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EPJ manuscript No.(will be inserted by the editor)

Feasibility studies of time-like proton electromagnetic formfactors at PANDA at FAIR

The PANDA Collaboration

B. Singh1, W. Erni2, B. Krusche2, M. Steinacher2, N. Walford2, B. Liu3, H. Liu3, Z. Liu3, X. Shen3,C. Wang3, J. Zhao3, M. Albrecht4, T. Erlen4, M. Fink4, F. Heinsius4, T. Held4, T. Holtmann4, S. Jasper4, I. Keshk4,H. Koch4, B. Kopf4, M. Kuhlmann4, M. Kümmel4, S. Leiber4, M. Mikirtychyants4, P. Musiol4, A. Mustafa4,M. Pelizäus4, J. Pychy4, M. Richter4, C. Schnier4, T. Schröder4, C. Sowa4, M. Steinke4, T. Triffterer4, U. Wiedner4,M. Ball5, R. Beck5, C. Hammann5, B. Ketzer5, M. Kube5, P. Mahlberg5, M. Rossbach5, C. Schmidt5, R. Schmitz5,U. Thoma5, M. Urban5, D. Walther5, C. Wendel5, A. Wilson5, A. Bianconi6, M. Bragadireanu7, M. Caprini7,D. Pantea7, B. Patel8, W. Czyzycki9, M. Domagala9, G. Filo9, J. Jaworowski9, M. Krawczyk9, F. Lisowski9,E. Lisowski9, M. Michałek9, P. Poznański9, J. Płażek9, K. Korcyl10, A. Kozela10, P. Kulessa10, P. Lebiedowicz10,K. Pysz10, W. Schäfer10, A. Szczurek10, T. Fiutowski11, M. Idzik11, B. Mindur11, D. Przyborowski11, K. Swientek11,J. Biernat12, B. Kamys12, S. Kistryn12, G. Korcyl12, W. Krzemien12, A. Magiera12, P. Moskal12, A. Pyszniak12,Z. Rudy12, P. Salabura12, J. Smyrski12, P. Strzempek12, A. Wronska12, I. Augustin13, R. Böhm13, I. Lehmann13,D. Nicmorus Marinescu13, L. Schmitt13, V. Varentsov13, M. Al-Turany14, A. Belias14, H. Deppe14, R. Dzhygadlo14,A. Ehret14, H. Flemming14, A. Gerhardt14, K. Götzen14, A. Gromliuk14, L. Gruber14, R. Karabowicz14, R. Kliemt14,M. Krebs14, U. Kurilla14, D. Lehmann14, S. Löchner14, J. Lühning14, U. Lynen14, H. Orth14, M. Patsyuk14,K. Peters14, T. Saito14, G. Schepers14, C. J. Schmidt14, C. Schwarz14, J. Schwiening14, A. Täschner14, M. Traxler14,C. Ugur14, B. Voss14, P. Wieczorek14, A. Wilms14, M. Zühlsdorf14, V. Abazov15, G. Alexeev15, V. A. Arefiev15,V. Astakhov15, M. Yu. Barabanov15, B. V. Batyunya15, Y. Davydov15, V. Kh. Dodokhov15, A. Efremov15,A. Fechtchenko15, A. G. Fedunov15, A. Galoyan15, S. Grigoryan15, E. K. Koshurnikov15, Y. Yu. Lobanov15,V. I. Lobanov15, A. F. Makarov15, L. V. Malinina15, V. Malyshev15, A. G. Olshevskiy15, E. Perevalova15,A. A. Piskun15, T. Pocheptsov15, G. Pontecorvo15, V. Rodionov15, Y. Rogov15, R. Salmin15, A. Samartsev15,M. G. Sapozhnikov15, G. Shabratova15, N. B. Skachkov15, A. N. Skachkova15, E. A. Strokovsky15, M. Suleimanov15,R. Teshev15, V. Tokmenin15, V. Uzhinsky15, A. Vodopianov15, S. A. Zaporozhets15, N. I. Zhuravlev15, A. G. Zorin15,D. Branford16, D. Glazier16, D. Watts16, M. Böhm17, A. Britting17, W. Eyrich17, A. Lehmann17, M. Pfaffinger17,F. Uhlig17, S. Dobbs18, K. Seth18, A. Tomaradze18, T. Xiao18, D. Bettoni19, V. Carassiti19, A. Cotta Ramusino19,P. Dalpiaz19, A. Drago19, E. Fioravanti19, I. Garzia19, M. Savrie19, V. Akishina20, I. Kisel20, G. Kozlov20,M. Pugach20, M. Zyzak20, P. Gianotti21, C. Guaraldo21, V. Lucherini21, A. Bersani22, G. Bracco22, M. Macri22,R. F. Parodi22, K. Biguenko23, K. Brinkmann23, V. Di Pietro23, S. Diehl23, V. Dormenev23, P. Drexler23, M. Düren23,E. Etzelmüller23, M. Galuska23, E. Gutz23, C. Hahn23, A. Hayrapetyan23, M. Kesselkaul23, W. Kühn23, T. Kuske23,J. S. Lange23, Y. Liang23, V. Metag23, M. Nanova23, S. Nazarenko23, R. Novotny23, T. Quagli23, S. Reiter23,J. Rieke23, C. Rosenbaum23, M. Schmidt23, R. Schnell23, H. Stenzel23, U. Thöring23, M. Ullrich23, M. N. Wagner23,T. Wasem23, B. Wohlfahrt23, H. Zaunick23, D. Ireland24, G. Rosner24, B. Seitz24, P.N. Deepak25, A. Kulkarni25,A. Apostolou26, M. Babai26, M. Kavatsyuk26, P. J. Lemmens26, M. Lindemulder26, H. Loehner26, J. Messchendorp26,P. Schakel26, H. Smit26, M. Tiemens26, J. C. van der Weele26, R. Veenstra26, S. Vejdani26, K. Dutta27, K. Kalita27,A. Kumar28, A. Roy28, H. Sohlbach29, M. Bai30, L. Bianchi30, M. Büscher30, L. Cao30, A. Cebulla30, R. Dosdall30,A. Gillitzer30, F. Goldenbaum30, D. Grunwald30, A. Herten30, Q. Hu30, G. Kemmerling30, H. Kleines30, A. Lehrach30,R. Nellen30, H. Ohm30, S. Orfanitski30, D. Prasuhn30, E. Prencipe30, J. Pütz30, J. Ritman30, S. Schadmand30,T. Sefzick30, V. Serdyuk30, G. Sterzenbach30, T. Stockmanns30, P. Wintz30, P. Wüstner30, H. Xu30, A. Zambanini30,S. Li31, Z. Li31, Z. Sun31, H. Xu31, V. Rigato32, L. Isaksson33, P. Achenbach34, O. Corell34, A. Denig34, M. Distler34,M. Hoek34, A. Karavdina34, W. L,34, Z. Liu34, H. Merkel34, U. Müller34, J. Pochodzalla34, S. Sanchez34,S. Schlimme34, C. Sfienti34, M. Thiel34, H. Ahmadi35, S. Ahmed35, S. Bleser35, L. Capozza35, M. Cardinali35,A. Dbeyssi35, M. Deiseroth35, F. Feldbauer35, M. Fritsch35, B. Fröhlich35, P. Jasinski35, D. Kang35, D. Khaneft35 a,R. Klasen35, H. H. Leithoff35, D. Lin35, F. Maas35, S. Maldaner35, M. Martínez35, M. Michel35, M. C. Mora Espí35,C. Morales Morales35, C. Motzko35, F. Nerling35, O. Noll35, S. Pflüger35, A. Pitka35, D. Rodríguez Piñeiro35,A. Sanchez-Lorente35, M. Steinen35, R. Valente35, T. Weber35, M. Zambrana35, I. Zimmermann35, A. Fedorov36,M. Korjik36, O. Missevitch36, A. Boukharov37, O. Malyshev37, I. Marishev37, V. Balanutsa38, P. Balanutsa38,

a e-mail: khaneftd@kph.uni-mainz.de

2

V. Chernetsky38, A. Demekhin38, A. Dolgolenko38, P. Fedorets38, A. Gerasimov38, V. Goryachev38, V. Chandratre39,V. Datar39, D. Dutta39, V. Jha39, H. Kumawat39, A.K. Mohanty39, A. Parmar39, B. Roy39, G. Sonika39, C. Fritzsch40,S. Grieser40, A. Hergemöller40, B. Hetz40, N. Hüsken40, A. Khoukaz40, J. P. Wessels40, K. Khosonthongkee41,C. Kobdaj41, A. Limphirat41, P. Srisawad41, Y. Yan41, M. Barnyakov42, A. Yu. Barnyakov42, K. Beloborodov42,A. E. Blinov42, V. E. Blinov42, V. S. Bobrovnikov42, S. Kononov42, E. A. Kravchenko42, I. A. Kuyanov42, K. Martin42,A. P. Onuchin42, S. Serednyakov42, A. Sokolov42, Y. Tikhonov42, E. Atomssa43, R. Kunne43, D. Marchand43,B. Ramstein43, J. van de Wiele43, Y. Wang43, G. Boca44, S. Costanza44, P. Genova44, P. Montagna44, A. Rotondi44,V. Abramov45, N. Belikov45, S. Bukreeva45, A. Davidenko45, A. Derevschikov45, Y. Goncharenko45, V. Grishin45,V. Kachanov45, V. Kormilitsin45, A. Levin45, Y. Melnik45, N. Minaev45, V. Mochalov45, D. Morozov45, L. Nogach45,S. Poslavskiy45, A. Ryazantsev45, S. Ryzhikov45, P. Semenov45, I. Shein45, A. Uzunian45, A. Vasiliev45, A. Yakutin45,E. Tomasi-Gustafsson46, U. Roy47, B. Yabsley48, S. Belostotski49, G. Gavrilov49, A. Izotov49, S. Manaenkov49,O. Miklukho49, D. Veretennikov49, A. Zhdanov49, K. Makonyi50, M. Preston50, P. Tegner50, D. Wölbing50, T. Bäck51,B. Cederwall51, A. K. Rai52, S. Godre53, D. Calvo54, S. Coli54, P. De Remigis54, A. Filippi54, G. Giraudo54,S. Lusso54, G. Mazza54, M. Mignone54, A. Rivetti54, R. Wheadon54, F. Balestra55, F. Iazzi55, R. Introzzi55,A. Lavagno55, J. Olave55, A. Amoroso56, M. P. Bussa56, L. Busso56, F. De Mori56, M. Destefanis56, L. Fava56,L. Ferrero56, M. Greco56, J. Hu56, L. Lavezzi56, M. Maggiora56, G. Maniscalco56, S. Marcello56, S. Sosio56,S. Spataro56, R. Birsa57, F. Bradamante57, A. Bressan57, A. Martin57, H. Calen58, W. Ikegami Andersson58,T. Johansson58, A. Kupsc58, P. Marciniewski58, M. Papenbrock58, J. Pettersson58, K. Schönning58, M. Wolke58,B. Galnander59, J. Diaz60, V. Pothodi Chackara61, A. Chlopik62, G. Kesik62, D. Melnychuk62, B. Slowinski62,A. Trzcinski62, M. Wojciechowski62, S. Wronka62, B. Zwieglinski62, P. Bühler63, J. Marton63, D. Steinschaden63,K. Suzuki63, E. Widmann63, and J. Zmeskal63

1 Aligarth Muslim University, Physics Department, Aligarth India2 Universität Basel, Basel Switzerland3 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing China4 Universität Bochum, Institut für Experimentalphysik I, Bochum Germany5 Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn Germany6 Università di Brescia, Brescia Italy7 Institutul National de C&D pentru Fizica si Inginerie Nucleara "Horia Hulubei", Bukarest-Magurele Romania8 P.D. Patel Institute of Applied Science, Department of Physical Sciences, Changa India9 University of Technology, Institute of Applied Informatics, Cracow Poland

10 IFJ, Institute of Nuclear Physics PAN, Cracow Poland11 AGH, University of Science and Technology, Cracow Poland12 Instytut Fizyki, Uniwersytet Jagiellonski, Cracow Poland13 FAIR, Facility for Antiproton and Ion Research in Europe, Darmstadt Germany14 GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt Germany15 Veksler-Baldin Laboratory of High Energies (VBLHE), Joint Institute for Nuclear Research, Dubna Russia16 University of Edinburgh, Edinburgh United Kingdom17 Friedrich Alexander Universität Erlangen-Nürnberg, Erlangen Germany18 Northwestern University, Evanston U.S.A.19 Università di Ferrara and INFN Sezione di Ferrara, Ferrara Italy20 Frankfurt Institute for Advanced Studies, Frankfurt Germany21 INFN Laboratori Nazionali di Frascati, Frascati Italy22 INFN Sezione di Genova, Genova Italy23 Justus Liebig-Universität Gießen II. Physikalisches Institut, Gießen Germany24 University of Glasgow, Glasgow United Kingdom25 Birla Institute of Technology and Science - Pilani , K.K. Birla Goa Campus, Goa India26 KVI-Center for Advanced Radiation Technology (CART), University of Groningen, Groningen Netherlands27 Gauhati University, Physics Department, Guwahati India28 Indian Institute of Technology Indore, School of Science, Indore India29 Fachhochschule Südwestfalen, Iserlohn Germany30 Forschungszentrum Jülich, Institut für Kernphysik, Jülich Germany31 Chinese Academy of Science, Institute of Modern Physics, Lanzhou China32 INFN Laboratori Nazionali di Legnaro, Legnaro Italy33 Lunds Universitet, Department of Physics, Lund Sweden34 Johannes Gutenberg-Universität, Institut für Kernphysik, Mainz Germany35 Helmholtz-Institut Mainz, Mainz Germany36 Research Institute for Nuclear Problems, Belarus State University, Minsk Belarus37 Moscow Power Engineering Institute, Moscow Russia38 Institute for Theoretical and Experimental Physics, Moscow Russia39 Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai India40 Westfälische Wilhelms-Universität Münster, Münster Germany

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41 Suranaree University of Technology, Nakhon Ratchasima Thailand42 Budker Institute of Nuclear Physics, Novosibirsk Russia43 Institut de Physique Nucléaire, CNRS-IN2P3, Univ. Paris-Sud, Université Paris-Saclay, 91406, Orsay cedex France44 Dipartimento di Fisica, Università di Pavia, INFN Sezione di Pavia, Pavia Italy45 Institute for High Energy Physics, Protvino Russia46 IRFU,SPHN, CEA Saclay, Saclay France47 Sikaha-Bhavana, Visva-Bharati, WB, Santiniketan India48 University of Sidney, School of Physics, Sidney Australia49 National Research Centre "Kurchatov Institute" B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina, St.

Petersburg Russia50 Stockholms Universitet, Stockholm Sweden51 Kungliga Tekniska Högskolan, Stockholm Sweden52 Sardar Vallabhbhai National Institute of Technology, Applied Physics Department, Surat India53 Veer Narmad South Gujarat University, Department of Physics, Surat India54 INFN Sezione di Torino, Torino Italy55 Politecnico di Torino and INFN Sezione di Torino, Torino Italy56 Università di Torino and INFN Sezione di Torino, Torino Italy57 Università di Trieste and INFN Sezione di Trieste, Trieste Italy58 Uppsala Universitet, Institutionen för fysik och astronomi, Uppsala Sweden59 The Svedberg Laboratory, Uppsala Sweden60 Instituto de Física Corpuscular, Universidad de Valencia-CSIC, Valencia Spain61 Sardar Patel University, Physics Department, Vallabh Vidynagar India62 National Centre for Nuclear Research, Warsaw Poland63 Österreichische Akademie der Wissenschaften, Stefan Meyer Institut für Subatomare Physik, Wien Austria

Received: date / Revised version: date

Abstract The results of simulations for future measurements of electromagnetic form factors at PANDA(FAIR) within the PandaRoot software framework are reported. The statistical precision at which theproton form factors can be determined is estimated. The signal channel p̄p → e+e− is studied on thebasis of two different but consistent procedures. The suppression of the main background channel, i.e.the p̄p → π+π−, is studied. Furthermore, the background versus signal efficiency, statistic and systematicuncertainties on the extracted proton form factors are evaluated using to the two different procedures.The results are consistent with those of a previous simulation study using an older, simplified framework.However, a slightly better precision is achieved in the PandaRoot study in a large range of momentumtransfer, assuming the nominal beam condition and detector performances.

PACS. 25.43.+t Antiproton-induced reactions – 13.40.Gp Electromagnetic form factors

3

1 Introduction

The PANDA [1] experiment at FAIR (Facility for Anti-proton and Ion Research, at Darmstadt, Germany) willdetect the products of the annihilation reactions inducedby high intensity antiproton beams with momentum from1.5 to 15 GeV/c. The comprehensive physics program in-cludes charmonium spectroscopy, search for hybrids andglueballs, search for charm and strangeness in nuclei, ba-ryon spectroscopy and hyperon physics, as well as nucleonstructure studies [2]. Here we focus on the extraction oftime-like (TL) proton electromagnetic form factors (FFs)through the measurement of the angular distribution ofthe produced electron (positron) in the annihilation ofproton-antiproton into an electron-positron pair.

Electromagnetic FFs are fundamental quantities whichdescribe the intrinsic electric and magnetic distributionsof hadrons. Assuming parity and time invariance, a had-ron with spin S is described by 2S + 1 independent FFs.Protons and neutrons (spin 1/2 particles) are then char-acterized by two FFs: the electric GE and the magnetic

GM . In the TL region, electromagnetic FFs have been as-sociated with the time evolution of these distributions [3].

Theoretically, the FFs enter in the parametrization ofthe proton electromagnetic current. They are experiment-ally accessible through measurements of differential andtotal cross sections for elastic ep scattering in the space-like (SL) region and p̄p ↔ e+e− in the TL region. It is as-sumed that the interaction occurs through the exchange ofone photon, which carries a momentum transfer squaredq2. In the time-like region, this corresponds to the totalenergy squared s.

Space-like (SL) form factors have been rigorously stud-ied since the 1960’s [4]. However, the polarization trans-fer method [5, 6] that was used for the first time in 1998,gave rise to new questions in the field. The recent access tohigh precision measurements over a large kinematic range,further contributed to the new interest. Elastic e−p →e−p data from the JLab-GEp collaboration, covering arange of momentum transfer squared up to Q2 = −q2 ≃8.5 (GeV/c)2 showed that the electric and magnetic dis-tributions inside the proton are not the same. This was

4

in contrast what was previously assumed: the ratio of theelectric and the magnetic FF, µpGE/GM (µp is the protonmagnetic moment) decreases almost linearly from unity asthe momentum transfer squared increases, approaching avalue of zero.

In the TL region, the precision of the proton FF meas-urements has been limited by the achievable luminosity ofthe e+e− colliders and p̄p annihilation experiments. At-tempts have been made at LEAR [7], BABAR [8] andmore recently at BESIII [9]. The FFs ratio shows a differ-ent tendency, somehow inconsistent in the limit of com-bined systematic and statistical uncertainties and defin-itely calls for more precise experiments. The results ofLEAR and BABAR disagree with each other up to 3σsignificance, while the BESIII measurements have largetotal uncertainties.

The PANDA experiment, designed with an averagepeak luminosity of L = 2 · 1032 cm−2s−1 in the so- calledhigh luminosity mode, will bring new information in tworespects: the precision measurement of the angular distri-bution for the individual determination of FFs, and themeasurement of the integrated cross section for the ex-traction of a generalized FF up to larger values of s. Thesedata are expected to set a stringent test of nucleon mod-els. In particular, the high s-region brings information onanalyticity properties of FFs and on the asymptotic q2

behavior predicted by perturbation Quantum ChromoDy-namics (pQCD).

The FAIR facility and the PANDA experiment are un-der construction in Darmstadt (Germany). Simulations ofthe different physics processes have been performed or arein progress. The feasibility of the FFs measurement withthe PANDA detector, as suggested in Ref. [10], has beeninvestigated in Ref. [11]. Since the latter paper was pub-lished, lots of progress has been made in the developmentof a new simulation framework (PandaRoot, see Ref. [12])with a much more realistic detector model and more elab-orated reconstruction algorithms. This is the motivationto reinvestigate this channel. In addition, recently a lotof progress has been made regarding the design of thedetector. Prototypes of sub-detectors have been built. Aseries of performance tests have been carried out whichprovided data for improvements in the design. The Tech-nical Design Reports of most of the detectors are available.Although the simulation and analysis software is still on-going, in parallel with the detector construction, a real-istic description of the sub-detectors and new algorithmsfor the tracking, digitization, and PID has been implemen-ted. The realistic magnetic field, calculated with TOSCAsoftware [13], has been implemented.

In the PandaRoot revision used for this work, the de-scription of most of the sub-detectors has been completedwith passive materials, pipe, magnet yoke, etc. Moreover,during recent years GEANT4 [14] has undergone continu-ous improvement, concerning in particular the shape ofthe electromagnetic shower.

The aim of this paper is to present a new simulation,based on a recent PandaRoot revision, in order to checkthe validity of the previously made assumptions and to

confirm the conclusions regarding the feasibility of thee+e− detection at a sufficient level of precision. In addi-tion, new and efficient analysis tools have been developedfor the extraction of the physical information, which willbe applied also to the treatment of the experimental data.Full scale simulations have been performed on the HIM-ster cluster of the Helmholtz-Institut Mainz.

The paper is organized as follows: The kinematics ofthe reactions of interest (signal and background) and theevaluation of the counting rates are described in Section 2.The detector is briefly described in Section 3. The stand-ard chain of the full simulation with PandaRoot and theprocedure to identify and analyze the signal and the back-ground channels are described in Section 3. In Section 4we present the results in terms of the protons FFs ratioR = |GE |/|GM |, individual FFs GE and GM , the angu-lar asymmetry A and the effective FF. Finally, in Section5 a number of sources of systematic uncertainties are dis-cussed. Conclusions contain a summary and final remarks.

2 Basic formalism

Let us consider the reactions:

p̄(p1) + p(p2) → ℓ−(k1) + ℓ+(k2), ℓ = e, µ, π, (1)

where the four-momenta of the particles are written withinparentheses. In the center-of-mass (c.m.) system, the four-momenta are:

p1 = (E,p), p2 = (E,−p),

k1 = (E,k), k2 = (E,−k), p · k = pk cos θ, (2)

θ is the angle between the negative emitted particle andthe antiproton beam.

The cylindrical symmetry around the beam axis ofthe unpolarized binary reactions originates an isotropicdistribution in the azimuthal angle φ. These reactionsare two-body final state processes. The final particles areemitted back to back in the c.m. system, and each ofthem, having equal mass, carries half of the total en-ergy of the system, E =

√s/2, where the invariant s is

s = q2 = (p1 + p2)2 = (k1 + k2)

2.All leptons in the final state (e, µ, τ) bring the same in-

formation on the electromagnetic hadron structure. How-ever, the experimental requirements for their detection arepeculiar to each particle species. In this work we focus onthe electron-positron pair production, denoted the signalreaction, and on the charged pion pair production, denotedthe background reaction. The cross section of hadron pro-duction is expected to be much larger than that of leptons:for charged pions it is 106 times larger than for e+e− pro-duction [15–17]. The signal and the background reactionshave very similar kinematics, because the mass of the elec-tron is sufficiently close to the pion mass in the energyscale of the PANDA experiment (for plab = 3.3 GeV/c theelectron momentum spread is 0.58-3.82 GeV/c, while forthe pions it is 0.92-3.5 GeV/c). Therefore, the kinematicsplays a minor role in the electron (positron)/pion sep-aration. The discrimination between electrons and pions

5

Figure 1: Tree-level contributing diagram to p̄p → l+l−.

requires high performance PID detectors and precise mo-mentum measurement. For example, the information fromthe electromagnetic shower induced by different chargedparticles in an electromagnetic calorimeter does play animportant role for the electron identification. The kin-ematic selection suppresses contributions from hadronicchannels with more than two particles in the final states,as well as events with secondary particles originating fromthe interaction of primary particles with the detector ma-terial. A kinematic selection is also very efficient in sup-pressing the neutral pions, as discussed in Refs. [11, 18].Note that the cross section of neutral pion pair produc-tion, π0π0, is ten times smaller than that of π+π−.

2.1 The signal reaction

The expression of the hadron electromagnetic current forthe p̄p annihilation into two leptons is derived assumingone-photon exchange. The diagram which contributes tothe tree-level amplitude is shown in Fig. 1. The internalstructure of the hadrons is then parametrized in termsof two FFs, which are complex functions of q2, the fourmomentum squared of the virtual photon. For the case ofunpolarized particles the differential cross section has theform [15]:

dσ

d cos θ=

πα2

2βs

[

(1 + cos2 θ)|GM |2 + 1

τsin2 θ|GE |2

]

,(3)

where β =√

1− 1/τ , τ = s/(4m2), α is the electromag-netic fine-structure constant, and m is the proton mass.This formula can be also written in equivalent form as [19]:

dσ

d cos θ= σ0

[

1 +A cos2 θ]

, (4)

where σ0 is the value of the differential cross section atθ = π/2 and A is the angular asymmetry which lies in the

range −1 ≤ A ≤ 1, and can be written as a function ofthe FFs ratio as:

σ0 =πα2

2βs

(

|GM |2 + 1

τ|GE |2

)

A =τ |GM |2 − |GE |2τ |GM |2 + |GE |2

=τ − R2

τ + R2, (5)

where R = |GE |/|GM |.The fit function defined in Eq. (4) can be reduced to a

linear function (instead of quadratic) where σ0 and A arethe parameters to be extracted from the experimental an-gular distribution. In the case of R = 0, the minimizationprocedure based on MINUIT has problems to converge,while the asymmetry A varies smoothly in the consideredq2 interval. Therefore, it is expected to reduce instabilit-ies and correlations in the fitting procedure. The angularrange where the measurement can be performed is usuallyrestricted to | cos θ| ≤ c̄, with c̄ = cos θmax.

The integrated cross section, σint, is:

σint =

∫ c̄

−c̄

dσ

d cos θd cos θ = 2σ0 c̄

(

1 +A3c̄2)

(6)

=πα2

2βsc̄

[(

1 +c̄2

3

)

|GM |2 + 1

τ

(

1− c̄2

3

)

|GE |2]

.

The total cross section, σtot, corresponds to c̄ = 1:

σtot = 2σ0

(

1 +A3

)

=2πα2

3βs

[

2|GM |2 + |GE |2τ

]

(7)

=2πα2|GM |2

3βs

[

2 +R2

τ

]

.

Being known the total cross section, one can define aneffective FF as:

|Fp|2 =3βsσtot

2πα2

(

2 +1

τ

) , (8)

or from the integrated cross section, as:

|Fp|2 =βs

πα2

σint

c̄

[(

1 +c̄2

3

)

+1

τ

(

1− c̄2

3

)] , (9)

which is equivalent to the value extracted from cross sec-tion measurements, assuming |GE | = |GM |.

Literature offers several parameterizations of the pro-ton FFs (see Refs. [20, 21]). The world data are illustratedin Fig. 2. In Ref. [11] two parameterizations were con-sidered. Cross section parameters are extracted from ex-perimental data of the integrated cross section. BABARdata [22, 23] suggest a steeper decrease with s.

The Quantum ChromoDynamics (QCD) inspired para-meterization of |GE,M | is based on an analytical extensionof the dipole formula from the SL to the TL region and

6

]2[(GeV/c)2q5 10 15 20 25 30

| p|F

0.1

0.2

0.3

(a)

]2[(GeV/c)2q10D

|/G p|F

5

10

15

20

BABAR

E835FenicePS170E760DM1DM2BESBESIIICLEO

(b)

Figure 2: q2 dependence of the world data for p̄p → e+e− and e+e− → p̄p. The effective proton TL FF, |FP |, isextracted from the annihilation cross sections assuming |GE | = |GM |: E835 [24, 25], Fenice [26], PS170 [7], E760 [27],DM1 [28], DM2 [29, 30], BES [31], BESIII [9], CLEO [32], BABAR [22, 23]. Different parameterizations are shownfrom Eq. (11) (black solid and red dashed lines) and from Eq. (10) (blue dash-dotted line), as described in the text.

corrected to avoid ’ghost’ poles in αs (the strong interac-tion running constant) [33]:

|GQCDE,M | = AQCD

q4[

log2(q2/Λ2QCD) + π2

] . (10)

The parameter AQCD = 89.45 (GeV/c)4 is obtained fromfitting to experimental data, and ΛQCD = 0.3 (GeV/c) isthe QCD cut-off parameter. It is shown in Fig. 2 as a bluedash-dotted line.

The existing data on the TL effective FF is well repro-duced by the function proposed in Ref. [19]:

|GE,M | = A

1 + q2/m2a

GD,

GD = (1 + q2/q20)−2, (11)

where the numerator A is a constant extracted from the fitto the TL data. It is illustrated by a black solid line withthe nominal parameters A = 22.5, m2

a = 3.6 (GeV/c)2,and q20 = 0.71 (GeV/c)2. Note that an updated global fitwith a data set including 85 points (starting from s = 4GeV2) gives A(fit) = 71.5 (GeV/c)4 and m2

a(fit) = 0.85(GeV/c)2, with a value of χ2/NDF = 1.4 (red dashed line),overestimating the low energy data. These parameteriza-tions reproduce reasonably well the data in the consideredkinematic region. In our calculations, we chose the para-metrization from Eq. (11) with the nominal parameters.

The expected count rates for an ideal detector are re-ported in Table 1, assuming R = |GE |/|GM | = 1 and us-ing the parametrization from Eq. (11), the angular range

| cos θ| ≤ 0.8 and Eq. (3). Due to the PANDA detectoracceptance, the electron identification efficiency becomesvery low above | cos θ| = 0.8 (see Section 4). For eachkinematic point, Nint(e

+e−) is reported assuming an in-tegrated luminosity of 2 fb−1. This corresponds to fourmonths of measurement with 100% efficiency at the max-imum luminosity of L = 2·1032 cm−2s−1). In the table, wealso list the cross sections and expected number of countsNint(π

+π−) of the most dominant background channel,i.e. p̄p → π+π−.

As already mentioned, time-like FFs are complex func-tions. However, the unpolarized cross section contains onlythe moduli squared of the FFs. An experiment with po-larized antiproton beam and/or polarized proton targetwould allow to access the phase difference of the protonFFs (Ref. [34]). The feasibility of implementing a trans-versely polarized proton target in PANDA is under in-vestigation.

2.2 The background reaction

In order to estimate the π+π− background in the interest-ing kinematic range, phenomenological parameterizationsfor the differential cross sections have been developed anda new generator has been developed (see Ref. [35]).

The difficulties for a consistent physics description arerelated to different aspects:

– The dominant reaction mechanism is changing withenergy and angle [36].

7

s plab σint(e+e−) Nint(e

+e−) σint(π+π−) Nint(π

+π−)σint(π

+π−)

σint(e+e−)[(GeV/c)2] [GeV/c] [pb] [µb] ·10−6

5.40 1.70 415 830·103 101 202·109 0.247.27 2.78 55.6 111·103 13.1 262·108 0.248.20 3.30 24.8 496·102 2.96 592·107 0.1211.12 4.90 3.25 6503 0.56 111·107 0.1712.97 5.90 1.16 2328 0.23 455·106 0.2013.90 6.40 0.73 1465 0.15 302·106 0.21∗16.69 7.90 0.21 428 0.05 101·106 0.24∗22.29 10.90 0.03 61 0.01 205·105 0.34∗26.03 12.90 0.01 21∗27.90 13.90 0.66·10−2 13

Table 1: Integrated cross section σint for the range | cos θ| ≤ 0.8 and number of counts Nint for p̄p → e+e−. Theprediction was made according to parameterization as in Ref. [11] The corresponding values for the p̄p → π+π−

channel are also listed. A 100% detector efficiency and an integrated luminosity L = 2 fb−1 was assumed for each datapoint, which corresponds to four months of data taking. For the s-values marked with an ’*’ the full simulation hasnot been performed and the numbers are given for future references. The last value, s = 27.9 (GeV/c)2, is the highestenergy where the upper limit of the cross section can be measured at PANDA.

– At low energy, the angular distribution of the finalstate pions is measured [16]. A baryon exchange modelwas developed in Ref. [37], restricted to

√s < 1 GeV.

– At high energy (plab ≥ 5), lack of statistics does notallow to better constrain the parameters [38, 39], todiscriminate among models.

– Model independent considerations based on crossingsymmetry or T-invariance, which help to connect therelevant reactions, in general can not be considered aspredictive [40].

As a consequence, the generator utilizes two differentparameterizations: in the low energy region, 0.79 ≤ plab ≤2.43 GeV/c (plab being the antiproton beam momentumin the laboratory system), the Legendre polynomial para-meters up to the order of ten have been fitted to the datafrom Ref. [16]. In the high energy region, 5 ≤ plab < 12GeV/c, the Regge inspired parametrization from Ref. [17],previously tuned on data from Refs. [38, 39, 41, 42], wasapplied. In the intermediate region, 2.43 GeV/c ≤ plab < 5GeV/c, where no data exist and the validity of the modelis questionable, a soft interpolation is applied.

Differential cross sections of the p̄p → π+π− reactionare displayed in Fig. 3, for different plab. The functionsused in the pion generator are shown in comparison tothe data sample.

The total cross section is shown in Fig. 4 as a functionof the antiproton momentum. The lack of data aroundplab = 4 GeV/c does not constrain the parameters andthey could therefore not be fixed to a precise value in thegenerator. The cross section measured at plab = 12 GeV/cfrom Ref. [42] should be considered as a lower limit. Forcomparison, the parametrization from the compilation inRef. [34] is also shown. This parametrization reproducesthe data.

The expected count rates for the background channel,as well as the total signal-to-background cross section ra-tio, are reported in Table 1. Within the range | cos θ| ≤ 0.8,

θcos-0.5 0 0.5

b]

µ [θ/d

cos

σd

-310

-210

-110

1

10

210

310

Figure 3: Data and modeling of angular distributionsfor the reaction p̄p → π+π−, as a function of cos θ,for different values of the beam momentum: plab = 1.7GeV/c [16] (green triangles and dash-triple dotted line)plab = 5 GeV/c [38] (blue full squares and dash-dottedline); plab = 6.21 GeV/c [39] (black full circles and solidline). The results of the generator [35] are also given atplab = 3 GeV/c (red dotted line) and plab = 10 GeV/c(magenta dashed line).

8

[GeV/c]lab

p1 10

b]

µ [σ

-410

-310

-210

-110

1

10

210

310

Figure 4: Total cross section for the reaction p̄p → π+π−,as a function of the beam momentum in laboratory sys-tem, plab. The selected data are from references: [7] (blackopen circle), [43] (blue full circle), [44] (cyan full cross),[45] (red full diamonds), [46] (brown full stars), and [47](black open star). The result from [42] (green full square)corresponds to total backward cross section and has tobe considered as a lower limit. The solid line is the resultfrom the generator [35]. The dashed line is the result ofthe compilation from Ref. [34].

the p̄p annihilation with the subsequent production of twocharged pions is about five to six orders of magnitudes lar-ger than the production of a lepton pair.

3 The PANDA experiment

3.1 The PANDA detector

The future PANDA experiment will offer a broad physicsprogram thanks to the foreseen 4π acceptance, high res-olution and tracking capability and excellent neutral andcharged particle identification in a high rate environment.The average interaction rate is expected to reach 20 MHz.The structure and the components of the detector havebeen optimized following the experience gained at highenergy experiments. An detailed overview of the PANDAdetector and its performance can be found in Ref. [1]. Inthe following, we will outline characteristics of the detect-ors which play an important role in the FFs measurements.

An overall picture of the PANDA detector is shown inFig. 5. The size of the detector is about 13 m along thebeam direction. PANDA is a compact detector with twomagnets: a central solenoid [48] and a forward dipole. The

(pellet or jet) target is surrounded by a number of detect-ors. The target spectrometer consists of the Micro VertexDetector (MVD) [49] and the Straw Tube Tracker (STT)[50] to ensure a precise vertex finding as well as a spatialreconstruction of the trajectories of charged particles. Inaddition, both sub-detector systems are able to measurethe energy loss to support the particle identification bydE/dx measurements. The DIRC (Detection of InternallyReflected Cerenkov light) is used for particle identificationat polar angles between 22◦ and 140◦, and momentum upto 5 GeV/c [51].

The barrel is completed by an electromagnetic calor-imeter (EMC), consisting of Lead Tungstate (PbWO4)crystals, to assure an efficient photon detection from 10 MeVto 14.6 GeV [52] with an energy resolution lower than 2%.Besides the cylindrical barrel (11360 crystals), a forwardendcap (3856 crystals) and a backward endcap (ca. 600crystals) are added [53].

Particles emitted at angles smaller than 22◦ are detec-ted by three planar stations of Gas Electron Multipliers(GEM) downstream of the target [54]. The muon iden-tification is performed by Iarocci proportional tubes andstrips, in the inner gap of the solenoid yoke and betweenthe hadron calorimeter planes, with a forward angular cov-erage up to 60◦ [55]. The Barrel DIRC is used for PID forparticles with momenta of 0.8 GeV/c up to about 5 GeV/c,at polar angles between 22◦ and 140◦ [56]. The achievedmomentum resolution is expected to be ∆p/p ≃ 1.5% at1 GeV/c.

A time-based data acquisition system capable of a fastcontinuous readout followed by an intelligent software trig-ger is under development.

3.2 Simulation and analysis software

The offline software for the PANDA detector simulationand event reconstruction is PandaRoot, which is developedin the frame of the common framework for the futureFAIR experiments, FairRoot [57]. It is mainly based onthe object oriented data analysis framework ROOT [58],and utilizes different transport models such as Geant4 [14],which is used in the present simulations. Different recon-struction algorithms for tracking and PID are under devel-opment and optimization in order to achieve the require-ments of the experiment.

A schematic view of the simulation and data analysischain is shown in Fig. 6.

3.3 Generated events

The signal and background events can be produced bydifferent event generators according to the physics case.As mentioned in the previous section, the generator fromRef. [35] was used for the p̄p → π+π− background sim-ulation. Taking into account the ratio of cross sectionsσ(p̄p → π+π−)/σ(p̄p → e+e−) < 106, in order to makea reliable proton FF measurement, we need to achieve abackground rejection factor on the order of 10−8. Monte

9

Figure 5: View of the PANDA detector.

θcos-1 -0.5 0 0.5 1

Eve

nts

0

2000

4000

6000

8000

10000310×

-π+π→pp

(a)

θcos-1 -0.5 0 0.5 1

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nts

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610×

-π+π→pp

(b)

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Eve

nts

0

5

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20

25

30

35

40

45

610×

-π+π→pp

(c)

Figure 7: Angular distribution of the p̄p → π+π− events generated using the model from Ref. [35] for different s values:(a) 5.4 GeV2, (b) 8.2 GeV2, and (c) 13.9 GeV2.

Carlo angular distributions of the π− mesons are shownin Fig. 7, for the three incident antiproton beam momentaconsidered in this work.

The EvtGen generator [59] was used to generate thep̄p → e+e− signal channel. The generated data were usedto determine the efficiency of the signal channel with highprecision. The final state leptons are produced accordingto two models of the angular distribution implemented inthe EvtGen (see Section 4).

3.4 PID and kinematic variable reconstruction

In order to separate the signal from the background, anumber of criteria have been applied to the reconstructedevents. For this purpose, the raw output of the PID andtracking sub-detectors as EMC, STT, MVD, and DIRChave been used. In this Section the most relevant recon-structed variables for the signal selection are described indetail.

The EMC provides many efficient variables that can beused in the event selection, e.g. the lateral moment andthe energy deposit in a specific crystal. In electromag-

10

Figure 6: Standard analysis chain in PandaRoot.

EMC lateral moment0 0.2 0.4 0.6 0.8 1

Eve

nts

0

20

40

60

80

100

120

140

160

180

200

220

310×

Figure 8: EMC lateral moment for the signal (bluesquares) and background (red circles) events.

netic showers, most of the energy is typically contained intwo to three crystals, while hadronic showers spread outover greater distances. Therefore the lateral moment istypically small for electromagnetic showers in comparisonwith hadronic showers. EMC lateral moment of signal andbackground events is shown in Fig. 8.

The ratio EEMC/preco of the shower energy depositedin the calorimeter to the reconstructed momentum of thetrack associated with the shower is another standard vari-able for electron selection (Fig. 9). Due to the very lowelectron mass, the EEMC/preco ratio is close to unity forthe signal (Fig. 9a). The discontinuities that appear in the

plot are due to the transition regions between the differ-ent parts of the EMC. For the background (Fig. 9b), thedistribution shows a double structure: one narrow peakat low EEMC/preco values, which is due to the energy lossby ionization, and another one around EEMC/preco = 0.4corresponding to the hadronic interactions. The tail of thelatter extends to much higher values, giving backgroundunder the electron peak.

Figures 9c and 9d show the momentum dependenceof the energy loss per unit length for signal and back-ground, respectively. Although the energy loss, denoteddE/dxSTT , show overlapping patterns for electrons andpions dE/dxSTT , a cut on deposited energy in the STTcan be applied in order to partially suppress the pion back-ground.

3.4.1 PID probabilities

Using the raw output of the EMC, STT, MVD and DIRCdetectors, the probabilities of a reconstructed particle be-ing an electron or positron have been calculated. Theprobabilities can be calculated for each detector individu-ally (PIDs) or as a combination of all of them (PIDc).

In this work, both types of probabilities PIDs and PIDc

have been used in order to increase the signal efficiencyand the background suppression factor. The distributionsof PIDc are shown in Fig. 10. For the signal, the distribu-tions of the PIDc (Fig. 10a) have a maximum at PIDc = 1,where the generated electron and positron events are wellidentified. The peak at 0.2 is related to events for whichno definitive type of the particle was assigned. In this case,the probability splits equally into the five particle types (e,µ, K, π, and p). The same explanation holds for the highlypopulated region around PIDc = 0.3, where two or threeparticle types have the same behavior in some detectors.For the generated background events, PIDc distributionsare shown in Fig. 10b. As expected, the distributions ofPIDc all have a maximum at zero.

3.4.2 Kinematic variables

The signal and background reactions are two-body finalstate processes. The electrons or pions are emitted backto back in the c.m. system. Since all final state particlesare detected, their total energy is equal to that of the p̄psystem.

In Fig. 11, the relevant kinematic variables are presen-ted for generated and reconstructed MC events, before theselection procedure, at q2 = 8.2 (GeV/c)2.

Unlike the momentum and energy of a charged particle,the average of neither the polar angle nor the azimuthalangle are affected by the Bremsstrahlung emission duringthe passage of the particle through the matter.

The kinematic selection improves the suppression ofthe hadronic background from the channels with morethan two particles in the final state. It also eliminates sec-ondary particles whose kinematics is different from that ofprimary particles. After applying a given set of cuts (see

11

1

10

210

310

p [GeV/c]0 1 2 3 4 5

[GeV

/(G

eV/c

)]re

co/p

EM

CE

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(a)

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/(G

eV/c

)]re

co/p

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[a.u

.]S

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x

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(c)

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[a.u

.]S

TT

dE/d

x

0

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14

16

18

20

-π+π→pp

(d)

Figure 9: Detector response to the signal (left column) and the background (right column): (a,b) is the ratio of theenergy deposited in the EMC over the reconstructed momentum; (c,d) is the energy loss per unit of length in the STTas a function of momentum at q2 = 8.2 (GeV/c)2.

Sections 4.2 and 4.3), the vertex fit and the kinematic fitsdo not contribute to rejection of the secondary particlesnor to the signal-background separation.

4 Simulation

Two independent simulation studies have been performedfor the signal and the background. In the following, wewill present both studies in detail and clarify the meth-ods in each one. The main differences between them are(i) the angular distribution model used as input for thesignal event generator, (ii) the determination of the effi-ciency and (iii) the fit of the reconstructed events afterefficiency correction. By comparing the two simulations

we can estimate the effect of the statistic fluctuations, theefficiency determination, and the extraction of the protonFFs using different fit functions. The two approaches aredenoted Method I and Method II. Both methods used thesame background samples.

4.1 Background events

The generator from Ref. [35] is used for the p̄p → π+π−

background simulation. 108 background events are gener-ated at three incident antiproton beam momenta, plab =1.7, 3.3 and 6.4 GeV/c (s = 5.4, 8.2, and 13.9 GeV2, re-spectively). Each method use its unique set of criteria forthe background suppression (see Section 4.2 and 4.3).

12

PID probability

0 0.2 0.4 0.6 0.8 1

Eve

nts

310

410

510

610 (a) )+|e+ (ecPID)-|e- (ecPID

PID probability

0 0.2 0.4 0.6 0.8 1

Eve

nts

510

610

710(b) )+|e+π (cPID

)-|e-π (cPID

Figure 10: (a) Total PID probability for e+ to be identified as e+ PIDc(e+|e+) (red dashed) and e− to be identified as

e− PIDc(e−|e−) (black solid). (b) Total PID probability for π+ to be identified as e+ PIDc(π

+|e+) (red dashed) andπ− to be identified as e− PIDc(π

−|e−) (black solid).

4.2 Method I1

All signal events are generated using the differential crosssection parametrized in terms of proton electromagneticFFs, according to Eq. (3). Eq. (11) is used as parameter-ization of |GM |, together with the hypothesis that |GE | =|GM |.

Assuming an integrated luminosity of L = 2 fb−1 (fourmonths of data taking) and 100% efficiency, the expectednumber of the produced e+e− pairs in the range −0.8 <cos θ < 0.8 can be calculated. Table 1 shows the total crosssection and the expected number of events for differentvalues of s. The signal events are generated using Eq. (3)and Table 1.

4.2.1 Particle identification

The event selection is performed in two steps. First, eventshaving exactly one positive and one negative reconstruc-ted charged track are selected for further analysis. Thenumber of reconstructed pairs of particles with an oppos-ite charge are shown in Fig. 12. Note that only in 10% ofthe cases, the multiplicity is larger than one. If an eventhas e.g. one positive and two negative particles, it is con-sidered as carrying a multiplicity of two, because the pos-itive particle could be associated with either of the twonegative particles.

1 This work is a part of D. Khaneft PhD thesis

Pair multiplicity0 2 4 6 8 10

Eve

nts

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310×

Figure 12: Number of reconstructed pairs of particles over106 generated events, with an opposite charge for p̄p →e+e− at q2 = 8.2 (GeV/c)2. Only events with one positiveand one negative reconstructed tracks, i.e., multiplicityequal to 1, were selected.

13

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(f)

Figure 11: Spectra of generated (black) and reconstructed (red) events of different kinematic variables for the signal(top row) and the background (bottom row) at q2 = 8.2 (GeV/c)2: (a,d) the sum of the polar angles; (b,e) the differenceof the azimuthal angles; (c,f) the invariant mass of two reconstructed particles.

Next, all events passing the above selection are filteredthrough a set of additional criteria listed in Table 2. Thesecriteria are chosen in order to maximize signal reconstruc-tion efficiency while suppressing as many background eventsas possible. Some cuts are fixed for all values of q2, whereasothers are optimized to fit the detector’s response at eachenergy.

Table 3 shows the reconstruction efficiency for the sig-nal (e+e−) selection and the background (π+π−) suppres-sion for each value of q2.

q2 [(GeV/c)2] e+e− π+π−

5.4 50.9% 6.8 · 10−6%7.3 53.5% -8.2 46.3% 2.0 · 10−6%11.1 46.2% -12.9 46.6% -13.9 38.7% 2.9 · 10−6%

Table 3: Reconstruction efficiency for the criteria de-scribed in Section 4.2.1 for the signal (e+e−) and the back-ground (π+π−) for each value of q2 (Method I).

4.2.2 Measurement of signal efficiency

An additional, significantly larger, sample of e+e− pairsis simulated for each q2. The signal efficiency is extractedfrom each sample and equals the ratio between number ofreconstructed events passing the dedicated selection, andthe generated events. The uncertainty of the efficiency wascalculated in the following way:

∆ǫi =

√

ǫi(1 − ǫi)

N recoi

, (12)

where ǫi in the efficiency and N recoi is the number of

reconstructed events in the i-th bin. The angular distri-bution of generated electrons, reconstructed and identifiedevents, and the reconstruction efficiency at s = 8.2 GeV2

are presented in Fig. 13.

4.2.3 Extraction of the ratio R

To extract the FFs ratio R, the reconstructed angular dis-tributions first need to be corrected using the efficiencycorrection method described in Section 4.2.2. As a second

14

q2 [(GeV/c)2] 5.4 7.3 8.2 11.1 12.9 13.9PIDc [%] >99 >99 >99 >99 >99 >99PIDs [%] >10 >10 >10 >10 >10 >10dE/dxSTT [a.u.] >5.8 >5.8 >5.8 >5.8 >5.8 >6.5EEMC/preco [GeV/(GeV/c)] >0.8 >0.8 >0.8 >0.8 >0.8 >0.8EMC LM - <0.75 <0.75 <0.75 <0.75 - -EMC E1 [GeV] >0.35 >0.35 >0.35 >0.35 >0.35 >0.35θ + θ′ [degree] 175 < θ + θ′ < 185|φ− φ′| [degree] 175 < |φ− φ′| < 185Minv [GeV/c2] - - >2.2 >2.2 >2.2 >2.7

Table 2: Criteria used to select the signal (e+e−) and suppress the background (π+π−) events for each q2 value(Method I).

θcos1− 0.5− 0 0.5 1

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econ

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ficie

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Figure 13: Angular distribution for p̄p → e+e− at q2 =8.2 (GeV/c)2 of generated (red circles) and reconstruc-ted/identified (blue squares) electrons. The reconstructionefficiency (green triangles) corresponds to the y-axis scaleon the right.

step of this procedure, the corrected angular distributionis fitted using the following equation:

dσ

d cos θ=

πα2

2βs|GM |2

[

(1 + cos2 θ) +R2

τsin2 θ

]

, (13)

where R is a free fit parameter. Equation (11) is used tocalculate the the value of GM for each q2. The reconstruc-ted and acceptance corrected angular distribution for theelectrons is shown together with the fitted curve in Fig. 14.For low q2, where the cross section is higher, the fittedcurve matches the shape of the angular distribution andthe uncertainties are relatively small. At higher q2, the re-constructed angular data points fluctuate and have largerstatistical uncertainties. For q2 = 13.9 (GeV/c)2 the fitrange is reduced to | cos θ| < 0.7, because of the large un-certainties at cos θ = ±0.8. The reduced χ2, i.e. χ2/NDFis close to unity for all q2, except the largest one whereχ2/NDF approaches 2.

4.2.4 Individual extraction of |GE | and |GM |

To extract |GE | and |GM | individually, the differentialcross sections are calculated assuming an integrated lu-minosity of L = 2 fb−1, and using:

σi =Ni

L · 1

Wi

, (14)

where Ni is the number of events in the i-th cos θ binand Wi is the bin width of the i-th bin. Cross sectionuncertainty ∆σ was calculated in the following way:

∆σi =1

Wi

∆Ni

L . (15)

Each differential cross section is fitted using Eq. (3), whichincludes |GE | and |GM | as free parameters.

4.2.5 Results

After the fitting procedure, the ratio R and the individualvalues and uncertainties of |GE | and |GM | are extractedfrom the fit. The extracted FFs ratio is shown in Fig. 15 to-gether with results of other experiments. From this we con-clude that the PANDA experiment will be able to measurethe FFs ratio with a high statistic precision around 1%, atlower q2 than other experiments. Furthermore, PANDAwill provide new measurements in the high q2 domain.

The extracted |GE | and |GM | are shown in Fig. 16along with their statistical uncertainties. |GM | can be meas-ured with uncertainties within the range 2%-9% whereas|GE | has uncertainties within 3%-45%. The difference inprecision between |GE | and |GM | is due to the factor τin the fit function. Table 4 shows the expected values anduncertainties of extracted |GE |, |GM |, and R.

4.3 Method II

The second method enables a verification of the resultsand to check systematic effects. The signal (p̄p → e+e−)is generated with the EvtGen generator in phase space(PHSP) angular distributions. An amount of 106 events isgenerated at each value of the three incident antiproton

15

θcos1− 0.5− 0 0.5 1

Eve

nts

0

10000

20000

30000

40000

50000

(a)

θcos1− 0.5− 0 0.5 1

Eve

nts

0

1000

2000

3000

4000

5000

6000

7000

8000

(b)

θcos1− 0.5− 0 0.5 1

Eve

nts

0

500

1000

1500

2000

2500

3000

3500

(c)

θcos1− 0.5− 0 0.5 1

Eve

nts

0

100

200

300

400

500

(d)

θcos1− 0.5− 0 0.5 1

Eve

nts

0

50

100

150

200

250

(e)

θcos1− 0.5− 0 0.5 1

Eve

nts

0

20

40

60

80

100

120

140

160

(f)

Figure 14: Reconstructed and corrected angular distributions of generated electrons (green squares) and the fit (redline) for different q2 values: (a) 5.4 (GeV/c)2, (b) 7.3 (GeV/c)2, (c) 8.2 (GeV/c)2, (d) 11.1 (GeV/c)2, (e) 12.9 (GeV/c)2,and (f) 13.9 (GeV/c)2.

16

4 6 8 10 12 140.01−

0.008−

0.006−

0.004−

0.002−

0

0.002

0.004

0.006

0.008

0.01

4 6 8 10 12 140.01−

0.008−

0.006−

0.004−

0.002−

0

0.002

0.004

0.006

0.008

0.01

Figure 16: Residual values of |GE | (left panel) and |GM | (right panel) for different q2 values with statistical uncertaintiesonly (Method I).

]2 [(GeV/c)2q4 6 8 10 12 14

R

0

0.5

1

1.5

2

2.5

3BaBar

LEAR

FENICE+DM2

E835

BESIII

PANDA Method I

Figure 15: Form factor ratio extracted from the presentsimulation (magenta circles) as a function of q2, com-pared with the existing data. Data are from Ref. [7] (redsquares), from Ref. [8] (black triangles), and from Ref. [60](green cross and blue star).

momenta plab = 1.7, 3.3 and 6.4 GeV/c (q2 = 5.4, 8.2,and 13.9 (GeV/c)2, respectively). This enables an optim-ization of the event selection with respect to backgroundsuppression, predominately from p̄p → π+π−.

The signal and background (p̄p → π+π−) are ana-lyzed in two steps. First, events with one positive andone negative particle are selected. If the event containsmore than one positive or negative track, e.g secondaryparticles produced by the interaction between generated

q2 R ±∆R |GE | ±∆|GE | |GM | ±∆|GM |(GeV/c)2

5.40 1.0065±0.0129 0.1216±0.0010 0.1208±0.00047.27 1.0679±0.0315 0.0620±0.0013 0.0580±0.00048.21 0.9958±0.0523 0.0435±0.0017 0.0437±0.000511.1 0.9617±0.1761 0.0189±0.0029 0.0197±0.000612.9 1.1983±0.3443 0.0148±0.0033 0.0123±0.000713.8 1.0209±0.5764 0.0108±0.0051 0.0106±0.0010

Table 4: Expected values and uncertainties of extractedR, |GE |, and |GM | (Method I).

primary particles and detector material, the best pair isidentified by one positive and one negative track emittedback-to-back with respect to the c.m. polar angle. Second,reconstructed variables e.g momentum, energy deposit,PID probabilities are studied for the selected events in or-der to achieve the best possible pion rejection. The latterstep is explained below.

4.3.1 PID probability and kinematic cuts

The applied selection criteria are listed in Table 5, forq2 = 5.4, 8.2, and 13.9 (GeV/c)2. The sequential effectof all cuts, i.e., when applied in a sequence one after theother, as well as the individual impact of each single cut,are reported for s = 8.2 GeV2 in Table 6.

The selected reconstructed signal events, as well as theundistorted generated events, are shown in Fig. 17 for q2 =8.2 (GeV/c)2. The intensity drop at cos θ = 0.65 (θlab ∼22.3◦) is due to the transition region between the forwardand the barrel EMC (Fig. 17).

17

q2 [(GeV/c)2] 5.4 8.2 13.9PIDc [%] >99 >99 >99.5PIDs [%] >10 >10 >10dE/dxSTT [a.u.] >6.5 >5.8 0 or >6.5EEMC/preco [GeV/(GeV/c)] >0.8 >0.8 >0.8EMC LM - <0.66 <0.75 <0.66EMC E1 [GeV] >0.35 >0.35 >0.35θ + θ′ [degree] 175 < θ + θ′ < 185|φ− φ′| [degree] 175 < |φ− φ′| < 185Minv [GeV/c2] - >2.2 >2.7

Table 5: Criteria used to select the signal (e+e−) and sup-press the background (π+π−) events for each q2 value(Method II).

Cut individual ǫ [%] sequential ǫ [%]background signal background signal

Acceptance/tracking 83.7 86.4 83.7 86.4PIDc +PIDs 3.5×10−4 70.1 2.9×10−4 60.5dE/dxSTT 15.5 95.1 1.9×10−4 59.3EEMC/preco 4.3×10−2 93.5 1.1×10−4 58.7EMC E1 + LM 2.0 83.7 1.9×10−5 52.6Kinematic cuts 94.9 72.9 9.8×10−7 44.6

Table 6: Individual and sequential efficiency for the signaland the background after the cuts, for s = 8.2 GeV2 and| cos θ| ≤ 0.8 (Method II).

4.3.2 Extraction of the signal efficiency

The signal efficiency was extracted from the events gen-erated according to PHSP, for each cos θ bin. The ratiobetween the PHSP reconstructed events after cuts R(cos θ),and the MC events M(cos θ) represents the signal effi-ciency as a function of the angular distribution. It can bewritten as

ǫ(cos θ) =R(cos θ)

M(cos θ). (16)

The uncertainty ∆R(cos θ) =√

R(cos θ) is attributed tothe center of each bin of R(cos θ). The uncertainty on theefficiency calculated as follows:

∆ǫ(cos θ) =

√

ǫ(cos θ)(1 − ǫ(cos θ))

M(cos θ). (17)

The efficiency distribution of the signal as a function ofcos θ is shown in Figs. 17 and 18 for q2=8.2, 5.4, and 13.9(GeV/c)2. As mentioned above, the reaction mechanismfor pion pair production changes as a function of energyand the shape of the efficiency varies with the energy. Dueto the drop of the efficiency in the region | cos θ| > 0.8, theanalysis for the proton FF measurements is limited to theangular range cos θ ∈ [−0.8, 0.8]. The integrated efficiencyin this region is given in Table 7.

θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Eve

nts

10000

20000

30000

40000

50000

60000

0

0.2

0.4

0.6

0.8

1

Figure 17: Angular distribution in the c.m. system for theMonte Carlo events for e− (green asterisk) and e+ (bluetriangles) generated according to the PHSP, for s = 8.2(GeV/c)2. The reconstructed events after the cuts are alsoshown for e− (black squares) and (e+) (red circles). Theright y-axis represents the efficiency values.

q2 [(GeV/c)2] ǫ(e+e−) ǫ(π+π−)

5.4 41% 1.9 · 10−6%8.2 44.6% 9.8 · 10−7%13.9 40.8% 1.9 · 10−6%

Table 7: Efficiency of the signal (e+e−) and the back-ground (π+π−) integrated over the angular range | cos θ| ≤0.8 for each value of q2 (Method II).

4.3.3 Simulation of events with a realistic angulardistribution

The PHSP events have a flat cos θ distribution in the c.m.system and do not contain the physics of the proton FFs.In reality, the angular distribution can be described ac-cording to Eq. (4). Therefore, the generated cos θ histo-grams are rescaled by the weight ω(cos θ) = 1 +A cos θ2,where A is given according to a model for GE and GM . Inthe following, these events will be referred to as realisticevents.

Note that we do not expect any difference betweenthe simulated electrons and positrons. In one-photon ex-change, the angular distribution in c.m. system is de-scribed by a forward-backward symmetric even functionin cos θ. Possible contributions from radiative corrections(i.e. two-photon exchange diagrams and interference betweeninitial and final state photon emissions) can in principleintroduce odd contributions in cos θ, leading to a forward-backward asymmetry. Being a binary process, in the ab-

18

θ cos

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

∈

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a)

θ cos-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

(b)

Figure 18: Signal efficiency for e+ (red circles) and e− (black squares) in the c.m. system: (a) q2 = 5.4 and (b) 13.9(GeV/c)2.

sence of odd contributions in the amplitudes the detec-tion of an electron at a definite value of cos θ is equival-ent to the detection of a positron at cos(π − θ). This isthe case in the present simulation: one-photon exchangeis implied and the photon emission calculated with thePHOTOS [61] package does not induce any asymmetry.

Once the PANDA experiment is up and running, theefficiency will be validated using experimental data. Inthis analysis we corrected for e+ and e− MC generatedevents separately, since some asymmetry appears at thelevel of reconstruction. This is attributed to the differentinteraction of positive and negative particles with matter.

To determine the number of the realistic undistortedMC events (P (cos θ)) and the realistic reconstructed events(W (cos θ)), three other samples for the reaction p̄p →e+e− are generated using the PHSP angular distributionsat each q2. The P (cos θ) and W (cos θ) histograms are res-caled by the weight w(cos θ) for the case R = 1, A =(τ − 1)/(τ + 1).

4.3.4 Normalization: observed events

The reconstructed realistic events, W (cos θ), are normal-ized according to the integrated count rate Nint(e

+e−)given in Table 1. The integrated count rate depends onthe energy of the system and on the luminosity. The num-ber of observed are expected to be

O(cos θ) = W (cos θ) · Nint(e+e−)

∫ 0.8

−0.8P (cos θ)d cos θ

, (18)

with an uncertainty ∆O(cos θ) =√

O(cos θ) since the ex-perimental uncertainty will finally be given by the accu-mulated statistics of the detected events.

4.3.5 Efficiency correction and fit

The fit procedure was applied to the observed events afterthe correction by the efficiency, F (cos θ):

F (cos θ) = O(cos θ)/ǫ(cos θ), (19)

∆F (cos θ)

F (cos θ)=

√

(

∆O(cos θ)

O(cos θ)

)2

+

(

∆ǫ(cos θ)

ǫ(cos θ)

)2

, (20)

For each q2 value, the distribution F (cos θ) as a func-tion of cos2 θ was fitted with a two-parameter function(straight line). The linear fit function is:

y = a0+a1x, with x = cos2 θ, a0 ≡ σ0, a1 ≡ σ0A, (21)

where a0 and a1 are the parameters to be determinedby minimization. They are related to the FFs, throughEq. (4) and Eq. (5). The number of observed events, be-fore (O(cos θ)) and after (F (cos θ)) efficiency correctionare shown in Fig. 19 for q2 = 5.4 (GeV/c)2.

From the measured angular asymmetry, the FF ratioR can be calculated using:

R =

√

τ1−A1 +A . (22)

19

θ2cos0 0.2 0.4 0.6 0.8

Eve

nts

4000

6000

8000

10000

12000

Figure 19: Observed events before efficiency correctionO(cos θ) (forward events (blue triangles) and backwardevents (orange stars)) and after efficiency correctionF (cos θ) (forward events (red circles) and backward events(green squares)), as a function of cos2 θ for q2 = 5.4(GeV/c)2, assuming R = 1. The line represents the linearfit. The abscissa for backward (forward) events is shiftedby +0.005 (-0.005) for better visualization.

In the limit of small uncertainties, provided first orderstatistic methods work, the uncertainty in R can be ob-tained from standard uncertainty propagation on A:

∆R =1

Rτ

(1 +A)2∆A. (23)

The results of the fit are reported in Table 8. The rel-ative uncertainty on the proton FF ratio increases fromabout 1.4% at the low energy point, to 40% at q2 = 13.9(GeV/c)2.

q2 [(GeV/c)2] R A ∆R ∆A5.40 1 0.210 0.014 0.0148.21 1 0.400 0.050 0.04213.90 1 0.590 0.407 0.264

Table 8: Expected statistical uncertainties on the angularasymmetry and the proton FF ratio, for different s val-ues assuming an integrated luminosity of 2 fb−1 (MethodII). The second and the third columns are the theoret-ical values (simulation inputs). The fourth and the fifthcolumns are the results of the fit on the uncertainties.The statistical uncertainties are extracted in the angularrange | cos θ| ≤ 0.8.

]2 [(GeV/c)2q4 6 8 10 12 14

R0

0.5

1

1.5

2

2.5

3BaBarLEAR

E835FENICE+DM2

PANDA Method IIBESIII

Figure 20: Expected statistic precision on the determin-ation of the proton FF ratio from the present simulation(magenta circles) as a function of q2, compared with theexisting data. Data are from Ref. [7] (red squares), fromRef. [8] (black triangles), and from Ref. [60] (green crossand blue stars). The statistical uncertainties are extractedin the angular range | cos θ| ≤ 0.8. Curves are the graphicrepresentations related to the theoretical predictions, asexplained in the text.

The advantage of the PHSP based generator methodis that the same simulation can be used to test differentmodels, by weighting each event according to the appliedmodel. The points obtained from the present simulationsand the published world data on the proton FF ratio areshown in Fig. 20, for the different values of R predicted bytheoretical models. The curves are theoretical predictionsfrom vector dominance model (solid green line), exten-ded Gary-Krümpelmann (blue dash-dotted), naive quarkmodel (red dashed line) model where the parameters havebeen adjusted as in Ref. [62], and Ref. [3] (black dotted).For a fixed energy point, the relative uncertainty in theproton FF ratio increases when the ratio is approach-ing zero, giving a meaningless value at the highest energypoint q2 = 13.9 (GeV/c)2.

4.3.6 statistical uncertainties on |GE | and |GM |

The individual determination of |GE | and |GM | was ob-tained from a two-parameter fit to the efficiency correctedhistograms, as defined in Section 4.3.5:

y = a+ b cos2 θ, (24)

where a and b are the two parameter fit. Based on Eq. (4),|GE | and |GM | are extracted from a = σ0L and b = σ0A

20

q2 [(GeV/c)2] |GM | = |GE | ∆|GM | ∆|GE |5.40 0.1212 0.0007 0.00108.21 0.0438 0.0007 0.000213.90 0.0109 0.0010 0.0035

Table 9: Expected statistical uncertainties in the protonFFs, for different s values (Method II). The second columnis the theoretical value (simulation input). The third andfourth columns are the results of the fit. The statisticaluncertainties are extracted in the angular range | cos θ| ≤0.8.

by:

|GM |2 =(a+ b)

2N , |GE |2 = τ(a− b)

2N , (25)

N =πα2

2βsL.

Their uncertainties are obtained by:

∆|GM |2 =1

2N√

(∆a)2 + (∆b)2,

∆|GE |2 =τ

2N√

(∆a)2 + (∆b)2, (26)

where (∆a) and (∆b) are the statistical uncertainties, ob-tained from the fit, on a and b respectively. The resultsof the fits are reported in Table 9. The input values of|GM | = |GE | (Eq. (11) is recovered within the uncertaintyranges).

4.3.7 Extrapolation and projected data

The real efficiency will be fine-tuned using real data whenthe experiment is taken into operation. Moreover, the countrates (i.e., the cross sections) have an additional uncer-tainty: different models give different values, especially atlarge s, where they are not very constrained by the data.The experimental uncertainty depends on the square rootof the efficiency.

Figure 21 shows the integrated efficiencies for q2 = 5.4,8.2, 13.9, 16.7, and 29 (GeV/c)2 over the angular range| cos θ| ≤ 0.8. The efficiency at q2 = 16.7 (GeV/c)2 isobtained from the simulation of the signal by applyingthe same cuts as for q2 = 13.9 (GeV/c)2. The efficiencyat q2 = 29 (GeV/c)2 is calculated by extrapolation of theefficiency points obtained from the present simulations atq2 = 5.4, 8.2, 13.9, 16.7 (GeV/c)2.

The predicted values of the effective proton FF, Fp (seeEq. (9)), assuming the hypothesis |GE | = |GM | and usingthe dipole parametrization for the magnetic FF, Eq. (11),are shown together with the world data in Fig. 22.

4.3.8 Results with the full and reduced luminosity mode

FAIR is designed to provide instantaneous luminosities upto 2 × 1032 cm−2 s−1. The results presented in this work

]2

[(GeV/c)2

Q0 10 20 30

[%]

�

0

20

40

Figure 21: Integrated efficiency of the signal p̄p → e+e−

for q2 = 5.4, 8.2, 13.9 (GeV/c)2 (black triangles) andcalculated efficiency (red square) at q2 = 16.7 and 29(GeV/c)2 (see text).

]2|[(GeV/c)2|q0 5 10 15 20 25 30 35 40

D|/G

p|F

1−10

1

10

GM(SL)

GE(SL)

GM(TL)

BaBarE835FenicePS170E760DM1DM2BESBESIIICLEOPANDA Method II

Figure 22: q2-dependence of the world data of the effect-ive proton TL FF, as extracted from the annihilation crosssection, assuming |GE | = |GM |, and on the SL FFs. Theuncertainties obtained or extrapolated from the presentsimulation, for an integrated luminosity of 2 fb−1, areshown by the black squares.

21

correspond to an integrated luminosity of 2 fb−1 which canbe accumulated in about 4 months of data taking. Withthis integrated luminosity, the proton FF ratio (R = 1)can be determined with a statistic precision of 1.4%, 5%and 40.7% at q2 = 5.4, 8.2, and 13.9 (GeV/c)2 respect-ively (Table 8). A separate measurement of |GE | and |GM |is possible. The relative uncertainty on |GE | (|GM |) in-creases from about 0.8% (0.6%) at q2 = 5.4 (GeV/c)2 to37% (9%) at q2 = 13.9 (GeV/c)2 (Table 9). The proton ef-fective FF can be measured over a large momentum rangewith a precision ranging from 0.3% at the lowest energypoint up to 62.4% at q2 = 27.9 (GeV/c)2 (see Fig. 22).

In the startup phase of the PANDA experiment, a lu-minosity of about 20 times lower than that proposed inthis paper is expected. With an integrated luminosity of0.2 fb−1 (8 months of data taking with the reduced scen-ario), the statistical uncertainty on the proton FF ratioincreases by a factor of ∼

√10 compared to the full lu-

minosity mode. As a consequence, the upper limit of themeasurable FF range will be reduced to ∼10 (GeV/c)2.At q2 = 10 (GeV/c)2 the expected precision of FF ratiowill be around 40% (estimated using Method II).

4.4 Comparison

Both methods demonstrate consistent results at q2 = 5.4and 8.2 (GeV/c)2 where the expected statistics is relat-ively high. At q2 = 13.9 (GeV/c)2, Method I gives a lar-ger statistical uncertainty than Method II: at this energypoint, the number of events is small and, as a consequence,the statistic fluctuations are important. This is not takeninto account in Method II, where about 106 events havebeen generated and rescaled. Indeed, the statistic fluctu-ations are arbitrary, and by repeating Method I multipletimes one can obtain a Gaussian distribution of the statist-ical uncertainty where the mean, the most probable value,is equal to the uncertainty value obtained with Method II.The uncertainty distribution of the angular asymmetry Ais verified to be Gaussian symmetric up to q2 = 13.9.

5 Systematic uncertainties

Since a full systematic study requires both experimentaldata and MC, we are limited in our ability to estimateevery possible source. Therefore, we will in the follow-ing discuss some of the sources of systematic uncertain-ties which can be tested with MC only. A more preciseestimate estimation of systematic uncertainties will notbe feasible until design and construction of the detector iscompleted.

When the detector design and construction has con-verged, a more precise estimation of systematic uncertain-ties will be feasible.

5.1 Luminosity measurement

The PANDA experiment will use p̄p elastic scattering forthe luminosity measurement. Based on Ref. [63] the sys-

tematic uncertainty on the luminosity measurement mightvary from 2% to 5%, depending on the beam energy, the p̄pelastic scattering parameterization, and p̄p inelastic back-ground contamination. We considered the relative system-atic luminosity uncertainty ∆L/L to be 4.0%, 2.5%, and2.2% for the q2 = 5.40, 8.21, and 13.9 (GeV/c)2, respect-ively. Table 11 shows the impact of the luminosity uncer-tainty on the precise extraction of GE and GM .

5.2 Detector alignment

Thanks to the almost 4π acceptance of the PANDA de-tector, misalignments of its different components will notaffect the determination of proton FFs. Known displace-ments will be corrected for during the reconstruction. Theeffect of small displacements up to a few hundred micro-meters will give rise to spatial uncertainties that are muchsmaller than the foreseen uncertainties from the trackingresolution.

5.3 Pion background

Using the achieved background rejection factor of 108, wecan estimate the effect of the misidentification of back-ground events contaminating the signal. The analysis inSection 4 was repeated with signal and background eventsmixed together. The number of added background eventswas calculated in accordance with achieved backgroundsuppression.

In addition, the cross section of the background chan-nel p̄p → π+π− will be measured at PANDA with a veryhigh precision due to its large cross section. Therefore, sys-tematic uncertainties due to the model of the backgrounddifferential cross section used in simulations is expected tobe negligible. The impact of the background on the FFsprecision is shown in Table 11.

5.4 Sensitivity to cos θ odd contributions

The analysis above assumes even cos θ angular distribu-tions, as expected from the one-photon exchange mech-anism. However, odd contributions may be present in thedata. One example of the latter is the presence of two-photon exchange (TPE) mechanism, which may play arole at large s and give rise to odd cos θ terms in the an-gular distribution (see Ref. [21]).

In the presence of TPE, the matrix element of the reac-tion p̄p → e+e− contains three complex amplitudes: G̃E ,G̃M and F3 [64], instead of two form factors. The dif-ferential cross section of p̄p → e+e− including the TPEcontributions, can be approximated as:

dσ

d cos θ=

πα2

2q2

√

τ

τ − 1D, (27)

22

with radiative corrections or TPE

D = ( 1 + cos2 θ)|GM |2 + 1

τsin2 θ|GE |2

+ 2

√

τ(τ − 1)

(

GE

τ−GM

)

F3 cos θ sin2 θ. (28)

Only the real part of the three amplitudes, denoted GE ,GM and F3, is taken into account since their relativephases are not known. In the following, we investigate thelimit of a detectable odd cos θ contribution. If present inthe data, the source of the asymmetry may either be morecomplicated underlying physics, e.g. radiative corrections,TPE, or experimental artifacts, e.g. non-symmetric detec-tion of leptons which haven’t been properly corrected for.The MC histograms (produced according to the PHSPmodel) are rescaled according to Eq. (27) with R = 1 andF3/GM = 0, 0.02, 0.05 and 0.20. We fitted the angulardistributions by the function:

y = a0 + a1 cos2 θ + a2 cos θ(1− cos2 θ), (29)

where a2, directly related to the ratio F3/GM , gives therelative size of odd contributions. The results of the fit arereported in Table 10. Below F3/GM = 0.05, a2 is compat-ible with zero, indicating a sensitivity to an asymmetrylarger than 5% at q2 = 5.4 (GeV/c)2. The extraction of Rand A is not affected by the relative size of F3/|GM |.

In case of the charge symmetric detection of electronsand protons, the interference term between the one- andtwo-photon-exchange channels will not contribute to thedifferential cross section [64]. Since PANDA will be able todetect both electrons and positrons (exclusive processes)and the contribution of TPE is symmetric between them,the TPE contribution can be eliminated by adding elec-tron and positron angular distributions.

5.5 Contribution to FFs

The contribution of the luminosity and background tothe precision of extracted values of FFs are reported inTable 11 together with the statistic contribution. The back-ground contamination is on the level of a few percent forall values of q2. The luminosity uncertainty affects onlyGE and GM , since the luminosity measurement is notneeded for R determination. At lower q2 values, where thenumber of signal events is relatively large, the total uncer-tainty is dominated by the background contamination andluminosity contributions. In the intermediate energy do-main, statistics decreases and affects the total uncertaintyon the same level as systematic uncertainties. At higherq2, the main contribution to the total uncertainty is givenby the statistical uncertainty due to the small signal crosssection.

6 Conclusion

Feasibility studies for the measurement of the process p̄p →e+e− at PANDA have been performed and reported in

q2 Stat Systematic[(GeV/c)2] Bg Lumi Total

∆GE/GE

5.40 0.9% 0.3% 2.0% 2.3%8.21 4.1% 2.9% 1.3% 5.2%13.9 48% 3.1% 1.1% 48%

∆GM/GM

5.40 0.4% 2.8% 2.0% 3.4%8.21 1.2% 1.1% 1.3% 2.1%13.9 9.4% 1.0% 1.1% 9.5%

∆ R/R5.40 1.3% 2.9% n/a 3.3%8.21 5.3% 4.0% n/a 6.6%13.9 56% 4.1% n/a 57%

Table 11: Effect of systematic and statistical uncertainties,as well as their total contribution, on the precision of GE ,GM , and R. (Method I)

]2 [(GeV/c)2q4 6 8 10 12 14

R

0

0.5

1

1.5

2

2.5

3BaBar

LEARFENICE+DM2

E835

BESIIIPANDA method I

PANDA method II

Figure 23: Expected statistic precision on the determina-tion of the proton FF ratio from the present simulations(magenta squares and blue down triangles) for R = 1 asa function of q2, compared with the existing data. Dataare from Ref. [7] (red squares), from Ref. [8] (black uptriangles), from Ref. [60] (green cross and blue star). Thestatistical uncertainties are extracted in the angular range| cos θ| ≤ 0.8.

this work. Full simulations for the processes p̄p → e+e−

have been carried out with the PandaRoot software. Anamount of 3 × 108 events of the main background pro-cess p̄p → π+π− has been generated and reconstructedat three energy points. A background suppression factorof the order of ∼ 108 has been achieved, keeping a largeand sufficient signal efficiency for the proton FF meas-urements at PANDA. Two independent simulations havebeen performed for the signal using i) two different modelsin the event generator, ii) different number of generatedevents iii) two sets of event selection criteria and iv) twofit functions to extract the proton electromagnetic FFs.The results from the two simulations, assuming R = 1,are shown in Fig. 23, together with the existing experi-mental data. For q2 = 5.4 and 8.2 (GeV/c)2, the resultsare consistent with each other.

23

q2 [(GeV/c)2] F3/|GM | [%] a0 a1 a2 ∆R ∆A5.4 0 10045±34 2080±130 76±83 0.014 0.015.4 2 10045±34 2080±130 -0.6±83 0.014 0.015.4 5 10045±34 2080±130 -115±83 0.014 0.015.4 20 10045±34 2080±130 -687±83 0.014 0.018.2 0 579±6 231±24 -3.5±14.8 0.05 0.048.2 2 579±6 231±24 -19.9±15 0.05 0.048.2 5 579±6 231±24 -44.3±15 0.05 0.048.2 20 579±6 231±24 -166±15 0.05 0.0413.9 0 17±1 10±4.2 -0.1±2.6 0.4 0.2613.9 2 17±1 10±4.2 -1.4±2.6 0.4 0.2613.9 5 17±1 10±4.2 -3.4±2.6 0.4 0.2613.9 20 17±1 10±4.2 -13.4±2.6 0.4 0.25

Table 10: The results from the fit of the angular distributions using Eq. (29), for q2 = 5.4, 8.2 and 13.9 (GeV/c)2.

The determination of the statistical uncertainties onR has been extended with Method II to different modelsof proton FF ratio. A larger relative uncertainty has beenobtained for R < 1 than for the case R = 1.

Compared to the previous analysis [11], the GEANT4description of the detector is more realistic. Furthermore,the MC data are digitized and reconstructed using real-istic pattern recognition and tracking algorithms whichwas not the case in the old software framework. The totalsignal efficiency is improved by 5–10% in comparison tothe previous studies [11] thanks to the new selection pro-cedure and new PID capabilities available in the Panda-Root framework. However, the experimental cuts will befine-tuned using the real data.

The PANDA experiment at FAIR will extend the know-ledge of the TL electromagnetic FFs in a large kinematicrange. The present results show that the statistical uncer-tainty at q2 ≥ 14 (GeV/c)2 will be comparable to the oneobtained by BABAR at ∼ 7 (GeV/c)2.

The study of the systematic uncertainties shows thatthe background misidentification and luminosity uncer-tainty dominate the total uncertainty at lower q2, whilein the high energy domain the total uncertainty is dom-inated by the statistical fluctuations due to the smallerp̄p → e+e− cross section. The total relative uncertaintyof individual FFs is expected to be around 2–48% andaround 3–57% for the ratio R.

The absolute cross section measurement depends es-sentially on the precision achieved in the luminosity meas-urement, which is expected to lie between 1% and 2%. ThePANDA experiment at FAIR will allow the individual de-termination of the FFs in the TL region, from 5.4 GeV/cto 13.9 GeV/c. This is essential for a global analysis of theFFs in the SL and TL regions, to test the models whichapply in the whole kinematic range and require analyticalcontinuation of FFs.

The moduli of the individual FFs, GE , and GM will bemeasured for the first time at BESIII using the data col-lected at 20 different beam energies between 4.0 and 9.5GeV2 [65]. A statistic precision on the FF ratio between9% and 35% is expected. Based on these numbers, it isclear that PANDA will extend these measurements up toa higher energy range (14 GeV2), with a precision bet-

ter than expected at BESIII or comparable in case of thereduced luminosity mode.

The modulus of the proton FFs ratio can be also meas-ured at Belle [66], using the initial state radiation (ISR)technique, with a comparable accuracy to the BABARdata . The Belle detector was operating on the KEKBe+e− collider [67]. An integrated luminosity of about 1040fb−1 was collected at KEKB between 1999 to 2010. Mostof the data were taken at the Y (4S) resonance. The up-graded facility of the KEKB collider (SuperKEKB) aimsto accumulate 50 ab−1 around 2025 [68]. The Belle II ex-periment at the superKEKB collider may provide the mostaccurate data on the proton FF ratio. So far, no estima-tion has been presented by the collaboration for the FFsmeasurement at Belle and Belle II. One disadvantage ofthe ISR technique is that it only allow extraction of FFsin wide bins of q2. This is in contrast to direct produc-tion, that PANDA will utilize, where the precision of q2

is given by the, in general, very precise beam momentumresolution.

7 Acknowledgments

The success of this work relies critically on the expertiseand dedication of the computing organizations that sup-port PANDA. We acknowledge financial support from theScience and Technology Facilities Council (STFC), Brit-ish funding agency, Great Britain; the Bhabha AtomicResearch Center (BARC) and the Indian Institute ofTechnology, Mumbai, India; the Bundesministerium fürBildung und Forschung (BMBF), Germany; the Carl-Zeiss-Stiftung 21-0563-2.8/122/1 and 21-0563-2.8/131/1,Mainz, Germany; the Center for Advanced RadiationTechnology (KVI-CART), Gröningen, the Netherlands;the CNRS/IN2P3 and the Université Paris-Sud, France;the Deutsche Forschungsgemeinschaft (DFG), Germany;the Deutscher Akademischer Austauschdienst (DAAD),Germany; the Forschungszentrum Jülich GmbH, Jülich,Germany; the FP7 HP3 GA283286, European Commis-sion funding; the Gesellschaft für SchwerionenforschungGmbH (GSI), Darmstadt, Germany; the Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF), Ger-many; the INTAS, European Commission funding; the

24

Institute of High Energy Physics (IHEP) and theChinese Academy of Sciences, Beijing, China; the Isti-tuto Nazionale di Fisica Nucleare (INFN), Italy; theMinisterio de Educación y Ciencia (MEC) under grantFPA2006-12120-C03-02, Spain; the Polish Ministry of Sci-ence and Higher Education (MNiSW) grant No. 2593/7,PR UE/2012/2, and the National Science Center (NCN)DEC-2013/09/N/ST2/02180, Poland; the State AtomicEnergy Corporation Rosatom, National Research CenterKurchatov Institute, Russia; the Schweizerischer Nation-alfonds zur Forderung der wissenschaftlichen Forschung(SNF), Switzerland; the Stefan Meyer Institut für Subato-mare Physik and the Österreichische Akademie der Wis-senschaften, Wien, Austria; the Swedish Research Coun-cil, Sweden.

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