Tagged)CMD3 BESFENICEE760E835PS170DM2 ] c p[GeV/ − F ( p ) /ndf=2.1401 χ FitBESIII 2020BESIII 2015BESIII(unTagged)BaBar(Tagged) [GeV]s ( s ) | E | G BESIII 2020 [GeV]s ( s ) | M ( s ) / G E | G BESIII 2020BESIII 2015BESIII(unTagged) BaBar(Tagged)CMD3PS170 [GeV]s ( s ) | M | G BESIII 2020BESIII 2015BaBar(unTagged)PS170 (a) (b)(c) (d)(f)(e) [GeV]s ( s ) [ pb ] pp σ /ndf=0.7340 χ FitBESIII 2020BESIII 2015BESIII(unTagged)BaBar(Tagged)BaBar(unTagged)CMD3 BESFENICEE760E835PS170DM2
FIG. 3. Results from this analysis (red solid squares) including statistical and systematic uncertainties for (a) the e þ e − → p ¯ p crosssection and a fit through the data (blue solid line); (b) the ratio j G E =G M j of the proton; (c) the electric FF of the proton j G E j ; (d) themagnetic FF of the proton j G M j ; (e) the effective FF of the proton j G eff j and a fit through the data (blue solid line) by Eq. (7) suggested inRef. [16]; (f) Proton effective FF values, after subtraction of the smooth function described by Eq. (7), as a function of the relativemomentum p . Also shown are previously published measurements from BESIII [8,15], BABAR [14], CMD3 [10], BES [4], FENICE [9],E760 [12], E835 [13], PS170 [11], and DM2 [6]. χ ¼ P i ½ f ð x i Þ − y i (cid:2) = err i , where err i is the error of the measured results includingstatistical and correlated systematic uncertainties, f is the fit function, ndf is the number of degrees of freedom. PHYSICAL REVIEW LETTERS124,
FIG. 3. Results from this analysis (red solid squares) including statistical and systematic uncertainties for (a) the e þ e − → p ¯ p crosssection and a fit through the data (blue solid line); (b) the ratio j G E =G M j of the proton; (c) the electric FF of the proton j G E j ; (d) themagnetic FF of the proton j G M j ; (e) the effective FF of the proton j G eff j and a fit through the data (blue solid line) by Eq. (7) suggested inRef. [16]; (f) Proton effective FF values, after subtraction of the smooth function described by Eq. (7), as a function of the relativemomentum p . Also shown are previously published measurements from BESIII [8,15], BABAR [14], CMD3 [10], BES [4], FENICE [9],E760 [12], E835 [13], PS170 [11], and DM2 [6]. χ ¼ P i ½ f ð x i Þ − y i (cid:2) = err i , where err i is the error of the measured results includingstatistical and correlated systematic uncertainties, f is the fit function, ndf is the number of degrees of freedom. PHYSICAL REVIEW LETTERS124, σ p ¯ p arefound to be in good agreement with previously publishedresults. The FF ratio j G E =G M j is measured with totaluncertainties around 10% for scan points ranging fromlow to intermediate energy. For the first time, the accuracyof the measured FF ratio in the TL region is comparable tothat of data in the SL region. We have obtained an update ofthe FF measurement, especially for the ratio j G E =G M j ,at c.m. energies of 2.2324 and 3.0800 GeV. We havetested the Coulomb enhancement factor hypothesiswhich depends on nonperturbative QCD. The oscillatingstructures in Refs. [15,26] are clearly seen in the j G eff j line shape.Our measurement strongly favors the result of BABAR [14] over that of PS170 [11]. The magnetic formfactor j G M j is measured for the first time over a widerange of energies with uncertainties of 1.6% to 3.9%,greatly improving the precision compared to previousmeasurements.The BESIII Collaboration thanks the staff of BEPCII andthe IHEP computing center for their strong support. Thiswork is supported in part by National Key Basic ResearchProgram of China under Contract No. 2015CB856700;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; theChinese Academy of Sciences (CAS) Large-ScaleScientific Facility Program; the CAS Center forExcellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CASunder Contracts No. U1532102, No. U1532257,No. U1532258, No. U1732263, No. U1832103; CASKey Research Program of Frontier Sciences underContracts No. QYZDJ-SSW-SLH003, No. QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC andShanghai Key Laboratory for Particle Physics andCosmology; German Research Foundation DFG underContract No. Collaborative Research Center CRC 1044,FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy;Koninklijke Nederlandse Akademie van Wetenschappen(KNAW) under Contract No. 530-4CDP03; Ministry ofDevelopment of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; TheSwedish Research Council; The Knut and AliceWallenberg Foundation (Sweden); U.S. Department ofEnergy under Contracts No. DE-FG02-05ER41374,No. DE-SC-0010118, No. DE-SC-0010504, No. DE-SC-0012069; University of Groningen (RuG) and theHelmholtzzentrum fuer Schwerionenforschung GmbH(GSI), Darmstadt. a Also at Bogazici University, 34342 Istanbul, Turkey. b Also at the Moscow Institute of Physics and Technology,Moscow 141700, Russia. c Also at the Functional Electronics Laboratory, Tomsk StateUniversity, Tomsk 634050, Russia. d Also at the Novosibirsk State University, Novosibirsk630090, Russia. e Also at the NRC “ Kurchatov Institute, ” PNPI, 188300Gatchina, Russia. f Also at Istanbul Arel University, 34295 Istanbul, Turkey. g Also at Goethe University Frankfurt, 60323 Frankfurt amMain, Germany. h Also at Key Laboratory for Particle Physics, Astrophysicsand Cosmology, Ministry of Education; Shanghai KeyLaboratory for Particle Physics and Cosmology; Instituteof Nuclear and Particle Physics, Shanghai 200240, People ’ sRepublic of China. i Also at Government College Women University, Sialkot — j Also at Key Laboratory of Nuclear Physics and Ion-beamApplication (MOE) and Institute of Modern Physics, FudanUniversity, Shanghai 200443, People ’ s Republic of China. k Also at Harvard University, Department of Physics,Cambridge, Massachusetts 02138, USA. l Also at State Key Laboratory of Nuclear Physics andTechnology, Peking University, Beijing 100871, People ’ sRepublic of China.[1] R. 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