Measurement of initial-state-final-state radiation interference in the processes e+e- →μ+μ-γ and e+e- →π+π-γ

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Measurement of initial-state

–final-state radiation interference in the

processes e

þ

_e

−

_{→ μ}

þ

_μ

−

_{γ and e}

þ

_e

−

_{→ π}

þ

_π

−

_γ

J. P. Lees,1 V. Poireau,1 V. Tisserand,1 E. Grauges,2 A. Palano,3a,3b G. Eigen,4 B. Stugu,4 D. N. Brown,5 L. T. Kerth,5

Yu. G. Kolomensky,5M. J. Lee,5 G. Lynch,5 H. Koch,6T. Schroeder,6 C. Hearty,7T. S. Mattison,7 J. A. McKenna,7

R. Y. So,7A. Khan,8V. E. Blinov,9a,9b,9cA. R. Buzykaev,9aV. P. Druzhinin,9a,9bV. B. Golubev,9a,9bE. A. Kravchenko,9a,9b

A. P. Onuchin,9a,9b,9cS. I. Serednyakov,9a,9b Yu. I. Skovpen,9a,9bE. P. Solodov,9a,9bK. Yu. Todyshev,9a,9bA. J. Lankford,10

B. Dey,11J. W. Gary,11O. Long,11M. Franco Sevilla,12T. M. Hong,12D. Kovalskyi,12J. D. Richman,12C. A. West,12

A. M. Eisner,13W. S. Lockman,13W. Panduro Vazquez,13B. A. Schumm,13A. Seiden,13 D. S. Chao,14C. H. Cheng,14

B. Echenard,14K. T. Flood,14D. G. Hitlin,14J. Kim,14T. S. Miyashita,14P. Ongmongkolkul,14F. C. Porter,14M. Röhrken,14

R. Andreassen,15Z. Huard,15B. T. Meadows,15B. G. Pushpawela,15M. D. Sokoloff,15L. Sun,15W. T. Ford,16J. G. Smith,16

S. R. Wagner,16R. Ayad,17,†W. H. Toki,17B. Spaan,18D. Bernard,19M. Verderi,19S. Playfer,20D. Bettoni,21aC. Bozzi,21a

R. Calabrese,21a,21bG. Cibinetto,21a,21bE. Fioravanti,21a,21bI. Garzia,21a,21bE. Luppi,21a,21bL. Piemontese,21aV. Santoro,21a

A. Calcaterra,22R. de Sangro,22G. Finocchiaro,22S. Martellotti,22P. Patteri,22I. M. Peruzzi,22M. Piccolo,22A. Zallo,22

R. Contri,23a,23bM. R. Monge,23a,23b S. Passaggio,23a C. Patrignani,23a,23b B. Bhuyan,24V. Prasad,24A. Adametz,25

U. Uwer,25H. M. Lacker,26U. Mallik,27C. Chen,28 J. Cochran,28S. Prell,28H. Ahmed,29A. V. Gritsan,30N. Arnaud,31

M. Davier,31D. Derkach,31G. Grosdidier,31F. Le Diberder,31A. M. Lutz,31B. Malaescu,31,‡ P. Roudeau,31A. Stocchi,31

L. L. Wang,31,§ G. Wormser,31D. J. Lange,32D. M. Wright,32 J. P. Coleman,33 J. R. Fry,33E. Gabathuler,33

D. E. Hutchcroft,33D. J. Payne,33C. Touramanis,33A. J. Bevan,34F. Di Lodovico,34R. Sacco,34G. Cowan,35

D. N. Brown,36C. L. Davis,36A. G. Denig,37M. Fritsch,37W. Gradl,37K. Griessinger,37A. Hafner,37K. R. Schubert,37

R. J. Barlow,38,¶G. D. Lafferty,38R. Cenci,39B. Hamilton,39A. Jawahery,39D. A. Roberts,39 R. Cowan,40R. Cheaib,41

P. M. Patel,41,* S. H. Robertson,41 N. Neri,42a F. Palombo,42a,42bL. Cremaldi,43R. Godang,43,** D. J. Summers,43

M. Simard,44P. Taras,44G. De Nardo,45a,45bG. Onorato,45a,45bC. Sciacca,45a,45bG. Raven,46C. P. Jessop,47J. M. LoSecco,47

K. Honscheid,48R. Kass,48M. Margoni,49a,49bM. Morandin,49a M. Posocco,49a M. Rotondo,49a G. Simi,49a,49b

F. Simonetto,49a,49bR. Stroili,49a,49bS. Akar,50E. Ben-Haim,50M. Bomben,50G. R. Bonneaud,50H. Briand,50G. Calderini,50

J. Chauveau,50Ph. Leruste,50G. Marchiori,50J. Ocariz,50M. Biasini,51a,51bE. Manoni,51aA. Rossi,51a C. Angelini,52a,52b

G. Batignani,52a,52bS. Bettarini,52a,52b M. Carpinelli,52a,52b,††G. Casarosa,52a,52bM. Chrzaszcz,52a F. Forti,52a,52b

M. A. Giorgi,52a,52bA. Lusiani,52a,52cB. Oberhof,52a,52b E. Paoloni,52a,52bM. Rama,52a G. Rizzo,52a,52bJ. J. Walsh,52a

D. Lopes Pegna,53J. Olsen,53 A. J. S. Smith,53F. Anulli,54a R. Faccini,54a,54b F. Ferrarotto,54aF. Ferroni,54a,54b

M. Gaspero,54a,54b A. Pilloni,54a,54bG. Piredda,54a C. Bünger,55S. Dittrich,55O. Grünberg,55 M. Hess,55T. Leddig,55

C. Voß,55R. Waldi,55 T. Adye,56E. O. Olaiya,56F. F. Wilson,56S. Emery,57G. Vasseur,57D. Aston,58D. J. Bard,58

C. Cartaro,58 M. R. Convery,58J. Dorfan,58G. P. Dubois-Felsmann,58W. Dunwoodie,58M. Ebert,58R. C. Field,58

B. G. Fulsom,58 M. T. Graham,58 C. Hast,58W. R. Innes,58P. Kim,58D. W. G. S. Leith,58S. Luitz,58V. Luth,58

D. B. MacFarlane,58D. R. Muller,58H. Neal,58T. Pulliam,58B. N. Ratcliff,58A. Roodman,58R. H. Schindler,58A. Snyder,58

D. Su,58M. K. Sullivan,58J. Va’vra,58W. J. Wisniewski,58H. W. Wulsin,58M. V. Purohit,59J. R. Wilson,59

A. Randle-Conde,60S. J. Sekula,60M. Bellis,61P. R. Burchat,61E. M. T. Puccio,61M. S. Alam,62J. A. Ernst,62

R. Gorodeisky,63N. Guttman,63D. R. Peimer,63A. Soffer,63S. M. Spanier,64J. L. Ritchie,65R. F. Schwitters,65J. M. Izen,66

X. C. Lou,66F. Bianchi,67a,67bF. De Mori,67a,67bA. Filippi,67a D. Gamba,67a,67bL. Lanceri,68a,68b L. Vitale,68a,68b

F. Martinez-Vidal,69A. Oyanguren,69J. Albert,70Sw. Banerjee,70A. Beaulieu,70F. U. Bernlochner,70H. H. F. Choi,70

G. J. King,70R. Kowalewski,70M. J. Lewczuk,70T. Lueck,70I. M. Nugent,70J. M. Roney,70R. J. Sobie,70N. Tasneem,70

T. J. Gershon,71P. F. Harrison,71 T. E. Latham,71H. R. Band,72S. Dasu,72Y. Pan,72R. Prepost,72and S. L. Wu72

(The BABAR Collaboration)

1_{Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3,}

F-74941 Annecy-Le-Vieux, France

2_{Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain} 3a_{INFN Sezione di Bari, I-70126 Bari, Italy}

3b_{Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy} 4_{University of Bergen, Institute of Physics, N-5007 Bergen, Norway}

5_{Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA} 6_{Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany}

7_{University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1} 8_{Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom} 9a_{Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia}

9b_{Novosibirsk State University, Novosibirsk 630090, Russia} 9c_{Novosibirsk State Technical University, Novosibirsk 630092, Russia}

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10_{University of California at Irvine, Irvine, California 92697, USA} 11_{University of California at Riverside, Riverside, California 92521, USA} 12_{University of California at Santa Barbara, Santa Barbara, California 93106, USA}

13_{University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA} 14_{California Institute of Technology, Pasadena, California 91125, USA}

15_{University of Cincinnati, Cincinnati, Ohio 45221, USA} 16_{University of Colorado, Boulder, Colorado 80309, USA} 17_{Colorado State University, Fort Collins, Colorado 80523, USA}

18_{Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany} 19_{Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France}

20_{University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom} 21a_{INFN Sezione di Ferrara, I-44122 Ferrara, Italy}

21b_{Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy} 22_{INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy}

23a_{INFN Sezione di Genova, I-16146 Genova, Italy}

23b_{Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy} 24_{Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India} 25_{Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany} 26_{Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany}

27_{University of Iowa, Iowa City, Iowa 52242, USA} 28_{Iowa State University, Ames, Iowa 50011-3160, USA}

29_{Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia} 30_{Johns Hopkins University, Baltimore, Maryland 21218, USA}

31_{Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,}

Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France

32_{Lawrence Livermore National Laboratory, Livermore, California 94550, USA} 33_{University of Liverpool, Liverpool L69 7ZE, United Kingdom} 34_{Queen Mary, University of London, London E1 4NS, United Kingdom} 35_{University of London, Royal Holloway and Bedford New College, Egham,}

Surrey TW20 0EX, United Kingdom

36_{University of Louisville, Louisville, Kentucky 40292, USA}

37_{Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany} 38_{University of Manchester, Manchester M13 9PL, United Kingdom}

39_{University of Maryland, College Park, Maryland 20742, USA} 40_{Massachusetts Institute of Technology, Laboratory for Nuclear Science,}

Cambridge, Massachusetts 02139, USA

41_{McGill University, Montréal, Québec, Canada H3A 2T8} 42a_{INFN Sezione di Milano, I-20133 Milano, Italy}

42b_{Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy} 43_{University of Mississippi, University, Mississippi 38677, USA}

44_{Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7} 45a_{INFN Sezione di Napoli, I-80126 Napoli, Italy}

45b_{Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy} 46_{NIKHEF, National Institute for Nuclear Physics and High Energy Physics,}

NL-1009 DB Amsterdam, The Netherlands

47_{University of Notre Dame, Notre Dame, Indiana 46556, USA} 48_{Ohio State University, Columbus, Ohio 43210, USA}

49a_{INFN Sezione di Padova, I-35131 Padova, Italy}

49b_{Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy} 50_{Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS,}

Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France

51a_{INFN Sezione di Perugia, I-06123 Perugia, Italy}

51b_{Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy} 52a_{INFN Sezione di Pisa, I-56127 Pisa, Italy}

52b_{Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy} 52c_{Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy} 53_{Princeton University, Princeton, New Jersey 08544, USA}

54a_{INFN Sezione di Roma, I-00185 Roma, Italy}

54b_{Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy} 55_{Universität Rostock, D-18051 Rostock, Germany}

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56_{Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom} 57_{CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France}

58_{SLAC National Accelerator Laboratory, Stanford, California 94309 USA} 59_{University of South Carolina, Columbia, South Carolina 29208, USA}

60_{Southern Methodist University, Dallas, Texas 75275, USA} 61_{Stanford University, Stanford, California 94305-4060, USA} 62_{State University of New York, Albany, New York 12222, USA} 63_{Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel}

64_{University of Tennessee, Knoxville, Tennessee 37996, USA} 65_{University of Texas at Austin, Austin, Texas 78712, USA} 66_{University of Texas at Dallas, Richardson, Texas 75083, USA}

67a_{INFN Sezione di Torino, I-10125 Torino, Italy}

67b_{Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy} 68a_{INFN Sezione di Trieste, I-34127 Trieste, Italy}

68b_{Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy} 69_{IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain} 70_{University of Victoria, Victoria, British Columbia, Canada V8W 3P6} 71_{Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom}

72_{University of Wisconsin, Madison, Wisconsin 53706, USA}

(Received 1 September 2015; published 28 October 2015)

Charge asymmetry in the processes eþ_e−_{→ μ}þ_μ−_{γ and e}þ_e−_{→ π}þ_π−_{γ is measured using 232 fb}−1_of

data collected with the BABAR detector at eþ_e−_{center-of-mass energies near 10.58 GeV. An observable is}

introduced and shown to be very robust against detector asymmetries while keeping a large sensitivity to the physical charge asymmetry that results from the interference between initial- and final-state radiation (FSR). The asymmetry is determined as a function of the invariant mass of the final-state tracks from production threshold to a few GeV=c2_{. It is compared to the expectation from QED for e}_þ_e−_{→ μ}þ_μ−_{γ, and}

from theoretical models for eþ_e−_{→ π}þ_π−_{γ. A clear interference pattern is observed in e}þ_e−_{→ π}þ_π−_γ,

particularly in the vicinity of the f2ð1270Þ resonance. The inferred rate of lowest-order FSR production is

consistent with the QED expectation for eþ_e−_{→ μ}þ_μ−_{γ, and is negligibly small for e}þ_e−_{→ π}þ_π−_γ.

DOI:10.1103/PhysRevD.92.072015 PACS numbers: 13.60.Hb, 13.66.Bc, 13.66.Jn

I. INTRODUCTION The radiative processes

eþ_e−_{→ Xγ} _ð1Þ

have been extensively studied by several eþ_e−_experiments

and the cross sections for eþ_e−_{→ X have been measured}

using the initial-state radiation (ISR) method [1–4]. At

BABAR[5], the cross sections have thus been determined in

large energy ranges below the total eþ_e− _{center-of-mass}

(c.m.) energy p ∼ 10.58GeV available at the SLACffiffiffis

PEP-II collider. The state X can be either fully described

by quantum electrodynamics (QED) such as μþ_μ−_{, or any}

hadronic state with JPC

¼ 1−−.

In reaction (1)at lowest order (LO) the photon can be

emitted from either the incoming electron or positron, or from the final state (final-state radiation, or FSR). At BABAR, the kinematic conditions are such that the process is dominated by ISR photons, which justifies the ISR method. The LO FSR contribution to the hadronic radiative process is neglected, as its theoretical estimates are well below the systematic uncertainties of the cross section

measurement. This is due to the fact that the available eþ_e−

c.m. energy is far beyond the domain of the hadronic resonances that dominate the cross section, so that hadronic form factors considerably reduce the probability that the photon is emitted from the final state. However, the theoretical estimations are model dependent, and it is thus important to have a direct experimental proof of the smallness of the FSR contribution to the hadronic cross sections when high precision is at stake, as for the

determination of the hadronic contribution to the g− 2

value of the muon[6]. Because of the point-like nature of

the muon, the FSR reduction does not occur for the eþ_e− _→

μþ_μ−_{γ process. The LO FSR contribution to the cross}

section is expected to vanish at threshold and to increase

*_Deceased.

†_{Now at: University of Tabuk, Tabuk 71491, Saudi Arabia.} ‡_{Now at: Laboratoire de Physique Nucléaire et de Hautes} Energies, IN2P3/CNRS, F-75252 Paris, France.

§_{Now at: Institute of High Energy Physics, Beijing 100049,} China.

¶_{Now at: University of Huddersfield, Huddersfield HD1 3DH,} United Kingdom.

**_{Now at: University of South Alabama, Mobile, Alabama} 36688, USA.

††_{Also at: Università di Sassari, I-07100 Sassari, Italy.}

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with the invariant mass of the muon pair (mμμ). Still, the FSR fraction remains small for low di-muon mass (less than

1% for m_μμ< 1 GeV=c2_{). For the e}_þ_e− _{→ μ}þ_μ−_{γ cross}

section measurement, a correction is applied for the LO

FSR contribution as a function of mμμ, which is so far

determined by turning off FSR in the Monte Carlo (MC) generation.

While it is not possible to distinguish ISR from FSR photons on an event-by-event basis, as the corresponding amplitudes are both present and interfere, a measurement of the interference provides a sensitive and quantitative determination of their relative strength. Measurement of the forward-backward asymmetry of the pions was first

proposed in Ref.[3], as a test of the underlying model for

final-state radiation. In this paper, the ISR-FSR interference

for eþ_e− _{→ μ}þ_μ−_{γ and e}þ_e− _{→ π}þ_π−_{γ is studied through}

the charge asymmetry of the production of these events at various decay plane angles. The comparison between the QED prediction and the measurement is done for the

charge asymmetry in eþ_e− _{→ μ}þ_μ−_{γ. Various FSR models}

are discussed for eþ_e− _{→ π}þ_π−_{γ, and the most realistic}

quark-FSR model is compared to the measurement of the charge asymmetry in that channel.

This paper reports the first measurement of charge

asymmetry in the eþ_e− _{→ μ}þ_μ−_{γ process. For e}þ_e−_→

πþ_π−_{γ, a preliminary measurement} _[7] _{of the}

forward-backward asymmetry has been reported at low energies

( ffiffiffisp ∼ 1GeV). No previous result exists at high energies.

II. ISR-FSR INTERFERENCE AND CHARGE ASYMMETRY

A. Charge asymmetry

The Feynman diagrams for the LO ISR and LO FSR

emission in the process eþ_e− _{→ x}þ_x−_{γ (where x ¼ μ or π),}

are illustrated in Fig.1. The total LO amplitude M is the

sum of the corresponding amplitudes MISRand MFSR, and

the cross section for eþ_e−_{→ x}þ_x−_{γ is}

σ_{∝ jMj}2

¼ jMISRj2þ jMFSRj2þ 2ReðMISRM%FSRÞ:

ð2Þ If the photon is emitted from the initial (final) state, the

xþ_x− _{pair is produced with charge parity C ¼ −1ðþ1Þ,}

which implies that the interference term changes sign if one

interchanges xþ _{and x}−_{. While the contribution of the}

interference term to the total cross section vanishes when one integrates over the kinematic variables of the final state, that term induces a significant observable charge asymme-try in the differential cross section.

Charge asymmetry is defined as

A ¼jMj2− jMxþ↔x−j2 jMj2_{þ jM} xþ_↔x−j2 ¼ 2ReðMISRM%FSRÞ jMISRj2þ jMFSRj2 ; ð3Þ

where xþ_{↔ x}− _{means that x}þ _{and x}− _{are interchanged.}

Although it is not possible to reconstruct MISRor MFSR

from the charge asymmetry and the cross section, as the relative phase between them remains unknown,

informa-tion on the ratio jMFSR=MISRj can be derived within the

framework of specific models.

B. Choice of kinematic variables

Aside from an overall azimuthal rotation about the beam

axis, the kinematic topology of the xþ_x−_{γ final state (where}

x ¼ μ or π) is described by four variables, which are the

muon-pair (pion-pair) invariant mass mxx(or equivalently

E%

γ, the energy of the radiated photon in the eþe−c.m.) and

three angular variables. At a given mxxmass, the

distribu-tion of the three angular variables contains all the available information on the ISR/FSR amplitudes.

At variance with the definition of forward-backward

asymmetry used in Ref.[8], which refers to the polar angle

of x−_{with respect to the incoming electron in the e}þ_e−_c.m.

system (c.m.s.), this analysis introduces the set of angular

variables illustrated in Fig.2. These are found to be more

sensitive observables to measure the ISR-FSR interference: (i) θ%

γ—polar angle of the radiated photon in the eþe−

c.m.s. (with respect to the eþ_e− _axis);

(ii) θ%_{—polar angle of x}−_{with respect to the photon axis}

in the xþ_x− _c.m.s.;

(iii) ϕ%_{—azimuthal angle of x}−_{with respect to the γe}þ_e−

plane in the xþ_x− _{c.m.s. (or the e}þ_e− _c.m.s.).

Since xþ_{↔ x}− _{interchange means reversal of the x}−

direction to its opposite in the xþ_x− _{c.m.s., the charge}

asymmetry, for fixed mxx and θ%γ, is equal to

FIG. 1. Feynman diagrams for eþ_e−_{→ x}þ_x−_{γ (x ¼ μ, π), where the photon is from LO ISR (left) or LO FSR (right).}

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Aðθ%_{; ϕ}%_{Þ ¼}σðθ%; ϕ%Þ − σðπ − θ%; π þ ϕ%Þ

σðθ%_{; ϕ}%_{Þ þ σðπ − θ}%_{; π þ ϕ}%_Þ: ð4Þ

For the eþ_e− _{→ μ}þ_μ−_{γ process, the charge asymmetry}

as a function of cos θ% _{and ϕ}%_{, studied with the} _AFKQED

generator (see Sec. IV B), is shown in Fig. 3. The FSR

amplitude is dominant at j cos θ%_{j ∼ 1, when one of the}

charged-particle tracks is very close to the radiated photon.

However, Fig.3shows that ϕ%_{is a more sensitive variable}

to measure the ISR/FSR content over the full phase space, with sign reversal of the charge asymmetry. After

integra-tion over cos θ%_γ _{and integration over symmetrical cos θ}%

intervals, the distribution of the integrated charge

asym-metry Aðcos ϕ%_{Þ suggests a simple linear dependence}

Aðcos ϕ%_{Þ ¼ A}

0cos ϕ%: ð5Þ

From the expressions of the differential cross section

detailed in the next section, it results that the slope A0 is

an estimator of the ISR-FSR interference, sensitive to the

ratio jMFSR=MISRj in each mμμ interval. Moreover, it will

be shown in Sec.V that the measurement of A0is barely

affected by detector charge asymmetries.

III. THEORETICAL PREDICTIONS FOR THE CHARGE ASYMMETRY

A. QED prediction for the eþ_e−_{→ μ}þ_μ−_{γ process}

In the massless limit[9], the differential cross section of

the QED eþ_e− _{→ μ}þ_μ−_{γ process, written as a function of}

the four kinematic variables defined above (Sec. II B),

implies that the differential charge asymmetry is propor-tional to cos ϕ%_: Aeþ_e−_→μþ_μ−_γðm_μμ; θ%_γ; θ%; ϕ%Þ ¼ −2p mffiffiffis μμsin θ % γsin θ%cos ϕ% s sin2_θ_% þ m2 μμsin2θ%γ : _ð6Þ

When the masses are taken into account, the effect from the electron/positron mass is found to be negligible for radiated photons away from the beams. The effect from the

muon mass is sizable, especially at large mμμ when the

radiated photon is close to one of the muons. Predictions for the charge asymmetry in the massive case are obtained by numerical integration of several variants of the QED

differential cross section[9–11]. The phase space

consid-ered in those calculations is limited to the experimental

acceptance 20° < θ%

γ < 160°, and the results are shown in

FIG. 2. Definition of the angular variables describing the kinematic topology of the final states of the process eþ_e−_{→ x}þ_x−_γ

ðx ¼ μ; πÞ at a given xþ_x− _{invariant mass (left) in the e}þ_e−_{c.m.s., and (right) in the x}þ_x− _c.m.s.

-1 -0.5 0 0.5 1 -2 0 2 -0.5 0 0.5 cosθ∗ φ* (rad ) Asymmetry -1 -0.5 0 0.5 1 -2 0 2 -0.5 0 0.5 cosθ∗ φ* (rad ) Asymmetry -1 -0.5 0 0.5 1 -2 0 2 -0.5 0 0.5 cosθ∗ φ* (rad ) Asymmetry

FIG. 3. Charge asymmetry at generator level in eþ_e−_{→ μ}þ_μ−_{γ simulation, as a function of cos θ}%_{and ϕ}%_{for the same m}_μμ_interval

(6.5 < m_μμ< 7.0 GeV=c2_{) and various cos θ}_%

γ ranges: (left) −1 < cos θ%γ<−0.6, (middle) −0.6 < cos θ%γ <−0.4, (right) −0.4 <

cos θ%_γ _{< 0.}

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Fig.4as a function of mμμ. Predictions differ at the physical

threshold (m_μμ _{¼ 2m}_μ), where only the charge asymmetry

based on Ref. [11] extrapolates to zero as expected,

suggesting that the validity of formulas in Refs. [9,10]

does not extend to small mμμ. At large mass (mμμ >

3 GeV=c2_{), the prediction from Ref.} _[9] _{differs from the}

others by up to a few percent. The formula of the

differ-ential LO cross section implemented in the AFKQED

generator, which is used in this analysis for simulation

(see Sec.IV B), is the one by Arbuzov et al.[11], which has

the most reliable behavior over the full mμμ range.

B. FSR models for the eþ_e− _{→ π}þ_π−_{γ process}

As in the eþ_e− _{→ μ}þ_μ−_{γ process, ISR and FSR}

con-tribute to eþ_e−_{→ π}þ_π−_{γ (Fig.} ₅_{). However, the charge}

asymmetry is expected to be much smaller in the latter process because the FSR contribution is strongly reduced

by the pion form factor at large ffiffiffisp . In addition, its estimate

is model dependent.

1. FSR from point-like pions (model 1)

In the FSR model shown in Fig. 5(b), the photon is

emitted from one of the final-state pions, where the pion is treated as a point-like particle. In this hypothesis, the FSR

amplitude MFSRis proportional to the pion form factor at

the collision energy squared s, namely FπðsÞ. The ISR

amplitude MISR shown in Fig.5(a) is proportional to the

pion form factor F_π_ðs0_{Þ at a reduced energy squared}

s0 _{¼ sð1 − 2E}%

γ= ffiffiffisp Þ. According to this FSR model, the

charge asymmetry to be measured at BABAR reflects the

relative magnitude of the pion form factor at pffiffiffis¼

10.58 GeV and at low energy. It is consequently negligibly

small, since Fπðs0Þ, dominated by the ρ resonance in the

s0 _{¼ m}2

ππ domain accessible to the experiment, is 3 orders

of magnitude larger than jFπð10.582GeV2Þj ∼ 0.01, as

estimated from an extrapolation of existing data [6,12]

using a 1=s dependence. This model is studied with the PHOKHARA4.0[13]generator, in which the FSR current has a point-like Lorentz structure, including a contact term,

m_µµ (GeV/c2) A0 GW BK AF -1 -0.8 -0.6 -0.4 -0.2 0 m_µµ(GeV/c2₎ ∆ A0 = A( φ * =0) - (A 0 )AfkQed (LO) GW BK AF -0.1 -0.05 0 0.05 0.1 0 2 4 6 0 2 4 6

FIG. 4. (Left) Charge asymmetry at ϕ%_{¼ 0, A}₀_{, as a function of m}_μμ_{, obtained by numerical integration according to three different}

theoretical predictions (see text), with the condition 20° < θ%_γ _{< 160° applied. (Right) The difference between the prediction and the}

AFKQEDLO value. Results labeled GW, BK, AF are obtained from Refs.[9–11], respectively.

(a) (b) (c)

FIG. 5. Feynman diagrams for eþ_e−_{→ π}þ_π−_{γ. (a) Initial state radiation, (b) Final state radiation with pions treated as point-like}

particles (FSR model 1), (c) Final state radiation at quark level (FSR model 2).

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globally multiplied by the pion form factor. In this model,

the A0ðmππÞ distribution is expected to increase

quadrati-cally with mass on the ρ resonance, with a change of sign at the ρ mass

A0∼ 2 × 10−3ðm2ππ− m2ρÞ; ð7Þ

with values well below the sensitivity of this analysis because of the large pion form factor suppression at 10.58 GeV.

2. FSR from quarks (model 2)

In the a priori more realistic FSR model for eþ_e−_→

πþ_π−_{γ depicted in Fig.} _5(c)_{, the FSR photon is emitted}

from the quarks, which subsequently hadronize into a pion

pair[14]. The dominant ISR1 _{and FSR contributions, and}

their interference, are written in terms of the variables

defined in Sec. II B: dσISR eþ_e−_→πþ_π−_γ dm2 ππd cos θ%γd cos θ%dϕ% ¼ α 3_β3 16πs2_m2 ππðs − m2ππÞjFπðm 2 ππÞj2 × " ðs2 þ m4 ππÞ 1þ cos2_θ_% γ sin2_θ% γ sin2_θ_% þ 4sm2 ππcos2θ% − 2pffiffiffismππðs þ m2ππÞðtan θ%γÞ−1sin 2θ%cos ϕ%

− 2sm2

ππsin2θ%cos 2ϕ% #

; ð8Þ

where α and β are the QED fine-structure constant and the pion velocity β ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− 4m2 π=m2ππ p , respectively. The FSR contribution is dσFSR eþ_e−_→πþ_π−_γ dm2 ππd cos θ%γd cos θ%dϕ% ¼α 3_{βðs − m}2 ππÞ 64πs3 ð1 þ cos 2_θ_% γÞjVðm2ππ; θ%Þj2; ð9Þ

and the interference term

dσI eþ_e−_→πþ_π−_γ dm2 ππd cos θ%γd cos θ%dϕ% ¼ α 3_β2 16πs2p mffiffiffis ππRefF % πðm2ππÞVðm2ππ; θ%Þg × "

−pffiffiffismππcos θγ%cos θ%þ ½ð1 þ cos2θ%γÞs

þ m2 ππsin2θ%γ' sin θ%_{cos ϕ}% 2 sin θ% γ # ; ð10Þ where V ¼X q e2 qVq¼ X q e2 q Z ₁ 0 dz 2z− 1 zð1 − zÞΦþqðz; m2ππ; cos θ%Þ ðq ¼ u; dÞ; ð11Þ and Φþ_q_{ðz; m}2

ππ; cos θ%Þ is the C-even part of the two-pion

generalized distribution amplitudes (GDA). The pion

time-like form factor Fπðm2ππÞ is taken from a fit to BABAR

data[6]with a vector dominance model.

So far, there is no implementation of this model in an MC generator to describe the ISR-FSR interference in the

eþ_e− _{→ π}þ_π−_{γ process. In order to predict the charge}

asymmetry numerically, we take the following GDA model, which is a modified version of the model found

in Ref.[16]: Φþ_u_{ðz; m}2 ππ; cos θ%Þ ¼ Φþdðz; m2ππ; cos θ%Þ ¼ 10zð1 − zÞð2z − 1Þ × $ c0 3− β2 2 e iδ0ðmππÞ_{þ c} 2β2BWðmππÞP2ðcos θ%Þ % ; ð12Þ

where c0 and c2 are the magnitudes of the S-wave and

D-wave contributions, respectively. As the scalar sector is known to involve wide resonances, the S-wave contri-bution is approximated by a constant amplitude with

a mass-dependent phase δ0ðmππÞ taken from pion-pion

phase-shift analyses[17]in the region below 1.6 GeV=c2_.

This model incorporates the rapid phase variation across

the f0ð980Þ resonance. Using c0¼ −0.5[16]yields an A0

value of about−1% near the ρ resonance and nearly flat

with mass. For the D-wave tensor contribution, we use a

Breit-Wigner (BW) form for the f2ð1270Þ resonance in

order to take properly into account the mass dependence of the amplitude, the phase variation being given by the BW

form in agreement with the measured δ2ðmππÞ values[17].

The angular dependence in the ππ center of mass is given

by the Legendre polynomial P2ðcos θ%Þ, which assumes the

dominance of helicity 0 for f2ð1270Þ production.

1_{We thank Leonard Lesniak for pointing out a sign mistake in}

the sin 2θ%_{cos ϕ}% _{term of Eq.} ₍₈₎ _{as given in the erratum of}

Ref.[14]. The correct sign has been checked with the formulas given in Ref.[15].

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C. Other sources of charge asymmetry

Next-to-leading-order (NLO) corrections including addi-tional photons (soft and hard) and loops are expected to affect the LO predictions for the charge asymmetry. For the

μþ_μ−_{γ process these corrections have been computed}

recently [18] and implemented in the PHOKHARA 9.0

generator[18]. As discussed in Sec. VI C, the effects are

found to be small, at the percent level for the experimental conditions of the present analysis, and to be well accounted for by the simpler structure function approach implemented inAFKQED. No exact NLO calculation is available for the

πþ_π−_{γ process. In this case, since the LO charge}

asym-metry is expected to be small because the FSR amplitude is suppressed, NLO corrections could play a relatively more important role. The soft and virtual photon contributions to

the Born process eþ_e−_{→ π}þ_π− _{are known} _[15,19] _to

generate an asymmetry of the pion production, with asymmetry values at the percent level at a ππ mass of

1 GeV=c2_{. However, it is unclear if the above result can be}

used in the conditions of the present process eþ_e−_→

πþ_π−_{γ, where one of the incoming electrons is highly off}

shell after emission of a hard ISR photon. Furthermore, such an asymmetry would vanish because of the

symmet-rical integration in cos θ%_{. NLO corrections as implemented}

inAFKQEDhave indeed no effect on the charge asymmetry. No correction on the measured charge asymmetry is

therefore applied for the πþ_π−_{γ process.}

Another potential source of charge asymmetry comes from Z exchange. This contribution is strongly suppressed by the Z propagator, especially for the ISR diagrams where

m2

xx=M2Z∼ 10−4. Therefore one expects this effect to be

negligible for πþ_π−_{γ. The contribution is larger for the FSR}

diagrams for μþ_μ−_{γ since here the relevant ratio is}

s=M2

Z¼ 1.4%. The contribution of Z exchange is studied

with theKKMCgenerator[20]. As reported in Sec.VI C, the

effect is at the level of a few per mille. IV. EXPERIMENTAL ANALYSIS A. The BABAR detector and data samples

The analysis is based on 232 fb−1_{of data}_[21]_collected

with the BABAR detector at the SLAC National Accelerator Laboratory at the PEP-II asymmetric-energy

eþ_e− _{collider operated at the} _{ϒð4SÞ resonance. About}

10% of the data was collected 40 MeV below the resonance. The BABAR detector is described in detail

elsewhere[22]. Charged-particle tracks are measured with

a five-layer double-sided silicon vertex tracker (SVT) together with a 40-layer drift chamber (DCH), both inside a 1.5 T superconducting solenoid. Photons are assumed to originate from the primary vertex defined by the charged-particle tracks of the event, and their energy and position are measured in a CsI(Tl) electromagnetic calorimeter (EMC). Charged-particle identification (PID) uses the ionization energy loss dE=dx in the SVT and DCH, the

Cherenkov radiation detected in a ring-imaging device

(DIRC), the shower energy deposit (Ecal) in the EMC, and

the shower shape in the instrumented flux return (IFR) of the magnet. The IFR system is made of modules of resistive plate chambers interspaced with iron slabs, arranged in a layout with a barrel and two end caps. Collision events are recorded and reconstructed if they pass three levels of trigger (hardware, online software, and offline filter), each using complementary information from the subdetectors.

B. Monte Carlo generators and simulation

Signal and background processes eþ_e− _{→ Xγ are}

simu-lated with theAFKQEDevent generator, which is based on

QED for eþ_e− _{→ μ}þ_μ−_{γ and Ref.} _[23] _{for hadronic}

production. LO ISR and FSR emission is simulated for

eþ_e− _{→ μ}þ_μ−_{γ, while LO FSR is neglected for hadronic}

processes. The main photon (hereafter called the “ISR”

photon) is emitted within the angular range 20° < θ%

γ <

160° in the eþe− c.m. system, bracketing the photon

detection range with a margin for resolution. Additional ISR photons are generated with the structure function

method [24], and additional FSR photons with the

PHOTOS[25]program. Additional ISR photons are emitted

along the eþ _{or e}− _{beam particle direction. A minimum}

mass mXγISR > 8 GeV=c

2_{is imposed at generation, which}

puts an upper bound on the additional ISR photon energy. Samples corresponding to 5 to 10 times the data are

generated for the signal eþ_e− _{→ μ}þ_μ−_{γ and e}þ_e− _→

πþ_π−_{γ channels, as well as large samples of backgrounds}

from the other two-prong and multihadron ISR processes.

Background processes eþ_e− _{→ q ¯q (q ¼ u; d; s; c) are}

gen-erated with the JETSET [26] generator, and eþe−→ τþτ−

with theKORALB[27]program. The response of the BABAR

detector is simulated using theGEANT4[28] package.

C. Event selection

Event selection follows the same procedure as the selection of two-charged particle ISR events used for cross

section measurements[6]. It requires a photon with energy

E%

γ > 3 GeV in the eþe− c.m. and laboratory polar angle

with respect to the e−_{beam in the range [0.35}_{–2.4] rad, and}

exactly two tracks of opposite charge, each with

momen-tum p > 1 GeV=c and within the angular range [0.40–

2.45] rad. If more than one photon is detected, the candidate

with the highest E%

γ is taken to be the “ISR” photon. To

ensure a rough momentum balance at an early stage of the selection, the “ISR” photon is required to lie within 0.3 rad of the missing momentum of the charged particles (or of the tracks plus the other photons). The tracks are required to have at least 15 hits in the DCH, to originate within 5 mm of the collision axis and within 6 cm from the beam spot along the beam direction, and to extrapolate to the DIRC and IFR active areas in order to exclude low-efficiency regions. Both tracks are required to be identified either as

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0 2000 4000 6000 8000 -1 -0.5 0 0.5 1 2mµ∼0.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 2000 4000 6000 -1 -0.5 0 0.5 1 0.5∼1GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 -1 -0.5 0 0.5 1 1∼1.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 -1 -0.5 0 0.5 1 1.5∼2GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 500 1000 1500 2000 -1 -0.5 0 0.5 1 2∼2.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 500 1000 1500 -1 -0.5 0 0.5 1 2.5∼3GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 500 1000 1500 -1 -0.5 0 0.5 1 3∼3.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 500 1000 1500 -1 -0.5 0 0.5 1 3.5∼4GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 -1 -0.5 0 0.5 1 4∼4.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 -1 -0.5 0 0.5 1 4.5∼5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 -1 -0.5 0 0.5 1 5∼5.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 4000 -1 -0.5 0 0.5 1 5.5∼6GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 4000 -1 -0.5 0 0.5 1 6∼6.5GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+ 0 1000 2000 3000 4000 -1 -0.5 0 0.5 1 6.5∼7GeV/c2 cosφ∗ Events / 0.05 µ -µ−_↔µ+

FIG. 6 (color online). The cos ϕ%_{distributions for e}þ_e−_{→ μ}þ_μ−_{γ data in 0.5 GeV=c}2_m

μμintervals. The points labeled “μ−” refer to

the configurations with ϕ%₋_{∈ ½0; π', while the points labeled “μ}−_{↔ μ}þ_{” correspond to ϕ}%

þ∈ ½0; π'.

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0 200 400 -1 -0.5 0 0.5 1 0.3∼0.4GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 250 500 750 1000 -1 -0.5 0 0.5 1 0.4∼0.5GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 500 1000 1500 2000 -1 -0.5 0 0.5 1 0.5∼0.6GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 2000 4000 -1 -0.5 0 0.5 1 0.6∼0.7GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 5000 10000 -1 -0.5 0 0.5 1 0.7∼0.8GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 2000 4000 -1 -0.5 0 0.5 1 0.8∼0.9GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 500 1000 -1 -0.5 0 0.5 1 0.9∼1GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 200 400 -1 -0.5 0 0.5 1 1∼1.1GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 100 200 300 -1 -0.5 0 0.5 1 1.1∼1.2GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 50 100 150 -1 -0.5 0 0.5 1 1.2∼1.3GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 25 50 75 100 -1 -0.5 0 0.5 1 1.3∼1.4GeV/c2 cosφ∗ Events / 0.1 π− π−_↔π+ 0 25 50 75 100 -1 -0.5 0 0.5 1 1.4∼1.5GeV/c2 cosφ∗ Events / 0.5 π− π−_↔π+ 0 20 40 60 -1 -0.5 0 0.5 1 1.5∼1.6GeV/c2 cosφ∗ Events / 0.5 π− π−_↔π+ 0 20 40 60 -1 -0.5 0 0.5 1 1.6∼1.7GeV/c2 cosφ∗ Events / 0.5 π− π−_↔π+ 0 25 50 75 100 -1 -0.5 0 0.5 1 1.7∼1.8GeV/c2 cosφ∗ Events / 0.5 π− π−_↔π+

FIG. 7 (color online). The cos ϕ%_{distributions for e}þ_e−_{→ π}þ_π−_{γ data in 0.1 GeV=c}2_m

ππintervals. The points labeled “π−” refer to

the configurations with ϕ%₋_{∈ ½0; π', while the points labeled “π}−_{↔ π}þ_{” correspond to ϕ}%

þ∈ ½0; π'.

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muons or as pions. To suppress the background to πþ_π−_{γ at}

threshold due to the eþ_e− _{→ γγ process followed by a}

photon conversion and misidentification of both electrons as pions, it is further required that the distance in the

transverse plane Vxy between the vertex of the two tracks

and the beam collision point be less than 0.5 cm for

mππ < 0.5 GeV=c2. Electron background to μþμ−γ is

negligible over the full mass range.

In order to suppress multihadron ISR events and reduce higher-order radiative processes, the selected two-prong candidates are subjected to a one-constraint kinematic fit to

the eþ_e−_{→ x}þ_x−_{γ hypothesis (x ¼ μ, π), in which only the}

two good charged-particle tracks are taken as input and the corresponding missing mass is constrained to the null

photon mass. The χ2_{value of the kinematic fit is required}

to be less than 15.

D. Charge asymmetry calculation

For a complete topology of the final states, the azimuth

ϕ%_{defined in Sec.}_{II B}_{should cover the 2π range. However,}

the event sample with x− _{azimuth ϕ}%

− ∈ ½0; π' is

comple-mentary to the sample with xþ _{azimuth ϕ}%

þ ∈ ½0; π', since

ϕ%

þ¼ π þ ϕ%−ðmod 2πÞ in every event. This allows us to

restrict ϕ% _{to the range ½0; π' with no loss of phase space.}

After integrating over θ%

γ and θ%, the total event sample in a

fixed mxx interval subdivides into two subsamples: one

with ϕ%₋_{∈ ½0; π' (N}₋_{), and the other with ϕ}%

þ∈ ½0; π' (Nþ).

We obtain separately the distributions in cos ϕ%_{of the N}obs

(

samples in data, namely Nobs

− ðcos ϕ%Þ with ϕ%−¼ ϕ% and

Nobs

þ ðcos ϕ%Þ with ϕ%þ¼ ϕ%. Distributions of background

events NBG

( ðcos ϕ%Þ are determined separately for each

subsample, as described below. Likewise, efficiencies are

split into ϵ₍_{ðcos ϕ}%_{Þ and computed using the full simulation}

of eþ_e−_{→ x}þ_x−_{γ (x ¼ μ, π) events, with corrections for the}

differences between data and simulation (see Secs.VI Cand

VII C). In a given mμμ (mππ) interval, the asymmetry at a

given cos ϕ% _{is derived from the difference between the}

N−ðcos ϕ%Þ and Nþðcos ϕ%Þ yields, corrected for efficiency,

and is obtained from the following expression:

Aðcos ϕ%_Þ

¼_NN−ðcos ϕ%Þ=ϵ−ðcos ϕ%Þ − Nþðcos ϕ%Þ=ϵþðcos ϕ%Þ −ðcos ϕ%Þ=ϵ−ðcos ϕ%Þ þ Nþðcos ϕ%Þ=ϵþðcos ϕ%Þ ; ð13Þ

where N₍ðcos ϕ%_{Þ ¼ N}obs

( ðcos ϕ%Þ − NBG( ðcos ϕ%Þ. 0 25 50 75 100 -1 -0.5 0 0.5 1 cosφ∗ Events / 0.05 0.5∼1GeV BGµ -BGµ−_↔µ+ 0 2 4 6 -1 -0.5 0 0.5 1 cosφ∗ Events / 0.05 1∼1.5GeV BGµ -BGµ−_↔µ+ 0 0.25 0.5 0.75 1 -1 -0.5 0 0.5 1 cosφ∗ Events / 0.05 3.5∼4GeV BGµ -BGµ−_↔µ+

FIG. 8 (color online). Backgrounds estimated with MC for eþ_e−_{→ μ}þ_μ−_{γ as a function of cos ϕ}%_{in selected m}_μμ_{intervals. The points}

labeled “μ−_{” refer to the configurations with ϕ}%₋_{∈ ½0; π', while the points labeled “μ}−_{↔ μ}þ_{” correspond to ϕ}%

þ∈ ½0; π'. 0 25 50 75 100 -1 -0.5 0 0.5 1 0.3∼0.4GeV/c2 cosφ∗ Events / 0.1 BGπ− BGπ−_↔π+ 0 10 20 -1 -0.5 0 0.5 1 0.7∼0.8GeV/c2 cosφ∗ Events / 0.1 BGπ− BGπ−_↔π+ 0 2.5 5 7.5 10 -1 -0.5 0 0.5 1 1.2∼1.3GeV/c2 cosφ∗ Events / 0.1 BGπ− BGπ−_↔π+

FIG. 9 (color online). Backgrounds estimated with MC for eþ_e−_{→ π}þ_π−_{γ as a function of cos ϕ}%_{in selected m}_ππ_{intervals. The points}

labeled “π−_{” refer to the configurations with ϕ}%₋_{∈ ½0; π', while the points labeled “π}−_{↔ π}þ_{” correspond to ϕ}%

þ∈ ½0; π'.

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Note that distributions of N₍ðcos ϕ%_{Þ can be obtained in}

each ðmxx; θ%γ; cos θ%Þ cell of the phase space, and the

asymmetry defined by Eq.(13)can be calculated. However,

this one-dimensional quantity Aðcos ϕ%_{Þ is a valid}

defini-tion of charge asymmetry only when the variable cos θ%_is

integrated within a symmetric range. This is a consequence

of the fact that the xþ _{↔ x}− _{interchange means both}

ϕ%_{→ π þ ϕ}% _{and cos θ}%_→_{− cos θ}%_{, and therefore a}

non-null value of Aðcos ϕ%_{Þ in an arbitrary cos θ}% _interval

is not an intrinsic signature of ISR-FSR interference. E. Event samples and backgrounds after selection

The cos ϕ%_{distributions N}obs

( ðcos ϕ%Þ for eþe− → μþμ−γ obtained in data after the overall event selection are shown in

Fig. 6, for μμ mass intervals ranging from threshold to

7 GeV=c2_{. The event distributions N}obs

( ðcos ϕ%Þ obtained for

eþ_e−_{→ π}þ_π−_{γ in data are shown in Fig.}₇_{, in 0.1 GeV=c}2

mass intervals ranging from 0.3 GeV=c2_{to 1.8 GeV=c}2_.

The backgrounds remaining after selection are estimated using the full simulation, normalized to the data luminosity, of the nonsignal two-prong ISR events, multihadron events

produced through ISR, eþ_e− _{→ q ¯q events, and τ}þ_τ−

events. The expected contamination for eþ_e− _{→ μ}þ_μ−_γ

as a function of cos ϕ%_{in typical m}

μμintervals is shown in

Fig. 8, where the total error is the quadratic sum of the

statistical error and 10% systematic uncertainty on

nor-malization [6]. Likewise, the estimated backgrounds for

eþ_e−_{→ π}þ_π−_{γ in typical m}_ππ_{intervals are shown in Fig.}₉_.

V. ACCEPTANCE AND DETECTOR EFFICIENCY EFFECTS ON THE CHARGE ASYMMETRY The charge asymmetry measurement is affected by the event reconstruction and selection. These experimental effects are investigated using the full simulation of signal events through changes of the raw charge asymmetry,

defined as Araw _{¼ ðN}

−− NþÞ=ðN−þ NþÞ, which are

observed after each selection step as a function of cos ϕ%_.

A. Study of the effects with the muon simulation 1. Kinematic acceptance

The kinematic acceptance includes the angular accep-tance for the primary photon and the two charged-particle tracks, and the momentum restriction (p > 1 GeV=c) applied to charged-particle tracks. Each kinematic selection is found to modify the slope of the raw charge asymmetry significantly, though the total effect on the slope from the kinematic requirements altogether turns out to be small due to accidental cancellations.

It is worthwhile to note that the kinematic selection in itself is charge symmetric. Hence the observed bias on the measured raw charge asymmetry is a cross effect of physical asymmetric kinematics and charge-symmetric detector acceptance. It does vanish for a null

physical charge asymmetry. As checked with a μþ_μ−_γ

simulated sample produced by ISR only, no fake charge asymmetry emerges from the kinematic selection.

2. Software trigger and tracking

Biases on the raw charge asymmetry measurement originate from the software trigger and the track reconstruction. They are observed in the low-mass region,

as illustrated in Fig. 10 and vanish at high mass

(mμμ> 1.5 GeV=c2).

The common origin of the mass-dependent trigger and tracking inefficiencies is geometrical and has been

thor-oughly studied for the πþ_π−_{cross-section measurement}_[6]_.

Converging trajectories in the DCH of oppositely deflected tracks emitted in close-by directions confuses the track reconstruction and causes both the software trigger and the final tracking inefficiencies. In the charge-conjugate con-figuration, in which the positive and negative tracks are interchanged, tracks diverge in the magnetic field and are well separated in the transverse plane, although with the same absolute azimuthal opening angle. The efficiencies are consequently charge asymmetric, sharply reduced for

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.2∼0.5GeV in acc.

in acc. and 2 good tracks

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.5∼1GeV in acc.

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 1∼1.5GeV in acc.

FIG. 10 (color online). Raw charge asymmetry as a function of cos ϕ%_{in selected m}_μμ_{intervals, for e}þ_e−_{→ μ}þ_μ−_{γ MC events with (}_∘)

and without (•) reconstruction of two good charged-particle tracks, where the events are already required to be within the kinematic acceptance.

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overlapping tracks in low-mμμregions, atΔϕ values close to

zero but always positive, whereΔϕ is the signed angular

difference between the azimuths of the positive and negative tracks

Δϕ ¼ ðϕþ− ϕ−Þ ∈ ½−π; π': ð14Þ

3. “ISR” photon reconstruction

The event selection requires that an “ISR” photon with E%

γ > 3 GeV be measured in the EMC. The raw charge

asymmetries for the fully simulated events with and with-out the requirement of the “ISR” photon being

recon-structed are shown in Fig.11, where the events are already

required to be within the kinematic acceptance. Effects are

observed in high-mμμ regions.

The origin of a charge-asymmetric photon reconstruction inefficiency is again geometrical. In case one of the charged-particle tracks and the “ISR” photon overlap in the EMC, the shower produced by the “ISR” photon is mistakenly associated to the charged-particle track, and the “ISR” photon is lost. In the charge-conjugate configuration, no overlap occurs because of the opposite deflection of the charged-particle track in the magnetic field. The overlap

happens at _{Δϕ ¼ −π þ ε where ε is a small positive}

quantity, and as a consequence, the “ISR” photon reconstruction efficiency is charge asymmetric, strongly

reduced aroundΔϕ ∼ −π. As the overlap of one

charged-particle track and the “ISR” photon occurs preferentially at high mass, due to phase space, the corresponding effects

are only observed at m_μμ > 3.5 GeV=c2_.

4. Muon identification

The charge asymmetry measurement is also affected by

muon identification, in the low- and high-mμμ regions, as

shown in Fig.12. Charge-asymmetric inefficiency of muon

identification results again from event topologies. The first

cause, which affects low-mμμ regions, is the overlap of the

two charged-particle tracks at the IFR, which confuses

the muon identification algorithm. The second cause is the partial overlap of one charged-particle track and the “ISR” photon at the EMC, which makes the track look unlike a muon. The latter effect is more pronounced at high mass, when a muon and the “ISR” photon are emitted in close-by directions. The efficiency of muon identification as a

function of Δϕ exhibits a sharp dip at positive Δϕ at

low mass, and atΔϕ≃ −π at high mass.

5. Summary of the acceptance and detector efficiency

effects in the μþ_μ−_{γ process}

The overall efficiencies ϵ₍needed to correct the N₍event

yields entering the charge asymmetry measurement [Eq.(13)]

are the overall result of the acceptance-induced and detector asymmetries discussed above. They are determined using the

full simulation, separately for the N₍ samples.

As previously discussed, the detector inefficiencies are mostly caused by the spatial overlap of trajectories occur-ring in the detector: two-track overlaps in the DCH and the

IFR, respectively for _{Δϕ ¼ 0.1 ( 0.1 and 0.5 ( 0.2 and}

affecting masses below 2 GeV=c2_{, and the photon-muon}

overlap in the EMC, for Δϕ≳ −π and affecting masses

above 4 GeV=c2_{. These various overlap effects contribute}

very asymmetrically to the two N₍ samples, due to a

complete correlation between the cos ϕ%

(andΔϕ variables.

This is demonstrated in Fig.13, which shows that the N_þ

(N₋) sample corresponds to Δϕ > 0 (Δϕ < 0). Since the

two-track overlaps occur forΔϕ > 0, one expects ϵ− > ϵþ.

For the photon-muon overlap with nearly opposite tracks the situation is reversed.

As a consequence, as summarized in Fig.14(Sec.VI), the

acceptance and detector inefficiencies induce a changeΔA in

the observed charge asymmetry magnitude and also distort

the linear dependence on cos ϕ%_{. The dominant effects are}

from geometric acceptance, “ISR” photon reconstruction and the track momentum requirement p > 1 GeV=c.

However, although the detector is not completely charge

symmetric, Fig. 13 shows that the effects producing

in acc. and 1 detected photon

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 5∼5.5GeV in acc.

and without (•) the requirement of the reconstruction of the “ISR” photon, where the events are already required to be within the kinematic acceptance.

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an asymmetry are nearly symmetric in cos ϕ%_{. The A} 0

observable introduced in Eq. (5) is thus expected to be

robust against such effects. Overall, the slope of the asymmetry is barely affected by detector inefficiencies

and event selection. In the μþ_μ−_{γ process, the maximum}

effects, of a few 10−2_{, take place around m}

μμ∼ 2–4 GeV=c2.

In the low-mass region ½0.5–1.0' GeV=c2_{, the effect from the}

overall selection is at the level of a few 10−3_.

B. Study of the effects with the pion simulation The acceptance and detector effects are also studied with

the simulated eþ_e− _{→ π}þ_π−_{γ events. The overall effect}

around the ρ resonance (mππ ∈ ½0.4; 1.2' GeV=c2) is

ð0.30 ( 0.07Þ × 10−2 on average.

As the charge asymmetry is null for eþ_e− _{→ π}þ_π−_{γ MC}

events, generated with no LO FSR, the acceptance effects on the slope of the charge asymmetry are quite small for any

selection requirement, including the kinematic ones. This is

in contrast with the μþ_μ−_{γ case, where the individual}

kinematic requirements induce large effects. However this conclusion holds only if the charge asymmetry in the data is actually null. If a sizable asymmetry is measured, the bias introduced by the cross effect of acceptance and asymmetry

has to be evaluated and corrected (Sec.VII B).

VI. RESULTS ON THE CHARGE ASYMMETRY

IN THE eþ_e−_{→ μ}þ_μ−_{γ PROCESS}

The measured raw charge asymmetry after the complete event selection for the data is obtained as a function of

cos ϕ%_{in various m}

μμ intervals, and shown in Fig.14. It is

consistent to the first order with the full simulation

of eþ_e− _{→ μ}þ_μ−_{γ events, except in the mass interval}

populated by the J=ψ resonance (3.0–3.5 GeV=c2_).

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.5∼1GeV no muID with muID -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 4∼4.5GeV no muID with muID -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 6.5∼7GeV no muID with muID

and without (•) the two-muon identification, where the events are already required to be within the kinematic acceptance and have two good charged-particle tracks reconstructed.

-1 -0.5 0 0.5 1 -2 0 2 ∆φ=φ₊-φ₋ cos φ * − 0.5-1GeV -1 -0.5 0 0.5 1 -2 0 2 ∆φ=φ₊−φ₋ cos φ * + 0.5-1GeV

FIG. 13. (Left) Distribution of cos ϕ%₋_vs_{Δϕ for the N}₋_{sample (ϕ}%₋_{∈ ½0; π') in the ð0.5–1.0Þ GeV=c}2_{mass interval for the μ}_þ_μ−_{γ MC.}

(Right) The same for the N_þ sample (ϕ%

þ∈ ½0; π').

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-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.2∼0.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.5∼1GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 1∼1.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 1.5∼2GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 2∼2.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 2.5∼3GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 3∼3.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 3.5∼4GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 4∼4.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 4.5∼5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 5∼5.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 5.5∼6GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 6∼6.5GeV data MC full simulation -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 6.5∼7GeV data MC full simulation

FIG. 14 (color online). Raw charge asymmetry as a function of cos ϕ%_{for μ}þ_μ−_{γ events in data (•) and MC (}_{∘), in various m}

μμintervals,

after the complete event selection.

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-0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.2∼0.5GeV

data without correction

data with correction MC truth generated -0.5 -0.25 0 0.25 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 0.5∼1GeV

data with correction

MC truth generated _-0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 1∼1.5GeV

data with correction MC truth generated -0.5 -0.25 0 0.25 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 1.5∼2GeV

MC truth generated -0.5 0 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 2∼2.5GeV

MC truth generated -0.5 0 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 2.5∼3GeV

data with correction MC truth generated -0.5 0 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 3∼3.5GeV

data with correction MC truth generated -0.5 0 0.5 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 3.5∼4GeV

data with correction MC truth generated -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 4∼4.5GeV

data with correction MC truth generated -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosφ∗ Asymmetry 4.5∼5GeV

data with correction MC truth generated

FIG. 15 (color online). Charge asymmetry in eþ_e−_{→ μ}þ_μ−_{γ data before (•) and after (}_{∘) background subtraction and efficiency}

corrections, and for MC (Δ) at generation level.

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This is expected as J=ψ production is not considered in AFKQED.

The physical charge asymmetry for eþ_e− _{→ μ}þ_μ−_{γ is}

obtained from the measured cos ϕ% _{distributions after}

background subtraction and efficiency correction. The

background dependence on cos ϕ% _{is estimated with the}

simulation for each of the samples ϕ%

( ¼ ϕ%∈ ½0; π', as

explained above. Similarly, the overall efficiency ϵ₍ is

obtained with fully simulated eþ_e−_{→ μ}þ_μ−_{γ events,}

and corrected for data/MC differences in detector response. The efficiency differences between data and simulation have been studied extensively for the cross-section

mea-surements of eþ_e−_{→ μ}þ_μ−_{γ and e}þ_e− _{→ π}þ_π−_γ_[6]_{. They}

are parametrized as a function of the azimuthal opening

angleΔϕ between the two muons, and projected onto the

cos ϕ% _{variable by sampling with MC.}

The charge asymmetry distributions for eþ_e− _{→ μ}þ_μ−_γ

data after background subtraction and efficiency

cor-rection, as well as the charge asymmetry for eþ_e−_→

μþ_μ−_{γ MC at generation level, are shown in Fig.}₁₅_{. While}

the cos ϕ% _{dependence of the measured raw charge}

asym-metry is not linear, the corrected data distributions are quite consistent with the MC distributions at generation level. The slopes of charge asymmetry in various mass intervals are obtained by fitting the background-subtracted

efficiency-corrected charge asymmetry distributions

to A0cos ϕ%.

A. Test of the charge asymmetry with J=ψ → μþ_μ−

events in data

Since eþ_e− _{→ γJ=ψ with on-shell J=ψ → μ}þ_μ− _{is a}

pure ISR process at ϒð4SÞ energies, the J=ψ → μþ_μ−

sample in the data provides a test of fake asymmetries that could arise in the analysis.

To overcome the limited statistics of the J=ψ → μþ_μ−

sample, a loosened event selection is applied, with the muon identification requirement removed, which provides a gain in statistics by a factor of about 4, with no significant

increase of the hadronic background. The m_μμ spectrum

shows a clear J=ψ peak, over a linear QED background.

Defining AJ=ψ₀ and AQED₀ , the respective slopes of the charge

asymmetry for J=ψ and underlying QED events, the slope

A0 measured in the vicinity of the J=ψ resonance is the

average

A0¼

AJ=ψ₀ NJ=ψþ AQED0 NQED

NJ=ψþ NQED

; _ð15Þ

where NJ=ψ and NQED are the yields from J=ψ and QED,

respectively. The quantities NQEDand NJ=ψare obtained by

fitting the mass spectrum with a sum of a linear QED component and a Gaussian J=ψ signal, with fixed width equal to the mass resolution at the J=ψ and centered at the

nominal J=ψ mass. The slope AQED₀ is obtained by fitting

the charge asymmetry in the J=ψ sidebands. The measured

slope of the charge asymmetry as a function of mμμ is

shown in Fig.16: the expected behavior is clearly observed,

with a smooth variation from the QED continuum with a large negative value and a sharp peak approaching a null

slope on the J=ψ resonance. The specific AJ=ψ₀ slope is

obtained as a function of m_μμ in Fig. 16 according to

Eq.(15): its value is stable across the J=ψ peak and a fit to a

constant between 3.07 and 3.12 GeV=c2_yields

AJ=ψ₀ ¼ ð0.3 ( 1.6Þ × 10−2; ð16Þ -0.6 -0.4 -0.2 0 2.8 3 3.2 3.4 m_µµ(GeV/c2) A0 -0.4 -0.2 0 0.2 0.4 3.05 3.1 3.15 3.2 m_µµ(GeV/c2) A0 J/ψ

FIG. 16. (Left) The measured charge asymmetry as a function of μμ mass near the J=ψ resonance. (Right) The derived charge asymmetry for μþ_μ−_{γ from e}þ_e−_{→ γJ=ψ.}

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which is consistent with zero, as expected from the ISR-only J=ψ production.

B. Comparison to QED

The final slope A0as a function of mμμ measured on the

data is shown in Fig.17, together with the asymmetry at the

MC generation level, and the difference between them. The mass interval containing the pure-ISR contribution

from the J=ψ (eþ_e− _{→ γ}

ISRJ=ψ, J=ψ → μþμ−), discussed

in detail in Sec.VI A, is excluded. The absolute difference

between data and MC ΔA0¼ Adata0 − AMC0 is at a few

percent level (0–3%).

The measured slope of charge asymmetry is negative throughout the mass range under study, and its magnitude

increases with mass, reaching values as large as −0.7 at

5 GeV=c2_{, in agreement with the trend predicted by}

QED. However, while data and LO QED agree within 10−2

at mass less than 1 GeV=c2 _{and above 5 GeV=c}2_{, a small}

but significant discrepancy shows up for intermediate

mass, reaching ∼3 × 10−2 _{between 1.5 and 4 GeV=c}2_.

Investigations of systematic uncertainties, both at the exper-imental and theoretical levels, are reported in the next section. Due to the asymmetry of the beam energies at PEP-II, independent charge asymmetry measurements in two different kinematic regimes are provided by splitting the

data into a forward (cos θ%_γ _{> 0) sample and a backward}

(cos θ%

γ < 0) sample. The full analysis, including

back-ground subtraction and efficiency correction, is redone on

each sample separately. The results are shown in Fig.18. A

significant discrepancy between data and AFKQED is

observed in the forward region, in the 1.5–4 GeV=c2_mass

region, while in the backward hemisphere data andAFKQED

are consistent. The differences are quantified in TableI. No

significant forward-backward difference is expected from the generator.

C. Systematic uncertainties 1. Experimental systematic effects

The primary sources of systematic uncertainty are the background estimation, data/MC differences in detector -0.8 -0.6 -0.4 -0.2 0 0 2 4 6 m_µµ (GeV/c2) A0

MC truth generated -0.02 0 0.02 0.04 0 2 4 6 m_µµ (GeV/c2) ∆ A0

FIG. 17 (color online). The slope A0of charge asymmetry as a function of mμμin eþe−→ μþμ−γ data and in MC at generation level

(left), and the absolute difference between them (right). The J=ψ mass region (3.0–3.2 GeV=c2_{) is excluded. Only statistical}

uncertainties are shown.

-0.04 -0.02 0 0.02 0.04 0.06 0 2 4 6 A₀-A₀(MC Truth) (cosθ_γ>0) A₀-A₀(MC Truth) (cosθγ<0) m_µµ (GeV/c2) ∆ A0 A₀(data)-A₀(MCTruth)

FIG. 18 (color online). The difference between the measured asymmetry slope in the μþ_μ−_{γ process and the} _AFKQED

pre-diction, as a function of mμμ, excluding the J=ψ 3.0–3.2 GeV=c2

region. Forward (cos θ%_γ _{> 0) and backward (cos θ}%_γ _{< 0)}

hemi-spheres are analyzed separately. Statistical uncertainties only.