application/xmlMeasurement of the running of the electromagnetic coupling at large momentum-transfer at LEPL3 CollaborationP. AchardO. AdrianiM. Aguilar-BenitezJ. AlcarazG. AlemanniJ. AllabyA. AloisioM.G. AlviggiH. AnderhubV.P. AndreevF. AnselmoA. ArefievT. AzemoonT. AzizP. BagnaiaA. BajoG. BaksayL. BaksayS.V. BaldewS. BanerjeeSw. BanerjeeA. BarczykR. BarillèreP. BartaliniM. BasileN. BatalovaR. BattistonA. BayF. BecattiniU. BeckerF. BehnerL. BellucciR. BerbecoJ. BerdugoP. BergesB. BertucciB.L. BetevM. BiasiniM. BigliettiA. BilandJ.J. BlaisingS.C. BlythG.J. BobbinkA. BöhmL. BoldizsarB. BorgiaS. BottaiD. BourilkovM. BourquinS. BracciniJ.G. BransonF. BrochuJ.D. BurgerW.J. BurgerX.D. CaiM. CapellG. Cara RomeoG. CarlinoA. CartacciJ. CasausF. CavallariN. CavalloC. CecchiM. CerradaM. ChamizoY.H. ChangM. ChemarinA. ChenG. ChenG.M. ChenH.F. ChenH.S. ChenG. ChiefariL. CifarelliF. CindoloI. ClareR. ClareG. CoignetN. ColinoS. CostantiniB. de la CruzS. CucciarelliR. de AsmundisP. DéglonJ. DebreczeniA. DegréK. DehmeltK. DeitersD. della VolpeE. DelmeireP. DenesF. DeNotaristefaniA. De SalvoM. DiemozM. DierckxsensC. DionisiM. DittmarA. DoriaM.T. DovaD. DuchesneauM. DudaB. EchenardA. ElineA. El HageH. El MamouniA. EnglerF.J. EpplingP. ExtermannM.A. FalaganS. FalcianoA. FavaraJ. FayO. FedinM. FelciniT. FergusonH. FesefeldtE. FiandriniJ.H. FieldF. FilthautP.H. FisherW. FisherI. FiskG. ForconiK. FreudenreichC. FurettaYu. GalaktionovS.N. GanguliP. Garcia-AbiaM. GataullinS. GentileS. GiaguZ.F. GongG. GrenierO. GrimmM.W. GruenewaldM. GuidaV.K. GuptaA. GurtuL.J. GutayD. HaasD. HatzifotiadouT. HebbekerA. HervéJ. HirschfelderH. HoferM. HohlmannG. HolznerS.R. HouB.N. JinP. JindalL.W. JonesP. de JongI. Josa-MutuberríaM. KaurM.N. Kienzle-FocacciJ.K. KimJ. KirkbyW. KittelA. KlimentovA.C. KönigM. KopalV. KoutsenkoM. KräberR.W. KraemerA. KrügerA. KuninP. Ladron de GuevaraI. LaktinehG. LandiM. LebeauA. LebedevP. LebrunP. LecomteP. LecoqP. Le CoultreJ.M. Le GoffR. LeisteM. LevtchenkoP. LevtchenkoC. LiS. LikhodedC.H. LinW.T. LinF.L. LindeL. ListaZ.A. LiuW. LohmannE. LongoY.S. LuC. LuciL. LuminariW. LustermannW.G. MaL. MalgeriA. MalininC. MañaJ. MansJ.P. MartinF. MarzanoK. MazumdarR.R. McNeilS. MeleL. MerolaM. MeschiniW.J. MetzgerA. MihulH. MilcentG. MirabelliJ. MnichG.B. MohantyG.S. MuanzaA.J.M. MuijsB. MusicarM. MusyS. NagyS. NataleM. NapolitanoF. Nessi-TedaldiH. NewmanA. NisatiT. NovakH. NowakR. OfierzynskiG. OrgantiniI. PalC. PalomaresP. PaolucciR. ParamattiG. PassalevaS. PatricelliT. PaulM. PauluzziC. PausF. PaussM. PedaceS. PensottiD. Perret-GallixD. PiccoloF. PierellaM. PioppiP.A. PirouéE. PistolesiV. PlyaskinM. PohlV. PojidaevJ. PothierD. ProkofievG. Rahal-CallotM.A. RahamanP. RaicsN. RajaR. RamelliP.G. RancoitaR. RanieriA. RasperezaP. RazisD. RenM. RescignoS. ReucroftS. RiemannK. RilesB.P. RoeL. RomeroA. RoscaC. RosemannC. RosenbleckS. Rosier-LeesS. RothJ.A. RubioG. RuggieroH. RykaczewskiA. SakharovS. SaremiS. SarkarJ. SalicioE. SanchezC. SchäferV. SchegelskyH. SchopperD.J. SchotanusC. SciaccaL. ServoliS. ShevchenkoN. ShivarovV. ShoutkoE. ShumilovA. ShvorobD. SonC. SougaP. SpillantiniM. SteuerD.P. SticklandB. StoyanovA. StraessnerK. SudhakarG. SultanovL.Z. SunS. SushkovH. SuterJ.D. SwainZ. SzillasiX.W. TangP. TarjanL. TauscherL. TaylorB. TelliliD. TeyssierC. TimmermansSamuel C.C. TingS.M. TingS.C. TonwarJ. TóthC. TullyK.L. TungJ. UlbrichtE. ValenteR.T. Van de WalleR. VasquezV. VeszpremiG. VesztergombiI. VetlitskyG. ViertelS. VillaM. VivargentS. VlachosI. VodopianovH. VogelH. VogtI. VorobievA.A. VorobyovM. WadhwaQ. WangX.L. WangZ.M. WangM. WeberS. WynhoffL. XiaZ.Z. XuJ. YamamotoB.Z. YangC.G. YangH.J. YangM. YangS.C. YehAn. ZaliteYu. ZaliteZ.P. ZhangJ. ZhaoG.Y. ZhuR.Y. ZhuH.L. ZhuangA. ZichichiB. ZimmermannM. ZöllerPhysics Letters B 623 (2005) 26-36. doi:10.1016/j.physletb.2005.07.052journalPhysics Letters BCopyright © 2005 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936231-28 September 20052005-09-0826-36263610.1016/j.physletb.2005.07.052http://dx.doi.org/10.1016/j.physletb.2005.07.052doi:10.1016/j.physletb.2005.07.052http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.2

PLB22229S0370-2693(05)01043-910.1016/j.physletb.2005.07.052Elsevier B.V.Experiments

Fig. 1

t-channel Feynman diagrams contributing to Bhabha scattering. Diagrams with virtual-fermion vacuum-polarisation insertions generate an electromagnetic coupling α(Q2). The sum of all diagrams including zero, one, two or more vacuum-polarisation insertions is denoted by the diagram to the left with the double-wavy photon propagator.

Fig. 2

Measured Bhabha differential cross sections for the ten |cosθ| intervals used in the study as a function of the centre-of-mass energy s. The Standard Model predictions are represented by the solid lines.

Fig. 3

Measured differential cross section as a function of |cosθ|. Data at different centre-of-mass energies are combined at the luminosity-averaged centre-of-mass energy 〈s〉=198 GeV. The predictions in case of a running electromagnetic coupling and for a constant value α=α0 are also shown.

Fig. 4

Ratio between the measured Bhabha differential cross section as a function of |cosθ| and the Standard Model expectations including a running electromagnetic coupling. Data at different centre-of-mass energies are combined at the luminosity-averaged centre-of-mass energy 〈s〉=198 GeV. The inner error bars represent statistical uncertainties and the full error bars the sum in quadrature of the statistical and systematic uncertainties. The predictions for a constant value of α=α0 are also shown.

Fig. 5

Evolution of the electromagnetic coupling with Q2 determined from the present measurement of C for 1800 GeV2<−Q2<21600 GeV2, yellow band, and from previous data for Bhabha scattering at 2.10 GeV2<−Q2<6.25 GeV2 and 12.25 GeV2<−Q2<3434 GeV2[10]. The open symbols indicate the values of Q2 where α(Q2) was fixed to the QED predictions [5] in order to infer the values of α(Q2) shown by the full symbols. These QED predictions are shown by the solid line.

Fig. 6

Table 1

Luminosity-averaged centre-of-mass energies, 〈s〉, and corresponding integrated luminosities, L, used in the analysis. The s spread in each point is of the order of 1 GeV. The numbers of observed events, ND, are given together with the total Monte Carlo expectations, NMC, and their breakdown into signal, NS, and background, NB, events. The last row lists the average centre-of-mass energy, the total integrated luminosity and the total numbers of events

〈s〉 (GeV)L (pb−1)NDNMCNSNB188.6156.4115611155911288271191.629.719761953190548195.683.7567756735539134199.583.5538253385201137201.839.123792417235562205.275.9425941654063102206.7130.4738875127339173208.28.744148447311198.0607.4390633910138163938Table 2

Measured, Meas., and expected, Exp., folded differential cross sections for the eight average centre-of-mass energies, 〈s〉, and the ten |cosθ| intervals, with expected average values 〈|cosθ|〉. The first uncertainty is statistical and the second systematic

|cosθ|〈|cosθ|〉dσ/d|cosθ| (pb)〈s〉=188.6 GeV〈s〉=191.6 GeV〈s〉=195.6 GeV〈s〉=199.5 GeVMeas.Exp.Meas.Exp.Meas.Exp.Meas.Exp.0.00–0.090.05212.8±0.9±0.210.49.3±1.9±0.210.18.6±1.0±0.19.610.3±1.1±0.29.20.09–0.180.13810.9±0.8±0.111.310.2±2.0±0.211.010.5±1.2±0.110.59.8±1.1±0.110.00.18–0.270.22714.0±1.0±0.213.411.5±2.1±0.213.012.0±1.2±0.212.412.2±1.3±0.211.90.27–0.360.31716.2±1.0±0.117.114.3±2.4±0.216.618.0±1.5±0.215.914.7±1.4±0.215.20.36–0.450.40725.0±1.3±0.223.723.6±3.0±0.322.920.6±1.6±0.221.920.0±1.6±0.221.00.45–0.540.49735.0±1.6±0.335.330.9±3.5±0.334.232.4±2.1±0.332.828.0±1.9±0.331.50.54–0.630.58857.9±2.0±1.157.761.2±5.0±1.255.951.2±2.6±1.053.649.3±2.6±1.051.50.63–0.720.678109.8±3.1±2.6105.8109.4±7.4±2.9102.699.3±4.0±2.598.598.9±4.1±2.594.60.72–0.810.770227.1±18.2±10.8232.2196.4±39.3±16.3225.1211.2±23.6±14.0216.2231.3±26.9±16.2207.70.81–0.900.862735.4±8.4±6.3735.9720.4±19.8±7.3713.5690.4±11.2±6.4685.1670.3±11.1±6.2658.4〈s〉=201.8 GeV〈s〉=205.2 GeV〈s〉=206.7 GeV〈s〉=208.2 GeV0.00–0.090.05211.0±1.7±0.19.08.7±1.2±0.28.89.0±0.9±0.18.63.9±2.3±0.18.50.09–0.180.13811.0±1.7±0.19.812.9±1.4±0.29.68.7±0.9±0.19.410.3±3.7±0.29.20.18–0.270.22711.9±1.8±0.211.612.3±1.4±0.211.410.4±0.9±0.211.25.4±2.7±0.110.90.27–0.360.31714.8±2.1±0.214.916.1±1.6±0.114.616.8±1.2±0.214.313.4±4.2±0.114.00.36–0.450.40721.2±2.5±0.220.620.0±1.8±0.220.223.1±1.4±0.319.823.4±5.7±0.319.40.45–0.540.49737.2±3.3±0.430.931.7±2.3±0.430.229.4±1.6±0.429.726.3±6.0±0.329.10.54–0.630.58855.5±4.1±1.050.548.0±2.8±0.949.544.5±2.0±0.848.537.6±7.2±0.747.60.63–0.720.67891.4±5.7±2.392.793.3±4.3±2.390.990.0±3.2±2.489.284.3±12.0±2.387.50.72–0.810.770243.7±39.0±18.7203.7252.2±29.3±15.5199.7170.0±17.5±14.5195.9280.3±88.6±24.0192.20.81–0.900.862618.3±15.8±6.0645.7628.9±11.9±5.7633.3604.4±8.7±6.2621.2565.3±33.0±5.8609.5Table 3

Combined differential cross sections for the luminosity-averaged centre-of-mass energy 〈s〉=198 GeV, compared with the Standard Model expectations, dσth(1)/d|cosθ|, and the expectations for the case in which α does not change with Q2, dσth(0)/d|cosθ|. The first uncertainties are statistical and the second systematic

Sources of systematic uncertainty and their effect, ΔC, on the determination of the C parameter

Source of uncertaintyΔCTheoretical uncertainty0.11Experimental systematic0.08Frad0.05Bin migration<0.01Total0.14Measurement of the running of the electromagnetic coupling at large momentum-transfer at LEP

L3 Collaboration

Achard

Adriani

Aguilar-Benitez

Alcaraz

Alemanni

Allaby

Aloisio

M.G.

Alviggi

Anderhub

V.P.

Andreev

Anselmo

Arefiev

Azemoon

Aziz

Bagnaia

Bajo

Baksay

S.V.

Baldew

Banerjee

Sw.

Banerjee

Barczyk

Barillère

Bartalini

Basile

Batalova

Battiston

Bay

Becattini

Becker

Behner

Bellucci

Berbeco

Berdugo

Berges

Bertucci

B.L.

Betev

Biasini

Biglietti

Biland

J.J.

Blaising

S.C.

Blyth

G.J.

Bobbink

Böhm

Boldizsar

Borgia

Bottai

Bourilkov

Bourquin

Braccini

J.G.

Branson

Brochu

J.D.

Burger

W.J.

Burger

X.D.

Cai

Capell

Cara Romeo

Carlino

Cartacci

Casaus

Cavallari

Cavallo

Cecchi

Cerrada

Chamizo

Y.H.

Chang

Chemarin

Chen

G.M.

Chen

H.F.

Chen

H.S.

Chen

Chiefari

Cifarelli

Cindolo

Clare

Coignet

Colino

Costantini

de la Cruz

Cucciarelli

de Asmundis

Déglon

Debreczeni

Degré

Dehmelt

Deiters

della Volpe

Delmeire

Denes

DeNotaristefani

De Salvo

Diemoz

Dierckxsens

Dionisi

Dittmar

Doria

M.T.

Dova

Duchesneau

Duda

Echenard

Eline

El Hage

El Mamouni

Engler

F.J.

Eppling

Extermann

M.A.

Falagan

Falciano

Favara

Fay

Fedin

Felcini

Ferguson

Fesefeldt

Fiandrini

J.H.

Field

Filthaut

P.H.

Fisher

Fisk

Forconi

Freudenreich

Furetta

Yu.

Galaktionov

S.N.

Ganguli

Garcia-Abia

Gataullin

Gentile

Giagu

Z.F.

Gong

Grenier

Grimm

M.W.

Gruenewald

Guida

V.K.

Gupta

Gurtu

L.J.

Gutay

Haas

Hatzifotiadou

Hebbeker

Hervé

Hirschfelder

Hofer

Hohlmann

Holzner

S.R.

Hou

B.N.

Jin

Jindal

L.W.

Jones

de Jong

Josa-Mutuberría

Kaur

M.N.

Kienzle-Focacci

J.K.

Kim

Kirkby

Kittel

Klimentov

A.C.

König

Kopal

Koutsenko

Kräber

R.W.

Kraemer

Krüger

Kunin

Ladron de Guevara

Laktineh

Landi

Lebeau

Lebedev

Lebrun

Lecomte

Lecoq

Le Coultre

J.M.

Le Goff

Leiste

Levtchenko

Likhoded

C.H.

Lin

W.T.

Lin

F.L.

Linde

Lista

Z.A.

Liu

Lohmann

Longo

Y.S.

Luci

Luminari

Lustermann

W.G.

Malgeri

Malinin

Maña

Mans

J.P.

Martin

Marzano

Mazumdar

R.R.

McNeil

Mele

Merola

Meschini

W.J.

Metzger

Mihul

Milcent

Mirabelli

Mnich

G.B.

Mohanty

G.S.

Muanza

A.J.M.

Muijs

Musicar

Musy

Nagy

Natale

Napolitano

Nessi-Tedaldi

Newman

Nisati

Novak

Nowak

Ofierzynski

Organtini

Pal

Palomares

Paolucci

Paramatti

Passaleva

Patricelli

Paul

Pauluzzi

Paus

Pauss

Pedace

Pensotti

Perret-Gallix

Piccolo

Pierella

Pioppi

P.A.

Piroué

Pistolesi

Plyaskin

Pohl

Pojidaev

Pothier

Prokofiev

Rahal-Callot

M.A.

Rahaman

Raics

Raja

Ramelli

P.G.

Rancoita

Ranieri

Raspereza

Razis

Ren

Rescigno

Reucroft

Riemann

Riles

B.P.

Roe

Romero

Rosca

Rosemann

Rosenbleck

Rosier-Lees

Roth

J.A.

Rubio

Ruggiero

Rykaczewski

Sakharov

Saremi

Sarkar

Salicio

Sanchez

Schäfer

Schegelsky

Schopper

D.J.

Schotanus

Sciacca

Servoli

Shevchenko

Shivarov

Shoutko

Shumilov

Shvorob

Son

Souga

Spillantini

Steuer

D.P.

Stickland

Stoyanov

Straessner

Sudhakar

Sultanov

L.Z.

Sun

Sushkov

Suter

J.D.

Swain

Szillasi

X.W.

Tang

Tarjan

Tauscher

Taylor

Tellili

Teyssier

Timmermans

Samuel C.C.

Ting

S.M.

Ting

S.C.

Tonwar

Tóth

Tully

K.L.

Tung

Ulbricht

Valente

R.T.

Van de Walle

Vasquez

Veszpremi

Vesztergombi

Vetlitsky

Viertel

Villa

Vivargent

Vlachos

Vodopianov

Vogel

Vogt

Vorobiev

A.A.

Vorobyov

Wadhwa

Wang

X.L.

Wang

Z.M.

Wang

Weber

Wynhoff

Xia

Z.Z.

Yamamoto

B.Z.

Yang

C.G.

Yang

H.J.

Yang

S.C.

Yeh

An.

Zalite

Yu.

Zalite

Z.P.

Zhang

Zhao

G.Y.

Zhu

R.Y.

Zhu

H.L.

Zhuang

Zichichi

Zimmermann

Zöller

aIII Physikalisches Institut, RWTH, D-52056 Aachen, Germany11

Supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie.

bNational Institute for High Energy Physics, NIKHEF, and University of Amsterdam, NL-1009 DB Amsterdam, The NetherlandscUniversity of Michigan, Ann Arbor, MI 48109, USAdLaboratoire d'Annecy-le-Vieux de Physique des Particules, LAPP, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux Cedex, FranceeInstitute of Physics, University of Basel, CH-4056 Basel, SwitzerlandfLouisiana State University, Baton Rouge, LA 70803, USAgInstitute of High Energy Physics, IHEP, 100039 Beijing, China66

Supported by the National Natural Science Foundation of China.

hUniversity of Bologna and INFN, Sezione di Bologna, I-40126 Bologna, ItalyiTata Institute of Fundamental Research, Mumbai (Bombay) 400 005, IndiajNortheastern University, Boston, MA 02115, USAkInstitute of Atomic Physics and University of Bucharest, R-76900 Bucharest, RomanialCentral Research Institute for Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary22

Supported by the Hungarian OTKA fund under contract numbers T019181, F023259 and T037350.

mMassachusetts Institute of Technology, Cambridge, MA 02139, USAnPanjab University, Chandigarh 160 014, IndiaoKLTE-ATOMKI, H-4010 Debrecen, Hungary33

Also supported by the Hungarian OTKA fund under contract number T026178.

pDepartment of Experimental Physics, University College Dublin, Belfield, Dublin 4, IrelandqINFN, Sezione di Firenze, and University of Florence, I-50125 Florence, ItalyrEuropean Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, SwitzerlandsWorld Laboratory, FBLJA Project, CH-1211 Geneva 23, SwitzerlandtUniversity of Geneva, CH-1211 Geneva 4, SwitzerlanduUniversity of Hamburg, D-22761 Hamburg, GermanyvChinese University of Science and Technology, USTC, Hefei, Anhui 230 029, China6wUniversity of Lausanne, CH-1015 Lausanne, SwitzerlandxInstitut de Physique Nucléaire de Lyon, IN2P3-CNRS, Université Claude Bernard, F-69622 Villeurbanne, FranceyCentro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT, E-28040 Madrid, Spain44

Supported also by the Comisión Interministerial de Ciencia y Tecnología.

zFlorida Institute of Technology, Melbourne, FL 32901, USAaaINFN, Sezione di Milano, I-20133 Milan, ItalyabInstitute of Theoretical and Experimental Physics, ITEP, Moscow, RussiaacINFN, Sezione di Napoli, and University of Naples, I-80125 Naples, ItalyadDepartment of Physics, University of Cyprus, Nicosia, CyprusaeRadboud University and NIKHEF, NL-6525 ED Nijmegen, The NetherlandsafCalifornia Institute of Technology, Pasadena, CA 91125, USAagINFN, Sezione di Perugia, and Università Degli Studi di Perugia, I-06100 Perugia, ItalyahNuclear Physics Institute, St. Petersburg, RussiaaiCarnegie Mellon University, Pittsburgh, PA 15213, USAajINFN, Sezione di Napoli, and University of Potenza, I-85100 Potenza, ItalyakPrinceton University, Princeton, NJ 08544, USAalUniversity of California, Riverside, CA 92521, USAamINFN, Sezione di Roma, and University of Rome “La Sapienza”, I-00185 Rome, ItalyanUniversity and INFN, Salerno, I-84100 Salerno, ItalyaoUniversity of California, San Diego, CA 92093, USAapCentral Laboratory of Mechatronics and Instrumentation, Bulgarian Academy of Sciences, BU-1113 Sofia, BulgariaaqThe Center for High Energy Physics, Kyungpook National University, 702-701 Taegu, South KoreaarNational Central University, Chung-Li, Taiwan, ROCasDepartment of Physics, National Tsing Hua University, Taiwan, ROCatPurdue University, West Lafayette, IN 47907, USAauPaul Scherrer Institut, PSI, CH-5232 Villigen, SwitzerlandavDESY, D-15738 Zeuthen, GermanyawEidgenössische Technische Hochschule, ETH Zürich, CH-8093 Zürich, Switzerland5

Also supported by CONICET and Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina.

Editor: L. Rolandi

Abstract

The evolution of the electromagnetic coupling, α, in the momentum-transfer range 1800 GeV2<−Q2<21600 GeV2 is studied with about 40 000 Bhabha-scattering events collected with the L3 detector at LEP at centre-of-mass energies s=189–209 GeV. The running of α is parametrised as: α(Q2)=α01−CΔα(Q2), where α0≡α(Q2=0) is the fine-structure constant and C=1 corresponds to the evolution expected in QED. A fit to the differential cross section of the e+e−→e+e− process for scattering angles in the range |cosθ|<0.9 excludes the hypothesis of a constant value of α, C=0, and validates the QED prediction with the result: C=1.05±0.07±0.14, where the first uncertainty is statistical and the second systematic.

Introduction

A fundamental consequence of quantum field theory is that the value of the electromagnetic coupling, α, depends on, or runs with, the squared momentum transfer, Q2. This phenomenon is due to higher momentum-transfers probing virtual-loop corrections to the photon propagator. This process of vacuum polarisation is sketched in Fig. 1

. In QED, the dependence of α on Q2 is described as [2]: (1)α(Q2)=α01−Δα(Q2), where the fine-structure constant, α0≡α(Q2=0), is a fundamental quantity of physics. It is measured with high accuracy in solid-state processes and via the study of the anomalous magnetic moment of the electron to be 1/α0=137.03599911±0.00000046[1]. The contributions to Δα(Q2) from lepton loops are precisely predicted [3], while those from quark loops are difficult to calculate due to non-perturbative QCD effects. They are estimated using dispersion-integral techniques [4] and information from the e+e−→hadrons cross section. At the scale of the Z-boson mass, recent calculations yield α−1(mZ2)=128.936±0.046[5]. Similar results, with smaller uncertainty, are found by other evaluations using stronger theoretical assumptions. For example, Ref. [6] obtains α−1(mZ2)=128.962±0.016.

The running of α was studied at e+e− colliders both in the time-like region, Q2>0, and the space-like region, Q2<0. The first measurement in the time-like region was performed by the TOPAZ Collaboration at TRISTAN for Q2=3338 GeV2 by comparing the cross sections of the e+e−→e+e− and e+e−→e+e−μ+μ− processes [7]. The OPAL Collaboration at LEP exploited the different sensitivity to α(Q2) of the cross sections of the e+e−→μ+μ−, e+e−→τ+τ− and e+e−→qq¯ processes above the Z resonance to determine α(37236 GeV2)[8]. Information on α(mZ2) is also extracted from the couplings of the Z boson to fermion pairs [9].Bhabha scattering at e+e− colliders, e+e−→e+e−, gives access to the running of α in the space-like region. In addition, like other processes dominated by t-channel photon exchange, it has little dependence on weak corrections. The four-momentum transfer in Bhabha scattering depends on s and on the scattering angle, θ: Q2=t≃−s(1−cosθ)/2<0. Small-angle and large-angle Bhabha scattering allow to probe the running of α in different Q2 ranges.LEP detectors were equipped with luminosity monitors, high-precision calorimeters located close to the beam pipe and designed to measure small-angle Bhabha scattering in order to determine the integrated luminosity collected by the experiments. The L3 Collaboration first established the running of α in the range 2.10 GeV2<−Q2<6.25 GeV2[10] by comparing event counts in different regions of its luminosity monitor. More recently, the OPAL Collaboration studied the similar range 1.81 GeV2<−Q2<6.07 GeV2[11].The running of α in large-angle Bhabha scattering was first investigated by the VENUS Collaboration at TRISTAN in the range 100 GeV2<−Q2<2916 GeV2[12]. Later, the L3 Collaboration studied the same process at s=189 GeV for scattering angles 0.81<|cosθ|<0.94, probing the range 12.25 GeV2<−Q2<3434 GeV2[10].This Letter investigates the running of α by studying the differential cross section for Bhabha scattering at LEP at s=189–209 GeV for scattering angles such that |cosθ|<0.9. Less than 1% of the events scatter backwards, cosθ<0, and this analysis effectively probes the region 1800 GeV2<−Q2<21600 GeV2, extending and complementing previous space-like studies.2

Analysis strategy

In the following, the running of α is described by a free parameter, C, defined according to (2)α(Q2)=α01−CΔα(Q2), where the parametrisation of Ref. [5] is used for the term Δα(Q2). A value of C consistent with C=1 would indicate that data follow the behaviour predicted by QED, while the hypothesis α=α0, with no dependence on Q2, corresponds to C=0.The value of C is derived by a study of the measured differential cross section of the e+e−→e+e− process, dσ/dcosθ. This quantity depends on C through the measured integrated luminosity, L(C), which is calculated from the expected cross section of the e+e−→e+e− process for small scattering angles. The measurements used in the following are obtained under the Standard Model hypothesis, C=1, as (3)dσ(1)dcosθ=N(cosθ)Δcosθ1L(1)ɛ(cosθ), where N(cosθ) is the number of events observed in a given cosθ range, of width Δcosθ, with average acceptance ɛ(cosθ). The measured integrated luminosity depends on C as (4)L(C)≡NLσL(C)ɛL(C)=L(1)σL(1)ɛL(1)σL(C)ɛL(C), where NL is the number of events observed in the fiducial volume of the luminosity monitor, σL(C) is the corresponding e+e−→e+e− cross section for a given value of C and ɛL(C) is the detector acceptance. This acceptance may depend on C due to the combined effect of small angular anisotropies of detector efficiencies and the dependence of the predicted differential cross section on C. These changes in the acceptance are found to have negligible impact on the results presented below.The value of the parameter C is extracted by comparing the measured differential cross section to the theoretical prediction as a function of C, dσth(C)/dcosθ, derived as (5)dσth(C)dcosθ≡dσth(1)dcosθL(1)L(C), where dσth(1)/dcosθ is the Standard Model prediction, discussed in Ref. [13]. The value of L(1) is derived by using the BHLUMI Monte Carlo program [14]. The dependence of dσth(C)/dcosθ and L(C) on C is implemented by means of the BHWIDE Monte Carlo program [15]. The differential cross section is factorised as (6)dσth(C)dcosθ≡dσBorn(C)dcosθFrad(cosθ), where dσBorn(C)/dcosθ is the tree-level differential cross section, which has a simple analytical form. The term Frad(cosθ) parametrises initial-state and final-state radiation effects, dominated by real-photon emission, as implemented in BHWIDE. It is verified that Frad(cosθ) has a negligible dependence on the spread of s considered in this analysis and, most important, on C.3

Cross section measurement

The data were collected at LEP by the L3 detector [16,17] in the years from 1998 through 2000. They correspond to an integrated luminosity of 607.4 pb−1 and are grouped in eight intervals of s with the average values and corresponding integrated luminosities listed in Table 1

Events from the e+e−→e+e− process are selected as described in Ref. [18]. Electrons and positrons are identified as clusters in the BGO electromagnetic calorimeter, matched with tracks in the central tracker. In the barrel region of the detector, |cosθ|<0.72, the energy of the most energetic cluster must satisfy E1>0.25s, while the energy of the other cluster must satisfy E2>20 GeV. In the endcap region, 0.81<|cosθ|<0.98, these criteria are relaxed to E1>0.2s and E2>10 GeV. Events with clusters in the transition region between the barrel and endcap regions, 0.72<|cosθ|<0.81, instrumented with a lead and scintillating-fiber calorimeter [17], are rejected. To suppress contributions from events with high-energy initial-state radiation, the complement to 180° of the angle between the two clusters, the acollinearity, ζ, is required to be less than 25°. The number of events observed at different values of s is shown in Table 1 together with the Monte Carlo expectations for signal and background.The e+e−→e+e− process is simulated with the BHWIDE Monte Carlo generator assuming C=1. Background processes are described with the following Monte Carlo generators: KORALZ [19] for e+e−→τ+τ−, KORALW [20] for e+e−→W+W−, PYTHIA [21] for e+e−→Ze+e−, DIAG36 [22] for e+e−→e+e−e+e−, GGG [23] for e+e−→γγγ and TEEGG [24] for e+e−→e+e−γ events where one fermion is scattered into the beam pipe and the photon is in the detector. The L3 detector response is simulated using the GEANT package [25], which describes effects of energy loss, multiple scattering and showering in the detector. Time-dependent detector inefficiencies, as monitored during the data-taking period, are included in the simulation.Systematic effects, such as charge confusion, are reduced by folding the differential cross section into dσ/d|cosθ|, which is defined as (7)dσd|cosθ|≡dσdcosθ|cosθ<0+dσdcosθ|cosθ>0. This differential cross section is measured in the fiducial volume defined by (8)12°<θe−,e+<168°,(9)|cosθ|<0.9,(10)ζ<25°, where θe− and θe+ are the polar angles of the electron and the positron, respectively. The value of cosθ is derived as (11)cosθ≡sin|θe+−θe−|sinθe−+sinθe+.Ten intervals of |cosθ| are considered for each of the eight values of s, for a total of 80 independent measurements. Table 2

and Fig. 2

present the measurements of dσ/d|cosθ| and the Standard Model expectations. The larger uncertainties in the interval 0.72–0.81 are due to the transition region between the barrel and the endcap regions.

Results

Figs. 3 and 4

compare the combined differential cross section at the average centre-of-mass energy 〈s〉=198 GeV with the Standard Model prediction, corresponding to C=1, and with a constant value of α, corresponding to C=0. The data favour the hypothesis C=1 over the hypothesis C=0, as also presented in Table 3

The value of C is extracted by comparing the 80 measurements of dσ/d|cosθ| with the theoretical expectations dσth(C)/dcosθ in a χ2 fit with the result C=1.06±0.07, where the quoted uncertainty is statistical only. Several sources of systematic uncertainties are then considered.

•The theoretical expectations for dσth(1)/dcosθ have an uncertainty which varies from 0.5% in the endcap region to 1.5% in the barrel region [13,15].

•The measurements of dσ/d|cosθ| are affected by a systematic uncertainty, dominated by the event-selection procedure, which varies between 1% and 10%, as listed in Table 2[18].

•An uncertainty between 0.2% and 1.5% is assigned to Frad(cosθ), as a function of cosθ, in order to account for possible higher-order effects not included in the BHWIDE parametrisation.

•Migration effects among the different cosθ bins are found to be negligible due to the large bin size and the good detector resolution.

Systematic uncertainties are conservatively treated as fully correlated and the fit is repeated including both statistical and systematic uncertainties with the result C=1.05±0.07±0.14, where the first uncertainty is statistical and the second systematic. A breakdown of the systematic uncertainty is presented in Table 4

. This result is in agreement with the Standard Model expectation, C=1. The quality of the fit is satisfactory, with a χ2 of 91.9 for 79 degrees of freedom, corresponding to a confidence level of 17%. The hypothesis of a value of α which does not depend on Q2, C=0, is totally excluded with a χ2 of 316 for 80 degrees of freedom, corresponding to a confidence level of 10−29.

Discussion

The result presented above establishes the evolution of the electromagnetic coupling with −Q2 in the range 1800 GeV2<−Q2<21600 GeV2. This finding extends and complements studies based on small-angle Bhabha scattering by the L3 [10] and OPAL [11] Collaborations, which studied the regions 2.10 GeV2<−Q2<6.25 GeV2 and 1.81 GeV2<−Q2<6.07 GeV2, respectively. The advantage of large-angle Bhabha scattering, investigated in this Letter, is to probe large values of −Q2, while studies of small-angle Bhabha scattering at lower values of −Q2 benefit from a larger cross section and thus statistical accuracy. The experimental systematic uncertainties of measurements in the two −Q2 regions are implicitly different. At large −Q2, they are dominated by the selection of Bhabha events in the large-angle calorimeters, while at low −Q2 they mostly arise from the event reconstruction in the luminosity monitors and from effects of the material traversed by electrons and positrons before their detection. Both studies, at large and low −Q2, are affected by theoretical uncertainties on the differential cross section of Bhabha scattering, although in different angular regions.

Figs. 5 and 6

present the evolution of the electromagnetic coupling with −Q2. A band for 1800 GeV2<−Q2<21600 GeV2 shows the 68% confidence level result from this analysis. It is derived by inserting the measured value of C with its errors in Eq. (2) together with the QED predictions of Ref. [5]. The results from previous L3 data for Bhabha scattering at 2.10 GeV2<−Q2<6.25 GeV2 and 12.25 GeV2<−Q2<3434 GeV2[10] are also shown. These two measurements are not absolute measurements of the electromagnetic coupling but differences between the values of α(Q2) at the extreme of the Q2 ranges [10]: (12)α−1(−2.10 GeV2)−α−1(−6.25 GeV2)=0.78±0.26,(13)α−1(−12.25 GeV2)−α−1(−3434 GeV2)=3.80±1.29. The results in Fig. 5 are obtained by fixing the values of α(−2.10 GeV2) and α(−12.25 GeV2) to the QED predictions of Ref. [5] in order to extract the values of α(−6.25 GeV2) and α(−3434 GeV2) from Eqs. (12) and (13). The results shown in Fig. 6 are obtained by first determining the values of α(−2.10 GeV2) and α(−12.25 GeV2) from the measured value of C and from Eq. (2) and then extracting the values of α(−6.25 GeV2) and α(−3434 GeV2) from Eqs. (12) and (13). This procedure relies on the assumption that the measured value of C also describes the running of the electromagnetic coupling for lower values of −Q2. Both figures provide an impressive evidence of the running of the electromagnetic coupling in the energy range accessible at LEP.

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