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→ K0Sπ+π – with the LHCb spectrometer

Marouen Baalouch

To cite this version:

Marouen Baalouch. Dalitz analysis of the three-body charmless decay B0 → K0Sπ+π– with the LHCb spectrometer. Other [cond-mat.other]. Université Blaise Pascal - Clermont-Ferrand II, 2015. English.

�NNT : 2015CLF22652�. �tel-01333554�

EDSF :853

UNIVERSITE BLAISE PASCAL

(U.F.R. Sienes etTehnologies)

ECOLE DOCTORALE DES SCIENCES FONDAMENTALES

THESE

présentée pour obtenir legrade de

DOCTEUR D'UNIVERSITE

(SPECIALITE PHYSIQUE DESPARTICULES)

par

Marouen BAALOUCH

DALITZ ANALYSIS OF THE THREE-BODY

CHARMLESS DECAY

B ⁰ → K _S ⁰ π ⁺ π ⁻

^WITH

THE LHCb SPECTROMETER

Thèsesoutenue le 14Deember 2015 devant la ommission d'examen:

Président : M. S. Desotes-Genon

Rapporteurs : Mme M.H. Shune

M T. Latham

Examinateurs : Mme H. Fonvieille

M O. Deshamps

Direteur de thèse : M S. Monteil

Studiesof harmlessthree-bodydeaysof the neutral

B

^{mesonswith a}

K _S ⁰

^{inthe nalstate}

are presented in this thesis. The analyses are performed with the full statistis reorded

by the LHCb spetrometer during the Run I of the LHC. The amplitude analysis of the

deay

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{represents the main part of this thesis analysis. A} time-integrated untagged Dalitz-Plot analysis of the deay is performed. The t frations of the quasi-two-

bodydeaysareobtained. Likewise,thediret

CP

asymmetriesofthequasi-two-bodydeays

B ⁰ → K ^∗+ (892)π ⁻

^,

B ⁰ → K ₀ ^∗+ (1430)π ⁻

^,

B ⁰ → K ₂ ^∗+ (1430)π ⁻

^and

B ⁰ → f ₀ (980)K _S ⁰

^are

obtained. Thelargestsensitivityisobtainedfor

A CP (B ⁰ → K ^∗+ (892)π ⁻ )

^{. This}measurement is the rst observation of the

CP

^{asymmetry with a signiane larger then ve standard}

deviations. The measurement is in agreement with the world average, with an improved

preision.

Keywords

LHC - CERN - LHCb detetor - Standard Model - Partile Physis - Heavy

Flavour Physis - CKM Triangle -

CP

^{Violation -}

B

^{Physis - Diret}

CP

Asymmetry - Branhing Ratio - Fit Fration - Charmless deay - Dalitz-Plot

-

B _d,s ⁰ → K _S ⁰ h ⁺ h ⁻

^-

B ⁰ → K _S ⁰ π ⁺ π ⁻

^-

B ⁰ → K ^∗+ (892)π ⁻

^-

B ⁰ → f 0 (980)K _S ⁰

^.

Le travail présenté dans ette thèse onerne l'étude des désintégrations en trois orps sans

quark harmé des mésons beaux neutres, dont l'état nal ontient un

K _S ⁰

^{. Ce travail de}

reherhe s'est réalisé dans le adre de l'éxpériene LHCb au LHC, en analysant un éhan-

tillon d'événements de 3 fb

−1

olleté dans le Run I du LHC. L'analyse d'amplitude de

la désintégration

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{représente la partie prinipale de e travail de thèse. La}

mesure des amplitudes est eetuée au moyen d'une étude du plan de Dalitz de la désinté-

grationintégréedans letemps sansétiquetage de lasaveur de lapartiulebelle. Nousavons

mesurélesrapportsd'embranhementsrelativesdesdésintégrationsquasi-deux-orpsàpartir

de ette analyse de Dalitz. Également,nous avons mesurél'asymétrie

CP

^{diretedes désin-}

tégrationsquasi-deux-orps

B ⁰ → K ^∗+ (892)π ⁻

^,

B ⁰ → K ₀ ^∗+ (1430)π ⁻

^,

B ⁰ → K ₂ ^∗+ (1430)π ⁻

^et

B ⁰ → f 0 (980)K _S ⁰

^{. Nous avons observé pour la première fois} l'asymétrie

CP

^{direte dans la}

désintégration

B ⁰ → K ^∗+ (892)π ⁻

^{ave unesignianesupérieureàinqdéviationsstandard.}

Cette mesureest en aord ave la moyenne mondiale,ave une préisionaméliorée.

Mots Clés

LHC - CERN - Deteteur LHCb - Physique des Partiules - Modèle Standard

- Physique des Saveurs Lourdes - Triangle CKM - Violation

CP

^{- Physique des}

mesons

B

^{- Asymétrie}

CP

^{Direte - Rapport} d'embranhement - Désintégration sans quark harm - Dalitz-Plot -

B ⁰ _d,s → K _S ⁰ h ⁺ h ⁻

^-

B ⁰ → K _S ⁰ π ⁺ π ⁻

^-

B ⁰ → K ^∗+ (892)π ⁻

^-

B ⁰ → f ₀ (980)K _S ⁰

^.

Depuis des mois j'attends d'érireette partie de la thèse qui vient typiquement à lan de

la rédation et... nous y voilà !

Ce travail dotoral n'aurait pas pu être réalisé sans le soutien d'un grand nombre de

personnes etsurtout mon direteur de thèse, M. Stéphane Monteil,professeur à l'université

BlaisePasal. Je ne pourrais jamaisle remerierassez pourtout e qu'ilm'a donné. Jesuis

trèsreonnaissantpourletempsonséquentqu'ilm'aaordéetpourl'aideompétentequ'il

m'a apportée, pour sa patiene etson enouragement. J'ai beauoup appris de ses qualités

pédagogiques et sientiques, sa franhise et sa sympathie ainsi de ses qualités humaine

exeptionnels.

Je tiens à exprimer toute magratitude aux membres du jury. Je remerie M. Sébastien

Desotes-Genonquiabien vouluprésider lejury. Meri àmesdeux rapporteursM.Thomas

Latham et Mme Marie-Hélène Shune pour leurs suggestions et orretions, ainsi que pour

leursonseilsetleursoutien. JesouhaiteremerieraussimesdeuxexaminateursMmeHélène

Fonvieille et M. Olivier Deshamps pour avoir partiipé à la soutenane et d'avoir apporté

un regard extérieur ritique àe travail.

Je tiens évidemment à remerier tout le groupe LHCb de Clermont-Ferrand pour leur

soutien etleurs onseils: M. Pasal PERRET responsablede l'équipe,M. Régis Lefèvre, M.

Ziad Ajaltouni, M. Olivier Deshamps, M. EriCogneras etM. ValentinNiess. Je souhaite

remerier aussi ledireteur de laboratoireLPC, M. AlainFalvard pour m'avoiraueilli au

sein de e laboratoire. Mes remeriements vont également à mes ollègues pour tous les

momentsinoubliablespartagés danslelaboratoire: MostafaHoballah,JanMaratas, Meriem

Ben Ali, Ibrahim El Rifai, Diego Roa Romero, Mohamed Kozeiha, Maxime Vernet, Giulio

Gazzouni, Arianna Batista Camejo, Luigi Ligioi et Christos Hadjivasiou. J'ai aussi une

penséepourtouteslespersonnesave lesquellesj'aipartagélespauses-afé pleinesd'humour

et onversations sientiques: enore Mostafa Hoballah (thanks khayi), Xavier Lopez mon

parrain, Alouane Selmi, Arnaud Rozes, Romano Marino, Siavah et Alexandre Claude. Je

remerie toutes les personnes formidables que j'ai renontrées par le biais de LPC ou de

CERN. Meri pour votre supportetvos enouragements.

Je souhaite remerier spéialement M. Adel Trabelsi, professeur à l'université Tunis El-

Manar, pour son enouragement, ses multiples onseils et son soutien, depuis mon master

en Tunisie jusqu'à la n de ma thèse. Enn, quatre personnes ont fait preuve d'un énorme

quem'aapportéettoutequem'adonné,meripoursessoutienssansfaillesdepuistoujours.

Meri à ma mère, elle était toujours à oté de moi malgré la distane et les engagements.

Toute queje suis ouaspireàdevenir, 'est àmamèrequeje ledois. Merià matrès hère

et"mybeloved" Marwa,pour m'avoirsupporté es dernierstemps, toujoursave doueuret

bonté. Sa présene et ses enouragements sont pour moi les piliers fondateurs de e que je

suis et de e queje fais.

Introdution 2

1 Charmless deays of

B

^{mesons in the Standard Model 3}

1.1

CP

^{violation inthe SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3}

1.1.1 Introdution toStandard Model . . . 3

1.1.2 CKM mixingmatrix . . . 4

1.1.3 CKM parameterisations and representations . . . 6

1.1.4

CP

^{Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7}

1.1.5

CP

^{violation in neutral}

B

^{setor . . . . . . . . . . . . . . . . . . . . . 8}

1.2 Constraints onCKM matrix elements . . . 15

1.2.1 Magnitudes of the matrix elements . . . 15

1.2.2 CKM angles . . . 16

1.3 Charmless three-body neutral

B

^{deays . . . . . . . . . . . . . . . . . . . . . 16}

1.3.1

β

^{angle and New Physis . . . . . . . . . . . . . . . . . . . . . . . . . 17}

2

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{and Dalitz Plot formalism 20} 2.1 Three-body kinematis: The Dalitz-plot . . . 20

2.2 Heliity angle . . . 21

2.3 Three-body dynamis: Isobar Model . . . 22

2.3.1 Angular distributions . . . 23

2.4 Massterm desription . . . 24

2.4.1 RelativistiBreit Wigner lineshape . . . 24

2.4.2 Gounaris-Sakurai(GS) lineshape . . . 26

2.4.3 Flattémass lineshape. . . 27

2.4.4 LASSmass lineshape . . . 28

2.4.5 Redued

K

^{-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29}

2.5 DP probability density funtion and

CP

Observables . . . 31

2.5.1 Probability density funtion . . . 32

2.5.2 The Square DalitzPlot . . . 32

3 Large Hadron Colliderand the LHCb experiment 34

3.1 The Large Hadron Colliderat CERN . . . 34

3.1.1 The LHC aeleratorsystem . . . 34

3.1.2 Experimentsat the LHC . . . 36

3.1.3 Luminosity and

b ¯ b

^{quark pair prodution . . . . . . . . . . . . . . . . 36}

3.2 The LHCb detetor . . . 37

3.2.1 Generaloverview . . . 37

3.2.2 Traking system . . . 39

3.2.3 Partile identiationdetetors . . . 47

3.2.4 Partile identiationtehniques . . . 51

3.2.5 Trigger system . . . 52

3.2.6 LHCb reonstrution and data stripping . . . 55

3.2.7 LHCb software . . . 55

4 Study of

B _(s) ⁰ → K _S ⁰ h ⁺ h ^′−

^{deays 57} 4.1 Dataset, triggerand stripping . . . 57

4.1.1 Trigger . . . 58

4.1.2 Stripping . . . 59

4.2 Seletion . . . 63

4.2.1 Preseletion . . . 63

4.2.2 Datasets for the MVA training. . . 65

4.2.3 Disriminatingvariables . . . 66

4.2.4 Training and validationof the BDT . . . 73

4.2.5 Optimisationof the BDT uts . . . 74

4.2.6 Partile Identiation. . . 75

4.3 Bakground studies . . . 84

4.4 Masst model . . . 85

4.4.1 Generalstrategy . . . 86

4.4.2 Signal model . . . 86

4.4.3 Models forrossfeed bakground. . . 88

4.4.4 Partially-reonstruted bakgrounds. . . 88

4.4.5 Combinatorialbakground . . . 91

4.5 Masst results . . . 94

4.5.1 Fit results for the loose BDT optimisation . . . 94

4.5.2 Fit results for the tightBDT optimisation . . . 94

4.5.3 Frationof signal in the

B ⁰

^{mass windowfor the tight BDT. . . . . 98}

5 Dalitz-plot analysis of

B ⁰ → K _S ⁰ π ⁺ π ⁻

⁹⁹ 5.1 Amplitudeanalysis formalism . . . 99

5.1.1 DalitzSignal p.d.f. . . 99

5.1.2 Likelihoodfuntion . . . 100

5.1.3 Physial observables fromDP t . . . 101

5.1.4 Analysismethod . . . 102

5.2 DP bakground . . . 103

5.2.1 Combinatorialbakground . . . 103

5.2.2

B _s ⁰ → K _S ⁰ K ^± π ^∓

^{ross-feed bakground. . . . . . . . . . . . . . . . . . 103}

5.3 Signaleieny variation aross the Dalitzplot . . . 110

5.3.1 Geometrial eieny . . . 110

5.3.2 Seletion eieny . . . 111

5.3.3 PID eieny . . . 114

5.3.4 Totaleieny . . . 115

5.4 Multiple solutions . . . 120

5.5 DalitzPlotFit . . . 121

5.5.1 Baseline modeland additionalresonanes. . . 121

5.5.2 Towards the nominalDP model . . . 125

5.6 DalitzPlot tresults . . . 134

5.6.1 Phase and t frationstatistial unertainties . . . 143

5.7 Fit validation . . . 154

5.7.1 Likelihoodsans. . . 154

5.7.2 Pseudo-experiments study from the tresults . . . 155

5.8 Systematis studies . . . 162

5.8.1 Experimentalsystematiunertainties . . . 162

5.8.2 Model systematiunertainties . . . 195

6 Result interpretation 202 6.1 Interpretation of the DP t results . . . 202

6.1.1 Isobar parameter and t frations measurements . . . 202

6.1.2 Diret

CP

asymmetries measurements . . . 205

6.1.3 First observation of diret

CP

^{asymmetry in}

B ⁰ → K ^∗ (892) ⁺ π ⁻

^{. . . 205}

A Dalitz plot kinematis 210 B Angular distribution in Dalitz plot 213 C Seletion - extra plots 216 D Fit model - extra plots 240 E CRAFT tter 249 E.1 Numerialintegration tehnique . . . 249

E.2 Generation tehnique of pseudo-experiments . . . 250

E.3 Eieny . . . 250

E.4 Fittingmahinery . . . 250

F Denition of a goodness-of-t estimator 252 G Eieny - extra plots 256 H Seond solution analysis 267 Referenes . . . 269

The

CP

^{-violating phase emerging from the} Cabibbo-Kobayashi-Maskawa paradigm [1,2℄ is enoughtodesribeall

CP

^-violatingobservables measuredsofarinpartilesystems[3℄. This isthe onlysoure of

CP

^{violation inthe StandardModel(SM) whih yieldsmeasurable}

CP

^-

violating phenomena to date. The existene of new soures of

CP

^{violation in addition to}

thatpreditedby theCKMmatrixismadeneessary toaountforthebaryoniasymmetry

in the Universe [4℄ and hene the searh for it onstitutes an importantgoal of the urrent

researhes inhigh energy physis.

Oneappealingapproahtosearhfornewsouresof

CP

^{violationonsistsinstudyingthe}

deay-time distribution of neutral

B

^{meson deays to}

CP

-eigenstates hadroni nal states mediated by a

b → s

^{loop amplitude (so-alled penguin} amplitude). Many measurements havebeen performedbythe BaBarand Belleexperimentsinthatrespet, suhas

B ⁰

^deays

to

φK _S ⁰

^or

η ^′ K _S ⁰

^{to ite only the most sensitive. Gathering all of these studies, the latest}

results [5℄ provide a onsistent piture with the SM preditions, demanding an improved

preision to inrease the sensitivity tonew

CP

^{-violating phases.}

The deays mentioned above into a nal

CP

^{eigenstate quasi-two body are often on-}

tributing to a three-body deay (

B ⁰ → f 0 (980)K _S ⁰

^{is one of the} ontributing amplitude to the

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{deay forinstane) andexperienefromprevious}experimentshasshown that full deay-time-dependent Dalitz plot analysis of a three-body deay is more sensitive

than aquasi-two-body approah,inpartiularinthe ase wherebroad resonanesare on-

tributingto the deay amplitude [68℄. On a similar note,the Dalitzplot analysis of these

deays are neessary inputs inmethods todetermine CKM phase

γ

^[913℄.

Theinlusivedeay

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{providesarihstrutureof}interferingamplitudes,in- volvingboth

CP

^{eigenstateamplitudes (}

B ⁰ → ρ ⁰ K _S ⁰

^,

B ⁰ → f 0 (980)K _S ⁰

^{,et.)andavour spe-}

i amplitudes (

B ⁰ → K ^∗+ (892)π ⁻

^,

B ⁰ → K ₀ ^∗+ (1430)π ⁻

^{et.). Full}deay-time-dependent Dalitz plot analyses of

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{have been performed by BaBar and Belle experi-}

ments [14,15℄. These amplitude analyses rely on model-dependent parameterization of the

deay amplitudes. Similarstudiesofthe deay

B ⁰ → K _S ⁰ π ⁺ π ⁻

reonstrutedwith theLHCb spetrometer are the ultimate goals of the analysis presented in this thesis. However, the

statistis of reonstruted deays in the light of the modest avour tagging w.r.t. the

B

^-

fatories experimentsmakethat attemptnot ompetitivewith the LHCRun I data set. On

the ontrary, the seletion of the reonstruted

B ⁰ → K _S ⁰ π ⁺ π ⁻

^{that we designed with the}

mer experiments. A time-integrated untagged analysis will hene provide a novel view of

the hadroni amplitudes model. On top of this,the study of avour spei quasi two-body

deaysbenetsaswellfromtheleanlinessofthesignaleventsseletion, allowinginpriniple

a ompetitive determinationof diret

CP

^-violating asymmetries.

Charmless deays of

B

^{mesons in the Standard Model}

In this hapter, we desribe the sienti ontext of this thesis work. We start with a brief

reviewofthe StandardModel(SM)desribingthe interationsbetween elementarypartiles.

Thereafter wedisuss the symmetriesin partilephysisto introduethe formalismused to

desribetheviolationof

CP

^{symmetryintheSMframework. Finally,wepresentthe physis}

interest of the three-bodyharmlesshadroni deays.

1.1

CP

^{violation in the SM}

The SM is a theory that desribes all the known phenomena at the subatomi sale. It

embodieseletromagneti, strongand weakinterations. The prinipleofloalgauge invari-

ane, whih keeps the Lagrangian of the theory invariant under loal transformation,plays

a ruial role in the onstrution of the SM. There are two soures of

CP

^{violation in the}

SM and we willexamine inthis Chapter the one provided by the weak interation.

1.1.1 Introdution to Standard Model

The SM is a renormalizable quantum eld theory onstruted under the priniple of loal

gaugeinvarianeunderthe

SU (3) c ⊗ SU(2) L ⊗ U(1) Y

^{symmetrygroup}transformations. These loal gauge invarianes generate strong, weak and eletromagneti interations between the

elementaryfermions,through the exhangeof gaugebosons: eight gluons,masslessandele-

trially neutral, for strong interation, one massless photon for eletromagneti interation

and three massive bosons, harged

W ^±

^{and neutral}

Z

^{for weak} interations. The strong interations are governed by the group

SU (3) C

^{(the subsript}

C

^{stands here for the olour,}

hargeoftheinteration),whereasthegroups

SU (2) L

^and

U (1) Y

^{giveaunieddesriptionof}

eletroweak interations.

SU (2) _L

^isanon-abeliangroupwiththe weak isospinastheharge of the interation and ats only on left-handed fermions.

U(1) Y

^{is the weak hyperharge}

group, dened by

Y

2 = I 3 + Q

^{, where}

I 3

^{is the third weak isospin omponent and}

Q

^{is the}

eletriharge.

The masses of both the fermions and mediating bosons are vanishing to preserve the

invariane under

SU(2) L ⊗ U (1) Y

^{. However, the} introdutionof adoubletof omplexsalar

elds of

SU (2) L

^breaks spontaneously the symmetry. Three degrees of freedoms an be used to provide masses to the

W ^±

^and

Z

^{bosons, while keeping the photon massless. The}

remaining degree of freedom is the Brout-Englert-Higgs fundamental salar [1618℄. The

disovery of a narrow bosoni state by the ATLAS and CMS experiments (CERN), so far

experimentallyonsistentwithboththeBEHbosonhypothesisandtheeletroweakpreision

observables [19℄, signs a tremendous suess of the SM to adequately desribe the Nature

up to an energy sale

O (100)

^{GeV. The Yukawa ouplings of the BEH boson with elemen-}

tary fermions are proportional to a mass and an be used to desribe the fermion masses

aordingly. Nothing in the symmetries is xing there values though. They are hene free

parameters of the theory.

The quarks andleptons are dividedinto threegenerations, eah of thembeing adoublet

of

SU (2) L

^{. Therst generationofquarksonsistsoftheup- and}down-quarks,the seondof theharm-and strange-quarks,andthe thirdgenerationofthe top-andbeauty-quarks. The

leptons and their assoiated lepton-neutrinos are divided into the eletron, muon and tau

generations. In addition,eahpartilehas anassoiated anti-partilewith opposite internal

quantum numbers. Anillustrationof the SMmatter ontents is given inTable 1.1.

Table 1.1: The three lepton andquark generations. The indies

L

^and

R

^{note the partile hirality}

state, left andright, respetively.

Generation Leptons Quarks

I

ν e

e

L

,

e R

u d

L

,

u R

^,

d R

II

ν µ

µ

L

,

µ R

c s

L

,

c R

^,

s R

III

µ _τ τ

L

,

τ _R t

b

L

,

t _R

^,

b _R

1.1.2 CKM mixing matrix

TheloalgaugeinvarianeintheSMforbidsfermionsandbosonstobemassive. Thefermion

masses are introdued after the spontaneous eletroweak symmetry breaking, via Yukawa

oupling of fermions, with left and right hirality, to Higgs eld, whih the Lagrangian

density isgiven by

L Y = − λ ^d _ij Q ¯ ^I _Li ³ φD ^I _Rj ³ − λ ^u _ij Q ¯ ^I _Li ³ φ ^∗ U _Rj ^{I 3} + h.c,

^(1.1)

ˆ

i

^and

j

^{are for the generation indies,}

ˆ

Q ^I _L ³ , D ^I _R ³ , U _R ^{I 3}

^{arethemultipletsof}

SU(2) L ⊗ SU (3) c ⊗ U (1) Y

^.

Q ^I _L ³ = (U, D) ^I _L ³

^aretheleft

hiralitydoubletsand

U _R ^{I 3} , D ^I _R ³

^{theouplesofrighthiralitysingletsinweakinteration}

eigenstates basis.

ˆ

φ

^{is the Higgseld.}

1.1

CP

^{violation in the SM 5}

ˆ

λ ^d,u _ij

^{are the omplexmatries}

3 × 3

^{of the quark-down and -up oupling,} respetively. When the Higgs eld aquires a value in the vauum (v.e.v.)

v = h 0 | φ | 0 i

^{, the fermion}

mass terms appear

− λ ^d _ij .v

√ 2 . D ¯ ^I _Li ³ D _Rj ^{I 3} − λ ^u _ij .v

√ 2 . U ¯ _Li ^{I 3} U _Rj ^{I 3} + h.c.

^(1.2)

Itisworthwhiletomovefromthebasisoftheweakinterationeigenstatestomasseigenstates,

wherein the oupling matries willbe diagonal of real values. This transformation is made

using unitary matries

U L ^u(d)

^and

U R ^u(d)

U L ^u(d)

λ ^u _ij .v

√ 2 U R ^u(d) =





m u(d) 0 0 0 m c(s) 0 0 0 m _t(b)



 .

The diagonalizationuses separate transformations for quarks of type up and down for the

same weak doublet, therefore it is ustomary to redene the transformations so that they

onlyapply totype down quark

Q ^I _L ³ = U _L ^{I 3}

D _L ^{I 3}

= ( U L ^u† ) j

U Lj

( U L ^u U L ^d† ) _jk D _Lk

,

where the so-alled Cabbibo,Kobayashi and Maskawa (CKM)matrix appears

V CKM = U L ^u U L ^d† =





V ud V us V ub

V cd V cs V cb

V _td V _ts V _tb



 .

Thus, the urrents responsible for weak interation are transformed under the inuene of

the hange of weak eigenstates basis to the mass eigenstates by making expliitly appear

theCKM matrixelements. TheorrespondingLagrangiandensityinvariantunderthe

SU(2)

transformationsis given by

L ^W = i g 1

2 Q ¯ ^I _Li ³ γ ^µ (~τ. ~ W ) µ Q ^I3 _Li ,

^(1.3)

where

g 1

^{is the weak oupling onstant,}

~τ

^{are the Pauli matries, generators of the}

SU(2)

group and

W ~

^{the three additional vetors eld brought by the} requirement of loal gauge invariane. This density beomesin the mass eigenstates basis

L ^W = i g 1

√ 2 ( ¯ U Li γ ^µ U ik ^u U kj ^d† D Lj W µ + + ¯ D Li γ ^µ U ik ^d U kj ^u† U Lj W µ − ) + ig 1

2 Q ¯ Li γ ^µ τ ³ W µ 3 Q Li .

^(1.4)

Itshouldbenotedthattheinterationsthroughneutralurrents(thethirdterminEq.(1.4))

arenotmodied. Thereisatuallynotree-levelproessofavorhangingby neutralurrent

inthe Standard Model(FCNC).

1.1.3 CKM parameterisations and representations

The CKM matrix is a

3 × 3

^{omplex unitary matrix and an as suh be} parameterised by onlyfour parameters: three mixingangles(rotation angles)and one phase

δ

V CKM = R ²³ (θ 23 , 0) ⊗ R ¹³ (θ 13 , δ 13 ) ⊗ R ¹² (θ 12 , 0) .

^(1.5)

Among the many possible onventions, a standard hoie, adopted by the Partile Data

Group [20℄ reads as

V CKM =





c 12 c 13 s 12 c 13 s 13 e ^{−iδ 13}

− s 12 c 23 − c 12 s 23 s 13 e ^{iδ 13} c 12 c 23 − s 12 s 13 s 23 e ^{iδ 13} s 23 c 13

s 12 s 23 − c 12 c 23 s 13 e ^{iδ 13} − c 12 s 23 − s 12 c 23 s 13 e ^{iδ 13} c 23 c 13





where

c ij = cos θ ij

^and

s ij = sin θ ij

^{, with}

i, j = 1, 2, 3

^.

There is an alternative popular parameterisation whih has been rst introdued by

Altomari and Wolfenstein [21,22 ℄. It is inspired by the experimentally observed hierarhy

between the matrix element magnitudes

s 13 ≪ s 23 ≪ s 12 ≪ 1

^{. The four} independent parameters are noted

λ

^{(whih is the sine of Cabibbo angle,}

λ = 0.22537 ± 0.00061

^[20℄),

A

^,

ρ

^and

η

^{and the} parameterisation onsists of developing the CKM matrix in order of

λ

poweraording to

s 12 = λ, s 23 = Aλ ² , s 13 e ^−iδ = Aλ ³ (ρ − iη) .

^(1.6)

This denition ensures the matrix unitarity at allorders. For example,at order

O (λ ⁴ )

^{, the}

CKM matrix reads

V CKM =





1 − λ ² /2 − 1/8λ ⁴ λ Aλ ³ (ρ − iη)

− λ 1 − λ ² /2 − 1/8λ ⁴ (1 + 4A ² ) Aλ ² Aλ ³ (1 − ρ − iη) − Aλ ² + Aλ ⁴ (1 − 2(ρ + iη))/2 1 − A ² λ ⁴ /2



 + O (λ ⁵ ).

The unitarity of the CKM matrix implies various relations between its elements. In

partiular, the relationsinvolvingthe

b

^{quark are}

V ud V _ub ^∗

V cd V _cb ^∗ + V cd V _cb ^∗

V cd V _cb ^∗ + V td V _tb ^∗

V cd V _cb ^∗ = 0 ,

^(1.7)

V td V _ud ^∗

V cd V _cb ^∗ + V ts V _us ^∗

V cd V _cb ^∗ + V tb V _ub ^∗

V cd V _cb ^∗ = 0 .

^(1.8)

Aonvenientwayofrepresentingtheunitarityrelationsistodisplaythemintheomplex

plane, hene as a triangle.Fig. 1.1 proposes suh a representation of the unitarity triangle

for

b

^-quark transitions. The triangleis dened by the angles

α

^,

β

^and

γ α = arg

− V td V _tb ^∗ V ud V _ub ^∗

, β = π − arg

V td V _tb ^∗ V cd V _cb ^∗

, γ = arg

− V ud V _ub ^∗ V cd V _cb ^∗

.

The apexof the triangleis dened by its oordinates

ρ ¯ + i¯ η = − _V

ud V _ub ^∗ V cd V _cb ^∗

, where

1.1

CP

^{violation in the SM 7}

Figure1.1: The unitaritytrianglewithsides of the same

λ

^{order with}

α

^,

β

^and

γ

^{angles assoiated.}

The real axis of the omplex plane is dened by

ℑ (V _cd V _cb ^∗ ) = 0

^{and the side lengths are normalized}

w.r.t.

| V _cd V _cb ^∗ |

^.

¯

ρ + i η ¯ =

√ 1 − λ ² (ρ + iλ)

√ 1 − A ² λ ² + A ² λ ⁴ √

1 − λ ² (ρ + iλ) .

^(1.9)

Any non-vanishing value of

η ¯

^{is synonymous of}

CP

^violation.

1.1.4

CP

^Symmetry

Inquantummehanis,the

CP

transformationombineshargeonjugation

C

^withparity

P

transformations. The parity operator,

P

^{, inverts the algebraisign of all spae oordinates}

usedinthedesriptionofaphysialproess. Asexample,if theparityoperatorisperformed

on a salar wavefuntion

ψ(x, y, z, t)

^{, the latter will transform it to}

ψ( − x, − y, − z, t)

^{. The}

parityonservationor

P

^{-symmetryimpliesthat anyphysialproess willproeedidentially}

when is transformed under parity operator. Before 1956, the general feeling was that all

physial proess would onserve parity. However, a number of experiments were performed

(

e.g.

^{Wu experiment [23℄) and showed that, for proesses involving weak} interation, the

P

^{-symmetry violated.}

Regarding the harge onjugation operator, this transformation hanges the sign of all

intrinsiadditivequantum numbers,astheeletriharge,thebaryonquantumnumber,the

lepton quantum number, the strangeness, et. The

C

^{-symmetry,as the}

P

^{-symmetry, means}

thesymmetry ofphysiallawsunderthe hargeonjugation transformation. Thissymmetry

is onserved by eletromagnetism, gravity and strong interation, but violated in the weak

interations [24℄.

Thus, ombiningthetwooperators

P

^and

C

^,the

CP

^{operatorwilltransform,forinstane,}

a left-handed eletron

e ⁻ _L

^{into a} right-handed positron

e ⁺ _R

^{1. Therefore, if}

CP

^{were an exat}

symmetry, the laws of Nature would be the same for matter and antimatter. The violation

of this symmetryissubtle and has been diulttoexplore. However, Croninand Fith[25℄

performeda beam experimentin1964 inwhih they measuredthe deayof neutralkaons in

1

Inthesamespaeoordinates,

P

^{operatorinvertstheheliity.}

W W d

¯ b

b

d ¯ t, c, u

¯ t, ¯ c, u ¯

B ⁰ B ¯ ⁰

Figure1.2: One of the two box diagrams desribing the

B ⁰

^-

B ⁰

^{mixing in the SM.}

two pions atthe end of long beamline. This experiment showed that there was a small

CP

violation,within weak interation, inthe neutral kaon mixing.

Toillustratethemanifestationof

CP

^{violationwithweakinterationintheSM,let'sapply}

theoperator

CP

^{tothersttermoftheLagrangiandensityshowninEq.(1.4)(}

L ⁽¹⁾ W

−→ L CP ^(1)′ W

⁾

L ⁽¹⁾ W = i g 1

√ 2 ( ¯ U Li γ ^µ U ik ^u U kj ^d† D Lj W µ + ) ,

^(1.10)

L ^(1)′ W = i g 1

√ 2 ( ¯ D Li γ ^µ U ik ^d U kj ^u† U Lj W µ − ) .

^(1.11)

Therefore if the matrix element

U ik ^d U kj ^u†

^{is omplex wewillhave}

L ⁽¹⁾ W 6 = L ^(1)′ W

^{, whihimplies a}

CP

^{violation. Then the}

δ

^{phase introdued in the CKM matrix is a soure of}

CP

^violation

inthe weak interation.

1.1.5

CP

^{violation in neutral}

B

^setor

Despitealargenumberof attemptstoobserve

CP

^{violationphenomena,ittookalmostforty}

years to reah a seond observation of it. Before addressing the

CP

^{violation in neutral}

B

^{mesons, a brief overview is given in the following subsetion disussing the quantum}

mehanisof neutral

B

^mesons.

1.1.5.1 The quantum mehanis of neutral

B

^{meson mixing}

The neutral

B

^{mesons are} pseudo-salarmesons whihan have twoavor states,

B ⁰

^made

of

d

^-quarkand

¯ b

^-quark,and

B _s ⁰

^{made of}

s

^-quarkand

¯ b

^{-quark. They aneahmixwith their}

respetive antipartile, as illustrated by the Feynman diagram (for

B ⁰

^-

B ⁰

^{mixing) given}

inFig. 1.2 (inthe followingonly

B ⁰

^{meson is}onsidered).

The

B ⁰

^and

B ⁰

^{mesons are dubbed the avour} eigenstates, whilst the eigenstates of the propagation Hamiltonian are dubbed the mass eigenstates, denoted by

B _H

^and

B _L

^{. Thus,}

the neutral

B

^{mesons an be desribed in term of two physial states ombination of the}

avoreigenstates

1.1

CP

^{violation in the SM 9}

| B _L i = p | B ⁰ i + q | B ¯ ⁰ i ,

| B _H i = p | B ⁰ i − q | B ¯ ⁰ i ,

^(1.12)

where

p

^and

q

^{are the linear omplex oeients satisfying the relation}

| p | ² + | q | ² = 1

^.

The states

| B L i

^and

| B H i

^{are the lighter and heavier mass} eigenstates, respetively. The time-dependent Shroedingerequation for these states reads

i ∂

∂t p

q

= H ^eff p

q

,

^(1.13)

where

H ^eff

^{isthe eetive} Hamiltoniandesribing the neutralmesons mixingasfollows

H ^eff = M − i Γ 2 =

M 11 M 12

M ₂₁ M ₂₂

− i 2

Γ 11 Γ 12

Γ ₂₁ Γ ₂₂

,

=

ω L 0 0 ω H

.

^(1.14)

M

and

Γ

are

2 × 2

^{Hermitian matries desribing the mass and deay rate omponent of}

H ^eff

^, respetively. We take note that the

H ^eff

^{matrix is on the ontrary not hermitian. In}

the mass eigenstates

{| B L i , | B H i}

^basis,

H ^eff

^{is diagonal with omplex} eigenvalues,

ω L

^and

ω H

^{, expressed as}

ω _L = m _L − i Γ _L

2 , ω _H = m _H − i Γ _H

2 ,

^(1.15)

where

m L

^and

m H

^{are the masses of the} eigenstates

| B L i

^and

| B H i

^, respetively, and

Γ L

and

Γ H

^{their deay rate} ounterpart. The 2-partile system

{ B ⁰ , B ¯ ⁰ }

^is haraterized by 5 physial observables (named also mixingobservables): the mass and deay rate averages,

the dierenes in mass and deay rate, and its "ompositionfration"

| q/p |

^{. The mass and}

deay rate averages are

m = m H + m L

2 , Γ = Γ H + Γ L

2 .

^(1.16)

The dierenes inmass and deay rate are given by

∆m = m H − m L , ∆Γ = Γ H − Γ L .

^(1.17)

∆m

^{is always positive in this denition, the sign of}

∆Γ

^{depends on whih mass eigenstate}

has the longer lifetime. The sign of

∆Γ

^{is predited,by the SM,tobe negative, but has not}

yet been established, while is well established in

B _s ⁰

^-

B ⁰ _s

^{mixing (}

∆Γ s = (0.091 ± 0.008) × 10 ¹² s

^{[20℄). Thevaluesfound fortheworldaverageof themassdierene}measurements[20℄, are

∆m B ⁰ = (3.337 ± 0.033) × 10 ⁻¹⁰ MeV

^and

∆m B _s ⁰ = (1.1691 ± 0.00014) × 10 ⁻⁸ MeV

^{. As}

mentioned above, the deay rate dierene has on the ontrary not yet been observed and

we onsider itnegligiblein the following study.

The parameters

p

^and

q

^{are relatedto the o-diagonalelements of}

H ^eff

^along

q p

2

= M ₁₂ ^∗ − ₂ ⁱ Γ ^∗ ₁₂ M 12 − ₂ ⁱ Γ 12

,

^(1.18)

If

CP

^{were asymmetry of}

H ^eff

^{, then}

Γ 12 /M 12

^{would bereal, leading to}

q p

2

= e ^{2iθ(B 0 )} ⇒ q p

= 1 ,

^(1.19)

where

θ(B ⁰ )

^{is an arbitrary phase ourring inthe ation of}

CP

^{operator onthe state}

| B ⁰ i

(

| B ⁰ i

^{)whihtransforms it to}

| B ⁰ i

⁽

| B ⁰ i

⁾

CP | B ⁰ i = e ^{2iθ(B 0 )} | B ⁰ i , CP | B ⁰ i = e ^{−2iθ(B 0 )} | B ⁰ i .

^(1.20)

1.1.5.2 Time evolution of

B ⁰

⁽

B ⁰

^{) meson}

The time evolution of the states

| B ⁰ (t) i

^and

| B ⁰ (t) i

^{an be expressed in terms of initially}

pure avorstates

| B ⁰ (t = 0) i ≡ | B ⁰ i

^and

| B ⁰ (t = 0) i ≡ | B ⁰ i

| B ⁰ (t) i = g ₊ (t) | B ⁰ i − q

p g ₋ (t) | B ⁰ i ,

| B ⁰ (t) i = g + (t) | B ⁰ i − q

p g − (t) | B ⁰ i ,

^(1.21)

with

g ± (t) = 1 2

e ^{−im H t− 1 2 Γ H t} ± e ^{−im L t− 1 2 Γ L t}

.

^(1.22)

Wethen nd

| g _± (t) | ² = 1 4

h e ^{−Γ H t} − e ^{−Γ L t} ± 2 Re

e ^{− 1 2 (Γ H +Γ L )−i(m H −m L )t} i ,

= 1 2 e ^−Γt

cosh

∆Γt 2

± cos(∆mt)

.

^(1.23)

and

g ₊ ^∗ (t)g ₋ ^∗ (t) = 1 4

h e ^{−Γ H t} − e ^{−Γ L t} − 2iIm

e ^{− 1 2 (Γ H +Γ L )−i(m H −m L )t} i ,

= − 1 2 e ^−Γt

sinh

∆Γt 2

+ i sin(∆mt)

.

^(1.24)

The deay rate of a

| B ⁰ i

^{meson produed at time}

t = 0

^{to a nal state}

f

^{at time}

t

^is

given by

1.1

CP

^{violation in the SM 11}

dΓ _B ⁰ _{→f( ¯ f)} (t)

dt = |h f ( ¯ f) |T | B ⁰ (t) i| ² , dΓ _B 0 → f(f) ¯ (t)

dt = |h f ¯ (f) |T | B ⁰ (t) i| ² ,

^(1.25)

where

T

^{is the transitionmatrix.}

The time-dependent deay rates of the initially produed avor eigenstates

| B ⁰ i

^and

| B ⁰ i

^{, assuming}

∆Γ = 0

⁽

cosh ^∆Γt ₂

= 1

^,

sinh ^∆Γt ₂

= 0

^{), are given by the four possible}

deay equations

dΓ B ⁰ →f (t)

dt = e ^−Γt

2 | A f | ² (1 + | λ f | ² )[1 + C f cos(∆mt) − S f sin(∆mt)] ,

^(1.26)

dΓ _B 0 →f (t)

dt = e ^−Γt 2

q p

2

| A _f | ² (1 + | λ _f | ² )[1 − C _f cos(∆mt) + S _f sin(∆mt)] ,

^(1.27)

dΓ _B 0 → f ¯ (t)

dt = e ^−Γt

2 | A ¯ f ¯ | ² (1 + | λ ¯ f ¯ | ² )[1 + C f ¯ cos(∆mt) − S f ¯ sin(∆mt)] ,

^(1.28)

dΓ _B ⁰ _→ f ¯ (t)

dt = e ^−Γt 2

q p

2

| A ¯ f ¯ | ² (1 + | λ ¯ f ¯ | ² )[1 − C f ¯ cos(∆mt) + S f ¯ sin(∆mt)] ,

^(1.29)

where

A f = h f |T | B ⁰ i

^and

A ¯ f ¯ = h f ¯ |T | B ⁰ i

^{are the deay amplitudes for}

| B ⁰ i

^and

| B ⁰ i

deaying to the nal state

| f i

^and

| f ¯ i

^, respetively, and

λ _f

^and

λ ¯ f ¯

^{are dened as}

λ f = 1 λ ¯ f

= q p

A ¯ f

A f

, λ ¯ f ¯ = 1 λ f ¯

= q p

A f ¯

A ¯ f ¯

.

^(1.30)

Similarly,

A ¯ f = h f |T | B ⁰ i

^and

A f ¯ = h f ¯ |T | B ⁰ i

^{. Here,}

C f

^,

S f

^,

C f ¯

^and

S f ¯

^{are the}

CP

^violation

observables,disussed indetails inthe following setion. they an be dened as

C f = 1 − | λ f | ²

1 + | λ f | ² , S f = 2Im(λ f ) 1 + | λ f | ² , C f ¯ = 1 − | λ f ¯ | ²

1 + | λ f ¯ | ² , S f ¯ = 2Im(λ f ¯ )

1 + | λ f ¯ | ² .

^(1.31)

The evaluationof the

CP

^{violation parametersis performed by the omparison between}

the deay rates

Γ(B ⁰ → f)

^and

Γ(CP (B ⁰ → f ))

^,where

CP (B ⁰ → f)

^{isthe proess}

B ⁰ → f

transformed under

CP

^{operator. The denition of the}

CP

^{asymmetry isgiven by}

A ^CP = Γ CP(B ⁰ →f ) − Γ B ⁰ →f

Γ _{CP (B} ⁰ _→f) + Γ _B ⁰ _→f .

^(1.32)

A CP 6 = 0

^{is a sign of}

CP

^{violation. In general, the} observation of

CP

^{violation relies on}

notieabledierenes amongproessesand theirorresponding

CP

-onjugates. The observa- tionof

CP

^{isrelatedtothe}interferene betweendierentamplitudesthatontributetothese proesses,manifestedbythe omplexphaseintheouplingthatbreaks

CP

^{invariane,more}

Figure 1.3: Diagrams showing the three type of

CP

^{violation: (A)}

CP

^{violation in deay, (B)}

CP

violation in mixing and(C)

CP

^{violation between deays withand without mixing.}

detailsare giveninSetion1.1.5.3. The possiblemanifestation of

CP

^{violationan belassi-}

ed in three ategories: (A)

CP

^{violation indeay, (B)}

CP

^{violationin mixingand (C)}

CP

violation between deays with and without mixing(Mixing-indued

CP

violation). Fig. 1.3 illustrates eah manifestation type of

CP

^{violation. In eah ase there is a} orresponding observable of

CP

^{violation. All}

CP

^violation observables in the proesses of

B ⁰

^/

B ⁰

^deay-

ingto the nal state

f( ¯ f )

^/

f ¯ (f)

^{an be expressed in terms of} phase-onvention-independent ombinationof

A f

^,

A ¯ f

^,

A f ¯

^and

A ¯ f ¯

^with

q/p

^.

1.1.5.3

CP

^{violation in deay}

This type of

CP

^{violationis adiret}

CP

^{violation,whihrequires a}avour-tagging informa- tion on the initialstate in the neutral

B

^deays,

i.e.

^{a distintion between the deays of}

B ⁰

and

B ⁰

^{to a nalstate}

f

^and

f ¯

^,respetively, where

CP | f i = e ^2iθ(f) | f ¯ i .

θ(f )

^{here is is an arbitrary phase. The} manifestation of

CP

^{violation in this ase ours if}

Γ(B ⁰ → f )

^{isdierentfrom}

Γ(B ⁰ → f ¯ )

^{. The terms}

λ _f

^and

λ ¯ f ¯

^{inequations(1.26)and (1.28)}

are zero. Thus the proess rate willbeproportionalto the total amplitude square. The

CP

asymmetry an be writtenas

A ^CP = | A ¯ f ¯ | ² − | A f | ²

| A ¯ f ¯ | ² + | A f | ² ,

^(1.33)

1.1

CP

^{violation in the SM 13}

hene, the

CP

^{violation in deay ourswhen}

| A ¯ f ¯ |

| A f | 6 = 1 = ⇒ CP violation.

^(1.34)

If several amplitudes

j

^{ontribute to the deay}

B ⁰ (B ⁰ ) → f ( ¯ f )

^{, the total amplitude}

A f

and its

CP

^{onjugate amplitude}

A ¯ f ¯

^{an be dened in term of a real magnitude}

a j

^{, weak}

phase

φ _j

^{and strong phase}

δ _j

^:

A f = X

j

a j e ^{i(δ j +φ j )} , A ¯ f ¯ = X

j

a _j e ^{i(δ j −φ j )} .

^(1.35)

The

CP

^{asymmetry beomes}

A ^CP = 2 P

jk a j a k sin(δ j − δ k ) sin(φ j − φ k ) P

jk a ² _j + a ² _k + 2a _j a _k cos(δ _j − δ _k ) cos(φ _j − φ _k ) ,

^(1.36)

From equation (1.36) it an been seen that

A ^CP

^{will have a non-zero value if the weak}

phases, as well as the strong phases, from the proesses that ontributes to the nal state

are dierent. The interferene is a key requirement for the manifestation of

CP

^violation,

whih the amplitude

A f

^{should have at least two} ontributing omplex amplitudes with dierent weak and strongphase, the reason forthat omes fromthe fat that

CP

^-onjugate

amplitude dier from the originalamplitudes at most by a phase fator. The

CP

^violation

in deay is most thoroughly studied in

b

^{-hadron deays to harmless two body nal states.}

An appropriate exampleis

B ⁰ → K ⁺ π ⁻

^[26℄.

1.1.5.4

CP

^{violation in mixing}

The

CP

^{violation in mixing is an indiret}

CP

^{violation, whih implies that the osillation}

from

B ⁰

^to

B ⁰

^{is dierent from the osillation}

B ⁰

^to

B ⁰

Γ(B ⁰ → B ⁰ ) 6 = Γ(B ⁰ → B ⁰ ) = ⇒ CP violation in mixing.

^(1.37)

The

CP

^{asymmetry an be writtenas}

A ^cp = Γ _B 0 →B ⁰ − Γ _B 0 →B ⁰

Γ _B 0 →B ⁰ + Γ _B 0 →B ⁰

,

=

h B ⁰ |H ^eff | B ⁰ i −

h B ⁰ |H ^eff | B ⁰ i

h B ⁰ |H ^eff | B ⁰ i +

h B ⁰ |H ^eff | B ⁰ i

.

^(1.38)

Tohekthatthedierenebetween

h B ⁰ |H ^eff | B ⁰ i

^and

h B ⁰ |H ^eff | B ⁰ i

^{isasign ofmixing}

CP

^{violation,we apply the}

CP

^{operator onthe twoterms}

h B ⁰ |H ^eff | B ⁰ i −−→ h ^CP B ⁰ | (CP ) ^† (CP ) H ^eff (CP ) ^† (CP ) | B ⁰ i

= h B ⁰ | (CP ) ^† H ^CP eff (CP ) | B ⁰ i

= e ^{−4iθ(B 0 )} h B ⁰ |H ^CP eff | B ⁰ i ,

^(1.39)

h B ⁰ |H ^eff | B ⁰ i = e ^{4iθ(B 0 )} h B ⁰ |H ^CP eff | B ⁰ i ,

^(1.40)

where

H ^CP eff = (CP ) H ^eff (CP ) ^†

^and

θ(B ⁰ )

^{is the arbitrary unphysial phase introdued in}

Eq. (1.20). So, if

CP

^{is a symmetry of}

H ^eff

^then

[ H ^eff , CP ] = 0

^{, whih implies}

H ^eff = H ^CP eff = ⇒

h B ⁰ |H ^eff | B ⁰ i =

h B ⁰ |H ^eff | B ⁰ i

.

^(1.41)

If the terms of Eq. (1.38) are desribed in the mass eigenstates basis {

| B L i

^,

| B H i

^{}, the}

CP

^{asymmetrybeomes}

A ^CP =

p q

−

q p

p q

+

q p

.

^(1.42)

Therefore,

CP

^{violation inmixingours if}

p q

6 = 1 = ⇒ CP violation in mixing.

^(1.43)

as wasintroduedearlier inthis Chapter.

The

CP

^{violation in mixingwas observed} experimentally in the neutral kaon system in 1964 [25℄.

CP

^{violation in the}

B ⁰

^-

B ⁰

^or

B _s ⁰

^-

B ⁰ _s

^{mixings is expeted to be negligible in the}

SM[2729℄. Ithas not been observed sofar. Inthe following, wewillassumethat

| q/p | = 1

^,

unless otherwise stated.

1.1.5.5 Mixing-indued

CP

^violation

CP

^{violation in the} interferene between deays with and without mixing ours for the deays of

B ⁰

^and

B ⁰

^{to anal state}

f

^{whih is a}

CP

-eigenstate

B ⁰ ( → B ⁰ ) → f ← (B ⁰ ← )B ⁰ , CP | f i = η CP | f i ,

where

η CP

^isa

CP

-eigenvalueequalto

1

^or

− 1

^{. Inthefollowing,thenalstate}

CP

-eigenstate will benoted as

f _CP

^.

This type of

CP

^{violation omes from the} interferene of mixing and deay amplitudes

A(B ⁰ → B ⁰ → f CP )

^and

A(B ⁰ → f CP )

^,respetively.

The time-dependentmixing-indued

CP

^{asymmetry reads}

A CP (t) =

dΓ _B 0 →f (t)

dt − ^{dΓ B 0} _dt ^{→f (t)}

dΓ _B 0 →f (t)

dt + ^{dΓ B 0} _dt ^{→f (t)}

.

^(1.44)

using Eq.(1.26) and (1.27) (orEq.(1.28) and (1.29)),the

CP

^{asymmetry reads}

A ^CP (t) = S sin(∆mt) − C cos(∆mt) ,

^(1.45)