Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://zebu.uoregon.edu/~uochep/docs/docs00/pr328_journal.pdf Äàòà èçìåíåíèÿ: Mon May 23 22:44:58 2005 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 11:42:20 2012 Êîäèðîâêà: Windows-1251 Ïîèñêîâûå ñëîâà: m 5

EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH
CERN-EP-2000-148 OPAL PR 328 30 Novemb er 2000

Precise Determination of the Z Resonance Parameters at LEP: "Zedometry"
The OPAL Collaboration

Abstract
This final analysis of hadronic and leptonic cross-sections and of leptonic forward-backward asymmetries in e+ e- collisions with the OPAL detector makes use of the full LEP 1 data sample comprising 161 pb-1 of integrated luminosity and 4.5 × 106 selected Z decays. An interpretation of the data in terms of contributions from pure Z exchange and from /Z interference allows the parameters of the Z resonance to b e determined in a model-indep endent way. Our results are in good agreement with lepton universality and consistent with the vector and axial-vector couplings predicted in the Standard Model. A fit to the complete dataset yields the fundamental Z resonance 0 parameters: mZ = (91.1852 + 0.0030) GeV , Z = (2.4948 + 0.0041) GeV , h = (41.501 + 0.055) nb, R = 20.823 + 0.044, and A0, = 0.0145 + 0.0017. Transforming these parameters gives a measurement FB of the ratio b etween the decay width into invisible particles and the width to a single sp ecies of charged lepton, inv / = 5.942 + 0.027. Attributing the entire invisible width to neutrino decays and assuming the Standard Model couplings for neutrinos, this translates into a measurement of the effective numb er of light neutrino sp ecies, N = 2.984 + 0.013. Interpreting the data within the context of the Standard Model allows the mass of the top quark, mt = (162+29 ) GeV, to b e determined through its -16 influence on radiative corrections. Alternatively, utilising the direct external measurement of mt as an additional constraint leads to a measurement of the strong coupling constant and the mass of the Higgs b oson: s (mZ ) = 0.127 + 0.005 and mH = (390+750 ) GeV. -280

To be submitted to Eur. Phys. J. C


The OPAL Collaboration

G. Abbiendi2 , C. Ainsley5 , P.F. œ esson3 , G. Alexander22 , J. Allison16 , G. Anagnostou1 , Ak K.J. Anderson9 , S. Arcelli17 , S. Asai23 , S.F. Ashby1 , D. Axen27 , G. Azuelos18,a , I. Bailey26 , A.H. Ball8 , E. Barb erio8 , R.J. Barlow16 , T. Behnke25 , K.W. Bell20 , G. Bella22 , A. Bellerive9 , G. Benelli2 , S. Bentvelsen8 , C. Beeston16 , S. Bethke32 , O. Bieb el32 , I.J. Bloodworth1 , O. Boeriu10 , P. Bock11 , J. BÅ me14,g , D. Bonacorsi2 , M. Boutemeur31 , S. Braibant8 , P. Bright-Thomas1 , L. Brigliadori2 , oh R.M. Brown20 , H.J. Burckhart8 , J. Cammin3 , P. Capiluppi2 , R.K. Carnegie6 , B. Caron28 , A.A. Carter13 , J.R. Carter5 , C.Y. Chang17 , D.G. Charlton1,b , P.E.L. Clarke15 , E. Clay15 , I. Cohen22 , J.E. Conb oy15 , O.C. Cooke8 , J. Couchman15 , R.L. Coxe9 , A. Csilling15,i , M. Cuffiani2 , S. Dado21 , G.M. Dallavalle2 , S. Dallison16 , C. Darling34 A. De Roeck8 , E.A. De Wolf8 , P. Dervan15 , K. Desch25 , B. Dienes30,f , M.S. Dixit7 , M. Donkers6 , J. Dubb ert31 , E. Duchovni24 , G. Duckeck31 , I.P. Duerdoth16 , P.G. Estabrooks6 , E. Etzion22 , F. Fabbri2 , M. Fanti2 , L. Feld10 , P. Ferrari12 , F. Fiedler8 , I. Fleck10 , M. Ford5 , M. Foucher17 , A. Frey8 , A. Furtjes8 , D.I. Futyan16 , P. Gagnon12 , J. Gascon18 , Å S.M. Gascon-Shotkin17 , J.W. Gary4 , G. Gaycken25 , C. Geich-Gimb el3 , G. Giacomelli2 , P. Giacomelli8 , R. Giacomelli2 , D. Glenzinski9 , J. Goldb erg21 , C. Grandi2 , K. Graham26 , E. Gross24 , J. Grunhaus22 , M. Gruw?25 , P.O. Gunther3 , C. Ha jdu29 , G.G. Hanson12 , K. Harder25 , A. Harel21 , M. Harin-Dirac4 , e Å P.A. Hart9 , M. Hauschild8 , C.M. Hawkes1 , R. Hawkings8 , R.J. Hemingway6 , C. Hensel25 , G. Herten10 , R.D. Heuer25 , M.D. Hildreth8 , J.C. Hill5 , S.J. Hillier1 , A. Hocker9 , K. Hoffman8 , R.J. Homer1 , A.K. Honma8 , D. Horvath29,c , K.R. Hossain28 , R. Howard27 , P. Huntemeyer25 , P. Igo-Kemenes11 , ? Å K. Ishii23 , F.R. Jacob20 , A. Jawahery17 , H. Jeremie18 , C.R. Jones5 , P. Jovanovic1 , T.R. Junk6 , N. Kanaya23 , J. Kanzaki23 , G. Karap etian18 , D. Karlen6 , V. Kartvelishvili16 , K. Kawagoe23 , T. Kawamoto23 , R.K. Keeler26 , R.G. Kellogg17 , B.W. Kennedy20 , D.H. Kim19 , J. Kirk8 , K. Klein11 , A. Klier24 , S. Kluth32 , T. Kobayashi23 , M. Kob el3 , T.P. Kokott3 , S. Komamiya23 , R.V. Kowalewski26 , T. Kress4 , P. Krieger6 , J. von Krogh11 , D. Krop12 , T. Kuhl3 , M. Kupp er24 , P. Kyb erd13 , G.D. Lafferty16 , R. Lahmann17 , W.P. Lai19 , H. Landsman21 , D. Lanske14 , J. Laub er15 I. Lawson26 , J.G. Layter4 , A.M. Lee34 , A. Leins31 , D. Lellouch24 , J. Letts12 , L. Levinson24 , R. Liebisch11 , J. Lillich10 , C. Littlewood5 , A.W. Lloyd1 , S.L. Lloyd13 , F.K. Loebinger16 , G.D. Long26 , M.J. Losty7 , J. Lu27 , J. Ludwig10 , A. Macchiolo18 , A. Macpherson28,l , W. Mader3 , M. Mannelli8 , S. Marcellini2 , T.E. Marchant16 , A.J. Martin13 , J.P. Martin18 , G. Martinez17 , T. Mashimo23 , P. Mattig24 , Å W.J. McDonald28 , J. McKenna27 , T.J. McMahon1 , R.A. McPherson26 , F. Meijers8 , P. Mendez-Lorenzo31 , W. Menges25 , S. Menke3 F.S. Merritt9 , H. Mes7 , A. Michelini2 , S. Mihara23 , G. Mikenb erg24 , D.J. Miller15 , W. Mohr10 , A. Montanari2 , T. Mori23 , U. Muller3 , K. Nagai13 , Å 23 , H.A. Neal33 , R. Nisius8 , S.W. O'Neale1 , F.G. Oakham7 , F. Odorici2 , A. Oh8 , I. Nakamura A. Okpara11 , N.J. Oldershaw16 , M.J. Oreglia9 , S. Orito23,n , F. Palmonari2 , G. Pasztor8,i , J.R. Pater16 , ? G.N. Patrick20 , P. Pfeifenschneider14,h , J.E. Pilcher9 , J. Pinfold28 , D.E. Plane8 , B. Poli2 , J. Polok8 , O. Pooth8 , M. Przybycien8,d , A. Quadt8 , G. Quast8 . K. Rabb ertz8 , B. Raith3 , C. Rembser8 , ? P. Renkel24 , H. Rick4 , N. Rodning28 , J.M. Roney26 , S. Rosati3 , K. Roscoe16 , A.M. Rossi2 , Y. Rozen21 , K. Runge10 , O. Runolfsson8 , D.R. Rust12 , K. Sachs6 , T. Saeki23 , O. Sahr31 , E.K.G. Sarkisyan8,m , C. Sbarra26 , A.D. Schaile31 , O. Schaile31 , P. Scharff-Hansen8 , B. Schmitt8 , M. Schroder8 , Å M. Schumacher25 , C. Schwick8 , W.G. Scott20 , R. Seuster14,g , T.G. Shears8,j , B.C. Shen4 , C.H. Shepherd-Themistocleous5 , P. Sherwood15 , G.P. Siroli2 , A. Skuja17 , A.M. Smith8 , T.J. Smith26 G.A. Snow17,n , R. Sobie26 , S. Soldner-Remb old10,e , S. Spagnolo20 , R.W. Springer17 , M. Sproston20 , Å A. Stahl3 , K. Stephens16 , K. Stoll10 , D. Strom19 , R. Strohmer31 , L. Stumpf26 , B. Surrow8 , Å 1 , S. Tarem21 , R.J. Taylor15 , R. Teuscher9 , M. Tecchio9 , J. Thomas15 , M.A. Thomson8 , S.D. Talb ot S. Towers6 , D. Toya23 , T. Trefzger31 , I. Trigger8 , Z. Trocs? yi30,f , T. Tsukamoto23 E. Tsur22 , ? an 1 , I. Ueda23 , B. Vachon26 , P. Vannerem10 , M. Verzo cchi8 , E.H. Vokurka16 , M.F. Turner-Watson H. Voss8 , J. Vosseb eld8 , A. Wagner25 , D.L. Wagner9 , D. Waller6 , C.P. Ward5 , D.R. Ward5 , P.M. Watkins1 , A.T. Watson1 , N.K. Watson1 , P.S. Wells8 , T. Wengler8 , N. Wermes3 , D. Wetterling11 J.S. White6 , G.W. Wilson16 , J.A. Wilson1 , T.R. Wyatt16 , S. Yamashita23 , V. Zacek18 , D. Zer-Zion8,k 1


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School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK Dipartimento di Fisica dell' Universit` di Bologna and INFN, I-40126 Bologna, Italy a 3 Physikalisches Institut, UniversitÅ Bonn, D-53115 Bonn, Germany at 4 Department of Physics, University of California, Riverside CA 92521, USA 5 Cavendish Lab oratory, 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, Europ ean Organisation for Nuclear Research, CH-1211 Geneva 23, Switzerland 9 Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago IL 60637, USA 10 Fakultat fur Physik, Alb ert Ludwigs UniversitÅ D-79104 Freiburg, Germany ÅÅ at, 11 Physikalisches Institut, UniversitÅ Heidelb erg, D-69120 Heidelb erg, Germany at 12 Indiana University, Department of Physics, Swain Hall West 117, Blo omington IN 47405, USA 13 Queen Mary and Westfield College, University of London, London E1 4NS, UK 14 Technische Ho chschule Aachen, I I I Physikalisches Institut, Sommerfeldstrasse 26-28, D-52056 Aachen, Germany 15 University College London, London WC1E 6BT, UK 16 Department of Physics, Schuster Lab oratory, The University, Manchester M13 9PL, UK 17 Department of Physics, University of Maryland, College Park, MD 20742, USA 18 Lab oratoire de Physique Nucl? eaire, Universit? de Montr? Montr? Queb ec H3C 3J7, Canada e eal, eal, 19 University of Oregon, Department of Physics, Eugene OR 97403, USA 20 CLRC Rutherford Appleton Lab oratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK 21 Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel 22 Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 23 International Centre for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo 113-0033, and Kob e University, Kob e 657-8501, Japan 24 Particle Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel 25 UniversitÅ Hamburg/DESY, I I Institut fur Exp erimental Physik, Notkestrasse 85, D-22607 Hamat Å burg, Germany 26 University of Victoria, Department of Physics, P O Box 3055, Victoria BC V8W 3P6, Canada 27 University of British Columbia, Department of Physics, Vancouver BC V6T 1Z1, Canada 28 University of Alb erta, Department of Physics, Edmonton AB T6G 2J1, Canada 29 Research Institute for Particle and Nuclear Physics, H-1525 Budap est, P O Box 49, Hungary 30 Institute of Nuclear Research, H-4001 Debrecen, P O Box 51, Hungary 31 Ludwigs-Maximilians-UniversitÅ Munchen, Sektion Physik, Am Coulombwall 1, D-85748 Garching, at Å Germany 32 Max-Planck-Institute fur Physik, Fohring Ring 6, 80805 Munchen, Germany Å Å Å 33 Yale University, Department of Physics, New Haven, CT 06520, USA 34 Duke University, Department of Physics, Durham, NC 27708-0305, USA

a b c d e f g h i j k

and and and and and and and now and now and

at TRIUMF, Vancouver, Canada V6T 2A3 Royal Society University Research Fellow Institute of Nuclear Research, Debrecen, Hungary University of Mining and Metallurgy, Cracow Heisenb erg Fellow Department of Exp erimental Physics, La jos Kossuth University, Debrecen, Hungary MPI Munchen Å at MPI fur Physik, 80805 Munchen Å Å Research Institute for Particle and Nuclear Physics, Budap est, Hungary at University of Liverp ool, Dept of Physics, Liverp ool L69 3BX, UK University of California, Riverside, High Energy Physics Group, CA 92521, USA 2


l n

m

and CERN, EP Div, 1211 Geneva 23 and Tel Aviv University, School of Physics and Astronomy, Tel Aviv 69978, Israel. deceased

Contents
1 Introduction 2 The OPAL detector and 2.1 LEP 1 data samples . 2.1.1 LEP op eration 2.1.2 OPAL detector its simulation .................................... .................................... stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8 9 9 10 10 11 11 12 12 13 14 15 15 16 17 18 18 19 20 21 21 21 22 23 23 23 24 25 26 26 27 28 29 29 30 32 33 33 3

3 Selection and analysis of Z decay channels 3.1 Exp erimental acceptance . . . . . . . . . . . 3.2 Kinematic acceptance for cross-sections and 3.3 Four-fermion processes and radiative photon 3.4 Monte Carlo event generators . . . . . . . . 4 LEP energy 5 The luminosity measurement 6 Me 6.1 6.2 6.3

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7 Measurement of leptonic events 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . 7.1.1 Corrections and systematic uncertainties . 7.1.2 Trigger efficiency for e+ e- + - events 7.2 Selection of e+ e- e+ e- events . . . . . . . . . 7.2.1 t-channel contribution to e+ e- e+ e- . 7.2.2 Selection criteria for e+ e- e+ e- . . . . 7.2.3 Selection efficiency for e+ e- e+ e- . . . 7.2.4 Background in the e+ e- e+ e- channel 7.3 Selection of e+ e- ÷+ ÷- events . . . . . . . . . 7.3.1 Selection criteria for e+ e- ÷+ ÷- . . . . 7.3.2 Selection efficiency for e+ e- ÷+ ÷- . . . 7.3.3 Background in the e+ e- ÷+ ÷- channel 7.4 Selection of e+ e- + - events . . . . . . . . . 7.4.1 Selection criteria for e+ e- + - . . . . 7.4.2 Selection efficiency for e+ e- + - . . . 7.4.3 Background in the e+ e- + - channel 7.5 Correlations among lepton sp ecies . . . . . . . . 8 Cross-section measurements

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9 The 9.1 9.2 9.3

asymmetry measurements e+ e- ÷+ ÷- forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . e+ e- + - forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . e+ e- e+ e- forward-backward asymmetry . . . . . . . . . . . . . . . . . . . . . . . . resonance .................................... .................................... to e+ e- e+ e- . . . . . . . . . . . . . . . . . . . . . . . . . . parameters ........ ........ ........ ........ ........ ........ angle . . . . ........ ........ ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Parametrisation of the Z 10.1 Lowest order formulae 10.2 Radiative corrections . 10.3 t-channel contributions

11 Determination of electroweak and Standard Model 11.1 C -Parameter fits . . . . . . . . . . . . . . . . . . . . 11.2 Results of the model-indep endent Z parameter fits . 11.2.1 Error comp osition and 2 . . . . . . . . . . . 11.2.2 Theoretical uncertainties . . . . . . . . . . . . 11.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Z decay widths . . . . . . . . . . . . . . . . . 11.3.2 Coupling parameters and the effective mixing 11.3.3 Vector and axial-vector couplings . . . . . . . 11.3.4 s from the Z resonance parameters . . . . . 11.4 Standard Model fits . . . . . . . . . . . . . . . . . . 12 Summary and conclusions

A Four-fermion pro cesses and radiative photon interference A.1 Treatment of four-fermion final states . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Initial-final state interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B t-channel contributions to e+ e
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C Fit covariance matrix and energy spread corrections D S-Matrix results

List of Tables
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Summary of the data samples used for the cross-section measurements Ideal kinematic cuts for cross-sections and asymmetries . . . . . . . . Correction factors for e+ e- qq cross-section . . . . . . . . . . . . . Systematic errors of the acceptance hole emulation . . . . . . . . . . . Correction factors for e+ e- e+ e- cross-section . . . . . . . . . . . . Correction factors for e+ e- ÷+ ÷- cross-section . . . . . . . . . . . . Correction factors for e+ e- + - cross-section . . . . . . . . . . . . The e+ e- qq cross-section . . . . . . . . . . . . . . . . . . . . . . . The e+ e- e+ e- cross-section . . . . . . . . . . . . . . . . . . . . . . The e+ e- ÷+ ÷- cross-section . . . . . . . . . . . . . . . . . . . . . The e+ e- + - cross-section . . . . . . . . . . . . . . . . . . . . . . The pseudo cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . The LEP centre-of-mass energy covariance matrix for 1990-1992 . . . The LEP centre-of-mass energy covariance matrix for 1993-1995 . . . The LEP energy spread covariance matrix . . . . . . . . . . . . . . . . The luminosity systematic covariance matrix . . . . . . . . . . . . . . The e+ e- qq cross-section systematic covariance matrix . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 60 61 62 63 64 65 66 67 68 69 70 71 71 72 72 73


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The e+ e- e+ e- cross-section systematic covariance matrix . . . . . . . . . . . . . The e+ e- ÷+ ÷- cross-section systematic covariance matrix . . . . . . . . . . . . . The e+ e- + - cross-section systematic covariance matrix . . . . . . . . . . . . . The inter-sp ecies cross-section systematic covariance matrix . . . . . . . . . . . . . . The e+ e- ÷+ ÷- asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The e+ e- + - asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The e+ e- e+ e- asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lepton asymmetry systematic error covariance matrix . . . . . . . . . . . . . . . Results of the C-parameter fits with and without assuming lepton universality . . . . Correlation matrix for the C-parameter fit which does not assume lepton universality Correlation matrix for the C-parameter fit which assumes lepton universality . . . . Results for the model-indep endent Z parameters . . . . . . . . . . . . . . . . . . . . 5 parameter fit correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 parameter fit correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for the mZ consistency test . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error sources for the Z resonance parameters . . . . . . . . . . . . . . . . . . . . . . Results for the Z partial decay widths . . . . . . . . . . . . . . . . . . . . . . . . . . 5x5 partial width correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 3x3 partial width correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Limits on partial widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for the coupling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling parameter correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . Results for the axial and vector couplings . . . . . . . . . . . . . . . . . . . . . . . . 6x6 leptonic coupling correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . s from the Z resonance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the SM fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainties in the electron t+ti cross-sections . . . . . . . . . . . . . . . . . . . . . Results of the 16 and 15 parameter S-Matrix fits . . . . . . . . . . . . . . . . . . . . Correlation matrix for the 16 parameter S-Matrix fit . . . . . . . . . . . . . . . . . .

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73 74 74 75 76 77 78 79 80 81 82 82 82 83 83 84 84 85 85 85 85 86 86 86 86 87 87 88 89

List of Figures
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 OPAL event examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two- and four-fermion diagrams . . . . . . . . . . . . . . . . . . . . . . Distributions of the fundamental hadronic selection variables . . . . . . Comparison of data and MC simulation near the x-axis hole . . . . . . . Rescaling of the MC simulated cluster energies . . . . . . . . . . . . . . Non-resonant background in the e+ e- qq channel . . . . . . . . . . . Separation of e+ e- + - events in Etotal / s vs. ptotal / s. . . . . . . Fundamental distributions of the e+ e- e+ e- event selection variables Angular and acollinearity distributions of the e+ e- e+ e- event sample ÷ Evis distributions for e+ e- ÷+ ÷- . . . . . . . . . . . . . . . . . . . . Distributions of the principal selection cut variables for e+ e- + - . Systematic checks of the e+ e- + - selection . . . . . . . . . . . . . eeq q and luminosity event sensitivity checks . . . . . . . . . . . . . . . . Tau angular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . The differential cross-section in cos for e+ e- ÷+ ÷- . . . . . . . . . . The differential cross-section in cos for e+ e- + - . . . . . . . . . . The differential cross-section in cos for e+ e- e+ e- . . . . . . . . . . Contribution of individual terms to the differential cross-section . . . . . C -parameters as a function of the Higgs mass . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108


20 21 22 23 24 25 26

Model-indep endent Z parameters as a function Z cross-sections . . . . . . . . . . . . . . . . . Z forward-backward asymmetries . . . . . . . Probability contours in the A0, - R plane . . FB Probability contours in the gV - gA plane . . Z resonance parameters as a function of s . Probability contours in the s - mH plane . .

of .. .. .. .. .. ..

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Higgs .... .... .... .... .... ....

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109 110 111 112 113 114 115

6


1

Intro duction

One of the principal goals of the Large Electron-Positron (LEP) collider pro ject [1] at CERN is to make precise measurements of the prop erties of the Z gauge b oson, which are basic parameters of nature. These include the Z mass and its total decay width, as well as the comp osition and angular distributions of Z decay products. In combination with other precise measurements [2, 3], these physical observables provide the most stringent tests of the Standard Model (SM) of electroweak interactions [4] yet p ossible, allowing it to b e investigated at the level of higher-order electroweak corrections [5]. These tests represent a unique prob e of the underlying gauge structure of electroweak physics. Possible new physics or new particles b eyond those of the SM might b e revealed through the subtle changes which they would induce in the precise results rep orted here. From 1989 to 1995 LEP produced e+ e- collisions with centre-of-mass energies, s, close to the Z mass, mZ . This is referred to as the LEP 1 programme. For much of this p eriod s was chosen to b e within ab out 200 MeV of mZ , close to the p eak of the Z resonance curve, in order to obtain the maximum numb er of Z decay events. These "on-p eak" data provide high-statistics samples for measurements of production cross-sections and Z decay prop erties, such as partial decay widths and forward-backward asymmetries. Data were also recorded at several centre-of-mass energy p oints up to 4 GeV ab ove and b elow the p eak of the Z resonance. These "off-p eak" data samples provide sensitivity to the lineshap e of the Z resonance and hence to its mass, mZ , and total decay width, Z . The OPAL collab oration has previously published measurements of Z prop erties based on LEP 1 data recorded up to the end of 1992 [6-9]. In this pap er we rep ort new OPAL measurements of hadronic and leptonic cross-sections and leptonic forward-backward asymmetries based on the higher luminosity LEP 1 runs which took place during the years 1993-1995 and resulted in a four-fold increase of our data set. In 1993 and 1995 LEP energy scans were p erformed with significant luminosity collected at the off-p eak p oints, in order to improve substantially the determination of mZ and Z . These off-p eak p oints were chosen to b e approximately 1.8 GeV b elow and ab ove mZ . In 1994 all data were collected on p eak. In the following we will refer to the energy p oints b elow, close to and ab ove the Z resonance p eak as "p eak-2", "p eak" and "p eak+2", resp ectively. In parallel to the large increase of the data set, significant progress has b een made in each of the many asp ects which affect the precision of the results and their interpretation, namely the LEP energy calibration, the luminosity determination, the selection of the Z decay products and the theoretical predictions of observable quantities. The measurements from the 1993-1995 data are combined with those from previous years in order to determine the OPAL values for Z prop erties based on the full LEP 1 data sample. We analyse our results by first interpreting the cross-section and forward-backward asymmetry measurements in a model-indep endent fashion, in which the Z couplings to hadrons and leptons are allowed to vary freely. This provides a useful phenomenological description of observable Z prop erties and allows basic predictions of the SM, such as lepton universality and the vector and axial-vector structure of the couplings, to b e verified. We then go on to make a fit within the full context of the SM, leading to a direct determination of the accessible SM parameters. In a forthcoming publication these OPAL measurements will b e combined with similar results from the ALEPH [10], DELPHI [11] and L3 [12] collab orations, in order to determine the final set of LEP results for Z parameters. The structure of this pap er is as follows. Section 2 contains a brief description of the OPAL detector, simulation program and LEP 1 data samples. Section 3 gives an overview of the essential concepts we use to define our measured sample of Z decays. The LEP centre-of-mass energy calibration is outlined in Section 4. The luminosity measurement is reviewed briefly in Section 5, full details b eing available in [13]. The details of the hadronic and leptonic event selections and analyses are given in Sections 6 and 7. The cross-section and leptonic forward-backward asymmetry measurements are describ ed in Sections 8 and 9. In Section 10 we introduce the basic formalism for the parametrisation of the Z resonance and discuss radiative corrections. The determination of Z prop erties, their interpretation within the context of the SM and the implications for SM parameters are presented in Section 11. The results are summarised in Section 12. 7


2

The OPAL detector and its simulation

The OPAL detector is describ ed in detail in [14]. Therefore only those asp ects which are relevant to the present analysis are mentioned briefly here. In the following, a right-handed coordinate system is used in which the origin is located at the geometrical centre of the tracking chamb ers, the z -axis is along the electron b eam direction, the x-axis p oints to the centre of the LEP ring, r is the radial coordinate, normal to z , and the angles and are resp ectively the p olar and azimuthal angles with resp ect to z . The x-axis defines = 0. We set c = h = 1 throughout. ? Charged particle tra jectories are reconstructed and their momenta are measured using cylindrical central tracking detectors (CT). These consist of a silicon micro-vertex detector [15], a high precision vertex wire chamb er, a large volume jet chamb er (CJ) and thin z -chamb ers. The jet chamb er is 400 cm in length and 185 cm in radius. Its 24 azimuthal sectors are formed by planes of anode and cathode wires stretched parallel to the z -axis. It provides up to 159 space p oints p er track and also measures the ionisation energy loss of charged particles, dE/dx [16]. The z -chamb ers, which improve the track measurements in , are situated immediately outside and coaxial with the jet chamb er. Quality cuts are made to select well-reconstructed tracks emanating from the interaction p oint for use in the analysis. In general no track is used which has a momentum comp onent transverse to the b eam axis less than 100 MeV. The numb er of hits associated with the track in the central tracking chamb ers must b e at least 20. The distance of closest approach of the track to the nominal b eam crossing p oint must b e less than 2 cm radially and less than 100 cm along the z -axis. Track finding is nearly 100% efficient within the angular region | cos | < 0.97. For the analyses presented here, slightly tighter tracking requirements are imp osed which are optimised for each of the Z decay channels. The whole central detector is contained within a pressure vessel which maintains a constant absolute pressure of 4 bar and a solenoid which provides a uniform axial magnetic field of 0.435 T. The solenoid is surrounded by a time-of-flight scintillation counter array. The electromagnetic calorimeter (ECAL), located outside the CT pressure vessel and the solenoid, measures the energies and p ositions of showering particles. The barrel consists of a presampler followed by a cylindrical ensemble of 9440 lead glass blocks arranged such that each block p oints towards the b eam collision p oint, but the inter-block gaps p oint slightly away from the origin. Mechanically the barrel lead glass calorimeter consists of 10 C-shap ed modules. The two endcaps each consist of a presampler followed by 1132 lead glass blocks aligned parallel to the b eam axis. The barrel covers the angular region | cos | < 0.82 while the endcaps cover 0.81 < | cos | < 0.98. Overall, the electromagnetic calorimeter provides complete coverage for the entire angular range of | cos | < 0.98. Only in the region where the barrel and endcaps overlap, and at the narrow b oundaries b etween the barrel modules, is the uniformity of resp onse slightly degraded. For use in the analysis electromagnetic clusters in the barrel are required to have a minimum energy of 100 MeV, and clusters in the endcaps must consist of energy dep osits in at least 2 adjacent lead glass blocks and must have a minimum energy of 200 MeV. Calorimeters close to the b eam axis, and located on b oth sides of the interaction p oint, measure the luminosity using small-angle Bhabha scattering events. They complete the geometrical acceptance down to 25 mrad from the b eam axis. These include the forward detectors (FD), which are leadscintillator sandwich calorimeters, and at smaller angles, silicon-tungsten calorimeters [13, 17] (SiW), which were installed in 1993, increasing the precision of the luminosity measurement by an order of magnitude. The iron return yoke of the magnet lies outside the electromagnetic calorimeter and is instrumented with streamer tub es as a hadronic calorimeter (HCAL). Four layers of muon detectors [18] (MU) are situated outside the hadronic calorimeter. Muons with momenta ab ove 3 GeV usually p enetrate to the muon detectors. In addition, up to nine hits may b e recorded for minimum ionising particles traversing the hadronic calorimeter, further aiding muon identification. The OPAL data-acquisition system [19] reads out and records data associated with particular events which are selected using a three-level system, consisting of a pretrigger [20], a trigger [21] and an online event filter [22]. These make use of a large numb er of indep endent signals from a variety 8


of detector comp onents and have very high efficiency and redundancy for the Z decay events which are of interest for the analyses rep orted in this pap er. The trigger inefficiency for Z events decaying to charged fermions within the geometrical acceptance is less than 0.1% and the trigger redundancy allows all efficiencies to b e measured from the data. The online event filter serves primarily for data quality monitoring, but also allows a small numb er of obvious non-physics events to b e identified through software reconstruction and rejected from the data stream. The fact that no good events are rejected by the filter has b een carefully tested using the redundancy of selection criteria and also in numerous samples where rejection by the filter was temp orarily disabled. Events selected by the filter are fully reconstructed online and written into an offline storage facility for further analyses [23]. Unless stated otherwise, all Monte Carlo event samples have b een processed using a full simulation of the OPAL detector [24] which treats in detail the detector geometry and material as well as the effects of detector resolution and efficiency. The simulated events have b een reconstructed using the same procedures that were used for the OPAL data. Monte Carlo event generators are further discussed in Section 3.

2.1

LEP 1 data samples

In Table 1 the integrated luminosities are given for all of the LEP 1 data samples which are used in this analysis. The total integrated luminosity is 161 pb-1 , which includes 43 pb-1 of data recorded off-p eak. The p eak data are dominated by the dedicated high-statistics running in 1992 and esp ecially 1994, while most of the off-p eak data were collected in the precision scans in 1993 and 1995, when the running was confined to three energies: on-p eak and p eak+2 GeV. During the "prescan" p eriods in 1993 and 1995 running was confined to the p eak while all the necessary elements of the LEP b eam energy calibration were commissioned. These p eriods also coincide with the commissioning of the SiW luminometer (1993) or the SiW bunch tagger (1995), describ ed b elow. The additional 117 pb-1 of luminosity from 1993 to 1995 has warranted a significantly improved analysis of the hadronic decay channel. This has reduced the systematic error in the hadronic acceptance by ab out a factor of three, which makes it comparable with the much reduced statistical error. In the analysis of the leptonic channels studies of systematic effects have b enefited from the greatly increased statistics. Except for the measurement of the ÷+ ÷- asymmetry in 1992 (see Section 9.1) the data presented in our previous publications has not b een reanalysed. However, we have retrosp ectively applied corrections for a few small effects (see Section 8). 2.1.1 LEP op eration

Many asp ects of the LEP exp erimental programme were optimised to reduce p otential systematic effects in measuring the parameters of the Z. The off-p eak data are essential for the measurements of mZ and Z . For the 1993 and 1995 scans, the choice was made to run at only three p oints to maximise the statistical precision in the measurement of these quantities: at the p eak and approximately +1.8 GeV from the p eak. The exact energies were chosen to allow a precise calibration of the LEP b eam energies by resonant dep olarisation (see Section 11). In each scan the cross-sections at the two off-p eak p oints were typically measured in adjacent LEP fills, intersp ersed with fills at the Z p eak. This reduces any p ossible systematic biases resulting from changes in LEP or OPAL op erating conditions, and also gives balanced data samples at the off-p eak p oints within each year. For the determination of mZ the crucial exp erimental measurement is the ratio of the cross-sections ab ove and b elow the Z p eak, and the impact of inter-year systematic effects are minimised by the balance of p eak+2 and p eak-2 measurements within each year. A check of the stability of the LEP energy calibration can also b e made by measuring mZ in each scan year (see Section 11). The determination of Z , however, chiefly dep ends on the measurement of the ratio of off-p eak to on-p eak cross-sections. The on-p eak cross-sections are essentially determined by the 1992 and esp ecially the 1994 data, while the off-p eak cross-sections are determined by the 1993 and 1995 data. The measurement of Z therefore enjoys little inherent protection from time-dep endent systematic shifts 9


in the scale of the cross-section measurements. Control of p otential inter-year systematic uncertainties from changes in LEP op eration or the OPAL detector configuration are essential. The op eration of the LEP collider has evolved considerably over the years to increase the luminosity delivered to the exp eriments and to improve the precision with which the centre-of-mass energy could b e calibrated (see Section 4). Initially LEP op erated in a mode in which four bunches of electrons and four bunches of p ositrons collided at the four interaction p oints every 22 ÷s. From 1992 to 1994 LEP ran in a mode with 8 bunches of electrons and p ositrons which collide every 11 ÷s. In 1995 the LEP collider was op erated in a new "bunch-train" mode, in which four equally spaced bunch-trains replaced the usual bunches, and crossed at each interaction p oint every 22 ÷s. Each train consisted of up to four (typically three) bunchlets, separated from each other by 247 ns. Preparations for the bunch-train mode required a return to 4-bunch running at the end of 1994 (p eriods p eak(c) and (d)). The impact of these changes on the OPAL set up are describ ed b elow. 2.1.2 OPAL detector stability

The configuration of the OPAL detector has changed slightly over the entire p eriod of these measurements. Of greatest significance was the installation of the precision SiW luminometer b efore the 1993 run. This yields absolute luminosity measurements for the 1993-1995 data samples which are an order of magnitude more precise than the earlier, systematics-limited measurements using the forward detectors. A silicon micro-vertex detector was first installed in OPAL b efore the 1992 running p eriod. The detector was augmented for the 1993 run. It was removed for repairs at the end of 1994 (p eriods p eak(c) and (d)), and replaced with an improved geometry for the 1995 run. Data from the microvertex detector are not used in the reconstruction of tracks for this analysis. The 0.015X0 of additional material it introduces affects the conversion of photons into electron pairs, and is adequately reproduced by the detector simulation program. Halving the interval b etween bunch crossings from 22 to 11 ÷sec in 1992 no longer allowed sufficient time to form the full trigger information from the tracking chamb ers. OPAL adopted a pretrigger scheme [20] in which very loose trigger conditions using fast signals identified a small fraction ( 1%) of all bunch crossings for which sensitivity to interactions in the next crossing would b e lost while waiting for the full track trigger information. The pretrigger inefficiency for all relevant events was determined to b e negligible. The new bunch-train mode of op eration in 1995 required several modifications. In particular, the SiW luminometer electronics were considerably modified to op erate prop erly under these conditions [13]. In addition, small corrections were applied to the electromagnetic calorimeter energy measurements on the basis of bunchlet timing information obtained from the time-of-flight detectors, or from the tracking chamb ers. For most events, the tracking detectors are able to determine the bunchlet crossing which produced the visible tracks, since only one p otential choice of the origin for the drift times yields good tracks passing through the interaction p oint. The misassignment of bunchlet numb er could p otentially cause problems in track reconstruction. The influence of such bunchlet effects on the event selection efficiencies has b een checked using redundant selections and was found to b e negligible.

3

Selection and analysis of Z decay channels

In the SM the Z is exp ected to decay into a fermion-antifermion pair. With three generations of fermions, there are eleven p ossible decay channels: five quark flavours (the top quark is too heavy), three neutrino sp ecies and three charged leptons. The approximate branching ratios are 70:20:10 to hadrons, neutrinos and charged leptons, resp ectively. Decays of the Z to neutrinos normally go undetected and are referred to as invisible decays. No attempt is made in this analysis to separate the different quark flavours, with all hadronic Z decays b eing classified as e+ e- qq events. Mea10


surements of Z partial decay widths and forward-backward asymmetries using hadronic Z decays in which the different quark flavours are distinguished have b een published in [25]. The analysis describ ed here is focused on selecting visible Z decay events in just four categories: e+ e- qq, e+ e- , ÷+ ÷- and + - , where, in each case, initial- and final-state radiation can lead to one or more additional photons in the final state. Also, the classification e+ e- qq is inclusive of all final-state QCD interactions, including hard gluon bremsstrahlung. In general, events in these categories can easily b e distinguished from each other and from the remaining background, which is very small compared to the Z resonance signal, leading to event selections of high efficiency and purity. Typical examples of the four event categories as observed in the OPAL detector are shown in Figure 1. Indep endent of the sp ecific decay channel, the detected particles in Z events generally exhibit momentum balance along the b eam direction, in contrast to background events from twophoton interaction processes (e+ e- e+ e- f f ). Events produced by the passage of cosmic rays through the detector can normally b e rejected since they are rarely consistent with the observed origin of true signal events in space and time. The e+ e- qq events are characterised by high-multiplicity final states, due to quark fragmentation and hadronisation, in contrast to the low-multiplicity lepton-pair events. The e+ e- e+ e- events have two high-energy dep osits from the final-state electrons in the electromagnetic calorimeter, associated with tracks reconstructed in the central detector. The e+ e- ÷+ ÷- events typically contain two high-momentum central detector tracks, with little energy dep osited in the electromagnetic calorimeter. The tracks are usually associated with track segments in the muon chamb ers or the hadron calorimeter strips, which provide evidence for p enetrating muons. The e+ e- + - events are distinguished by two low-multiplicity `jets', each consistent with the decay of a , and typically a lower measured energy than the other lepton-pair final states, due to the undetected neutrinos from the decays. The detailed selection criteria for events in each of these four categories are describ ed in Sections 6 and 7. The selections have very little overlap, in particular for the three leptonic selections it is ensured that no event is classified in more than one category (see Section 7.5).

3.1

Experimental acceptance

The limited coverage of the tracking chamb ers in the forward direction prevents us from detecting leptons close to the b eam direction, but the simple top ologies of the lepton-pair final states allow a precise exp erimental acceptance to b e defined in a restricted region of cos . In the case of the e+ e- ÷+ ÷- and + - channels the cross-sections in these regions (| cos | < 0.95 and 0.90 resp ectively) can then b e extrap olated to the full angular acceptance. In addition to the Z s-channel annihilation diagram, the process e+ e- e+ e- has contributions from t-channel diagrams, dominated by photon exchange, which lead to a divergence of the forward cross-section. This makes a similar extrap olation of the e+ e- e+ e- cross-sections to the full angular acceptance meaningless. We therefore restrict the e+ e- e+ e- measurements to a region well within the detector, | cos | < 0.70, which limits the t-channel contributions to a manageable level (15% at the p eak), and do not extrap olate them. For e+ e- qq events, the hadronic jets are broad enough to ensure high efficiency even for events with decay axes close to the b eam direction. Hence we have almost 100% acceptance for Z events decaying hadronically, and directly measure events produced over essentially the full angular region. The approximately 0.5% inefficiency is dominated by very narrow 2-jet events oriented along the b eam line.

3.2

Kinematic acceptance for cross-sections and asymmetries

In order to interpret the measured cross-section and asymmetry for each reaction the limits of kinematic phase space must b e sp ecified in a precise manner, which is adapted to available theoretical calculations, in order to take into account prop erly the effect of initial- and final-state photon radiation. We therefore correct our raw measurements to corresp ond to simple, ideal, kinematical limits, which 11


are chosen to corresp ond reasonably closely to the exp erimental acceptances to reduce the resulting acceptance extrap olations. Table 2 summarises the limits of the ideal kinematic phase space within which we define our cross-sections and asymmetries for each sp ecies. The exp erimentally observed numb er of events in each channel is corrected for selection inefficiencies and background to give the total numb er of events produced within the kinematic acceptance defined by these idealised cuts. We sp ecify the kinematic phase space for the cross-sections for all sp ecies, except electrons, in terms of a lower b ound on either s /s or m2f /s. Here s is the squared centre-of-mass energy after initial-state f radiation and mf f is the invariant mass of the final-state fermion pair. Both definitions suffer from some degree of ambiguity. In the case of e+ e- qq, QCD effects obscure the precise meaning of m2f , while s suffers from the imp ossibility of distinguishing initial- from final-state radiation. There f is, however, little quantitative difference in our regime b etween m2f and s . Switching b etween using f m2f and s in the acceptance definition changes the hadronic cross-section, for example, by only a few f parts in 105 . The exp erimental acceptance b ecomes very small for events with low s or m2f , but the numb er f of produced events in this region is also small. The calculated numb er of hard radiative events which fall b etween our ideal and exp erimental acceptance limits in prop ortion to all accepted events is ab out 1.5 × 10-4 for hadrons and 5 × 10-3 for ÷+ ÷- and + - . For e+ e- ÷+ ÷- and + - the dominant acceptance extrap olations (resp ectively 7% and 15%) are due to the limited angular region of the exp erimental event selection. For the process e+ e- e+ e- the ideal kinematic acceptance is chosen very close to the exp erimental acceptance, b oth b eing defined by the same range of p olar angles allowed for the final-state e- and the maximum e+ e- acollinearity. No additional constraint is placed on the angle of the e+ The total acceptance correction required is less than 1%. For measuring the asymmetries we treat the kinematic acceptance for all the leptons similarly to the electrons, and limit the range of p olar angle of the final-state fermions to the nominal exp erimental acceptance. To reduce effects associated with strong initial-state radiation, we also limit the maximum acollinearity b etween the pair of leptons to 10 or 15 , as shown in Table 2.

3.3

Four-fermion processes and radiative photon interference

Four-fermion diagrams, as shown in Figure 2, also contribute at a small level to the selected event samples. Some of these, such as the conversion of initial- or final-state photons to fermion pairs (Figure 2(b) and Figure 2(c)) are prop erly considered as radiative corrections to fermion pair production. We therefore apply corrections to the selection efficiency where necessary to ensure that such events are fully counted as part of the measured signal. Other four-fermion processes, such as multi-p eripheral diagrams (two-photon processes, Figure 2(d)) are clearly unrelated to the Z and we therefore subtract them as background. A third class of four-fermion processes (Figure 2(e) and Figure 2(f )) are pair corrections to scattering in the t-channel. Such events are treated as signal only in the e+ e- e+ e- channel. More details concerning the treatment of four-fermion final states are given in App endix A.1. Our measured cross-sections and asymmetries include the effects of interference b etween initial- and final-state photon radiation. Since our primary Monte Carlo programs, JETSET and KORALZ1 do not include such interference, we adjusted the calculated event selection efficiency for our cross-section measurements by corrections of O(10-4 ) to account for this deficiency, as describ ed in App endix A.2. For the asymmetries there is a close corresp ondence b etween the exp erimental and ideal acceptances, and no such acceptance correction is necessary.

3.4

Monte Carlo event generators

The following event generator programs have b een used to simulate signal and background processes: e+ e- qq events have b een generated using the programs JETSET, version 7.3 [26], and HERWIG,
1

KORALZ can include initial-final-state interference, but only when radiative corrections are treated to O().

12


version 5.8 [27], with hadronisation parameters tuned using a sample of hadronic events selected from OPAL LEP 1 data [28]. Cross checks have b een made using version 7.4 of the JETSET program [29] with an up dated set of parameters [30] tuned using a larger sample of OPAL LEP 1 data. The KORALZ program, version 4.02 [31], has b een used for e+ e- ÷+ ÷- and e+ e- + - events and the BHWIDE program, version 1.00 [32], for e+ e- e+ e- events. Background events produced by twophoton interaction processes, e+ e- e+ e- f f , have b een studied using the programs PHOJET [33], HERWIG and the program of Vermaseren [34], while those from e+ e- have b een calculated using the RADCOR program [35]. The four-fermion signal processes in the s-channel have b een studied using the FERMISV [36] generator. Background from t-channel four-fermion events has b een evaluated using PYTHIA [29] for e+ e- e+ e- qq and grc4f [37] for e+ e- e+ e- + - .

4

LEP energy

The LEP energy scale is one of the crucial ingredients in the determination of Z resonance parameters. The initial energy calibration of LEP [38] was p erformed in 1989 and 1990 by circulating protons in the LEP ring at 20 GeV and using magnetic measurements to obtain the centre-of-mass energy in physics conditions. This resulted in errors on s, and hence mZ , of 28 MeV and 22 MeV for the 1989 and 1990 data, resp ectively. Using the technique of resonant dep olarisation [39], a precise calibration of the LEP energy scale was achieved in 1991, resulting in a systematic uncertainty of 6 MeV on mZ and approximately 5 MeV on Z [40]. The recognition of energy shifts due to the alignment of the copp er RF cavities had particular imp ortance at the two interaction p oints equipp ed with cavities (OPAL and L3). The calibration of the LEP energy scale in 1992 [41] was p erformed using a similar procedure. In 1992, however, calibrations with resonant dep olarisation were successful only late in the year and showed a large spread. The central value of the energy derived from the p olarisation measurements was found to b e in good agreement with the one obtained from methods based on measuring the magnetic field in the LEP dip oles. The quoted error of 18 MeV on s arises predominantly from the scatter of the dep olarisation measurements and the extrap olation to the b eginning of the year using magnetic measurements. Since the 1992 data have all b een collected at the p eak of the Z resonance, this larger error has an insignificant impact on the precision of the derived Z resonance parameters. In 1993 a more complete understanding of the time-dep endent parameters that influence the LEP energy, such as tidal deformations of the Earth and changes in magnet temp eratures, as well as more frequent p olarisation measurements, led to a further increase in the precision of the energy calibration. A concerted effort to ensure the complete logging of all LEP parameters relevant to the energy measurement allowed this inherent precision to b e extended to a large fraction of all fills. This resulted in a systematic uncertainty of 1.4 MeV on the absolute centre-of-mass energy of LEP [42], and the p oint-to-p oint energy errors, including the error on the centre-of-mass energy spread, contributed an uncertainty of 1.5 MeV on Z . The quality of measurement was maintained throughout the 1994 running. In 1995 vertical disp ersion of the b eam energy at the interaction p oint was introduced by bunchtrain op eration, which required frequent vernier scans to control p ossible shifts in the mean centreof-mass collision energy [43]. More significantly, additional instrumentation installed in the LEP tunnel allowed the observation of an unexp ected change in the b eam energy during LEP fills. Leakage currents from the electric railway system in the Geneva area flow through the LEP tunnel and p erturb the magnetic field of the LEP dip oles, which causes the b ending field (and hence, the b eam energy) to rise steadily during a fill. The interplay b etween the rise introduced by the leakage currents and that due to temp erature variations in the dip ole magnets necessitated a more detailed study of the temp erature-dep endence of the b ending field, resulting in a much improved model of this b ehaviour. This knowledge provides the definitive description of the LEP energy calibration for 1993-1995 [43]. In light of this improved understanding, the 1993 energies and their errors, first published in [42], have b een revised, and the errors significantly increased. The data needed for up dating the less critical calibrations for years earlier than 1993 are unavailable. However, these errors are uncorrelated with 13


the errors for 1993-1995, so the data can easily b e combined. Taking these final calibration results into account, the uncertainties in the LEP energy contribute errors of 1.8 MeV and 1.3 MeV to the OPAL determination of mZ and Z resp ectively, as describ ed in Section 11.2.1. At a scale a few times larger than the inherent precision of the LEP energy calibration an imp ortant and completely indep endent test of its consistency over time can b e made by studying the stability in the value of mZ measured over the six years of data-taking (see Table 32). The values for the spread of the centre-of-mass energies, due to the energy spread of the particles in the b eams, for the different running p eriods and energy p oints are summarised in Tables 20 and 21 of reference [43]. For the 1993-1995 data they range b etween 54.6-56.7 MeV, increasing with s, and with uncertainties of 1.1 MeV in 1993-1994 and 1.3 MeV in 1995. The centre-of-mass energy spreads for data from years b efore 1993 have b een re-evaluated using measurements made in subsequent years. They are b etween 43-53 MeV, with an error of 3 MeV. We quote our cross-sections and asymmetries b oth as measured and after correction for the energy spread, as describ ed in App endix C.

5

The luminosity measurement

All OPAL cross-section measurements rely on normalising the observed numb er of events in any given final state to the integrated luminosity measured by counting the numb er of Bhabha scattering events at angles small enough that t-channel photon exchange dominates the cross-section, and the influence of the Z is reduced to a small correction (< 1%). The SiW luminometer and the associated luminosity selection, which are fully describ ed in [13], have b een optimised to exploit the characteristics of Bhabha scattering, e+ e- e+ e- . The small-angle cross-section of e+ e- e+ e- is dominated by the t-channel exchange of a photon, leading to a 1/ 3 sp ectrum, where is the angle of the out-going electron and p ositron with resp ect to the incoming b eams. By instrumenting a region at very small angles, the accepted e+ e- e+ e- cross-section for SiW is approximately 79 nb, sufficiently larger than the total cross-section for Z production to limit the imp ortance of the statistical uncertainty in the luminosity measurement. The forward-p eaked 1/ 3 Bhabha sp ectrum requires that the detector and luminosity analysis define the inner edge of the acceptance with particularly high precision in order to reduce the systematic error. For example, to achieve a precision of 10-3 , the inner edge of the acceptance must b e known to 10 ÷rad, corresp onding to ab out 25 ÷m in the radial coordinate of the showers produced in the calorimeters, which are located at a distance of approximately 2.5 m on each side of the b eam crossing p oint. For the measurement of the absolute luminosity, the exp erimental error achieved by the SiW luminometer is 3.4 × 10-4 , which includes all intrinsic and time-dep endent sources of exp erimental uncertainty, such as detector geometry, gain variations, energy and p ositional biases in the detector resp onse to electromagnetic showers, variations in the LEP b eam geometry, backgrounds and other environmental influences. In 1995 the introduction of bunch-trains in LEP led us to install a "wagon tagger" [13] which allowed the luminometer to measure Bhabha-scattering events in all bunchlet crossings in each bunch-train with no compromise in p erformance. For the first four years of the LEP 1 programme OPAL used the forward detectors [14] for measuring the luminosity with a resulting systematic uncertainty of 4.1 × 10-3 . This uncertainty dominates the total uncertainty on the hadronic p ole cross-section obtained from the 1990-1992 data. The larger systematic error of the FD measurement leads to a reduced weight from the earlier data ( 4%) in determining the hadronic cross-section. Since the SiW luminometer was installed in front of FD, obscuring its inner edge, no direct exp erimental cross-calibration of the two detectors is p ossible. A retrosp ective check of the hadronic p eak cross-sections measured from the 1992 and 1994 data shows good agreement (0.4 + 0.5)%. The luminosity measurement requires that the theoretical cross-section for small-angle Bhabha scattering within the exp erimental acceptance b e accurately calculated. At the time of our last Z resonance publication [9], techniques based on YFS exp onentiation yielded an error of 2.5 × 10-3 [44]. These calculations have b een extended to include second-order next-to-leading log terms [45], and cal14


culations of third-order indicate that the error is 6.1 × 10-4 [46]. Improved calculations of light pair corrections within the OPAL SiW acceptance result in a final total theoretical error of 5.4 × 10-4 [47]. The scale of our previously published cross-sections, normalised with the FD luminometer, have b een corrected in this analysis to incorp orate these latest theoretical results in small-angle Bhabha scattering.

6

Measurement of e+ e- qq events

In the SM ab out 87% of visible Z decays are exp ected to result in quark-antiquark pairs, leading to hadronic final states. These events therefore provide the most accurate determination of mZ and Z . In general, hadronic events can b e clearly distinguished from leptonic Z decays, or other background processes, due to their high multiplicity and large visible energy of final-state particles and the balance of visible momentum along the b eam direction. Problems mainly arise for events consisting of two narrow jets in which most of the final-state particles have tra jectories with only a very small angle relative to the b eam axis. The geometrical coverage of the detector is incomplete in these regions, b ecause of the b eam pip e and associated apparatus, and therefore only a fraction of the particles from such an event are registered. The evaluation of the selection inefficiency which results from the loss of such events dep ends on the modelling of QCD effects in the process qq hadrons, which we subsequently refer to as "hadronisation" modelling. Since the treatment of hadronisation in Monte Carlo event generators is limited by phenomenological uncertainties, we have develop ed a new technique which uses the centre of the barrel region to emulate the setup and resp onse of the detector close to the b eam axis. This allows us to estimate the inefficiency using real data events and results in a factor of almost three reduction in the systematic error for the acceptance calculation compared to our previous publications [8, 9].

6.1

Selection criteria

The selection of e+ e- qq events uses the information from tracks reconstructed in the central detector (CT) and clusters of energy reconstructed in b oth the lead glass (ECAL) and the forward detector (FD) electromagnetic calorimeters. Tracks are typically reconstructed with good efficiency down to = 15o , ECAL clusters extend down to = 11o and the FD covers the range from 3o < < 9o . The selection is based on the following event parameters. Ntrack is the total numb er of tracks. i Ncluster and NFD are the total numb ers of ECAL clusters and FD clusters, resp ectively. Ecluster is the j energy of ECAL cluster i, (i = 1,Ncluster ). EFD is the energy of FD cluster j , (j = 1,NFD ), where a valid FD cluster must have at least 3 GeV. Each event is divided into two hemispheres with resp ect to the thrust axis of the event, which is determined using all the tracks and clusters in the event. The invariant mass of each hemisphere is calculated from the tracks and clusters which lie within it, assigning the pion mass to tracks and zero mass to clusters. No attempt is made to eliminate the implicit double counting of energy when using b oth track momenta and calorimeter energy for charged particles, such as electrons. Such a simple treatment is sufficient for a good separation of hadronic and leptonic final states. The sum of these two hemisphere invariant masses is mhemi . Five selection criteria are used to define a candidate e+ e- qq event: ž Charged multiplicity: Ntr
ack

2. +N
cluster

ž Total multiplicity: Nall N

track

+N

FD

11 .
hemi

ž Sum of the invariant masses of the two hemispheres: m ž Visible energy in ECAL and FD:
mh Ecal / s i i Ecl

> 4.5 GeV.
j

uster

+(

j EFD /3) / s > 0.1 .

15


ž Energy balance along the b eam direction: R
energy bal



i

i (Ecl

uster i

cos i )+
uster

j j

j (EFD cos j ) j EFD )

(

i Ecl

+

< 0.75 ,

where i and j are the p olar angles of ECAL cluster i and FD cluster j , resp ectively. Most Z decays to leptonic final states are rejected by the cuts on Nall and mhemi . The additional loose cut on Ntrack reduces the background arising from events induced by cosmic ray muons or ener mh showers. The requirements on Ecal / s and Rbal gy suppress contributions from two-photon interaction processes. The factor 1/3, by which the weight of the FD energy is reduced in the construction of mh Ecal / s, is chosen to optimise the separation b etween e+ e- qq signal events and two-photon background. A comparison of distributions of some of the cut variables b etween data and Monte Carlo simulation is shown in Figure 3. The Monte Carlo prediction, based on the JETSET [26] event generator, for the e+ e- qq selection inefficiency on-p eak is MC = (48.1 + 1.1) × 10-4 within s /s > 0.01. The inefficiency of the ? selection is approximately constant as a function of s down to s /s = 0.25. For events with still harder initial-state radiation (ISR) it rises rapidly and approaches 100% for s /s = 0.01, most events b eing ener rejected by the requirement on Rbal gy . At the p eak energy events with hard radiation (s /s < 0.25) comprise only a small fraction (4 × 10-4 ) of the total cross-section and ab out half of these fail the selection. At centre-of-mass energies away from the Z p eak the relative rate of hard-radiation events is higher, which slightly reduces the efficiency compared to the p eak. Changes of -4.7 × 10-4 and -1.4 × 10-4 were calculated using dedicated Monte Carlo samples at p eak-2 and p eak+2, resp ectively. In addition, a small contamination (0.3 × 10-4 at the p eak) from selected events with s /s < 0.01 is subtracted. Effects from multiple ISR, which are not simulated in the JETSET Monte Carlo generator, have b een checked and were found to b e negligible for determining the efficiency. The total numb er of e+ e- qq events selected from the 1993-1995 data samples is 2 908 566, giving a relative statistical error of 6 × 10-4 . In order to reduce the uncertainties on systematic effects to a corresp onding level one must carefully disentangle the p otential error sources and apply new techniques which base the corrections largely on the observed data prop erties. In the following subsections we first describ e a method to determine the inefficiency due to events lost along the b eam axis based on wellmeasured data events in the barrel region of the detector. We then discuss corrections and systematic errors associated with the selection of e+ e- qq events and the background estimation. Unless sp ecified otherwise the quoted correction factors and uncertainties refer to the 1994 p eak data. Full details for all data p eriods are given in Table 3. Correlations b etween energy p oints and data-taking years are sp ecified in Table 17.

6.2

Acceptance hole emulation

The overall inefficiency of the event selection is approximately 5 × 10-3 . Most events which are lost have narrow, two-jet final states p ointing into the very small p olar-angle region (large | cos |, close to the b eam axis), where there is a "hole" in the acceptance. In contrast, the inefficiency in the entire barrel region is ab out 1 × 10-4 . One of the principal systematic errors in the Monte Carlo calculation of the selection inefficiency arises from uncertainties in the physics modelling of this class of events and hence in the evaluation of the rate of events lost in the acceptance hole. In order to match the goal of 5 × 10-4 acceptance uncertainty set by the statistical precision, the rate of such lost events must b e understood at the 10 % level. In previous OPAL analyses [8, 9] this hadronisation uncertainty was assessed by comparing the inefficiencies predicted by different Monte Carlo generators, JETSET and HERWIG, and by using different parameter settings within the JETSET program, which led to a systematic error assignment of 0.11% in the cross-section. To reduce substantially this source of uncertainty a new technique was develop ed, which uses data events collected in the barrel region of the detector as a control sample. 16


In this technique, which we refer to as the "acceptance hole emulation", we modify the resp onse of b oth the real detector and its Monte Carlo simulation in the region surrounding the x-axis, i.e. p erp endicular to the b eam axis, to corresp ond closely to the actual gaps and imp erfections in detector coverage around the z -axis. The first step of the acceptance hole emulation is to identify a sample of hadronic Z events whose thrust axes lie within a well-defined cone around the x-axis, chosen to b e large enough to safely cover the corresp onding problematic region around the z -axis (| cos thr | > 0.90). This sample is complete and unbiased due to the almost complete detection efficiency in the barrel region. We then emulate the detector resp onse which would have b een observed if the defining cone of this sample had b een oriented along the z -axis by degrading the actual detector resp onse corresp ondingly. Since CT plays a minor role in the selection, the edge of the CT acceptance is emulated simply by rejecting tracks which p oint within the corresp onding narrow cone ab out the x-axis. For the most crucial detector comp onent, ECAL, in addition to using such a geometric rejection, we emulate the p osition-dep endent energy resp onse and cluster separation near the edge of the hole. Similarly, the FD is emulated using the data from the ECAL clusters in the barrel by mapping the appropriate angular range, cluster size and energy threshold. Small adjustments of the selection cuts are made, such that the exclusive inefficiency of each cut2 due to the emulated hole matches the corresp onding inefficiency due to the endcap hole. We then re-apply the event selection criteria3 to determine the inefficiency of the selection for the emulated, x-axis hole, x . ?data This entire procedure is rep eated with a sample of Monte Carlo events to determine the corresp onding inefficiency of the selection in the x-axis hole, x . To the extent that the emulated holes ?MC in the Monte Carlo simulation and in the data are identical, the differences in x ?data and x will ?MC b e exclusively due to departures in the Monte Carlo model of hadronisation. The ratio b etween the two emulated inefficiencies, x /x , therefore provides a correction factor to the Monte Carlo cal?data ?MC culation, which accounts for the imp erfections in the hadronisation modelling. The overall corrected e+ e- qq event selection inefficiency, corr , is calculated as ? ?
corr

=

x ?data ? x ?MC

MC

,

(1)

where MC is the overall selection inefficiency predicted by the Monte Carlo simulation. We obtain ? corr = (56.8 + 1.3) × 10-4 , which is 8.7 × 10-4 larger than MC . Accounting for detector simulation ? ? deficiencies in the barrel region, as discussed in Section 6.3.1, reduces this difference to 5.5 × 10-4 . Note that it is sufficient to calibrate the rate of lost events with this method at the level of several p er-cent. It is not necessary, or exp ected, that either x or x ?MC ?data gives an accurate estimate of the true inefficiency close to the z -axis. Small differences b etween the comp osition of the event samples along the x- and z -axes, such as introduced by initial-state radiation, are completely negligible for this purp ose.

6.3

Selection uncertainties for e+ e- qq events

In this subsection the various sources of systematic uncertainties in the selection of e+ e- qq events are discussed. The acceptance hole emulation strongly reduces the dep endence on the hadronisation modelling. However, it also introduces new systematic errors which are related to the quality of the hole emulation and the detector simulation in the barrel for determining x . The direct reliance on ?MC MC is the other main source of uncertainty. Here the simulation detector simulation for determining ? of the endcap detectors is of particular imp ortance. In our previous analysis [9] a detector simulation uncertainty of 0.14% was assigned, based on the global comparison of data and Monte Carlo energy distributions. Such broad checks cannot distinguish b etween detector simulation problems
The inefficiency due to events which fail just the cut under discussion. en In applying this event selection all axis-sensitive quantities, such as Rbaergy , are transformed from the z -axis to the l x-axis.
3 2

17


and deficiencies in the physics modelling of the generator. They are also insensitive to p ossible local inhomogeneities in detector resp onse. In order to disentangle these effects a new approach has b een adopted which investigates in more detail the simulation of individual hadronic and electromagnetic showers in the calorimeters. Finally, the p erformance of the detector was carefully checked. Each asp ect of triggering, data taking and data quality which could bias the selection has b een examined. 6.3.1 Systematic errors in the acceptance hole emulation

Three different sources contribute to systematic uncertainties of the acceptance hole emulation: (i) the residual hadronisation dep endence, (ii) limitations of the acceptance hole emulation program, and (iii) the quality of the detector simulation in the barrel region for determining x . Table 4 summarises ?MC the corresp onding uncertainties. Residual hadronisation dep endence: The sensitivity of the method to uncertainties in the modelling of hadronisation has b een assessed by varying the parameters of the fragmentation model in the JETSET [26] Monte Carlo event generator and by employing an alternative Monte Carlo program, HERWIG [27], which uses a different fragmentation mechanism. As an example, we changed the JETSET parameter Q0 , the invariant mass cut-off b elow which gluon radiation stops, from its default value of 1.0 GeV to 1.8 GeV. This change corresp onds to one standard deviation as determined from a global tuning of the JETSET parameters to OPAL LEP 1 data [28], and shifts MC by (7 + 1) × 10-4 . Using ? the acceptance hole emulation, however, the corresp onding shift in corr is only (0.8 + 1.1) × 10-4 , ? which illustrates the effectiveness of the method. The variation of other fragmentation parameters yielded effects of similar size. The value of corr calculated using the sample of HERWIG Monte ? Carlo events differs by (2.3 + 2.0) × 10-4 compared to the reference value obtained using JETSET Monte Carlo events. We take this difference as the systematic error due to the residual sensitivity to hadronisation. A further uncertainty arises from the small inefficiency in the barrel region which is not addressed by the hole extrap olation. From similar Monte Carlo studies of the hadronisation dep endence, i.e., variation of fragmentation parameters and comparison of models, the largest change in this inefficiency is found to b e (2.2 + 0.5) × 10-4 , which is taken as an additional systematic error. The total direct hadronisation uncertainty is 3.2 × 10-4 . Limitations of the acceptance hole emulation procedure: The detector setup and resp onse differs substantially b etween barrel and endcap, therefore the emulation of the endcap hole in the barrel is only approximate. As discussed ab ove the selection cuts for the barrel hole are adjusted to match the exclusive inefficiency in the endcap hole. This scaling of the cuts changes corr by ? -4 which we take as a systematic error. Furthermore, the radius defining the edge of the 1.5 × 10 ECAL acceptance in the hole emulation program was varied over the full size of an ECAL block, which leads to an additional uncertainty of 0.4 × 10-4 . Detector simulation in the barrel region: The acceptance hole emulation procedure relies on the quality of the barrel detector simulation for determining x . Figure 4 shows distributions of the ?MC emulated variables used in the cuts for events close to the x-axis. There are small offsets b etween the data and Monte Carlo prediction for the energy and multiplicity distributions, although the shap es of the distributions are well simulated. Rescaling the Monte Carlo energy and multiplicity distributions in order to correct for these differences changes the value of corr by 3.3 × 10-4 . Switching on and ? off the correction methods describ ed in Section 6.3.2, which are indep endent of assumptions ab out hadronisation, gives a consistent difference in corr . We take this value as a correction and assign to ? it a 100 % uncertainty. In summary, the acceptance hole emulation leads to an efficiency correction of (5.5 + 4.8) × 10-4 . 6.3.2 Detector simulation uncertainties

In addition to hadronisation uncertainties, which are reduced by the hole emulation study, the event selection efficiency also relies directly on the accuracy of the detector simulation for determining MC . ? Since the main source of selection inefficiency occurs for events with jets close to the b eam axis, the 18


simulation of the detector resp onse in the endcap region is more critical than that in the barrel region. mh The cut on Ecal / s causes the largest inefficiency and consequently the most imp ortant issue is the simulation of the energy resp onse in the ECAL and the FD. The electromagnetic resp onse of these calorimeters has b een examined using lepton-pair events with an identified low-energy radiated photon. The measured photon energy is compared to the energy predicted from the momenta of the two leptons, assuming a three-particle final state. Checks are made for global differences b etween data and Monte Carlo distributions and for local inhomogeneities. Most crucial for the e+ e- qq event selection inefficiency are the regions just inside the edges of the acceptance, at small angles to the b eam axis. The stability of the energy resp onse of the ECAL endcap inner rings has b een found to b e good and the simulation is accurate to the 5% level. This corresp onds to an uncertainty of 2.0 × 10-4 on the overall inefficiency. There are further small differences b etween the data and detector simulation for the remainder of the ECAL endcaps, leading to a further correction to the inefficiency of (2.0 + 2.0) × 10-4 . A similar check for the FD electromagnetic resp onse shows good consistency b etween data and Monte Carlo simulation. The statistical precision of this check translates into an uncertainty of 2.0 × 10-4 on the overall inefficiency. The studies describ ed ab ove provide verification of the ECAL resp onse to electromagnetic showers. Also imp ortant is the ECAL resp onse to hadrons. The energy sp ectra of single ECAL clusters associated with isolated tracks in hadronic events were studied in narrow bins of track momentum in b oth data and Monte Carlo simulation. The simulation did not reproduce the resp onse sp ectra exactly, although the mean detector b ehaviour was modelled adequately. The barrel, endcap, and overlap regions of the ECAL were studied separately. In each momentum bin and detector region a correction function dep ending on energy was then constructed to adjust the detailed shap e of the simulated calorimeter resp onse to match that observed in the data. When this correction is applied in the Monte Carlo event simulation to all ECAL energy dep osits which arise from charged or neutral hadrons (using the measured and true particle momentum, resp ectively), the overall selection efficiency shifts by (2.0 + 2.0) × 10-4 , which we take as a correction. The effect of the correction on the barrel ECAL resp onse is used in deriving the hole emulation systematic uncertainty. Figure 5 shows the distribution of ECAL cluster energies for all tracks summed over all momenta from the data and from the Monte Carlo simulation b efore and after correction. The resp onse of FD to hadrons plays a minor role for the selection since most hadronic showers remain b elow the threshold of 3 GeV required for a valid FD cluster. Therefore no additional uncertainty is assigned. Overall, general improvements in the Monte Carlo simulation and detailed studies of the calorimeter resp onse to electromagnetic and hadronic showers have reduced the direct detector simulation uncertainty, and result in an efficiency correction of (4.0 + 4.0) × 10-4 . 6.3.3 Detector p erformance

The OPAL trigger system [21] uses a large numb er of indep endent signals from a variety of detector comp onents. This information is combined to form many different event selection criteria, any one of which is sufficient to trigger the OPAL detector to b e read out. The relatively low b eam-induced background conditions of LEP 1 p ermit the choice of very loose settings for these trigger conditions. In general, e+ e- qq events satisfy many indep endent trigger criteria. This redundancy allows the trigger inefficiency to b e determined directly from the data. It is found to b e less than 10-5 . Each subsequent step in the data-recording chain, from data acquisition, through online reconstruction, to the final writing of the data samples into the offline storage facility, was investigated. Possible effects, such as incorrect logging of detector status, b ookkeeping discrepancies at various stages or failures of the event reconstruction have b een examined. Only one problem has b een found: very rarely, individual events suffer from high electronic noise levels in the central tracking detector, which causes the track reconstruction to fail. These events are not classified as e+ e- qq candidates, which causes an inefficiency of ab out 0.8 × 10-4 . Both the short-term and long-term stability of each of the detector comp onents used for the selection of e+ e- qq events have b een examined, as have distributions of the cut variables themselves. Only one significant effect has b een found: for the FD small offsets in 19


the energy distribution of the 1995 data give rise to an additional inefficiency of (5 + 3) × 10-4 . The stability of the selection was checked by using alternative selection criteria, each based on data from only a single detector comp onent. No notable effect is seen. In total an uncertainty of 2 × 10-4 is assigned due to irregularities in the detector p erformance during the 1993 and 1994 data-taking. For the 1995 data further studies were p erformed to search for p ossible effects on the e+ e- qq selection inefficiency due to the bunch-train mode of op eration of LEP. As discussed in Section 2.1 the bunch-train mode could p otentially affect the CT reconstruction and the ECAL resp onse. In order to check for such problems a tracking-indep endent selection has b een develop ed, which is designed to overlap as much as p ossible with the standard selection. Within statistics no bunchlet-dep endent effects are observed. The 3 × 10-4 precision of the check is assigned as a systematic uncertainty. Together with the FD energy offset discussed ab ove and the general 2 × 10-4 data-taking uncertainty, the overall detector p erformance error for 1995 is 4.7 × 10-4 .

6.4

Background in the e+ e- qq channel

The largest backgrounds to the e+ e- qq event selection arise from e+ e- + - events and from two-photon interaction processes, e+ e- e+ e- qq. The background fraction from e+ e- + - events is estimated to b e 14.1 × 10-4 using the KORALZ Monte Carlo event generator [31]. Detailed studies of variables sensitive to e+ e- + - events indicate that the Monte Carlo simulation underestimates this background by 13%. Part of this underestimate is caused by deficiencies in the simulation of the conversion of radiated photons into electron-p ositron pairs. Taking this into account results in a corrected background estimate of (15.9 + 2.0) × 10-4 , where the assigned systematic error is based on the observed discrepancy. The background from two-photon interaction processes has b een estimated directly from the data by making use of the characteristics of two-photon events. In general they have low visible enener mh ergy, Ecal / s, and large energy imbalance along the b eam-axis, Rbal gy (see Figure 3). The crosssection for two-photon processes is prop ortional to log s and therefore the cross-section changes by less than 1 % in the energy range b etween p eak-2 and p eak+2. Figure 6 shows, for the three enener mh ergy p oints, the cross-sections for low Ecal / s (0.10-0.18) or high Rbal gy (0.50-0.75) events versus ener mh the cross-section for events with high Ecal / s (> 0.18) and low Rbal gy (< 0.50). Most events from two-photon processes fall into the former category, whereas the latter is completely dominated by hadronic Z decays. The constant cross-section from non-resonant background sources, mainly two-photon events, is obtained from the intercept of a straight-line fit to these data, which yields 0.051 + 0.006 nb. The error is dominated by the data statistics available. This estimate is corrected for the small fraction, (7 + 5)%, of accepted two-photon events which ener mh fall outside the defined region of low Ecal / s or high Rbal gy using the Monte Carlo generators PHOJET [33] and HERWIG [27], which simulate two-photon interactions. Furthermore, the estimate based on the data contains a small bias due to the larger fraction of events with high-energy initial-state ener radiation photons at the off-p eak energy p oints. These events tend to have a high value of Rbal gy and therefore give a small contribution from signal events to the fitted non-resonant background. This bias is estimated using Monte Carlo simulation to b e 0.004 + 0.002 nb. Overall, the non-resonant background is 0.051 + 0.007 nb, which amounts to (16.7 + 2.3) × 10-4 at the p eak energy p oint. The background from four-fermion processes other than two-photon interactions is evaluated using four-fermion event generators, FERMISV [36] for s-channel + - f f and PYTHIA [29] for t-channel e+ e- f f. Part of the t-channel four-fermion events are implicitly included in the two-photon subtraction. Adding the residual fraction and the s-channel four-fermion events results in a small contamination at the 0.4 × 10-4 level which is subtracted. Other background sources are even lower; contributions from e+ e- e+ e- and cosmic-ray induced events are estimated to b e 0.2 × 10-4 each. There is no indication of any backgrounds induced by b eam interactions with the residual gas in the LEP vacuum or the wall of the b eam pip e ab ove our sensitivity of 10-5 . 20


7

Measurement of leptonic events

The branching fraction of the Z b oson to charged leptons is approximately 10%. Consequently the leptonic decays provide less information on mZ and Z than the hadronic decays of the Z. Since the Z b osons we observe at LEP are all produced through their coupling to electrons in the initial state, however, the measurement of the leptonic decay channels is of particular interest. By measuring the cross-sections for all visible Z decays, including electrons, the absolute branching fraction to invisible final states can b e determined. Assuming these states are exclusively neutrinos coupling to the Z according to the SM then allows the effective numb er of such light neutrino sp ecies to b e determined. In contrast to the quarks in hadronic events, the charge of the leptons can b e determined almost unambiguously, thereby allowing the forward-backward charge asymmetry, AFB , to b e measured. Measurements of AFB determine the relative strengths of the vector and axial-vector couplings of the Z to each of the three charged lepton sp ecies, gV /gA . In the SM, this ratio is related to the weak mixing angle, sin2 W . When combined with the + - cross-sections, which are prop ortional to 2 2 gV + gA , the couplings gA and gV are determined. In addition, comparisons among the partial Z decay widths to the three charged lepton sp ecies and the resp ective forward-backward asymmetries provide a precise test of the lepton universality of the neutral current.

7.1

Introduction

Leptonic decays of the Z b oson, e+ e- + - , result in low multiplicity events. The different leptonic sp ecies are distinguished from each other mainly using the sum of the track momenta, ptotal , and the energy dep osited in the electromagnetic calorimeter, Etotal . The total momentum is calculated as a scalar sum over the individual track momenta, ptotal = pi ack . Similarly the total energy is the tr i sum of the energies of the individual clusters in the electromagnetic calorimeter, Etotal = Ecluster . Figure 7 shows the distribution of (ptotal ,Etotal ) for a small sample of Monte Carlo e+ e- + - events. The e+ e- e+ e- events have b oth Etotal and ptotal concentrated around the centre-of-mass energy, s. However, due to final-state and Bremsstrahlung photons the distribution of ptotal extends to lower values. The e+ e- ÷+ ÷- events are concentrated around ptotal = s and have low values of Etotal . Final-state radiation has the effect of producing a small tail of events at higher values of Etotal and corresp ondingly lower values of ptotal . Due to the undetected neutrinos from decays in e+ e- + - events the distributions of b oth Etotal and ptotal are broad but well separated from the other leptonic decays of the Z, with only a few events with Etotal or ptotal close to s. Cuts on the global quantities Etotal and ptotal provide the basis for separating the different leptonic decay modes. Detailed descriptions of the selection criteria for the three different e+ e- + - categories are describ ed in Sections 7.2-7.4. The event selections use a series of cuts to reject background. The selections are exclusive, with no event allowed to b e classified in more than one category. The largest backgrounds in the selected e+ e- e+ e- , ÷+ ÷- and + - event samples arise from cross-contamination b etween the three lepton sp ecies. The treatment of four-fermion final states is discussed in App endix A.1. Backgrounds from the processes, e+ e- qq, e+ e- e+ e- + - , e+ e- and cosmic ray events, are all relatively small. The background from e+ e- qq is rejected by cuts on the numb er of observed tracks and the numb er of clusters in the electromagnetic calorimeter. In two-photon processes, e+ e- e+ e- + - , the scattering angles of the incident electron and p ositron tend to b e close to the b eam direction, b eyond the exp erimental acceptance. This source of background is rejected by lower b ounds on variables such as Etotal or ptotal . Background from cosmic rays is removed by requiring the event to originate from the e+ e- interaction region and to b e in time with the e+ e- b eam crossing. 7.1.1 Corrections and systematic uncertainties

Estimates of selection efficiencies and accepted background cross-sections are obtained from Monte Carlo samples generated at p eak and off-p eak centre-of-mass energies. These efficiencies are corrected to the s acceptance sp ecified in Section 3 and are also corrected to account for interference b etween 21


initial- and final-state radiation diagrams (see App endix A.2). Data and Monte Carlo distributions are compared to assess the quality of the simulation for each cut used in the event selection and where necessary, corrections are derived. Dep ending on the exact nature of the systematic check, a single correction factor is often adequate to describ e all 1993-1995 data. Otherwise, where the effect under consideration is related to the p erformance of the detector, indep endent corrections are determined for each year. Similarly, when the effect dep ends on the centre-of-mass energy, different corrections are derived for each of the different centre-of-mass energy p oints. The derived corrections greatly reduce the dep endence on the accuracy of the Monte Carlo simulation. There are two imp ortant sources of detector-related systematic uncertainty which affect each of the three lepton channels. The largest p otential source of systematic bias arises from imp erfect simulation of the detector resp onse to charged particles whose paths lie near one of the wire planes of the jet chamb er. The wire planes, which create the drift field, are situated at 7.5 intervals in azimuth around the chamb er, alternating b etween the anode and cathode planes which form the 24 sectors of the jet chamb er. The reconstruction of tracks within +0.5 of an anode wire plane can b e problematic due to field distortions. This does not cause problems for low momentum tracks, which b end significantly in the axial magnetic field, since only a small fraction of the tra jectory will b e close to a wire plane. For high momentum particles with relatively straight tra jectories, such as muons and electrons from e+ e- + - , a serious degradation of track quality can occur in a small fraction of the events where most of the track lies near an anode wire plane. This can result in a degraded momentum measurement, single tracks b eing reconstructed as two tracks (split across a wire plane) or, in the worst instance, failure to reconstruct the track at all, resulting in `lost' tracks. Field distortions close to the wire planes are simulated in the Monte Carlo but the numb ers of tracks affected is significantly underestimated. These effects are particularly imp ortant in the selection of e+ e- ÷+ ÷- . Events where the measured momentum of one of the muons is anomalously low, due to wire plane effects, result in the Monte Carlo prediction of 0.3% for the selection inefficiency b eing under-estimated by approximately 0.4% compared to the true value. Significant corrections to the Monte Carlo selection efficiencies, relating to these tracking problems, are therefore obtained from the data. Since e+ e- ÷+ ÷- events measured to have anomalously low total momenta are classified as e+ e- + - , the uncertainties in these corrections lead to anti-correlated errors in the ÷+ ÷- and + - selections. Details are given in Sections 7.3.2 and 7.5. The measurement of energy in the electromagnetic calorimeter plays the primary role in the selection of e+ e- e+ e- events and in discriminating them from e+ e- + - . An uncertainty in the energy scale or in the resp onse of the ECAL can therefore introduce correlated uncertainties in the e+ e- and + - cross-section measurements. The Monte Carlo simulation underestimates the fraction of e+ e- events which fail the selection due to imp erfect Monte Carlo modelling of the resp onse of the electromagnetic calorimeter near the mechanical b oundaries b etween calorimeter modules, at = + 90 and | cos | = 0.22 and 0.60. In addition, there is an unsimulated problem in the electronic gain calibration for two out of the 9440 barrel lead glass blocks for the 1994 and 1995 data samples. As a result the Monte Carlo estimates of the e+ e- e+ e- selection efficiency need to b e corrected by 0.1 - 0.2%. Details are given in Sections 7.2.3 and 7.5. 7.1.2 Trigger efficiency for e+ e
-



+-

events

Low multiplicity events from e+ e- + - produce relatively few energy dep osits in the detector. As a result, there is the p otential that events are lost due to trigger inefficiency. The trigger efficiency for e+ e- + - events is determined using indep endent sets of trigger signals [21] from the electromagnetic calorimeters, from the tracking chamb ers, from the muon chamb ers and from the time-of-flight counters. The efficiencies are determined in bins of p olar and azimuthal angle and then combined into an overall efficiency. When averaged over the three years of data analysed here, the trigger efficiencies for the e+ e- e+ e- , ÷+ ÷- and + - selections are (> 99.99)%, (99.96 + 0.01)% and (99.98 + 0.01)% resp ectively. The online event filter [22] is found to b e 100% efficient for e+ e- + - events and no systematic error is assigned. 22


7.2

Selection of e+ e- e+ e- events

The selection of e+ e- e+ e- events is accomplished with high efficiency and purity by requiring low-multiplicity events with large total electromagnetic energy and by requiring at least two electron4 candidates. In order to maintain high efficiency, an electron is identified simply as a high energy dep osit in the electromagnetic calorimeter which is associated with a track in the central detector. The main exp erimental systematic uncertainties are from the energy resp onse of the calorimeter, the edge of the p olar angle acceptance, the track reconstruction quality and the background from e+ e- + - events. 7.2.1 t-channel contribution to e+ e
-

e+ e

-

The analysis of s-channel Z decays into e+ e- pairs is complicated by indistinguishable contributions from t-channel scattering processes, and interference b etween s- and t-channels. The t-channel amplitude is non-resonant and is dominated by photon-exchange. These contributions are accounted for when fitting the e+ e- e+ e- data (see App endix B), but statistical and systematic uncertainties on the magnitude of the t-channel contribution result in an overall reduction of the sensitivity of the e+ e- e+ e- analysis to the Z prop erties. The relative size of the t-channel amplitude dep ends on the scattering angle (p olar angle) of the electron, e- , and b ecomes the ma jor comp onent at high values of cos e- . In order to enhance the s-channel comp onent of the selected data sample, a cut is made, for the e+ e- e+ e- analysis only, constraining the p olar angle to lie well within the barrel region of the detector (| cos e- | < 0.70, see Section 7.2.2). The relative size of the t-channel comp onent is larger for the off-p eak data samples than the 15% it represents on-p eak, due to the reduction of the s-channel contribution from the Z resonance. This results in small differences in efficiencies, backgrounds and systematic uncertainties amongst the e+ e- e+ e- data samples at different energy p oints. Because of the t-channel diagram the differential cross-section rises steeply in the forward direction, resulting in an increased sensitivity to the precise definition of the edge of the acceptance in cos e- . 7.2.2 Selection criteria for e+ e
-

e+ e

-

The selection criteria for e+ e- e+ e- events make use of information from the electromagnetic calorimeter and the central tracking detectors by requiring: ž Low multiplicity: 2 Ntr where Ntr
ack

8 and 2 Ncl

uster

8,
uster

ack

is the numb er of tracks and Ncl

is the numb er of clusters.

ž High energy clusters: The energies of the highest energy, E1 , and second highest energy, E2 , clusters must satisfy E1 > 0.2 s and E2 > 0.1 s. ž Total electromagnetic energy: i Etotal Ecluster > 0.80 s. ž Two electrons: At least two of the three highest energy clusters of energy ab ove 2 GeV must b e associated with a track, which is required to p oint to the cluster p osition to within < 5 in azimuth and to within < 10 in p olar angle. These clusters are identified as electron candidates. ž Geometrical and kinematic acceptance: | cos e- | < 0.70 and
4

acol

< 10 ,

Throughout this section the word "electron" should b e understood to imply "electron or p ositron". Where a distinction is required, the symb ols e- and e+ are used.

23


where acol is the acollinearity angle of the e+ e- pair, defined as 180 - , where is the op ening angle b etween the directions of the two tracks. These criteria select 96 669 events from the 1993-1995 data sample. The efficiency and background are first estimated using a sample of Monte Carlo events. Corrections to these estimates are obtained by studying the data. The correction factors and their associated systematic errors are summarised in Table 5 for the seven data samples recorded during 1993-1995 at the three energy p oints. The systematic errors for the seven data samples are strongly correlated, as shown in Table 18. The main corrections and systematic uncertainties are describ ed b elow. 7.2.3 Selection efficiency for e+ e
-

e+ e

-

The Monte Carlo prediction for the e+ e- e+ e- selection efficiency within the geometrical acceptance of | cos e- | < 0.70 is (99.44 + 0.02)% for p eak data. The main corrections to this efficiency and related systematic errors are describ ed b elow. Electromagnetic energy cuts: The main effect of the cut on Etotal > 0.8 s, shown in Figure 8(a), is to remove background from e+ e- + - . From the Monte Carlo simulation only 11 × 10-4 of e+ e- e+ e- events are rejected by this cut. There is a discrepancy b etween data and Monte Carlo in the vicinity of the cut, due to imp erfect Monte Carlo modelling of the electromagnetic calorimeter resp onse near the mechanical b oundaries b etween calorimeter modules and a problem with the electronic gain calibration for two lead glass blocks in 1994 and 1995. These two effects introduce additional inefficiencies for the data, estimated to b e (9 + 10) × 10-4 , (22 + 7) × 10-4 and (17 + 8) × 10-4 for 1993, 1994 and 1995, resp ectively. These efficiency corrections are obtained from a detailed study of events which fail only the cut on electromagnetic energy. For these events, the acoplanarity, acop ||e- - e+ |- 180 |, and the sum of the track momenta, ptotal , are used to discriminate e+ e- e+ e- events from the dominant e+ e- + - background. The distribution of acop for events which fail only the cut on electromagnetic energy is shown in Figure 8(b). The acop distribution for e+ e- e+ e- events is strongly p eaked at 0 . The distribution for e+ e- + - events is broader due to the momentum transverse to the tracks which is carried away by the unobserved neutrinos from the decays. The excess of data events compared to Monte Carlo near acop = 0 indicates that the excess of events in the region of the cut arises from e+ e- e+ e- . This interpretation is corrob orated by the distribution of ptotal / s for events in the region of small acop , shown in Figure 8(c), where the excess of data over Monte Carlo events is clustered near ptotal / s = 1.0. The efficiency corrections are derived from the distributions of acop and ptotal in the region 0.6 s < Etotal < 0.8 s. Electron identification: Electron candidates are defined as electromagnetic clusters of energy greater than 2 GeV which are associated with a track. An electron can fail these requirements if the track associated with the cluster is of very low momentum, due to hard bremsstrahlung in the material in front of the tracking detectors. It can also fail due to the emission of a hard final state photon or if, due to p oor track reconstruction, the track fails the geometrical matching conditions or the track quality requirements. The inefficiency arising from the demand for at least two electron candidates is estimated from Monte Carlo simulation to b e (39 + 2) × 10-4 for e+ e- e+ e- . The inefficiency in the data is assessed from a sample of events where only one electron candidate is found. This control sample is obtained by assuming the electrons in the event corresp ond to the two highest energy clusters, only one of which is identified as an electron candidate on the basis of the tracking requirements. The background from the non-resonant QED process e+ e- where one of the photons has converted to an e+ e- pair is reduced by removing events where two tracks are associated to the same cluster. Within this sample there are more events in the data than the Monte Carlo predicts. The excess is concentrated in the regions of near the wire planes of the jet chamb er. In this region the p olar angle resolution for reconstructed tracks can b e degraded sufficiently such that the track no longer p oints to the cluster. The excess of data over the Monte Carlo prediction is used to evaluate a correction to the Monte Carlo 24


estimate for these inefficiencies. The size of the correction differs for each year of data-taking. For example in the 1994 data an efficiency correction of (26 + 5) × 10-4 is derived. Acceptance definition: The geometrical and kinematic acceptance for e+ e- e+ e- events is defined by cuts on | cos e- | and on acol , shown in Figure 9. The p olar angle cut is made with resp ect to e- , the direction of the negatively charged lepton as measured in the electromagnetic calorimeter. This is determined from an energy-weighted average of the p ositions of the lead glass blocks which form the cluster, which is then corrected for biases caused by showering in the material in front of the calorimeter. The existence of any systematic offset b etween the reconstructed cluster p osition and the actual tra jectory of the electron was studied by measuring the displacement b etween well measured electron tracks and their associated clusters near the critical cos b oundaries. These studies indicated that the effective edge of the acceptance is offset symmetrically from the nominal cut value towards cos e- = 0 by 0.0004 + 0.0006 at cos e- = +0.70. Consistent displacements were obtained in b oth the data and Monte Carlo simulation. The central value of the offset is obtained from the highstatistics Monte Carlo samples and the uncertainty is taken as the statistical precision of the study based on data. For the p eak data the uncertainty in the location of the cos e- = +0.70 b oundary results in an uncertainty of +9 × 10-4 in the measured cross-section. Further corrections and uncertainties: The inefficiency arising from the multiplicity cuts is estimated to b e (1 + 1) × 10-4 . This estimate is obtained from the simulation and checked by examining the electromagnetic calorimeter energy distribution for the events which fail only the multiplicity cuts. The trigger inefficiency is determined to b e less than 5 × 10-5 and no correction is applied. Corrections related to four-fermion events, determined as describ ed in App endix A.1, are negligible and a 2 × 10-4 uncertainty is assigned. 7.2.4 Background in the e+ e
-

e+ e

-

channel

The exp ected background in the p eak data samples is (34 + 6) × 10-4 , dominated by e+ e- + - . The energy-dep endence of the e+ e- e+ e- cross-section is different from the other s-channel fermionpair production processes due to the t-channel photon exchange contributions. Consequently the background fraction is dep endent on centre-of-mass energy. Background from e+ e- + -: The Monte Carlo estimate of this background for the p eak energy p oint is (32 + 1) × 10-4 . The ma jority of the e+ e- + - background is rejected by the cut Etotal > 0.8 s, shown in Figure 8(a). The distribution of Etotal / s is well reproduced by the Monte Carlo simulation in the region dominated by the e+ e- + - background, Etotal < 0.70 s. To investigate the level of the background within the e+ e- e+ e- sample, events just ab ove the energy cut are selected, 0.8 < Etotal / s < 0.9. For these events the data and Monte Carlo distributions of ptotal and acop are again used to distinguish e+ e- e+ e- events from e+ e- + - . The Monte Carlo estimate of the e+ e- + - background has b een found to b e consistent with the observations from the data, to within the statistical errors of the comparisons made. The systematic uncertainty on the e+ e- + - background for the p eak energy p oint is estimated to b e 6 × 10-4 . Other backgrounds: The energy dep osits in the electromagnetic calorimeter from e+ e- events are similar to those from e+ e- e+ e- events. However, for e+ e- events to pass the electron identification requirements the two electromagnetic calorimeter clusters b oth must have associated tracks. The probability that b oth the photons convert is ab out 1% and the e+ e- cross-section is relatively small. Consequently, the background fraction in the e+ e- e+ e- sample is small ( 1 × 10-4 ). The background from hadronic events is estimated to b e ab out the same size. A 100% relative uncertainty is assigned to the background from hadronic events and to that from e+ e- events. Backgrounds from two-photon interaction processes, e+ e- e+ e- f f , and from cosmic ray events are less than 1 × 10-4 . No correction is applied, and this estimate is taken as a systematic error. 25


7.3

Selection of e+ e- ÷+ ÷

-

events

Of the three e+ e- + - channels, the ÷+ ÷- final state provides the cleanest environment for precise measurements of the + - cross-sections and asymmetries. The ÷+ ÷- channel does not suffer from the theoretical uncertainties associated with t-channel corrections in the e+ e- final state nor the systematic uncertainties arising from the less well defined exp erimental signature of the + - final state. However, of the three lepton channels, ÷+ ÷- events are the most sensitive to systematic uncertainties arising from track reconstruction. The e+ e- ÷+ ÷- events are separated from other Z decays and background processes by requiring exactly two tracks to b e reconstructed in the central detector b oth of which are identified as muons. The cross-sections are measured within the phase space region defined by m2 /s > 0.01. The selection criteria are summarised b elow. ÷÷ 7.3.1 Selection criteria for e+ e
-

÷+ ÷-

The selection criteria for e+ e- ÷+ ÷- events are: ž Two tracks: Exactly two tracks are required each of which satisfies p
track

> 6 GeV ,

| cos | < 0.95,

where ptrack is the track momentum and is the reconstructed p olar angle. Tracks identified as coming from photon conversions are not counted. ž Azimuthal separation: The azimuthal angular separation b etween the two tracks must satisfy cos() < 0.95, i.e. > 18 , to avoid difficulties in the muon identification of two closely separated tracks. ž Muon identification: Both tracks must satisfy at least one of the following three muon identification criteria: (a) At least two muon chamb er hits are associated with the track. (b) At least four hadronic calorimeter strips are associated with the track. The average numb er of strips in layers containing hits has to b e less than two, in order to reject hadronic showers. For | cos | < 0.65, where tracks traverse all nine layers of strips in the barrel calorimeter, at least one hit in the last three layers of strips is required. (c) The track has ptrack > 15 GeV and the sum of the energy dep osited in the electromagnetic calorimeter within a cone of half-angle 63 mrad ab out the track is less than 3 GeV. ž Visible energy: Backgrounds from e+ e- + ÷ Evis > 0.6 s,
-

and two-photon interaction events are reduced by requiring

÷ where the visible energy, Evis , is the scalar sum of the two track momenta and the energy of the highest energy cluster found in the electromagnetic calorimeter.

ž Despite the fact that the OPAL detector is situated in a cavern 100 m underground there is still a large flux of cosmic ray particles. The ma jority of these are muons which traverse the detector volume. Those which happ en to pass close to the b eam interaction p oint and which are synchronous with a bunch crossing can resemble e+ e- ÷+ ÷- events. Cosmic ray background is rejected by requiring that the selected events originate from the average e+ e- interaction p oint and are coincident in time with the b eam crossing. These criteria select 128 682 events which enter the cross-section analysis from the 1993-1995 data sample. Control samples from the data are used to check and, where necessary, correct the Monte Carlo estimates. The dominant corrections are due to tracking losses and the residual background from 26


e+ e- + - events. The resulting correction factors and their systematic errors are summarised in Table 6 for the seven data samples recorded during 1993-1995 at the three energy p oints. The systematic errors of the seven data samples are strongly correlated as shown in Table 19. Unless otherwise sp ecified, the illustrative errors quoted in the following text refer to the 1994 sample. 7.3.2 Selection efficiency for e+ e
-

÷+ ÷-

The Monte Carlo prediction for the e+ e- ÷+ ÷- selection efficiency is (91.34 + 0.05)%. This corresp onds to a selection efficiency of (98.40 + 0.03)% within the geometric acceptance of | cos ÷- | < 0.95. Small corrections are then applied to account for the Monte Carlo events generated b elow the ideal kinematic acceptance limit of m2f /s > 0.01, and for interference b etween photons radiated in f the initial and final states, as describ ed in App endix A.2. Using the 1994 data as an example, the selection efficiency from the Monte Carlo simulation is then corrected by (-76 + 8) × 10-4 through comparisons with the data. The main efficiency corrections and systematic errors are describ ed b elow. Tracking losses: The selection of e+ e- ÷+ ÷- events relies heavily on track reconstruction in the ÷ central detector, which is required to measure Evis and to associate tracks with activity in the outer detectors used for muon identification. As discussed in Section 7.1.1, of particular concern are tracks whose paths lie within +0.5 of an anode wire plane of the jet chamb er. Figure 10(a) shows the ÷ distribution of Evis for events passing all other selection cuts. There is a clear discrepancy b etween data and Monte Carlo. The origin of this discrepancy is the imp erfect Monte Carlo simulation of tracking in the region close to the jet chamb er wire planes. The discrepancy b ecomes more apparent ÷ in Figure 10(b) which shows the corresp onding Evis distribution for tracks within 0.5 of the anode wire planes. The wire plane effect was investigated using an alternative e+ e- ÷+ ÷- selection procedure. This selection is indep endent of the central detector, relying instead on back-to-back signals in the muon chamb ers (within 30 mrad) and electromagnetic calorimeters (within 50 mrad). By using relatively tight cuts on the acollinearity measured in the outer detectors the background from e+ e- + - events is strongly suppressed. This is mainly due to the greater deflection in the magnetic field exp erienced by the lower momentum charged particles from decays. Background is suppressed further by requiring at least one identified muon in the event. The efficiency of the tracking-indep endent selection is approximately 79%. The events which are selected by the tracking-indep endent selection but fail the default ÷+ ÷- selection are concentrated in regions where one of the muons passes near a jet-chamb er wire plane. The numb ers of "lost" e+ e- ÷+ ÷- events, selected by these cuts, are corrected for the inefficiency of the tracking-indep endent selection. The corrected numb ers are compared b etween data and Monte Carlo simulation in bins of cos for each year of data taking. There is an excess of lost events in the data, indicating that the Monte Carlo estimate of 30×10-4 for the inefficiency due to tracking problems is too low. The difference is (42 + 4) × 10-4 for the 1994 data. The uncertainties are the combined statistical errors of the data and Monte Carlo control samples. The correction factors obtained in this manner are applied to the Monte Carlo inefficiency estimates. Track multiplicity cuts: In addition to the tracking losses discussed ab ove, 0.5% of e+ e- ÷+ ÷- events are rejected by the requirement of exactly two tracks due to additional tracks from converted final-state photons which have failed to b e classified as such, or if a track is p oorly reconstructed and split into two tracks. A correction to the Monte Carlo prediction for the inefficiency due to this cut is made on the basis of a visual scan of b oth data and Monte Carlo events containing three, four or five tracks, but which otherwise pass the e+ e- ÷+ ÷- selection. The resulting corrections, typically less than 10-3 , are listed in Table 6, with the errors reflecting the data and Monte Carlo statistics and the uncertainty of the scanning procedure. Muon identification: Both tracks in a selected e+ e- ÷+ ÷- event are required to b e identified as muons using data from at least one of three indep endent detector subsystems; the muon chamb ers, 27


the hadronic calorimeter and the electromagnetic calorimeter. The Monte Carlo estimate of the inefficiency introduced by the muon identification requirement is (79 + 5) × 10-4 . Most of this inefficiency occurs in geometrical regions where either the muon chamb er coverage or the hadronic calorimeter coverage are incomplete. In particular, for one sixth of the total azimuth in the p olar angle range 0.65 < | cos | < 0.85 these gaps overlap due to supp ort structures, leaving coverage only from the electromagnetic calorimeter. For collinear e+ e- ÷+ ÷- events there is a high degree of correlation b etween the muon identification inefficiencies of the two tracks due to the symmetry of the detectors. The redundancy of the three muon identification requirements is used to determine the single track muon identification efficiency in bins of azimuthal and p olar angle. For each muon identification criterion, the muon identification criteria from the other two outer detectors are used to define a control sample of tagged muons. The single track efficiencies, determined in bins of (cos , ), are used to calculate overall muon identification inefficiencies for e+ e- ÷+ ÷- events accounting for exp ected angular distributions. The total inefficiency determined in this way is compared b etween data and Monte Carlo prediction separately for each year of data. There is good agreement for the 1993 and 1995 data, and no correction is applied. For the 1994 data, it is found that the inefficiency from the corresp onding Monte Carlo sample needs to b e corrected by (15 + 4) × 10-4 where the errors reflect the statistical p ower of the checks. The correction arises due to inadequate simulation of the resp onse of the hadron calorimeter in the Monte Carlo sample used to simulate data from the 1994 op eration of the OPAL detector. Acceptance definition: Both muon tracks are required to lie within the geometrical acceptance | cos | < 0.95. The measurement of cos in this region therefore affects the overall event selection inefficiency and any discrepancy b etween data and Monte Carlo simulation must b e corrected. A 1 mrad bias in angle at | cos | = 0.95 corresp onds to a bias in the acceptance of 5 × 10-4 . For muon tracks there are generally three separate detector comp onents which can b e used to measure cos ; the track reconstructed in the central detector, the energy cluster in the electromagnetic calorimeter and the track found in the muon chamb ers. For muons close to | cos | = 0.95 the end-cap muon chamb ers have the b est p olar angle resolution, approximately 1 mrad, and this measurement is used when available, otherwise the angle from the reconstructed central detector track is used (ab out 10% of cases). The changes to the numb ers of e+ e- ÷+ ÷- events selected when using alternative measurements of the p olar angle are investigated and compared b etween data and Monte Carlo simulated events, separately for each year of data. Based on the scatter of these comparisons a systematic error of 10 × 10-4 is assigned for the 1993 data sample, and 5 × 10-4 for the 1994 and 1995 data samples. Further efficiency corrections: The trigger inefficiency for e+ e- ÷+ ÷- events is be (6 + 2) × 10-4 , (5 + 2) × 10-4 and (2 + 2) × 10-4 for the 1993, 1994 and 1995 data, Corrections to the selection efficiency related to four-fermion events are determined as App endix A.1. Finally, the inefficiency associated with the cosmic ray veto is found to 1 × 10-4 . 7.3.3 Background in the e+ e
-

estimated to resp ectively. describ ed in b e less than

÷+ ÷- channel

The background in the e+ e- ÷+ ÷- event selection is approximately 1%, dominated by misclassified e+ e- + - events. The backgrounds from two-photon interaction processes, such as e+ e- e+ e- ÷+ ÷- , and from events induced by the passage of cosmic ray muons through the detector are small. The Monte Carlo background estimates are corrected for discrepancies b etween simulated and real data. These corrections are describ ed b elow and the resulting background estimates are summarised in Table 6. e+ e- + - background: The lifetime of the lepton is sufficiently short that only the decay products of the are registered in the detector. For e+ e- + - events to b e selected as e+ e- ÷+ ÷- b oth tau decays must result in an identified muon and the visible energy requirement 28


÷ must b e satisfied, Evis > 0.6 s. Approximately 17% of leptons decay into a muon and neutrinos, consequently ab out 3% of e+ e- + - events result in a visible ÷+ ÷- final state. In addition, other decays of the to single charged particles, in particular , can b e misidentified as muons. Two control samples are selected from the data to check the simulation of the + - background. The first is sensitive to problems with the visible energy distribution and the second is sensitive to ÷ p ossible problems with muon identification. For the first control sample the Evis cut in the ÷+ ÷- ÷ ÷ selection is relaxed to Evis > 0.5 s and the low visible energy region considered, Evis < 0.8 s. Events with tracks within 0.5 in azimuth of a wire plane are rejected to remove p oorly measured e+ e- ÷+ ÷- events. The + - background is further enhanced by rejecting events, predominantly ÷+ ÷- , which have low acoplanarity < 20 mrad. A loose cut is also made to eliminate events with high energy radiated photons by requiring the acollinearity to b e less than 150 mrad. This selected control sample is predicted by Monte Carlo simulation to b e made up of approximately 97% e+ e- + - events with the remainder b eing e+ e- ÷+ ÷- events. The data are in good agreement with the Monte Carlo prediction. ÷ A second, higher statistics control sample is selected by relaxing the Evis requirement of the ÷ e+ e- ÷+ ÷- selection to 0.5 < Evis < 0.8, and requiring exactly one of the tracks to b e identified as a muon. Approximately 99.2% of Monte Carlo generated events selected in this way are e+ e- + - events, in the ma jority of which one decays to a muon and the other decays to a single charged particle which is not a muon. From the 1993 and 1995 samples there is good agreement b etween the numb ers of events selected in the data and the Monte Carlo exp ectation. In the 1994 samples an excess of ab out 4% of Monte Carlo over data events is observed. On the basis of these two studies the Monte Carlo prediction for the e+ e- + - background of 1.00% for the 1994 e+ e- ÷+ ÷- data sample is corrected to (0.97 + 0.04)%. No corrections are made for the 1993 or 1995 samples, and a systematic uncertainty of 2 × 10-4 is assigned. Other background: Most events from the two-photon process e+ e- e+ e- ÷+ ÷- are rejected by ÷ the cut on visible energy, Evis > 0.6 s. The Monte Carlo predicts the remaining background to b e approximately 0.05% on-p eak and 0.1% off-p eak, at which energies the resonant Z production is less dominant. The two-photon background is evaluated separately for each year and systematic errors of 1 × 10-4 are assigned to these calculations. The backgrounds from cosmic ray events, estimated from time-of-flight and vertex information, were (2 + 2) × 10-4 in 1993 and 1994 and (3 + 2) × 10-4 in 1995.

7.4

Selection of e+ e- +

-

events

Tau leptons produced in the process e+ e- + - decay b efore entering the sensitive volume of the detector. The branching ratio for the decay - - is ab out 35%, and the resulting electron or muon has, in general, less momentum than a directly produced fermion from Z + - due to the two associated neutrinos. In decays to hadrons the associated single neutrino also reduces the total visible energy of the final state. The -decay branching ratio to three or more charged hadrons is ab out 15%, and electrons from the conversion of photons from 0 decays further increase the average charged multiplicity of the final state. The exp erimental signature for e+ e- + - events is therefore less well defined than that for either e+ e- e+ e- or e+ e- ÷+ ÷- events. Consequently, the cuts used to select e+ e- + - events are relatively involved. 7.4.1 Selection criteria for e+ e
-

+

-

The selection criteria used to identify e+ e- + - events remain unchanged with resp ect to our previous publications where a more detailed description is provided [7-9]. The selection is summarised below: ž Multiplicity cuts, shown in Figures 11(a) and 11(b), to reject hadronic Z decays: 2 Ntr
ack

6 and Ntr

ack

+N

cluster

15. 29


ž e+ e- +

-

event top ology:

Events are reconstructed using a cone jet-finding algorithm [7] with a cone half-angle of 35 . The sum of the electromagnetic calorimeter energy and scalar sum of track momenta in each cone has to b e more than 1% of the b eam energy. Exactly two cones containing tracks are required, not counting cones which contain only tracks associated with photon conversions. The direction of each candidate is taken to b e the total momentum vector reconstructed from the tracks and electromagnetic clusters in its cone. ž e+ e- +
acol -

event acollinearity: , is 180 minus the angle b etween the directions of the two

< 15
acol

where the acollinearity angle, candidates. ž Geometrical acceptance: | cos | < 0.9.

where is the p olar angle of the event axis, defined using the vectorial difference b etween the momenta of the two candidates. ž Rejection of e+ e- ÷+ ÷- events: Events selected as e+ e- ÷+ ÷- by the criteria describ ed in Section 7.3.1 are rejected. ž Rejection of e+ e- e+ e- events: i Etotal Ecluster < 0.8 s, shown in Figure 11(c). For the region | cos | > 0.7, where there is additional material in front of the electromagnetic calorimeter, it is also required that: Evis < 1.05 s or Etotal < 0.25 s, (for | cos | > 0.7),
where Evis is the total visible energy, Evis = Etotal + p total

.

ž Rejection of non-resonant e+ e- e+ e- Evis > 0.18 s.

+-

events, see Figure 11(d):

ž Cosmic ray background is rejected by requiring that the selected events originate from the average e+ e- interaction p oint and are coincident in time with the b eam crossing. These criteria select 107 340 events from the 1993-1995 data sample. The selection efficiency and background contributions from processes other than e+ e- + - are estimated using Monte Carlo events. These efficiencies and backgrounds are corrected for the observed differences b etween data and Monte Carlo. The corrections and systematic errors are summarised in Table 7 and are describ ed b elow. The systematic errors of the seven data samples are strongly correlated, as shown in Table 20. Unless otherwise sp ecified, the numb ers quoted in the text corresp ond to the 1994 data. 7.4.2 Selection efficiency for e+ e
-

+

-

The Monte Carlo prediction for the e+ e- + - selection efficiency is (75.18 + 0.07)%. This corresp onds to a selection efficiency of (87.70 + 0.05)% within the geometric acceptance of | cos | < 0.90. Small corrections are then applied to account for the Monte Carlo events generated b elow the ideal kinematic acceptance limit of m2f /s > 0.01, and for interference b etween photons radiated in f the initial and final states, as describ ed in App endix A.2 and listed in Table 7. Within the angular acceptance, the largest source of inefficiency arises from the cuts used to reject background from e+ e- e+ e- . To estimate the efficiency for data, the inefficiency introduced by each cut exclusively is first estimated from the Monte Carlo simulation. These estimates are corrected by comparing the data and Monte Carlo. The decomp osition into exclusive inefficiencies is appropriate since only 0.6% 30


of events fail more than one of the classes of selection cuts. Using the 1994 data as an example, the selection efficiency from the Monte Carlo simulation is corrected by (-1.30 + 0.28)% to give an estimated selection efficiency of (73.88 + 0.29)%. Details of the main corrections to the efficiency and the main sources of systematic uncertainty are given b elow. Multiplicity cuts: The multiplicity requirements exclusively reject (1.32 + 0.02)% of Monte Carlo e+ e- + - events. However, a numb er of effects which can influence multiplicity are not p erfectly modelled, such as the simulation of the material of the detector which affects the rate of photon conversions and the two-track resolution. To assess the effect using the data, the multiplicity cuts are removed from the e+ e- + - selection yielding a sample dominated by e+ e- + - and e+ e- qq. In this sample, e+ e- + - events are identified by requiring that one of the two cones is consistent with b eing a ÷÷ decay. This requirement removes essentially all e+ e- qqevents. Backgrounds from e+ e- ÷+ ÷- and e+ e- e+ e- ÷+ ÷- are rejected using cuts on acoplanarity and momentum. In this way a sample of e+ e- + - events is isolated using only the multiplicity information from a single cone. The other cone is used to provide an unbiased estimator for the multiplicity distribution for a single cone. By convolving this measured single cone multiplicity distribution with itself, the e+ e- + - multiplicity distribution is estimated using data alone. The convolution is p erformed in two dimensions, track and total multiplicity. The multiplicity cuts are applied to the convolved distribution to determine the exclusive inefficiency of (1.69 + 0.11)% for the 1994 data sample5 . The validity of the procedure is verified using different Monte Carlo samples. Acollinearity and cone cuts: The acollinearity cut and the requirement that there b e exactly two charged cones in the event reject (3.29 + 0.03)% of Monte Carlo e+ e- + - events. These cuts reject background from e+ e- e+ e- and e+ e- qq. To study the effect of these cuts, the acollinearity and cone requirements are replaced by cuts using particle identification information such as dE/dx. Relatively hard cuts are necessary to b e able to study the acollinearity distribution for e+ e- + - events since, in the region of the cut, the background from e+ e- e+ e- dominates. Figure 12(a) shows the acollinearity distribution for this alternative selection which has an efficiency for e+ e- + - of approximately 33% but has little background (less than 0.5%). For Monte Carlo e+ e- + - events it is verified that the alternative selection does not significantly bias the acollinearity distribution. In the region acol < 15 the good agreement b etween data and Monte Carlo indicates that the modelling of the efficiency of the particle identification cuts is reasonable. The relative normalisations of the data and Monte Carlo in the region 15 < acol < 45 are used to determine corrections to the Monte Carlo efficiencies for the different centre-of-mass energies. A similar check of the inefficiency related to the cone requirements is made. The corrected inefficiency due to the acollinearity and cone cuts is (3.55 + 0.14)%. Definition of | cos |: A systematic uncertainty on the selection efficiency of 0.1% is assigned due to the uncertainty in the definition of the geometrical acceptance, | cos | < 0.9. This estimate is obtained by comparing relative numb ers of selected events in data and Monte Carlo using various definitions of | cos |, e.g. calculated using tracks, using clusters in the electromagnetic calorimeter or using b oth tracks and clusters. e+ e- e+ e- rejection cuts: The Monte Carlo simulation predicts an exclusive inefficiency due to the e+ e- e+ e- rejection cuts of (3.40 + 0.03)%. To investigate p otential biases, the energybased e+ e- e+ e- rejection is replaced by cuts using electron identification information. Imp erfect modelling of the detector resp onse in the region | cos | > 0.7, shown in Figure 12(b), results in the Monte Carlo underestimating the true inefficiency. The inefficiency for the data is estimated to b e (3.92 + 0.17)%.
We give here, by way of example for all these corrections, the corresp ondence b etween this correction to the exclusive inefficiency and the multiplicity correction factor, f , in Table 7. f = 1.0 + (0.0169 - 0.0132)/(0.7388 + 0.0169), where 0.7388 is the overall corrected efficiency, i.e. 1.3536-1 .
5

31


Further corrections and uncertainties : By using lepton identification information the effect of the cuts used to reject e+ e- ÷+ ÷- and e+ e- e+ e- + - events are found to b e adequately modelled by the Monte Carlo, however corrections, consistent with unity, and corresp onding uncertainties are obtained from the data. The trigger efficiency for e+ e- + - events is estimated to be (99.98 + 0.02)%. During 1995, when LEP op erated in bunch-train mode, no discernible effect on the e+ e- + - selection is observed and no systematic error is assigned. Uncertainties in the branching ratios of the lepton result in a 0.05% uncertainty in the e+ e- + - selection efficiency. The uncertainty on the mean p olarisation [48] has a negligible effect on the e+ e- + - selection efficiency (< 0.01%). Corrections to the selection efficiency related to four-fermion events are determined as describ ed in App endix A.1. A 10% uncertainty is assigned to the Monte Carlo exp ectation that 0.6% of events fail more than one of the classes of selection cuts. 7.4.3 Background in the e+ e
-

+

-

channel

For p eak data approximately 2.6% of the events passing the e+ e- + - selection come from background processes. This is significantly larger than the corresp onding backgrounds in the e+ e- e+ e- and e+ e- ÷+ ÷- selections. The main backgrounds are from e+ e- e+ e- (0.4%), e+ e- ÷+ ÷- (1.1%), e+ e- qq (0.4%) and e+ e- e+ e- + - (0.6%). The background fractions from e+ e- e+ e- and e+ e- e+ e- + - are higher for the off-p eak samples. To estimate the size of the background contributions using the data, cuts are applied to the selected e+ e- + - sample to enhance the various background sources. The resulting background estimations and systematic uncertainties are given in Table 7 and are summarised b elow (for p eak data). Background from e+ e- e+ e-: The Monte Carlo exp ectation for the e+ e- e+ e- background fraction is (0.26 + 0.02)%. For the reasons discussed in Section 7.2.3, the Monte Carlo underestimates the e+ e- e+ e- background in the barrel region of the detector. The techniques describ ed in Section 7.2.3 are used to estimate the effect on the background level in the e+ e- + - selection. Similar studies indicate that the e+ e- e+ e- background in the end-cap region, (| cos | > 0.7), is well modelled. The corrected total background from e+ e- e+ e- is then (0.47 + 0.07)% Background from e+ e- ÷+ ÷-: The e+ e- ÷+ ÷- background is estimated from Monte Carlo simulation to b e (0.98 + 0.02)%. For e+ e- ÷+ ÷- events to enter the e+ e- + - sample either ÷ one of the tracks must fail the muon identification criteria of Section 7.3 or Evis must b e less than 0.6 s. The e+ e- ÷+ ÷- background in the e+ e- + - sample is enhanced by requiring at least one identified muon and by imp osing loose momentum and electromagnetic energy cuts. The acoplanarity distribution of the surviving events is used to estimate corrections to the Monte Carlo exp ectation for the e+ e- ÷+ ÷- background. Discrepancies b etween data and Monte Carlo are observed for tracks near the anode planes of the central jet chamb er as indicated in Figure 12(c). As a result of these comparisons, the Monte Carlo background estimate for the 1994 data is corrected to (1.17 + 0.09)%. Background from e+ e- qq: The Lund string model as implemented in the JETSET program is used to describ e the hadronisation process for the generated Monte Carlo samples. The parameters within JETSET are tuned to describ e the prop erties of OPAL e+ e- qq events [28, 30]. Although these Monte Carlo samples provide a good description of the global prop erties of e+ e- qq, e.g. event shap e variables, there is no guarantee that they adequately describ e the low multiplicity region. This is illustrated by the comparison of the e+ e- qq background exp ectations obtained from Monte Carlo samples using two different JETSET tunes, [28] and [30], which give resp ective background estimates of (0.40 + 0.02)% and (0.88 + 0.02)%. To assess the e+ e- qq background fraction from the data, the multiplicity cuts of the e+ e- + selection are relaxed and events where one of the cones is consistent with a leptonic decay are removed from the selected e+ e- + - sample. Events in the predominant + - track multiplicity top ologies, 1 - 1, 1 - 3 and 3 - 3, are also rejected. In this way e+ e- + - events are removed whilst retaining a large fraction of the e+ e- qq background. For example, Figure 12(d) shows the 32

-


distribution of track multiplicity after the e+ e- qq enhancement. The resulting background enriched sample allows the e+ e- qq background fraction to b e estimated from the data by fitting the track and total multiplicity distributions with contributions from e+ e- + - and e+ e- qq where the shap es are taken from Monte Carlo and the normalisations left free. In this manner the e+ e- qq background is estimated to b e (0.41 + 0.11)%. The systematic error reflects the sensitivity to the Monte Carlo model used to describ e the shap e of the multiplicity distributions. Background from e+ e- e+ e- + -: At the Z p eak, the Monte Carlo prediction of the background from e+ e- e+ e- + - two-photon processes is (0.53 + 0.02)%. The Monte Carlo predictions for the background from e+ e- e+ e- e+ e- and e+ e- e+ e- ÷+ ÷- backgrounds are checked by requiring b oth cones in selected e+ e- + - events to b e consistent with b eing electrons and muons resp ectively. Distributions of Evis / s and missing transverse momentum are used to estimate the background directly from data. No deviations from the Monte Carlo exp ectation are found. The statistical precision of the check is taken as the systematic error. A 100% systematic uncertainty is assigned to the background from e+ e- e+ e- + - . The resulting estimate of the background from two-photon processes for the 1994 data is (0.58 + 0.07)%. Other backgrounds: The background from four-fermion processes (see App endix A.1) is estimated to b e (4 + 1) × 10-4 for the p eak data. The cosmic ray background is estimated from the data and found to b e small, (2 + 2) × 10-4 for the 1994 data.

7.5

Correlations among lepton species

Although the cross-sections for each lepton sp ecies are measured indep endently, the requirement that there is no overlap in the selections leads to anti-correlations b etween the systematic errors for each sp ecies. These anti-correlations reduce the precision with which the inter-sp ecies ratios can b e measured. However, comp onents of the total systematic uncertainties which are anti-correlated b etween lepton sp ecies do not contribute significantly to the uncertainty on the leptonic cross-section from all sp ecies taken together. The separation of e+ e- e+ e- and e+ e- + - events is made primarily on the basis of the cut on the total dep osited electromagnetic energy, Etotal . Almost all e+ e- events failing this cut are accepted by the + - selection, while + - events which exceed this cut app ear as background in the e+ e- sample. Consequently, the covariance b etween e+ e- and + - cross-section measurements is taken as the product of the systematic uncertainty in e+ e- efficiency due to the electromagnetic energy cut and the uncertainty of + - background in the e+ e- sample. A second source of anti-correlation arises from e+ e- ÷+ ÷- events b eing explicitly excluded from the e+ e- + - sample. The covariance is taken as the product of the uncertainty in the estimated ÷+ ÷- background in the + - sample and the uncertainty in the loss of + - events improp erly identified as ÷+ ÷- . The covariance b etween the ÷+ ÷- and + - selections and b etween the + - and e+ e- selections are of similar size, with the exact values listed in Table 21. Due to their very different exp erimental signatures, there is no cross-talk b etween the e+ e- e+ e- and e+ e- ÷+ ÷- selections.

8

Cross-section measurements

Table 1 gives the numb er of selected events in each channel used in the cross-section analysis of the entire OPAL LEP 1 data sample. For measuring the cross-section of a particular final state b oth the detector sub-systems relevant for the selection and the luminosity detector are required to b e fully op erational. Since the detector sub-systems used in the different event selections are not identical, the luminosity and mean b eam energy differ slightly for each final state. Within these `detector status' requirements, the numb er of selected final state events, the numb er of luminosity events, the luminosity weighted b eam energy and the b eam-energy spread are calculated for each data sample. The crosssections in the idealised phase space describ ed in Section 3 are then calculated by multiplying the 33


numb er of selected events by the correction factors given in Tables 3, 5, 6 and 7, and dividing by the integrated luminosity. Details of the luminosity calculation entering these tables can b e found in [13]. Tables 8, 9, 10, and 11 give the measured fermion-pair cross-sections within the kinematic cuts describ ed in Section 3, for hadrons, electrons, muons, and taus resp ectively. The "measured" crosssections have b een corrected for all effects except the spread of collision energies. The "corrected" cross-sections have b een corrected for this spread according to the exp ected curvature of the crosssection as a function of energy, and corresp ond to the true cross-section at the central b eam energy, as describ ed in App endix C. The cross-section measurements from 1990-1992 corresp ond to the results published in [7-9], except for two corrections. The small effect of initial-final state interference (see App endix A) has b een applied retrosp ectively to the lepton cross-sections. In addition, the improved theoretical calculations (see Section 5) of the small-angle Bhabha scattering cross-section causes a slight change of the FD luminosity used for the 1990-1992 cross-sections. The SiW luminosity detector was fully op erational for 94% of the 1994 p eak running and the energy scans in 1993 and 1995 (but not during the 1993 and 1995 prescan p eriods). During the p eriods when the SiW luminosity detector was not fully functional, the data are used to measure ratios of cross-sections. For these measurements the integrated luminosity is determined from the numb er of selected e+ e- qq events using the exp ected cross-section. A common 10 % normalisation uncertainty is assigned to these "pseudo-cross-sections". This arbitrary large uncertainty ensures that these pseudo-cross-section measurements only contribute to the measurement of R , but do not affect 0 the determination of mZ , Z or h (see Section 11). Table 12 gives the pseudo-cross-sections, where the errors given are statistical only. The systematic errors, and their correlations, are taken to b e the same as those for the p eak cross-sections measured in the same year, except for the correlated normalisation scale error. For the 1990 data the systematic uncertainties on the absolute luminosity and centre-of-mass energy scales are significantly larger than in subsequent years (see Section 4). For this reason the 1990 cross-section measurements are treated as pseudo-cross-sections with a common luminosity as well as LEP energy scale error imp osed across all decay channels and energy p oints. Since the 1990 data includes off-p eak as well as on-p eak p oints, it contributes to the measurement of R and Z , but not 0 to mZ or h . The quoted errors on the measured cross-sections listed in Tables 8-11 are the statistical errors from the counting of signal and luminosity events. Systematic errors, which arise from uncertainties of the LEP centre-of-mass energy, the luminosity measurement and the event selections, have varying degrees of correlation b etween the different final-state sp ecies and the various data-taking p eriods. Tables 13-15 give the covariance matrices for the LEP energy and its spread. The covariance matrix for the luminosity is sp ecified in Table 16 and Tables 17-21 show the covariance matrices for the hadron and lepton event selections. App endix C describ es in detail how the full covariance matrix for the cross-sections (and asymmetries) is constructed. Many checks were made to ensure the consistency of the event sample over widely varying time scales. One example is shown in Figure 13, which shows the distributions of the numb er of luminosity events observed b etween consecutive e+ e- qq events and vice versa. If OPAL had temp orarily lost sensitivity to small angle Bhabha scattering events in the SiW detector or e+ e- qq decays (but not b oth) for a continuous p eriod of a few minutes in the approximately six months of OPAL live time sp ent running at the p eak during 1993-1995, the failure would b e visible as a tail in one of these distributions. Similar tests involving the lepton sp ecies also revealed no such problems. Other checks probing the constancy of event typ e ratios at time scales varying from hours to months revealed no statistically significant effect. For the 1995 data, the cross-section for each final state was determined, sp ecific to each bunchlet in the bunch-train. No significant variation was observed. 34


9

The asymmetry measurements
F - B , F + B

The forward-backward asymmetry AFB is defined as AFB = (2)

where F and B are the cross-sections integrated over the forward (0 < cos - < 1) and backward (-1 < cos - < 0) hemispheres resp ectively, and - is the angle b etween the final state - and the e- b eam direction. In the SM the s-channel contribution to the differential cross-section for the reaction e+ e- + - with unp olarised b eams is exp ected to have an approximately quadratic dep endence on the cosine of the lepton scattering angle: d d(cos - ) 8 1+ AFB cos 3
-

+cos2

-

.

(3)

For e+ e- e+ e- events, the angular dep endence is more complex due to the large contributions from the t-channel and interference b etween s- and t-channels, discussed in App endix B. The magnitude and energy-dep endence of AFB are sensitive to the vector and axial-vector couplings b etween leptons and the Z (see Section 11). Within the SM, the asymmetry is predicted to b e ab out 0.01 at s = mZ (after unfolding initial-state radiation) and to increase with s. In principle the measurement of AFB is simple and suffers from few systematic uncertainties. The statistically most p owerful method of extracting a measurement of the forward-backward asymmetry is to p erform an unbinned maximum likelihood fit of the differential cross-section with AFB as the only free parameter using the logarithm of the product of the event weights, wi : wi = 3 (1 + (cos i- )2 )+ AFB cos i- , 8 (4)

where i- is the p olar angle of the negatively charged lepton in the ith event. This method, which is used for the e+ e- ÷+ ÷- and e+ e- + - channels, has the advantage of b eing insensitive to any event selection inefficiencies, or any variation of the efficiency with cos - , provided there is no forward-backward charge asymmetry in the efficiency function itself. Technically, the inclusion of a selection efficiency term in the event weights results in an additive constant when taking the logarithm of the likelihood function provided the efficiency can b e expressed in terms of | cos i- |. This is in contrast to the event-counting method (see b elow) which must b e corrected for selection inefficiencies. The fit yields a value for AFB over the full p olar angle range. This can b e corrected to a restricted range | cos | < k using a simple geometrical factor AFB (| cos | < k) = AFB (| cos | < 1) 4k . 3+ k2 (5)

This region corresp onds to the ideal acceptance definition of Table 2 and is chosen to corresp ond closely to the geometric acceptance of the event selection. In addition, small corrections to the measured asymmetry are required to take account of background and other biases. The measurement of the forward-backward asymmetry in the e+ e- e+ e- channel is more complicated. Due to the t-channel contributions, the angular distribution in e+ e- e+ e- events can not b e parametrised as a simple function of the forward-backward asymmetry, Aee . For this reason FB Aee is measured from the observed numb ers of events in the forward and backward hemispheres, NF FB and NB : Aee = FB NF - NB . NF + N B (6)

This counting method is also used as a cross check of the unbinned maximum likelihood method in the e+ e- ÷+ ÷- and e+ e- + - channels and yields consistent results. 35


In each of the three leptonic channels the choice of which particle ( + or - ) is used to define the p olar angle of the event is varied at random from event to event. This reduces many p ossible geometrical biases to the asymmetry measurement arising from an asymmetry in the detector. In the cases where the + track is used, the angle is measured relative to the e+ b eam direction. The measured asymmetries are corrected for the energy spread of the colliding b eams as describ ed in App endix C. Corrections are required to account for charge misassignment and for differences b etween the exp erimental acceptance and the ideal acceptance of Table 2. Details of these corrections and the main systematic uncertainties are given b elow. Note that our measured asymmetries implicitly include the effects of initial-final-state interference. We make no correction to remove these effects from our measurements, but calculate the predicted asymmetries accounting for this interference using ZFITTER. The asymmetry measurements are dominated by statistical rather than systematic uncertainties even when all data sets are combined.

9.1

e+ e- ÷+ ÷

-

forward-backward asymmetry

The forward-backward asymmetry of muon pairs, A÷÷ , is a conceptually simple measurement. Muon FB pair events are selected with high efficiency and the asymmetry measurement itself is robust against many systematic effects. Compared to our previous publication [9] the analysis presented here represents a significant improvement in the understanding of the exp erimental biases. To b enefit from the resulting reduction in systematic uncertainties, the asymmetry in the 1992 data sample has b een re-evaluated. In addition to the e+ e- ÷+ ÷- selection cuts of Section 7.3.1, the acollinearity angle b etween the directions of the two muon tracks is required to b e less than 15 . This removes ab out 1.3% of the total sample by rejecting events with high-energy radiated photons. Such events distort the angular distribution and bias the asymmetry measurement due to the lower value of the effective centre-of-mass energy, s . Equivalent acceptance cuts are made inside the ZFITTER program [49] when evaluating the theoretical predictions used to fit the data, as discussed in Section 11. Ab out 1.2% of the selected e+ e- ÷+ ÷- events have the same sign assigned to b oth candidate muon tracks. These events are concentrated in the region where the tracks are close to the jet chamb er wire planes or are in the forward region, | cos | > 0.90. The probability that a track is assigned the incorrect charge is found to b e dep endent b oth on the p olar angle of the track and on its direction of curvature. Both p ositive tracks with cos < 0 and negative tracks with cos > 0 are more likely to b e assigned the wrong charge. Rejection of same-sign events would therefore lead to a selection inefficiency which is b oth asymmetric and charge-dep endent, removing more events in the backward hemisphere than in the forward hemisphere and producing a p ositive bias in A÷÷ . The origin of this effect is FB not understood. The size of this bias is found to b e 0.0010. These "like-charge" events are recovered by determining which track was the most likely to have b een assigned the wrong charge, based on numb ers of assigned track hits, measured momentum and its uncertainty and the acoplanarity angle for tracks close to the anode wire planes of the central detector. The efficiency of this procedure is determined using the sample of events where the charge can b e almost unambiguously determined from the acoplanarity as measured in the muon chamb ers. The resulting charge measurement is estimated to b e correct for 93% of like-charge events. A small correction is made for the 7% of like-charge events where the assignment is incorrect. The fraction of events where b oth muons are assigned the wrong charge is small, 2 × 10-4 , and has negligible impact on the measurement. ÷ For Monte Carlo events the cut on Evis introduces a small additional bias, approximately 1 × 10-4 , in the measured asymmetry with resp ect to the asymmetry within the ideal acceptance of Table 2. However there is an additional bias for the data, related to the detector resp onse, from events ÷ which fail the Evis cut due to b eing mismeasured. These events are concentrated in two regions either | cos | > 0.90 or within 0.5 of an anode wire plane. Additional selection criteria were applied ÷ to recover these mismeasured events. The main background in the region Evis < 0.6 arises from + e- + - . Cuts based on acoplanarity and the reconstructed momenta of the two muon tracks e reduced this background to a manageable level. The inclusion of the recovered events in the event 36


sample changes the measured value of A÷÷ by (6 + 2) × 10-4 . FB Figure 15 shows the observed angular distribution of e+ e- ÷+ ÷- events from the 1993-1995 data samples, after applying small corrections for inefficiency and background. The asymmetries are obtained from unbinned maximum likelihood fits to the observed distributions of cos ÷- and corrected to the restricted acceptance of | cos | < 0.95 using Equation 5. In the event selection b oth reconstructed muons are required to b e within | cos | < 0.95. In contrast, the acceptance for the muon pair asymmetry is defined constraining only | cos ÷- | < 0.95, thus including a larger fraction of events with significant initial state radiation. The measured asymmetries are therefore corrected for the resulting bias of ab out 0.0003, which was estimated using KORALZ Monte Carlo events. The forward-backward asymmetry measurements for the e+ e- ÷+ ÷- channel, fully corrected to the acceptance definition of Table 2, are summarised in Table 22. For each of the data sets the systematic errors are less than 10% of the statistical uncertainties. The total systematic uncertainty on the measured asymmetry for the highest statistics data sample is estimated to b e 0.0004. The largest uncertainty is related to charge misassigned events, describ ed ab ove, varying b etween +0.0002 and +0.0010 dep ending on the data sample. The second largest systematic error, +0.0002, comes from limits on the p ossible inadequacies of the fitting function in the maximum likelihood fit. In addition, small systematic uncertainties, each less than 0.0002, are assigned due to the definition of the angular acceptance, the residual tau pair background and biases from the event selection. The systematic errors for the different data samples and their correlations are given in Table 25.

9.2

e+ e- +

-

forward-backward asymmetry

Measurement of A b egins with the definition of a suitable event sample. In addition to the FB e+ e- + - selection cuts of Section 7.4.1, events in which the sums of the charges in b oth cones have the same sign are rejected and the sum of the charges of the tracks in at least one of the tau cones must b e either +1. These additional requirements reject approximately 2% of the selected e+ e- + - events. Unlike in the muon-pair asymmetry measurement, the softer momentum sp ectrum of charged particles from tau decays allows like-charged events to b e rejected without introducing a significant bias. The charge misassignment in e+ e- + - events mainly arises from track reconstruction ambiguities for tau decays producing three or more closely separated tracks. Another difference with resp ect to the muon-pair asymmetry measurement is that the tau-pair asymmetry is sub ject to a p otentially imp ortant background from electron-pair events. Due to the t-channel contribution, e+ e- e+ e- background events have a large forward-backward asymmetry. To reduce the sensitivity of the A measurement to uncertainties in the e+ e- e+ e- background, additional cuts FB are applied. These cuts, based on dE/dx, acoplanarity and Etotal , reduce the e+ e- e+ e- background by over 90% and remove only 1.5% of e+ e- + - events. Figure 14 shows the angular distributions of the events selected for the e+ e- + - asymmetry measurement. Figure 16 shows the corresp onding distributions after corrections for background and selection efficiency are applied. Unbinned maximum likelihood fits to the uncorrected observed data distributions are used to measure A . The measured asymmetries are then corrected for the presence of background FB AFB = (A - Abkg )f FB FB
bkg

,

where fbkg is the background fraction and Abkg is the asymmetry of the background. The resulting FB corrections are small, corresp onding to less than 10% of the statistical error. The measured values of A corresp ond to the asymmetry in the selected sample extrap olated FB to the full cos acceptance. These are corrected to the acceptance definition | cos - | < 0.9 and acol < 15 given in Table 2. The corrections are obtained using Monte Carlo e+ e- + - events which are treated in the same manner as data. The corrections also take into account other biases such as the effect of the non-zero average p olarisation of the lepton. The visible energy sp ectra from decays of p ositive and negative helicity leptons are different. The e+ e- + - event selection is 37


approximately 1.5% more efficient for the negative helicity final states than for p ositive helicity final states and a bias arises since the forward-backward asymmetries for the two helicity states are different. For each energy p oint, the measured asymmetry from the Monte Carlo sample is compared with the true asymmetry within the acceptance | cos - | < 0.9 and acol < 15 obtained from the Monte Carlo generator information. The differences are used to obtain corrections to the measured asymmetries of -0.0027 + 0.0017, -0.0014 + 0.0011 and -0.0013 + 0.0015 for the p eak-2, p eak and p eak+2 data samples, resp ectively, where the errors include statistical and systematic comp onents. These corrections and uncertainties implicitly include effects due to p ossible inadequacies of the function used in the maximum likelihood fit. The corrected forward-backward asymmetry measurements for the e+ e- + - channel are summarised in Table 23. The overall systematic uncertainties on A are 0.0018, 0.0012 and 0.0016 for the FB peak-2, p eak and p eak+2 data p oints, resp ectively. The errors for the different data sets and their correlations are listed in Table 25. The systematic errors are dominated by the uncertainties on the Monte Carlo correction from the measured asymmetries to the asymmetry definition of Table 2. The systematic error associated with the measurement of cos is studied by using 12 different definitions of this angle, e.g. using tracks and clusters, using tracks alone, taking the average value from the two tau cones. The shifts in the measured asymmetries compared to the default result are reasonably well reproduced by the Monte Carlo and a systematic error of 0.0005 is assigned. The knowledge of the absolute angular scale, or length-width ratio of the detector, has negligible impact on these measurements. As a consequence of the additional e+ e- e+ e- rejection cuts, the systematic errors arising from the uncertainties in the background fractions are small (typically 0.0002).

9.3

e+ e- e+ e- forward-backward asymmetry

Aee is measured from the observed numb ers of events in the forward and backward hemispheres: FB Aee = FB NF - NB . NF + N B (7)

Here NF and NB are the numb ers of events within 0.00 < cos e- < 0.70 and -0.70 < cos e- < 0.00, resp ectively, and e- is the e- scattering angle. In 1.5% of the selected e+ e- e+ e- events the signs of reconstructed charges of the tracks are the same. These events are retained for the asymmetry measurement by using the acoplanarity angle b etween the electromagnetic calorimeter clusters associated with the two electron candidates to distinguish their charges. This method is limited by the exp erimental resolution of the acoplanarity measurement giving the correct assignment in ab out 92% of events. This p erformance is studied in b oth data and Monte Carlo by applying the cluster-based acoplanarity method to selected events in which the two tracks have b een assigned opp osite charges. We therefore correct the measured asymmetry for residual bias due to the fact that we make the wrong charge assignment for an estimated 1.2 × 10-3 of all events when we recover the like-sign events using the acoplanarity method. The resulting corrections to Aee are 0.0010 + 0.0005, 0.0004 + 0.0003 and FB 0.0004 + 0.0006 for the p eak-2, p eak and p eak+2 energy p oints, resp ectively. In the process e+ e- e+ e- , there is a small difference in the event selection efficiency for forward and backward events due mainly to the cut on the electromagnetic energy. This is caused by the softer energy sp ectra of ISR photons from s-channel Z than from t-channel photon exchange processes. ISR photons tend to b e produced along the b eam direction and are therefore likely to remain undetected. Consequently, events with harder ISR are more likely to b e rejected by the cut on the total visible electromagnetic energy in the event, Etotal > 0.80 s. The t-channel contribution is relatively more imp ortant in the forward direction while the contribution from the Z dominates in the backward direction. As a result a small forward-backward asymmetry in the e+ e- e+ e- event selection efficiency arises. The resulting s-dep endent effect on Aee is evaluated using the Monte Carlo simulation to b e FB -0.0002 + 0.0004, -0.0003 + 0.0003 and -0.0007 + 0.0008 for the p eak-2, p eak and p eak+2 energy p oints, resp ectively. 38


Unlike the unbinned maximum likelihood fit, the counting method used to obtain Aee is sensitive to FB variations in the selection efficiency as a function of | cos |. The efficiency variations for e+ e- e+ e- , however, are small enough to have negligible impact on the asymmetry measurement. The background in the e+ e- e+ e- sample is dominated by e+ e- + - events for which the exp ected asymmetry is different from that of e+ e- e+ e- events. Small corrections for background are obtained in the same manner as used for A . The corresp onding systematic uncertainties are negligible. FB The offset of 0.0004 + 0.0005 in the effective edge of the acceptance in p olar angle relative to the nominal value (see Section 7.2.3) causes a small bias in the measured asymmetry. The size of this effect has b een evaluated numerically, using the SM prediction for the differential cross-section at the acceptance edges. Corrections of 0.0003 + 0.0009, 0.0002 + 0.0004 and 0.0002 + 0.0005 are obtained for the p eak-2, p eak and p eak+2 energy p oints, resp ectively. The systematic errors corresp ond to the uncertainty on the edge of the acceptance. The accuracy of the definition of the b oundary at cos = 0.0 is also imp ortant for the asymmetry measurement, since this is used to separate forward events from backward events. However, by randomly choosing either the e- or the e+ to classify each event, the effect of any p ossible small bias is largely cancelled and is reduced to a negligible level. As a check of the asymmetry measurement, Aee has also b een evaluated using several different methods to FB classify events as forward or backward: using only e- clusters and tracks; using only e+ clusters and tracks; using only measurements from tracks rather than from clusters when a high quality track is found; and using only events in which the two tracks have b een assigned opp osite charges. There are no significant differences b etween these results, b eyond what is exp ected from statistical fluctuations. The forward-backward asymmetry measurements for the e+ e- e+ e- channel are summarised in Table 24. The systematic errors for the different data samples and their correlations are given in Table 25. Figure 17 shows the electron differential cross-sections at the three main energy p oints, fully corrected for the effects discussed ab ove.

10

Parametrisation of the Z resonance

Before proceeding to interpret the measurements we first discuss the basic formalism used to describ e cross-sections and asymmetries at the Z resonance. Then we give a brief overview of radiative corrections and the programs which we use for their calculation. Finally, we describ e the t-channel corrections which need to b e accounted for in e+ e- final states.

10.1

Lowest order formulae

The process e+ e- f f can b e mediated in the s-channel by two spin-1 b osons, a massless photon and a massive Z b oson. In lowest order and neglecting fermion masses the differential cross-section for this reaction can b e written as: 2s df f (8) = 2 Q2 (1 + cos2 ) f Nc dcos
s a + 8 Re Qf (s) [ C Z (1 + cos2 )+ 2C Z cos ] s + 16 |(s)|2 [ CZZ (1 + cos2 )+ 8C a ZZ

cos ]

with (s) = and
s C Z = s CZZ

GF m2 s Z 2 + im 8 2 s - mZ Z

(9)
Z

gVe gVf ^^ (gVe ^2 + gAe ^2 )(gVf ^2 + gAf ^2 )

a , C Z = gAe gAf , ^^

(10) (11)

=

,

a CZZ

= gVe gAe gVf gAf . ^^^^

Here is the electromagnetic coupling constant, GF is the Fermi constant and Qf is the electric charge of the final state fermion f. The colour factor Nc is 1 for leptons and 3 for quarks, mZ is the mass 39


of the Z b oson and Z its total decay width.6 gAf and gVf are the axial-vector and vector couplings ^ ^ b etween the participating fermions and the Z b oson. The first term in Equation 8 accounts for pure photon exchange, the second term for /Z interfers a s a ence and the third for pure Z exchange. The four coefficients C Z , C Z , CZZ and CZZ parametrise the terms symmetric (`s') and antisymmetric (`a') in cos . The relative size and the energy dep endence of the five comp onents of the differential cross-section (three symmetric terms for , /Z and Z , and two anti-symmetric terms for /Z and Z ) as exp ected in the SM are shown in Figure 18. The symmetric terms are clearly dominated by pure Z exchange. In contrast, the antisymmetric pure Z term is much smaller in magnitude than the corresp onding contribution from /Z interference, except very close to the p ole of the Z resonance, where the latter crosses zero. Integrated over the full angular phase space, the Z exchange term can b e further expressed as a Breit-Wigner resonance: (s) =
Z ff 0 f

sZ (s - m2 )2 + m2 Z Z
2 Z

2

,

(12)

0 where f is the "p ole cross-section", i.e. the total cross-section at s = m2 . It is given in terms of the Z s partial widths or the CZZ parameter:

12 ee ff Cs N = 2 = ZZ 6 mZ 2 Z
0 f

c

m2 GF Z Z

2

,

(13)

where ee and ff are the partial decay widths of the Z to electron pairs and the fermion pair f f, resp ectively. In terms of couplings the partial width is given by: ff = GF Nc m 6 2
3 Z

gVf + gAf ^2 ^2

.

(14)

The anti-symmetric terms in Equation 8 give rise to the forward-backward asymmetry AFB defined in Equation 2. At centre-of-mass energies close to mZ it can b e approximated by: AFB a 3 2Qf C Z s - m (s) = - s GF m2 CZZ s Z
2 Z

+

a 3CZZ . s CZZ

(15)

The first term arises from the /Z interference and the second from pure Z exchange. The former gives the dominant contribution to AFB except very close to the p ole and causes a strong s dep endence, as illustrated in Figure 18 (b). The latter is termed the "p ole forward-backward asymmetry", A0,f , FB and can b e conveniently expressed in terms of the coupling parameters Af 3 A0,f = Ae A FB 4
f

with Af = 2

gVf gAf ^^ 2 + g2 gVf ^Af ^

.

(16) exchange of a massive the heavy b oson and vector-typ e coupling are given at tree level universal electroweak (17)

One should note that the relations presented so far are generally valid for the spin-1 b oson, indep endent of the sp ecific form or size of the couplings b etween the fermions. Only the well-tested prediction of QED for the strength of the b etween photon and fermions is assumed. Within the SM, however, gAf and gVf ^ ^ 3 , the electric charge Q and the by the third comp onent of the weak isospin If f mixing angle, sin2 W : gAf = If3 , gVf = If3 - 2Qf sin2 W . ^ ^
6

The bars on mZ and Z distinguish quantities defined in terms of the Breit-Wigner parametrisation with an s- indep endent total width from mZ and Z which are defined in terms of an s-dep endent total width, as discussed in connection with Equation 21.

40


10.2

Radiative corrections

Radiative corrections significantly modify the e+ e- f f cross-sections and forward-backward asymmetries with resp ect to the tree level calculation. One can distinguish four main categories: ž Photon vacuum p olarisation: Vacuum p olarisation leads to a scale dep endence of the electromagnetic coupling constant : (0) - (s) = (0) . 1 - (s) (18)

Due to uncertainties in the hadronic contribution to (s) the value of at s = m2 has a Z considerable uncertainty ((m2 )-1 = 128.886 + 0.090 [50]), despite the high precision of at Z s = 0 (0.04 ppm). There is also a small imaginary comp onent of (s) which leads to an additional contribution to the /Z interference in conjunction with the imaginary part of the Z propagator in Equation 9. ž Initial-state radiation: Photonic radiation in the initial state has a profound effect on the Z lineshap e. It reduces the p eak height by ab out 25%, shifts the p eak upward by ab out 100 MeV and increases the apparent (FWHM) width by ab out 500 MeV, as illustrated in Figure 18 (e). Initial-state radiative corrections can b e implemented in terms of a radiator function, H (s, s ), obs which relates the electroweak cross-section, f f , to the observable cross-section, f f , in terms of an integral over s , the squared invariant mass available to the hard electroweak interaction where s
min obs ff

(s) =
s

s
min

f f (s )H (s, s )ds ,

(19)

is the minimum invariant mass squared of the system after initial state radiation.

ž Final-state radiation: The radiation of photons or gluons (only for qq) in the final state increases the partial widths in first order by factors QED = 1 + 3 Q2 (m2 ) f Z , QCD 4 1+ s (m2 ) Z , (20)

where s is the strong coupling constant. For hadronic final states QCD causes a sizeable correction which allows a precise determination of s (m2 ) from the inclusive hadronic width, Z had . ž Electroweak corrections: Quantum-loop effects in the Z propagator and vertex corrections involving the Higgs and the top quark give rise to radiative corrections, which in leading order dep end quadratically on the mass of the top quark, mt , and logarithmically on the Higgs b oson mass, mH . These effects give sensitivity to physics at much larger scales than m2 . Z When higher order corrections are considered mixed corrections arise; in particular QED/QCD and QCD/electroweak corrections are significant. In addition to and Z exchange, b ox diagrams also make small contributions to the process e+ e- f f , with a relative size 10-4 close to the Z resonance. After unfolding the effects of initial and final-state radiation one can still retain the form of the differential cross-section expressed in Equation 8 with a few modifications: ž (0) must b e replaced by the (complex) (m2 ). Z ž Electroweak corrections can b e absorb ed by replacing the tree-level axial-vector and vector couplings, gAf and gVf , by the corresp onding effective couplings GAf and GV f , which are complex ^ ^ numb ers with small imaginary comp onents. In general, the effect of these imaginary comp onents, termed "remnants", on the different terms of the differential cross-section is small. Most notable is their contribution to the symmetric part of the /Z interference, as shown in Figure 18 (c). 41


ž Loop corrections to the Z propagator lead to an s-dep endent decay width [51]. This can b e accounted for by replacing ( Z Z (s) sZ /m2 ) in Equation 9, resulting in: Z (s) = GF m2 s Z 2 +i 8 2 s - mZ
s mZ



.
Z

(21)

One should note that this does not alter the form of the resonance curve but corresp onds to an exact algebraic transformation which redefines b oth the Z mass, mZ = mZ 1+ Z /mZ , and width, Z = +1 MeV.
Z

1+ Z /mZ . Numerically, mZ is shifted by ab out +34 MeV and Z by

The Z resonance measurements are conveniently interpreted in terms of "model-indep endent" parameters, such as mass, total width, partial widths or p ole cross-section, and p ole asymmetry. In the presence of radiative corrections these parameters are no longer direct observables but dep end on the sp ecific choice of which radiative corrections to unfold or include and are therefore termed "pseudoobservables". In this pap er we define the Z mass and width, mZ and Z , in terms of the s-dep endent Breit-Wigner of Equation 21. Following the standard convention adopted by the LEP exp eriments, the partial decay widths absorb all final-state and electroweak corrections, such that their sum equals the total decay width: ff = GF Nc m 6 2
3 Z f f f f |GV |2 RV + |GA |2 RA +

ew/QCD

.

(22)

f f Here, RV and RA account for final-state corrections and fermion masses and ew/QCD accounts for non-factorisable mixed electroweak/QCD contributions. After unfolding initial-state radiation the total cross-section from Z exchange can then b e written as:

Z f f (s) =



QED (s - m

0 f

s
2 )2 Z

2 Z

+

s2 m2 Z



2 Z

0 with f =

12 ee m2 2 Z Z

ff

.

(23)

The factor 1/QED used to describ e th In contrast, for nary remnants are

(Equation 20) is needed to cancel the final-state radiative corrections in ee when e couplings to the initial state. the p ole asymmetries (Equation 16), the effects of final-state radiation and imagiexcluded; they are interpreted in terms of the real parts of the effective couplings:
f

3 A0,f = Ae A FB 4

, Af = 2

gVf gAf 2 + g2 gVf Af

f f with gVf Re(GV ) , gAf Re(GA ) .

(24)

f f The imaginary remnants (Im(GA ), Im(GV ) ) are evaluated in the SM and the corresp onding corrections are applied. For the calculation of radiative corrections we used the programs ZFITTER [49] and TOPAZ0 [52]. Initial-state radiation is implemented completely to O(2 ) [53] with leading O(3 ) corrections [54]. The radiation of fermion pairs in the initial state is also included to O(3 ) [55]. Photon radiation in the final state as well as initial and final state interference is treated to O(). The calculation of final-state radiation deserves a further comment. For the asymmetries and e+ e- e+ e- cross-section, which are measured within tight kinematic cuts, we use (0) as recommended in [56]. In contrast, for the other cross-sections, which are measured within inclusive cuts, we use (m2 ), which implicitly Z covers the dominant part of small effects from final-state pair production. In addition to the one-loop level electroweak corrections, TOPAZ0 and ZFITTER include the leading (O(m4 )) and sub-leading (O(m2 m2 )) two-loop corrections [57]. QCD corrections are calculated t tZ to O(3 ) [58], with mixed terms included to O(s ) and the leading O(m2 s ) terms [59, 60]. s t

42


10.3

t-channel contributions to e+ e- e+ e-

For e+ e- e+ e- events not only s-channel annihilation contributes but also the exchange of and Z in the t-channel. Since t-channel exchange is not included in our main fitting program, ZFITTER, an external treatment is needed to account for it. We use the program ALIBABA [61] to calculate the exp ected contribution from t-channel and s-t interference and add them to the pure s channel contributions calculated with ZFITTER. The details of how we treat these contributions in the fit and the associated uncertainties are describ ed in App endix B. The uncertainties on the fitted e+ e- partial width, ee , and p ole asymmetry, A0,e , due to the t-channel are 0.11 MeV and 0.0015, resp ectively, FB with a correlation coefficient of 0.85.

11

Determination of electroweak and Standard Mo del parameters

The 211 measurements of cross-sections and asymmetries listed in Tables 8 - 12 and Tables 22 - 24 form the basis of our tests of SM exp ectations, and our determination of electroweak parameters. The cross-section measurements at the three energy p oints of the precision scan determine the basic 0 parameters of the Z resonance, its mass, mZ , width, Z , and its p ole production cross-sections, f , for each of the four final states. By combining the measured p ole cross-sections one can determine directly the absolute branching ratios (ff /Z ) to hadrons and leptons, as can b e seen from Equation 23. The branching ratio to invisible states, such as neutrinos, can b e inferred from the difference b etween unity and the sum of the visible branching ratios. Combining the branching ratios with the total width yields the decay widths to each sp ecies. For leptons these widths provide a precise check of SM predictions for the leptonic couplings. The hadronic width, through QCD effects, provides a measurement of s . In contrast to almost all other measurements of this quantity, no hadronisation corrections are needed to compare with QCD calculations, which are available in O(3 ). Therefore QCD uncertainties are s small. The charge asymmetries of the leptons produced at the p eak allow the ratios of the leptonic vector and axial-vector couplings to b e determined. Finally, through radiative effects on the effective couplings, limits can b e placed on the mass of the Higgs b oson. The interpretation of our measured cross-sections and asymmetries in terms of electroweak parameters proceeds in three stages. First, we use a parametrisation based on Equation 8, where the C coefficients are treated as indep endent parameters, not imp osing the constraints of Equations 10 and 11. Secondly, we p erform the analysis in terms of the "model-indep endent Z parameters", which are based on mass, total width, p ole cross-sections and p ole asymmetries of the Z. The main difference with resp ect to the first approach is that in deriving the model-indep endent Z parameters the /Z interference is constrained to the theoretical prediction. Finally, we compare our measured cross-sections and asymmetries directly with calculations made in the context of the SM. This allows us to test the consistency of our measurements with the theoretical prediction and also to determine the SM parameters, mZ , mt , mH and s . For all our fits we use ZFITTER, version 6.21, with the default settings7 to calculate the crosssections and asymmetries within our ideal kinematic acceptance as functions of the fit parameters. The fit parameters are obtained from a 2 minimisation based on MINUIT [62], which takes into account the full covariance matrix of our data. In App endix C we describ e how the various sources of uncertainty are combined to construct the covariance matrix.

Except for had , which we sp ecify directly (flag ALEM = 2), and the correction of [60], which is used only for the SM fits in Section 11.4 (flag CZAK) as recommended by the authors of ZFITTER.
7

(5)

43


The SM calculations require the full sp ecification of the fundamental SM parameters. The main parameters are the masses of the Z b oson (mZ ), the top quark (mt ) and the Higgs b oson (mH ), and the strong and electromagnetic coupling constants, s and . Unless explicitly sp ecified otherwise, we f f use for the calculation of imaginary remnants (Im(GA ), Im(GV ) ), non-factorisable corrections and for comparison with the SM predictions the following values and ranges: mZ = 91.1856 + 0.0030 GeV m
H

, mt = 175 + 5 GeV , , s = 0.119 + 0.002 [2] , .

(25)

=

150+850 -60

GeV

had =

(5)

0.02804 + 0.00065

The values and ranges of mt and s differ slightly from the most recent evaluations [63, 64]. They were chosen for consistency with the corresp onding publications of the other LEP exp eriments; the small differences are completely negligible for the results presented here. The range of mH corresp onds approximately to the lower limit from direct searches [65] and the upp er limit from theoretical (5) considerations [66]. The electromagnetic coupling is expressed here in terms of had , the contribution of the five light quark flavours to the running of at s = m2 as describ ed in Equation 18. Z (5) The quoted value for had corresp onds to (m2 )-1 = 128.886 + 0.090 [50] when the contributions Z from the top quark and the leptons are also included. A further crucial parameter is the Fermi constant GF , which can b e determined precisely from the muon lifetime. We use the most recent value, GF = (1.16637 + 0.00001) × 10-5 GeV-2 [67], which includes two-loop QED corrections. Other input parameters of the SM, such as the masses of the five light quarks8 and the leptons, are fixed to their present values [2].

11.1

C -Parameter fits

Our first analysis is based on a parametrisation according to Equation 8, treating the C -coefficients as indep endent fit parameters. In this ansatz, which we first introduced in the analyses of our 1990/1991 data samples [7, 8], the differential cross-section is describ ed in terms of four indep endent parames a s a ters, C Z , C Z , CZZ and CZZ , which express the symmetric and anti-symmetric contributions from /Z interference and pure Z exchange, resp ectively. Sp ecifically we do not imp ose the constraints expressed in Equations 10 and Equations 11, which are in principle generally valid for the exchange of a massive vector b oson with arbitrary axial-vector and vector couplings interfering with the photon. Contributions from new physics, however, such as an additional heavy vector b oson, would lead to additional terms in Equation 8. In particular one would exp ect contributions from the interference b etween the Z and the extra b oson, which would have the same form as the /Z interference. In the case of leptonic final states, the measurements of cross-sections and asymmetries are sensitive to all four parameters indep endently. Fitting the data with indep endent interference terms retains the s sensitivity to such new physics effects. In addition, from the indep endent determination of C Z and a C Z one can distinguish gVf and gAf and determine the relative signs of the couplings, as discussed s a b elow in Section 11.3.3. This is not p ossible from the pure Z terms CZZ and CZZ alone, which are symmetric in gVf and gAf . In this analysis we do not include any hadronic forward-backward asymmetry data and hence s s only the terms symmetric in cos , C Z and CZZ , are accessible. However, the sensitivity to the /Z s interference (C Z ) is small at centre-of-mass energies close to mZ , since a shift of the interference term is essentially equivalent to a shift in mZ . The fact that high statistics measurements of the cross-section are available only for three different centre-of-mass energies further aggravates the lack of discrimination b etween shifts in mZ and the interference term. In the following analysis we therefore use the C -parametrisation only for leptons. The hadrons are parametrised in terms of an s-dep endent 0 Breit-Wigner resonance (Equation 23) with mZ , Z and h as fit parameters. The hadronic /Z interference term is fixed to the SM prediction. This constraint allows a precise determination of mZ .
8

See [49] for details on the treatment of quark masses.

44


With the leptonic /Z interference terms left free in the fit, mZ is determined from the hadronic data s and the C Z parameters for leptons therefore dep end on the assumed interference term for hadrons. For the precise definition of the C -parameters several choices could b e made. We employ a definition such that the parameters can b e interpreted directly in terms of the real parts of the effective couplings, i.e. the meaning of the C -parameters in the SM corresp onds to Equations 10 and 11 with gAf and gVf replaced by gAf and gVf as in Equation 24. We use ZFITTER to correct for initial and ^ ^ final-state radiation and the running of and to calculate the imaginary remnants of the couplings. Since these remnants are small and essentially indep endent of the centre-of-mass energy in the LEP 1 region, this SM correction does not compromise the indep endence of the parametrisation from theoretical assumptions, but it preserves a transparent interpretation of the C -parameters in terms of SM couplings. 0 If lepton universality is not assumed, there are a total of 15 parameters: mZ , Z ,h and four s a s a C -parameters - C Z , C Z , CZZ , CZZ - for each lepton sp ecies. These fitted parameters are shown in column two of Table 26. The values obtained from the different lepton sp ecies for the C -parameters are consistent with one another, compatible with lepton universality. Column three of Table 26 gives the results for a 7 parameter fit when lepton universality is imp osed by requiring each of the four C -parameters to b e equal across the three lepton sp ecies. The SM predictions are shown in the last column. The error correlation matrices are given in Tables 27 and 28. Figure 19 compares the parameters measured assuming lepton universality with the SM predictions as a function of mH . We see agreement for all parameters. The largest discrepancy, of ab out two a standard deviations, occurs in the parameter C Z . The same tendency was observed in our previous analysis using only the data collected from 1990 to 1992 [9]. This parameter corresp onds to the energy dep endence of the leptonic forward-backward asymmetry. As a check we also p erformed a fit releasing the SM constraint for the hadronic /Z interference by introducing a scale factor for the /Z interference contribution (f/Z = 1 in the SM) which was treated as an additional free parameter. The fit resulted in mZ = 91.190 + 0.011 GeV and f/Z = 0.0 + 3.0, with a correlation coefficient of -0.96. This is in good agreement with the mZ result in Table 26 and with f/Z = 1. The large increase of the mZ uncertainty and the high anti-correlation with f/Z illustrates the fact that LEP 1 measurements alone give marginal discrimination b etween hadronic /Z interference effects and mZ shifts. In combination with measurements at energies further away from the Z resonance, e.g., the fermion-pair cross-sections at LEP 2, this problem can b e resolved [3]. An alternative model-indep endent parametrisation of the Z resonance is the S-Matrix formalism [68]. In practice this approach is equivalent to the C -parameters. Since the S-Matrix parametrisation is the standard framework for combined LEP 1 and LEP 2 cross-section and asymmetry measurements we also give the S-Matrix results in App endix D.

11.2

Results of the model-independent Z parameter fits

0 Our second fit is based on the model-indep endent Z parameters, which consist of mZ , Z , h , and A0, FB as given in Equations 23 and 24, as well as the ratios of hadronic to leptonic widths:

R

had

( = e,÷, ) .

(26)

The partial widths include final-state and electroweak corrections as defined in Equation 22. This set of pseudo-observables is closely related to the exp erimental measurements and sufficient to parametrise the Z prop erties; correlations b etween the parameters are small. For this reason it has b een adopted by the four LEP collab orations to facilitate the comparisons and averaging of the Z resonance measurements. The essential difference b etween this fit and the C -parameter fit ab ove is that now, in addition to s a the hadronic interference term, the leptonic /Z interference terms (C Z and C Z ) are also constrained
0 to the SM prediction. The expression of resonant lepton production in terms of R , h and A0, is FB s a equivalent to the CZZ and CZZ formulation.

45


The results of the 9 parameter fit (without lepton universality) and the 5 parameter fit (assuming lepton universality) are shown in Table 29, together with the SM predictions in the last column. The error correlations are given in Tables 30 and 31. For the fits where lepton universality is not imp osed the large mass of the is exp ected to reduce by ab out 0.2%. Lepton universality would therefore b e reflected in R b eing 0.047 larger than Re and R÷ . In the 5 parameter fit, where lepton universality is imp osed, the mass effect is corrected; refers to the partial decay width of the Z into massless charged leptons. Figure 20 compares the parameters fit assuming lepton universality with the SM prediction as a function of the Higgs mass. The agreement for all parameters is good. In the context of this parameter set, we also make a test of the consistency of the LEP energy calibration. To make this test, we replace the single parameter mZ with three indep endent parameters for the three p eriods which distinguish the stages in the evolution of the LEP calibration; these are 1990-92, 1993-94 and 1995. The results are given in Table 32. The errors of the three mZ values are largely uncorrelated, and the stability of the LEP energy calibration is verified within the precision of our measurements. The value of mZ we obtain in the fit to the model-indep endent Z parameters is ab out 1 MeV smaller than the value we obtain in the fit to the C -parameters. This difference is due to the fact that in the C -parameter fit the leptons contribute much less to the mZ measurement, since the leptonic /Z interference terms are allowed to vary freely. One should also note that mZ shifts by -0.5 MeV when lepton universality is imp osed, in b oth the fit with C -parameters and the fit with model-indep endent Z parameters. This is due to a subtle effect in the e+ e- e+ e- t-channel correction, which gives an additional weak constraint on mZ (see App endix B). Figures 21 and 22 compare the cross-section and asymmetry measurements with the results of the 9-parameter model-indep endent fit. Figure 23 shows the contours in the A0, - R plane, separately FB for the three lepton sp ecies as well as for a universal charged lepton. There is a large anti-correlation b etween A0,e and Re , due to the appreciable non-s-channel contributions, as discussed in App endix B. FB 11.2.1 Error comp osition and 2

Table 33 shows the approximate error comp osition for the model-indep endent Z parameters and for 0 the derived parameters which are discussed b elow. Only in the case of h and R do systematic uncertainties exceed the statistical errors. For Z and the asymmetries the statistical errors are still much larger than the systematics. This parameter set also b enefits from a division of the ma jor systematic uncertainties. The LEP energy mainly affects mZ and Z , the luminosity uncertainty enters 0 nearly exclusively in h , and t-channel uncertainties are isolated in Re and A0,e . Such a separation of FB effects greatly facilitates the combination of the results with the other LEP exp eriments, since these are the uncertainties which are common to all exp eriments. For other p ossible parametrisations, e.g., in terms of partial widths, C -parameters or the S-matrix, the systematic effects on the parameters are much more p oorly separated. The 2 values of our fits are somewhat lower than exp ected, for example in the 9-parameter modelindep endent fit 2 = 155.6 with 194 degrees of freedom.9 The probability to obtain 2 155.6 is 2%. The low value of 2 can b e attributed predominantly to anomalously low statistical fluctuations in the data with lower precision taken in the early phase of LEP running (1990-1992). It was already present in our earlier publications [7-9]. A fit to the 1993-1995 data alone results in 2 /d.o.f . = 34.2/40, where the probability to obtain a lower 2 is 27%. 11.2.2 Theoretical uncertainties

The relatively large effects from theoretical uncertainties in the luminosity determination and the e+ e- e+ e- t-channel correction are discussed elsewhere in this pap er; these are already treated in
For determining the numb er of degrees of freedom one needs to account for the six sets of "pseudo-cross-sections" (Section 8) and for the fact that we allow the absolute normalisation of the cross-sections and the energy scale to float in the 1990 data. Therefore the numb er of effective measurements is reduced from 211 to 203.
9

46


the fit procedure and included in the quoted parameter errors. Additional uncertainties arise from the deconvolution of initial-state radiation (ISR), p ossible ambiguities in the precise definition of the pseudo-observables and residual dep endence on the values of the SM parameters which we have assumed. ISR corrections substantially affect the Z lineshap e. In the fitting programs ZFITTER and TOPAZ0, their implementation is complete to O(2 ) and includes the leading O(3 ) terms. In a recent evaluation [69] different schemes have b een compared and the effect of missing higher order 0 terms estimated; the residual uncertainties are limited to 0.004 nb for h , 0.1 MeV for mZ and Z , and are negligible for other parameters. A related effect is the correction for fermion-pair radiation in the initial state. Although this correction is ab out two orders of magnitude smaller than the ISR correction it gives rise to somewhat larger uncertainties. Comparing different schemes and implementations10 0 leads to uncertainties of 0.3 MeV for mZ , 0.2 MeV for Z and 0.006 nb for h . We further investigated p ossible differences in the definition of pseudo-observables or their implementation by comparing the programs ZFITTER and TOPAZ0. In a very detailed comparison [56] the authors of the two packages demonstrated the good overall consistency in their implementations of radiative corrections and SM calculations. In addition, we evaluated differences b etween the two programs by using one to calculate a set of cross-sections and asymmetries and using the other to re-fit this set. The effective differences in terms of pseudo-observables are 0.2 MeV for mZ , 0.1 MeV for Z , 0 0.003 nb for h , 0.004 for R , and 0.0001 for A0, . The difference in R is the only notable effect; it FB corresp onds to 10% of the exp erimental error. Further effects from missing higher order electroweak corrections have also b een studied. Such corrections are small and affect the pseudo-observables only through the tiny SM remnants. Numerically, the changes are found to b e negligible. Finally, we evaluated the dep endence of the fitted pseudo-observables on the SM parameters (m2 ), Z s (m2 ), mt and mH . The values assumed for these parameters affect the fitted pseudo-observables Z mainly through the /Z interference which is taken from the SM. The effects, however, are small; the only notable impact is on mZ , which changes by 0.2 MeV when varying mH from 90 to 1000 GeV. The overall theoretical uncertainties for the Z resonance parameters from all these sources are given in the last column of Table 33. These have a negligible effect on our results, since in all cases 0 the theoretical error is much less than the total uncertainty; the largest effects are for h with 15% and mZ and R with 10% of the total error.

11.3

Interpretation

The model-indep endent Z parameters were chosen to b e closely related to the measured quantities and have minimal correlations with each other. These parameters can easily b e transformed to other equivalent sets of physical quantities for the interpretation of our results. 11.3.1 Z decay widths

0 From the model-indep endent Z parameters Z , h , Re , R÷ , and R the partial Z decay widths inv , ee , ÷÷ , , and had can b e derived, and are shown in Table 34. Here inv is the width of the Z to final states not accounted for in the analysis of e+ e- e+ e- , ÷+ ÷- , + - and qq processes discussed in the previous sections



inv

Z - ee -

÷÷

-



-

had

.

(27)

The correlations b etween the partial widths are large and are given in Tables 35 and 36. In Table 34 good agreement is seen among the measured lepton partial widths and with the SM exp ectations. By assuming lepton universality, our measurement of inv b ecomes more precise.
We rep eated the fits varying the ZFITTER flags ISPP=2,3,4 which select the parametrisations of [55] and two variants of [70]. The maximal difference is taken as the error.
10

47


A quantity sensitive to a p ossible deviation of the data from the SM prediction for the invisible width is the ratio inv Rinv . (28) In this ratio exp erimental errors partially cancel and theoretical uncertainties due to the assumed values of mt and mH are strongly reduced. In the SM only neutrinos contribute to inv . The first Z resonance measurements at SLC and LEP in 1989 [71] demonstrated the existence of three generations of light neutrinos. For three generations one exp ects
SM Rinv = 3

= 5.974 + 0.004 ,
H

where the uncertainty corresp onds to variations of mt = 175 + 5 GeV and 90 < m Taking into account the correlation b etween inv and our measured value is Rinv = 5.942 + 0.027 . Dividing Rinv by the SM exp ectation for a single generation, / , gives N = 2.984 + 0.013 .

< 1000 GeV.

The measurements of the total and partial Z decay widths give imp ortant constraints for extensions of the SM which predict additional decay modes of the Z into new particles. From the difference b etween measured widths and the SM predictions (assuming three neutrino sp ecies) one can derive upp er limits for such contributions from new physics. New particles could contribute either to one of the visible decay channels in the partial widths or to the invisible width, dep ending on their sp ecific prop erties. In order to calculate the upp er limits for such extra contributions to the widths we added the exp erimental and theoretical errors in quadrature. For the latter we evaluated the change of the predicted widths when varying the SM input parameters, mZ , mt , s (m2 ) and (m2 ), within their Z Z exp erimental precision (Equation 25). For the mass of the Higgs b oson we used 1000 GeV. This results in the smallest theoretical predictions for the widths and therefore in the most conservative limits. We obtain new < 14.8MeV and new < 3.7 MeV (29) Z inv as one-sided upp er limits at 95% confidence level for additional contributions to the total (new ) and Z the invisible width (new ). The limits for mH = 150 GeV and for the visible partial widths are given inv in Table 37. All limits are Bayesian with a prior probability which is uniform for p ositive new . x 11.3.2 Coupling parameters and the effective mixing angle

From the measured p ole asymmetries A0, it is straightforward (Equation 24) to determine the couFB pling parameters A , which quantify the asymmetry of the Z-lepton coupling for each lepton sp ecies. The results for the three lepton sp ecies are consistent with each other and agree well with the SM predictions, as shown in Table 38. However, since the measured A0,÷ and A0, are products of Ae with FB FB either A÷ or A , large anti-correlations (Table 39) arise b etween Ae and A÷ , A . An equivalent formulation of the coupling asymmetries can b e made in terms of the effective weak lep mixing angle sin2 eff t . Assuming lepton universality we obtain sin2
lept eff



1 4

1-

gV gA

= 0.2325 + 0.0010 ,

(30)

which is in good agreement with the world average of 0.23151 + 0.00017 [3]. 48


11.3.3

Vector and axial-vector couplings

Using the set of the model-indep endent Z parameters one can determine the leptonic vector and axialvector couplings gV and gA for the neutral currents. The results are given in Table 40. There are strong anti-correlations b etween the e+ e- couplings on one side and the ÷+ ÷- and + - couplings on the other, and also b etween vector and axial-vector couplings for each lepton sp ecies. This is shown in Table 41 and illustrated in Figure 24. When lepton universality is imp osed the anti-correlation b etween gV and gA is reduced to -29%. Good agreement is found with the prediction of the SM. Being determined primarily from the lepton cross-sections, gA has a small relative error. Ratios of couplings provide a p owerful test of lepton universality for the axial-vector couplings at the 0.5% level: gA÷ +0. = 0.9989-0. gAe
0033 0058

,

gA +0. = 1.0003-0. gAe

0037 0055

,

gA +0. = 1.0014-0. gA÷

0036 0034

.

(31)

For the vector couplings the relative uncertainties are much larger. They are also very asymmetric and strongly correlated b etween lepton sp ecies. Because the observables A0,÷ or A0, are each products of FB FB gV÷ or gV and gVe , the derived values of gV÷ and gV diverge if gVe approaches zero. This pathology is visible in the strongly asymmetric errors in gV÷ and gV and the corresp onding tails in Figure 24. When we form the vector coupling ratios the non-Gaussian b ehaviour of the uncertainties is further enhanced for gV÷ /gVe and gV /gVe , while gV /gV÷ has much smaller and rather symmetric errors: gV÷ +1. = 1.79-0. gVe
84 64

,

gV +1. = 1.63-0. gVe

72 61

,

gV +0. = 0.91-0. gV÷

25 21

.

(32)

One should emphasise that the non-linearities in the determination of gV are driven by the observed, statistically limited, measurement of A0,e . Our value of A0,e is small and only two standard deviations FB FB ab ove zero, although compatible with the SM prediction. The problem is less apparent and the propagated errors in the vector couplings are significantly reduced when A0,e happ ens to b e large, as FB is the case for example in [10]. The pure Z pseudo-observables R and A0, are symmetric in gV and gA , as can b e seen from FB Equations 14 and 16. The results would not change if the values of gV and gA were interchanged. Our measurements of the C -parameters describing the /Z interference (Equation 10) can b e used to s a resolve this ambiguity. C Z parametrises the interference contribution symmetric in cos and C Z the strong s dep endence of the asymmetries (Equation 15). The results of the fit, as given in Table 26, a a distinguish gV and gA unambiguously. C Z and CZZ also determine the signs of gA÷ and gA relative to gAe (and the signs of gV÷ and gV relative to gVe ). The last remaining ambiguity, which the lineshap e and asymmetry data alone cannot resolve, is the relative sign11 of gV and gA . This can b e determined by measurements of the polarisation asymmetries [72] or the left-right asymmetry [73], which determine directly the ratio gV /gA for a sp ecific flavour. 11.3.4 s from the Z resonance parameters

As discussed in Section 10.2 the hadronic partial decay width, had , is increased by final state QCD corrections, in first order by a factor 1 + s . In ZFITTER corrections up to O(3 ) are included. s The largest contribution to Z is from had . As a result several Z resonance parameters are sensitive to s , namely the total width Z directly, the ratio of hadronic to leptonic partial width 0 R , the hadronic p ole cross-section h and finally the leptonic p ole cross-section 0 (obtained by 0 transforming the results in Table 29, 0 = 1.9932 + 0.0043 nb). From the three observables R , h , 0 only two are indep endent and the b est constraint on is obtained in a simultaneous fit to all s parameters. This is discussed in Section 11.4. However, from the exp erimental p oint of view rather different effects dominate the uncertainty of each of these observables, and it is therefore instructive to examine the value of s derived from each individual observable. The measurement via Z is statistics
The absolute sign of one of the axial-vector or vector couplings needs to b e defined. By convention one takes g negative.
11 Ae

as

49


limited and free of normalisation errors, that via R is indep endent of the luminosity and limited by 0 lepton statistics and systematics, while for the measurement via h statistics, selection systematics and luminosity uncertainties contribute ab out equally, and finally, the measurement via 0 is free of any hadronisation uncertainties. Figure 25 shows these observables and the SM prediction as implemented in ZFITTER as a function of s . Also shown are the effects of varying the parameters mt , mH and (m2 ). The resulting values Z for s are listed in Table 42. They are in good agreement within the exp erimental errors. 0 Since 0 , h and R are ratios of partial or total widths, the values of s derived through these parameters are rather insensitive to variations of the SM parameters, which affect , had and Z similarly, while the value derived through Z dep ends more strongly on mt and mH . It is interesting to note that s from 0 , a measurement relying entirely on leptonic final states, has by far the smallest uncertainty. This somewhat counter-intuitive result is due to the fact that 0 dep ends quadratically on Z and is therefore most sensitive to the effect of s on had , through the dominant contribution of the latter to Z . QCD uncertainties on the determination of s (m2 ) from the Z resonance parameters are small. Z The dominant effect comes from the unknown terms of O(4 ) and higher. However, several evaluations s exist in the literature, which result in rather different error estimates. In Reference [58], which is the basis of the ZFITTER QCD corrections, the renormalisation scale dep endence (1/4 < ÷2 /m2 < 4) of Z the massless non-singlet terms entering had translates into -0.0002 < s < +0.0013 for s = 0.125. Other contributions, such as the renormalisation scheme or the uncertainty of the b quark mass, are below +0.0005. In Reference [64] the renormalisation scale dep endence of R is evaluated based on an effective parametrisation of R as function of s . Using a wider range (1/16 < ÷2 /m2 < 16) results in Z -0.0004 < s < +0.0028. In Reference [74] a similar scale dep endence of s (m2 ) from R is found. Z However, the authors then resum additional terms (" 2 terms") in the p erturbative expansion of R , which leads to a slight correction of +0.6% for the resulting s (m2 ) and reduces the renormalisation Z scale uncertainty to +0.0005. Given the small size of this correction, and to remain consistent with the other LEP collab orations, we do not correct our result obtained from ZFITTER and assign a conservative systematic error of +0.002 as the QCD uncertainty for s derived from the Z resonance parameters, which is still significantly smaller than the exp erimental precision. In principle, s dep endent propagator corrections enter differently in the various Z resonance observables.12 However, since these corrections are much smaller than the dominating s correction to had , these differences can b e safely neglected for the overall QCD uncertainty.

11.4

Standard Model fits

As a last step we compare our measured cross-sections and asymmetries directly with the full SM calculations implemented in ZFITTER, using as fit parameters mZ , s , mt and mH . In addition, the (5) electromagnetic coupling constant is constrained by had = 0.02804 + 0.00065 [50]. The lineshap e and asymmetry measurements alone are not sufficient to determine simultaneously mt , mH and s . The leading electroweak radiative corrections in terms of m2 and - log(mH ) have the same form for all t fermion partial widths and lepton asymmetries. The only exception is the b-quark partial width, which has unique mt dep endent vertex corrections. This leads to a somewhat different mt dep endence for the inclusive hadronic width, had . However, this p otential discrimination p ower is absorb ed by s when it can vary freely in the fit. Therefore additional constraints on these parameters or supplementary electroweak precision measurements are needed for a simultaneous fit to mt , mH and s . We choose to restrict ourselves to three scenarios: (i) determination of s and mt from the lineshap e and asymmetry measurements alone, restricting mH to the range 90-1000 GeV, with a central value of 150 GeV; (ii) determination of mH and s , using the direct Tevatron measurement of the top quark
The residual differences are caused by QCD effects in terms involving the top-quark mass which almost completely cancel in quantities that dep end on the ratio of widths. At present, uncertainties in these effects [75] are equivalent to an error 1 GeV in mt , and are much smaller than the uncertainty on the measured top mass.
12

50


mass, mt = 174.3 + 5.1 GeV [63], as an external constraint; and (iii) determination of mH alone, using in addition to the mt constraint also the recent world average of s = 0.1184 + 0.0031 [64]. The results of these fits are shown in Table 43. Our measured cross-sections and asymmetries are consistent with the SM predictions as indicated b oth by the absolute 2 /d.o.f . and by its small change with resp ect to the model-indep endent fits. In fit (i) we determine a value of mt = 162 + 15+25 (mH ) GeV through the indirect effect of radiative corrections, which agrees well -5 with the direct measurement. This agreement is a sensitive validation of the electroweak loop corrections. In fit (ii) we obtain s (m2 ) = 0.127 + 0.005 + 0.002 ( QCD) (33) Z for the strong coupling constant, which is consistent within ab out 1.5 standard deviations with the world average. Also in fit (ii) we find mH = 390+750 GeV as the mass of the Higgs b oson. In fit -280 (iii), when we further use the external value of s , the fitted result for the Higgs mass moves lower, to mH = 190+335 GeV, due to the correlation b etween s and mH . The correlation is illustrated in -165 Figure 26 which shows the 68% C.L. contour of s and mH for fit (ii). Since the leading radiative corrections dep end logarithmically on mH the uncertainty of mH is very asymmetric. But even in terms of log mH the error is asymmetric. This is due to the fact that the fit allows a wide range for mH but only for mH mW is the logarithmic dep endence a good approximation. It is imp ortant to verify that p erforming a SM fit on the level of pseudo-observables is equivalent to fitting the cross-sections and asymmetries directly. Using the results of the 9-parameter modelindep endent fit (Tables 29 and 31) as input to the SM fit yields consistent results: the central values for the SM parameters agree within 10% of the error; uncertainties and correlations are indistinguishable within the quoted precision. This is illustrated in Figure 26 which also shows the 68% contour from the pseudo-observable fit. When these results are combined with other electroweak measurements [3] to determine SM parameters, the fits are p erformed at the level of pseudo-observables.

12

Summary and conclusions

We present here the OPAL analysis of the Z cross-section and lepton forward-backward asymmetry measurements. This analysis includes the entirety of the OPAL data sample taken near the Z resonance from 1990 to 1995. In addition to the nearly four-fold increase in statistics since our last publication [9] many improvements have b een made in the exp erimental and theoretical systematic uncertainties of the measurements. One imp ortant contribution is the reduction of the exp erimental error of the luminosity determination [13] by more than a factor of ten. New techniques to evaluate systematic effects of the event selections more than halved this source of uncertainty for the hadronic cross-section. Similar progress has b een made in the precision of the external inputs to our measurements, namely the precise determination of the LEP centre-of-mass energy [43] and the theoretical calculation of the luminosity cross-section [46, 47], as well as in the theoretical tools and programs used to evaluate and interpret our results [49, 52, 56]. Overall, these coherent efforts allowed us to exploit most of the inherent statistical precision of our 4.5 × 106 measurable Z decays. In a model-indep endent ansatz we parametrise indep endently the contributions from pure Z exchange and from /Z interference. Our results are in good agreement with lepton universality and consistent with the vector and axial-vector couplings b etween the Z and fermions as predicted in the SM. Our main results in terms of Z resonance parameters, assuming lepton universality, can b e summarised as: m
Z

= 91.1852 + 0.0030 GeV , = 2.4948 + 0.0041 GeV , = 41.501 + 0.055 nb , = 20.823 + 0.044 , = 0.0145 + 0.0017 . 51

Z 0 h

R A0, FB


Transforming these parameters yields the ratio of invisible to leptonic decay width,
inv

/ = 5.942 + 0.027 .

Assuming the SM couplings for leptons this can b e converted into a measurement of the effective numb er of light neutrino sp ecies N = 2.984 + 0.013 . Alternatively, one can use the SM prediction of inv for three neutrino generations and derive an upp er limit for additional contributions from new physics to the invisible or the total Z width:
new inv

< 3.7MeV

or



new Z

< 14.8 MeV

at 95 % C.L.

Finally, we compare our measured cross-sections and asymmetries with the full SM calculations. Radiative corrections are sensitive to the parameters mt , mH and s (m2 ), allowing our measurements Z to determine: m
t

= 162 + 15+25 (mH ) GeV , -5
004 001

+0. s (m2 ) = 0.125 + 0.005-0. Z

(mH ) + 0.002 ( QCD) ,

where we fixed mH = 150+850 GeV. Including the direct measurement of the top quark mass (mt = -60 174.3 + 5.1 GeV [63]) as an additional constraint we obtain results for s (m2 ) and the mass of the Z Higgs b oson: s (m2 ) = 0.127 + 0.005 + 0.002 ( QCD) Z m
H

= 390+750 GeV . -280

These Z lineshap e and asymmetry measurements test and confirm the SM at the level of quantum loop corrections and set tight constraints on new physics.

Acknowledgements
We particularly wish to thank the SL Division for the efficient op eration of the LEP accelerator, for the painstaking calibration of the b eam energy and for their continuing close coop eration with our exp erimental group. We would also like to extend sp ecial thanks to the community of theorists whose contributions to precise electroweak calculations we cite, and whose efforts have made our own meaningful. We thank our colleagues from CEA, DAPNIA/SPP, CE-Saclay for their efforts over the years on the time-of-flight and trigger systems which we continue to use. In addition to the supp ort staff 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, Japanese Society for the Promotion of Science (JSPS), German Israeli Bi-national Science Foundation (GIF), Bundesministerium fur Bildung und Forschung, Germany, Å National Research Council of Canada, Research Corp oration, USA, Hungarian Foundation for Scientific Research, OTKA T-029328, T023793 and OTKA F-023259.

52


App endices A Four-fermion pro cesses and radiative photon interference

Small radiative corrections to fermion-pair production due to four-fermion final states and the interference b etween initial- and final-state photon radiation are not treated by the fermion-pair Monte Carlo generators, JETSET and KORALZ, which we used in determining our event selection efficiencies. We describ e here, in some technical detail, how we derived the small acceptance corrections arising from these effects.

A.1

Treatment of four-fermion final states

The selection criteria for hadronic and lepton-pair events define samples which are primarily the result of fermion-pair production processes e+ e- f f . Studies have b een made of the much smaller contributions arising from four-fermion final states e+ e- f fFF, as shown in Figure 2. Some classes of four-fermion events have little connection to Z production, and can prop erly b e considered as background, for example multi-p eripheral (d) or pair corrections to t-channel scattering (e,f ). Other four-fermion events (b,c) however, must b e considered as radiative corrections to the fermion-pair production (a) which interests us. In the following we use the designation f f to refer to the primary fermion pair of interest and FF to refer to the radiated pair, although a rigid distinction b etween the pairs cannot always b e made. From the theoretical p oint-of-view it is desirable to design the selection of fermion pairs in such a way that no stringent cuts are imp osed on events with photon radiation and pair production. One then b enefits from cancellations b etween real and virtual corrections leading to smaller uncertainties in the theoretical calculations. In practice, however, the event selection efficiency for events with radiated pairs can b e exp ected to b e different from the efficiency for events without such radiation, and this efficiency cannot b e evaluated using the standard Monte Carlo event generators for fermionpair production since they do not in general include such four-fermion final states. Studies have b een made following the approach describ ed in [76] using a sp ecially generated sample of Monte Carlo four-fermion events. An unambiguous separation of signal and background four-fermion events is not p ossible due to the interference of amplitudes which lead to the same final states. We therefore adopt a practical signal definition based on kinematics. Typically the fermion pair (f f ) from Z decay is of high invariant mass and the radiated pair (FF) is of low mass. Four-fermion final states from t-channel and multip eripheral diagrams tend to occupy a region of phase space well separated from the signal events of interest. We ignore the interference b etween s-channel and t-channel diagrams and generate separate four-fermion Monte Carlo samples: four-fermion events from s-channel diagrams are generated using the FERMISV program, while those from the t-channel diagrams are generated using the grc4f (version 1.11) and Pythia programs. For channels other than e+ e- e+ e- , (e+ e- f f, with f = e), we identify four-fermion events from s-channel diagrams as part of the fermion-pair signal if they satisfy mf f > mFF and m2f /s > 0.01. f The second requirement is common to our definition of the fermion-pair signal phase space. All other four-fermion events arising from s-channel diagrams failing these requirements, as well as those from t-channel diagrams and the two photon process, are regarded as background. In case of e+ e- e+ e- , four-fermion final states are considered as signal if the final-state electronpair meets the normal requirements for the e+ e- e+ e- signal in terms of electron energy, acollinearity and electron p olar angle. Here, four-fermion events from t-channel diagrams are treated as signal. In principle even contributions from Figure 2(e) where the lower b oson is a virtual Z are considered as signal. The numb er of such events falling within our ideal acceptance, however, is negligible. Including four-fermion events leads to a correction of the effective signal efficiency of = -
f fFF



(f f - f

fFF

),

(34)

53


where is the total cross-section for f f production including the effects of pair-production, and f fFF is the cross-section for four-fermion events f fFF defined as signal. The selection efficiency calculated using the fermion-pair Monte Carlo is denoted by f f , and the selection efficiency calculated for the four-fermion signal contributions is f fFF . We exp ect the largest effect in the case of the e+ e- ÷+ ÷- selection, where the requirement that the numb er of tracks b e exactly two explicitly excludes all the visible four-fermion final states, even those which need to b e counted as part of the signal. Here we find that the efficiency for signal four-fermion final states is typically 20-30% for e+ e- ÷+ ÷- ÷+ ÷- , ÷+ ÷- qq, ÷+ ÷- + - , and ab out 80% for e+ e- ÷+ ÷- e+ e- , but the resulting correction to the e+ e- ÷+ ÷- efficiency is only ab out 0.1%. Background from the t-channel diagrams gives a very small contribution of 2×10-5 , and background four-fermion events from s-channel diagrams give an even smaller contribution. We find a correction of similar size for the e+ e- + - selection. In the e+ e- e+ e- selection no tight multiplicity requirement is made. The efficiency for signal four-fermion events is found to b e high (over 90%). As a result, the efficiency correction is nearly cancelled by the small background contributions, and no overall correction due to four-fermion final states is necessary. In the case of e+ e- qq the event selection is very inclusive so that there is no notable effect on the efficiency. We subtract a very small four-fermion background contribution of 0.4 × 10-4 . The theoretical programs ZFITTER and TOPAZ0 which we use to compute exp ected cross-sections and asymmetries treat pair radiation inclusively in the pair mass and do not include contributions from virtual Z b osons. Therefore the four-fermion signal definition as describ ed here cannot b e exactly mapp ed to these calculations, e.g. the separation b etween signal and background based on the mass of the pairs is not made in these programs. However the differences are quantitatively negligible [77].

A.2

Initial-final state interference

The effect of interference b etween initial and final state photon radiation is in general strongly suppressed at the Z resonance. In view of the high precision measurements presented here, however, we studied several p ossible effects. The main effect of initial-final state interference is a change in the angular distribution, prop ortional to the value of the differential cross-section, but of opp osite sign in the forward and backward hemispheres, giving an increasing effect towards cos = +1. The size of the effect dep ends on the cuts, mainly on s or m : in general the tighter the cut on the energy of radiated photon, the larger the effect. Initial-final state interference is not included in the Monte Carlo event samples used to calculate the event selection efficiency for the e+ e- qq, ÷+ ÷- , or + - processes, while it is included in our use of ZFITTER. We therefore do not remove its effects in our measured cross-sections and asymmetries but need to make small corrections to our calculated selection efficiencies to account for its absence in the Monte Carlo samples. Since the ma jor source of the event selection inefficiency is due to limited angular acceptance near | cos | = 1 and reduced acceptance at small s (m ), the missing initial-final state interference in these Monte Carlo samples can cause some bias in the acceptance extrap olation. The effect of this missing initial-final state interference was evaluated using the program ZFITTER, for which this effect can b e switched on and off. To incorp orate the event selection efficiency in the calculation, a matrix of efficiency resolved in bins of cos and the invariant mass of the final state lepton pair, m2 , was calculated using the KORALZ Monte Carlo sample. The exp ected observed cross-section was obtained by multiplying the efficiency in each bin by the corresp onding differential cross-section calculated using ZFITTER, and then summing over the full phase space. The overall efficiency was calculated as the ratio of the observed to the total cross-section. Two calculations were p erformed, one with and one without initial-final state interference, and the two results compared. For b oth e+ e- ÷+ ÷- and e+ e- + - the effect of missing initial-final state interference is found to b e small (at 10-4 level) as a result of the large acceptance (in b oth cos and m2 ) of b oth these selections. For e+ e- qq, the effect of initial-final state interference is more complicated to calculate due to the presence of a mixed QED and QCD parton shower in the final state. The size of 54


the effect is strongly suppressed, however, by the almost complete acceptance of the e+ e- qq event selection in b oth cos and s (the selection inefficiency is only 0.5%), as well as the smaller size of the quark charges. Without considering the effect of the parton shower, we evaluated the effect to b e less than 10-5 , which we neglect. In the e+ e- e+ e- selection, the exp erimental acceptance and ideal kinematical acceptance are very close. No appreciable phase-space extrap olation which could introduce a significant correction for the effect of initial-final state interference is needed. Therefore no correction is applied to the e+ e- e+ e- selection efficiency.

B

t-channel contributions to e+ e- e+ e-

For the reaction e+ e- e+ e- not only s-channel annihilation but also the t-channel exchange of and Z contribute. Since our primary fitting program (ZFITTER) includes only s-channel processes an external correction is needed to account for the contributions from t-channel exchange diagrams and their interference with the s-channel. To ensure a consistent treatment of the s-channel in all final states we continue to calculate the e+ e- e+ e- s-channel terms using ZFITTER, but proceed as follows. We use the program ALIBABA [61] to calculate the SM prediction for b oth the pure s-channel and the full s+t channel, separately for forward and backward cross-sections. The contributions from t-channel and s-t interference are obtained by subtracting the ALIBABA s-channel cross-section from AL,t AL,s AL,s the full ALIBABA s+t cross-section, F(B) = F(B)+t - F(B) . To these t and s-t contributions we then add the s-channel cross-sections calculated with ZFITTER ( the total cross-section and asymmetry:
ee ZF,s F

,

ZF,s B

) to obtain the predictions for

=

Aee = FB

ZF,s AL ZF AL + F ,t + B ,s + B ,t F ZF AL ZF AL (F ,s + F ,t ) - (B ,s + B ,t ZF AL ZF AL (F ,s + F ,t )+(B ,s + B ,t

) )

.

(35)

The separation into forward and backward cross-sections ensures a correct propagation of errors to the fitted s-channel observables Re and A0,e . The non s-channel contributions lead to a statistical FB correlation of -11 % b etween Re and A0,e . FB The size of the t-channel corrections changes rapidly as a function of centre-of-mass energy, since the s-t interference is prop ortional to s - mZ in the vicinity of the Z p ole. We therefore parametrise AL AL the corrections F ,t and B ,t as a function of ( s - mZ ) in order to account for their variation as mZ converges in the fit. In this way our results for Re and A0,e prop erly resp ect the uncertainty of FB our fitted mZ and the corresp onding correlations. However, one should note a subtle side effect of this treatment. When lepton universality is assumed in the fit the measurement of ee at the p eak contributes to the determination of mZ . The imp osition of lepton universality introduces a tension in the fit since fluctuations induce discrepancies b etween the measured and predicted cross-sections for each lepton sp ecies at the p eak p oint. This tension can b e relaxed in the case of the electron cross-section by shifting mZ since the non-s contributions in the electron channel have a non-zero slop e with energy at the p ole. This effect is resp onsible for the shift of 0.5 MeV of mZ b etween the fits with and without lepton universality (Section 11.2). Theoretical uncertainties of the t-channel correction have b een estimated in [78] and the uncertainties in the forward and backward cross-section have also b een determined separately [79]. Adjusting these estimates to the smaller angular acceptance used in our e+ e- selection results in the uncertainties listed in Table 44 which are separated into three classes according to the centre-of-mass energy (b elow, at and ab ove the p eak). For the fit these uncertainties are translated into the corresp onding uncertainties of the measured ee and Aee and included in the covariance matrix. It is not known to FB what extent the errors b etween the energy p oints and the forward-backward regions are correlated. We tested several p ossibilities - uncorrelated, fully correlated and fully anti-correlated - and found that our fitted parameters are insensitive within the effective t-channel errors to the scenario chosen. 55


For the results presented here we assumed no correlation b etween forward and backward regions, full correlation b etween data p oints within each centre-of-mass energy class and no correlation b etween data p oints of different classes. The effective t-channel theory uncertainties on Re and A0,e amount FB to 0.027 and 0.0015, resp ectively, with a correlation of -0.85. Overall, statistical errors and the theoretical t-channel uncertainties lead to a correlation of -0.20 b etween Re and A0,e . FB

C

Fit covariance matrix and energy spread corrections
-1

For all our fits we p erform a 2 minimisation using the MINUIT [62] package. The 2 is defined as 2 = T C , (36)

where is the vector of residuals b etween the measured and predicted cross-sections and asymmetries and C is the covariance matrix describing the statistical and systematic uncertainties of the measurements and their correlations. In total we have 211 measured p oints, 124 cross-sections and 87 asymmetries. Therefore 22366 comp onents Cij need to b e determined to sp ecify the full matrix C . Due to the large variety of error sources entering the measurements and non-trivial correlations, the construction of the covariance matrix is a rather complex task. In general, each Cij is comp osed of
se Cij = Cij l ,stat

+C

lumi,stat ij

se + Cij l

,syst

lu + Cij mi

,syst

t + Cij-

chan

+C

E ij

cm

+C

E ij

.

(37)

XX The elements Cij are in general constructed from `small' covariance matrices V XX , which describ e the year, energy p oint and final state dep endencies. In the following we give a brief overview of how each uncertainty is treated. The indices i and j refer to the 211 measured p oints; k and l to the k sp ecific element in the V matrices. So the notation i refers to the i-th cross-section which is related to the row or column k in the matrix V . Statistical errors of the cross-sections (i ) are determined by the numb er of selected events, Nsel , and corrected for estimated background, Nbg . They enter only the diagonal elements


se Cii l,stat

=

N

sel

Nsel -N

2


bg

i

.

(38)

Luminosity statistical errors are common to all cross-sections for a sp ecific running p eriod. They are calculated as lu lu kk Cij mi,stat = (k mi,stat )2 i j , (39)
lu k where k mi,stat is the relative statistical luminosity error and i,j refers to the four cross-sections in each running p eriod k. Systematic errors from the event selection affect mostly a sp ecific final state. A large fraction of this error is fully correlated but there are also comp onents which are indep endent or only partially correlated for different running p eriods or energy p oints. Therefore these errors are themselves sp ecified in a covariance matrix of relative errors, V sel . We use three such matrices, one for the hadron crosssections (Table 17), one for the three leptonic cross-sections (Tables 18-21), and one for the three lepton asymmetries (Table 25). In this way correlations among the three lepton sp ecies are accounted se for. Cij l,syst is given by

C

sel,syst ij

=V

sel, k,l

k i

l j

or

se Cij l

,syst

=V

sel,A k,l

FB

.

(40)

There is no correlation b etween the exp erimental errors for the cross-sections and those for the asymmetries. Luminosity errors include the exp erimental systematics and the theoretical uncertainties of the luminosity measurement. They affect all cross-section measurements, but similar to the selection 56


errors there are comp onents which are only partially correlated b etween running p eriods or energy p oints. Therefore a covariance matrix of relative luminosity errors, V lumi , is also used (Table 16):
lu Cij mi ,syst

=V

lumi k,l

kl i j .

t-channel errors refer to the theoretical uncertainty in the t-channel correction for e+ e- crosssections and asymmetries. These are sp ecified in terms of forward and backward cross-sections, F and B (Table 44). Different data p oints are only correlated if b oth energies are either b elow, at, or t ab ove the p eak (App endix B). Then Cij-chan is given by:
i ee



j ee

:

t Cij- t Cij-

chan chan

k k = (F )2 +(B )2

(41)
k (F )2 +

Aee,i Aee,j : FB FB
i ee

= =

1 1



Aee,j FB

:

t Cij-chan

- Aee,i FB i ee - Aee,j FB j ee

1

- Aee,j FB j ee
k2 F)

1+ Aee,i FB i ee
k (B )2

1+ Aee,j FB j ee

k (B )2

(

-

1+ Aee,j FB
j ee

Centre-of-mass energy errors have b een determined in [43] and are sp ecified as a covariance
matrix V Ecm (see Tables 13 and 14). They are transformed into the corresp onding cross-section or asymmetry errors using the derivatives C
E ij
cm

=V

Ecm kl

l k d Oi d Oj dE dE

(O = or AFB ).

(42)

The slop es d/dE and dAFB /dE are determined numerically using ZFITTER ( ZF ) in an iteration during the fit. For cross-sections the dep endence of the luminosity on the centre-of-mass energy must also b e taken into account. Since this is a tiny effect it is sufficient to consider only the dominant 1/E 2 dep endence of the low-angle Bhabha cross-section, neglecting small distortions due to the /Z interference. Combining the predicted slop e and the luminosity dep endence yields
ZF d i d i i . = +2 dE dE Ei

(43)

Beam-energy spread of the electrons and p ositrons in LEP leads to a disp ersion of the centre-ofmass energy with a width Ecm 50 MeV. Therefore the measured cross-sections and asymmetries do not corresp ond to a sharp energy, Ei , but form a weighted average around Ei + Ecm which can b e shifted from the value exactly at Ei . A similar effect is caused by the fact that many LEP fills are combined for each data p oint. These fills are not at precisely the same energy but scatter by typically 10 MeV around the average. Moreover, within a fill the energy also varies by several MeV. These two effects need to b e added in quadrature with the intrinsic LEP Ecm . Correction terms are determined according to
spr i

=- =-

1 d2 2 d E2

2 E i

cm

(44)
2 E i

spr Ai FB

1 d2 AFB 1 d d AFB + 2 d E2 dE dE

cm

and added to the measured i and Ai . The first and second derivatives of and AFB are again FB determined numerically with ZFITTER during the fit. The spread Ecm has typically an uncertainty of ab out 1.2 MeV, which is largely correlated b etween years and energy p oints and sp ecified in detail in the energy spread covariance matrix V E (Table 15). It enters the fit covariance matrix as
CijE = 4 spr i



spr j

E kl k l Ecm E

V

.
cm

(45)



spr i

refers to the corrections

spr i

and

spr Ai FB

in Equation 44. 57


D

S-Matrix results

The S-Matrix formalism [68] is an alternative phenomenological approach to describ e the s-channel reaction e+ e- f f by the exchange of two spin-1 b osons, a massless photon and a massive Z boson. 0 The lowest-order total cross-section, tot , and forward-backward asymmetry, A0 , are given as: fb
0 a (s) = a a 4 2 gf j a (s - m2 )+ rf s Z +f 2 3 s (s - m2 )2 + m2 Z Z Z 0 3 fb (s) , 0 4 tot (s)

for a = tot, fb

(46) (47)

A0 (s) = fb

where s is the centre-of-mass energy. The S-Matrix ansatz uses a Breit-Wigner denominator with s-indep endent width for the Z resonance, s - m2 + imZ Z . As discussed in Section 10.2 the two Z Breit-Wigner forms with s-indep endent width and s-dep endent width, resp ectively, are equivalent. They differ only in that the definition of the mass and width, mZ and Z are shifted with resp ect to mZ and Z . Note that in order to avoid confusion with different definitions we quote values for mZ and Z , rather than mZ and Z , by applying the resp ective transformations. The S-Matrix parameters rf , jf and gf , which are real numb ers, describ e the Z exchange, Z interference and photon exchange contributions, resp ectively. For the latter (gf ) the QED prediction is in general used. The parameters rf and jf are identical at tree-level to the C -parameters introduced in Section 10.1, apart from constant factors:
tot = 2 C s rf ZZ tot = 2C s jf Z G m2

, ,

r j

fb f fb f

a = 42 CZZ a, = 2C Z

(48)

FZ where = 22 1.50. When radiative corrections are considered the relation is less direct; the SMatrix parameters absorb by definition all electroweak and final state corrections. Further differences are caused by the treatment of the imaginary comp onents in (m2 ) and the couplings GAf and GV f . Z In the S-Matrix approach the qq final states are also parametrised in terms of rq and jq . For the inclusive hadronic final state these are summed over all colours and op en quark flavours to yield the corresp onding parameters rhad and jhad . The results of the full fit with 16 parameters are given in Table 45. The 12 parameters describing the three lepton flavours are consistent with lepton universality and we find overall good agreement with the SM exp ectations. The interpretation suffers, however, from the large correlations b etween the fitted parameters (Table 46). As discussed in Section 11.1 the precision of the fitted mZ is much reduced when the hadronic interference, jhad , is treated as a fit parameter; the correlation coefficient b etween mZ and jhad is -0.96. For comparison, we p erformed a fit with the hadronic interference fixed to the SM prediction. The results are shown in Table 45. This parametrisation is in practice equivalent to the C -parameters (Table 26); we transformed the C -parameters into S-Matrix parameters, accounting for the differences in the treatment of radiative corrections, and obtained consistent results at the level of 1 - 2 % of the errors.

58


Year

1990

1991

1992 1990-1992

1993

1994

1995

1993-1995 Grand

Data sample peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 p eak Total prescan(a) prescan(b) p eak-2 peak peak+2 peak(ab) p eak(c) peak(d) prescan(a) prescan(b) p eak-2 peak peak+2 Total Total

s (GeV) 88.22 89.23 90.23 91.22 92.21 93.22 94.22 91.25 88.48 89.47 90.23 91.22 91.97 92.97 93.72 91.30 91.14 91.32 89.45 91.21 93.04 91.22 91.43 91.22 91.80 91.30 89.44 91.28 92.97



L dt (pb-1 ) 0.5 0.6 0.4 3.5 0.5 0.6 0.6 5.1 0.7 0.8 0.9 3.0 0.8 0.6 0.9 24.9 44.4 0.3 5.3 8.5 8.8 9.0 50.1 0.4 2.3 0.2 9.9 8.4 4.6 8.9 116.7 161.1

Nhad 2229 5322 7045 103664 10412 6848 4373 156592 3646 7991 16011 92025 20353 8356 9404 733059 1187330 9905 162218 85727 265494 125320 1520277 11255 69062 5941 300676 84236 140749 127707 2908566 4095896

N

ee

N

÷÷

N



169 306 320 3363 271 203 128 5624 297 451 683 3365 566 325 284 23998 40353 345 5256 4595 8766 3549 49142 345 2170 178 9642 4407 4623 3651 96669 137022

109 231 316 4834 527 308 202 7563 176 363 744 4422 916 478 404 32492 54085 454 7139 3884 10871 5521 67791 500 3060 259 13401 3768 6338 5696 128682 182767

81 214 221 3563 364 260 161 6059 166 289 569 3603 734 436 359 27036 44115 370 6002 3336 9712 4612 55886 381 2478 191 11049 3185 5262 4876 107340 151455

Table 1: Summary of the data samples used for the cross-section measurements, showing the numb ers of selected events for each final state at each energy p oint and the integrated luminosities ( L dt) for the e+ e- qq analyses. The integrated luminosities for the other final states vary within ab out 1% due to different requirements on the status of the detector p erformance for the various event selections. For the leptonic forward-backward asymmetry measurements the data samples available for analysis are generally larger since there is no reliance on the op eration of the luminometers. The p eak data from 1994, and from the 1993 and 1995 prescan data samples, have b een divided into subsets corresp onding to data-taking p eriods characterised by significantly different mean values of s.

59


Measurement cross-sections e+ e+ e+ e+ e+ e+ e+

asymmetries

process - qq e e- ÷+ ÷- e- + - e- e+ e- e- e+ e- e- ÷+ ÷- e- + -

Emin (GeV) - - - 0.2 0.2 6.0 6.0



- - - 10 10 15 15

max acol

| cos - |max 1.00 1.00 1.00 0.70 0.70 0.95 0.90

(s /s)min 0.01 - - - - - -

(m2f /s)min f - 0.01 0.01 - - - -

Table 2: The measured cross-sections and asymmetries are corrected to corresp ond to ideal regions of phase-space adapted to theoretical calculations. The phase-space is defined by the maximum | cos - | in which the fermion must fall, and, either the minimum final-state energy fraction, or a combined max requirement on the minimum fermion energy, Emin (GeV), and acol where acol is the acollinearity - , where is the op ening angle b etween the directions of angle of the fermion pair, defined as 180 the two fermions. The final-state energy fraction is defined in terms of either the squared centre-ofmass energy available after initial-state radiation, (s /s), or the final-state fermion pair mass squared, (m2f /s). The symb ol (-) indicates that no requirements are made on the indicated quantity. f

60


e+ e- qq Monte Carlo ISR effects Acceptance Hadronisation Detector simulation Detector p erformance Corrected Acceptance Backgrounds e+ e- + - Non-resonant (0.051 + 0.007 nb) Four fermion e+ e- e+ e- plus Cosmics Background Sum Total Correction

peak-2 f f/f (%) 1.00481 0.011 1.00037 0.030 1.00055 0.99960 1.00008 1.00542 0.99841 0.99490 0.99994 0.99988 0.99314 0.99852 0.048 0.040 0.020 0.071 0.020 0.070 0.004 0.009 0.073 0.102

1993 peak f 1.00481 0.99997 1.00055 0.99960 1.00008 1.00501 0.99841 0.99833 0.99996 0.99996 0.99666 1.00166 f/f (%) 0.011 0.003 0.048 0.040 0.020 0.066 0.020 0.023 0.003 0.003 0.031 0.073

peak+2 f f/f (%) 1.00481 0.011 1.00007 0.010 1.00055 0.99960 1.00008 1.00511 0.99841 0.99637 0.99992 0.99992 0.99463 0.99971 0.048 0.040 0.020 0.067 0.020 0.050 0.006 0.006 0.055 0.086

1994 peak f 1.00481 0.99997 1.00055 0.99960 1.00008 1.00501 0.99841 0.99833 0.99996 0.99996 0.99666 1.00166 f/f (%) 0.011 0.003 0.048 0.040 0.020 0.066 0.020 0.023 0.003 0.003 0.031 0.073

peak-2 f f/f (%) 1.00481 0.011 1.00037 0.030 1.00055 0.99960 1.00058 1.00593 0.99841 0.99490 0.99994 0.99988 0.99314 0.99902 0.048 0.040 0.047 0.083 0.020 0.070 0.004 0.009 0.073 0.111

1995 peak f 1.00481 0.99997 1.00055 0.99960 1.00058 1.00551 0.99841 0.99833 0.99996 0.99996 0.99666 1.00216 f/f (%) 0.011 0.003 0.048 0.040 0.047 0.079 0.020 0.023 0.003 0.003 0.031 0.085

peak+2 f/f (%) 1.00481 0.011 1.00007 0.010 f 1.00055 0.99960 1.00058 1.00561 0.99841 0.99637 0.99992 0.99992 0.99463 1.00021 0.048 0.040 0.047 0.079 0.020 0.050 0.006 0.006 0.055 0.096

61

Table 3: Summary of the correction factors, f , and their relative systematic errors, f/f , for the e+ e- qq cross-section measurements. These numb ers, when multiplied by the numb er of events actually selected, give the numb er of signal events which would have b een observed in the ideal acceptance describ ed in Table 2. ISR effects encompass the off-p eak acceptance change due to initial-state radiation and the contamination from events with s /s < 0.01. Hadronisation refers to the full correction and uncertainty resulting from the acceptance hole emulation (Table 4). The error correlation b etween the energy p oints and data-taking years is sp ecified in Table 17.


Uncertainty (×10-4 ) Residual hadronisation model dep endence: Inefficiency correction Inefficiency from barrel region Adjustment of cut variables Change of ECAL acceptance radius Barrel detector simulation Total 2.3 2.2 1.5 0.4 3.3 4.8

Table 4: Systematic errors on the selection inefficiency for e+ e- qq events arising from uncertainties of the acceptance hole emulation.

62


peak-2 f f/f (%) Monte Carlo e+ e- e+ e- Monte Carlo Acceptance Correction Electromagnetic energy Electron identification Acceptance definition Low multiplicity Other Corrections Four-fermion events Signal Correction Backgrounds e+ e- + - e+ e- e+ e- qq e+ e- e+ e- + - Background Correction Total Correction Factor 1.0063 1.0009 1.0025 1.0000 1.0001 1.0000 1.0098 0.9982 0.9999 0.9999 1.0000 0.9979 1.0078 0.06 0.10 0.08 0.14 0.01 0.03 0.20 0.04 0.01 0.01 0.01 0.04 0.21

1993 peak f f/f (%) 1.0056 1.0009 1.0025 1.0000 1.0001 1.0000 1.0091 0.9968 0.9999 0.9999 1.0000 0.9966 1.0057 0.02 0.10 0.08 0.09 0.01 0.02 0.16 0.06 0.01 0.02 0.01 0.06 0.17

peak+2 f f/f (%) 1.0061 1.0009 1.0025 1.0000 1.0001 1.0000 1.0096 0.9965 0.9999 0.9998 1.0000 0.9961 1.0057 0.04 0.10 0.08 0.10 0.01 0.03 0.17 0.07 0.02 0.02 0.01 0.08 0.19

1994 peak f f/f (%) 1.0056 1.0022 1.0026 1.0000 1.0001 1.0000 1.0105 0.9968 0.9999 0.9999 1.0000 0.9966 1.0071 0.02 0.07 0.05 0.09 0.01 0.02 0.13 0.06 0.01 0.02 0.01 0.06 0.14

peak-2 f f/f (%) 1.0063 1.0017 1.0031 1.0000 1.0001 1.0000 1.0112 0.9982 0.9999 0.9999 1.0000 0.9979 1.0091 0.06 0.09 0.08 0.14 0.01 0.03 0.20 0.04 0.01 0.01 0.01 0.04 0.20

1995 peak f f/f (%) 1.0056 1.0017 1.0031 1.0000 1.0001 1.0000 1.0105 0.9968 0.9999 0.9999 1.0000 0.9966 1.0070 0.02 0.08 0.08 0.09 0.01 0.02 0.15 0.06 0.01 0.02 0.01 0.06 0.16

peak+2 f f/f (%) 1.0061 1.0017 1.0031 1.0000 1.0001 1.0000 1.0110 0.9965 0.9999 0.9998 1.0000 0.9961 1.0070 0.04 0.09 0.08 0.10 0.01 0.03 0.17 0.07 0.02 0.02 0.01 0.08 0.18

63

Table 5: Summary of the correction factors, f , and their relative systematic errors, f/f , for the e+ e- e+ e- cross-section measurements. The Monte Carlo correction factor corresp onds to the efficiency for events within the ideal phase space definition. The factors listed under acceptance corrections take into account the observed discrepancies b etween the data and Monte Carlo. The total correction factor, when multiplied by the numb er of events actually selected, gives the numb er of signal events which would have b een observed in the ideal acceptance describ ed in Table 2. The error correlation matrix is given in Table 18.


peak-2 f f/f (%) Monte Carlo e+ e- ÷+ ÷- Monte Carlo s cut correction Initial/final state interference Acceptance Correction Tracking losses Track multiplicity cuts Muon identification Acceptance definition Other Corrections Trigger efficiency Four-fermion events Signal Correction Backgrounds e+ e- + - e+ e- e+ e- ÷+ ÷- Cosmic rays Background Correction Total Correction Factor 1.0995 0.9971 1.0003 1.0046 0.9999 1.0000 1.0000 1.0006 1.0009 1.1032 0.9914 0.9988 0.9998 0.9900 1.0922 0.10 - - 0.06 0.05 0.05 0.10 0.02 0.01 0.17 0.02 0.01 0.02 0.03 0.17

1993 peak f
f/f

(%) 1.0955 0.9990 1.0002 1.0046 1.0007 1.0000 1.0000 1.0006 1.0011 1.1022 0.9914 0.9995 0.9998 0.9907 1.0920 0.07 - - 0.06 0.04 0.05 0.10 0.02 0.01 0.15 0.02 0.01 0.02 0.03 0.16

peak+2 f f/f (%) 1.0986 0.9980 1.0001 1.0046 1.0000 1.0000 1.0000 1.0006 1.0011 1.1034 0.9914 0.9991 0.9998 0.9903 1.0927 0.10 - - 0.06 0.04 0.05 0.10 0.02 0.01 0.17 0.02 0.01 0.02 0.03 0.17

1994 peak f
f/f

(%) 1.0948 0.9990 1.0002 1.0042 1.0004 1.0015 1.0000 1.0005 1.0011 1.1024 0.9903 0.9996 0.9998 0.9897 1.0910 0.04 - - 0.04 0.02 0.04 0.05 0.02 0.01 0.09 0.04 0.01 0.02 0.05 0.10

peak-2 f f/f (%) 1.1032 0.9971 1.0003 1.0043 1.0007 1.0000 1.0000 1.0002 1.0009 1.1071 0.9905 0.9987 0.9997 0.9889 1.0948 0.12 - - 0.06 0.09 0.06 0.05 0.02 0.01 0.18 0.02 0.01 0.02 0.03 0.18

1995 peak f
f/f

(%) 1.0970 0.9990 1.0002 1.0043 1.0010 1.0000 1.0000 1.0002 1.0011 1.1034 0.9905 0.9995 0.9997 0.9897 1.0920 0.05 - - 0.06 0.04 0.06 0.05 0.02 0.01 0.12 0.02 0.01 0.02 0.03 0.12

peak+2 f f/f (%) 1.1001 0.9980 1.0001 1.0043 1.0013 1.0000 1.0000 1.0002 1.0011 1.1056 0.9905 0.9990 0.9997 0.9892 1.0937 0.10 - - 0.06 0.08 0.06 0.05 0.02 0.01 0.16 0.02 0.01 0.02 0.03 0.17

64

Table 6: Summary of the correction factors, f , and their relative systematic errors, f/f , for the e+ e- ÷+ ÷- cross-section measurements. These numb ers, when multiplied by the numb er of events actually selected, give the numb er of signal events which would have b een observed in the ideal acceptance describ ed in Table 2. The effects tracking losses, track multiplicity cuts and muon identification were, in principle, simulated by the Monte Carlo. The quoted corrections were introduced to take into account the observed discrepancies b etween the data and Monte Carlo for these effects. The error correlation matrix is given in Table 19.


peak-2 f f/f (%) Monte Carlo e+ e- + - Monte Carlo s cut correction Initial/final state interference Acceptance Correction Multiplicity cuts Acollinearity and cone cuts Definition of | cos | e+ e- e+ e- rejection e+ e- ÷+ ÷- rejection e+ e- e+ e- + - rejection Cosmic ray cuts Combinations of cuts Other Corrections Trigger efficiency Tau branching ratios Four-fermion events Signal Correction Backgrounds e+ e- e+ e- e+ e- ÷+ ÷- e+ e- qq e+ e- e+ e- + - Four-fermion Cosmic rays Background Correction Total Correction Factor 1.3384 0.9976 1.0005 1.0018 0.9999 1.0000 1.0037 0.9997 1.0019 1.0001 1.0000 1.0002 1.0000 1.0012 1.3473 0.9921 0.9900 0.9961 0.9848 0.9991 0.9991 0.9618 1.2960 0.22 - - 0.16 0.25 0.10 0.25 0.08 0.12 0.01 0.09 0.01 0.05 0.04 0.50 0.16 0.12 0.10 0.22 0.02 0.05 0.32 0.59

1993 peak f
f/f

(%) 1.3302 0.9992 1.0004 1.0017 1.0002 1.0000 1.0038 0.9997 1.0018 1.0001 1.0000 1.0002 1.0000 1.0013 1.3414 0.9966 0.9902 0.9960 0.9948 0.9994 0.9997 0.9768 1.3105 0.09 - - 0.16 0.23 0.10 0.26 0.08 0.11 0.01 0.08 0.01 0.05 0.04 0.45 0.07 0.11 0.11 0.07 0.02 0.02 0.19 0.48

peak+2 f f/f (%) 1.3388 0.9984 1.0001 1.0019 1.0007 1.0000 1.0037 0.9997 1.0019 1.0001 1.0000 1.0002 1.0000 1.0018 1.3502 0.9963 0.9902 0.9961 0.9893 0.9993 0.9994 0.9708 1.3108 0.19 - - 0.17 0.28 0.10 0.25 0.08 0.12 0.01 0.10 0.01 0.05 0.04 0.51 0.08 0.11 0.10 0.16 0.02 0.04 0.24 0.56

1994 peak f
f/f

(%) 1.3302 0.9992 1.0004 1.0049 1.0034 1.0000 1.0068 0.9998 1.0014 1.0001 1.0000 1.0002 1.0000 1.0013 1.3536 0.9953 0.9886 0.9960 0.9943 0.9994 0.9996 0.9733 1.3178 0.09 - - 0.14 0.19 0.10 0.23 0.05 0.07 0.01 0.08 0.01 0.05 0.04 0.39 0.07 0.09 0.11 0.07 0.02 0.02 0.18 0.42

peak-2 f f/f (%) 1.3384 0.9976 1.0005 1.0021 1.0008 1.0000 1.0044 0.9993 1.0000 1.0001 1.0000 1.0002 1.0000 1.0012 1.3467 0.9906 0.9892 0.9961 0.9838 0.9991 0.9993 0.9587 1.2913 0.22 - - 0.16 0.25 0.10 0.25 0.08 0.10 0.01 0.09 0.01 0.05 0.04 0.50 0.16 0.13 0.10 0.22 0.02 0.06 0.32 0.59

1995 peak f
f/f

(%) 1.3302 0.9992 1.0004 1.0021 1.0008 1.0000 1.0045 0.9994 1.0000 1.0001 1.0000 1.0002 1.0000 1.0013 1.3409 0.9959 0.9893 0.9960 0.9947 0.9994 0.9998 0.9752 1.3077 0.09 - - 0.16 0.23 0.10 0.26 0.08 0.10 0.01 0.08 0.01 0.05 0.04 0.44 0.07 0.12 0.11 0.07 0.02 0.02 0.19 0.48

peak+2 f f/f (%) 1.3388 0.9984 1.0001 1.0022 1.0012 1.0000 1.0043 0.9993 1.0000 1.0001 1.0000 1.0002 1.0000 1.0018 1.3490 0.9957 0.9894 0.9961 0.9887 0.9993 0.9996 0.9690 1.3074 0.19 - - 0.17 0.28 0.10 0.25 0.08 0.10 0.01 0.10 0.01 0.05 0.04 0.50 0.08 0.13 0.10 0.16 0.02 0.04 0.25 0.56

65

Table 7: Summary of the correction factors, f , and their relative systematic errors, f/f , for the e+ e- + - cross-section measurements. These numb ers, when multiplied by the numb er of events actually selected, give the numb er of signal events which would have b een observed in the ideal acceptance describ ed in Table 2. The error correlation matrix is given in Table 21.


Sample peak-3 peak-2 peak-1 1990 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 1991 p eak peak+1 peak+2 peak+3 1992 p eak peak-2 1993 p eak peak+2 peak(ab) 1994 p eak(c) peak(d) peak-2 1995 p eak peak+2

s(GeV) mean rms 88.2510 0.0481 89.2510 0.0490 90.2490 0.0500 91.2440 0.0510 92.2350 0.0520 93.2380 0.0529 94.2350 0.0539 91.2540 0.0471 88.4810 0.0441 89.4720 0.0451 90.2270 0.0461 91.2230 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 89.4505 0.0564 91.2063 0.0570 93.0351 0.0570 91.2199 0.0565 91.4287 0.0562 91.2195 0.0557 89.4415 0.0568 91.2829 0.0578 92.9715 0.0581



e+ e- qq cross-section measured corrected 4.669 + 0.110 4.667 8.501 + 0.130 8.494 18.899 + 0.281 18.890 30.445 + 0.130 30.488 21.400 + 0.271 21.394 12.434 + 0.180 12.427 7.947 + 0.130 7.944 30.355 + 0.099 30.391 5.326 + 0.095 5.323 10.087 + 0.126 10.080 18.243 + 0.171 18.234 30.370 + 0.129 30.407 24.603 + 0.215 24.605 14.058 + 0.178 14.051 9.916 + 0.115 9.912 30.566 + 0.045 30.609 10.053 + 0.037 10.042 30.352 + 0.070 30.407 13.856 + 0.043 13.847 30.379 + 0.029 30.433 30.308 + 0.339 30.353 30.598 + 0.138 30.650 9.989 + 0.037 9.978 30.559 + 0.097 30.614 14.282 + 0.044 14.272

(nb) fit 4.605 8.687 18.713 30.500 21.529 12.450 8.045 30.510 5.258 10.210 18.398 30.469 24.836 14.304 9.949 30.514 10.048 30.433 13.808 30.463 30.163 30.462 9.981 30.520 14.278

Table 8: The e+ e- qq production cross-section near the Z resonance. The cross-section is corrected to the simple kinematic acceptance region defined by s /s > 0.01. For each data sample, we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- qq cross-section. The errors shown are statistical only. The cross-section measurements are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical cross-section at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

66


Sample peak-3 peak-2 peak-1 1990 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 1991 p eak peak+1 peak+2 peak+3 1992 p eak peak-2 1993 p eak peak+2 peak(ab) 1994 p eak(c) peak(d) peak-2 1995 p eak peak+2

s(GeV) mean rms 88.2510 0.0481 89.2510 0.0490 90.2490 0.0500 91.2420 0.0510 92.2350 0.0520 93.2380 0.0529 94.2350 0.0539 91.2540 0.0471 88.4810 0.0441 89.4720 0.0451 90.2270 0.0461 91.2230 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 89.4505 0.0564 91.2063 0.0570 93.0351 0.0570 91.2197 0.0565 91.4286 0.0562 91.2195 0.0557 89.4415 0.0568 91.2829 0.0578 92.9715 0.0581



e+ e- e+ e- cross-section measured corrected 0.3520 + 0.0281 0.3519 0.4850 + 0.0281 0.4848 0.8110 + 0.0451 0.8109 1.0120 + 0.0180 1.0132 0.6010 + 0.0371 0.6007 0.3650 + 0.0261 0.3648 0.2310 + 0.0210 0.2309 0.9875 + 0.0140 0.9886 0.3623 + 0.0239 0.3622 0.5628 + 0.0280 0.5626 0.7617 + 0.0299 0.7616 1.0093 + 0.0190 1.0104 0.6885 + 0.0300 0.6884 0.4165 + 0.0270 0.4163 0.3020 + 0.0180 0.3019 1.0062 + 0.0065 1.0074 0.5414 + 0.0080 0.5411 1.0064 + 0.0109 1.0080 0.3945 + 0.0067 0.3942 1.0047 + 0.0046 1.0063 0.9422 + 0.0512 0.9434 0.9676 + 0.0210 0.9691 0.5260 + 0.0080 0.5257 1.0074 + 0.0150 1.0089 0.4088 + 0.0068 0.4085

(nb) fit 0.3403 0.4795 0.7762 1.0054 0.6307 0.3594 0.2429 1.0036 0.3636 0.5282 0.7677 1.0079 0.7422 0.4114 0.2920 0.9958 0.5231 1.0098 0.3973 1.0083 0.9635 1.0083 0.5209 0.9988 0.4107

Table 9: The e+ e- e+ e- production cross-section near the Z resonance. The cross-section is corrected to the simple kinematic acceptance region defined by | cos e- | < 0.70 and acol < 10 . For each data sample, we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- e+ e- cross-section. The errors shown are statistical only. The crosssection measurements are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical cross-section at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

67


Sample peak-3 peak-2 peak-1 1990 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 1991 p eak peak+1 peak+2 peak+3 1992 p eak peak-2 1993 p eak peak+2 peak(ab) 1994 p eak(c) peak(d) peak-2 1995 p eak peak+2

s(GeV) mean rms 88.2510 0.0481 89.2510 0.0490 90.2490 0.0500 91.2440 0.0510 92.2350 0.0520 93.2380 0.0529 94.2350 0.0539 91.2540 0.0471 88.4810 0.0441 89.4720 0.0451 90.2270 0.0461 91.2230 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 89.4505 0.0564 91.2072 0.0568 93.0351 0.0570 91.2200 0.0565 91.4286 0.0562 91.2195 0.0557 89.4415 0.0568 91.2827 0.0578 92.9715 0.0581



e+ e- ÷+ ÷- cross-section measured corrected 0.2450 + 0.0243 0.2449 0.4181 + 0.0284 0.4178 0.8432 + 0.0477 0.8427 1.4833 + 0.0221 1.4854 1.0841 + 0.0477 1.0838 0.5961 + 0.0345 0.5958 0.3970 + 0.0284 0.3969 1.4851 + 0.0182 1.4869 0.2319 + 0.0202 0.2318 0.5171 + 0.0273 0.5167 0.9082 + 0.0353 0.9078 1.4859 + 0.0232 1.4877 1.2447 + 0.0424 1.2448 0.6836 + 0.0354 0.6833 0.4794 + 0.0242 0.4792 1.4781 + 0.0083 1.4802 0.4964 + 0.0081 0.4959 1.4563 + 0.0142 1.4589 0.6681 + 0.0091 0.6677 1.4752 + 0.0058 1.4778 1.4786 + 0.0674 1.4808 1.4762 + 0.0272 1.4787 0.4912 + 0.0081 0.4907 1.5085 + 0.0193 1.5111 0.6894 + 0.0093 0.6889

(nb) fit 0.2345 0.4302 0.9116 1.4779 1.0467 0.6102 0.3983 1.4784 0.2657 0.5034 0.8965 1.4764 1.2057 0.6994 0.4899 1.4785 0.4956 1.4748 0.6755 1.4761 1.4617 1.4761 0.4923 1.4789 0.6981

Table 10: The e+ e- ÷+ ÷- production cross-section near the Z resonance. The cross-section is corrected to the simple kinematic acceptance region defined by m2f /s > 0.01. For each data sample, f we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- ÷+ ÷- cross-section. The errors shown are statistical only. The cross-section measurements are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical cross section at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

68


Sample peak-3 peak-2 peak-1 1990 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 1991 p eak peak+1 peak+2 peak+3 1992 p eak peak-2 1993 p eak peak+2 peak(ab) 1994 p eak(c) peak(d) peak-2 1995 p eak peak+2

s(GeV) mean rms 88.2510 0.0481 89.2510 0.0490 90.2490 0.0500 91.2440 0.0510 92.2350 0.0520 93.2380 0.0529 94.2350 0.0539 91.2540 0.0471 88.4810 0.0441 89.4720 0.0451 90.2270 0.0461 91.2230 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 89.4505 0.0564 91.2060 0.0570 93.0351 0.0570 91.2198 0.0565 91.4286 0.0562 91.2195 0.0557 89.4415 0.0568 91.2827 0.0578 92.9715 0.0581



e+ e- + - cross-section measured corrected 0.2152 + 0.0245 0.2151 0.4304 + 0.0296 0.4301 0.9328 + 0.0633 0.9323 1.4566 + 0.0245 1.4587 1.0292 + 0.0552 1.0289 0.6518 + 0.0409 0.6515 0.4095 + 0.0327 0.4094 1.4325 + 0.0194 1.4343 0.2769 + 0.0235 0.2768 0.4845 + 0.0297 0.4841 0.8331 + 0.0368 0.8327 1.4384 + 0.0256 1.4402 1.1892 + 0.0450 1.1893 0.6953 + 0.0409 0.6950 0.4989 + 0.0276 0.4987 1.4734 + 0.0092 1.4755 0.5074 + 0.0092 0.5069 1.5048 + 0.0158 1.5074 0.6660 + 0.0102 0.6656 1.4812 + 0.0065 1.4838 1.3708 + 0.0724 1.3730 1.4477 + 0.0301 1.4502 0.4892 + 0.0091 0.4887 1.4988 + 0.0213 1.5014 0.7089 + 0.0105 0.7084

(nb) fit 0.2343 0.4299 0.9109 1.4767 1.0459 0.6097 0.3980 1.4771 0.2655 0.5029 0.8957 1.4751 1.2047 0.6988 0.4895 1.4773 0.4951 1.4734 0.6750 1.4749 1.4605 1.4748 0.4919 1.4776 0.6976

Table 11: The e+ e- + - production cross-section near the Z resonance. The cross-section is corrected to the simple kinematic acceptance region defined by m2f /s > 0.01. For each data sample, f we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- + - cross-section. The errors shown are statistical only. The cross-section measurements are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical cross section at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

69


Sample 1993

s (GeV) mean rms



input (nb) e+ e- qq

pseudo-cross-sections (nb) e+ e- e+ e- e+ e- ÷+ ÷- e+ e- +

-

prescan(a) 91.1386 0.0867 30.176 + 0.306 1.056 + 0.057 prescan(b) 91.3211 0.0566 30.479 + 0.076 0.988 + 0.014 peak 91.1964 0.0576 30.383 + 0.198 1.024 + 0.036

1.499 + 0.071 1.444 + 0.078 1.461 + 0.018 1.471 + 0.020 1.420 + 0.045 1.435 + 0.050 1.469 + 0.028 1.426 + 0.031 1.278 + 0.080 1.129 + 0.084 1.481 + 0.013 1.463 + 0.014

1994
peak(ab)

91.2183 0.0568 30.437 + 0.122 1.051 + 0.023

1995
prescan(a) 91.7990 0.0568 26.910 + 0.352 0.810 + 0.061 prescan(b) 91.3030 0.0576 30.501 + 0.056 0.983 + 0.010

Table 12: For some p eriods of running at the Z p eak a precise luminosity measurement is not available. We nevertheless use this data to measure inter-sp ecies cross-section ratios. To make these data compatible with other cross-section measurements, we fix each e+ e- qq cross-section to its exp ected value, and normalise the cross-sections for the three lepton sp ecies measured in the same p eriod to this arbitrary hadron cross-section. We term such measurements pseudo-cross-sections, and allow the absolute scale of each set of four to float by 10% in the fit using appropriate error matrices. For each data sample we list here the mean s of the colliding b eams, its rms spread, and the observed pseudo-cross-sections. The errors shown are statistical only.

70


Centre-of-mass energy errors -3 193 194 195 196 197 198 199 6.1 7.4 7.0 6.6 6.1 5.7 5.1 4.7 -2 194 195 196 197 198 199 200 5.8 6.7 6.4 6.1 5.8 5.5 5.2 4.9 -1 195 196 197 198 199 200 201 5.5 5.9 5.8 5.6 5.5 5.4 5.2 5.0 1990 0 196 197 198 199 200 202 203 5.1 5.0 5.1 5.1 5.1 5.2 5.2 5.2 +1 197 198 199 200 202 203 204 4.8 4.0 4.3 4.5 4.8 5.0 5.2 5.4 +2 198 199 200 202 203 204 205 4.4 2.4 3.3 3.8 4.4 4.8 5.2 5.6 +3 199 200 201 203 204 205 206 4.0 -1.9 1.8 2.9 3.9 4.6 5.3 5.8 0 6.1 5.8 5.5 5.1 4.8 4.4 4.0 20.3 6.6 6.3 6.1 5.8 5.6 5.2 5.0 -3 7.4 6.7 5.9 5.0 4.0 2.4 -1.9 6.6 9.7 7.8 7.3 6.6 6.0 5.2 4.4 -2 7.0 6.4 5.8 5.1 4.3 3.3 1.8 6.3 7.8 8.8 6.9 6.3 5.9 5.2 4.6 1991 -1 6.6 6.1 5.6 5.1 4.5 3.8 2.9 6.1 7.3 6.9 8.2 6.1 5.7 5.2 4.8 0 6.1 5.8 5.5 5.1 4.8 4.4 3.9 5.8 6.6 6.3 6.1 7.6 5.6 5.2 4.9 +1 5.7 5.5 5.4 5.2 5.0 4.8 4.6 5.6 6.0 5.9 5.7 5.6 7.3 5.2 5.1 +2 5.1 5.2 5.2 5.2 5.2 5.2 5.3 5.2 5.2 5.2 5.2 5.2 5.2 7.3 5.3 +3 4.7 4.9 5.0 5.2 5.4 5.6 5.8 5.0 4.4 4.6 4.8 4.9 5.1 5.3 7.4

-3 -2 -1 1990 0 +1 +2 +3 pre -3 -2 -1 1991 0 +1 +2 +3

Table 13: The signed square-root of the covariance matrix elements for systematic errors in the LEP centre-of-mass energy calibration for 1990-1991. The column and row headings lab el the centre-ofmass energies relative to the p eak in GeV. The entries are the absolute errors in units of MeV. The energy measurement in 1992 is uncorrelated with other years. Its error is 18.0 MeV.

1993 1993 1993 1994 1995 1995 1995

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

Centre-of-mass en 1993 1993 1993 pk-2 p eak pk+2 3.55 2.88 2.73 2.88 6.76 2.77 2.73 2.77 3.09 2.36 2.49 2.28 1.31 1.16 1.25 1.21 1.22 1.27 1.22 1.17 1.35

ergy errors 1994 1995 p eak pk-2 2.36 1.31 2.49 1.16 2.28 1.25 3.78 1.25 1.25 1.83 1.32 1.28 1.26 1.26

1995 p eak 1.21 1.22 1.27 1.32 1.28 5.41 1.38

1995 p k+ 2 1.22 1.17 1.35 1.26 1.26 1.38 1.74

Table 14: The signed square-root of the covariance matrix elements for systematic errors in the LEP centre-of-mass energy in 1993 - 1995. These errors are sp ecific to the OPAL interaction p oint. The entries are the absolute errors in units of MeV.

71


1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 3.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

Centre-of-mass energy spread errors 1991 1992 1993 1993 1993 1994 pk-2 p eak pk+2 p eak 3.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 1.8 1.8 1.8 1.8 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1 0.0 1.8 1.1 1.1 1.1 1.1

1995 pk-2 0.0 0.0 1.8 1.1 1.1 1.1 1.1 1.3 1.3 1.3

1995 p eak 0.0 0.0 1.8 1.1 1.1 1.1 1.1 1.3 1.3 1.3

1995 p k+ 2 0.0 0.0 1.8 1.1 1.1 1.1 1.1 1.3 1.3 1.3

Table 15: The signed square-root of the covariance matrix elements for systematic errors in the LEP centre-of-mass energy spread. The entries are the absolute errors in units of MeV.

1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 3000.00 30.91 30.91 5.40 5.40 5.40 5.40 5.40 5.40 5.40

1991 30.91 60.25 30.91 5.40 5.40 5.40 5.40 5.40 5.40 5.40

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

Luminosity errors 1992 1993 1993 pk-2 p eak 30.91 5.40 5.40 30.91 5.40 5.40 41.82 5.40 5.40 5.40 6.52 6.27 5.40 6.27 6.52 5.40 6.09 6.27 5.40 6.26 6.44 5.40 6.44 6.26 5.40 6.26 6.44 5.40 6.08 6.26

1993 pk+2 5.40 5.40 5.40 6.09 6.27 6.52 6.26 6.08 6.26 6.44

1994 p eak 5.40 5.40 5.40 6.26 6.44 6.26 6.49 6.26 6.44 6.26

1995 pk-2 5.40 5.40 5.40 6.44 6.26 6.08 6.26 6.59 6.33 6.15

1995 p eak 5.40 5.40 5.40 6.26 6.44 6.26 6.44 6.33 6.58 6.33

1995 p k+ 2 5.40 5.40 5.40 6.08 6.26 6.44 6.26 6.15 6.33 6.58

Table 16: The signed square-root of the covariance matrix elements for systematic errors in the luminosity measurement. The entries are the relative errors in units of 10-4 . The theoretical error in the calculated luminometer acceptance is included. The large error for the 1990 luminosity has b een artificially inflated, as explained in the text.

72


1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 40.00 19.00 19.00 10.51 10.51 10.51 10.51 10.51 10.51 10.51

1991 19.00 19.00 19.00 7.27 7.27 7.27 7.27 7.27 7.27 7.27

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

e+ e- qq cross-section errors 1992 1993 1993 1993 pk-2 p eak pk+2 19.00 10.51 10.51 10.51 19.00 7.27 7.27 7.27 19.00 7.27 7.27 7.27 7.27 10.24 8.05 9.28 7.27 8.05 7.32 7.75 7.27 9.28 7.75 8.65 7.27 7.80 7.04 7.49 7.27 10.04 7.80 9.07 7.27 7.80 7.04 7.49 7.27 9.07 7.49 8.41

1994 p eak 10.51 7.27 7.27 7.80 7.04 7.49 7.32 7.80 7.04 7.49

1995 pk-2 10.51 7.27 7.27 10.04 7.80 9.07 7.80 11.08 9.10 10.21

1995 p eak 10.51 7.27 7.27 7.80 7.04 7.49 7.04 9.10 8.46 8.84

1995 p k+ 2 10.51 7.27 7.27 9.07 7.49 8.41 7.49 10.21 8.84 9.63

Table 17: The signed square-root of the covariance matrix elements for systematic errors in the measurement of the e+ e- qq cross-section. The hadrons are not appreciably correlated with any of the leptons. The entries are the relative errors in units of 10-4 . They do not include errors in the luminosity measurement, which are sp ecified separately in Table 16.

1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 Table measu which errors 70.0 22.0 22.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

e+ e- e+ e- cross-section errors 1991 1992 1993 1993 1993 1994 pk-2 p eak pk+2 p eak 22.0 22.0 14.0 14.0 14.0 14.0 22.0 22.0 14.0 14.0 14.0 14.0 22.0 23.0 14.0 14.0 14.0 14.0 14.0 14.0 20.0 17.0 18.0 15.0 14.0 14.0 17.0 17.0 17.0 14.0 14.0 14.0 18.0 17.0 19.0 15.0 14.0 14.0 15.0 14.0 15.0 14.0 14.0 14.0 18.0 15.0 16.0 15.0 14.0 14.0 15.0 14.0 15.0 14.0 14.0 14.0 16.0 15.0 17.0 15.0

1995 pk-2 14.0 14.0 14.0 18.0 15.0 16.0 15.0 19.0 17.0 17.0

1995 p eak 14.0 14.0 14.0 15.0 14.0 15.0 14.0 17.0 16.0 16.0

1995 p k+ 2 14.0 14.0 14.0 16.0 15.0 17.0 15.0 17.0 16.0 18.0

18: The signed square-root of the covar rement of the e+ e- e+ e- cross-section. are given in Table 21. The entries are the in the luminosity measurement, which are

iance matrix elements for systematic errors in the Additional terms arise b etween electrons and taus, relative errors in units of 10-4 . They do not include sp ecified separately in Table 16.

73


1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 Table measu which errors 50.0 16.0 16.0 9.0 9.0 9.0 9.0 9.0 9.0 9.0

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

e+ e- ÷+ ÷- cross-section errors 1991 1992 1993 1993 1993 1994 pk-2 p eak pk+2 p eak 16.0 16.0 9.0 9.0 9.0 9.0 25.0 16.0 9.0 9.0 9.0 9.0 16.0 16.0 9.0 9.0 9.0 9.0 9.0 9.0 17.0 13.3 13.3 8.4 9.0 9.0 13.3 16.0 13.3 8.4 9.0 9.0 13.3 13.3 17.0 8.4 9.0 9.0 8.4 8.4 8.4 10.0 9.0 9.0 9.8 9.8 9.8 8.5 9.0 9.0 9.8 9.8 9.8 8.5 9.0 9.0 9.8 9.8 9.8 8.5

1995 pk-2 9.0 9.0 9.0 9.8 9.8 9.8 8.5 18.0 10.6 10.6

1995 p eak 9.0 9.0 9.0 9.8 9.8 9.8 8.5 10.6 12.0 10.6

1995 p k+ 2 9.0 9.0 9.0 9.8 9.8 9.8 8.5 10.6 10.6 17.0

19: The signed square-root of the covariance matrix elements for systematic errors in the rement of the e+ e- ÷+ ÷- cross-section. Additional terms arise b etween muons and taus, are given in Table 21. The entries are the relative errors in units of 10-4 . They do not include in the luminosity measurement, which are sp ecified separately in Table 16.

1990 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 130. 40. 40. 20. 20. 20. 20. 20. 20. 20.

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

e+ e- + - cross-section errors 1991 1992 1993 1993 1993 1994 pk-2 p eak pk+2 p eak 40. 40. 20. 20. 20. 20. 76. 40. 20. 20. 20. 20. 40. 43. 20. 20. 20. 20. 20. 20. 60. 48. 54. 33. 20. 20. 48. 49. 47. 35. 20. 20. 54. 47. 57. 32. 20. 20. 33. 35. 32. 42. 20. 20. 45. 34. 34. 33. 20. 20. 34. 37. 33. 35. 20. 20. 34. 34. 43. 33.

1995 pk-2 20. 20. 20. 45. 34. 34. 33. 60. 47. 55.

1995 p eak 20. 20. 20. 34. 37. 33. 35. 47. 48. 47.

1995 p k+ 2 20. 20. 20. 34. 34. 43. 33. 55. 47. 58.

Table 20: The signed square-root of the covariance matrix elements for systematic errors in the measurement of the e+ e- + - cross-sections. The -ee terms and - ÷÷ terms are given in Table 21. The entries are the relative errors in units of 10-4 . They do not include errors in the luminosity measurement, which are sp ecified separately in Table 16.

74


1990 -20.0 -4.8 -4.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -40.0 -8.0 -7.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1991 -4.8 -18.0 -4.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -8.0 -24.0 -5.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

ee ee ee ee ee ee ee ee ee ee ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷

inter-sp ecies cross-section 1992 1993 1993 pk-2 p eak -4.4 0.0 0.0 -4.1 0.0 0.0 -15.0 0.0 0.0 0.0 -14.0 -9.0 0.0 -9.0 -10.0 0.0 -9.4 -7.9 0.0 -2.9 -2.4 0.0 -3.1 -2.6 0.0 -2.9 -2.4 0.0 -3.0 -2.5 -7.3 0.0 0.0 -5.3 0.0 0.0 -18.0 0.0 0.0 0.0 -11.0 -8.3 0.0 -8.3 -11.0 0.0 -8.3 -8.3 0.0 -2.4 -2.4 0.0 -3.1 -3.1 0.0 -2.9 -2.9 0.0 -3.0 -3.0

errors 1993 pk+2 0.0 0.0 0.0 -9.4 -7.9 -11.0 -2.5 -2.8 -2.5 -2.6 0.0 0.0 0.0 -8.3 -8.3 -11.0 -2.4 -3.1 -2.9 -3.0

1994 p eak 0.0 0.0 0.0 -2.9 -2.4 -2.5 -9.0 -2.5 -2.3 -2.4 0.0 0.0 0.0 -2.4 -2.4 -2.4 -8.0 -2.8 -2.5 -2.6

1995 pk-2 0.0 0.0 0.0 -3.1 -2.6 -2.8 -2.5 -11.0 -7.5 -7.9 0.0 0.0 0.0 -3.1 -3.1 -3.1 -2.8 -14.0 -9.8 -10.1

1995 p eak 0.0 0.0 0.0 -2.9 -2.4 -2.5 -2.3 -7.5 -9.0 -7.1 0.0 0.0 0.0 -2.9 -2.9 -2.9 -2.5 -9.8 -12.0 -9.4

1995 p k+ 2 0.0 0.0 0.0 -3.0 -2.5 -2.6 -2.4 -7.9 -7.1 -10.0 0.0 0.0 0.0 -3.0 -3.0 -3.0 -2.6 -10.1 -9.4 -13.0

Table 21: The signed square-root of the covariance matrix elements for systematic errors correlated b etween sp ecies in the measurement of the lepton cross-sections. The entries are the relative errors in units of 10-4 . They do not include errors in the luminosity measurement, which are sp ecified separately in Table 16.

75


1990

1991

1992

1993

1994

1995

Sample peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 p eak prescan(a) prescan(b) p eak-2 peak peak+2 peak(ab) p eak(c) peak(d) prescan(a) prescan(b) p eak-2 peak peak+2

s(GeV) mean rms 88.2600 0.0481 89.2550 0.0490 90.2570 0.0500 91.2430 0.0510 92.2420 0.0520 93.2450 0.0529 94.2420 0.0539 91.2540 0.0471 88.4800 0.0441 89.4720 0.0451 90.2270 0.0461 91.2240 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2989 0.0520 91.1354 0.0868 91.3209 0.0566 89.4502 0.0564 91.2063 0.0570 93.0348 0.0570 91.2198 0.0565 91.4285 0.0561 91.2194 0.0558 91.7994 0.0568 91.3032 0.0575 89.4414 0.0568 91.2826 0.0578 92.9716 0.0581



N÷÷ 130 258 391 4963 576 354 231 7563 176 363 744 4422 916 478 404 33733 449 7325 3937 12066 5628 70493 637 3157 256 14619 3739 6510 5688

- - -

- - - -

- - -

-

e+ e- ÷+ ÷- asymmetry measured corrected fit 0.1590 + 0.0830 -0.1591 -0.2720 0.2780 + 0.0570 -0.2782 -0.1781 0.0770 + 0.0480 -0.0772 -0.0825 0.0100 + 0.0130 0.0100 0.0047 0.0490 + 0.0400 0.0491 0.0735 0.0940 + 0.0510 0.0941 0.1218 0.0830 + 0.0610 0.0830 0.1561 0.0020 + 0.0110 0.0020 0.0056 0.2280 + 0.0700 -0.2281 -0.2515 0.1060 + 0.0500 -0.1061 -0.1573 0.0690 + 0.0340 -0.0691 -0.0853 0.0220 + 0.0140 -0.0220 0.0032 0.0020 + 0.0310 0.0021 0.0571 0.1520 + 0.0420 0.1521 0.1101 0.0850 + 0.0460 0.0851 0.1393 0.0088 + 0.0051 0.0088 0.0092 0.0439 + 0.0442 -0.0439 -0.0041 0.0081 + 0.0109 0.0081 0.0109 0.1503 + 0.0146 -0.1505 -0.1594 0.0020 + 0.0085 -0.0020 0.0018 0.1001 + 0.0124 0.1002 0.1130 0.0040 + 0.0035 0.0040 0.0028 0.0182 + 0.0371 0.0183 0.0193 0.0096 + 0.0166 0.0096 0.0028 0.0523 + 0.0585 0.0524 0.0460 0.0094 + 0.0077 0.0094 0.0095 0.1414 + 0.0151 -0.1416 -0.1603 0.0198 + 0.0115 0.0198 0.0079 0.1170 + 0.0123 0.1171 0.1103

Table 22: The e+ e- ÷+ ÷- forward-backward charge asymmetry near the Z resonance. The measured asymmetry is corrected to the simple kinematic acceptance region defined by | cos - | < 0.95 and acol < 15 , with the energy of each fermion required to b e greater than 6.0 GeV. For each data sample we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- ÷+ ÷- asymmetry. The errors shown are statistical only. The asymmetries are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical asymmetry at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

76


1990

1991

1992

1993

1994

1995

Sample peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 p eak prescan(a) prescan(b) p eak-2 peak peak+2 peak(ab) p eak(c) peak(d) prescan(a) prescan(b) p eak-2 peak peak+2

s(GeV) mean rms 88.2610 0.0481 89.2540 0.0490 90.2630 0.0500 91.2460 0.0510 92.2420 0.0520 93.2480 0.0529 94.2400 0.0539 91.2540 0.0471 88.4800 0.0441 89.4720 0.0451 90.2270 0.0461 91.2240 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 91.1361 0.0867 91.3211 0.0566 89.4504 0.0564 91.2052 0.0570 93.0348 0.0570 91.2196 0.0565 91.4277 0.0561 91.2194 0.0558 91.7987 0.0568 91.3032 0.0575 89.4413 0.0568 91.2823 0.0578 92.9715 0.0581



N 102 224 281 3722 416 302 183 6059 166 289 569 3603 734 436 359 28037 347 5745 3255 10374 4551 56230 451 2471 195 11765 3060 5242 4711

- - - -

- - - -

- - - - - - -

e+ e- + - asymmetry measured corrected fit 0.3467 + 0.0800 -0.3468 -0.2651 0.0667 + 0.0620 -0.0669 -0.1743 0.0954 + 0.0550 -0.0956 -0.0810 0.0084 + 0.0150 -0.0084 0.0034 0.0686 + 0.0460 0.0687 0.0700 0.1277 + 0.0550 0.1278 0.1171 0.0887 + 0.0720 0.0887 0.1503 0.0146 + 0.0120 0.0146 0.0040 0.2547 + 0.0680 -0.2548 -0.2454 0.1017 + 0.0540 -0.1018 -0.1541 0.0714 + 0.0390 -0.0715 -0.0843 0.0004 + 0.0160 -0.0004 0.0017 0.0386 + 0.0350 0.0387 0.0540 0.0947 + 0.0440 0.0948 0.1056 0.1697 + 0.0480 0.1697 0.1340 0.0152 + 0.0056 0.0152 0.0075 0.0191 + 0.0508 -0.0191 -0.0053 0.0084 + 0.0136 -0.0084 0.0092 0.1497 + 0.0167 -0.1499 -0.1561 0.0060 + 0.0094 -0.0060 0.0002 0.1106 + 0.0142 0.1107 0.1085 0.0008 + 0.0040 0.0008 0.0013 0.0926 + 0.0438 -0.0925 0.0172 0.0193 + 0.0189 -0.0193 0.0013 0.1095 + 0.0673 0.1096 0.0432 0.0006 + 0.0088 0.0006 0.0078 0.1334 + 0.0171 -0.1336 -0.1569 0.0227 + 0.0132 0.0227 0.0062 0.0938 + 0.0139 0.0939 0.1058

Table 23: The e+ e- + - forward-backward charge asymmetry near the Z resonance. The measured asymmetry is corrected to the simple kinematic acceptance region defined by | cos - | < 0.90 and acol < 15 , with the energy of each fermion required to b e greater than 6.0 GeV. For each data sample we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- + - asymmetry. The errors shown are statistical only. The asymmetries are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical asymmetry at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

77


1990

1991

1992

1993

1994

1995

Sample peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 prescan peak-3 peak-2 peak-1 p eak peak+1 peak+2 peak+3 p eak prescan(a) prescan(b) p eak-2 peak peak+2 peak(ab) p eak(c) peak(d) prescan(a) prescan(b) p eak-2 peak peak+2

s(GeV) mean rms 88.2610 0.0481 89.2550 0.0490 90.2610 0.0500 91.2460 0.0510 92.2450 0.0520 93.2470 0.0529 94.2420 0.0539 91.2540 0.0471 88.4790 0.0441 89.4690 0.0451 90.2270 0.0461 91.2200 0.0471 91.9690 0.0481 92.9680 0.0490 93.7170 0.0500 91.2990 0.0520 91.1447 0.0857 91.3213 0.0566 89.4502 0.0565 91.2055 0.0571 93.0346 0.0570 91.2196 0.0565 91.4282 0.0562 91.2195 0.0558 91.7987 0.0567 91.3032 0.0575 89.4417 0.0568 91.2823 0.0578 92.9715 0.0581

Nee 194 321 384 3719 308 236 160 5624 297 451 683 3365 566 325 284 25280 343 5254 4750 9628 3664 51939 462 2281 261 10741 4451 4812 3706

e+ e- e+ e- asymmetr measured corrected 0.3730 + 0.0670 0.3731 0.3290 + 0.0530 0.3291 0.2260 + 0.0500 0.2261 0.0850 + 0.0160 0.0849 0.0790 + 0.0570 0.0789 -0.0070 + 0.0650 -0.0070 0.2180 + 0.0780 0.2181 0.0900 + 0.0130 0.0899 0.4700 + 0.0510 0.4701 0.2750 + 0.0450 0.2751 0.1820 + 0.0380 0.1821 0.1070 + 0.0170 0.1069 0.1180 + 0.0420 0.1179 0.0780 + 0.0550 0.0780 0.0580 + 0.0590 0.0580 0.0996 + 0.0063 0.0995 0.2141 + 0.0528 0.2139 0.0930 + 0.0137 0.0928 0.2619 + 0.0140 0.2621 0.1052 + 0.0101 0.1051 0.0958 + 0.0164 0.0958 0.1046 + 0.0044 0.1045 0.1312 + 0.0461 0.1310 0.0999 + 0.0208 0.0998 0.0894 + 0.0617 0.0892 0.0943 + 0.0096 0.0941 0.2840 + 0.0144 0.2842 0.0736 + 0.0144 0.0735 0.0878 + 0.0164 0.0878

y fit 0.3877 0.2977 0.1911 0.1004 0.0691 0.0994 0.1588 0.0999 0.3697 0.2759 0.1948 0.1022 0.0705 0.0866 0.1256 0.0969 0.1076 0.0955 0.2778 0.1032 0.0894 0.1022 0.0893 0.1022 0.0740 0.0967 0.2787 0.0980 0.0867

Table 24: The e+ e- e+ e- forward-backward charge asymmetry near the Z resonance. The measured asymmetry is corrected to the simple kinematic acceptance region defined by | cos - | < 0.70 and acol < 10 , with the energy of each fermion required to b e greater than 0.2 GeV. For each data sample we list here the mean s of the colliding b eams, its root-mean-square (rms) spread, and the observed e+ e- e+ e- asymmetry. The errors shown are statistical only. The asymmetries are also shown after b eing corrected for the b eam energy spread to corresp ond to the physical asymmetry at the central value of s. The fit values are the result of the 9-parameter model-indep endent fit.

78


1990 ee 5.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ÷÷ 1.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 3.0 3.0 2.0 1.8 1.2 1.6 1.2 1.8 1.2 1.6

1991 ee 1.0 3.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ÷÷ 0.4 1.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 3.0 3.0 2.0 1.8 1.2 1.6 1.2 1.8 1.2 1.6

1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995 1990 1991 1992 1993 1993 1993 1994 1995 1995 1995

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

ee ee ee ee ee ee ee ee ee ee ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

p p p p p p p

k-2 eak k+ 2 eak k-2 eak k+ 2

lepton asymmetry errors 1992 1993 1993 1993 pk-2 p eak pk+2 ee ee ee ee 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0 1.0 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ÷÷ ÷÷ ÷÷ ÷÷ 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.4 0.4 0.4 0.4 1.1 0.4 0.4 0.4 0.4 0.7 0.4 0.4 0.4 0.4 1.0 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 2.0 1.8 1.2 1.6 2.0 1.8 1.2 1.6 2.0 1.8 1.2 1.6 1.8 1.8 1.2 1.6 1.2 1.2 1.2 1.2 1.6 1.6 1.2 1.6 1.2 1.2 1.2 1.2 1.8 1.8 1.2 1.6 1.2 1.2 1.2 1.2 1.6 1.6 1.2 1.6

1994 p eak ee 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ÷÷ 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

1995 pk-2 ee 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.0 1.0 ÷÷ 0.4 0.4 0.4 0.4 0.4 0.4 0.4 1.2 0.4 0.4 1.8 1.8 1.8 1.8 1.2 1.6 1.2 1.8 1.2 1.6

1995 p eak ee 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ÷÷ 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.9 0.4 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

1995 p k+ 2 ee 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.2 ÷÷ 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.9 1.6 1.6 1.6 1.6 1.2 1.6 1.2 1.6 1.2 1.6

Table 25: The signed square-root of the covariance matrix elements for systematic errors in the measurement of the lepton asymmetries. There are no appreciable inter-sp ecies correlations in the asymmetry measurements. The entries are the absolute errors in units of 10-3 .

79


Parameter mZ (GeV) Z (GeV) 0 h (nb)
s +- ZZ (e e ) s +- ZZ (÷ ÷ ) s +- ZZ ( ) s CZZ ( + - ) a CZZ (e+ e- ) a CZZ (÷+ ÷- ) a CZZ ( + - ) a CZZ ( + - ) a C Z (e+ e- ) a C Z (÷+ ÷- ) a C Z ( + - ) a C Z ( + - ) s C Z (e+ e- ) s C Z (÷+ ÷- ) s C Z ( + - ) s C Z ( + - ) 2 /d.o.f

Interpretation in terms of couplings

Without Lepton Universality 91.1866 + 0.0031 2.4942 + 0.0042 41.505 + 0.055

With Lepton Universality 91.1861 + 0.0030 2.4940 + 0.0041 41.505 + 0.055

Standard Model Prediction input +0.0021 2.4949-0.0074 +0.012 41.480-0.011

C C C

2 2 2 2 (gAe + gVe )(gAe + gVe ) 2 2 2 2 (gAe + gVe )(gA÷ + gV÷ ) 2 2 2 2 (gAe + gVe )(gA + gV ) 2 2 (gA + gV )2

0.06330 + 0.00031 0.06359 + 0.00025 0.06366 + 0.00033 0.06353 + 0.00022 0.000189 + 0.000094 0.000320 + 0.000049 0.000293 + 0.000064 0.000294 + 0.000036 0.242 + 0.022 0.232 + 0.011 0.234 + 0.013 0.2350 + 0.0080 -0.0262 + 0.0139 -0.0029 + 0.0093 -0.0011 + 0.0106 -0.0075 + 0.0064 146.6 / 188 151.8 / 196
+0. 0.00134-0. 00006 00015 +0. 0.25130-0. 00013 00047 +0. 0.000336-0. 000014 000039 +0. 0.06382-0. 00009 00031

gAe gVe gAe gVe gAe gVe gA÷ gV÷ gAe gVe gA gV (gA gV )2 gAe gAe gAe gA÷ gAe gA
2 gA

gVe gVe gVe gV÷ gVe gV
2 gV

Table 26: Results with and without than those on the section 11.2.2). In variations given in

of the C-parameter fits to the measured cross-section and lepton asymmetry data imp osing lepton universality (7 and 15 parameters). Theory uncertainties, other t-channel for the electrons and the luminosity, are not included in the errors (see the last column we give the values predicted by the SM assuming the parameter Equation 25.

80


Parameters 1 mZ 2 Z 0 3 h s 4 CZZ (e+ e- ) s 5 CZZ (÷+ ÷- ) s 6 CZZ ( + - ) a 7 CZZ (e+ e- ) a 8 CZZ (÷+ ÷- ) a 9 CZZ ( + - ) a 10 C Z (e+ e- ) a 11 C Z (÷+ ÷- ) a 12 C Z ( + - ) s 13 C Z (e+ e- ) s 14 C Z (÷+ ÷- ) s 15 C Z ( + - ) 81

1 1.000 .043 .034 -.092 -.008 -.005 -.054 .067 .052 .009 .009 .009 -.088 -.142 -.122

2 .043 1.000 -.354 .572 .710 .542 .003 .024 .017 .000 .049 .043 -.024 .022 .021

3 .034 -.354 1.000 -.098 -.120 -.090 .006 .008 .007 -.004 -.007 -.006 -.011 -.018 -.016

4 -.092 .572 -.098 1.000 .457 .338 .171 -.007 -.007 -.003 .029 .024 .055 .033 .030

5 -.008 .710 -.120 .457 1.000 .413 .009 .032 .014 -.003 .059 .032 -.018 .158 .020

6 -.005 .542 -.090 .338 .413 1.000 .005 .014 .027 -.001 .028 .075 -.012 .017 .127

7 -.054 .003 .006 .171 .009 .005 1.000 -.014 -.011 .080 -.001 -.001 -.004 .010 .008

8 .067 .024 .008 -.007 .032 .014 -.014 1.000 .016 .001 .208 .003 -.008 .008 -.010

9 .052 .017 .007 -.007 .014 .027 -.011 .016 1.000 .001 .002 .184 -.006 -.009 .008

10 .009 .000 -.004 -.003 -.003 -.001 .080 .001 .001 1.000 .000 .000 .217 -.001 -.001

11 .009 .049 -.007 .029 .059 .028 -.001 .208 .002 .000 1.000 .002 -.003 -.084 .000

12 .009 .043 -.006 .024 .032 .075 -.001 .003 .184 .000 .002 1.000 -.002 .000 -.089

13 -.088 -.024 -.011 .055 -.018 -.012 -.004 -.008 -.006 .217 -.003 -.002 1.000 .025 .020

14 -.142 .022 -.018 .033 .158 .017 .010 .008 -.009 -.001 -.084 .000 .025 1.000 .033

15 -.122 .021 -.016 .030 .020 .127 .008 -.010 .008 -.001 .000 -.089 .020 .033 1.000

Table 27: Error correlation matrix for the C-parameters in Table 26 which do not assume lepton universality.


Parameters 1 mZ 2 Z 0 3 h s 4 CZZ ( + a 5 CZZ ( + a 6 C Z ( + s 7 C Z ( +

- - - -

) ) ) )

1 1.000 .011 .059 -.052 .064 .004 -.207

2 .011 1.000 -.353 .796 .028 .064 .025

3 .059 -.353 1.000 -.135 .009 -.010 -.032

4 -.052 .796 -.135 1.000 .051 .078 .119

5 .064 .028 .009 .051 1.000 .185 .005

6 .004 .064 -.010 .078 .185 1.000 -.031

7 -.207 .025 -.032 .119 .005 -.031 1.000

Table 28: Error correlation matrix for the C-parameters in Table 26 which assume lepton universality

Without lepton universality mZ (GeV) Z (GeV)
0 h (nb)

With lepton universality 91.1852 + 0.0030 2.4948 + 0.0041 41.501 + 0.055

SM prediction input
+0. 2.4949-0. 0021 0074 +0.012 41.480-0.011 +0.015 20.738-0.023 +0.015 20.738-0.023 +0.015 20.785-0.023 +0.015 20.738-0.023

91.1858 + 0.0030 2.4948 + 0.0041 41.501 + 0.055 20.902 + 0.084 20.811 + 0.058 20.832 + 0.091

R R R R

e ÷

20.823 + 0.044 0.0089 + 0.0044 0.0159 + 0.0023 0.0145 + 0.0030 0.0145 + 0.0017 155.6/ 194 158.3/ 198

A0,e FB A0,÷ FB A0, FB A0, FB 2 /d.o.f.

+0. 0.0158-0.

0007 0018

Table 29: Results of fitting the model-indep endent Z parameters to the measured cross-sections and asymmetries. The theory uncertainties given in Table 33, other than those on the t-channel for the electrons and the luminosity, are not included in the errors. In the last column we give the values calculated in the context of the SM assuming the parameter variations given in Equation 25.

m mZ Z 0 h R A0, FB

Z



Z



0 h

R 0.04 0.02 0.29 1.00 -0.01

A0, FB 0.07 -0.01 0.02 -0.01 1.00

1.00 0.05 0.03 0.04 0.07

0.05 1.00 -0.35 0.02 -0.01

0.03 -0.35 1.00 0.29 0.02

Table 30: Error correlation matrix for the 5 parameter fit assuming lepton universality given in Table 29.

82


m mZ Z 0 h Re R÷ R A0,e FB A0,÷ FB A0, FB

Z



Z



0 h

R

e

R

÷

R



A0,e FB -0.05 0.00 0.01 -0.20 0.00 0.00 1.00 -0.02 -0.01

A0,÷ FB 0.08 0.00 0.01 0.03 0.01 0.00 -0.02 1.00 0.02

A0, FB 0.06 0.00 0.01 0.02 0.00 0.01 -0.01 0.02 1.00

1.00 0.05 0.03 0.11 0.00 0.00 -0.05 0.08 0.06

0.05 1.00 -0.35 0.01 0.02 0.01 0.00 0.00 0.00

0.03 -0.35 1.00 0.15 0.22 0.14 0.01 0.01 0.01

0.11 0.01 0.15 1.00 0.09 0.04 -0.20 0.03 0.02

0.00 0.02 0.22 0.09 1.00 0.06 0.00 0.01 0.00

0.00 0.01 0.14 0.04 0.06 1.00 0.00 0.00 0.01

Table 31: Error correlation matrix for the 9 parameter fit without assuming lepton universality given in Table 29.

mZ (GeV) 1990-2 1993-4 1995 91.1851 + 0.0091 91.1870 + 0.0046 91.1851 + 0.0039

1990-2 1.00 0.02 0.01

1993-4 0.02 1.00 0.09

1995 0.01 0.09 1.00

Table 32: Results of the model-indep endent Z parameters fit with indep endent mZ for the three main data-taking phases, 1990-2, 1993-4 and 1995. The correlations are given in columns 3-5.

83


Parameter mZ (GeV) Z (GeV) 0 h (nb) Re R÷ R A0,e FB A0,÷ FB A0, FB R A0, FB 0 (nb) had (MeV) (MeV) inv (MeV) inv / s log10 (mH /GeV)

statistics 0.0023 0.0036 0.031 0.067 0.050 0.055 0.0038 0.0022 0.0026 0.034 0.0015 0.0032 2.8 0.12 1.9 0.017 0.0035 0.41

Error sources included systematics selection lumi t-chan. Ebeam 0.0001 0.0010 0.033 0.040 0.027 0.071 0.0016 0.0004 0.0014 0.027 0.0006 0.0021 1.8 0.05 1.6 0.018 0.0026 0.06 0.0003 0.0007 0.029 0.009 0.008 0.010 0.0000 0.0000 0.0000 0.008 0.0000 0.0018 0.7 0.03 0.7 0.010 0.0026 0.13 0.0001 0.0000 0.0000 0.027 0.000 0.000 0.0018 0.0000 0.0000 0.005 0.0003 0.0005 0.3 0.01 0.2 0.002 0.0008 0.05 0.0018 0.0013 0.011 0.014 0.000 0.000 0.0004 0.0003 0.0003 0.003 0.0002 0.0004 0.8 0.04 0.5 0.004 0.0006 0.07

addtl. theory 0.0003 0.0002 0.008 0.004 0.004 0.004 0.0001 0.0001 0.0001 0.004 0.0001 0.0005 0.3 0.01 0.3 0.003

Table 33: Contribution of the various error sources to the uncertainty in the direct and derived Z resonance parameters. "Statistics" include the event counting and luminosity p oint-to-p oint statistical errors; "selection" are the systematic errors associated with the event selection; "lumi" errors contain the exp erimental systematic uncertainties and the theoretical error of the luminosity cross-section; "t-channel" are the theoretical uncertainties of the t-channel correction for e+ e- e+ e- ; "Ebeam " includes the uncertainties in the LEP centre-of-mass energy and centre-of-mass energy spread; "addtl. theory" gives the uncertainties related to the determination of the fit parameters from the measured cross-sections and asymmetries which are not included in the tables.

Without lepton universality
inv

With lepton universality 498.1 + 2.6

Standard Model prediction
26 93 +0.058 83.977-0.206 +0.058 83.976-0.206 +0.058 83.786-0.206 +0.058 83.977-0.206 +1.8 1741.5-5.9 +0. 501.64-0.

(MeV)

494.4 + 4.2 83.66 + 0.20 84.03 + 0.30 83.94 + 0.41

ee (MeV) ÷÷ (MeV) (MeV) (MeV)
had

83.82 + 0.15 1748.8 + 4.6 1745.4 + 3.5

(MeV)

Table 34: Z partial decay widths obtained from a parameter transformation from the fitted modelindep endent Z parameters given in Table 29. In the last column we give the values of the widths calculated in the context of the SM assuming the parameter variations given in Equation 25. 84




inv ee ÷÷ had

inv 1.00 0.81 -0.40 -0.35 -0.63

ee 0.81 1.00 -0.19 -0.15 -0.25

÷÷ -0.40 -0.19 1.00 0.33 0.64

-0.35 -0.15 0.33 1.00 0.47

had -0.63 -0.25 0.64 0.47 1.00

Table 35: Error correlation matrix for the measurements of the partial widths, without assuming lepton universality, presented in Table 34. inv 1.00 0.60 -0.27 0.60 1.00 0.36 had -0.27 0.36 1.00



inv

had

Table 36: Error correlation matrix for the measurements of the partial widths, assuming lepton universality, presented in Table 34. mH (GeV)
Z inv had exp

-

SM

(MeV) 1000 7.1 -2.6 9.6 0.05 -0.11 0.25 0.36

errors (MeV) +4.1 +2.6 +3.5 +0.15 +0.20 +0.30 +0.41 +2.1 +0.2 +1.9 +0.05 +0.05 +0.05 +0.05



new 95

(MeV) 1000 14.8 3.7 16.2 0.34 0.35 0.78 1.08

150 -0.1 -3.5 3.9 -0.15 -0.31 0.05 0.16

150 9.0 3.4 10.8 0.22 0.26 0.63 0.93

ee ÷÷

Table 37: Upp er limits for total and partial Z widths. The second column gives the difference b etween the measured width and the exp ected width in the SM for mH = 150 GeV, and the third column for mH = 1000 GeV. These two results share the same errors, which are given in the fourth column. The first error is exp erimental and the second error reflects parametric uncertainties in the SM input (5) parameters s , mt and had as sp ecified in Equation 25. The last two columns show the corresp onding upp er limits (one-sided Bayesian limits at 95 % C.L.) for new contributions, b eyond the SM. For , inv and had the exp erimental results with lepton universality imp osed have b een used. Both exp erimental and theoretical uncertainties are correlated b etween the different widths, therefore the limits cannot b e used simultaneously. Without lepton universality A A A A
e ÷ +0. 0.109-0. +0. 0.194-0. +0. 0.177-0. 025 032 086 044 083 047 +0. 0.1392-0. 0078 0082 +0. 0.1450-0. 0030 0084

With lepton universality

SM prediction

Table 38: The leptonic coupling parameters obtained from a parameter transformation from the modelindep endent Z parameters given in Table 29. In the last column we give the value of the parameter calculated in the context of the SM assuming the parameter variations given in Equation 25. 85


A A A

e ÷

Ae 1.000 -0.869 -0.777

A÷ -0.869 1.000 0.681

A -0.777 0.681 1.000

Table 39: Error correlation matrix for the measurements of the leptonic coupling parameters, presented in Table 38. Without lepton universality gAe gA gA gA gVe gV gV gV
÷ +0. -0.027-0. +0. -0.049-0. +0. -0.045-0. 008 006 011 022 012 021 +0. -0.0350-0. 0021 0020 +0. -0.0365-0. 0022 0008 ÷

With lepton universality

Standard Model prediction

- - -

+0. 0.5009-0. +0. 0.5004-0. +0. 0.5011-0.

0007 0007 0026 0013 0024 0016 +0. -0.50095-0. 00046 00046 +0. -0.50130-0. 00047 00013

Table 40: Axial-vector and vector couplings obtained from a parameter transformation from the standard LEP parameter set given in Table 29. In the last column we give the values of the couplings calculated in the context of the SM assuming the parameter variations given in Equation 25. gAe gAe gA÷ gA gVe gV÷ gV 1.00 -0.37 -0.29 -0.41 0.34 0.31 gA gA gVe -0.41 0.70 0.52 1.00 -0.87 -0.77 gV gV

÷



÷



-0.37 1.00 0.49 0.70 -0.83 -0.54

-0.29 0.49 1.00 0.52 -0.45 -0.70

0.34 -0.83 -0.45 -0.87 1.00 0.68

0.31 -0.54 -0.70 -0.77 0.68 1.00

Table 41: Error correlation matrix for the measurements of the axial vector and vector couplings, without assuming lepton universality, presented in Table 40. Observable R
Z 0 h 0

s
+0. 0.132 + 0.007-0. +0. 0.119 + 0.008-0. +0. 0.114 + 0.010-0. +0. 0.127 + 0.005-0. 003 001 017 004 002 001 003 001

Table 42: Determination of s from the Z resonance parameters. The central value is obtained with (5) the SM parameters mt , mH and had as sp ecified in Equation 25. The second error reflects the effect on s when these are varied within the given ranges. In all cases an additional uncertainty of +0.002 arises from QCD uncertainties on had , and must b e included. 86


OPAL Z resonance measurements alone mZ (GeV) mt (GeV)
(5) had s (m2 Z

external mt = (174.3 + 5.1)GeV 91.1851+ 0.0030 174.3+5.1

external mt = (174.3 + 5.1)GeV s = 0.1184 + 0.0031 91.1852+ 0.0030 173.4+5.1 2.800+0.064 0.121+0.003
+0. 2.28-0. 44 89 190+335 -165

91.1851+0.0030 162+ 2.803+ 0.125+ 15+25 -5 -0. 0.064+0. +0. 0.005-0. - 150 (fixed) 159.7/200
003 001 004 001

(×102 ) )

2.802+0.065 0.127+0.005
+0.46 2.59-0.55 390+750 -280

log10 (mH /GeV ) mH (GeV) /d.o.f .
2

159.7/200

161.7/201

Table 43: Results of the full SM fit to the measured cross-sections and asymmetries. In the second column mH is fixed to 150 GeV. The second errors show the variation for mH = 90 GeV (lower) and mH = 1000 GeV (upper). In the remaining columns mH is determined from the data, with additional (5) external constraints, as indicated. In all cases the electromagnetic coupling constant had was used as additional fit parameter with the constraint given in Equation 25.



ee F ee F ee F ee B ee B ee B

(p (p (p (p (p (p

k k k k k k

-) 0) +) -) 0) +)

(pb) 1.28 1.12 1.20 0.32 0.32 0.32

ee ee Table 44: Uncertainties in the forward (F ) and backward (B ) electron cross-section for t-channel plus s-t interference diagrams. The designations (pk -), (pk 0), and (pk +) refer to the energy p oints resp ectively b elow, at and ab ove the Z resonance, where the p eak region is taken to lie within +0.9 GeV of mZ .

87


Parameter mZ (GeV) Z (GeV)
tot rhad

fitting j

tot had

fixing j

tot had

SM Prediction

91.1901 + 0.0115 2.4936 + 0.0047 2.962 + 0.010 0.01 + 0.650 0.14122 + 0.00085 0.14205 + 0.00061 0.14221 + 0.00078 -0.085 + 0.052 -0.013 + 0.042 -0.007 + 0.045 0.00134 + 0.00086 0.00265 + 0.00046 0.00238 + 0.00059 0.763 + 0.070 0.732 + 0.036 0.740 + 0.042 146.6 / 187

91.1866 + 0.0031 2.4943 + 0.0041 2.963 + 0.009 0.14138 + 0.00069 0.14212 + 0.00056 0.14229 + 0.00074 -0.076 + 0.044 -0.003 + 0.030 0.003 + 0.034 0.00140 + 0.00084 0.00261 + 0.00044 0.00234 + 0.00057 0.763 + 0.070 0.732 + 0.036 0.740 + 0.042 146.7 / 188

input
+0. 2.4949-0. +0. 2.9627-0. +0. 0.2181-0. 0021 0074 0051 0173 0048 0139

j

tot had

tot re tot r÷ tot r

+0. 0.14260-0.

00020 00070

j j j r r r j j j

tot e tot ÷ tot fb e fb ÷ fb fb e fb ÷ fb

+0. 0.0043-0.

0002 0005

+0. 0.00300-0.

00013 00035

+0. 0.7985-0.

0007 0016

2 /d.o.f.

Table 45: Results of the 16 and 15 parameter S-Matrix fits to the measured cross-section and lepton asymmetry data. The uncertainties on the LEP energy are included in the errors quoted. In the last column we give the predictions of the SM assuming the parameters and variations given in Table 25.

88


Parameters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 mZ Z tot rhad tot j had tot re tot r÷ tot r tot je j tot ÷ j tot r fb e r fb ÷ r fb j fb e j fb ÷ j fb

1 1.000 -.441 -.428 -.964 -.578 -.357 -.281 -.537 -.709 -.658 -.252 .295 .237 .024 .002 .007

2 -.441 1.000 .934 .466 .684 .757 .595 .232 .344 .320 .116 -.117 -.096 -.011 .043 .035

3 -.428 .934 1.000 .457 .679 .755 .594 .222 .333 .310 .117 -.110 -.090 -.013 .043 .036

4 -.964 .466 .457 1.000 .585 .377 .296 .536 .708 .657 .247 -.289 -.232 -.023 .000 -.005

5 -.578 .684 .679 .585 1.000 .561 .433 .338 .431 .400 .277 -.175 -.141 -.027 .024 .017

6 -.357 .757 .755 .377 .561 1.000 .475 .186 .359 .259 .098 -.079 -.075 -.011 .059 .028

7 -.281 .595 .594 .296 .433 .475 1.000 .147 .218 .277 .076 -.073 -.043 -.008 .027 .074

8 -.537 .232 .222 .536 .338 .186 .147 1.000 .395 .365 .130 -.161 -.129 .189 -.002 -.004

9 -.709 .344 .333 .708 .431 .359 .218 .395 1.000 .482 .181 -.190 -.170 -.017 -.041 -.003

10 -.658 .320 .310 .657 .400 .259 .277 .365 .482 1.000 .168 -.197 -.138 -.016 .000 -.051

11 -.252 .116 .117 .247 .277 .098 .076 .130 .181 .168 1.000 -.084 -.068 .054 -.001 -.002

12 .295 -.117 -.110 -.289 -.175 -.079 -.073 -.161 -.190 -.197 -.084 1.000 .082 .008 .181 .004

13 .237 -.096 -.090 -.232 -.141 -.075 -.043 -.129 -.170 -.138 -.068 .082 1.000 .006 .002 .162

14 .024 -.011 -.013 -.023 -.027 -.011 -.008 .189 -.017 -.016 .054 .008 .006 1.000 .000 .000

15 .002 .043 .043 .000 .024 .059 .027 -.002 -.041 .000 -.001 .181 .002 .000 1.000 .002

16 .007 .035 .036 -.005 .017 .028 .074 -.004 -.003 -.051 -.002 .004 .162 .000 .002 1.000

89

Table 46: Error correlation matrix for the S-Matrix fit in Table 45 without assuming lepton universality, and without fixing j Z interference in the hadron channel.

tot had

, which controls the


e+ e- e+ e-
MU HCA L ECA L

e+ e- ÷+ ÷
HCA L ECA L

-

CT

CT

e+ e- +
HCA L ECA L

-

e+ e- qq
HCA L ECA L

CT

CT

Figure 1: Typical examples for the four event categories. These views of the four final states measured in this analysis all show the detector pro jected along the b eam axis, parallel to the magnetic field generated by the solenoid located b etween CT and ECAL. The approximately radial lines within the volume of the central tracker (CT) represent the reconstructed tracks of inonising particles. The dark trap ezoids in the volumes of the electromagnetic calorimeter (ECAL) and hadronic calorimeter (HCAL) represent corresp onding observed energy dep osits. The arrows within the volume of the multiple-layer muon chamb ers (MU) represent reconstructed track segments of p enetrating particles.

90


e

-

f

e

-

F F e f

e

-

f f Z F (c) F e f f e (f)
+ -

Z e+ (a) e
-

f e f f e (d)
+ -

e+ e
-

(b) Z e

f e e f
+

e+ e
-

f

e

+

e

+

(e)



f

e

+

e

Figure 2: Two- and four-fermion diagrams. (a) The basic fermion-pair s-channel diagram which is resp onsible for almost all of the f f signal. (b) Radiative correction to (a) with a fermion pair FF in the initial state. Treated as signal. (c) Radiative correction to (a) with a fermion pair FF in the final state. Treated as signal. (d) Multip eripheral (two-photon) diagram. Treated as background. (e) Initial-state pair production in the t-channel, treated as background except for e+ e- e+ e- . (f ) Final-state pair production in the t-channel, treated as background except for e+ e- e+ e- . Only the dominant b oson is indicated. Additional contributions arise by substituting Z and .

91


OPAL
Events
(a) 10 4 10 3 10 2 0 10 5 (c) 10 4 10
3

(b) 10 4 10 3 10 2 80 0

20

40

60

20

40

60

Nall Events
10 5 10 4 10 3 10 2 10 2 0 0.2 0.4 0.6
mh

mhemi (GeV)
(d)

0.8

0

0.2 0.4 0.6 0.8

1

Ecal / s

Rbal

energy

Figure 3: The e+ e- qq event selection. A comparison of the cut variables b etween data and Monte Carlo simulation: (a) the total multiplicity, Nall , (b) the hemisphere invariant mass sum, mhemi , (c) the ener mh visible energy, Ecal / s, (d) the energy imbalance along the b eam direction, Rbal gy . The p oints are + e- qq Monte Carlo simulation. The shaded histograms the data and the op en histograms the e show the background Monte Carlo prediction which is dominated in the case of (a) and (b) by e+ e- + - and in the case of (c) and (d) by e+ e- e+ e- f f . The cuts are indicated by the arrows. In each case events are plotted only if they pass all the other selection cuts.

92


OPAL
Events
10 10 10 1 0 10 20 30 40
3 2

(a)

10 3 10 2 10 1 0

(b)

2.5

5

7.5 10 12.5

Nall Events
10 3 10 2 10 10 1 0 0.1 0.2 0.3 0.4 0.5 1 0.6 (c) 10
2

mhemi (GeV)
(d)

0.8

1

Ecal / s
mh
Figure 4: Distributions x-axis, which have b een text. The data (p oints) plots events are required

Rbal

energy

of the emulated cut variables for e+ e- qq events at small angles to the processed by the acceptance hole emulation program, as describ ed in the and the JETSET Monte Carlo events (histogram) are compared. For these to pass the standard e+ e- qq selection b efore the hole emulation.

93


Probability

1 10 10 10 10
-1 -2 -3 -4

OPAL
Endcap

0

0.2

0.4

0.6

0.8

isol Eclus

1

1.2

(GeV)

Probability

10 10 10

-1 -2 -3

Barrel

0

0.2

0.4

0.6

0.8

isol Eclus

1

1.2

(GeV)

Figure 5: Detector simulation study for the e+ e- qq selection. The plots show the energy sp ectra (normalised to the numb er of tracks) for single ECAL clusters near isolated tracks for data (histogram), for the standard Monte Carlo simulation (closed circles) and for the corrected Monte Carlo simulation (op en circles). The upp er plot is for tracks p ointing into the endcaps of the electromagnetic calorimeter. The lower plot is for tracks p ointing into the barrel region. Both plots are summed over all momenta. A clear improvement in the modelling of the distributions can b e seen after correction.

94


OPAL low (nb)
1 0.8 0.6 peak+2 0.4 0.2 0 peak-2 peak

0

5

10

15

20

high (nb)

25

30

Figure 6: The non-resonant background evaluation for the e+ e- qq selection. The plot shows mh the cross-section for events having a low visible energy, 0.10 < Ecal / s < 0.18, or a large energy ener imbalance along the b eam direction, 0.50 < Rbal gy < 0.75, (low ) versus the cross-section for events ener mh having a high Ecal / s (> 0.18) and a small Rbal gy (< 0.50) (high ). The intercept of the fitted ener mh straight line yields the non-resonant background estimate. The distributions for Ecal / s and Rbal gy are shown in Figure 3 (c) and (d), resp ectively.

95


ptotal/s

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

OPAL

÷÷

+-

ee
+-

+-

0.2

0.4

0.6

0.8

1.0

1.2

E

total/

s

1.4

Figure 7: The separation of Monte Carlo e+ e- + - events using the energy dep osited in the electromagnetic calorimeter summed over all clusters, Etotal versus the scalar sum of the momenta of the reconstructed tracks in the event, ptotal . Both quantities are scaled to the centre-of-mass energy s. The triangles show e+ e- e+ e- events, the solid squares show e+ e- ÷+ ÷- events and the op en circles show e+ e- + - events.

96


OPAL
Events
10 4 10 3 10 2 10 1 0 0.2 0.4 0.6 50 (b) 40 30 20 10 0 2 4 0 0 0.25 0.5 0.75 1 (c) 0.8 1 1.2 (a)

Etotal/s

Events

140 120 100 80 60 40 20 0

acop(degrees)

ptotal/s

Figure 8: The e+ e- e+ e- event selection. (a) Distribution of the sum of electromagnetic energy, Etotal / s, after all the other cuts have b een applied, in the angular range | cos e- | < 0.70. The arrow indicates the selection cut used. (b) Distribution of the acoplanarity angle of e- and e+ tracks for events satisfying 0.7 < Etotal / s < 0.8. (c) Distribution of the scalar sum of the track momenta, ptotal / s, for the events satisfying 0.7 < Etotal / s < 0.8 and acop < 0.2 . In each case the p oints are the on-p eak data, the op en histogram shows the Monte Carlo exp ectation and the shaded histogram shows the contribution from background processes.

97


OPAL
Events
2500 2000 1500 1000 500 0 1 -1 -0.5 0 0.5 peak-2 10 3 10 2 10 peak-2

cos

1

0

5

10



15
acol

20

25

(degrees)

Events

15000 peak 10000 10 5000 0 10
4 3

peak

10 2 10 -1 -0.5 0 0.5

cos

1

0

5

10



15
acol

20

25

(degrees)

Events

1500 peak+2 1000 500 0 10 3 10 2 10 1 -1 -0.5 0 0.5 peak+2

cos

1

0

5

10



15
acol

20

25

(degrees)

Figure 9: The e+ e- e+ e- event acceptance. Angular distributions and acollinearity distributions from data samples at three different centre-of-mass energies. In each case the p oints are the data, the op en histogram shows the Monte Carlo exp ectation and the shaded histogram shows the contribution from background processes. The arrows indicate the acceptance cuts used.

98


Events

10 10 10 10 10

5 4 3 2

OPAL
(a)

0

0.2 (b)

0.4

0.6

0.8

Events

10 10 10 10

4 3 2

Evis/s

1

÷

1.2

0

0.2

0.4

0.6

0.8

Evis/s

1

÷

1.2

÷ Figure 10: Distribution of Evis for events passing all of the e+ e- ÷+ ÷- selection cuts with the ÷ exception of Evis > 0.6 s. The 1993-1995 data are shown by the p oints and the histograms indicate the Monte Carlo exp ectation. The shaded histograms represent the background (mainly e+ e- + - ). Figure (a) gives the distribution for all events, while (b) shows those events with tracks within 0.5 of the anode wire planes of the central tracking chamb er.

99


Events

8.10 6.10 4.10 2.10

4

3.10 (a) 2.10 1.10

4

OPAL
(b)

4 4

4 4

4

0

0

4

8

12

16

20

0

0

5

10 15 20 25 30

Ntrack Events
10 6 10 10
5 4

Ntrack+Ncluster
10 4

(c)
10 3 10 2

(d)

10 3 10 2 0 0.2 0.4 0.6 0.8 1 1.2

Etotal/s

0.05 0.15 0.25 0.35 0.45 0.55 Evis/s

Figure 11: Cuts used in the e+ e- + - event selection. Distributions of (a) and (b) the charged and total multiplicity, Ntrack and Ntrack + Ncluster used to reject background from e+ e- qq, (c) Etotal for events with | cos | < 0.7 showing the cut used to reject the background from e+ e- e+ e- in this barrel region of the detector and (d) Evis / s showing the cut used to reject two-photon interaction events. In all cases all other selection cuts have b een applied. The p oints represent the 1993-1995 data (on-p eak and off-p eak), the histograms show the Monte Carlo exp ectation and the shaded histograms indicate the background comp onent, predominantly e+ e- qq in (a) and (b), e+ e- e+ e- in (c) and e+ e- e+ e- + - in (d). The cuts are indicated by the arrows.

100


Events

10 5 10 10 10 10 0 10 20 30 40 50 60
4 3 2

1800 (a) 1500 1200 900 600 300 0 0.5 2000

OPAL
(b)

acol (degrees) (c)

1

1.5

Evis/s (d)



2

Events

400 300 200 100 0 0 5 10 15

1500 1000 500 0 3 4 5 6 7 8 9

MOD(,15 )
o

Ntrack

Figure 12: Systematic checks of the e+ e- + - event selection. Each plot corresp onds to one of the control samples used to check the efficiency and background of the e+ e- + - event selection. Plot a) corresp onds to the sample used to assess the effect of the acollinearity cut, b) corresp onds to the check of the inefficiency due to the cut on Evis in the + e- ÷+ ÷- background events in the region of the jet region | cos | > 0.7, c) shows the excess of e chamb er anode planes, and d) shows the distribution of Ntrack for the control sample used to assess the e+ e- qq background. In all plots the data are shown by the p oints, the total Monte Carlo exp ectations are shown by the histograms and the contributions from events other than e+ e- + - are shown by the shaded histograms. Details are given in the text.

101


10 10 10 10 10 10 10 1

Occurrences

7 6 5 4 3 2

OPAL

0 10 6 10 5 10 4 10 3 10 2 10 1 0

2

4

6

8

10

12

14

16

18

20

Nhad Occurrences

10

20

30

40

50

60

70

Nlumi
Figure 13: The distribution of the numb er of events of typ e (a) observed b etween adjacent events of typ e (b) forms a sensitive test for the constancy of the a/b rate ratio. When the ratio truly remains constant, the resulting distribution is a pure exp onential, indicated by the line, whose logarithmic slop e dep ends on the event typ e ratio. The upp er plot shows the numb er of e+ e- qq events observed b etween adjacent luminosity events in the 1993-1995 p eak data. The lower plot shows the reverse: the numb er of luminosity events observed b etween adjacent e+ e- qq events. The tails of these distributions are particularly sensitive to any interruption in the exp erimental sensitivity to events of typ e (b), at the level of a few minutes in the sample of ab out six months of livetime shown. There were no overflows in either of these distributions.

102


Events

2000 1500 1000 500 0 -1 -0.8 -0.6 -0.4 -0.2 0 1000 peak-2 400 500 200 0 0 (b) 0.2 0.4 peak

OPAL
(a)

0.6

0.8

1 cos(c)

Events

600

peak+2

-1

-0.5

0

0.5

1 cos-

-1

-0.5

0

0.5

1 cos-

Figure 14: The e+ e- + - distribution of p olar angle. Distributions of cos - for the combined 1993-1995 data (p oints) separated into p eak-2, p eak and p eak+2 energy p oints. For these plots | cos - | is determined from the average of the p olar angles of the p ositive and negative cones determined using tracks and electromagnetic clusters. The additional cuts used for the asymmetry analysis have b een applied. The Monte Carlo exp ectation is shown by the histogram and the background contribution is shown as the shaded comp onent.

103


d/dcos÷- (nb)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -1

e e ÷ ÷
+-

+peak-2 peak peak+2

OPAL

-0.5

0

0.5

cos

1
÷-

Figure 15: Observed differential cross-sections as a function of cos for the process e+ e- ÷+ ÷- at the three centre-of-mass energies in the 1993-1995 data. Corrections have b een applied for inefficiency and background. Only statistical errors are shown, bin-by-bin systematic uncertainties are not included. The curves corresp ond to fits to a simple parametrisation of the form a(1 + cos2 )+ b cos .

104


d/dcos- (nb)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -1

e e
+-

+peak-2 peak peak+2

OPAL

-0.5

0

0.5

cos

1
-

Figure 16: Observed differential cross-sections as a function of cos for the process e+ e- + - at the three centre-of-mass energies in the 1993-1995 data. Corrections have b een applied for inefficiency and background. Only statistical errors are shown, bin-by-bin systematic uncertainties are not included. The curves corresp ond to fits to a simple parametrisation of the form a(1 + cos2 )+ b cos .

105


d/dcose- (nb)

1.2 1.0 0.8 0.6 0.4 0.2

e e e e
+-

+peak-2 peak peak+2

OPAL
0.0 -1 -0.5 0 0.5

cose

1
-

Figure 17: Observed differential cross-sections as a function of cos for the process e+ e- e+ e- at the three centre-of-mass energies in the 1993-1995 data. Corrections have b een applied for inefficiency and background. Only statistical errors are shown, bin-by-bin systematic uncertainties are not included. The curves show the predictions of ALIBABA.

106


d/dz(z=1) (pb)

10 0 5 -0.2 0 88 90 92

s (GeV)

94

A

(a)

FB

0.2

(b)

88 A
FB

90

92

s (GeV)

94

symmetric term

of Z and ZZ terms

d/dz(z=1) (pb)

1 0.5 0 -0.5 -1 88

(c)

d/dz(z=1) (pb)

100 0 -100 88

(d)

90

92

s (GeV)

94

90

92

s (GeV)

94

Z symmetric term

Z anti-symmetric term

d/dz(z=1) (pb)

1500 1000 500 0 88

(e)

d/dz(z=1) (pb)

(f)
20 0 88 90 92

90

92

s (GeV)

94

s (GeV)

94

ZZ symmetric term

ZZ anti-symmetric term

Figure 18: The contribution at tree level of the five terms of equation 8 (solid lines) to the differential cross-section of e+ e- ÷+ ÷- (d/ dz (z = 1), z cos , in (pb)) as a function of s. Plots (a), (c) and (e) give the symmetric terms from , /Z and Z exchange resp ectively; (d) and (f ) the antisymmetric /Z and Z exchange terms. The dashed line in (c) demonstrates the effect when the imaginary parts of the couplings are taken into account. The dashed line in (e) illustrates the profound change of the lineshap e due to initial-state radiation. The dashed curve in (f ) shows the /Z term of (d) sup erp osed to illustrate how rapidly it dominates the more interesting ZZ term as the energy moves away from the p eak. (b) shows the forward-backward asymmetry which results from the /Z (solid) and ZZ (dashed) terms when the cross-sections are integrated over -1 < cos < +1. The dotted vertical line in each plot indicates s = mZ . 107


mH (GeV)

10 3

10 3

OPAL

10 2 63.2 63.4 63.6 63.8 64
s Czz

10 2

× 1000

0.2 10 3

0.25

0.3

a Czz

0.35

× 1000

0.4

mH (GeV)

10 3

10 2 -0.02 -0.01 0
s Cz

10 2 0.01 0.2 0.22 0.24 0.26

C

a z

mt=175+5 GeV

s =0.119+0.002

(5) h =0.02804+0.00065

Figure 19: Comparison of the results from the C-parameter fit with the SM prediction as a function of Higgs mass mH . The vertical bands indicate the fit results and their horizontal widths corresp ond to one standard deviation error intervals. The shaded area shows (linearly) the variation of the SM (5) prediction for mt , s , and had in the indicated ranges. These parameters are insensitive to s , and its variation band is therefore invisible.

108


mH (GeV)

10 3

10 3

OPAL

10 2 2.48 10 3 2.49

10 2

Z (GeV)

2.5

2.51 10 3

41.4

41.5

0 h (nb)

41.6

mH (GeV)

10 2 20.6 20.7 20.8 20.9
mt=175+5 GeV

10 2 21 0.01 0.015
0,l AFB
(5) h =0.02804+0.00065

Rl
s =0.119+0.002

Figure 20: Comparison of the results of the 5-parameter model-indep endent fit with the SM prediction as a function of Higgs mass mH . The vertical bands indicate the fit results and their horizontal widths corresp ond to one standard deviation error intervals. The shaded area shows (linearly) the variation (5) of the SM prediction for mt , s , and had in the indicated ranges.

109


Cross-section (nb)

35 30 25 20 15

hadrons

1990 1991 1992 1993 1994 1995

Cross-section (nb)

1.2 1 0.8 0.6 0.4

OPAL

ee

+-

s+t s non-s

10 5 0 0.01 0 88 89 90 91 92 93 94 95 0.2 0 0.05 0

s(GeV)

88

89

90

91

92

93

s(GeV)

94

95

data/fit - 1

-0.01

89.4

89.5

91.2

91.3

92.9

93

93.1

data/fit - 1

-0.05

89.4

89.5

91.2

91.3

92.9

93

93.1

Cross-section (nb)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 88

÷÷

Cross-section (nb)

1.8

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

+-



+-

89

90

91

92

93

s(GeV)

94

95

0 0.05 0

88

89

90

91

92

93

s(GeV)

94

95

data/fit - 1

0

-0.05

89.4

89.5

91.2

91.3

92.9

93

93.1

data/fit - 1

0.05

-0.05

89.4

89.5

91.2

91.3

92.9

93

93.1

Figure 21: The measured cross-sections for hadronic and leptonic final states as a function of centre-ofmass energy. The errors shown are statistical only. The solid line is the result of the 9-parameter modelindep endent fit to the combined leptonic and hadronic data (without assuming lepton universality) describ ed in Section 11.2. The lower plots show the residuals to the fit. For the electrons, the dashed curves show the contributions of the pure s-channel, and t-channel plus s - t interference, resp ectively. 110


Asymmetry

0.4 0.3 0.2

ee

+-

1990 1991 1992 1993 1994 1995

Asymmetry

0.5

0.3 0.2 0.1 0 -0.1

OPAL

÷÷

+-

0.1 -0.2 0 -0.1 0.05 -0.3 88 89 90 91 92 93 94 95 -0.4 0.05 88 89 90 91 92 93 94 95

s(GeV)

s(GeV)

data - fit

0

data - fit
89.4 89.5 91.2 91.3 92.9 93 93.1

0

-0.05

-0.05

89.4

89.5

91.2

91.3

92.9

93

93.1

Asymmetry

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 88 89 90 91 92 93 94 95



+-

s(GeV)

0.05

data - fit

0

-0.05

89.4

89.5

91.2

91.3

92.9

93

93.1

Figure 22: The measured forward-backward asymmetry in leptonic final states as a function of centreof-mass energy. The errors shown are statistical only. The solid line is the result of the 9-parameter model-indep endent fit to the combined leptonic and hadronic data (without assuming lepton universality) describ ed in Section 11.2. The lower plots show the residuals to the fit. 111


OPAL
0,l FB

A

0.02
mt s mH

0.015

e +e ÷+ ÷ + l l-

+-

0.01

0.005

0 20.6

20.7

20.8

20.9

21

21.1

R

l

Figure 23: Contours of 68% probability in the A0, - R plane for each of the three lepton sp ecies FB (dotted, dashed). The solid contour results from a fit assuming lepton universality. In this plot, the results for the are corrected for the mass effect so that a direct comparison with other lepton sp ecies (5) is p ossible. The SM prediction with s , mt , mH and had as sp ecified in Equation 25 is also shown as the intersection of the three arrows. The arrows indicate the range of the variation when s , mt and mH are varied within the ranges sp ecified in Equation 25.

112


OPAL
V

g

-0.03

-0.02
m
-0.035
H

-0.04
m
-0.04 -0.502
t

-0.501

-0.5

-0.06 -0.08 -0.1 -0.505

e+e- + ÷+ ÷ + l l-0.5 -0.495 -0.49

g

A

Figure 24: Contours of 68% probability in the gV - gA plane for each of the three lepton sp ecies (dotted, dashed). The solid contour results from a fit assuming lepton universality and is also shown enlarged in the inset figure. Here, the band indicates the SM prediction when mt and mH are varied as sp ecified in equation 25.

113


s

0.14 0.13 0.12 0.11 0.1 20.6 20.8

0.14 0.13 0.12 0.11 0.1 2.48

OPAL

Rl s
0.14 0.13 0.12 0.11 0.1 41.2 41.4 41.6 0.14 0.13 0.12 0.11 0.1
+850 -60

Z (GeV)

2.5

0 h (nb)

1.98

2

0 (nb) l

2.02

mt=175+5 GeV

mH = 150

GeV

(5) h =0.02804+0.00065

Figure 25: Comparison of Z resonance parameters with the SM prediction as a function of s . The vertical bands indicate the fit results and their horizontal widths corresp ond to one standard deviation error intervals. The hatched area shows (linearly) the variation of the SM prediction when mt , mH (5) and had are varied as sp ecified in Equation 25 and indicated in the figure.

114


OPAL
s

0.135

fit to 9 pars fit to , AFB

0.13

0.125

0.12

0.115 10

2

10

3

mH (GeV)
Figure 26: Contours of 68 % probability in the s - mH plane with mt constrained to 174.3 + 5.1 GeV [63]. The undashed contour is obtained from the SM fit to cross-sections and asymmetries. For the dashed contour the fit was made on the results of the 9-parameter model-indep endent fit. The small circle and the star give the central values for the two fits, resp ectively. In b oth cases a numerical evaluation of the 68 % C.L. contour was p erformed which accounts for asymmetric or non-parab olic parameter dep endencies (MINUIT contour [62]).

115


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