Measurement of CP-Violating Asymmetries in B0-(rho pi)0 Using a Time-Dependent Dalitz Plot

August2004

MeasurementofCP-ViolatingAsymmetriesinB0→(ρπ)0

UsingaTime-DependentDalitzPlotAnalysis

arXiv:hep-ex/0408099v1 20 Aug 2004TheBABARCollaboration

February7,2008

Abstract

WepresentthepreliminarymeasurementofCP-violatingasymmetriesinB0→(ρπ)0→π+π−π0decaysusingatime-dependentDalitzplotanalysis.Theresultsareobtainedfromadatasampleof213millionΥ(4S)→B

WorksupportedinpartbyDepartmentofEnergycontractDE-AC03-76SF00515.

TheBABARCollaboration,

B.Aubert,R.Barate,D.Boutigny,F.Couderc,J.-M.Gaillard,A.Hicheur,Y.Karyotakis,J.P.Lees,

V.Tisserand,A.Zghiche

LaboratoiredePhysiquedesParticules,F-74941Annecy-le-Vieux,France

A.Palano,A.Pompili

Universit`adiBari,DipartimentodiFisicaandINFN,I-70126Bari,Italy

J.C.Chen,N.D.Qi,G.Rong,P.Wang,Y.S.ZhuInstituteofHighEnergyPhysics,Beijing100039,China

G.Eigen,I.Ofte,B.Stugu

UniversityofBergen,Inst.ofPhysics,N-5007Bergen,Norway

G.S.Abrams,A.W.Borgland,A.B.Breon,D.N.Brown,J.Button-Shafer,R.N.Cahn,E.Charles,

C.T.Day,M.S.Gill,A.V.Gritsan,Y.Groysman,R.G.Jacobsen,R.W.Kadel,J.Kadyk,L.T.Kerth,Yu.G.Kolomensky,G.Kukartsev,G.Lynch,L.M.Mir,P.J.Oddone,T.J.Orimoto,M.Pripstein,

N.A.Roe,M.T.Ronan,V.G.Shelkov,W.A.Wenzel

LawrenceBerkeleyNationalLaboratoryandUniversityofCalifornia,Berkeley,CA94720,USAM.Barrett,K.E.Ford,T.J.Harrison,A.J.Hart,C.M.Hawkes,S.E.Morgan,A.T.Watson

UniversityofBirmingham,Birmingham,B152TT,UnitedKingdom

M.Fritsch,K.Goetzen,T.Held,H.Koch,B.Lewandowski,M.Pelizaeus,M.SteinkeRuhrUniversit¨atBochum,Institutf¨urExperimentalphysik1,D-44780Bochum,GermanyJ.T.Boyd,N.Chevalier,W.N.Cottingham,M.P.Kelly,T.E.Latham,F.F.Wilson

UniversityofBristol,BristolBS81TL,UnitedKingdom

T.Cuhadar-Donszelmann,C.Hearty,N.S.Knecht,T.S.Mattison,J.A.McKenna,D.Thiessen

UniversityofBritishColumbia,Vancouver,BC,CanadaV6T1Z1

A.Khan,P.Kyberd,L.Teodorescu

BrunelUniversity,Uxbridge,MiddlesexUB83PH,UnitedKingdom

A.E.Blinov,V.E.Blinov,V.P.Druzhinin,V.B.Golubev,V.N.Ivanchenko,E.A.Kravchenko,

A.P.Onuchin,S.I.Serednyakov,Yu.I.Skovpen,E.P.Solodov,A.N.Yushkov

BudkerInstituteofNuclearPhysics,Novosibirsk630090,Russia

D.Best,M.Bruinsma,M.Chao,I.Eschrich,D.Kirkby,A.J.Lankford,M.Mandelkern,R.K.Mommsen,

W.Roethel,D.P.Stoker

UniversityofCaliforniaatIrvine,Irvine,CA92697,USA

C.Buchanan,B.L.Hartfiel

UniversityofCaliforniaatLosAngeles,LosAngeles,CA90024,USA

S.D.Foulkes,J.W.Gary,B.C.Shen,K.Wang

UniversityofCaliforniaatRiverside,Riverside,CA92521,USA

2

D.delRe,H.K.Hadavand,E.J.Hill,D.B.MacFarlane,H.P.Paar,Sh.Rahatlou,V.Sharma

UniversityofCaliforniaatSanDiego,LaJolla,CA92093,USA

J.W.Berryhill,C.Campagnari,B.Dahmes,O.Long,A.Lu,M.A.Mazur,J.D.Richman,W.Verkerke

UniversityofCaliforniaatSantaBarbara,SantaBarbara,CA93106,USA

T.W.Beck,A.M.Eisner,C.A.Heusch,J.Kroseberg,W.S.Lockman,G.Nesom,T.Schalk,

B.A.Schumm,A.Seiden,P.Spradlin,D.C.Williams,M.G.WilsonUniversityofCaliforniaatSantaCruz,InstituteforParticlePhysics,SantaCruz,CA950,USAJ.Albert,E.Chen,G.P.Dubois-Felsmann,A.Dvoretskii,D.G.Hitlin,I.Narsky,T.Piatenko,

F.C.Porter,A.Ryd,A.Samuel,S.Yang

CaliforniaInstituteofTechnology,Pasadena,CA91125,USAS.Jayatilleke,G.Mancinelli,B.T.Meadows,M.D.SokoloffUniversityofCincinnati,Cincinnati,OH45221,USA

T.Abe,F.Blanc,P.Bloom,S.Chen,W.T.Ford,U.Nauenberg,A.Olivas,P.Rankin,J.G.Smith,

J.Zhang,L.Zhang

UniversityofColorado,Boulder,CO80309,USA

A.Chen,J.L.Harton,A.Soffer,W.H.Toki,R.J.Wilson,Q.Zeng

ColoradoStateUniversity,FortCollins,CO80523,USA

D.Altenburg,T.Brandt,J.Brose,M.Dickopp,E.Feltresi,A.Hauke,H.M.Lacker,R.M¨uller-Pfefferkorn,R.Nogowski,S.Otto,A.Petzold,J.Schubert,K.R.Schubert,R.Schwierz,B.Spaan,J.E.Sundermann

TechnischeUniversit¨atDresden,Institutf¨urKern-undTeilchenphysik,D-01062Dresden,GermanyD.Bernard,G.R.Bonneaud,F.Brochard,P.Grenier,S.Schrenk,Ch.Thiebaux,G.Vasileiadis,M.Verderi

EcolePolytechnique,LLR,F-91128Palaiseau,FranceD.J.Bard,P.J.Clark,D.Lavin,F.Muheim,S.Playfer,Y.XieUniversityofEdinburgh,EdinburghEH93JZ,UnitedKingdom

M.Andreotti,V.Azzolini,D.Bettoni,C.Bozzi,R.Calabrese,G.Cibinetto,E.Luppi,M.Negrini,

L.Piemontese,A.Sarti

Universit`adiFerrara,DipartimentodiFisicaandINFN,I-44100Ferrara,Italy

E.Treadwell

FloridaA&MUniversity,Tallahassee,FL32307,USA

F.Anulli,R.Baldini-Ferroli,A.Calcaterra,R.deSangro,G.Finocchiaro,P.Patteri,I.M.Peruzzi,

M.Piccolo,A.Zallo

LaboratoriNazionalidiFrascatidell’INFN,I-00044Frascati,Italy

A.Buzzo,R.Capra,R.Contri,G.Crosetti,M.LoVetere,M.Macri,M.R.Monge,S.Passaggio,

C.Patrignani,E.Robutti,A.Santroni,S.Tosi

Universit`adiGenova,DipartimentodiFisicaandINFN,I-16146Genova,Italy

S.Bailey,G.Brandenburg,K.S.Chaisanguanthum,M.Morii,E.Won

HarvardUniversity,Cambridge,MA02138,USA

3

R.S.Dubitzky,U.Langenegger

Universit¨atHeidelberg,PhysikalischesInstitut,Philosophenweg12,D-69120Heidelberg,GermanyW.Bhimji,D.A.Bowerman,P.D.Dauncey,U.Egede,J.R.Gaillard,G.W.Morton,J.A.Nash,

M.B.Nikolich,G.P.Taylor

ImperialCollegeLondon,London,SW72AZ,UnitedKingdom

M.J.Charles,G.J.Grenier,U.MallikUniversityofIowa,IowaCity,IA52242,USA

J.Cochran,H.B.Crawley,J.Lamsa,W.T.Meyer,S.Prell,E.I.Rosenberg,A.E.Rubin,J.Yi

IowaStateUniversity,Ames,IA50011-3160,USA

M.Biasini,R.Covarelli,M.Pioppi

Universit`adiPerugia,DipartimentodiFisicaandINFN,I-06100Perugia,Italy

M.Davier,X.Giroux,G.Grosdidier,A.H¨ocker,S.Laplace,F.LeDiberder,V.Lepeltier,A.M.Lutz,

T.C.Petersen,S.Plaszczynski,M.H.Schune,L.Tantot,G.Wormser

Laboratoiredel’Acc´el´erateurLin´eaire,F-918Orsay,FranceC.H.Cheng,D.J.Lange,M.C.Simani,D.M.Wright

LawrenceLivermoreNationalLaboratory,Livermore,CA94550,USA

A.J.Bevan,C.A.Chavez,J.P.Coleman,I.J.Forster,J.R.Fry,E.Gabathuler,R.Gamet,D.E.Hutchcroft,R.J.Parry,D.J.Payne,R.J.Sloane,C.Touramanis

UniversityofLiverpool,LiverpoolL6972E,UnitedKingdom

J.J.Back,1C.M.Cormack,P.F.Harrison,1F.DiLodovico,G.B.Mohanty1

QueenMary,UniversityofLondon,E14NS,UnitedKingdom

C.L.Brown,G.Cowan,R.L.Flack,H.U.Flaecher,M.G.Green,P.S.Jackson,T.R.McMahon,

S.Ricciardi,F.Salvatore,M.A.WinterUniversityofLondon,RoyalHollowayandBedfordNewCollege,Egham,SurreyTW200EX,

UnitedKingdom

D.Brown,C.L.Davis

UniversityofLouisville,Louisville,KY40292,USA

J.Allison,N.R.Barlow,R.J.Barlow,P.A.Hart,M.C.Hodgkinson,G.D.Lafferty,A.J.Lyon,

J.C.Williams

UniversityofManchester,ManchesterM139PL,UnitedKingdom

A.Farbin,W.D.Hulsbergen,A.Jawahery,D.Kovalskyi,C.K.Lae,V.Lillard,D.A.Roberts

UniversityofMaryland,CollegePark,MD20742,USA

G.Blaylock,C.Dallapiccola,K.T.Flood,S.S.Hertzbach,R.Kofler,V.B.Koptchev,T.B.Moore,

S.Saremi,H.Staengle,S.Willocq

UniversityofMassachusetts,Amherst,MA01003,USA

R.Cowan,G.Sciolla,S.J.Sekula,F.Taylor,R.K.Yamamoto

MassachusettsInstituteofTechnology,LaboratoryforNuclearScience,Cambridge,MA02139,USA

D.J.J.Mangeol,P.M.Patel,S.H.RobertsonMcGillUniversity,Montr´eal,QC,CanadaH3A2T8

A.Lazzaro,V.Lombardo,F.Palombo

Universit`adiMilano,DipartimentodiFisicaandINFN,I-20133Milano,Italy

J.M.Bauer,L.Cremaldi,V.Eschenburg,R.Godang,R.Kroeger,J.Reidy,D.A.Sanders,D.J.Summers,

H.W.Zhao

UniversityofMississippi,University,MS38677,USA

S.Brunet,D.Cˆot´e,P.Taras

Universit´edeMontr´eal,LaboratoireRen´eJ.A.L´evesque,Montr´eal,QC,CanadaH3C3J7

H.Nicholson

MountHolyokeCollege,SouthHadley,MA01075,USA

N.Cavallo,2F.Fabozzi,2C.Gatto,L.Lista,D.Monorchio,P.Paolucci,D.Piccolo,C.SciaccaUniversit`adiNapoliFedericoII,DipartimentodiScienzeFisicheandINFN,I-80126,Napoli,Italy

M.Baak,H.Bulten,G.Raven,H.L.Snoek,L.Wilden

NIKHEF,NationalInstituteforNuclearPhysicsandHighEnergyPhysics,NL-1009DBAmsterdam,

TheNetherlands

C.P.Jessop,J.M.LoSecco

UniversityofNotreDame,NotreDame,IN46556,USA

T.Allmendinger,K.K.Gan,K.Honscheid,D.Hufnagel,H.Kagan,R.Kass,T.Pulliam,A.M.Rahimi,

R.Ter-Antonyan,Q.K.Wong

OhioStateUniversity,Columbus,OH43210,USA

J.Brau,R.Frey,O.Igonkina,C.T.Potter,N.B.Sinev,D.Strom,E.Torrence

UniversityofOregon,Eugene,OR97403,USA

F.Colecchia,A.Dorigo,F.Galeazzi,M.Margoni,M.Morandin,M.Posocco,M.Rotondo,F.Simonetto,

R.Stroili,G.Tiozzo,C.Voci

Universit`adiPadova,DipartimentodiFisicaandINFN,I-35131Padova,Italy

M.Benayoun,H.Briand,J.Chauveau,P.David,Ch.delaVaissi`ere,L.DelBuono,O.Hamon,

M.J.J.John,Ph.Leruste,J.Malcles,J.Ocariz,M.Pivk,L.Roos,S.T’Jampens,G.TherinUniversit´esParisVIetVII,LaboratoiredePhysiqueNucl´eaireetdeHautesEnergies,F-75252Paris,

France

P.F.Manfredi,V.Re

Universit`adiPavia,DipartimentodiElettronicaandINFN,I-27100Pavia,Italy

P.K.Behera,L.Gladney,Q.H.Guo,J.PanettaUniversityofPennsylvania,Philadelphia,PA19104,USA

C.Angelini,G.Batignani,S.Bettarini,M.Bondioli,F.Bucci,G.Calderini,M.Carpinelli,F.Forti,M.A.Giorgi,A.Lusiani,G.Marchiori,F.Martinez-Vidal,3M.Morganti,N.Neri,E.Paoloni,M.Rama,

G.Rizzo,F.Sandrelli,J.WalshUniversit`adiPisa,DipartimentodiFisica,ScuolaNormaleSuperioreandINFN,I-56127Pisa,Italy

M.Haire,D.Judd,K.Paick,D.E.Wagoner

PrairieViewA&MUniversity,PrairieView,TX77446,USA

N.Danielson,P.Elmer,Y.P.Lau,C.Lu,V.Miftakov,J.Olsen,A.J.S.Smith,A.V.Telnov

PrincetonUniversity,Princeton,NJ08544,USA

F.Bellini,G.Cavoto,4R.Faccini,F.Ferrarotto,F.Ferroni,M.Gaspero,L.LiGioi,M.A.Mazzoni,

S.Morganti,M.Pierini,G.Piredda,F.SafaiTehrani,C.Voena

Universit`adiRomaLaSapienza,DipartimentodiFisicaandINFN,I-00185Roma,Italy

S.Christ,G.Wagner,R.Waldi

Universit¨atRostock,D-18051Rostock,Germany

T.Adye,N.DeGroot,B.Franek,N.I.Geddes,G.P.Gopal,E.O.OlaiyaRutherfordAppletonLaboratory,Chilton,Didcot,Oxon,OX110QX,UnitedKingdom

R.Aleksan,S.Emery,A.Gaidot,S.F.Ganzhur,P.-F.Giraud,G.HameldeMonchenault,W.Kozanecki,M.Legendre,G.W.London,B.Mayer,G.Schott,G.Vasseur,Ch.Y`eche,M.Zito

DSM/Dapnia,CEA/Saclay,F-91191Gif-sur-Yvette,FranceM.V.Purohit,A.W.Weidemann,J.R.Wilson,F.X.Yumiceva

UniversityofSouthCarolina,Columbia,SC29208,USA

D.Aston,R.Bartoldus,N.Berger,A.M.Boyarski,O.L.Buchmueller,R.Claus,M.R.Convery,

M.Cristinziani,G.DeNardo,D.Dong,J.Dorfan,D.Dujmic,W.Dunwoodie,E.E.Elsen,S.Fan,R.C.Field,T.Glanzman,S.J.Gowdy,T.Hadig,V.Halyo,C.Hast,T.Hryn’ova,W.R.Innes,M.H.Kelsey,P.Kim,M.L.Kocian,D.W.G.S.Leith,J.Libby,S.Luitz,V.Luth,H.L.Lynch,H.Marsiske,R.Messner,D.R.Muller,C.P.O’Grady,V.E.Ozcan,A.Perazzo,M.Perl,S.Petrak,B.N.Ratcliff,A.Roodman,A.A.Salnikov,R.H.Schindler,J.Schwiening,G.Simi,A.Snyder,A.Soha,

J.Stelzer,D.Su,M.K.Sullivan,J.Va’vra,S.R.Wagner,M.Weaver,A.J.R.Weinstein,

W.J.Wisniewski,M.Wittgen,D.H.Wright,A.K.Yarritu,C.C.Young

StanfordLinearAcceleratorCenter,Stanford,CA94309,USAP.R.Burchat,A.J.Edwards,T.I.Meyer,B.A.Petersen,C.Roat

StanfordUniversity,Stanford,CA94305-4060,USA

S.Ahmed,M.S.Alam,J.A.Ernst,M.A.Saeed,M.Saleem,F.R.Wappler

StateUniversityofNewYork,Albany,NY12222,USA

W.Bugg,M.Krishnamurthy,S.M.SpanierUniversityofTennessee,Knoxville,TN37996,USAR.Eckmann,H.Kim,J.L.Ritchie,A.Satpathy,R.F.Schwitters

UniversityofTexasatAustin,Austin,TX78712,USA

J.M.Izen,I.Kitayama,X.C.Lou,S.Ye

UniversityofTexasatDallas,Richardson,TX75083,USA

F.Bianchi,M.Bona,F.Gallo,D.Gamba

Universit`adiTorino,DipartimentodiFisicaSperimentaleandINFN,I-10125Torino,ItalyL.Bosisio,C.Cartaro,F.Cossutti,G.DellaRicca,S.Dittongo,S.Grancagnolo,L.Lanceri,P.Poropat,5

L.Vitale,G.Vuagnin

Universit`adiTrieste,DipartimentodiFisicaandINFN,I-34127Trieste,Italy

R.S.Panvini

VanderbiltUniversity,Nashville,TN37235,USA

Sw.Banerjee,C.M.Brown,D.Fortin,P.D.Jackson,R.Kowalewski,J.M.Roney,R.J.Sobie

UniversityofVictoria,Victoria,BC,CanadaV8W3P6

H.R.Band,B.Cheng,S.Dasu,M.Datta,A.M.Eichenbaum,M.Graham,J.J.Hollar,J.R.Johnson,

P.E.Kutter,H.Li,R.Liu,A.Mihalyi,A.K.Mohapatra,Y.Pan,R.Prepost,P.Tan,J.H.von

Wimmersperg-Toeller,J.Wu,S.L.Wu,Z.Yu

UniversityofWisconsin,Madison,WI53706,USA

M.G.Greene,H.Neal

YaleUniversity,NewHaven,CT06511,USA

1INTRODUCTION

Measurementsoftheparametersin2β[1,2]haveestablishedCPviolationintheB0mesonsystemandprovidestrongsupportfortheKobayashiandMaskawamodelofthisphenomenonasarisingfromasinglephaseinthethree-generationCKMquark-mixingmatrix[3].Wepresentinthisletterpreliminaryresultsfromatime-dependentanalysisoftheB0→π+π−π0Dalitzplot(DP)thatisdominatedbytheρintermediateresonances.Thegoaloftheanalysisisthesimultaneousextraction

∗/VV∗]oftheofthestrongtransitionamplitudesandtheweakinteractionphaseα≡arg[−VtdVtbudub

UnitarityTriangle.IntheStandardModel,anon-zerovalueforαwouldberesponsiblefortheoccurrenceofmixing-inducedCPviolationinthisdecay.TheBABARandBelleexperimentshaveobtainedconstraintsonαfromthemeasurementofeffectivequantitiessin2αeffinBdecaystoπ+π−[4,5]andBABARfromBdecaystoρ+ρ−[6],usinganisospinanalysis[7]thatinvolvestheotherchargesofthesefinalstatestoobtainboundsonα−αeff.

Unlikeπ+π−,ρ±π∓isnotaCPeigenstate,andfourflavor-chargeconfigurations(B0(

(2π)3

|A3π|2

spondingtothetransitionsB0→π+π−π0and

A3π,corre-

threeresonancesρ+,ρ−andρ0.Theρresonancesareassumedtobethesumofthegroundstateρ(770)andtheradialexcitationsρ(1450)andρ(1700),withaninitialsetofresonanceparame-tersandrelativeamplitudesdeterminedbyacombinedfittoτ+→

B0mixingparameterq/pintothe

A−+f0

A3π=f+

B0)isgivenby

2

|A±3π(∆t)|

=

e−|∆t|/τB0

A3π|∓|A3π|2−|

󰀏

2

󰀉

A3πA∗3πsin(∆md∆t),

󰀒

(6)

whereτB0isthemeanB0lifetime,∆mdistheB0

B0mixingisabsent(|q/p|=1),∆ΓBd=0andCPTisconserved.

Insertingtheamplitudes(4)and(5),oneobtainsforthetermsinEq.(6)|A3π|2±|

A3πA⋆3π

with

±

Uκ=|Aκ|2±|

κ

κ∗

󰀌

=

κ∈{+,−,0}

󰀅

|fκ|Iκ+

2

κ<σ∈{+,−,0}

󰀅

󰀉

∗Im

Re[fκfσ]Iκσ

+

∗Re

Im[fκfσ]Iκσ

󰀌

,(7)

Re

IκσImIκσ

=Re=Im

󰀍

AA

󰀏

Aκ

,AA

σσ

κ∗κ∗

∗bilinears[16].The27coefficients(8)–(12)arereal-valuedparametersthatmultiplythefκfσ

Theyaretheobservablesthataredeterminedbythefit.Eachofthecoefficientsisrelatedinauniquewaytothephysicallymoreintuitivequantities,liketree-levelandpenguin-typeamplitudes,theangleα,orthequasi-two-bodyCPanddilutionparameters[10](cf.Section6).

+coefficientsarerelatedtoresonancefractions(branchingfractionsandchargeasymme-TheUκ

−determinetherelativeabundanceoftheB0decayintoρ+π−andρ−π+(dilution)tries),theUκ

andthetime-dependentdirectCPasymmetries.TheIκmeasuremixing-inducedCPviolationand

±,Re(Im)Re(Im)

aresensitivetostrongphaseshifts.Finally,theUκσandIκσdescribetheinterferencepatternintheDalitzplot.Theirpresencedistinguishesthisanalysisfromthepreviousquasi-two-bodyone[10].Theyrepresenttheadditionaldegreesoffreedomthatallowsonetodeterminetheunknownpenguinpollutionandtherelativestrongphases.However,becausetheoverlapregionsoftheresonancesaresmallandbecausetheeventreconstructionintheseregionssuffersfromlargemisreconstructionratesandbackground,asubstantialdatasampleisneededtoperformafitthatconstrainsallamplitudeparameters.

ThechoicetofitfortheUandIcoefficientsratherthanfittingforthecomplextransitionamplitudesandtheweakphaseαdirectlyismotivatedbythefollowingtechnicalsimplifications:(i)incontrasttotheamplitudes,thereisauniquesolutionfortheUandIcoefficientsrequiringonlyasinglefittotheselecteddatasample7,(ii)inthepresenceofbackground,theUandIcoefficientsareapproximatelyGaussiandistributed,whichingeneralisnotthecasefortheamplitudes,and(iii)thepropagationofsystematicuncertaintiesandtheaveragingbetweendifferentmeasurementsarestraightforwardfortheU’sandI’s.

Wedeterminethequantitiesofinterestinasubsequentleast-squaresfittothemeasuredUandIcoefficients.

󰀍

AA

󰀏

(10)

,.

(11)(12)

󰀏

Aκisconserved.

10

1.3NORMALIZATION

Thedecayrate(6)isusedasprobabilitydensityfunction(PDF)inamaximum-likelihoodfitandmustthereforebenormalized:

122

|A±(13)|A±(∆t)|−→3π(∆t)|,3π2A3π|󰀒where

󰀑|A3π|2+|

π

arccos2

󰀑

m0−mmin0

π

θ0,(16)

=2mπ+=mB0−mπ0andmminwherem0istheinvariantmassbetweenthechargedtracks,mmax00

arethekinematiclimitsofm0,θ0istheρ0helicityangle,andJistheJacobianofthetransformationthatzoomsintothekinematicboundariesoftheDalitzplot.Thenewvariableshavevalidityrangesbetween0and1.ThedeterminantoftheJacobianisgivenby

∗

|detJ|=4|p∗+||p0|m0·

∂m0

∂θ′

,(17)

∗−m2,andwheretheenergiesE∗andE∗areintheE0where=+0π0

π+π−restframe.

Figure1showstheoriginal(lefthandplot)andthetransformed(righthandplot)DalitzplotsforMonteCarloB0→π+π−π0eventsgeneratedaccordingtoEqs.(4)and(5)withequalabundanceofallρchargesandwithvanishingrelativestrongphase.TheplotsillustratethehomogenizationoftheDalitzplotobtainedafterthetransformation(15).

|p∗+|

󰀊

11

3025Interference54321

Interference0.8

s– (GeV2/c4)201√s+ = 1.5 GeV/c20.6

151050θ'400300200100

10222324252627√s– = 1.5 GeV/c20510152

Figure1:Nominal(left)andsquare(right)B0→π+π−π0DalitzplotsobtainedfromMonteCarlogeneratedeventswithoutdetectorsimulation.ThegeneratingamplitudesusedareA+=A−=A0=1sothattheyinterferedestructivelyatequalρmasses.Thehatchedareasindicatethemainoverlapregionsbetweenthedifferentρbands.Thecontourlinesinbothplotscorrespondto√

BpairscollectedattheΥ(4S)resonance(“on-resonance”),andanintegratedluminosityof11.6fb−1collectedabout40MeVbelowtheΥ(4S)(“off-resonance”).

AdetaileddescriptionoftheBABARdetectorispresentedinRef.[17].Thetrackingsystemusedfortrackandvertexreconstructionhastwomaincomponents:asiliconvertextracker(SVT)andadriftchamber(DCH),bothoperatingwithina1.5Tmagneticfieldgeneratedbyasuperconductingsolenoidalmagnet.Photonsareidentifiedinanelectromagneticcalorimeter(EMC)surroundingadetectorofinternallyreflectedCherenkovlight(DIRC),whichassociatesCherenkovphotonswithtracksforparticleidentification(PID).Muoncandidatesareidentifiedwiththeuseoftheinstrumentedfluxreturn(IFR)ofthesolenoid.

√s 1. =00.4

s+ (GeV/c)

5 G/ceV20.2

204

2530θ'0.500

0.510

m'00.20.40.60.81m'

3ANALYSISMETHOD

TheUandIcoefficientsandtheB0→π+π−π0eventyieldaredeterminedbyamaximum-likelihoodfitofthesignalmodeltotheselectedcandidateevents.Kinematicandeventshapevariablesexploitingthecharacteristicpropertiesoftheeventsareusedinthefittodiscriminatesignalfrombackground.Welimitthesizeofthedatasamplethatentersthefitbytighteningtheacceptancerequirementsforthediscriminantvariablescomparedtosimilaranalyses,becausethefitmodelwithatleast17physicalparametersisratherinvolved.WiththesamegoalandbecausethemodelingofthedistributionofthecontinuumeventsintheDalitzplotisdelicate,weremove

12

thecenteroftheDalitzplotfromtheanalysis.Thisrequirementdoesnotaffectthesignal.

3.1EVENTSELECTIONANDBACKGROUNDSUPPRESSION

WereconstructB0→π+π−π0candidatesfrompairsofoppositely-chargedtracks,formingagoodqualityvertex,andaπ0candidate.Weuseinformationfromthetrackingsystem,EMC,andDIRCtoremovetracksforwhichthePIDisconsistentwiththeelectron,kaon,orprotonhypotheses.Inaddition,werequirethatatleastonetrackhasasignatureintheIFRthatisinconsistentwiththemuonhypothesis.Theπ0candidatemassmustsatisfy0.11TworequirementsareappliedontheDalitzplot.Firstly,theinvariantmassofthetwotracksm0

0(→π+π−)π0backgroundmustbelargerthan0.52GeV/c2.Thisrejectsabout80%oftheB0→KS

0wouldrequireadedicated∆ttreatment.Thiscutevents,whichduetothelonglifetimeoftheKS

retains98%(100%)ofsignalB0→ρ0π0(B0→ρ±π∓)events.Secondly,weremovethecenteroftheDalitzplotbyrequiringthatatleastoneofthethreeinvariantmasses,m0,m+orm−,islowerthan1.5GeV/c2.

AB-mesoncandidateischaracterizedkinematicallybytheenergy-substitutedmassmES=

112andenergydifference∆E=E∗−s,where(EB,pB)and(E0,p0)[(B

arethefour-vectorsoftheB-candidateandtheinitialelectron-positronsystem,respectively.TheasteriskdenotestheΥ(4S)frame,andsisthesquareoftheinvariantmassoftheelectron-positronsystem.Werequire5.272)2,quantity∆E′=(2∆E−∆E+−∆E−)/(∆E+−∆E−),with∆E±(m0)=c±−(c±∓c¯)(m0/mmax0

wherem0monitorstheπ0-energydependence.Weusethevaluesc¯=0.045GeV,c−=−0.140GeV,

=5.0GeV,andrequire−1<∆E′<1.Thesesettingshavebeenob-c+=0.080GeV,mmax0

tainedfromMonteCarlo(MC)simulationandaretunedtomaximizetheselectionofcorrectlyreconstructedovermisreconstructedsignalevents.Thecutretains75%(25%)ofthesignal(con-tinuum).

Backgroundsariseprimarilyfromrandomcombinationsincontinuumevents.Toenhancedis-criminationbetweensignalandcontinuum,weuseaneuralnetwork(NN)tocombinefourdiscrim-inatingvariables:theangleswithrespecttothebeamaxisoftheBmomentumandBthrustaxisintheΥ(4S)frame,andthezerothandsecondordermonomialsL0,2oftheenergyflowaboutthe

󰀋

Bthrustaxis.ThemonomialsaredefinedbyLj=ipi×|cosθi|j,whereθiistheanglewithrespecttotheBthrustaxisoftrackorneutralclusteri,piisitsmomentum,andthesumexcludestheBcandidate.TheNNistrainedinthesignalregionwithoff-resonancedataandsimulatedsignalevents.ThefinalsampleofsignalcandidatesisselectedwithacutontheNNoutputthatretains77%(8%)ofthesignal(continuum).

Thetimedifference∆tisobtainedfromthemeasureddistancebetweenthezpositions(along

+−0andB0thebeamdirection)oftheB3tagdecayvertices,andtheboostβγ=0.56oftheeeπ

0weusethetaggingalgorithmofRef.[18].Thissystem8.TodeterminetheflavoroftheBtag

producesfourmutuallyexclusivetaggingcategories.WealsoretainuntaggedeventsinafifthcategorytoimprovetheefficiencyofthesignalselectionandbecausetheseeventscontributetothemeasurementofdirectCPviolation.EventswithmultipleBcandidatespassingthefullselection

occurin16%(ρ±π∓)and9%(ρ0π0)ofthecases.Ifthemultiplecandidateshavedifferentπ0’s,wechoosethecandidatewiththereconstructedπ0massclosesttothenominalone;ifnot,onecandidateisselectedatrandom.

ThesignalefficiencydeterminedfromMCsimulationis24%forB0→ρ±π∓andB0→ρ0π0events,and11%fornon-resonantB0→π+π−π0events.

Oftheselectedsignalevents,22%(B0→ρ±π∓),13%(B0→ρ0π0),and6%(non-resonant)aremisreconstructed,mostlyduetocombinatorialbackgroundfromlow-momentumtracksandphotons.TheyconcentrateinthecornersoftheDalitzplot.Thefractionofmisreconstructedeventsstronglyvariesacrossthetaggingcategories.

3.2BACKGROUNDFROMOTHERBDECAYS

WeuseMC-simulatedeventstostudythebackgroundfromotherBdecays.TheexclusiveB-backgroundmodesaregroupedintoeighteenclasseswithsimilarkinematicandtopologicalprop-erties.Morethanhundreddecaychannelshavebeenconsideredofwhichthirty-sixareretainedinthelikelihoodmodel.ThemostsignificantonesareB+→ρ+ρ0withlongitudinalpolarization(27±18eventsexpected),B+→π+ρ0(48±6),B+→π0ρ+(43±7),B0→ρ+ρ−withlongitudinalpolarization(50±10),B0→(a1π)0(29±11),B0→ρ−K+(61±11),andB0→higherkaonreso-nances(6±1).ThecharmedmodesB0→D−(→π−π0)π+andB0→

D0(→K+π−)π0

andB0→J/ψ(→ℓ+ℓ−)π0.Intotalweexpect49±15exclusiveb→cevents.Twoadditionalclassesaccountforinclusiveneutralandchargedb→cdecays,whereweexpect82±6and181±9events,respectively.

3.3THEMAXIMUM-LIKELIHOODFIT

Weperformanunbinnedextendedmaximum-likelihoodfittoextracttheinclusiveB0→π+π−π0eventyieldandtheUandIcoefficientsdefinedinEqs.(8)–(12).ThefitusesthevariablesmES,∆E′,theNNoutput,andtheDalitzplottodiscriminatesignalfrombackground.The∆tmea-surementallowstodeterminemixing-inducedCPviolationandprovidesadditionalcontinuum-backgroundrejection.

Theselectedon-resonancedatasampleisassumedtoconsistofsignal,continuum-backgroundandB-backgroundcomponents,separatedbytheflavorandtaggingcategoryofthetagsideBdecay.Thesignallikelihoodconsistsofthesumofacorrectlyreconstructed(“truth-matched”,TM)componentandamisreconstructed(“self-cross-feed”,SCF)component.

Theprobabilitydensityfunction(PDF)Picforaneventiintaggingcategorycisthesumoftheprobabilitydensitiesofallcomponents,namely

Pic

≡

c

N3πf3πc+Nqq¯

1

󰀍

(1−

cc

fSCFP3π−SCF,i

󰀏

2

󰀁

c

1+qtag,iAB+,tag,jPB+,ij

󰀃

14

BNclass

0

+

j=1

cisthefractionwhere:N3πisthetotalnumberofπ+π−π0signaleventsthedatasample;f3π

ofsignaleventsthataretaggedincategoryc;

󰀅

cc

NB0jfB0jPB0,ij,

(18)

B0¯,tagparameterizespossibletagasymmetryintag;Aqq

B+(NB0)isthenumbercisthecontinuumPDFfortaggingcategoryc;Ncontinuumevents;Pqq¯,iclassclass

ofcharged(neutral)B-relatedbackgroundclassesconsideredinthefit;NB+j(NB0j)isthenumber

ccofexpectedeventsinthecharged(neutral)B-backgroundclassj;fB+j(fB0j)isthefractionof

charged(neutral)B-backgroundeventsofclassjthataretaggedincategoryc;AB+,tag,jdescribesapossibletagasymmetryinthecharged-Bbackgroundclassj;correlationsbetweenthetagandthepositionintheDalitzplot(the“charge”)areabsorbedintag-flavor-dependentDalitzplot

c+PDFsthatareusedforcharged-Bandcontinuumbackground;PB+,ijistheB-backgroundPDF

cfortaggingcategorycandclassj;finally,PB0,ijistheneutral-B-backgroundPDFfortagging

categorycandclassj.

caretheproductofthefourPDFsofthediscriminatingvariables,x=mThePDFsPX1ES,

′′′x2=∆E,x3=NNoutput,andthetripletx4={m,θ,∆t}:

c

PX,i(j)

≡

k=1

Theextendedlikelihoodoveralltaggingcategoriesisgivenby

L≡

c

5󰀇

4󰀇

c

PX,i(j)(xk).

(19)

e−

c=1

Nisthetotalnumberofeventsexpectedincategoryc.

Atotalof39parameters,includingtheinclusivesignalyieldandtheparametersfromEq.(6),arevariedinthefit.3.3.1

THE∆tANDDALITZPLOTPDFS

SignalParameterization.TheDalitzplotPDFsrequireasinputtheDalitzplot-dependentrelativeselectionefficiency,ǫ=ǫ(m′,θ′),andSCFfraction,fSCF=fSCF(m′,θ′).BothquantitiesaretakenfromMCsimulation.TheyaregiveninFig.2(leftplotforǫandrightplotforfSCF),wherethesymmetryoftheDalitzplothasbeenusedtofoldtheupperθ′halfintothelowerone.AwayfromtheDalitzplotcornerstheefficiencyisuniform,whileitdecreaseswhenapproachingthecorners,whereoneoutofthethreebodiesinthefinalstateisclosetorestsothattheacceptancerequirementsontheparticlereconstructionbecomeincisive.CombinatorialbackgroundsandhenceSCFfractionsarelargeinthecornersoftheDalitzplotduetothepresenceofsoftneutralclustersandtracks.

Foraneventi,wedefinethetime-dependentDalitzplotPDFs

2

P3π−TM,i=εi(1−fSCF,i)|detJi||A±3π(∆t)|,2P3π−SCF,i=εifSCF,i|detJi||A±3π(∆t)|,

(21)(22)

15

0.50.40.3

0.50.450.40.350.30.250.20.150.10.0500

0.2

0.4

0.6

0.8

1

0.50.40.3

10.90.80.70.60.50.40.30.20.100

0.2

0.4

0.6

0.8

1

θ'0.20.10

θ'0.20.10

m'm'

Figure2:SelectionefficiencyofB0→π+π−π0events(left)andfractionofmisreconstructedevents(right)inthe(symmetrized)squareDalitzplotforMC-simulatedevents.

whereP3π−TM,iandP3π−SCF,iarenormalized.Thecorrespondingphasespaceintegrationinvolvestheexpectationvalues󰀑ε(1−fSCF)|detJ|fκfσ∗󰀒and󰀑εfSCF|detJ|fκfσ∗󰀒forTMandSCFevents,wheretheindicesκ,σrunoverallresonancesbelongingtothesignalmodel.Theexpectationvaluesaremodel-dependentandarecomputedwiththeuseofMCintegrationoverthesquareDalitzplot:󰀎1󰀎1

κσ∗dm′dθ′

κσ∗00ε(1−fSCF)|detJ|ff󰀑ε(1−fSCF)|detJ|ff󰀒=fSCF≡󰀑fSCF|detJ|fκfσ∗󰀒.As

forthePDFnormalization,itisdecay-dynamics-dependent.Ithastobecomputediteratively,thoughtheremainingsystematicuncertaintyafteroneiterationstepissmall.WedeterminetheaverageSCFfractionsseparatelyforeachtaggingcategoryfromMCsimulation.

Thewidthofthedominantρ(770)resonanceislargecomparedtothemassresolutionforTMevents(about8MeV/c2coreGaussianresolution).WecanthereforeneglectresolutioneffectsintheTMmodel.MisreconstructedeventshaveapoormassresolutionthatstronglyvariesacrosstheDalitzplot.Itisdescribedinthefitbya2×2-dimensionalresolutionfunction

′′′

RSCF(m′r,θr,mt,θt),

(24)

′whichrepresentstheprobabilitytoreconstructatthecoordinate(m′r,θr)aneventthathasthe

′truecoordinate(m′t,θt).Itobeystheunitaritycondition

󰀈1󰀈1

00

′′′′′RSCF(m′r,θr,mt,θt)dmrdθr=1,′

∀(m′t,θt)∈SDP,

(25)

andisconvolvedwiththesignalmodel.TheRSCFfunctionisobtainedfromMCsimulation.

Figure3showstheresolutionfunctionofTM(left)andSCFevents(right)fortwocoordinatesdepictedbytheopenstars.

WeusethesignalmodeldescribedinSection1.1.Itcontainsthedynamicalinformationandisconnectedwith∆tviathematrixelement(6),whichservesasPDF.Itisdilutedbytheeffects

16

0.510.51TM events0.40.3

0.90.80.70.60.30.4

SCF events0.90.80.70.6generated0.20.100

0.2

0.4

0.6

0.8

1

0.50.40.30.20.10generated0.20.100

0.2

0.4

0.6

0.8

1

0.50.40.30.20.10θ'm'

θ'm'

Figure3:ResolutionforTM(left)andSCFevents(righthandplot)inthesquareDalitzplotfortwocoordinatesindicatedbytheopenstars.

ofmistaggingandthelimitedvertexresolution[10].The∆tresolutionfunctionforsignalandB-backgroundeventsisasumofthreeGaussiandistributions,withparametersdeterminedbyafittofullyreconstructedB0decays[18].

BackgroundParameterization.TheDalitzplot-and∆t-dependentPDFsfactorizeforthecharged-B-backgroundmodes,butnot(necessarily)fortheneutral-BbackgroundduetoB0

isparameterizedasthesumofthreeGaussiandistributionswithcommonmeanandthreedistinctwidthsthatscalethe∆tper-eventerror.Thisyieldssixshapeparametersthataredeterminedbythefit.Themodelismotivatedbytheobservationthatthe∆taverageisindependentofitserror,andthatthe∆tRMSdependslinearlyonthe∆terror.3.3.2

PARAMETERIZATIONOFTHEOTHERVARIABLES

ThemESdistributionofTMsignaleventsisparameterizedbyabifurcatedCrystalBallfunc-tion[20],whichisacombinationofaone-sidedGaussianandaCrystalBallfunction.Themeanofthisfunctionisdeterminedbythefit.Anon-parametricfunctionisusedtodescribetheSCFsignalcomponent.

The∆E′distributionofTMeventsisparameterizedbyadoubleGaussianfunction,whereallfiveparametersdependlinearlyonm20.MisreconstructedeventsareparameterizedbyabroadsingleGaussianfunction.

BothmESand∆E′PDFsareparameterizedbynon-parametricfunctionsforallB-backgroundclasses.

ThemESand∆E′PDFsforcontinuumeventsareparameterizedwithanArgusshapefunc-tion[21]andasecondorderpolynomial,respectively,withparametersdeterminedbythefit.

Weusenon-parametricfunctionstoempiricallydescribethedistributionsoftheNNoutputsfoundintheMCsimulationforTMandSCFsignalevents,andforB-backgroundevents.WedistinguishtaggingcategoriesforTMsignaleventstoaccountfordifferencesobservedintheshapes.

ThecontinuumNNdistributionisparameterizedbyathirdorderpolynomialthatisdefinedtobepositive.Thecoefficientsofthepolynomialaredeterminedbythefit.ContinuumeventsexhibitacorrelationbetweentheDalitzplotcoordinateandtheshapeoftheeventthatisexploitedintheNN.ThetightrequirementthateliminatesthecenteroftheDalitzplothasthepurposetoreducesuchcorrelation.Tocorrectforresidualeffects,weintroducealineardependenceofthepolynomialcoefficientsonthedistanceoftheDalitzplotcoordinatetothekinematicboundariesoftheDalitzplot.Theparametersdescribingthisdependencearedeterminedbythefit.

4SYSTEMATICSTUDIES

ThecontributionstothesystematicerroronthesignalparametersaresummarizedinTable1.

Theuncertaintiesassociatedwith∆mdandτareestimatedbyvaryingtheseparameterswithintheuncertaintiesontheworldaverage[19].

ThesystematiceffectsduetothesignalPDFs(“Signaldescription”fieldinTable1)compriseuncertaintiesinthePDFparameterization,thetreatmentofmisreconstructedevents,thetaggingperformance,andthemodelingofthesignalcontributions.

WhenthesignalPDFsaredeterminedfromfitstoacontrolsampleoffullyreconstructedBdecaystoexclusivefinalstateswithcharm,theuncertaintiesareobtainedbyvaryingtheparameterswithinthestatisticaluncertainties.Inothercases,thedominantparametershavebeenleftfreetovaryinthefit,andthedifferencesobservedinthesefitsaretakenassystematicerrors.

TheaveragefractionofmisreconstructedsignaleventspredictedbytheMCsimulationhasbeenverifiedwithfullyreconstructedB→Dρevents[10].Nosignificantdifferencesbetweendataandthesimulationwerefound.Wevary

∆mdandτB0

SignaldescriptionTagging

SignalmodelBBackground

ContinuumparametrizationFixing10ρ0π0parametersFitBias0.0030.0050.0030.0040.0110.0040.0030.0150.0010.0040.0020.0050.0130.0040.0030.0140.0000.0010.0010.0110.0070.0400.0250.0060.0040.0110.0100.0130.0290.0020.0190.0220.0000.0130.0010.0040.0150.0120.0070.0150.0010.0100.0100.0200.0350.0040.0390.0200.0030.0420.0420.2020.1180.0270.0420.1770.0070.0560.0190.1540.0710.0310.0220.161

∆mdandτB0

SignaldescriptionTagging

SignalmodelBBackground

ContinuumparametrizationFixing10ρ0π0parametersFitBias0.0020.0140.0090.0790.0650.0310.0190.0780.0010.0430.0110.1370.0380.0340.0100.0760.0100.1080.0620.6290.1590.0970.0240.2210.0410.0630.0410.3330.1790.0990.0040.2400.0010.0130.0040.1290.0120.0190.1100.0510.0010.0290.0060.0960.0400.0590.0690.0480.0010.0480.0050.1070.0380.0120.1020.0550.0010.0160.0070.1610.0380.0280.0000.047

Table1:Summaryofsystematicuncertainties.

Taggingefficiencies,dilutionsandbiasesforsignaleventsarevariedwithintheirexperimentaluncertainties.

ThemostimportantcontributiontothesystematicuncertaintystemsfromthesignalmodelingoftheDalitzplotdynamics.Wevarythemassandwidthoftheρ(770)andρ(1450)withinrangesthatexceedtwicetheerrorsfoundfortheseparametersinthefitstoτande+e−data[11].SincesomeoftheUandIcoefficientsexhibitsignificantdependenceontheρ(1450)contribution,weleaveitsamplitude(phaseandfraction)freetovaryinthenominalfit.Wevarytherelativeamountofρ(1700)by30%withrespecttothenominalmodel,anditsphaseby8◦,toassignasystematicerror.Wehaveperformedafitwheretheρ(1700)amplitudeparameters(magnitudeandphase)arefreetovary.Wefindresultsthatareinagreementwiththenominalmodel.ThevariationsfortheUandIcoefficientsobservedinthisfitcomparedtothenominalonearesmallerthanthesystematicuncertaintiesweassignduetotheρ(1700)amplitudeuncertainty.

Toestimatethecontributionfromnon-resonanceB0→π+π−π0events,wehaveperformedanindependentanalysiswhereweapplythecontraryoftheDalitzplotrequirementthatisusedinthenominalanalysis;toremovetheρsignalweretainonlythoseeventsforwhichtheminimuminvariantmassexceeds1.5GeV/c2.Forsimplicity,weassumeauniformDalitzdistributionforthe

19

non-resonanceevents.Thefitfindsnonon-resonanteventssothatwecandetermineapreliminaryupperlimitof1.4×10−6at90%confidencelevel(statisticalerrorsonly).Accordingtothislimit,weaddsimulatednon-resonanteventstothenominaldatasampletoestimatethesystematicuncertainty.WehavealsosearchedforthepresenceofB0→f0(980)π0eventswithoutfindingevidenceforasignal.

AmajorsourceofsystematicuncertaintyistheB-backgroundmodel.Theexpectedeventyieldsfromthebackgroundmodesarevariedaccordingtotheuncertaintiesinthemeasuredorestimatedbranchingfractions.SinceB-backgroundmodesmayexhibitCPviolation,thecorrespondingparametersarevariedwithinappropriateuncertaintyranges.AsisdoneforthesignalPDFs,wevarythe∆tresolutionparametersandtheflavor-taggingparameterswithintheiruncertaintiesandassignthedifferencesobservedintheon-resonancedatafitwithrespecttonominalfitassystematicerrors.

Theparametersforthecontinuumeventsaredeterminedbythefit.Noadditionalsystematicuncertaintiesareassignedtothem.AnexceptiontothisistheDalitzplotPDF:toestimatethesystematicuncertaintyfromthemESsidebandextrapolation,weselectlargesamplesofoff-resonancedatabylooseningtherequirementson∆EandtheNN.Wecomparethedistributionsofm′andθ′betweenthemESsidebandandthesignalregion.Nosignificantdifferencesarefound.WeassignassystematicerrortheeffectseenwhenweightingthecontinuumDalitzplotPDFbytheratioofbothdatasets.Thiseffectismostlystatisticalinorigin.Toaccountforpossibleinaccuraciesintheempiricalparameterization,weaddtestcomponentswithfloatingeventyieldstothefitthatconsistofcontinuum-background-likereferencedistributionsinallfitvariablesbuttheDalitzplot,forwhichsignal-likedistributionsareused.Inaddition,weleavetheB-backgroundclassesfreetovaryinthefit.Thisallowsthefittoabsorbeventsthat,duetopossibleproblemsinthecontinuumDalitzplotdescription,wouldbiasthesignalyield.Theshiftsinthesignalparametersobservedwhenaddingthesetestcomponentsaretakenassystematicuncertainties(“continuumparameterization”inTable1).Itleadstosignificanteffectsintheρ0π0regionoftheDalitzplot.

Weassessthedependenceoftheresultsonwhetherallthe27UandIcoefficientsarefreeinthefitoronlythe16mostsignificantones.Tostudythiseffect,wegenerateMCsampleswithvaluesfortheUandIcoefficientsaccordingtoournominal16parameterfitresult.Wethenperformafullamplitudefittothese16coefficients(cf.Section6)andcomputefromthebestfittheexpectedvaluesforthemissingcoefficients.MonteCarlosamplesaregeneratedwiththeuseofall27coefficients,whicharefitwiththenominalmodelwhere10coefficientsaresettozero.TheobservedsystematiceffectonthemeasuredUandIcoefficientsdependonthebranchingfractionforB0→ρ0π0forwhichweuseourupperbound[22](whichisinagreementwiththefindinginthisanalysis).

Finally,tovalidatethefittingtool,weperformfitsonlargeMCsampleswiththemeasuredproportionsofsignal,continuumandB-backgroundevents.Nosignificantbiasesareobservedinthesefits.Thestatisticaluncertaintiesonthefitparametersaretakenassystematicuncertainties(“Fitbias”).

Thesystematicerrorsfortheparametersthatmeasureinterferenceeffectsaredominatedbytheuncertaintyinthesignalmodel,mainlythetaildescriptionoftheρresonance.Fortheotherparameters,theuncertaintyonthefitbiasandtheB-backgroundcontaminationareimportant.

20

80

100

2BABARpreliminaryEvents/1 MeV/c60

Events/0.17550250

20

0-1

-0.75-0.5-0.25

0

0.25

0.50.75

1

40

5.2755.285.285

2

∆E'

100

10

75

2mES (GeV/c)

Events/0.150

Events/1 10

25

1

0

-0.2

0

0.2

0.4

0.6

0.8

1

-15-10-5 051015NN output

150

100

∆t/σ(∆t)Events/0.06Events/0.05100

50

50

00

0.2

0.4

0.6

0.8

00

0.2

0.4

0.6

0.8

1

m'θ'

Figure4:Distributionsof(clockwisefromtopleft)∆E′,mES,NNoutput,∆t/σ(∆t),m′andθ′forsamplesenhancedinB0→π+π−π0signal.Thedotswitherrorbarsgivetheon-resonancedata.Thesolidhistogramshowstheprojectionofthefitresult.Thedark,mediumandlightshadedareasrepresentrespectivelythecontributionfromcontinuumevents,thesumofcontinuumeventsandtheB-backgroundexpectation,andthesumoftheseandthemisreconstructedsignalevents.

21

I−I++U0−U−+U−−U+−,ImU+−−,ReU+−+,ImU+−+,ReU+−ImI+−ReI+−+,ImU+0+,ReU+0+,ImU−0+,ReU−0

CoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientCoefficientofofofofofofofofofofofofofofofof

|f−|2sin(∆md∆t)|f+|2sin(∆md∆t)|f0|2

|f−|2cos(∆md∆t)|f−|2

|f+|2cos(∆md∆t)

∗]cos(∆m∆t)Im[f+f−d

∗]cos(∆m∆t)Re[f+f−d∗]Im[f+f−∗]Re[f+f−

∗]sin(∆m∆t)Im[f+f−d∗Re[f+f−]sin(∆md∆t)∗]Im[f+f0∗]Re[f+f0∗]Im[f−f0∗]Re[f−f0

−0.19±0.11±0.02

0.06±0.11±0.020.16±0.05±0.050.22±0.16±0.051.19±0.12±0.030.50±0.17±0.060.3±1.4±0.32.0±1.2±0.20.16±0.70±0.14−0.26±0.65±0.17−5.2±1.9±0.7−0.3±2.0±0.50.25±0.35±0.18−0.34±0.39±0.150.34±0.43±0.17−0.98±0.44±0.18

I−I++U0−U−+U−−U+−,ImU+−−,ReU+−+,ImU+−+,ReU+−ImI+−ReI+−+,ImU+0+,ReU+0+,ImU−0+,ReU−0

1.000.000.160.030.01−0.060.07−0.02−0.10−0.030.09−0.02−0.07−0.000.02−0.05

1.000.210.010.01−0.03−0.04−0.01−0.010.02−0.050.00−0.00−0.01−0.01−0.05

1.000.01−0.050.010.07−0.04−0.09−0.020.060.01−0.180.040.13−0.05

1.000.00−0.03−0.100.020.050.02−0.100.060.03−0.190.010.01

1.00−0.06−0.050.010.13−0.080.040.030.030.01−0.050.01

1.00−0.040.010.130.07−0.00−0.02−0.010.000.010.04

1.00−0.05−0.05−0.010.27−0.02−0.110.130.03−0.06

1.000.15−0.01−0.04−0.040.08−0.00−0.010.03

1.00

−0.031.00

−0.14−0.101.00

0.05−0.03−0.011.00

0.060.02−0.100.121.00

−0.06−0.040.130.01−0.161.00

−0.040.040.00−0.15−0.19−0.001.000.130.01−0.070.120.14−0.05−0.01

+U+

,C

−

=

−U−

+U+

,

S−=

2I−

++U++U−

,(26)

andwhereC=(C++C−)/2,∆C=(C+−C−)/2,S=(S++S−)/2,and∆S=(S+−S−)/2.In

contrasttoourpreviousanalysis[10],thedefinitionsofEq.(26)explicitlyaccountforthepresenceofinterferenceeffects,andarethusexactevenforaρwithfinitewidth,aslongastheUandIcoefficientsareobtainedwithaDalitzplotanalysis.Thistreatmentleadstoadilutionoftheresultandhencetoslightlyincreasedstatisticaluncertaintiescomparedtoneglectingtheinterferenceeffects.

FortheCP-violationparameters,weobtain

Aρπ=−0.088±0.049±0.013,C=

0.34±0.11±0.05,

S=−0.10±0.14±0.04,

Γ(

(27)(28)(29)

B0→ρ+π−)]/[Γ(B0→ρ+π−)+

B0→ρ−π+)]/[Γ(B0→ρ−π+)+Γ(

1BABARpreliminary0.5Aρπ+ –0-0.5-1-1

-0.5

0

0.5

1

Aρπ

– +

−+−Figure5:ConfidencelevelcontoursforthedirectCPasymmetriesA+ρπversusAρπ.Theshaded

areasrepresent1σ,2σand3σcontours,respectively.

wherethefirsterrorsgivenarestatisticalandthesecondsystematic.Fortheotherparametersinthequasi-two-bodydescriptionoftheB0(

|κ+−|2+1|κ−+|2+1

A−/A+)andκ−+=(q/p)(

=−=

Aρπ+C+Aρπ∆C

Aρπ−C−Aρπ∆C

10.8

BABARpreliminary10.8

BABARpreliminaryC.L.0.40.20-180

C.L.180

0.6

using isospinwithout isospin0.60.40.20

-120-600601200306090120150180

δ+ – (deg)

α (deg)

Figure6:Confidencelevelfunctionsforδ+−(left)andα(right).IndicatedbythedashedhorizontallinesaretheC.L.valuescorrespondingto1σand2σ,respectively.

Themeasurementoftheresonanceinterferencetermsallowsustodeterminetherelativephase

δ+−=argA+∗A−,

󰀁

󰀃

(35)

betweentheamplitudesofthedecaysB0→ρ−π+andB0→ρ+π−.Throughthedefinitions(8)–(12),wecanderiveaconstraintonδ+−fromthemeasuredUandIcoefficients10byperformingaleast-squaresminimizationwiththesixcomplexamplitudesasfreeparameters.Inthisfit,onecomplexamplitudecanbefixedduetoanarbitraryglobalphaseandthenormalizationcondi-+

tionU+=1,leaving10real-valuedunknowns.Weobtaintheconfidencelevel(C.L.)functionrepresentedbythedashedlineinthelefthandplotofFig.6.Thefunctionincludessystematicerrors.

Thisresultdoesnotrequireassumptionsbeyondtheonesoutlinedintheintroduction.Theconstraintcanbeimprovedwiththeuseofstrongisospinsymmetry.TheamplitudesAκrepresentthesumoftree-levelandpenguin-typeamplitudes,whichhavedifferentCKMfactors:thetree-level(Tκ)B0→ρκπ

Aκ=T

κ

,(36)

κ.ThewherethemagnitudesoftheCKMfactorshavebeenabsorbedintheTκ,Pκ,T

Eqs.(36)represent13unknownsofwhichtwocanbefixedduetoanarbitraryglobalphaseandthe

+

=1.Usingstrongisospinsymmetryandneglectingisospin-breakingnormalizationconditionU+0effects,onecanidentifyP=−(P++P−)/2,whichreducesthenumberofunknownstobe

determinedbythefitto9.Thissetofparametersprovidestheconstraintonδ+−,givenbythe

κthechargeconjugateofκ,where

solidlineontheleftplotofFig.6.Thefitreturnsaminimumχ2of7.4,whichhasasignificancelevelof0.39for7degreesoffreedom.Wefindforthesolutionthatisfavoredbythefit

δ+−=

󰀉

−67+28−31

±7

wherethefirsterrorsarestatisticalandthesecondsystematic.Onlyamarginalconstraintonδ+−

isobtainedforC.L.<0.05.

Finally,followingthesameprocedure,wecanalsoderiveaconstraintonαfromthemeasuredUandIcoefficients.TheresultingC.L.functionversusαisgivenintherighthandplotofFig.6.Itincludessystematicuncertainties.Ignoringthemirrorsolutionatα+180◦,wefind

α=

󰀉

󰀌◦

,(37)

113+27−17

±6

wherethesystematicerrorisdominatedbytheuncertaintiesinthesignalmodel.Onlyamarginal

constraintonαisobtainedforC.L.<0.05.

󰀌◦

,(38)

7SUMMARY

WehavepresentedthepreliminarymeasurementofCP-violatingasymmetriesinB0→π+π−π0decaysdominatedbytheρresonance.Theresultsareobtainedfromadatasampleof213millionsΥ(4S)→B

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27