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Measurement of the angle gamma of the Unitarity
Triangle with BaBar experiment
D. Derkach
To cite this version:
Juin 2010
Universite Paris-sud 11
TH
ESE
presentee pourobtenirlegrade de
Do teur en S ien es de l'Universite Paris-Sud 11
Spe ialite : physiquedesparti ules
par
Denis DERKACH
Mesure de l'angle du triangle d'unitarite ave le
dete teur BABAR
Soutenue le25 juin2010 devant leJury ompose de :
M. A.Bevan
Mme E.Kou
M. J.-P.Lees Rapporteur
M. S.Monteil Rapporteur
M. A.Sto hi Dire teurde these
1 Introdu tion 13
2 CKM Matrix and CP Violation in the Standard Model 15
2.1 Symmetries . . . 16
2.2 CP Violationand theCKMMatrix . . . 16
2.3 Current StatusoftheUnitarityTriangle Measurements . . . 19
3 Measurement of the Unitarity Triangle Angle 27 3.1 GeneralFormalism . . . 28
3.1.1 The Gronau-London-Wyler Method . . . 30
3.1.2 The Atwood-Dunietz-Soni Method . . . 31
3.1.3 The Giri-Grossman-So er-ZupanMethod . . . 34
3.2 State-of-the-art inthe Measurements . . . 38
4 The DK system 45 5 The BABAR experiment 53 5.1 The PEP-IIA elerator . . . 53
5.2 The BABAR Dete tor . . . 55
5.2.1 The Sili onVertexTra ker . . . 56
5.2.3 The
CerenkovDete tor . . . 61
5.2.4 The Ele tromagneti Calorimeter . . . 62
5.2.5 The InstrumentedFlux Return . . . 65
5.2.6 The BABARTrigger. . . 66
6 Event Re onstru tion and Ba kground Chara terization 69 6.1 Charged Tra k Re onstru tionand Identi ation . . . 70
6.2 Charged Tra ks Identi ation . . . 71
6.3 Photon Sele tion . . . 75
6.4 Composite Parti leRe onstru tion . . . 75
6.4.1 0 Re onstru tion . . . 76 6.4.2 K 0 S Re onstru tion . . . 76 6.4.3 K 0 Re onstru tion . . . 77 6.4.4 D + Re onstru tion . . . 78 6.4.5 D 0 Re onstru tion . . . 79
6.4.6 The Re onstru tionof theB harged mesons . . . 79
6.5 Kinemati Variables: m ES andE . . . 79
6.6 Event Shape Variables . . . 83
7 Sear h for B + !D + K ()0 De ays 87 7.1 Introdu tionto theAnalysis . . . 87
7.2 Data and Monte CarloSamples . . . 88
7.3 Event Sele tionand Ba kground Chara terization . . . 89
7.3.1 Event Re onstru tionand Presele tionCriteria . . . 89
7.3.4 The PeakingBa kgroundStudies . . . 92
7.3.5 The BestCandidateChoi e . . . 94
7.3.6 Final Sele ted Sample . . . 94
7.4 MaximumLikelihoodFit. . . 95
7.4.1 The FitModel . . . 95
7.4.2 Parameterizations of theDistributionsUsedin theFit . . . 96
7.4.3 Fit Validation UsingParameterized Pseudo-experiments . . . 97
7.4.4 Fit Validation UsingFullySimulatedSamples. . . 99
7.4.5 Flavor Tagging . . . 100
7.5 Bran hing Fra tionMeasurements and SensitivityStudies . . . 115
7.6 CharmlessPeaking BB Ba kground . . . 117
7.7 Results . . . 118
7.7.1 Fit to theData . . . 118
7.7.2 Fit to theB 0 !D + and B 0 !D + samples . . . 125
7.7.3 Systemati ErrorEvaluation . . . 128
7.7.4 Final Results . . . 130 8 ADS analysis of B + !D 0 K + 133 8.1 Introdu tionto Analysis . . . 133 8.1.1 Motivation . . . 133
8.1.2 Previous Resultson theB + !D 0 K + withD 0 !K + 0 analysis . . . 133
8.2 Sele tionand Ba kground Chara terization . . . 134
8.2.1 Data Samples . . . 134
8.2.2 Presele tion Criteria . . . 134
8.2.5 The BestCandidateChoi e . . . 141
8.2.6 ContinuumBa kground Reje tion . . . 141
8.2.7 Crossfeed betweenSame Signand OppositeSignEvents . . . 143
8.2.8 OppositeSignto Same Sign EÆ ien yRatio . . . 144
8.2.9 Comparison betweenDataand SimulatedEvents . . . 145
8.3 MaximumLikelihoodFit. . . 145
8.3.1 Stru ture of theFitModel . . . 145
8.4 PDF Parameterizations ofthem ES andF Distributions . . . 147
8.4.1 PDF parameterization ofSignalEvents . . . 148
8.4.2 PDF parameterization fortheBB ba kgroundevents . . . 148
8.4.3 PDF Parameterizationforthe ContinuumBa kground Events. . . 149
8.4.4 PDF ParameterizationforPeaking Ba kground . . . 152
8.5 FitValidationStudies . . . 152
8.5.1 Fit Validation UsingParameterized Pseudo-experiments . . . 152
8.5.2 Fit Validation UsingFullySimulatedSamples. . . 154
8.5.3 Fit Validation forR Variables . . . 160
8.5.4 Charmless Peaking Ba kground . . . 161
8.6 FitResults on Data . . . 163
8.6.1 Results on R ADS . . . 163 8.6.2 Results on R . . . 165 8.7 Systemati Un ertainties . . . 166 8.7.1 Final Results . . . 171
9 Phenomenologi al Impa t of the Measurements 173 9.1 The Impa t ofthe ADSAnalysis . . . 173
9.1.2 The r
B
and Extra tion from Data . . . 176
9.2 The DK System Des ription . . . 178
9.2.1 The B !DK De ayMode System . . . 180
9.2.2 The B !DK De ayMode System . . . 182
10Con lusions 185 A B measurement additional information 193 A.1 ObservableDistributions . . . 193
A.2 GlobalEvent Variables. . . 199
A.3 Peakingba kgroundStudies . . . 201
A.4 Comparisonbetweendata andsimulatedevents . . . 204
A.5 Parameterizations . . . 207
A.6 ControlSampleParameterizations . . . 217
Dans ette these nouspresentonsdesetudessurles mesons B e e tuesen utilisantles donnees
enregistreespar l'experien eBABARaupresde PEP-IIaSLAC.
D'abord nous presentons lare her he desdesintegrationsrare B
+ ! D + K ()0 . Cesmodes
dedesintegrationsont interessants arils'agit depro essusd'annihilationquifournitdes
infor-mationsimportantessurladynamiquedeladesintegrationdesmesons beauxetleselementsde
lamatri eCKM,V
ij
. Lesresultatsobtenussur esmodesdedesintegrationpeuvent^etreutilises
dans des ajustements phenomenologiques. Cela permet de traduireles mesures sur les
ampli-tudes hargees B
+
! D
+
K ()0
en estimations sur les amplitudes B
0 ! D 0 K ()0 supprimees par V ub
. L'analyse experimentale est e e tuee en utilisant plusieursmodes de desintegration
du meson D harge. Nous n'avons obtenu au uneeviden e signi ative de signalet les limites
superieuressurles rapports d'embran hement suivantsont ete etablies
B(B + !D + K 0 )<2:910 6 a90% prob.; B(B + !D + K 0 )<3:010 6 a90% prob.
Dans ladeuxiemepartie delathesenous presentonsdesetudessur laviolationde CP dans
lesystemedesmesonsBetenparti ulierlamesuredel'angle duTriangled'Unitarite. L'angle
est laphase relative entre leselements V
ub
etV
b
de la matri eCKM. Un parametre ru ial
qui determine la sensibilite a est le rapport r entre les amplitudes de transition b ! u et
b ! . Dans ette these nous presentons une analyse du anal de desintegration des mesons
B harges: B + ! D 0 K +
. Ces desintegrations sont etudiees en utilisant la methode ADS et
le meson neutre D est re onstruitdans sonetat nal K
0
. En ombinant ette analyse ave
une analyse similaire quiutilise l'etat nal K des D
0
le rapport r(DK) et l'angle ont ete
Siles resultats experimentaux ontenusdans ette thesesont utiliseesdansles systemes
ex-primantlesamplitudesdedesintegrationsdesB !DK et B !DKdesresultatsinteressants
peuventtreobtenus. Lapre isionsurlerapportr(DK
)pourlesmesonsB hargesestamelioree
d'un fa teur trois : r(DK
) = 0:080:03. Le rapport entre les modules des amplitudes V
ub
d'annihilation (A) et supprimee de ouleur (C) est jA=Cj < 0:6 (a 90% du probabilite).
Fi-nalement lerapportr(DK
0
) pourles mesons neutres vaut0:270:09. La grande valeur de e
rapportest parti ulierement interessante pourles analysesfuturesquiont eubutde mesurerla
In this thesis, we present studies of the B mesons system performed using the full dataset
olle tedbytheBABARexperimentat the PEP-II olliderat SLAC.
The rst analysis presented is the sear h of the rare V
ub mediatedde ays B + ! D + K ()0 .
Thesede aysareparti ularlyinterestingbe ausethey areexpe tedto bedominatedbythe
an-nihilationpro essesand anprovideinsightto theinternaldynami softheB mesons. Another
pointof interest is omingfrom thefa t thattheratesofthese de ays an be usedto onstrain
the annihilation amplitudes in phenomenologi al ts. This allows the translation of the
mea-surements of the harged B
+
! D
+
K ()0
amplitudes into estimations of the V
ub suppressed amplitudes of B 0 ! D 0 K ()0
. The experimental analysis is performed looking at several D
+
de ay modes. No signalshave beenfoundand upperlimitshavebeenset to be:
B(B + !D + K 0 )<2:910 6 a 90%prob.; B(B + !D + K 0 )<3:010 6 a 90%prob.:
In these ondpartof thethesiswe present theCP violationstudiesintheB-mesonsystem,
and in parti ular measurements of the angle of the Unitarity Triangle. The angle is the
relativeweakphasebetweentheV
ub
andV
b
elementsoftheCKMmatrix. A ru ialparameter,
whi h drivesthesensitivityto ,is theratior betweenb!u and b! transitionamplitudes.
We present and des ribe theanalysis usingthe harged Bmeson de ays: B
+ !D 0 K + . These
de ays are studied through the ADS method, where the neutral D mesons are re onstru ted
into K
0
nal states. Combining thisanalysis witha similarone that used K as a D
0
nal
state theratior(DK) andtheangle have beenobtainedto be
r(DK)=0:083 +0:028 0:043 ; =86 Æ+51 Æ 45 Æ : (2)
Iftheresultsof thisthesis areusedinthefullsystemoftheB !DK andB !DK
de ay
amplitudes, other interesting results an be obtained. The error on the ratio r(DK
hargedB de aysisimprovedbyafa tor ofthreeresultinginr(DK)=0:080:03. The ratio
betweentheV
ub
mediatedannihilation(A)andthe olorsuppressed(C)amplitudesisobtained
tobejA=Cj<0:6(at90%probability). Finally,theratior(DK
0
)forneutralB de aysisfound
to be 0:270:09. The large value for this ratio is parti ularly interesting for future analyses
Introdu tion
The StandardModel of elementary parti les(SM) hasgiven agood e e tive des riptionof the
physi al pro esses that have been tested until 2010. This model has provided well on rmed
predi tionsofthee e ts onne tedtothreeoutofthefourfundamentalintera tionsina oherent
framework.
The omprehensive test of the Standard Model is themain goal of urrent physi al
exper-iments in the parti le physi s. These experiments an use di erent approa hes: elementary
parti le ollisions atdi erent energies, heavy ion ollisions,astroparti leexperiments,et . One
ofthewaytoexploreSMinthepartdes ribingCP violationistodevelopexperimentsaimedat
studyB mesonde ays,whereamultitudeofCP violatinge e tsareexpe ted. Forthesereasons
thetwo \B fa tory experiments"were onstru ted: BABAR,based inSLAC, Menlo Park,
Cali-fornia,USAand Belle,basedinKEK,Tsukuba,Japan. TheseB fa torieshave jointly olle ted
datasample withmore than10
9
BB mesonpairsforabout10 years of running.
The results of these experiments have played a ru ial role in the study of CP violation,
whi hisdes ribedintheSMwiththeuseoftheCKMmatrix(asdes ribedinChapter2). This
thesisisdedi atedtothestudyofCP violatione e tsandinparti ularthemeasurementsofthe
angle of the Unitarity triangle(Chapter 3) using the nal dataset olle ted with the BABAR
dete tor (des ribedinChapter 5).
Sin e thisangle is therelative phase betweenV
ub
and V
b
elements of theCKMmatrix, its
value an bea essedstudyingpro essesthatinvolveb!uandb! transitions. Thesimplest
way to a ess these transitions is to study in details the B ! DK de ays (as des ribed in
Chapters3and4). Inthere entyears,di erentmethodstomeasuretheangle weredeveloped
andalotofe ortsofs ienti ommunitywere on entratedontheexperimentaldetermination
it is that the sensitivity to is driven by the value of the ratios r between b ! u and b !
amplitudes. Thenumeri alvaluesoftheseratios arede aydependent. Theyareexpe tedtobe
smalland haveto be determinedon data.
Theexperimentalworkpresentedinthisthesisis omposedoftwoanalyses. The rstanalysis
(des ribedinChapters6and7) des ribesthesear hfortherarede aysB
+ !D + K ()0 . These
de ays have never been observed before and are quite interesting sin e they allow to a ess
annihilationamplitudesthatenterin thedeterminationof theratios r.
The se ond analysis on erns the measurements of the B
+ ! D 0 ( D 0 )K + de ays analyzed
throughtheADSmethod allowingthedetermination ofr andthe angle (Chapter8).
In Chapter9thephenomenologi alimpa tsofthemeasurementspresentedinthethesisare
CKM Matrix and CP Violation in
the Standard Model
Inthepast de ade,one of themajor hallengesofparti lephysi shasbeento gainanin-depth
understandingoftherole ofquark avor. Inthistimeframe, measurementsand thetheoreti al
interpretation of their results have advan ed tremendously. A mu h broader understanding of
avor parti les has been a hieved, apart from their masses and quantum numbers, there now
exist detailed measurements of the hara teristi s of their intera tions allowing stringent tests
of StandardModel predi tions.
The Standard Model (SM) [2 ℄ has a ri h stru ture in its avor se tor, mainly be ause it
ontainsthreegenerations ofquarksand leptons. Inthequarkse tor, itiswellestablishedthat
the misalignment between the weak intera tion eigenstates and mass eigenstates leads to the
Cabibbo-Kobayashi-Maskawa (CKM)matrix[3 ℄, whi his thesour eofthetransitions between
di erent generations. Even more importantly, it o ers the sour e of the CP violation. This
avorstru turehasbeen on rmedbymanyexperimental measurements to agoodpre ision.
The SM is a low energy e e tive theory and one of the main tasks of modern high-energy
physi sisthesear hfor ontradi tionsofmeasurementswiththeSMpredi tions. Theexisten e
of su h a di eren e would imply thepresen e of \new physi s" (NP) e e ts. The avorse tor
isone of themainfrontiers intheNP e e ts sear h[4 ℄.
This Chapter isdevotedto a briefintrodu tionto avor physi s. The urrent status of the
2.1 Symmetries
Symmetriesplayanimportantroleinthemodernphysi s. Thesymmetrypropertiesofaphysi al
system are intimately related to the onservation laws hara terizing that system. Noether's
theoremgivesapre isedes riptionof thisrelation[1℄. Thetheorem statesthatea h ontinuous
symmetryofaphysi alsystemimpliesthatsome physi alpropertyof thatsystemis onserved.
The parti le physi s has three related dis rete symmetries. These state that the spa e is
indistinguishablefromone under thefollowingtransformations:
paritytransformation P,whi his the ipinthesign of one ofspatial oordinates;
timereversaltransformation T,whi h reverses thetimevariablesof thesystem;
harge- onjugationtransformationC,whi hinversesallthe hargesofparti lestoopposite
(i.e. hangingparti les and anti-parti les).
The CPTtheorem requires the preservation of the dis reteCPT symmetry by all physi al
phenomena. Theviolationofthisruleisbeingsear hedbut,untilthetimeofthisthesis
publi a-tion,allobservation are onsistent withexa t CPT symmetry. Thestrongandele tromagneti
intera tions show no experimental eviden e for C, P or T violation, while weak intera tions
violate C and P separately, onserving, in rst approximation, their produ t CP. The rst
eviden e of CP violation was found in 1964 in rare pro esses in the kaon system [5 ℄. In 2001
thiswas on rmed inB mesonde ays [6 ℄studied inthededi ated experiments.
2.2 CP Violation and the CKM Matrix
The Standard Model is based on SU(3)
C
SU(2)U(1) gauge symmetry, where SU(3)
C
des ribesthe olorsymmetryofstrongintera tions,SU(2)theweakisospinsymmetry,andU(1)
thesymmetryunderhyper hargetransformations. CP violationinweakpro essesarisesfroma
singleirremovable omplexphaseinthemixingmatrixforquarks,whi hgovernsthe hargedW
gaugebosonintera tionwiththequarks: thisis alledtheCabibbo-Kobayashi-Maskawa(CKM)
me hanism[3℄. Su h harged urrent weakintera tion an be writtenas:
L W = g p 2 (u; ;t) L 0 d s b 1 A L W + + h. . ; (2.1) where g is theSU(2) L oupling onstant, W +
is the W boson eld operator, and (u; ;t)
L
Q = 1=3, respe tively. The matrix of the ouplings, V, alled Cabibbo-Kobayashi-Maskawa
(CKM) matrix [3 ℄ is in prin iple a unitary, 33 omplex matrix. The matrix elements are
normallynotated a ordingto the quarksthey orrespond to:
V = 0 V ud V us V ub V d V s V b V td V ts V tb 1 A : (2.2)
This CKM matrix thus depends on nine parameters, three real angles and six phases. It
an alwaysbeparameterizedwiththreeEulerangles(realparameters)andsixphases( omplex
parameters). Fiveofthesesixphasesdisappearundertransformationsthatrede nethephaseof
thequark eldsinthequarkmasseigenstatebasisandleavethediagonalmassmatrixun hanged.
Oneofthesixphasesisirredu ible. Thepresen eofthisphasea ountsfortheCP violation in
theStandardModel.
An expli itparameterization interms ofthree mixingangles
12 , 13 , 23 ,and aphaseÆ [8 ℄,
witha parti ular quark eldsphase onvention an bewritten:
V CKM = 0 12 13 s 12 13 s 13 e iÆ s 12 23 12 s 23 s 13 e iÆ 12 23 s 12 s 23 s 13 e iÆ s 23 13 s 12 s 23 12 23 s 13 e iÆ s 23 12 s 12 23 s 13 e iÆ 23 13 1 A ; (2.3) where ij = os ij and s ij =sin ij
(i;j =1;2;3, j>i). Theexperimental values suggestthat
13 23 12 .
Considering that the mixing angles are small, the Wolfenstein parametrization [9 ℄ an be
introdu edforthedes riptionoftheV matrixelementsmagnitude. Inthisparametrization,the
matrixelements aretheresult of an expansioninterms of a smallparameter=jV
us
j0:22.
The fourindependent realparameters an bede ned:
s 12 =; s 23 =A 2 ; = s 13 s 12 s 23 osÆ; = s 13 s 12 s 23 sinÆ: (2.4)
Forthese variables, theCKMmatrix anbe expressed
A more a urateexpressionoftheWolfensteinparametrization, in ludingO( 4 ) andO( 5 ) terms,gives[10 ℄: 0 1 1 2 2 1 8 4 A 3 ( i) + 1 2 A 2 5 [1 2(+i)℄ 1 1 2 2 1 8 4 (1+4A 2 ) A 2 A 3 (1 i) A 2 + 1 2 A 4 [1 2(+i)℄ 1 1 2 A 2 4 1 A +O( 6 ) (2.6)
whereand arerelated to and by:
=(1 2 2 ); =(1 2 2 ): (2.7)
The unitarityof theV matriximpliesseveral relations betweenits elements,
3 X i=1 V ij V ik =Æ jk and 3 X i=1 V ij V kj =Æ ik : (2.8)
Sixofthem are parti ularlyinteresting
Ea h of these relations an be represented asa triangle in the( ; ) plane, wherethe ones
obtained by produ t of neighboring rows or olumns are nearly degenerated. The areas of all
these trianglesare equalto halfof theJarlskog invariant J [14 ℄ de nedby:
J 12 23 2 13 s 12 s 23 s 13 sinÆ: (2.9)
ThevalueofJ anestimatetheamountofCPViolation. ThemaximumvalueJ,1=(6
p
3 )0:1,
gives the fully violated CP symmetry. The re ent value of determinant is J = (3:00:1)
10 5
[17 ℄.
The interest isdrivenbyequation
V ud V ub +V d V b +V td V tb =0; (2.10)
with ea h item approximately proportionalto
3
. This equation is onne ted to theB meson
de aysduetothepresen eofV
ub
andV
b
matrixelements. Dividingallthetermsoftherelation
byV d V b ,one obtains V ud V ub V d V b +1+ V td V tb V d V b =0: (2.11)
This equation an be graphi ally represented on the plane. Figure 2.1 shows the
triangle, whi hsides,usuallynoted R
b
andR
t
, an be al ulatedas:
ρ+
i
η
1−ρ−
i
η
β
γ
α
C=(0,0)
B=(1,0)
A=(
ρ,η)
Figure 2.1: UnitarityTriangleinthe plane.
The three angles, denoted by , ,and ,are
1 : =arg V td V tb V ud V ub ; =arg V d V b V td V tb ; =arg V ud V ub V d V b = (2.13)
In theWolfenstein parametrizationtheonly omplexelements, up to termsof order O(
5 ), areV ub and V td
and the phases and an be dire tly relatedto them:
V td = jV td je i ; (2.14) V ub = jV ub je i : (2.15)
2.3 Current Status of the Unitarity Triangle Measurements
AmongexperimentaltestsoftheCP violation,measurementsofthemixingindu edCP violation
in the neutral B meson system played a entral role at the present B fa tories. The angle
of the unitarity triangle has been measured very pre isely, and pre ision measurement of the
angle isalsopossiblebya umulatingmorestatisti satthefutureB fa tories. ThepresentB
fa tories have also demonstrated the sensitivityinother measurements investigating the avor
stru ture. Dire t CP violationin B mesonde ays hasbeenmeasured inthe de aymodes su h
as B ! and B ! K. The angle an be measured through the interferen e of de ay
1
amplitudes involving intermediate D mesons. Several other FlavorChanging Neutral Current
(FCNC) pro esses that are sensitive to possible new heavy parti les ex hanged in theloopsof
Feynman diagramshave alsobeeninvestigated.
Several methodshave beendeveloped forthe datastatisti al treatment. By thetimeof this
thesis,themosta tivegroupsareCKM tter[15 ℄ thatusesfrequentistapproa h,andUT t[17 ℄
using Bayesian approa h. The aim of ea h method is to obtain the onstraints in the ( )
plane.
The Bayesianapproa hoftheUTtriangle t developedbytheUT t ollaborationrelieson
thefollowingarguments. Ea hobservablegivesrisetotheequationthatrelatesa onstraint
j to
theCKMtriangleparameters viathesetofan illaryparametersx,wherex=x
1
;x
2
;:::;x
N
stands for all experimentally determined or theoreti ally al ulated quantities from whi h the
various
j
depend.
In an ideal ase of exa t knowledge of
j
and x, ea h of the onstraints provides a urve
in the plane. In su h a ase, there would be no reason to favor any of the points on
the urve, unless we have some further informationorphysi alprejudi e, whi h might ex lude
points outsidea determined physi alregion, or, ingeneral, assigndi erent weights to di erent
points. Inthereal experimental asewedeal withtheparameters, whi hare knownwith some
pre ision, whi h in general leadsto assigning di erent weights to di erent points. This means
that, instead of asingle urve inthe plane, we have afamily of urves whi h dependson
thedistributionofthesetf
j
;xg. Asaresult,thepointsinthe planegetdi erentweights
(even ifthey weretaken to be equally probablea priori)and our on den eon the valuesof
and lustersina regionof theplane be omes di erent.
One of themethods that takes into a ount the experimental and theoreti al un ertainties
and des ribes them in terms of a probability density fun tion f (PDF), whi h quanti es our
on den eon thevaluesof a given quantity,is the Bayesian approa h. The inferen e of and
be omesthena straightforwardappli ation of probabilitiestheory.
Theprobabilisti approa h anbeimplementedde ninganidealizedPDFforea h onstraint:
f(;j j ;x)Æ( j j (; ; x)); (2.16)
whereÆ isDira delta fun tion.
obtainedbymaking useof thestandardprobabilityrules: f(; ) = Z f(;j j ;x)f( j ;x)d j dx / Z Æ( j j (; ; x))f( j )f(x)d j dx / Z Æ( j j (; ; x)) 1 p 2( j ) exp ( j ^ j ) 2 2 2 ( j ) f(x)d j dx / Z 1 p 2( j ) exp ( j (; ; x) ^ j ) 2 2 2 ( j ) f(x 1 )f(x 2 )f(x N )dx (2.17) where^ j
istheexperimentalbestestimateof
j
,withun ertainty( j). Thejointpdff(
j
;x)
hasbeensplittedasaprodu toftheindividualpdfassumingtheindependen e ofthedi erent
quantities,whi h isa very good approximation forthe ase understudy.
This formula may be represented in a di erent approa h introdu ing a global interferen e
relating, ,
j
,and x(integrating overnotinterestingparameters). Inthis ase, withthehelp
of Bayes' theorem,one an obtain
f(; j^ j )/f(^ j j j ;; ) f( j ;; ; x) /f(^ j j j )f( j j ;; x)f(; ; x) /f(^ j j j )Æ( j j (;; x))f(x)f Æ (;); (2.18) wheref Æ
(; ) denotestheprior distribution.
The extension of relation 2.18 to several onstraints, assuming these onstraints are
un or-related, an bewritten as
f(;j^ 1 ;:::; ^ M )/ Y j=1;M f j (^ j j;; x) Y j=1;M f i (x i )f Æ ( ;): (2.19)
Integrating thisequation overx we an rewritethe interferen e s heme inthefollowingway:
f(; j^ ;f)/L(^ ;; ; f)f
Æ
(;); (2.20)
where^ standsfortheset of measured onstraints,and
L(^ ;; ; f)= Z Y j=1;M f j (^ j j ;; x) Y j=1;M f i (x i ) (2.21)
is the e e tive overall likelihood, whi h takes into a ount all possible values of x
j
, properly
weighted.
In on lusion, the nal (unnormalized) pdf is obtained starting from a at distribution of (; ) is f(; ) / Z Y j=1;M f j (^ j j;; x) Y j=1;M f i (x i ); (2.22)
The integration an bedonebyMonteCarlo methods.
Thefollowingmeasurements,resultingindi erent onstraintsonthe plane,arein luded
intheUnitarityTriangle analysis:
jV
ub
jand jV
b
j. Their values measuredin in lusiveorex lusive semileptoni B !X
u; l
l .
The relative rate of harmless over harmed B mesonsemileptoni de ays is proportional
to thesquare oftheratio:
jV ub j jV b j = 1 2 2 p 2 + 2 : (2.23)
This orrespondsinthe(; ) planetoaring enteredin(0;0)withradiusR
b = p 2 + 2 . m d
. Theboxdiagram withtheex hangeofatopquarkgivesthedominant ontribution
to the B
0
B 0
os illations. The time os illation frequen y an be related in the SM to
themassdi eren e betweenthelight andheavy masseigenstates of theB
0 B 0 system: m d = G 2 F 6 2 m 2 W b S(x t )m B d f 2 B d ^ B B d jV b j 2 2 ((1 ) 2 + 2 : (2.24) where S(x t
) is the Inami-Lim fun tion [11 ℄ and x
t = m 2 t =M 2 W , m t
is the top quark
mass and
b
is the perturbative QCD short-distan e NLO orre tion. The s ale for the
evaluation ofthose orre tions entering into
b
and therunning ofthe tquark masshave
to bede nedin a onsistentway. Thevalueof
b
=0:550:01 hasbeenobtained in[12 ℄
and,inordertobe onsistent,themeasuredvalueofthepoletopquarkmass, obtainedby
CDFandD0 ollaborations,m
t
=(172:61:4) GeV=
2
,hastobe orre teddownwardsby
(7 1) GeV=
2
. Theremainingfa tor,f
2 B d ^ B B d
,en odestheinformationofnon-perturbative
QCD. The onstant f
2
B
d
translates the size of the B mesonwave fun tion at the origin.
The bag fa tor
^
B
B
d
is also introdu ed to take into a ount all possible deviations from
va uum.
The m
d
onstraint an berepresented bya ring entered at (1,0).
m
d =m
s
. Theratiobetweenm
d
de nedinpreviousbulletand m
s
,whi hisde ned
inthesame waylikem
d butforB s B s system. K
,thephenomenologi alparameterdes ribing \indire t"CP violation intheK
0
K 0
sys-ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
γ
β
α
)
γ
+
β
sin(2
s
m
∆
d
m
∆
∆
m
d
K
ε
cb
V
ub
V
ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
Figure2.2: Allowedregions for( ). The losed ontoursat95%probabilityareshown. The
fulllines orrespond to 95%probabilityregions forthe onstraints, given bythe measurements
of jV ub j=jV b j, K ,m d ,m d =m s , , , ,2 + .
, ,and (orthe2 + ombination. Informationon theangles an be obtainedfrom
themeasurement ofCP violating B
+
and B
0
de ays. Methods to extra t aredes ribed
inChapter 3.
The t is performed assuming the validity of the Standard Model. Figure 2.2 shows the
graphi alresultsof the tinusingtheapproa h des ribedin[17 ℄. Withthere ent pre isionon
inputstheSM predi tionsareingood agreement withtheexperimentaldata.
The resultsforand parameters are:
=0:1300:019;
=0:3510:012;
(2.25)
witherrors giving68% probabilityregions.
Thenewphysi smodels(i.e. themodelsdes ribingthepro essesnotin ludedinSM)usually
predi t the deviationsfrom SM s enarios inthe pro esses that an be des ribed by Feynmann
diagramswithat leastoneloop. Thus,one an separatethe ontributionsfrom theobservables
that an be determinedfrom the\tree-level" pro esses( and jV
ub
ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
γ
cb
V
ub
V
ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
β
s
m
∆
d
m
∆
∆
m
d
K
ε
ρ
-1
-0.5
0
0.5
1
η
-1
-0.5
0
0.5
1
Figure 2.3: Allowed regions for ( ) from the \tree-level" (left) and \loop-level" (right)
variables. The losed ontoursat 68% and95% probabilityareshown.
The resulting tsare showninFigure 2.3. The resulting tsfrom thetree-levelpro esses:
=0:116 +0:060 0:077 ; =0:374 +0:031 0:029 : (2.26)
whi h hasgothigh relative error, sin ethe\tree-level" ontributionis onstrainedonlybytwo
measurements. The pre ision ofthese measurementsthus plays a ru ialrole in the sear h for
new physi s.
Anotherwayto performthesear h forthenewphysi sisthe omparisonoftheinputvalues
of the t (i.e. observablesobtained experimentally)with the predi tionsobtained after the t
(i.e. predi tions assuming the validity of SM). The predi tions an be also obtained without
insertingthisparti ularmeasurementinthe t. Any ontradi tionbetweenthesemeasurements
would implytheNP e e ts. Theresultsof thisstudyisshowninTable2.1.
The t also produ es thepredi tionsoftheelementsof theCKMmatrix.
V CKM = 0 0:974080:00031 0:22610:0013 0:003580:00011 0:22590:0013 0:973260:00030 0:041210:00044 0:008750:00019 0:040420:00043 0:9991450:000015 1 A (2.27)
Parameter Inputvalue Full t Predi tion | 0:1300:019 | | 0:3510:012 | A | 0:8060:013 | | 0:22610:0013 | jV ub j 0:003670:00020 0:003580:00011 0:003580:00011 jV b j 0:040820:00045 0:041210:00044 | m s ;ps 1 17:770:12 17:760:11 17:41:3 ;[ Æ ℄ 91:46:1 88:22:9 86:33:7 ;[ Æ ℄ | 21:970:75 24:21:4 sin(2 ) 0:6540:026 0:6940:018 | os (2 ) 0:860:12 0:7200:018 | 2 + ;[ Æ ℄ 9452 113:83:1 114:03:1 ;[ Æ ℄ 7411 69:63:0 69:23:1 j" K j 0:0023550:000049 0:0023400:000047 0:0023590:000049
Table 2.1: Theinputvaluesused inthe t,theirvalueafter the t and thepredi tionsof these
Measurement of the Unitarity
Triangle Angle
Theangle isde nedastheweakphaseoftheCKMelement V
ub =jV ub je i . Variousmethods related to B + ! D ()0 K ()+
de ays have been proposed to determine the UT angle . These
methods exploit thefa t that the neutral D meson de ay produ t an be either a D
0 (from a b ! us transition), or a D 0 (from a
b ! u s transition; or vi e versa for b de ays). If the
nal state is hosen su h that both D
0
and
D 0
an ontribute, the interferen e between these
amplitudesissensitivetothephase ,allowing tobedeterminedwithessentiallynotheoreti al
assumptions. Choi esfor the nalstate in ludeD
0
mesonde ayingto:
a singly Cabibbo-suppressed CP eigenstate, like D
0
! h
+
h (h = ;K) for
Gronau-London-Wyler(GLW) method[60 ℄;
adoublyCabibbo-suppressed avoreigenstate,likeD
0
!K
+
forAtwood-Dunietz-Soni
(ADS) method[61 ℄;
a Cabibbo-allowed self- onjugate 3-body state, like D
0 ! K 0 S + for Giri-Grossman-So er-Zupan(GGSZ) method[62 ℄.
Ifwenow onsiderthe ounterpartofneutralmesonde aysthesituationisdi erent. Infa t,
sin e neutral B mesons mix, interferen e e e ts between b ! and b ! u de ay amplitudes
in B
0
de ays (for instan e into D
()
nal states) are studied for the determination of the
ombination of UT angles 2 + . In this ase the tagging te hnique and a time dependent
analysis are required [20 ℄. In ontrast, B
0
! D
()0
K 0
de ay modes an be used to dire tly
measure . Infa t, inthis ase, tagging is notneeded and we an unambiguously identify ifa
B 0 or B 0
hasde ayed throughthe sign of the ele tri harge of thekaon from the K
0
de ay 1
.
An exampleof su h pro esses is shown in Figure3.1. is the relative weak phase between
thetwodiagrams,and an bea essedbymeasuringCP violatinge e tsinB de ayswherethe
two amplitudesinterfere. This type of interferen e an be seen in both harged an neutral B
mesonde ays. K 0 D 0 B 0 b d u s K 0 D 0 B 0 b u d s
Figure3.1: FeynmandiagramsforB
0 !D 0 K 0 and B 0 !D 0 K 0
. Therelative phasebetween
these de ays isproportionalto the CKMangle .
3.1 General Formalism
Keeping in mind that V
ub
= jV
ub je
i
one an de ne the following amplitudesfor B meson to
two bodyde ays:
A(B !D 0 K )=jA B je i B ; A(B !D 0 K )=j A B je i B e i ; A(B + !D 0 K + )=jA B je i B ; A(B + !D 0 K + )=j A B je i B e i ; (3.1) with B and B
being thestrongphase ofthe B de ay. The same an bedonefor thede ays
D 0 !f: A(D 0 !f)=jA D je i D ; A(D 0 ! f)=j A D je i D ; A( D 0 ! f)=jA D je i D ; A( D 0 !f)=j A D je i D ; (3.2) with D and D
beingthestrong phaseof theDde ay.
1
Thus, the amplitude of thede ayB 