輻射レプトニック崩壊 τ → ℓννγを用いたタウ粒子のミシェルパラメータ ηとξκの測定

学位論文

Measurement of the tau Michel parameters

𝜼

̅

and

𝝃𝜿

in the radiative leptonic decay

𝝉 → 𝓵𝝂𝝂̅𝜸

（

輻射レプトニック崩壊

_{𝜏 → ℓ𝜈𝜈̅𝛾}

を用いた

タウ粒子のミシェルパラメータ

_𝜂̅

と

_𝜉𝜅

の測定

）

平成 28 年 12 月博士（理学）申請

東京大学大学院理学系研究科

物理学専攻 清水 信宏

WepresentthemeasurementoftheMi helparametersoflepton¯ andintheradiativeleptoni

de ay  ! ` ¯ using 703fb

1

of ollision data olle ted withthe Belledete tor atthe KEKB

e +

e ollider. The Mi hel parameter is a fundamental property of unstable harged leptons and

hara terizes thedynami sofleptoni de ays. Theexperimentalvaluesof¯andparametersmay

revealthepresen eofnewphysi sbeyondtheStandardModel.

TheMi helparametersaremeasuredbyanunbinnedmaximumlikelihoodmethodwhere¯ and

 aretted to the kinemati distributionof e

+ e !  +  ! ( +  0 ¯ )(`  )¯ (` = e or). Using

the muon mode, ¯ and are simultaneouslytted tothe spe trato be ¯



= 1:31:50:8and

( ) 

= 0:80:5 0:3. In theele tron mode, taking intoa ount thesuppressionof ¯ sensitivity

from thesmallmassofdaughterele tron,we extra t( )

e

byxing¯ valuetotheStandardModel

predi tionof¯

SM

= 0. Themeasuredvalueis ( )

e

= 0:40:80:9. Thersterrorisstatisti al

and the se ondis systemati . Thisis the rstmeasurement ofthese parameters. Theseresults are

onsistentwiththeStandardModelpredi tionswithintheirun ertaintiesandgivea onstraintonthe

oupling oeÆ ientofthegeneralizedweakintera tion.

We alsomeasuredthe bran hingratioof theradiativeleptoni de aysunder thephotonenergy

thresholdof E



> 10MeVinthe restframetobeB(

 ! e   )¯ = (1:820:020:10)10 2 and B(  !    )¯ = (3:68 0:020:15) 10 3

. These resultsare onsistentwith the leading

order StandardModel predi tion. In thenext-leadingorder,there areee tsfrom multiplephoton

emission,whi hisnotimplementedinthe urrenteventgenerator. Animprovementofgeneratoris

1 Introdu tion 5

1.1 TheStandardModel . . . 5

1.1.1 Sear hforphysi sbeyondtheStandardModel . . . 6

1.2 Sear hforphysi sbeyondtheStandardModelin hargedleptons . . . 6

1.3 Mi helParameters . . . 7

1.3.1 Historyoftestofthe harged urrent . . . 8

1.3.2 Mi helformalism . . . 8

1.4 FurthertestsoftheV Aintera tioninde ays . . . 9

1.5 Physi smotivation . . . 10

1.6 Produ tionofleptons . . . 11

2 Radiativeleptoni de ay !`  ¯ 13 2.1 Denitionoftheradiativede ayanditsdistribution . . . 13

2.2 Spin-spin orrelationof +  andtwo-bodyde ay + ! + ¯ ! +  0 ¯ ) . . . 15

2.3 Bran hingratioof !`  ¯ de ays . . . 17

2.4 Ee toftheMi helparameteronthedistribution . . . 18

2.5 Determinationofdire tion . . . 19

3 Experimental Apparatus 23 3.1 TheKEKBa elerator . . . 23

3.1.1 Denitionofframe . . . 25

3.2 TheBelledete tor . . . 26

3.2.1 Sili onVertexDete tor(SVD) . . . 28

3.2.2 CentralDriftChamber(CDC) . . . 28

3.2.3 Ele tromagneti Calorimeter(ECL) . . . 31

3.2.4 AerogelCerenkovCounter(ACC) . . . 32

3.2.5 Time-Of-Flight ounter(TOF) . . . 34

3.2.6 K L andmuondete tor(KLM) . . . 36

3.2.7 Trigger . . . 37

3.2.8 Dataa quisitionsystem(DAQ) . . . 37

3.2.9 Parti leidenti ations . . . 40

3.3 OperationofBelledatataking . . . 44

3.4 MonteCarlosimulation . . . 44

4 Eventsele tion 46 4.1 Presele tion . . . 46

4.4.2   !(  )(¯  )¯ de ay andidates . . . 64

4.5 TotaleÆ ien y . . . 70

5 MethodofthemeasurementoftheMi helparameters 71 5.1 Notationsand onventions . . . 71

5.2 Unbinnedmaximumlikelihoodmethod . . . 71

5.3 AverageeÆ ien yandnormalization . . . 72

5.4 Implementationofprobabilitydensityfun tions . . . 73

5.4.1 Des riptionofthesignalPDF . . . 73

5.4.2 Des riptionofthemajorba kgroundPDFs . . . 78

5.4.3 Des riptionofotherba kgroundmodes . . . 80

5.4.4 Implementationoftheee tof ollinearISR . . . 85

5.4.5 Implementationoftheee tofdete torresolution . . . 86

5.5 Fitting . . . 87

5.6 Validationoftter . . . 87

5.6.1 Linearityoftter . . . 87

5.6.2 Dependen eofsensitivityonsele tion riteria . . . 87

5.6.3 FittingMi helparameterswithba kgroundPDFs . . . 92

6 Analysisoftheexperimental data 97 6.1 TriggereÆ ien y orre tions  . . . 97

6.2 Parti lesele tioneÆ ien y orre tions. . . 98

6.2.1  0 IDand IDeÆ ien y orre tions . . . 102

6.3 Re onstru tioneÆ ien y orre tions . . . 105

6.4 Binningof orre tionfa tors . . . 109

6.5 Conrmationofthe orre tionR . . . 110

7 Evaluationofun ertainties 115 7.1 Statisti alun ertainties . . . 115

7.2 Systemati errors . . . 115

7.2.1 Systemati un ertaintyfrombran hingratios . . . 115

7.2.2 Un ertaintyfromtherelativenormalization . . . 115

7.2.3 Un ertaintyfromtheabsolutenormalization . . . 119

7.2.4 Un ertaintiesfrom orre tionfa torsandineÆ ien ies . . . 119

7.2.5 Un ertaintyduetoimperfe tformulationofPDFs . . . 121

7.2.6 Un ertaintyfromthesimulationofoverlapintheECL lusters . . . 122

7.2.7 Un ertaintyfromthedete torresolution . . . 123

7.2.8 Un ertaintyfromthebeamEnergyspread . . . 123

7.2.9 Un ertaintyfromE distribution . . . 123

8 Resultsanddis ussion 125 8.1 Fitresult . . . 125

8.2 Goodnessoft . . . 125

8.3 Upperlimitson ouplingsg N ij . . . 129

8.4 Couplingswithright-handedlepton . . . 131

9.1 Eventsele tion . . . 134

9.2 Method . . . 138

9.3 Evaluationofsystemati un ertainties . . . 139

9.4 Result . . . 140

9.4.1 Ratioofbran hingratioQ=B( !e ¯ )=B( !  )¯ . . . 140

9.5 Dis ussion . . . 143

9.5.1 Treatmentofdoublephotons . . . 143

9.5.2 Anomalousfour-pointintera tion . . . 144

10 Futureprospe tsand on lusion 146 10.1 Futureexperimentandexpe tedimprovements . . . 146

10.2 Con lusion . . . 148

A Measurementofthebran hingratioB( ! ` ¯ )(validation) 157 A.1 Methodandevaluationofsystemati un ertainties . . . 157

A.2 Result . . . 157

A.3 Dis ussionand on lusions . . . 157

A.3.1 E LAB extra dependen e . . . 157 A.3.2 Con lusions. . . 158 B Des riptionofba kgroundPDFs 160 B.1 Ordinaryleptoni de ay+beamba kground . . . 160

B.2 Des riptionofPDFfor3events . . . 161

B.2.1 Extra tionoftheineÆ ien ies . . . 164

B.3 Des riptionofba kground . . . 164

B.3.1 Extra tionofineÆ ien y . . . 168

B.4 Des riptionofISRphoton+ordinaryleptoni de ayevents. . . 168

B.5 Des riptionof3-2de ayevents . . . 170

B.5.1 extra tionofineÆ ien y . . . 171

B.6 Des riptionofanordinaryleptoni de ay+bremsstrahlungevents . . . 173

C Cal ulationofJa obians 176 C.1 Ja obianforLorentz-transformation . . . 176

C.2 2-bodyde ay . . . 176 C.2.1       (P  0 ;  0 ;  ) (P ; ; 0 )       . . . 177 C.3       (P a 1 ; a 1 ;m 2 a 1 ; ˆ  ) (P  0 lost ;  0 lost ;P  ;  )       . . . 178 C.4     (   1 ;   2 ;) (P 1 ; 1 ;P 2 ; 2 )     . . . 179 C.4.1       (   ;  ) (P  ;  ;  )       . . . 179 C.5        (P ; ) ( e ;)        . . . 180

E.1 des riptionofenergyresponse . . . 186

Introdu tion

1.1 The Standard Model

Everythinginouruniverseisbelievedtobemadefromfundamentalparti les. Theirintera tionsor

for esaredes ribedbyanex hangeofotherparti les. Su hparti lesaredes ribedsoasnottohave

their sizesas wellasinternal stru turestherebytheyare alled elementaryparti les. Thequantum

eldtheory(QFT)isaphysi alframeworkwhi htreatsanentityofsu haparti leasanex itation

of eldin thespa e-time, relyingon boththequantumme hani s andthespe ial relativity—most

su essfultheoriesofphysi sinthetwentieth entury.

Inprin iple, in theframework ofQFT,people an freelybuildnew theories: arbitrarytypes of

parti les andrulesof intera tions anformone theory. However,thereare fewtheories whi h an

reasonablypredi trealbehaviorsofknownparti les. TheStandardModel(SM)isknowntobethe

strongestpredi tabletheoriesofQFT,inwhi htwelvetypesoffermions( orrespondingtomatters)

aregovernedbythreetypesoffor es.Thefor esaremediatedby orrespondingbosons.Themasses

oftheseparti lesareuniquelydeterminedbystrengthsofea h ouplingtotheeldofHiggsboson.

BelowwegiveasummaryoftheSM.

Typesofelementaryparti les

ˆ Higgsbosonisaspin-0parti letogiveotherparti lesmasses.

ˆ There arethreetypes offor es: ele tromagneti intera tion, hargedand neutralweak

inter-a tions and strong intera tion. These for esare mediated by spin-1 parti lesand play roles

in an ellationsofposition-dependentphases. Theinvarian eunderthephasetransformation

is alled gauge invarian e, hen e these parti les are also alled gauge bosons. These gauge

bosons are named photon for the ele tromagneti , W



and Z bosons for the harged and

neutralweakintera tionsandgluongforthestrongfor e.

ˆ Matters are made from

spin-1

2

parti les whi h are ategorized into two groups: six types of

quarks and six types of leptons. The quark has harges of all for es above and is able to

parti ipateinallintera tions. Whereastheleptondoesnothavea hargeofstrongfor ebuthas

aweak harge,a ordinglyitparti ipatesintheweakintera tions. Thethreequarkshave+2=3

ele tromagneti hargesandotherthreehave 1=3.Threeleptonswhi hhaveele tromagneti

harges+1are alled hargedleptonsandareabletointera tviaele tromagneti for ewhile

ˆ Parti les havea property alled hirality,whoseeigenvalue is1 or-1. Inthe masslesslimit,

it iswellknownthat the hiralityequalsto heli itythatisdened ash =

ˆ

S
n,where

ˆ

S isa

normalizedve torofspinand nisaunityve toroftheparti lemovement. The positiveand

negativeheli itiesare alledright-handedandleft-handed,respe tively.

ˆ Of all for es, only harged weak intera tion an hange the avor of parti le. Moreover, it

violates thesymmetryof hirality,i.e.,onlynegative- hiralityparti lesand anti-parti lesare

a tiveinthe hargedweakintera tion.

ˆ Strongfor eshaveapotentialproportionaltodistan eV(r) /kr: inotherwords,thestrength

of ouplingbe omeslargeinlowenergyorweakinhighenergy,so alledasymptoti freedom.

This means that a system whi h has two free distant quarks is unstable, hen e, in terms of

energy,itismorebene ialto reateqq¯ pair(qrepresentaquark)fromva uumtoformtwo

qq¯ bindingstates(ormesons). Forthisreason, neitherthefreequarknoritsfra tional harge

hasnotbeendis overedyet(quark onnement).

ˆ In addition, be ause of the asymptoti freedom, theoreti al al ulations using perturbation

te hnique are less a urate for low energy behaviors of strong intera tion. In su h energy

s ale,therefore,apre ise omparisonbetweenavalueobservedbyexperimentandtheoreti al

predi tionisdiÆ ult.

1.1.1 Sear h for physi s beyondthe StandardModel

In 2012, at Conseil Europ´een pour la Re her he Nu l´eaire (CERN), Higgs boson was dis overed

by experiments at the large hadron ollider (LHC) from proton-proton ollision data [1, 2℄. The

existen eoftheHiggsboson,thoughmanyresear her hadbelievedinit,madeavalidityoftheSM

de isive. TheSM anexplainalmostall ofparti lephenomenathato urinouruniverse. Various

quantumbehaviorsofparti lesarewithinapredi tionofthisframework. Manyphysi ist,however,

believethattheSMtobeneither ompletenorultimatetheorywhi hdes ribesnaturebe ausethere

areseveralstrongfa tsthatarein onsistentwiththeSM. Theobservationofnonzeromassof

neu-trinosdis overedbytheneutrinoos illation[3,4℄,theunknownsour eofthegravitationalpotential

(dark matter), theasymmetryof amountsbetweenmatterand antimatterandthe unnaturallysmall

mass of Higgs boson (so alled hierar hy problem)[5℄, all of them are not well explained in the

frameworkoftheSM.

Forthereason notedabove,physi ists aretryingto ndan in onsisten yof theSM orphysi s

beyondtheSM(BSM).Atleastfromexistingobservations,theee tfromphysi sBSMinvarious

behaviorsofparti lesappearstobesmall. Thismayimplythatanewparti le, whi hisresponsible

for phenomena BSM, has a very large mass. In fa t, using the LHC, people a hieved very

high-energeti environmentof10TeVor10

14

Kbya eleratingand ollidingprotonsandareattempting

to dire tlyunveilthe appearan e BSM. Anotherapproa his topre iselymeasurethe propertiesof

already knownphenomena. Based onobservationsof ahugenumberofintera tions ofparti lesat

relativelylowenergy,possibleee tsfromthephysi sBSMarepre iselyveried.

1.2 Sear h for physi s beyond the Standard Model in harged

leptons

In the SM, there are three avors of harged leptons: e; and . The ele tron e has the smallest

their properties. The muon  and tau  have masses (105:65837545  0:0000024) MeV= 2

and

(1776:86 0:12) MeV=

2

,respe tively[7℄, and an de ayintolighter parti les. Thetests ofthese

de aysalsogiveusadditionalinformationfromthephysi sBSM.

Intermsofsear hBSMbasedonthepre isionmeasurementofparti leproperties,experiments

usingthe hargedleptonsturnouttooerbeautifullaboratories. Theina tivityof hargedleptonsto

thestrongintera tionenablesustopursueex ellentpre isioninthetheoreti al al ulation. Various

propertiesofthesede ays, des ribedbytheele troweakse toroftheSM, arepre isely al ulated,

therefore, experimentalresults anbe denitely omparedwith theoreti alpredi tions. Moreover,

unlikequarks,the hargedleptons anexistinbarestatesandweareabletodire tlytestthenatureof

elementaryparti les.Thoughneutrinosalsosharethisnature,itisdiÆ ulttodosimilarmeasurement

duetothesmallrea tionrate.

The parti le

Therehavebeenvarietiesofexperimentstomeasureproperties. Mostnotably,atBrookhaven

Na-tional Laboratory(BNL),the E821experimentmeasuredananomalousmagneti momentof the

usingpolarizedbeamwithamazingpre ision(0.7ppm!)[6℄andasaresultexhibitedasigni ant

de-viationfromtheSMpredi tionby3level. Notonlytheanomalousmagneti momentbutavariety

ofpropertiesofhavebeenmeasuredformorethanone entury.Itsrelativelylonglifetime(2s)

and availabilityoftherebylarge numberofpure(moreoversometimespolarized)sampleenables

usto performex ellentpre isionexperimentsfor: itmaynotbeoverstatethatweunderstandthe

muonverywell.

The parti le

On the other hand, inspite of its equally interesting hara teristi s, various properties of  lepton

arenotsopre iselymeasured,parti ularlyduetoitste hni aldiÆ ultiesofexperiment. Theoreti al

treatment of  is assimple as that of  ase, but theshort lifetime of  ( 0:3 ps) does notallow

ompetitivemeasurementintermsofabsolutepre ision.

Nevertheless, measurements of the  de ay is one of the most sensitive probes to the ee ts

BSM.Thelargemassoftheallowsustoexpe tanenhan ementofthesensitivityontheBSM.For

instan e,thetwoHiggsdoubletmodel(2HDM),oneofthebran hesofthesupersymmetri models,

predi ts an existen e ofthe harged Higgs andthe magnitudeof their ouplings is proportionalto

massofalepton.Asaresult,in omparisonwithde ays,we anexpe tthegainofsensitivityby

afa torof(m  =m  ) 2  300.

Thelargemassofthemakesitpossibletode ayintobothleptonsandhadrons. Theformerone

is alledleptoni de ay anda ounts forapproximately35%of alltau de ays. Therest de aysof

the ontainhadronsinthenalstateandare alledhadroni de ay.

Takingintoa ountthesensitivitiestotheee tsfromphysi sBSM, we hosethe leptonfor

thethemeofstudy.Inthisthesis,wedes ribethemethodindetail.

1.3 Mi hel Parameters

ThemeasurementofMi helparametersisoneofthemostestablishedstrategiesfortheveri ation

Theweakintera tionwasrstproposedbyFermi[8,9℄toexplainthebetade ayofthenu leus. He

in orporatedan ideaoftheneutrino, whi hhadbeensuggestedbyPauli,andsu eededtoexplain

the ontinuousmomentumspe trumofthedaughterele tron. In1957,C.S.Wufoundthattheweak

for edidnotrespe tthesymmetryoftheparityinthebetade ayfrom

60

Co[10℄. Theangular

distri-butionoftheele tronfromthepolarized obaltnu leisuggestedthemaximalviolationofparityin

the ouplings,i.e.,theintera tionresultsintheasymmetri ouplingsbetweenleft-handedand

right-handed parti les. The stru ture of the oupling ontains the ve tor and axial-ve tor ontributions

almostinthesamemagnitudeswithoppositesigns,soitis alledV Aintera tion.

Be auseof itsunique properties, overmore thanone enturythere have beenvariousattempts

to reveal the nature of the weak intera tion. In 1949, Ruderman and Finkelstein predi ted that a

ratio ofde ay ratesB(

+ ! e + )=B( + !  +

) wassuppressedby fourorder ofmagnitudeif the

weak intera tion o urs through the V A stru ture [11℄. The V A type urrent permits only

negative-heli ity parti les to parti ipate in the weak intera tion, whi h results in the violation of

angular momentum onservation in 

+

! `

+

 in the massless limit m

`

! 0 (` = e, or ). This

well known me hanism is often alled heli ity suppression. In 1958, the ele tron de ay of pion

 +

! e

+

 wasrst observed [12℄ andthena re entexperimentalvalueusingstopped

+ ,B( + ! e + )=B( + ! + )=(1:23460:00350:0036)10 4

[13℄wellsupportsitstheoreti alpredi tion

(1:2330:004)10

4 [14℄.

Moregeneraltests ofthe Lorentzstru tureof theweak intera tionhave beenperformedin the

de ayof !e ¯and !` ¯bythemeasurementofMi helparameters.

1.3.2 Mi hel formalism

ThemostgeneralLorentz-invariantderivative-freematrixelementofleptoni de ay

  ! ` ¯ y isrepresentedas[17℄ M=     ` ` = 4G F p 2 X N=S;V;T i;j=L;R g N ij h u i (` ) N v n ( ` ) ih u m (  ) N u j () i ; (1.1) whereG F

istheFermi onstant,iand jarethe hiralityindi esforthe hargedleptons,nandmare

the hiralityindi esoftheneutrinos,`iseor,

S = 1, V =  and T =i(     )=2 p 2are,

respe tively,the s alar,ve torand tensorLorentzstru turesin termsofthe Dira matri es



,and

g N ij

are the orresponding dimensionless ouplings. The hirality indi es nand m arenot summed

in Eq. (1.1) be ausethey are uniquelyxed for given i, j and theintera tion type. In the SM, 

de ays into` ex lusivelyviatheW ve torbosonwith theV A Lorentzstru ture, i.e., theonly

non-zero ouplingisg

V LL

=1. Experimentally,onlythesquaredmatrixelementisobservableandso

bilinear ombinationsoftheg

N ij

area essible. Ofallsu h ombinations,fourMi helparameters—

, ,Æand— anbemeasuredbytheleptoni de ayofthe whenthenalstateneutrinosarenot



Thedis ussionhereholdsalsoforwhenthedaughterlepton`is hangedtoe.

y

  ` `   ` `   ` `

Figure 1.1: Radiativede ay. Thelast diagram arises from the radiationfrom W bosonbut this is

suppressedbytheverysmallfa torof(m

 =m W ) 2 510 4 . observed[18℄:  = 3 4 3 4    g V LR    2 +   g V RL    2 +2   g T LR    2 +2   g T RL    2 +<  g S LR g T LR +g S RL g T RL   ; (1.2)  = 1 2 <  6g V RL g T LR +6g V LR g T RL +g S RR g V LL +g S RL g V LR +g S LR g V RL +g S LL g V RR  ; (1.3)  = 4<  g S LR g T LR g S RL g T RL  +    g V LL    2 +3    g V LR    2 3    g V RL    2    g V RR    2 +5   g T LR    2 5   g T RL    2 + 1 4    g S LL    2   g S LR    2 +   g S RL    2   g S RR    2  ; (1.4) Æ = 3 16    g S LL    2   g S LR    2 +   g S RL    2   g S RR    2  3 4    g T LR    2   g T RL    2   g V LL    2 +   g V RR    2 <  g S LR g T LR +g S RL g T RL   : (1.5)

Parametrizedbythesevalues,thedierentialde aywidthof !` ¯ isexpli itlygivenby

d ( ! ` ¯ ) dE  ` d  ` = 4G 2 F m  E 3 max (2) 4 q x 2 x 2 0 " x(1 x)+ 2 9 (4x 2 3x x 2 0 ) +x 0 (1 x)  n  l
S   3 q x 2 x 2 0 1 x+ 2Æ 3  4x 4+ q 1 x 2 0  !# ; (1.6) where E max = (m 2  +m 2 ` )=2m 

isthemaximum energyoflepton inthetau restframe, x = E

 `

=E max

is a normalized lepton energy, x

0 = m ` =E max , and n  `
S  

is the osine of angle between the tau

spinandleptondire tion. ThustheMi helparameters hara terizespe traofleptonmomentumand

dire tion. Moreover,asEq. (1.6) shows and Æ appearwith n

 l

S

 

,itis thusthesetwovariables

determinetheleptonangulardependen evstau-spindire tion.

1.4 Further tests of the V A intera tion in  de ays

TheFeynmandiagramsdes ribingtheradiativeleptoni de ayofthearepresentedinFig1.1. The

lastamplitudeturnedouttobesuppressedbytheverysmallfa torof(m

 =m W ) 2 510 4 [26℄and

anbenegle ted.Then,asshowninRefs.[27,28,29℄,thepresen eofaradiativephotoninthenal

state (orsometimes alledinnerbremsstrahlung)exposesthreemore Mi helparameters, ,¯ 

00 and

 ,whi hareexpli itlygivenby

¯  =   g V RL    2 +   g V LR    2 + 1 8    g S RL +2g T RL    2 +   g S LR +2g T LR    2  +2    g T RL    2 +   g T LR    2  ; (1.7) 00 n V S T V S T V S V S o

Name SM Spin Experimental CommentsandRef.

value orrelation result

y [7℄  0 no 0:0570:034 [19℄  3=4 no 0:749790:00026 [20℄  1 yes 1:0009 +0:0016 0:0007 [21℄ Æ 3=4 yes 0:750470:00034 [20℄  0 no 0:020:08 [22℄

 0 yes 0:000:01 al . from

0

value[23℄

y

Experimentalresultsrepresentaveragevaluesobtainedbytheparti ledatagroup(PDG)[7℄.Themost

pre iseresultsarereferen edhere.

The formula of dierentialde ay width for the radiativede ay, whi h orrespondsto Eq. (1.6) in

 ! ` ¯ ase,be omesmore ompli atedandwepostponeitsdes riptionuntilChapter5.

Never-theless,thesenewMi helparametersalsoae tthespe traofdaughterparti les.

Similarlyto  and , both ¯ and 

00

appear as spin-independentterms in the dierentialde ay

width. Sin e allterms in Eq. (1.7) arenon-negative, theupper limiton ¯ provides a onstraint on

ea h oupling onstant. Thevalueof

00 issuppressedbyafa torofm ` =m   0:03%foranele tron

daughter and 6% foramuon daughterandso diÆ ulttomeasurewith thestatisti savailableso

far.Inthisstudy,weusetheSMvalue

00

= 0.

To measure  , whi h appears in the spin-dependent part of the dierential de ay width, we

must determine the spin dire tion of the . This spin dependen e is extra ted using the spin-spin

orrelationwiththepartnerintheevent(itisexplainedindetailinthenext hapter).

Theinformation onMi hel parameters issummarized inTables1.1 and1.2 formuon andtau,

respe tively. ¯ and parameters havebeenalready measuredin de ay (notethatparameter

isindu edfrom

0

parameter). Usingthestatisti allyabundantdatasetofordinaryleptoni de ays,

previousmeasurementshaddeterminedtheMi helparameters,,Æandtoana ura yofafew

per entandinagreementwiththeSMpredi tion. Takingintoa ountthismeasuredagreement,the

smallerdatasetoftheradiativede ayanditslimitedsensitivity,wefo usinthisanalysisonlyonthe

extra tionof¯ andbyxing,,ÆandtotheSMvalues. Thisrepresentstherstmeasurement

ofthe¯ andparametersofthelepton.

1.5 Physi s motivation

Asintrodu edinSe .1.3,therelationshipsbetweenthe oupling onstantsg

N ij

andtheMi hel

param-etersintri atelyintertwineea hother.Consequently,anintuitiveunderstandingofthe onne tionto

aspe i modelBSMisaroomfordis ussion.Forexample,itisknownthatisdire tlyasso iated

with the hargedHiggsmodel. In theSM,onlyg

V LL

= 1isnonzeroandotherg

N ij

beingzero, hen e

fromEq. (1.3)weobtain  0:5
<fg

S RR

g. Sin ethe hargedHiggsmediatestheradiativeleptoni

de ayoftheasas alar-typeintera tion,themeasurementofisregardedastheveri ationofthe

ouplingof Higgstotheright-handed. Thesameanalogy holdsfor

00 :  00  8
<fg S RR g. Onthe

ontrary,otherMi helparametersappearasthe omplex ombinationsofmany ontributionsBSM.

Nevertheless, there are a few omments for the new Mi hel parameters, ¯ and  . First, the

ordinary Mi helparameters(, ,Æ and) anbe measuredblindlytothepolarizationofoutgoing

Name SM Spin Experimental CommentsandRef.

value orrelation result

y [7℄

 0 no 0:0130:020 [24℄

 3=4 no 0:7450:008 [25℄

 1 yes 0:9950:007 measuredinhadroni de ays[24℄

Æ 3=4 yes 0:7460:021 [25℄

 0 no notmeasured fromradiativede ay(RD)

 0 yes notmeasured fromRD

 00

0 no notmeasured fromRD,suppressedbym

` =m

y

Experimentalresultsrepresentaveragevaluesobtainedbytheparti ledatagroup(PDG)[7℄.Themost

pre iseresultsarereferen edhere.

totheveri ationofthe ouplingsofea h hiralityofthedaughterlepton.Theangulardistribution

of thephoton vsthe movementof thedaughterlepton provides theinformationof thepolarization

of the lepton. In fa t, a ording to Ref. [30℄, the  is related to another Mi hel-like parameter

 0

=  4+ 8Æ=3. Be ausethe probability that the  de ays into the right-handed harged

daughter lepton Q  ` R is given by Q  ` R = (1  0

)=2 [31℄, the measurement of  provides a further

onstraintontheV Astru tureoftheweak urrent.

y

Itisknownthatveri ationoftheasymmetri

natureofthe hiralityhasastrongimpa tonthetheoryBSMlikeright-leftsymmetri model[32,33℄.

Se ond,asismentionedbefore,the¯isasumofnon-negativeterms,hen etheupperlimitofthe

¯

 onstrainsthevalueofea h omponent.Assummarizedin-Leptonde ayparametersinRef.[7℄,

someoftheg

N ij

in ludedinEq.(1.7)arenotwellmeasuredforthede ay:

jg V RL j<0:52 (95%C:L ); (1.10) jg T RL j<0:51 (95%C:L ); (1.11) jg S RL j<2:01 (95%C:L ); (1.12) jg S LR j<0:95 (95%C:L ): (1.13)

The measurementof the ¯ is very powerful wayto onstrainthese ouplings. Moreover, ¯ is also

relatedtoanotherMi hel-likeparameter

00

= 16=3 4¯ 3,whi hrepresentstheangular

depen-den eofthelongitudespinofthedaughterlepton(seee.g.Ref[34℄). Although

00

hasbeenalready

measuredforde ay,thatofisnotyetknown.

Finally,thesesixMi helparametersdeliverindependentinformation. Figure1.2summarizesthe

matrixofthe orrelation oeÆ ientsoftheseMi helparameters al ulatedbyttingtheparameters

to thespe traof MonteCarloeventsfor  ! e  ¯ (thedetailedmethodofthis evaluationis

ex-plainedinChapter5). The orrelationsoftheMi helparametersbetweentheordinaryandradiative

ones,i.e.,,,Æ,and,¯ aresuÆ ientlysmallandthisimpliesapotentialimpa tonthe onstraint

ofg

N ij

intermsofthe onstru tionoftheories.

η

_η''

ξκ

ρ

η

ξ

ξδ

η

η''

ξκ

ρ

η

ξ

Figure1.2:Correlation oeÆ ientsbetweentheMi helparameters.

Table1.3:Listofavailabledata

Experiment Integratedluminosity(fb

1 ) Beamenergies ARGUS 0.5 E ee =9.4-10.6GeV CLEO-II 4.7 E ee =10.6GeV CLEO- 0.8 E ee =3.8GeV Babar 467 E ee =10.0-10.6GeV Belle 980 E ee =9.5-10.9GeV LHCb > 2:0 E pp =13TeV(2015-2016)

numberofde aysand leanenvironmentinthedete tionofdaughterparti les. A ountingfornot

onlynumberofeventsbutalso leanenvironmentoflepton ollider,theBelleexperimentpossesses

thebestdataforitspre isionmeasurement.

TheBelleexperiment,whi hwasoperatedformorethantenyearsfrom1999to2010atTsukuba

IbarakiJapan,isaproje tusinganele tron-positron olliderKEKBandBelledete tor.Theproje t

wasoriginallyorganizedtoaimforanobservationofthesour eofCPviolationinthede aysof B

mesonsbasedonhugenumberofevents. Indeed,Bellesu eededtoun overtheme hanismofthe

CPasymmetryinthe ontextoftheSM. Atthesametime,however,theBelleexperiment olle ted

data from huge number of  de ays produ ed by e

+

e ! 

+

 pro ess. We use this ex ellent

Radiative leptoni de ay  ! `  ¯

InordertomeasuretheMi helparameters,¯and ,theprobabilitydensityfun tion(PDF)istted

tothede ayspe traof ! ` ¯ de ay(` = eor). Using

+ !  +  0 ¯

de ayasaspinanalyzer

forthepartnersideof

+

ine

+

e ! 

+

 produ tion,informationofpolarizationisextra ted. Inthis

se tion,wereviewthe hara teristi softhesignalde ay.Detailedmethodaboutthetpro edureis

explainedinChapter5.

2.1 Denition of the radiative de ay and its distribution

Twokineti parameters hara terizetheradiativeleptoni de ay ! ` ¯ . Firstoneisanenergy

oftheradiativephotonE

. Figure2.1showstheE

distributionsimulatedbyKKMCandTAUOLA

generators. 

Here, the E

is dened in the enter of mass system (CMS) of e

+

e beam.

y

As the

histograms show,the distribution of the photonenergy divergesin the limit E

! 0. This omes

fromthefa tthatthed =dE

 hasasingularityatE  !0,whereE 

representsthephotonenergyin

therestframe.

For the reason noted above, the ordinary leptoni de ay (no photon) and the radiative de ay

annot be naturallydistinguished. Thatis tosay, theenergythreshold is on eptuallyrequired: if

E 

ex eedsa ertainthreshold,theeventisregardedastheradiativede ay. A onventional hoi e

E 

= 10 MeV is determined in su h a way that is realisti ally measured by experiment and at

thesametimebran hingratiobe omesreasonablefra tion. Inaddition,ifweapplytypi alphoton

energy threshold 100 MeVin thelaboratoryframe (su hveto isne essaryto ex ludevarietyof

ba kgrounds),asoftradiativeeventswhosephotonenergyislessthanE



<10MeVisrarelysele ted

(order of1%). Weusethisspe i valueinthewholeanalysisto deneeÆ ien yof ourradiative

de ay. z

TheenergythresholdofE



=10isalsousedtodenethebran hingratioofradiativede ay,

whi hisexplainedinnextsubse tion.

Inreality,itisalsorequiredtodeterminelowerthresholdtogeneratetheradiativede aysbyMC

simulation.TheTAUOLAgeneratoradoptsthegenerating-energythresholdE

 gen

=m

=1000,whi h

should obviouslysatisfy E

 gen

< E 

. Figure 2.2showsthe fra tionof the radiative pro essout of

total amounts ofgenerated leptoni de aysas a fun tionof E



threshold. These plots tellthat the

fra tionofradiativeevents(usedtodetermineeÆ ien y)are10:6%and2:6%forele tronandmuon

modes,respe tively.

A osineofanglebetweentheoutgoingleptonandphoton os

`

isanotherimportantvariablein

thisanalysis. Be ausethede ayamplitudeisapproximatelyexpressedasasumof

h  2 ` +m 2 l =E 2 ` i n 

(GeV)

γ

E

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1

Nevents/0.01 GeV

0

5000

10000

15000

20000

25000

30000

γ

ν

ν

e

→

τ

(GeV)

γ

E

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1

Nevents/0.01 GeV

0

1000

2000

3000

4000

5000

6000

7000

8000

γ

ν

ν

µ

→

τ

Figure2.1:EnergydistributionoftheradiativephotonontheCMSgeneratedbyKKMC.

threshold (GeV)

*

γ

E

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Ratio (%)

0

2

4

6

8

10

12

14

16

)] (n=0,1)

γ

)(e+n

π

0

π

(

→

τ

τ

)]/N[

γ

)(e+1

π

0

π

(

→

τ

τ

N[

threshold (GeV)

*

γ

E

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Ratio (%)

0

0.5

1

1.5

2

2.5

3

3.5

4

)] (n=0,1)

γ

+n

µ

)(

π

0

π

(

→

τ

τ

)]/N[

γ

+1

µ

)(

π

0

π

(

→

τ

τ

N[

Figure 2.2: Fra tion of event having a photon energy above threshold (out of generated leptoni

de ays): (left) ! e ¯ and(right) !  ¯ . Thehorizontalaxisrepresentsphotonenergy

thresholdonthe-restframeandtheverti alaxisindi atestheratio. If onventionaldenition,E



=

10 MeV,is used, thefra tionsare10:6% and 2:6%forthe ele tronandmuon modes,respe tively.

Theatshapeofsmall-energyregion omesfromthegenerating-energythresholdE

 gen

=m

 =1000.

γ

l

θ

cos

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

-1

Nevents/0.002 GeV

0

20

40

60

80

100

3

10

×

τ

→

e

ν

ν

γ

γ

l

θ

cos

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

-1

Nevents/0.002 GeV

0

500

1000

1500

2000

2500

3000

γ

ν

ν

µ

→

τ

Figure 2.3: Distribution of an angle between lepton and photon: (left)  ! e  ¯ and (right)

 !  ¯ . Thehorizontalis os

` .

foranintegern,theheaviermassofmuonexhibitsabroaddistributionas anbeseeninFig.2.3.The

requirement of maximum-allowedangle betweenlepton and photon is usedto dis riminate signal

fromba kground ontamination.

2.2 Spin-spin orrelationof

+

 andtwo-bodyde ay

+ !  + ¯  !  +  0 ¯ )

τ

τ

‐

e

‐

_e

+

RH

RH

LH

LH

Figure2.4: Spin-spin orrelationine

+

e !

+

 pro ess. Theheli itiesof

+

 pairarepreferably

anti- orrelatedea hother.Same olorindi atessame ombination.

As mentioned in Se . 1.3, the measurement of the  requires the information of the spin of

mother.Thisisextra tedthroughthe orrelationoftheanditspartnerine

+

e ! 

+

produ -tion. AsdrawninFig.2.4, theheli itiesof

+

 pairareanti- orrelated(against)ea hother. Sin e

thispro esso ursthroughanex hangeof (spin-1parti le),theangular onservationpermitsonly

either  + R  L or + L  R

statesinthehigh energylimit E

! 1,where LandR denotetheheli itiesof

taus. In aseofbeamenergyofKEKBa elerator(approximatelyE

 5GeV),95%of

+

 pairs

areanti- orrelatedwhile5%are orrelated.

Intheother sideof, orsometimes alledtag-side, we use

+ !  +  0 ¯

 de ay. Ingeneral, the

hadroni de ayofthe with two pseudo-s alermesonshavea quantumnumber J

P

ofeither0

+ or

(GeV)

π

π

m

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-1

Nevents 500/1.777 GeV

0

20

40

60

80

100

120

140

160

180

3

10

×

π

π

Invariant mass distribution m

_π

_π

Invariant mass distribution m

Figure2.5:Invariantmassdistributionforthetwo-pionsystemgeneratedbyKKMCandTAUOLA.

τ

+

ρ

+

ν

θ

ρ

*

*

ρ

θ

cos

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4 0.6 0.8

1

Nevents/0.01

0

10

20

30

40

50

60

70

3

10

×

direction

ρ

Distribution of

Figure 2.6: Angular dependen eof 

+ movementin + !  + ¯  de ay: (a)   

is the anglebetween

spindire tionoftauand

+

inthe

+

restframe(b)distributionof os

 

. Thebluearrowrepresents

spinof

+ .

Thespindire tionof

+

ae tstheangulardistributionof

+

parti le. AsFig.2.6shows, the

+

are preferablygeneratedintotheoppositedire tionofthetauspin. Thissituation anbeexplained

byasuperpositionoftwoamplitudesofaandb:

jai=j0i      1 2 + : A a =haj+i; (2.1) jbi=j1i      1 2 + : A b =hbj+i; (2.2)

where thebra ketsin theright handsiderepresent heli itiesof

+

meson and,¯ j+i representsthe

initialstateof

+

polarizedin+zdire tion,andA

a

andA

b

arethe orrespondingamplitudesofea h

hannel whose maximums have a relationgiven by jA

max a =A max b j = p 2m  =m  [35℄. As illustrated

in Fig. 2.7, theamplitudesof aand bbe ome maximum(minimum)at 

  = (0) and   = 0 (),

τ

+

ρ

+

ν

θ

*

_ρ

RH

RH

a

τ

+

ρ

+

ν

θ

_ρ

RH

b

*

Figure 2.7: Two spin ongurations of 

+

and :¯ (a) the angular momentum perfe tly onserves

when



 =  whileviolateswhen



 = 0: (b)thesituationbe omesopposite. Asaresult,(a)and

(b)haveangulardependen esofsin

  =2and os   =2,respe tively.

respe tively,andinfa titisknownthattheangulardependen esaregivenbysin

  =2and os   =2.

Observedprobabilityisthus al ulatedtobe

P( )/ 1 jA max a j 2 jA max b j 2 jA max a j 2 +jA max b j 2 os   = 1 m 2  2m 2  m 2  +2m 2  os    1 0:43 os   : (2.3)

Thislineardependen eon os

 

isseeninthegure.

This rho de ay is hosen be ause of its large bran hing fra tion B(

+ !  +  0 ¯ ) = (25:52

0:09)% [7℄ andrelativelysimpleform-fa tor,whi h resultsin aneasy implementationofthe PDF.

As a matter of fa t, taking into a ount the magnitude of polarizations and bran hing fra tions,

Ref.[35℄reportsthat

+ ! +  0 ¯

exhibitsthelargestsensitivitiesofallde aysonthepolarization

measurement.

Asexplainedabove,throughthespin-spin orrelationine

+

e !

+

 produ tionandtheangular

distributionofpionsfromrhode ay, informationof spinisindire tlyextra tedonlytomeasure

theparameter.

2.3 Bran hingratio of  ! `  ¯ de ays

Beforestartingthisproje ttomeasuretheMi helparameters,themosta urateexperimentalvalues

of thebran hingratio of ! `  ¯ de aywerethe measurementbytheCLEO experiment[36℄.

Using4:68fb

1

ofe

+

e annihilationdata,theCLEOobtained

B EX: CLEO ( !e  )¯ E  >10MeV = (1:750:060:017)10 2 ; (2.4) B EX: CLEO ( !   )¯ E  >10MeV = (3:610:160:35)10 3 ; (2.5)

wheretherstun ertaintyisstatisti alandse ondissystemati . Thismeasurementwasrenewedin

2015 by BaBarexperiment usingmu h moreabundant statisti sof 431fb

1 e + e ollisiondata to give[37℄, B EX: BaBar ( !e  )¯ E  >10MeV =(1:8470:0150:052)10 2 ; (2.6) B EX: BaBar ( !  )¯ E  >10MeV =(3:690:030:10)10 3 : (2.7)

Thesemeasurementsareingoodagreementwiththetheoreti al al ulations,whi hrelyonthe

for-mulagivenby[38,39℄.

diers fromthatof singleemission: a ombinationof onevisiblephoton andoneinvisiblephoton

( soft

; vis:

)is ategorizedasanex lusivemodewhilea ombinationwhereatleastonevisiblephoton

exists( vis: ; vis: )+( soft ; vis:

), is ategorizedas anin lusivemode(bothvisiblemode(

vis: ;

vis: )is

alsodistinguishedasadoublyde ay). Interestingly,themeasurementofmentionedbran hingratios

for !e  ¯ de ay,whi hisinfa tapproximatelytheex lusivemode,deviatesfromtheex lusive

SMpredi tionby3:5.A ordingtothereferen e,theleadingorder(LO) al ulationpredi ts

B Th: LO ( !e ¯ ) E  >10MeV =1:83410 2 ; (2.8) B Th: LO ( ! ¯ ) E  >10MeV =3:66310 3 ; (2.9)

whereasthenext-leadingorder(NLO)predi ts

dB Th: NLO ( ! e  )¯ E  >10MeV =1:645(19)10 2 ; (2.10) B Th: NLO ( !  )¯ E  >10MeV =3:572(3)10 3 : (2.11)

Herein, the errors for the NLO al ulationarise from a next-next-leading order ee ts, numeri al

al ulationandanexperimentalvalueofthelifetimeofthe.

As a byprodu t of this analysis, we also measure the bran hing ratio. The pro edures are

de-s ribedindetailinChapter9.

2.4 Ee tof theMi hel parameteron the distribution

Inthisse tion,wedemonstratetheee toftheMi helparameteronthespe traofdaughterparti les.

Asweshallexplain,everyeventofsignal

+  !( +  0 ¯ )(`  )¯ isrepresentedasa orresponding

point in the twelve-dimension phasespa e. Due to itslarge dimension,it isdiÆ ult to intuitively

observethe hangeofdistribution.However,we anglimpsethedependen eofspe traofthelepton

andphotonvariablesontheMi helparameterbyobservingdistributionsproje tedon1D-axis.

Thedependen eondisappearswhenweintegrateisotropi allyinthephasespa ebe ause

isin ludedinthespin-dependenttermofthedierentialde aywidthas:

d (!`  )¯ dPS S  
V   ; (2.12) where V 

is ave torfun tion, whi hdoes notdependonS

 

and iswrittenas alinear ombination

ofthedire tionofleptonn

 `

andphotonn



.Integrationsoverthedire tionsofleptonandphoton(n

 `

and n



)giveanet ontributionofzero. Thusitisrequiredtoadoptsome asymmetri manipulation

tovisualizeee ts. Toseparatetheoverallphasespa e,weuseaheli itysensitiveparameter!

h ,

whi hrepresentspolarizationoftheandis al ulatedonlyfromobservables. By onstru tion,!

h

variesinanopeninterval: !

h

2( 1;1). Thepositivevalueof!

h

impliesitisprobablethatthespin

ofthe + (!  +  0 ¯

)ispointingtothesame(oppositewhen de aysto 

0

) dire tionasthatof

 +

movement. The detaileddenition of !

h

isintrodu ed inSe . 6.1. To observe theasymmetri

ee t,weintegratethedierentialde aywidthinthephasespa eonlywhere!

h

be omespositive.

Figures2.8and2.9showthedependen eof theshapeofmomentaofleptonandphotononthe

Mi hel parameters. Ea h distributionis al ulated for a ertainvalue of theMi hel parameter by

theintegrationofthedierentialde aywidthswithothervariables. Fordemonstrationpurpose,the

rangeofvariationoftheMi helparametersare hosentobelargerthanphysi ally-realisti values.

Asexplainedabove,only!

h

>0eventsareusedfortheintegrationtodrawFig.2.9. Weobserve

that themagnitudeof themomentumoflepton ismorestronglyae tedby theMi helparameters

BSMisenhan edbyafa torproportionaltom `

=m 

astheexpli itformulaisintrodu edinSe .5.4.1.

Here,weshowthevariationofdistributionassumingverylargeMi helparameters,thereal

pos-siblevaluesare,however,oforderof1andthisimpliesthatmeasurementoftheseMi helparameters

requires the pre iseveri ation ofthe small variation ofspe trum shape. Thatis whywe need to

observelargenumberofevents.

2.5 Determination of  dire tion

Duetotheshortlifetimeof,itisdiÆ ulttodire tlymeasurethede aydire tion. Nevertheless,in

the 

+

 restframe, we an onstrain theirdire tion assumingthe masses ofneutrinos to bezero.

Whentheleptoni de ayo urs,twoneutrinosappearinthenalstate. Be ausethetwo-bodysystem

of¯mustnothaveanegativeinvariantmass,aninequalityholds:

0  M 2 ¯ = p 2 ¯ =(p  p ` ) 2 , os `  2E  E ` M 2  M 2 ` 2P  P ` ; (2.13)

whi hmeansthatthede aysintheregionen losedbya onearoundleptondire tion. Ontheother

hand,ifthede ayshadroni ally,oneneutrinoisprodu edandgivesanequality:

x 0 = M 2  = p 2  =(p  p h ) 2 , os h = 2E  E h M 2  M 2 h 2P  P h ; (2.14) where p h

isasumoffourve torsforthehadroni daughtersandM

h

isitsinvariantmass.Thismeans

that the de aysinsidethe surfa e ofa onedeterminedfrom thedire tionofhadron momentum.

Depending on the onditions,through whi h typetwo tausde ay,we an divide thesituation into

three ategories:(h;h),(` ;h)and(` ;` ),where(a;b)witha;b=l;hmeanstwotaude ayleptoni ally

(l)orhadroni ally(h). AsFig2.10shows,(h;h)de ayenablesustoxthedire tionofthetauinto

two andidates. If either of the de ays leptoni ally, the dire tionis no more xed and be omes

a region: (` ;h) onstrains on aline and (` ;` ) onstrains on a region. In the ase of signal of this

analysis— ! `  ¯ and + !  +  0 ¯

—the andidatebe omesaline.Therefore,weparametrize

thedire tionusingoneparameter2[

1

;

2

℄. Asdes ribedlater,thisdeterminationofdire tion

(GeV)

l

P

0

1

2

3

4

5

-1

Nevents/0.189 GeV

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

=0

η

=-120

η

=-80

η

=-40

η

=40

η

=80

η

=120

η

γ

ν

ν

e

→

τ

(GeV)

l

P

0

1

2

3

4

5

-1

Nevents/0.189 GeV

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

=0

η

=-120

η

=-80

η

=-40

η

=40

η

=80

η

=120

η

γ

ν

ν

µ

→

τ

(GeV)

γ

P

0

0.2

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₌₀

=-120

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=-80

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=-40

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=40

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=80

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=120

η

γ

ν

ν

e

→

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(GeV)

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=120

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→

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=0

=-120

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=-80

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=-40

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=40

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=80

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=120

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ν

ν

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→

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l

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1.5

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η

=0

=-120

η

=-80

η

=-40

η

=40

η

=80

η

=120

η

γ

ν

ν

µ

→

τ

Figure2.8: Dependen eofmomentaandangleson:¯ leftgures(a)( )(d) representdependen eof

theshapeofP ` ,P and `

spe traon¯for!e ¯ de ayandrightgures(b)(d)(f)representthose

(GeV)

l

P

0

1

2

3

4

5

-1

Nevents/0.189 GeV

0

0.05

0.1

0.15

0.2

0.25

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κ

(+)=0

(+)=-120

κ

ξ

(+)=-80

κ

ξ

(+)=-40

κ

ξ

(+)=40

κ

ξ

(+)=80

κ

ξ

(+)=120

κ

ξ

γ

ν

ν

e

→

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(GeV)

l

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0

1

2

3

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5

-1

Nevents/0.189 GeV

0

0.05

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0.15

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(+)=0

κ

ξ

(+)=-120

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ξ

(+)=-80

κ

ξ

(+)=-40

κ

ξ

(+)=40

κ

ξ

(+)=80

κ

ξ

(+)=120

κ

ξ

γ

ν

ν

µ

→

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(GeV)

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(+)=0

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ξ

(+)=-120

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(+)=80

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(+)=120

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→

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(GeV)

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(+)=-40

κ

ξ

(+)=40

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ξ

(+)=80

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ξ

(+)=120

κ

ξ

γ

ν

ν

µ

→

τ

Figure 2.9: Dependen e ofmomentaand angleson  : left gures(a)( )(d) representdependen e

oftheshapeof P ` ,P and `

spe traonfor !e ¯ de ayandrightgures(b)(d)(f)represent

Figure2.10:Kinemati sofde ayfor(a)(h;h),(b)(` ;h)and( )(` ;` ).ConesAandBaresurfa es

whi hsatisfy ondition: p

2

miss

=0.Inthe aseof(h;h)de ay,the andidateofthedire tionbe omes

generally two pointsdeterminedby rossedpointsofthe reversal one Aand oneB. Similarlyin

(` ;h) ase,the andidatebe omeslineas oloredbyredand(` ;` ) onstrainsontoaregionen losed

byred urve. Inthe aseofsignalde ay,

+  !( +  0 ¯

Experimental Apparatus

Wedes ribetheexperimentalapparatuswhi hrealizesthemeasurementoftheMi helparameters,¯

and ,usinge + e !  +  ! ( +  0 ¯

)(` ¯ )pro ess. Eventsareprodu edbytheKEKBa elerator

andobserved/re ordedbytheBelledete tor.

3.1 The KEKB a elerator

The KEKBa eleratoris anasymmetri energy olliderofe

+

ande . Thebeamenergies ofE

e =

8:0GeVand E

e +

= 3:5GeVare hosen su hthatthe enterofmass energy oin ides withamass

of resonan e state of (4S):

p

s = 10:58 GeV where s is the Mandelstam variable. The (4S)

state,whi h onsistsofb

¯

bquarkpair,su essivelyde aysintoB

¯

Bpairs. Meanwhile,viaavirtual

inter hange,thee

+

ande pairalsoannihilatesinto

+

 and ¯ pairs,et . Theasymmetryofbeam,

 = 0:425,isintendedtoenlargethede aylengthsofBmesonsinthelaboratoryframetogainan

ee tivetimeresolutionforthemeasurementoftheirde ayrates.

AkeygoalofKEKBa eleratoristoprodu eBandparti lesofinterestasmanyaspossible.In

fa t,KEKBa hievedthemaximuminstantaneousluminosityL= 2:1110

34 m 2 s 1 ,whi histhe

world-largestinstantaneous luminosity atthe time ofwriting.



Forthis reason, KEKBa elerator

is alled B-fa tory or-fa tory. To realize the pre isemeasurement of  ! `  ¯ de ay (order

-suppressed relativetotheordinaryleptoni de ay ! ` ),¯ thelargenumberoftausthanksto

the-fa toryarene essary.

Not only did the KEKB a umulated e

+

e annihilation data at (4S) energy, but it also

ol-le ted data at dierent energy settings su h as mass resonan es of (1S) (9:46 GeV=

2 ), (2S) (10:02GeV= 2 )and(5S)(10:86GeV= 2

). Attheseenergies,thee

+

e !

+

 pro essstillo urs,

however,thesituationsofeventsele tionandtriggerarenotne essarilysameasthatof(4S),and

moreoverthedierentbeamenergiesmakethedes riptionofPDF (whi hisexplainedlater)

om-plex.Forthisreason,weuseonly(4S)resonan edata,whi hamountsto703fb

1

and orresponds

to70%ofalldata.

Figure 3.1shows an overallview of the KEKB a elerator. The ele trons are generated from

a thermalele tronwhilepositronsareobtainedby olliding4 GeVe beaminto ahigh-Zmaterial

(tungsten) in whi h a gamma onversion ! e

+

e generates the positrons. Both e

+

and e are

a elerated by alinear a elerator (LINAC)and inje ted intoa lowenergyring (LER) anda high

energyring(HER),respe tively.AtTsukubaarea,thee

+

ande ollideatintera tionpoint(IP)with

a rossingangleof22mraden losedbytheBelledete tor. Intable3.1,thema hineparametersof

Lin

ac

TSUKUBA

OHO

FUJI

NIKKO

HE

R

LE

R

H

ER

LE

R

IP

R

F

R

F

RF

RF

e-e+

e+

/e

-HER

LER

R

F

R

F

W

IG

G

LE

R

W

IG

G

LE

R

(TRISTAN

BY

PA

SS

Figure3.1: AdrawingofKEKBa elerator. Ele tronsandpositrons ir ulatethehighandlow

en-ergyringsin lo kwiseandanti- lo kwise,respe tively.TheBelledete torislo atedattheTsukuba

hall[42℄.

Table3.1:KEKBa eleratorma hineparameter

Item HER(e ) LER(e

+ )

Cir umferen e(m) 3016

Beamenergy(GeV) 8.0 3.5

Beam urrent(A) 1.6 1.2

Beam-beamparameter y (mm)y 0.09 0.129 Betafun tionatIP  y (mm)y 5.9 BeamsizeatIP x = y (m/m) 1:9=77 1:9=77 Numberinbun hesy 1584

Crossingangle(mrad) 22

e

+

e

-z

x

y

LAB

Figure3.2: Denitionofaxisinthelaboratoryframe. Thedire tionofz-axisisdenedasreversal

wayofpositronbeam. Theele tronandpositronmovementformsxzplane.

3.1.1 Denitionof frame

The dire tions ofthe ele tron andpositron beamsare not pre iselyba k-to-ba k inthe laboratory

frame: thetiltangleis

LAB

=22mrad. ThissituationisshowninFig.3.2. xyz-axisinthelaboratory

framearedenedbyusingbeamdire tion:thepositrondire tionisdenedasareversalwayof+z,

the plane,inwhi hbothele tronandpositronsettle, is xz-plane. Therefore,thefourve torsofthe

ele tronandpositronareparametrizedinthelaboratoryframeas:

p LAB e = (E LAB e ;P LAB e sin LAB ;0;P LAB e os LAB ) (3.1) and p LAB e + = (E LAB e + ;0;0; P LAB e + ): (3.2)

Thesumofthesemomenta p

LAB CMS

isthatoftheCMSinthelaboratoryframeandthevelo ity

LAB CMS = P LAB CMS =E LAB CMS

allowsusto onvertfourve torsin bothframesea hother. Whenthe beammomenta

areboostedtotheCMSwiththis

LAB CMS

,thedire tionofz-axisdoesnot oin idewiththatofele tron.

For this reason, we rotate frame around y-axisby  su h thatboth beams be ome ollinear along

z-axis,whereisapproximately13:24mrad.TherotatedframeisthedenitionofourCMSframe.

Here,wesummarizethedenitionofthe oordinatesystemandnotations.

ˆ Dire tionsofzinboththelaboratoryandCMSframesaredenedusinge

+

beamwhi hpoints

-zdire tion.

ˆ Dire tion of x inboth thelaboratoryand CMSframes aredeterminedby rotating

aforemen-tioned z dire tion by 90

Æ

in the plane formed by the laboratory movements of ele tron and

positron(-plane).

ˆ Dire tion of x in CMS frameis determinedby rotating thedened zdire tion by 90

Æ

in the

-plane.

ˆ Dire tionof yisdenedbythe rossprodu tofve torse

y = e z e x ,wheree i (iis x, y orz)

standsfortheunitve torofidire tion.

ˆ standsforthepolaranglefrom+zdire tion

ˆ standsfortheazimuthalanglearoundzaxis

ˆ rstandsforthetransversedistan e al ulatedasr =

p x 2 +y 2 .

Dete tor Type Conguration Performan e

SVD-1 DoublesidedSi-strip3-layers r=30:0; 45:5;60:5mm 

z =4244=psin 5=2 m[44℄ Strippit h25(p)/42(n)m 23 Æ <<139 Æ  r =1954=psin 3=2 m (pinGeV= )

SVD-2 DoublesidedSi-strip4-layers r=20:0; 43:5;70:0;88:0mm 

z

=2633=psin 5=2

m[44℄

Strippit h50(p)/75(n)m(lay.1-3) 17 Æ <<150 Æ  r =1734=psin 3=2 m 65(p)/73(n)m(lay.4) (pinGeV= )

CDC Wiredrift hamber r=8:3-87:4 m(SVD1),10:4-87:4 m(SVD2) 

r =130m Anode:50layers 77<z<160 m  z =200 1400m Cathode:3layers 17 Æ <<150 Æ  pt =p t =0:2%p t 0:3%=  dE=dx =6%

ECL CsIS intillator Barrel:r=125-162 m,32:2

Æ <<128:7 Æ  E =E=1:3%= p E

# rystalsinbarrel6624 End ap:z= 102 m,130:7

Æ <<155:1 Æ  pos =0:5 m= p E # rystalsinend ap2112 :z=196 m,12:4 Æ <<31:4 Æ (EinGeV)

ACC Sili aaerogel Barrel:r=89-117 m P(jK)<10%;P(KjK)>80%

#aerogelinbarrel960 End ap:z=1660 m for1:2GeV/ <P<3:5GeV/

#aerogelinend ap228

TOF Plasti S intillator r=120 m 2K=separation

128segmentation forP<1:2GeV/

 t

=100ps

KLM Resistiveplate ounter End ap:20

Æ <<45 Æ
=
=30mrad. 14layers :125 Æ <<155 Æ Barrel:45 Æ <<125 Æ

3.2 The Belle dete tor

The Belle dete tor is a general-purpose measurement system whi h is omposed of several

sub-dete tors. Thedete toris onguredby1:5Tsuper ondu tingsolenoidanden losestheIP ofthe

e +

e beam.

Figure3.3showstheoverallviewoftheBelledete tor. Thede ayverti esaremeasuredbythe

sili onvertexdete tor(SVD)lo atedjustoutsideofa ylindri albeampipe. Tra kingofthe harged

parti les are performedby the entral drift hamber (CDC). Energy of ele tromagneti shower is

measuredbytheele tromagneti alorimeter(ECL).Parti leidenti ationisprovidedbythe

infor-mationofdE=dxmeasurementsbytheCDC,ashapeofshowerinthe lustersandE=pmeasurement

in the ECL, anaerogel Cherenkov ounter(ACC) anda time-of-ight ounter (TOF). TheK

L and

muonsareidentiedbyarraysoftheresistiveplate ountersandironplateslo atedattheoutermost

partoftheBelledete tornamedK

L

andmuonsdete tor(KLM).Alloftheseinformationispro essed

andre ordedbyadataa quisition(DAQ)systemwheneventsaresele tedbyatrigger.Thegeneral

informationandperforman esofthesub-dete torsaresummarizedinTable3.2. Inthisse tion,we

SVD

ECL

CDC

TOF

ACC

KLM

0

1

2

3 (m)

z

(b)

r [44℄.

3.2.1 Sili on VertexDete tor(SVD)

The maingoaloftheBelleexperimentistoverifytheme hanismofthe CPviolationinBde ays,

where the violation of the CP appears as a time dependent asymmetry of the de ay rate between

B!f CP (t) and ¯ B!f CP (t) (f CP

stands for a CP eigenstate). Sin e the dieren e of the de ay rate of

B= ¯

Bmesonsismeasuredasthatoftheightlength,thepre isemeasurementofthevertexposition

is ru ial. The SVDplays a rolein lo ating thevertexposition of B mesons. Furthermore, alow

momentum tra k, whi h doesnotrea h theCDC innerwall,is re onstru tedonly bythe SVD. In

thisanalysis,theSVDhelpstheCDCinthe hargedtra kre onstru tion.

TherearetwotypesofSVDs. Therstversionis alledSVD1andworkeduntil2003.Be auseof

aprobleminthefront-end hip,theSVD1wasupgradedtoSVD2. TheSVD1(SVD2)is omposed

ofthree(four)layerslo atedatradiir = 30:0; 45:5; 60:5mm(r = 20:0; 43:5; 70:0; 88:0mm)and

overs23 Æ <  < 139 Æ (17 Æ <  < 150 Æ

), whi his onstru tedfrom8,10,14(6,12,18,18)ladder

stru tures, respe tively. Ea hlayeris madeof double-sidedSi-stripdete tors (DSSD). TheDSSD

has rossedlinearee tiveareas(strip)ontopandbottomsides,whi hareorthogonallysegmented

alongrandzdire tions,respe tively,andea hstripismadebyap-typeorn-typesemi ondu tor.

Whena hargedparti lepassesthroughthep-njun tion,theionizedele tron-holepairis

sepa-ratedbyanappliedhighbiasvoltageandreadoutseparatelyfrompandn-sidestripsofthedete tor.

Thefront-end ir uitnamedVA1 hipprovidesanampli ationofthe urrentandashapingofthe

signal.Figure3.4showsthea hievedimpa tparameterresolutionoftheSVD1andSVD2asa

fun -tionofpseudo-momentum, whi htakesintoa ounttheee tivein reaseofthepasslengthinside

materialanddenedby p˜ = psin

5=2

and p˜ = psin

3=2

forzandrdire tions,respe tively.The

informationoftheSVD1andSVD2issummarizedinTable3.3.

3.2.2 Central Drift Chamber(CDC)

The CDC plays a role in the tra king of harged parti le and a pre ise determination of the

mo-mentum. Sin etheBelle dete torisin themagneti eldof B = 1:5T,themomentumof harged

parti le isdetermineda ordingto p = 0:3B,where p isamomentum of harged tra kinGeV/

andistheobserved urvatureinmeter. Thetraje toryofthe hargedtra kisparametrizedbyve

freeparameters(alsoknownasahelixparameter)andttedtoamapofdete tedenergydeposition.

Item SVD1 SVD2 #layers 3 4 r(mm) 30.0,45.5,60.5 20.0,43.5,70.0,88.0 overage 23 Æ <<139 Æ 17 Æ <<150 Æ #DSSD#ladders layer1: 28 26 layer2: 310 312 layer3: 414 518 layer4: - 618 DSSDn-strips 42m640 50m512(layer1-3) 65m512(layer4) DSSDp-strips 25m640 75m1024(layer1-3) 73m1024(layer4) DSSDThi kness 300m 300m

Totalnumberof hannel 81920 110592

parameter,whi histhedistan eofthe losestapproa htotheintera tionpointanddenotedasdrand

dzintransverseandbeamdire tions,respe tively.Theimpa tparametersareusefultoredu e

ba k-groundssu hasse ondaryparti lesfrombeamand osmi rays. Moreover,theCDCalsoprovides

informationoftheparti leidenti ationbasedondE=dxandreliabletriggersignals.

Asthestru tureofCDCisshowninFig.3.5,theCDCisa ylindri alwiredrift hamberwhi h

liesintheregion83mm< r<880mmforSVD1termand104mm< r <880mmforSVD2term,

respe tively,and overs17

Æ

<<150 Æ

angle.Theasymmetri alstru tureinz-dire tionisoptimized

fortheboostofbeam. The hamberhas8400drift ells, allofwhi haregroupedasaxialorstereo

super-layers. Thestereowiresaretiltedandallowustodeterminez-position.Agasmixtureof50%

Heand50%C

2 H

6

was hosenbe auseofitssmalllow-Zsoastoredu ethemultiples atteringfor

lowmomentumtra ks.

Thereadoutsignalsfromthe hamberareampliedbyRadeka-typepre-amplier[46℄andsent

to the shaperand dis riminator. The dataare nally pro essedby a harge-to-time onverterwith

retaining the information of the drift time and pulse height. With an aid of SVD, the ombined

harged-tra kmomentumresolutionisgivenby:

 p T p T = 0:19p T  0:30  ! %; (3.3) where p T

isinGeV= andthetra kingeÆ ien yof hargedpionisapproximately90%for1GeV/

tra k.

Figure 3.6 shows a s atter plot of dE=dx vs momentum for various parti le types. It an be

understoodthattheparti letypesarewellseparateda ordingtoea hexpe ted urve. Theresolution

747.0

790.0

1589.6

880

702.2

1501.8

BELLE Central Drift Chamber

5

10

r

2204

294

83

Cathode part

Inner part

Main part

Forward

Backward

e

e

Interaction Point

17°

150°

y

x

100mm

y

x

100mm

-

+

765

770

Figure3.5:DimensionoftheCDC[45℄forSVD1 onguration. Theinnerwallwasextendedfrom

83mmto104mmwhenSVD2wasinstalled.

0.5

1

1.5

2

2.5

3

3.5

4

-1.5

-1

-0.5

0

0.5

1

log

₁₀

( p (GeV/c) )

dE/dx

_K

_P

e

π

Figure3.6: S atterplotof

dE

dx

vsmomentum. Thered urvesareexpe tedenergylossvaluesfor,

Figure3.7:GeometryoftheECL[47℄.

3.2.3 Ele tromagneti Calorimeter(ECL)

The mainpurposeof theBelleECLis tomeasurean energyofphotonwhi hisoftengeneratedby

as ade de ays of B mesonas well as the leptons. Be ause the energy of photonsgenerated by

daughterofthe(4S)tendtoberelativelysmall( 1GeV),itisrequiredtoprovideagoodenergy

and positionresolutions forsu h photons. Onthe otherhand, the ECLis alsodesigned to

a om-modate highenergyphotons ( 4 GeV)produ edfromlow-multipli itypro esseslikeforbidden

de ay ! ` . Furthermore,theECLplaysanimportantroleintheele tronidenti ationbasedon

theshowershapeinside rystalsandE=pvalue.

The Belle ECL is omposed of three se tions—ba kward and forward end aps and a barrel

region—whi hseparately over12:4

Æ <  < 31:4 Æ ,130:7 Æ <  < 155:1 Æ and32:2 Æ <  < 128:7 Æ ,

respe tively. Figure3.7showsthe ongurationofthe ECL. Allregions onsistofCsI (TI)arrays

andamountto8736 rystalsintotal.Ea h rystalhasatrapezoidalshapeandpointstotheintera tion

region.Thetypi aldimensionofthe rystalis5555mm

2

(frontfa e),6565mm

2

(rearfa e)and

30 mlong(i.e,16:2radiationlength)butslightlyvariesdependingonitslo ation. Thes intillation

photonsaredete tedbytwoPINphoto-diodes,whosea tiveareaare10 m20 m,gluedontheend

surfa e ofa rystal. Thepulse fromthePIN photo-diodesis ampliedbya pre-amplieratta hed

nearbyandsenttoashaping ir uit. Theseparatetwoshapedsignalsaresummedandpro essedby

a harge-to-time onverter.TheenergyandpositionresolutionoftheECLare

 E E = 1:34 0:066 E  0:81 E 1=4 ! %; (3.4)  pos = 0:27 3:4 p E  1:8 E 1=4 ! mm; (3.5) whereE isinGeV.