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Multi-photon final states in e^+e^

EUROPEAN LABORATORY FOR PARTICLE PHYSICS

CERN-PPE/97-109 11 August 1997

arXiv:hep-ex/9709002v1 5 Sep 1997

Multi-photon ?nal states in e+e? collisions at √ s = 130 – 172 GeV

The OPAL Collaboration

Abstract The process e+ e? → γγ(γ) is studied using data recorded with the OPAL detector at LEP. The data sample corresponds to a total integrated luminosity of 25.38 pb?1 taken at centre-of-mass energies of 130 - 172 GeV. The measured cross-sections agree well with the expectation from QED. In a combined ?t using data from all centre-of-mass energies, the angular distribution is used to obtain improved limits on the cut-o? parameters: Λ+ > 195 GeV and Λ? > 210 GeV (95% CL). In addition, limits on non-standard e+ e? γ couplings and contact interactions, as well as a 95% CL mass limit for an excited electron, Me? > 194 GeV for an e+ e? γ coupling κ = 1, are determined.

(Submitted to Zeit. Phys. C)

The OPAL Collaboration

K. Ackersta?8 , G. Alexander23 , J. Allison16 , N. Altekamp5 , K.J. Anderson9 , S. Anderson12 , S. Arcelli2 , S. Asai24 , D. Axen29 , G. Azuelos18,a , A.H. Ball17 , E. Barberio8 , R.J. Barlow16 , R. Bartoldus3 , J.R. Batley5 , S. Baumann3 , J. Bechtluft14 , C. Beeston16 , T. Behnke8 , A.N. Bell1 , K.W. Bell20 , G. Bella23 , S. Bentvelsen8 , S. Bethke14 , O. Biebel14 , A. Biguzzi5 , S.D. Bird16 , V. Blobel27 , I.J. Bloodworth1 , J.E. Bloomer1 , M. Bobinski10 , P. Bock11 , D. Bonacorsi2 , M. Boutemeur34 , B.T. Bouwens12 , S. Braibant12 , L. Brigliadori2 , R.M. Brown20 , H.J. Burckhart8 , C. Burgard8 , R. B¨ rgin10 , P. Capiluppi2 , R.K. Carnegie6 , A.A. Carter13 , u 5 17 J.R. Carter , C.Y. Chang , D.G. Charlton1,b , D. Chrisman4 , P.E.L. Clarke15 , I. Cohen23 , J.E. Conboy15 , O.C. Cooke8 , M. Cu?ani2 , S. Dado22 , C. Dallapiccola17 , G.M. Dallavalle2 , R. Davis30 , S. De Jong12 , L.A. del Pozo4 , K. Desch3 , B. Dienes33,d , M.S. Dixit7 , E. do Couto e Silva12 , M. Doucet18 , E. Duchovni26 , G. Duckeck34 , I.P. Duerdoth16 , D. Eatough16 , J.E.G. Edwards16 , P.G. Estabrooks6 , H.G. Evans9 , M. Evans13 , F. Fabbri2 , M. Fanti2 , A.A. Faust30 , F. Fiedler27 , M. Fierro2 , H.M. Fischer3 , I. Fleck8 , R. Folman26 , D.G. Fong17 , M. Foucher17 , A. F¨ rtjes8 , D.I. Futyan16 , P. Gagnon7 , J.W. Gary4 , J. Gascon18 , u 17 S.M. Gascon-Shotkin , N.I. Geddes20 , C. Geich-Gimbel3 , T. Geralis20 , G. Giacomelli2 , P. Giacomelli4 , R. Giacomelli2 , V. Gibson5 , W.R. Gibson13 , D.M. Gingrich30,a , D. Glenzinski9 , J. Goldberg22 , M.J. Goodrick5 , W. Gorn4 , C. Grandi2 , E. Gross26 , J. Grunhaus23 , M. Gruw?8 , e 32 12 8 13 7 9 C. Hajdu , G.G. Hanson , M. Hansroul , M. Hapke , C.K. Hargrove , P.A. Hart , C. Hartmann3 , M. Hauschild8 , C.M. Hawkes5 , R. Hawkings27 , R.J. Hemingway6 , M. Herndon17 , G. Herten10 , R.D. Heuer8 , M.D. Hildreth8 , J.C. Hill5 , S.J. Hillier1 , P.R. Hobson25 , R.J. Homer1 , A.K. Honma28,a , D. Horv?th32,c , K.R. Hossain30 , R. Howard29 , P. H¨ ntemeyer27 , a u D.E. Hutchcroft5 , P. Igo-Kemenes11 , D.C. Imrie25 , M.R. Ingram16 , K. Ishii24 , A. Jawahery17 , P.W. Je?reys20 , H. Jeremie18 , M. Jimack1 , A. Joly18 , C.R. Jones5 , G. Jones16 , M. Jones6 , U. Jost11 , P. Jovanovic1 , T.R. Junk8 , D. Karlen6 , V. Kartvelishvili16 , K. Kawagoe24 , T. Kawamoto24 , P.I. Kayal30 , R.K. Keeler28 , R.G. Kellogg17 , B.W. Kennedy20 , J. Kirk29 , A. Klier26 , S. Kluth8 , T. Kobayashi24 , M. Kobel10 , D.S. Koetke6 , T.P. Kokott3 , M. Kolrep10 , S. Komamiya24 , T. Kress11 , P. Krieger6 , J. von Krogh11 , P. Kyberd13 , G.D. La?erty16 , R. Lahmann17 , W.P. Lai19 , D. Lanske14 , J. Lauber15 , S.R. Lautenschlager31 , J.G. Layter4 , D. Lazic22 , A.M. Lee31 , E. Lefebvre18 , D. Lellouch26 , J. Letts12 , L. Levinson26 , S.L. Lloyd13 , F.K. Loebinger16 , G.D. Long28 , M.J. Losty7 , J. Ludwig10 , A. Macchiolo2 , A. Macpherson30 , M. Mannelli8 , S. Marcellini2 , C. Markus3 , A.J. Martin13 , J.P. Martin18 , G. Martinez17 , T. Mashimo24 , P. M¨ttig3 , W.J. McDonald30 , J. McKenna29 , E.A. Mckigney15 , T.J. McMahon1 , a R.A. McPherson8 , F. Meijers8 , S. Menke3 , F.S. Merritt9 , H. Mes7 , J. Meyer27 , A. Michelini2 , G. Mikenberg26 , D.J. Miller15 , A. Mincer22,e , R. Mir26 , W. Mohr10 , A. Montanari2 , T. Mori24 , M. Morii24 , U. M¨ ller3 , S. Mihara24 , K. Nagai26 , I. Nakamura24 , H.A. Neal8 , B. Nellen3 , u 8 R. Nisius , S.W. O’Neale1 , F.G. Oakham7 , F. Odorici2 , H.O. Ogren12 , A. Oh27 , N.J. Oldershaw16 , M.J. Oreglia9 , S. Orito24 , J. P?link?s33,d , G. P?sztor32 , J.R. Pater16 , a a a G.N. Patrick20 , J. Patt10 , M.J. Pearce1 , R. Perez-Ochoa8 , S. Petzold27 , P. Pfeifenschneider14 , J.E. Pilcher9 , J. Pinfold30 , D.E. Plane8 , P. Po?enberger28 , B. Poli2 , A. Posthaus3 , D.L. Rees1 , D. Rigby1 , S. Robertson28 , S.A. Robins22 , N. Rodning30 , J.M. Roney28 , A. Rooke15 , E. Ros8 , A.M. Rossi2 , P. Routenburg30 , Y. Rozen22 , K. Runge10 , O. Runolfsson8 , U. Ruppel14 , D.R. Rust12 , R. Rylko25 , K. Sachs10 , T. Saeki24 , E.K.G. Sarkisyan23 , C. Sbarra29 , A.D. Schaile34 , O. Schaile34 , F. Scharf3 , P. Schar?-Hansen8 , P. Schenk34 , J. Schieck11 , P. Schleper11 , 1

B. Schmitt8 , S. Schmitt11 , A. Sch¨ning8 , M. Schr¨der8 , H.C. Schultz-Coulon10 , M. Schumacher3 , o o C. Schwick8 , W.G. Scott20 , T.G. Shears16 , B.C. Shen4 , C.H. Shepherd-Themistocleous8 , P. Sherwood15 , G.P. Siroli2 , A. Sittler27 , A. Skillman15 , A. Skuja17 , A.M. Smith8 , G.A. Snow17 , R. Sobie28 , S. S¨ldner-Rembold10 , R.W. Springer30 , M. Sproston20 , K. Stephens16 , J. Steuerer27 , o B. Stockhausen3 , K. Stoll10 , D. Strom19 , P. Szymanski20 , R. Ta?rout18 , S.D. Talbot1 , S. Tanaka24 , P. Taras18 , S. Tarem22 , R. Teuscher8 , M. Thiergen10 , M.A. Thomson8 , E. von T¨rne3 , S. Towers6 , I. Trigger18 , Z. Tr?cs?nyi33 , E. Tsur23 , A.S. Turcot9 , M.F. Turner-Watson8 , o o a 11 12 P. Utzat , R. Van Kooten , M. Verzocchi10 , P. Vikas18 , E.H. Vokurka16 , H. Voss3 , F. W¨ckerle10 , A. Wagner27 , C.P. Ward5 , D.R. Ward5 , P.M. Watkins1 , A.T. Watson1 , a N.K. Watson1 , P.S. Wells8 , N. Wermes3 , J.S. White28 , B. Wilkens10 , G.W. Wilson27 , J.A. Wilson1 , G. Wolf26 , T.R. Wyatt16 , S. Yamashita24 , G. Yekutieli26 , V. Zacek18 , D. Zer-Zion8

1 2

School of Physics and Space Research, University of Birmingham, Birmingham B15 2TT, UK Dipartimento di Fisica dell’ Universit` di Bologna and INFN, I-40126 Bologna, Italy a 3 Physikalisches Institut, Universit¨t Bonn, D-53115 Bonn, Germany a 4 Department of Physics, University of California, Riverside CA 92521, USA 5 Cavendish Laboratory, Cambridge CB3 0HE, UK 6 Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada 7 Centre for Research in Particle Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada 8 CERN, European Organisation for Particle Physics, CH-1211 Geneva 23, Switzerland 9 Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago IL 60637, USA 10 Fakult¨t f¨ r Physik, Albert Ludwigs Universit¨t, D-79104 Freiburg, Germany a u a 11 Physikalisches Institut, Universit¨t Heidelberg, D-69120 Heidelberg, Germany a 12 Indiana University, Department of Physics, Swain Hall West 117, Bloomington IN 47405, USA 13 Queen Mary and West?eld College, University of London, London E1 4NS, UK 14 Technische Hochschule Aachen, III Physikalisches Institut, Sommerfeldstrasse 26-28, D-52056 Aachen, Germany 15 University College London, London WC1E 6BT, UK 16 Department of Physics, Schuster Laboratory, The University, Manchester M13 9PL, UK 17 Department of Physics, University of Maryland, College Park, MD 20742, USA 18 Laboratoire de Physique Nucl?aire, Universit? de Montr?al, Montr?al, Quebec H3C 3J7, e e e e Canada 19 University of Oregon, Department of Physics, Eugene OR 97403, USA 20 Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK 22 Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel 23 Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 24 International Centre for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo 113, and Kobe University, Kobe 657, Japan 25 Brunel University, Uxbridge, Middlesex UB8 3PH, UK 26 Particle Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel 27 Universit¨t Hamburg/DESY, II Institut f¨ r Experimental Physik, Notkestrasse 85, D-22607 a u 2

Hamburg, Germany 28 University of Victoria, Department of Physics, P O Box 3055, Victoria BC V8W 3P6, Canada 29 University of British Columbia, Department of Physics, Vancouver BC V6T 1Z1, Canada 30 University of Alberta, Department of Physics, Edmonton AB T6G 2J1, Canada 31 Duke University, Dept of Physics, Durham, NC 27708-0305, USA 32 Research Institute for Particle and Nuclear Physics, H-1525 Budapest, P O Box 49, Hungary 33 Institute of Nuclear Research, H-4001 Debrecen, P O Box 51, Hungary 34 Ludwigs-Maximilians-Universit¨t M¨ nchen, Sektion Physik, Am Coulombwall 1, D-85748 a u Garching, Germany
a b c d e

and at TRIUMF, Vancouver, Canada V6T 2A3 and Royal Society University Research Fellow and Institute of Nuclear Research, Debrecen, Hungary and Department of Experimental Physics, Lajos Kossuth University, Debrecen, Hungary and Department of Physics, New York University, NY 1003, USA

1

Introduction

This paper reports a study of the annihilation process e+ e? → γγ(γ) using data recorded with the OPAL detector at LEP. At LEP energies, this is one of the few processes having negligible contributions from the weak interaction. Since the QED di?erential cross-section is precisely predicted in theory, deviations from the expected angular distribution are a sensitive test for non-standard physics processes contributing to these photonic ?nal states. The OPAL collaboration has previously published a study of photonic ?nal states, with and √ without missing energy, at s = 130 - 140 GeV [1]. The present analysis concentrates on ?nal states with two or more detected photons, but no missing transverse momentum, to study only the QED process. Photonic ?nal states with missing energy have been analysed separately [2]. Any non-QED e?ects described by the general framework of e?ective Lagrangian theory should increase with centre-of-mass energy. Existing OPAL limits on deviations from QED can be improved by using the data at centre-of-mass energies of 161.3 GeV and 172.1 GeV. A small amount of data taken at 170.3 GeV is included in the 172 GeV sample. The corresponding integrated luminosities of these data sets are 9.97 and 10.13 pb?1 , respectively. Since the selection criteria have changed, previously analysed data taken at centre-of-mass energies of 130.3 GeV (2.69 pb?1 ) and 136.2 GeV (2.59 pb?1 ) are reanalysed here to allow for a coherent treatment. The 136 GeV sample includes a small amount of data taken at 140.2 GeV. The error on the luminosity di?ers slightly for the di?erent energies and is approximately 0.5% . These measurements test QED at the highest centre-of-mass energies. Possible deviations are conveniently parametrised by cut-o? parameters Λ± . A comparison of the measured photon angular distribution with the QED expectation leads to limits on the QED cut-o? parameters Λ± , contact interactions (e+ e? γγ) and non-standard e+ e? γ-couplings as described in section 3. 3

The possible e?ects of an excited electron, e? , which would also change the angular distribution, are investigated. In addition, the possible production of a resonance X via e+ e? → Xγ, followed by the decay X → γγ, is studied in the invariant mass spectrum of photon pairs in three-photon ?nal states. The following section contains a brief description of the OPAL detector and the Monte Carlo simulated event samples. Section 3 describes the QED di?erential cross-sections for e+ e? → γγ(γ), as well as those from several models containing extensions to QED. In sections 4 - 6 the analysis is described in detail. The results are presented in section 7.

2

The OPAL detector and Monte Carlo samples

A detailed description of the OPAL detector can be found in [3]. OPAL uses a right-handed coordinate system in which the z axis is along the electron beam direction and the x axis is horizontal. The polar angle, θ, is measured with respect to the z axis and the azimuthal angle, φ, with respect to the x axis. For this analysis the most important detector component is the electromagnetic calorimeter (ECAL) which is divided into two parts, the barrel and the endcaps. The barrel covers the polar angle range of | cos θ| < 0.81 and consists of 9440 lead-glass blocks. The endcaps cover the polar angle range of 0.81 < | cos θ| < 0.98 and consist of 1132 blocks. In this analysis, the central tracking detector is used primarily to reject events inconsistent with purely photonic ?nal states. Raw hit information from the vertex drift chamber (CV) and jet drift chamber (CJ) is used to reject events with tracks coming from the interaction point. CV is divided into 36 φ-sectors and its inner 12 (6) axial layers cover an angular range of | cos θ| < 0.95(0.97). CJ is divided into 24 φ-sectors and covers an angular range of | cos θ| < 0.97 with its inner 16 layers. Incorporated in the surrounding magnet yoke is the hadronic calorimeter (HCAL) covering 97% of the solid angle. The outermost detectors are the muon chambers, shielded from the interaction point by at least 1.3 m of iron and covering the polar angle range of | cos θ| < 0.985. Di?erent Monte Carlo samples are used to study e?ciency and background. For the signal process e+ e? → γγ(γ) the RADCOR [4] generator is used. It provides a O(α3 ) cross section up to | cos θ| = 1 for the photon angle. No Monte Carlo with complete fourth order is currently available. The only program that generates four-photon ?nal states neglects the mass of the electron and therefore does not correctly include photons in the far forward range. The Bhabha process is studied with two di?erent programs. BHWIDE [5] generates both electron and positron in the acceptance of the detector. In contrast, TEEGG [6] allows one of them to have very low energy or escape along the beam-pipe, in addition, one photon is scattered into the detector. The process e+ e? → ννγ(γ) is studied with NUNUGPV [7]. Both e+ e? → ?+ ?? and e+ e? → τ + τ ? are simulated using KORALZ [8] and PYTHIA [9] is used for hadronic events. All samples were processed through the OPAL detector simulation program [10] and reconstructed as for real data.

4

3

Cross section for the process e+e? → γγ

The di?erential cross-section for the process e+ e? → γγ in the relativistic limit of lowest order QED is given by [11]: dσ α2 1 + cos2 θ = . (1) d? Born s 1 ? cos2 θ where s denotes the square of the centre-of-mass energy, α is the electromagnetic coupling constant and θ the polar angle of one photon. Since the two photons cannot be distinguished the event angle is de?ned such that cos θ is positive. In Ref. [12] possible deviations from the QED cross-section for Bhabha and M?ller scattering are parametrized in terms of cut-o? parameters. These parameters correspond to a short range exponential term added to the Coulomb potential. This ansatz leads to a modi?cation of the photon angular distribution as given in Eq. (2). dσ d? =
Λ±

dσ d?

Born



s2 sin2 θ 2Λ4 ±

(2)

Alternatively, in terms of e?ective Lagrangian theory, a gauge invariant operator may be added to QED. Depending on the dimension of the operator di?erent deviations from QED can be formulated [13]. Contact interactions (γγe+ e? ) or non-standard γe+ e? couplings described by dimension 6, 7 or 8 operators lead to angular distributions with di?erent mass scales Λ (see Eqs. 3 - 5). The subscripts (QED+6 etc.) follow the notation in Ref. [13]. dσ d? dσ d? dσ d? =
QED+6

dσ d? dσ d? dσ d?

1+
Born

s2 sin2 θ αΛ4 6

(3) (4) (5)

=
QED+7

+
Born

s2 1 32π Λ6 7 s2 m2 1 e 32π Λ8 8

=
QED+8

+
Born

The de?nition of Eq. (3) is identical to the standard de?nition (Eq. 2) if Λ4 = α Λ4 . Similarly ± 2 6 Eq. (4) is equivalent to Eq. (5) if Λ8 = m2 Λ6 . Therefore only the parameters of Eq. (2) and 8 e 7 (4) are determined by a ?t to obtain limits on deviations from QED. The limits on the other parameters can easily be derived from these results. The existence of an excited electron e? with an e? eγ coupling would contribute to the photon dσ depends on production process via t-channel exchange. The resulting deviation from d? Born ? ? the e mass Me? and the coupling constant κ of the e eγ vertex [14]: dσ d? α2 =
e?

dσ d?

+
Born κ 4

(6)

1 q ′4 q4 2 E 2 sin2 θ + Me? + ′2 2 2 2 Me? (q 2 ? Me? )2 (q ? Me? )2 2 Me? E 4 sin2 θ κ 4 + 4 2 2 Me? (q 2 ? Me? )(q ′2 ? Me? ) q ′2 1 q2 1 κ 2 + ′2 + E 2 sin2 θ 2 + ′2 + 2 2 2 2 Me? q 2 ? Me? q ? Me? q ? Me? q ? Me? 5

,

√ with the beam energy E = s/2, q 2 = ?2E 2 (1 ? cos θ) and q ′2 = ?2E 2 (1 + cos θ). In the limit √ √ Me? ? s, the mass is related to the cut-o? parameter by Me? = κ Λ+ .

4

Event angle de?nition and radiative corrections

For the process e+ e? → γ1 γ2 the polar angle θ of the event is de?ned by the angle between either of the two photons and the beam direction since | cos θ1 | = | cos θ2 |. This is a good approximation for most of the events under consideration, since additional photons tend to be soft. For many events, however, there is a third energetic photon and thus | cos θ1 | = | cos θ2 | in general. Several angle de?nitions are possible to characterize an event. The following two are considered: cos θav = cos θ? | cos θ1 | + | cos θ2 | , 2 θ1 ? θ2 θ1 + θ2 = sin , sin 2 2 (7) (8)

where θ1 and θ2 are the polar angles of the most energetic photons. Both cos θav and cos θ? are identical to | cos θ| for two-photon ?nal states. For three-photon events in which the third photon is along the beam direction, θ? is equivalent to the scattering angle in the centre-of-mass system of the two observed photons. Fig. 1 shows the ratio of the angular distributions using both cos θav and cos θ? , relative to the Born cross section as derived using an O(α3 ) e+ e? → γγ(γ) Monte Carlo generator [4]. The event angles are calculated from the two photons with the highest generated energy. The comparison is made at the generator level, i.e. without detector simulation and e?ciency e?ects. It can be seen that the distribution of cos θav (Eq. 7) shows large deviations from the lowest order (Born) distribution for much of the cos θ range. For this analysis cos θ? (Eq. (8)) is chosen because it better matches the shape of the Born distribution over the range cos θ? < 0.9 considered in this analysis.

5

Event selection

Events are selected by requiring two or more clusters in the electromagnetic calorimeter (ECAL). A cluster is selected as a photon candidate if it is within the polar angle range | cos θ| < 0.97. The cluster must consist of at least two lead-glass blocks, with a combined ECAL energy deposit exceeding 1 GeV uncorrected for possible energy loss in the material before the ECAL. Events with a photon candidate having ?ve or more reconstructed clusters within a cone with a half-angle of 11.5? are rejected. This isolation criterion helps to reduce some instrumental background. There are two major classes of background remaining to the γγ(γ) signature. The ?rst can be identi?ed by the presence of primary charged tracks. Bhabha events, for example, have similar electromagnetic cluster characteristics as γγ(γ) events, but are normally easily distinguished 6

by the presence of charged tracks. The second class consists of events without primary charged tracks. Certain cosmic ray events and the Standard Model process e+ e? → ν νγγ contribute to ? this background.

5.1

Neutral events

Events having only photons in the ?nal state are classi?ed as ‘neutral events’. They should not have any charged track consistent with coming from the interaction point. The rejection of all events having tracks in the central tracking chambers CV or CJ would lead to an e?ciency loss because of converted photons. Nevertheless, contributions from any channel with primary charged tracks should be reduced to a negligible level. To reject events with primary charged tracks while retaining e?ciency for converted photons, only the inner part of the drift chambers are considered. First, the correlation between the observed clusters and charged hit activity in both drift chambers is used. Hits are counted in the φ-sectors of CV and CJ which are geometrically associated to each cluster. A correlation is assigned to a cluster if there are more than a given number of wires with hits in the associated φ-sector. ? A CV correlation is assigned if there are at least m wires with hits in the n CV layers nearest to the beam-pipe (denoted by m/n), depending on cos θ of the cluster: cos θ region Cut on m/n 6/12 0. < | cos θ| < 0.75 5/8 0.75 < | cos θ| < 0.95 4/6 or 5/8 0.95 < | cos θ| < 0.97 ? A CJ correlation is assigned if there are at least 12 wires with hits in the inner 16 CJ layers, independent of the cluster polar angle. Two vetoes are de?ned using combinations of these hit activity correlations in CV and CJ. A third veto tests for reconstructed charged tracks not correlated with either of the clusters. Any of the three vetoes rejects the event. ? The single veto requires that both the CV and CJ correlation are assigned for any cluster. ? The double veto requires that for each of the highest energy clusters either the CV or CJ correlation is assigned. ? The unassociated track veto requires that there be no reconstructed track with a transverse momentum of more than 1 GeV and at least 20 hits in CJ, separated by more than 10? in φ from all photon candidates.

7

5.2

Cosmic ray events

A cosmic ray particle can pass through the hadronic and electromagnetic calorimeters without necessarily producing a reconstructed track in the central tracking chambers. Events of this type are rejected if there are 3 or more hits in the muon chambers. In the case of 1 or 2 muon hits the event is rejected if the highest energy HCAL cluster with at least 1 GeV is separated from each of the photon candidates by more than 10? in φ. Events are rejected if the cluster extent in cos θ is larger than 0.4. This cut is primarily to reject beam halo events.

5.3

Kinematic selection

The event sample is divided into three classes I, II and III. The classes are distinguished by the number of photon candidates and the acollinearity angle ζ, de?ned as ζ = 180? ? ξ, where ξ is the angle between the two highest energy clusters. Di?erent selections are applied to each class separately, to make use of the di?erent kinematics. Only events with cos θ? < 0.9 are selected to avoid systematic errors due to large e?ciency and radiative corrections. All events having an acollinearity angle ζ < 10? (i.e. the two highest energy clusters are almost collinear) belong to class I independent of the number of photon candidates. For true e+ e? → γγ(γ) events in this class, the sum of the two highest cluster energies ES = E1 + E2 √ should almost be equal to the centre-of-mass energy s. The distribution of ES is shown in √ Fig. 2. Events having ES > 0.6 s are selected. This cut is well below the tail of the energy distribution for e+ e? → γγ(γ) Monte Carlo events as shown in the ?gure. Class II contains acollinear events (ζ > 10? ) with exactly two observed photon candidates. Events of this class typically contain an energetic photon that escapes detection near the beampipe (| cos θ| > 0.97). If the polar angle of this photon is approximated as | cos θ| = 1, its energy, Elost (Eq. 9), can be estimated from the angles of the observed photons θ1 and θ2 . The energy sum ES is then de?ned as the sum of the two observed cluster energies and the lost energy. Elost = ES The imbalance B, de?ned as B = (sin θ1 + sin θ2 ) cos φ1 ? φ2 2 , (11) sin θ1 + sin θ2 | sin (θ1 + θ2 )| = E1 + E2 + Elost s 1+ √
?1

(9) (10)

provides an approximate measure of the scaled transverse momentum of the event without using the cluster energies. Fig. 3 shows the distributions of B and ES . It can be seen that the in background (mainly ννγγ) is uniformly distributed√ B whereas the signal is peaked at low values. Events are selected if B < 0.2 and ES > 0.6 s. Since the angular de?nition discussed in section 4 uses the two highest energy photons, events are rejected if Elost exceeds the energy of either observed photon. Class III contains acollinear events (ζ > 10? ) having 3 or more observed photon candidates. To calculate the transverse and longitudinal momenta (pt , pl ) of the system the cluster energies 8

have to be used in addition to the photon angles. Since a non-zero longitudinal momentum could correspond to an additional photon along the beam direction, the energy sum ES is calculated as sum of the cluster energies Ei and pl :
n

ES =
i=1

Ei + pl .

(12)

√ √ Fig. 4 shows the distribution of ES / s versus pt / s for e+ e? → γγ(γ) Monte Carlo and for the data. In the data the e+ e? → γγ(γ) events are clearly separated from the background by the fact that they have small transverse momenta and an energy sum around the centre-of-mass energy. The main part of the background originates from cosmic ray events without hits in the √ √ muon chambers. The selection requirements ES > 0.6 s and pt < 0.1 s easily reject these events. No event with more than three clusters is observed. The angle sum planar three-photon events: α = αij + αik + αkj , α is used to identify (13)

with αij the angle between photons i and j. Seven planar events with α > 350? are accepted as three-photon events and included in the sample of e+ e? → γγ(γ). Two other events are consistent with three detected photons and an additional photon along the beam direction. The kinematic requirements used to select each of the event classes are summarised in Tab. 1. Event class all I II Requirements cos θ? < 0.9 ζ < 10?√ ES > 0.6 s ζ > 10?√ ES > 0.6 s E1 , E2 > Elost B < 0.2 2 photon candidates ζ > 10?√ ES > 0.6√s pt < 0.1 s ≥ 3 photon candidates α > 350? α < 350?

III

planar nonplanar

Table 1: Summary of the kinematic cuts. For de?nition of the variables see the text.

6

Corrections and systematic errors

Since the deviations from QED (Eqs. 2 - 6) are given with respect to Born level, the observed angular distributions need to be corrected to Born level. The e?ect of radiative corrections to the 9

Born level calculation is quanti?ed by R, the ratio of the angular distribution of e+ e? → γγ(γ) Monte Carlo and the Born cross-section as shown in Fig. 1: R= dσ d? (cos θ? )
MC

dσ d?

.
Born

(14)

The ratio R is used to correct the data bin by bin to the Born level. A 1% error on the total cross-section from higher order e?ects is assumed. No error on the slope of the distribution is included in the results. Since the O(α3 ) radiative corrections are small, the O(α4 ) e?ects are assumed to be negligible. The e?ciency and angular resolution of the reconstruction is determined using a Monte Carlo sample with full detector simulation. The e?ciency is reasonably constant for cos θ? < 0.9, but drops rapidly for cos θ? > 0.9. The overall e?ciency for cos θ? < 0.9 is 91.9% with a maximum of 95% in the barrel of the detector. A polynomial parametrisation E(cos θ? ) is used for the e?ciency correction. Due to uncertainties of the photon conversion probability and the Monte Carlo statistics a 1% systematic error is assumed for the e?ciency. The agreement between generated and reconstructed angles is very good. An angular resolution of 0.3? full width at half maximum is obtained. Background is studied using Monte Carlo events from the processes shown in Tab. 2. The expected ratio of background to signal is less than 0.4% and is neglected. Process Generator e+ e? → e+ e? (BHWIDE) e+ e? → e+ e? (TEEGG) e+ e? → ν νγγ ? + ? e e → ?+ ?? e+ e? → τ + τ ? e+ e? → qq ? Background events < 0.24 < 0.67 < 0.02 < 0.04 < 0.05 < 0.02

Table 2: Estimated 95 % CL upper limits for expected background processes from Monte Carlo √ at s = 130 - 172 GeV. The probability that a signal event is rejected by the neutral event selection due to random instrumental background causing a veto is studied with randomly-triggered events. For the single veto the probability is 4 × 10?4 and it is 1 × 10?4 for both the double veto and the track veto for. The small overall veto probability of 5 × 10?4 is therefore neglected. The systematic errors on the total cross-section are summarized in Tab. 3. Luminosity Radiative correction R Selection e?ciency E Background Total Uncertainty 0.5% 1.0% 1.0% < 0.4% 1.6%

Table 3: Summary of systematic errors on the cross-section

10

7

Results

In Tab. 4 the numbers of observed events in each class are compared to the QED expectations. The derived total cross-section σ in the range cos θ? < 0.9 is plotted in Fig. 5 as a function of the centre-of-mass energy. The numerical results for the cross-section are given in Tab. 4. They are corrected for e?ciency loss and O(α3 ) e?ects. All numbers agree well with QED expectations. The measured di?erential cross-sections at 130, 136, 161 and 172 GeV centre-of-mass energies dσ are shown in Figs. 6 and 7 together with a ?t of the function d? (Eq. 2). The ?t to the Λ± distribution is performed using the binned log likelihood method. The likelihood function L is based on Poisson statistics and de?ned as: Li = ?ni ??i i e ni ! (15)

with ni the number of observed and ?i the number of expected events per cos θ bin i. To dσ determine ?i the model dependent cross-section function d? is not integrated over the bin. Instead a simple procedure is applied in which the central value xi of the bin is determined as de?ned in Ref. [15]: dσ d? (xi ) =
Born

1 xu ? xl

xu xl

dσ d?

(y)dy,
Born

(16)

where xl and xu are the lower and upper boundaries of the bin i. In this way the di?erential dσ function d? can be directly compared to the integrated number of events presented as a histogram. The mean e?ciency Ei and radiative corrections Ri are included in the expectation ?i Energy 130 Expected Observed Class I 34.3±0.9 33 Class II 4.0±0.3 2 Class III planar 1.0±0.2 2 Class III nonplanar – 0 Born σtot 15.7 14.9±2.5 √ Energy s [GeV] 161 Expected Observed Class I 85.8±0.6 90 Class II 9.0±0.2 8 Class III planar 1.9±0.1 3 Class III nonplanar – 1 Born σtot 10.2 10.9±1.1 √ s [GeV] 136 Expected Observed 30.1±0.8 26 3.5±0.3 3 0.9±0.2 0 – 0 14.3 12.2±2.2 172 Expected Observed 77.6±0.7 75 8.0±0.2 14 1.8±0.1 2 – 1 9.0 9.7±1.0

Table 4: Comparison of number of observed events and Monte Carlo prediction. For nonplanar events no expectation is given, since the O(α3 ) Monte Carlo does not include these events. The two observed class III nonplanar events are kinematically compatible with a fourth photon along the beam-direction. In addition the total cross section corrected to the Born level is given. 11

for each bin. To allow the total number of expected events to vary within the systematic error, a normalization factor ? is added: ?i = ? dσ (xi ) (xu ? xl )Ei Ri L d? 1 ni
ni j=1

(17) (18)

Ei =

E(cos θj ),

where L is the integrated luminosity and cos θj is the angle of the j-th event. An estimator function P is de?ned which includes a Gaussian term with mean 1 and width δ = 0.016 (see Tab. 3) to account for the error of the normalization ?. The routine MINOS [16], which provides asymmetric errors, is used to minimize P : P = = (? ? 1)2 + δ2 (? ? 1)2 + δ2 ?2 ln Li 2 (?i ? ni ln ?i ) . (19)

i

i

The ?t is performed with two free parameters: the normalization ? and the model dependent parameter λ (see Tab. 5 and Eqs. 2, 4 and 6). To obtain the limits at 95% con?dence level the probability is normalized to the physically allowed region, i.e. λ+ > 0 and λ? < 0 as described in Ref. [17]. Results for the di?erent parameters are obtained from a simultaneous ?t to the angular distibutions for each centre-of-mass energy. A ?t is also performed for each centre-of-mass energy separately, with the results for Λ± given as an example in Tab. 5. The limits for the combined ?t are summarised in Tab. 6. To determine the limit on the mass of an excited electron Me? a dσ ?t is performed using d? ? (Eq. 6). For the results given in Tab. 5 the coupling constant κ
e

Model

λ



Fit result s [GeV] 130 136 λ ? 4.3 +20.6 · 10?10 GeV?4 0.999 ± 0.016 ?18.0 1.53 +5.12 · 10?10 GeV?4 1.001 ± 0.016 ?4.70 0.74 +3.17 · 10?10 GeV?4 1.000 ± 0.016 ?2.97 5.57 +35.9 · 10?18 GeV?6 1.000 ± 0.016 ?47.0 8.4 +11.0 · 10?6 GeV?2 1.000 ± 0.016 ?27.9 7.1 +19.3 · 10?10 GeV?4 0.999 ± 0.016 ?16.7

dσ d? Λ±

±1/Λ4 ±

161 172 130 - 172

?0.36 +4.13 · 10?10 GeV?4 1.000 ± 0.016 ?3.76

dσ d? QED+7 dσ d? e?

|1/Λ6|

130 - 172

2 |1/Me? | 130 - 172

Table 5: Results for ?t parameters λ and ?. For Λ± the results for all energies are shown seperately. The error on the normalisation ? refects the assumed systematic error. 12

for the (e? eγ)-vertex is ?xed at κ = 1. Fig. 8 shows the upper limit (95 % CL) on κ2 versus the mass of an excited electron Me? . The angular distributions for all energies agree well with the QED expectation. The lower limits obtained from the combined ?t on Λ± , Λ and Me? are higher than existing published results using lower energies (see Ref. [18], [19], [20] and [21]) and are in agreement with other results obtained at this centre-of-mass energy [22]. Previous limits on the excited electron mass with κ = 1 are Me? > 129 GeV [21], 136 GeV [20] and 147 GeV [1]. A resonance X produced in the process e+ e? → Xγ and decaying photonically X → γγ would be seen in the two photon invariant mass spectrum since this process leads to a three-photon ?nal state without missing energy. This search has been performed previously at the Z0 -peak [23] and at higher energies [21]. The invariant mass of each photon pair is shown in Fig. 9 for all events of classes II and III. There are three entries for events with three clusters. Since the angular resolution is very precise, the energies of the three photons are calculated from the angles assuming three photon kinematics: √ Ek ∝ sin αij ; E1 + E2 + E3 = s, (20) with Ek the energy of one photon and αij the angle between the two other photons. For class II events | cos θ| = 1 is assumed for the unobserved photon. A typical mass resolution for photon pairs of about 0.5 (0.7) GeV can be achieved for class III (II). The distribution agrees well with the Monte Carlo expectation from the QED process e+ e? → γγ(γ), with no enhancement due to a resonance is observed. From the class III distribution an upper limit on the total production cross-section times the photonic branching ratio of an isotropically produced resonance is calculated using the method of Bock [24]. Combining the data of all centre-of-mass energies and subtracting the e+ e? → γγ(γ) background the limits shown in Fig. 10 are obtained. The mass range is de?ned by the phase space of the selection and limited due to the acollinearity restriction. Parameter [GeV] Λ+ 195 Λ? 210 Λ6 793 Λ7 483 Λ8 15.5 Me? 194

√ Table 6: Summary of 95% CL lower limits obtained from the combined ?t to the s = 130, 136, 161 and 172 GeV angular distributions. The results are for the cut-o? parameters Λ± and mass scales Λ according to QED+6, QED+7 and QED+8 expectation (Eqs. 3 - 5). Λ6 and Λ8 are derived from Λ+ and Λ7 respectively. The lower limit for the mass of an excited electron is also determined with the coupling constant κ assumed to be κ = 1.

8

Conclusions

The QED process e+ e? → γγ(γ) has been studied using data taken with the OPAL detector at LEP energies above the Z0 resonance. Both the angular distributions and the total crosssection measurement agree well with QED predictions. Limits are set on cut-o? parameters, mass scales for contact interactions (γγe+ e? ) and for non-standard γe+ e? couplings, as well 13

as on the mass of an excited electron coupling to eγ. These limits are listed in Tab. 6. In the γγ invariant mass spectrum of events with three ?nal state photons, no evidence is found for a resonance X decaying to γγ. No photonic event with four or more detected photons is observed.

9

Acknowledgements

We particularly wish to thank the SL Division for the e?cient operation of the LEP accelerator at all energies and for their continuing close cooperation with our experimental group. We thank our colleagues from CEA, DAPNIA/SPP, CE-Saclay for their e?orts over the years on the time-of-?ight and trigger systems which we continue to use. In addition to the support sta? at our own institutions we are pleased to acknowledge the Department of Energy, USA, National Science Foundation, USA, Particle Physics and Astronomy Research Council, UK, Natural Sciences and Engineering Research Council, Canada, Israel Science Foundation, administered by the Israel Academy of Science and Humanities, Minerva Gesellschaft, Benoziyo Center for High Energy Physics, Japanese Ministry of Education, Science and Culture (the Monbusho) and a grant under the Monbusho International Science Research Program, German Israeli Bi-national Science Foundation (GIF), Bundesministerium f¨ r Bildung, Wissenschaft, Forschung und Technologie, Germany, u National Research Council of Canada, Hungarian Foundation for Scienti?c Research, OTKA T-016660, T023793 and OTKA F-023259.

References
[1] OPAL Collaboration, G. Alexander et al., Phys. Lett. B377 (1996) 222 [2] OPAL Collaboration, K. Ackersta? et al., Phys. Lett. B391 (1997) 210 [3] OPAL Collaboration, K. Ahmet et al., Nucl. Instr. and Meth. A305 (1991) 275 [4] F.A. Berends and R. Kleiss, Nucl. Phys. B186 (1981) 22 [5] S. Jadach et al., Phys. Lett. 390 (1997) 298 [6] D. Karlen, Nucl. Phys. B289 (1987) 23 [7] G. Montagna et al., Nucl. Phys. B452 (1996) 161 [8] S. Jadach et al., Comp. Phys. Comm. 66 (1991) 276

14

[9] T. Sj¨strand and M. Bengtsson, Comp. Phys. Comm. 43 (1987) 367 o T. Sj¨strand, Comp. Phys. Comm. 39 (1986) 347 o [10] OPAL Collaboration, J. Allison et al., Nucl. Instr. and Meth. A317 (1992) 47 [11] I. Harris and L.M. Brown, Phys. Rev. 105 (1957) 1656 F.A. Berends and R. Gastmans, Nucl. Phys. B61 (1973) 414 [12] S.D. Drell, Ann. Phys. 4 (1958) 75 ? [13] O.J.P. Eboli, A.A. Natale and S.F. Novaes, Phys. Lett. B271 (1991) 274 [14] A. Litke, Ph.D.Thesis, Harvard University, unpublished (1970) [15] G.D. La?erty and T.R. Wyatt, Nucl. Instrum. Meth. A355 (1995) 541 [16] MINUIT Reference Manual, F. James and M. Roos, CERN Program Library D506 [17] Review of Particle Physics, R.M. Barnett et al., Phys. Rev. D54 (1996) 1 [18] PLUTO Collaboration, C. Berger et al., Phys. Lett. B59 (1980) 87 JADE Collaboration, W. Bartel et al., Z. Phys. C19 (1983) 197 MARKJ Collaboration, B. Adeva et al., Phys. Rev. Lett. 53 (1984) 134 TASSO Collaboration, M. Altho? et al., Z. Phys. C26 (1984) 337 CELLO Collaboration, H.J. Behrend et al., Phys. Lett B168 (1986) 420 HRS Collaboration, M. Derrick et al., Phys. Rev. D34 (1986) 3286 MAC Collaboration, E. Fernandez et al., Phys Rev. D35 (1987) 1 AMY Collaboration, H.J. Kim, et al., KEK preprint 89-52 (1989) VENUS Collaboration, K. Abe et al., Z. Phys. C45 (1989) 175 TOPAZ Collaboration, K. Shimozawa et al., Phys. Lett. B284 (1992) 144 ALEPH Collaboration, D. Buskulic et al., Z. Phys. C59 (1993) 215 DELPHI Collaboration, P. Abreu et al., Phys. Lett. B268 (1991) 296 L3 Collaboration, O. Adriani et al., Phys. Lett. B288 (1992) 404 [19] OPAL Collaboration, M.Z. Akrawy et al., Phys. Lett B257 (1991) 531 [20] ALEPH Collaboration, D. Buskulic et al., Phys. Lett. B384 (1996) 333 [21] L3 Collaboration, M. Acciarri et al., Phys. Lett. B384 (1996) 323 [22] L3 Collaboration, M. Acciarri et al., CERN-PPE/97-77, submitted to Phys. Lett. [23] OPAL Collaboration, P.D. Acton et al., Phys. Lett B311 (1993) 391 L3 Collaboration, M. Acciarri et al., Phys. Lett. B345 (1995) 609 [24] P. Bock, Determination of Exclusion Limits for Particle Production Using Di?erent Decay Channels with Di?erent Energies, Mass Resolutions and Backgrounds, submitted to Nucl. Instrum. Meth. (1997)

15

Ratio R

1.3 1.2 1.1 1 0.9 0.8 0.7 0 0.2 0.4 0.6
cosθ
*

cosθav

0.8

1 cosθ

Figure 1: Ratio of the di?erential cross-section for the e+ e? → γγ(γ) Monte Carlo sample dσ dσ / d? . R is shown here for both angular relative to the Born cross-section, R = d? MC Born ? de?nitions cos θav (Eq. 7) and cos θ (Eq. 8). The event angle is calculated from the two photons with the highest generated energy.

16

Events / 0.025

80

60

OPAL
rejected

40

20

0

0

0.5

1

1.5 ES ? √s

Figure 2: Scaled sum of the two highest cluster energies for all events with an acollinearity angle ζ < 10? (corresponding to class I). The points with error bars represent the data, the histogram the e+ e? → γγ(γ) Monte Carlo expectation. The cut on this quantity is indicated.

17

Events / 0.05

10 1 10 10 10
-1

rejected

OPAL
a)

-2

-3

0

0.5

1

1.5

2 B

Events / 0.05

15

OPAL
b)
rejected

10

5

0

0

0.5

1

1.5 ES ? √s

√ Figure 3: Event distributions for data in class II for s = 130, 136, 161 and 172 GeV. Plot a) shows the distribution of the imbalance B, a measure of the scaled transverse momentum (for de?nition see Eq. 11) together with the selection cut. Plot b) shows the scaled sum of both cluster energies plus Elost after the cut on B. The cut is indicated. The points represent the data, the solid histogram the Monte Carlo expectation from e+ e? → γγ(γ) and the dashed histogram the Monte Carlos expectation from background (mainly ννγγ). 18

ES ? √s

1.5

1

0.5

a)
e e
+ -

γγ(γ) Monte Carlo
0.05 0.1 0.15 0.2 pt ? √s

0

0

ES ? √s

1.5

OPAL
b)
1

130 GeV 136 GeV 161 GeV 172 GeV

rejected
0.5

rejected

0

0

0.05

0.1

0.15

0.2 pt ? √s

Figure 4: The scaled energy sum versus the scaled transverse momentum for class III events, for a) e+ e? → γγ(γ) Monte Carlo and b) the OPAL data. The box indicates the selected region. The background comes mainly from cosmic ray events.

19

σ [pb]

40 30 20 10 0

OPAL
80 100 120 140 160 180 √s [GeV]

Figure 5: Total cross-section for the process e+ e? → γγ with cos θ < 0.9. The data are corrected for e?ciency loss and higher order e?ects and correspond to a Born level measurement. The result at the Z0 is taken from Ref. [19]. The curve corresponds to the Born level QED prediction.

20

dσ ? d? [pb ? sr]

12.5 10 7.5

OPAL
130 GeV

10

5 5 2.5 0 0 0.2 0.4 0.6 0.8 1 ? cos(θ ) 0

dσ ? d? [pb ? sr]

10 7.5

OPAL
136 GeV
5

5 2.5 0 0

0

0.2

0.4

0.6

0.8

1 ? cos(θ )

Figure 6: The measured angular distribution for the process e+ e? → γγ(γ) as selected in the √ three classes at s = 130 and 136 GeV. The data points show the e?ciency-corrected number of events; radiative corrections are also included. The solid curve corresponds to the Born level dσ QED prediction. The dotted lines represent 95% CL intervals of the ?t to the function d? .
Λ±

21

Corrected events / 0.05

12.5

10

Corrected events / 0.05

dσ ? d? [pb ? sr]

8 6 4

OPAL
161 GeV
20

10 2 0 0

0

0.2

0.4

0.6

0.8

1 ? cos(θ )

dσ ? d? [pb ? sr]

6

OPAL
172 GeV

20

4 10 2

0

0

0.2

0.4

0.6

0.8

1 ? cos(θ )

0

Figure 7: The measured angular distribution for the process e+ e? → γγ(γ) as selected in the √ three classes at s = 161 and 172 GeV. The data points show the e?ciency-corrected number of events; radiative corrections are also included. The solid curve corresponds to the Born level dσ QED prediction. The dotted lines represent 95% CL intervals of the ?t to the function d? .
Λ±

22

Corrected events / 0.05

Corrected events / 0.05

30

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

κ

2

OPAL
e lud d

ex

c

80

100 120 140 160 180 200 220 240 Mass e* [ GeV ]

Figure 8: Upper limit (95 % CL) on the square of the coupling constant κ2 as a function of the mass of an excited electron Me? .

23

Photon pairs / 1 GeV

4 3 2 1 0 80 100 120 140 160 Mass (γi γj) [ GeV ]

OPAL

a)

Photon pairs / 1 GeV

4 3 2 1 0 0 60 120 180 Mass (γi γj) [ GeV ]

OPAL

b)

Figure 9: The invariant mass of photon pairs for a) class II events and b) class III events. The points are the data, the histogram the e+ e? → γγ(γ) Monte Carlo expectation. There is one entry per event for class II events and three entries per event for class III events.

24

σXγ × BR( X → γγ) [pb]

0.5 0.4 0.3 0.2 0.1 0
excluded

OPAL
0 50 100 150 Mass X [ GeV ]

Figure 10: Lower limits (95 % CL) for the cross section times branching ratio for the process e+ e? → Xγ, X → γγ as a function of the mass of the resonance X. The e+ e? → γγ(γ) background is subtracted. The step at 120 GeV comes from the phase space limit of the 130 and 136 GeV data.

25



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