Dalitz analysis of the three-body charmless decay B0 → K0Sπ+π− with the LHCb spectrometer

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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

EDSF :853

UNIVERSITE BLAISE PASCAL

(U.F.R. S ien es etTe hnologies)

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 14De ember 2015 devant la ommission d'examen:

Président : M. S. Des otes-Genon

Rapporteurs : Mme M.H. S hune

M T. Latham

Examinateurs : Mme H. Fonvieille

M O. Des hamps

Studiesof harmlessthree-bodyde aysof the neutral

B

mesonswith a

K

0

S

inthe nalstate

are presented in this thesis. The analyses are performed with the full statisti s re orded

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

de ay

B

0

_{→ K}

0

S

π

+

_π

−

represents the main part of this thesis analysis. A time-integrated

untagged Dalitz-Plot analysis of the de ay is performed. The t fra tions of the

quasi-two-bodyde aysareobtained. Likewise,thedire t

CP

asymmetriesofthequasi-two-bodyde ays

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

−

₎

. Thismeasurement

is the rst observation of the

CP

asymmetry with a signi an e larger then ve standard

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

pre ision.

Keywords

LHC - CERN - LHCb dete tor - Standard Model - Parti le Physi s - Heavy

Flavour Physi s - CKM Triangle -

CP

Violation -

B

Physi s - Dire t

CP

Asymmetry - Bran hing Ratio - Fit Fra tion - Charmless de ay - Dalitz-Plot

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

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

K

0

S

. Ce travail de

re her he s'est réalisé dans le adre de l'éxpérien e LHCb au LHC, en analysant un

é han-tillon d'événements de 3 fb

−1

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

la désintégration

B

0

_{→ K}

0

S

π

+

_π

−

représente la partie prin ipale de e travail de thèse. La

mesure des amplitudes est ee tuée au moyen d'une étude du plan de Dalitz de la

désinté-grationintégréedans letemps sansétiquetage de lasaveur de laparti ulebelle. Nousavons

mesurélesrapportsd'embran hementsrelativesdesdésintégrationsquasi-deux- orpsàpartir

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

CP

dire tedes

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

dire te dans la

désintégration

B

0

_{→ K}

∗+

_(892)π

−

ave unesignian esupérieureà inqdéviationsstandard.

Cette mesureest en a ord ave la moyenne mondiale,ave une pré isionaméliorée.

Mots Clés

LHC - CERN - Dete teur LHCb - Physique des Parti ules - Modèle Standard

- Physique des Saveurs Lourdes - Triangle CKM - Violation

CP

- Physique des

mesons

B

- Asymétrie

CP

Dire te - Rapport d'embran hement - Désintégration

sans quark harm - Dalitz-Plot -

B

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

la réda tion et... nous y voilà !

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

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

BlaisePas al. Je ne pourrais jamaisle remer ierassez pourtout e qu'ilm'a donné. Jesuis

trèsre onnaissantpourletemps onséquentqu'ilm'aa ordéetpourl'aide ompétentequ'il

m'a apportée, pour sa patien e etson en ouragement. J'ai beau oup appris de ses qualités

pédagogiques et s ientiques, sa fran hise et sa sympathie ainsi de ses qualités humaine

ex eptionnels.

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

Des otes-Genonquiabien vouluprésider lejury. Mer i àmesdeux rapporteursM.Thomas

Latham et Mme Marie-Hélène S hune pour leurs suggestions et orre tions, ainsi que pour

leurs onseilsetleursoutien. Jesouhaiteremer ieraussimesdeuxexaminateursMmeHélène

Fonvieille et M. Olivier Des hamps pour avoir parti ipé à la soutenan e et d'avoir apporté

un regard extérieur ritique à e travail.

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

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

Ziad Ajaltouni, M. Olivier Des hamps, M. Eri Cogneras etM. ValentinNiess. Je souhaite

remer ier aussi ledire teur de laboratoireLPC, M. AlainFalvard pour m'avoira ueilli au

sein de e laboratoire. Mes remer iements 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 s ientiques: en ore Mostafa Hoballah (thanks khayi), Xavier Lopez mon

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

remer ie toutes les personnes formidables que j'ai ren ontrées par le biais de LPC ou de

CERN. Mer i pour votre supportetvos en ouragements.

Je souhaite remer ier spé ialement M. Adel Trabelsi, professeur à l'université Tunis

El-Manar, pour son en ouragement, ses multiples onseils et son soutien, depuis mon master

quem'aapportéettout equem'adonné,mer ipoursessoutienssansfaillesdepuistoujours.

Mer i à ma mère, elle était toujours à oté de moi malgré la distan e et les engagements.

Tout e queje suis ouaspireàdevenir, 'est àmamèrequeje ledois. Mer ià matrès hère

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

bonté. Sa présen e et ses en ouragements sont pour moi les piliers fondateurs de e que je

Introdu tion 2

1 Charmless de ays of

B

mesons in the Standard Model 3

1.1

CP

violation inthe SM . . . 3

1.1.1 Introdu tion 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

se tor . . . 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

de ays . . . 16

1.3.1

β

angle and New Physi s . . . 17

2

B

0

_{→ K}

0

S

π

+

_π

−

and Dalitz Plot formalism 20 2.1 Three-body kinemati s: The Dalitz-plot . . . 20

2.2 Heli ity angle . . . 21

2.3 Three-body dynami s: Isobar Model . . . 22

2.3.1 Angular distributions . . . 23

2.4 Massterm des ription . . . 24

2.4.1 Relativisti Breit 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 Redu ed

K

-matrix . . . 29

2.5 DP probability density fun tion and

CP

Observables . . . 31

2.5.1 Probability density fun tion . . . 32

3 Large Hadron Colliderand the LHCb experiment 34

3.1 The Large Hadron Colliderat CERN . . . 34

3.1.1 The LHC a eleratorsystem . . . 34

3.1.2 Experimentsat the LHC . . . 36

3.1.3 Luminosity and

b¯b

quark pair produ tion . . . 36

3.2 The LHCb dete tor . . . 37

3.2.1 Generaloverview . . . 37

3.2.2 Tra king system . . . 39

3.2.3 Parti le identi ationdete tors . . . 47

3.2.4 Parti le identi ationte hniques . . . 51

3.2.5 Trigger system . . . 52

3.2.6 LHCb re onstru tion and data stripping . . . 55

3.2.7 LHCb software . . . 55

4 Study of

B

0

(s)

→ K

S

0

h

+

h

′−

de ays 57 4.1 Dataset, triggerand stripping . . . 57

4.1.1 Trigger . . . 58

4.1.2 Stripping . . . 59

4.2 Sele tion . . . 63

4.2.1 Presele tion . . . 63

4.2.2 Datasets for the MVA training. . . 65

4.2.3 Dis riminatingvariables . . . 66

4.2.4 Training and validationof the BDT . . . 73

4.2.5 Optimisationof the BDT uts . . . 74

4.2.6 Parti le Identi ation. . . 75

4.3 Ba kground studies . . . 84

4.4 Masst model . . . 85

4.4.1 Generalstrategy . . . 86

4.4.2 Signal model . . . 86

4.4.3 Models for rossfeed ba kground. . . 88

4.4.4 Partially-re onstru ted ba kgrounds. . . 88

4.4.5 Combinatorialba kground . . . 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 Fra tionof signal in the

B

0

mass windowfor the tight BDT. . . 98

5 Dalitz-plot analysis of

B

0

_{→ K}

0

S

π

+

_π

−

99 5.1 Amplitudeanalysis formalism . . . 99

5.1.1 DalitzSignal p.d.f. . . 99

5.1.2 Likelihoodfun tion . . . 100

5.1.3 Physi al observables fromDP t . . . 101

5.3 Signale ien y variation a ross the Dalitzplot . . . 110 5.3.1 Geometri al e ien y . . . 110 5.3.2 Sele tion e ien y . . . 111 5.3.3 PID e ien y . . . 114 5.3.4 Totale ien y . . . 115 5.4 Multiple solutions . . . 120

5.5 DalitzPlotFit . . . 121

5.5.1 Baseline modeland additionalresonan es. . . 121

5.5.2 Towards the nominalDP model . . . 125

5.6 DalitzPlot tresults . . . 134

5.6.1 Phase and t fra tionstatisti al un ertainties . . . 143

5.7 Fit validation . . . 154

5.7.1 Likelihoods ans. . . 154

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

5.8 Systemati s studies . . . 162

5.8.1 Experimentalsystemati un ertainties . . . 162

5.8.2 Model systemati un ertainties . . . 195

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

6.1.1 Isobar parameter and t fra tions measurements . . . 202

6.1.2 Dire t

CP

asymmetries measurements . . . 205

6.1.3 First observation of dire t

CP

asymmetry in

B

0

_{→ K}

∗

₍₈₉₂₎

+

_π

−

. . . 205

A Dalitz plot kinemati s 210 B Angular distribution in Dalitz plot 213 C Sele tion - extra plots 216 D Fit model - extra plots 240 E CRAFT tter 249 E.1 Numeri alintegration te hnique . . . 249

E.2 Generation te hnique of pseudo-experiments . . . 250

E.3 E ien y . . . 250

E.4 Fittingma hinery . . . 250

The

CP

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

enoughtodes ribeall

CP

-violatingobservables measuredsofarinparti lesystems[3℄. This

isthe onlysour e of

CP

violation inthe StandardModel(SM) whi h yieldsmeasurable

CP

-violating phenomena to date. The existen e of new sour es of

CP

violation in addition to

thatpredi tedby theCKMmatrixismadene essary toa ountforthebaryoni asymmetry

in the Universe [4℄ and hen e the sear h for it onstitutes an importantgoal of the urrent

resear hes inhigh energy physi s.

Oneappealingapproa htosear hfornewsour esof

CP

violation onsistsinstudyingthe

de ay-time distribution of neutral

B

meson de ays to

CP

-eigenstates hadroni nal states

mediated by a

b → s

loop amplitude (so- alled penguin amplitude). Many measurements

havebeen performedbythe BaBarand Belleexperimentsinthatrespe t, su has

B

0

de ays to

φK

0

S

or

η

′

_K

0

S

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

results [5℄ provide a onsistent pi ture with the SM predi tions, demanding an improved

pre ision to in rease the sensitivity tonew

CP

-violating phases.

The de ays mentioned above into a nal

CP

eigenstate quasi-two body are often

on-tributing to a three-body de ay (

B

0

_{→ f}

0

(980)K

S

0

is one of the ontributing amplitude to

the

B

0

_{→ K}

0

S

π

+

_π

−

de ay forinstan e) andexperien efrompreviousexperimentshasshown

that full de ay-time-dependent Dalitz plot analysis of a three-body de ay is more sensitive

than aquasi-two-body approa h,inparti ularinthe ase wherebroad resonan esare

on-tributingto the de ay amplitude [68℄. On a similar note,the Dalitzplot analysis of these

de ays are ne essary inputs inmethods todetermine CKM phase

γ

[913℄.

Thein lusivede ay

B

0

_{→ K}

0

S

π

+

_π

−

providesari hstru tureofinterferingamplitudes,

in-volvingboth

CP

eigenstateamplitudes (

B

0

_{→ ρ}

0

_K

0

S

,

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 .). Fullde ay-time-dependent

Dalitz plot analyses of

B

0

_{→ K}

0

S

π

+

_π

−

have been performed by BaBar and Belle

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

de ay amplitudes. Similarstudiesofthe de ay

B

0

_{→ K}

0

S

π

+

_π

−

re onstru tedwith theLHCb

spe trometer are the ultimate goals of the analysis presented in this thesis. However, the

statisti s of re onstru ted de ays in the light of the modest avour tagging w.r.t. the

B

-fa tories experimentsmakethat attemptnot ompetitivewith the LHCRun I data set. On

the ontrary, the sele tion of the re onstru ted

B

0

_{→ K}

0

S

π

+

_π

−

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

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

de aysbenetsaswellfromthe leanlinessofthesignaleventssele tion, allowinginprin iple

Charmless de ays of

B

mesons in the Standard Model

In this hapter, we des ribe the s ienti ontext of this thesis work. We start with a brief

reviewofthe StandardModel(SM)des ribingthe intera tionsbetween elementaryparti les.

Thereafter wedis uss the symmetriesin parti lephysi sto introdu ethe formalismused to

des ribetheviolationof

CP

symmetryintheSMframework. Finally,wepresentthe physi s

interest of the three-body harmlesshadroni de ays.

1.1

CP

violation in the SM

The SM is a theory that des ribes all the known phenomena at the subatomi s ale. It

embodiesele tromagneti , strongand weakintera tions. The prin ipleoflo algauge

invari-an e, whi h keeps the Lagrangian of the theory invariant under lo al transformation,plays

a ru ial role in the onstru tion of the SM. There are two sour es of

CP

violation in the

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

1.1.1 Introdu tion to Standard Model

The SM is a renormalizable quantum eld theory onstru ted under the prin iple of lo al

gaugeinvarian eunderthe

SU(3)

c

⊗SU(2)

L

⊗U(1)

Y

symmetrygrouptransformations. These

lo al gauge invarian es generate strong, weak and ele tromagneti intera tions between the

elementaryfermions,through the ex hangeof gaugebosons: eight gluons,masslessand

ele -tri ally neutral, for strong intera tion, one massless photon for ele tromagneti intera tion

and three massive bosons, harged

W

±

and neutral

Z

for weak intera tions. The strong

intera tions are governed by the group

SU(3)

C

(the subs ript

C

stands here for the olour,

hargeoftheintera tion),whereasthegroups

SU(2)

L

and

U(1)

Y

giveaunieddes riptionof

ele troweak intera tions.

SU(2)

L

isanon-abeliangroupwiththe weak isospinasthe harge

of the intera tion and a ts only on left-handed fermions.

U(1)

Y

is the weak hyper harge

group, dened by

Y

2

= I

3

+ Q

, where

I

3

is the third weak isospin omponent and

Q

is the

ele tri harge.

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

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 s alar [1618℄. The

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

experimentally onsistentwithboththeBEHbosonhypothesisandtheele troweakpre ision

observables [19℄, signs a tremendous su ess of the SM to adequately des ribe the Nature

up to an energy s ale

O(100)

GeV. The Yukawa ouplings of the BEH boson with

elemen-tary fermions are proportional to a mass and an be used to des ribe the fermion masses

a ordingly. Nothing in the symmetries is xing there values though. They are hen e free

parameters of the theory.

The quarks andleptons are dividedinto threegenerations, ea h of thembeing adoublet

of

SU(2)

L

. Therst generationofquarks onsistsoftheup- anddown-quarks,the se ondof

the harm-and strange-quarks,andthe thirdgenerationofthe top-andbeauty-quarks. The

leptons and their asso iated lepton-neutrinos are divided into the ele tron, muon and tau

generations. In addition,ea hparti lehas anasso iated anti-parti lewith opposite internal

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

Table 1.1: The three lepton andquark generations. The indi es

L

and

R

note the parti le hirality

state, left andright, respe tively.

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

Thelo algaugeinvarian eintheSMforbidsfermionsandbosonstobemassive. Thefermion

masses are introdu ed after the spontaneous ele troweak symmetry breaking, via Yukawa

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

density isgiven by

L

Y

= −λ

d

ij

Q

¯

I

Li

3

φD

I

3

Rj

− λ

u

ij

Q

¯

I

Li

3

φ

∗

_U

I

3

Rj

+ h.c,

(1.1)

ˆ

i

and

j

are for the generation indi es,

ˆ

Q

I

3

L

, D

I

R

3

, U

R

I

3

arethemultipletsof

SU(2)

L

⊗SU(3)

c

⊗U(1)

Y

.

Q

I

3

L

= (U, D)

I

L

3

aretheleft

hiralitydoubletsand

U

I

3

R

, D

I

3

R

the ouplesofright hiralitysingletsinweakintera tion

eigenstates basis.

1.1

CP

violation in the SM 5

ˆ

λ

d,u

ij

are the omplexmatri es

3 × 3

of the quark-down and -up oupling, respe tively.

When the Higgs eld a quires a value in the va uum (v.e.v.)

v = h0|φ|0i

, the fermion

mass terms appear

−

λ

d

ij

.v

√

2

. ¯

D

I

3

Li

D

I

3

Rj

−

λ

u

ij

.v

√

2

. ¯

U

I

3

Li

U

I

3

Rj

+ h.c.

(1.2)

Itisworthwhiletomovefromthebasisoftheweakintera tioneigenstatestomasseigenstates,

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

using unitary matri es

U

u(d)

L

and

U

u(d)

R

U

L

u(d)

λ

u

ij

.v

√

2

U

u(d)

R

=





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

3

L

=

 U

I

3

L

D

I

3

L



= (U

L

u†

)

j

 U

Lj

(U

u

L

U

L

d†

)

jk

D

Lk



,

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

V

CKM

= U

L

u

U

d†

L

=





V

ud

V

us

V

ub

V

cd

V

cs

V

cb

V

td

V

ts

V

tb





.

Thus, the urrents responsible for weak intera tion are transformed under the inuen e of

the hange of weak eigenstates basis to the mass eigenstates by making expli itly appear

theCKM matrixelements. The orrespondingLagrangiandensityinvariantunderthe

SU(2)

transformationsis given by

L

W

= i

g

1

2

Q

¯

I

3

Li

γ

µ

(~τ. ~

W )

µ

Q

I3

Li

,

(1.3)

where

g

1

is the weak oupling onstant,

~τ

are the Pauli matri es, generators of the

SU(2)

group and

W

~

the three additional ve tors eld brought by the requirement of lo al gauge

invarian e. This density be omesin the mass eigenstates basis

L

W

= i

g

1

√

2

( ¯

U

Li

γ

µ

U

ik

u

U

d†

kj

D

Lj

W

µ

+

+ ¯

D

Li

γ

µ

U

ik

d

U

u†

kj

U

Lj

W

µ

−

) +

ig

1

2

Q

¯

Li

γ

µ

_τ

3

_W

µ

3

Q

Li

.

(1.4)

Itshouldbenotedthattheintera tionsthroughneutral urrents(thethirdterminEq.(1.4))

arenotmodied. Thereisa tuallynotree-levelpro essofavor hangingby neutral urrent

1.1.3 CKM parameterisations and representations

The CKM matrix is a

3 × 3

omplex unitary matrix and an as su h 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 hoi e, adopted by the Parti le 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 whi h has been rst introdu ed by

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

between the matrix element magnitudes

s

13

≪ s

23

≪ s

12

≪ 1

. The four independent

parameters are noted

λ

(whi h 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

powera ording 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

parti ular, 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)

A onvenientwayofrepresentingtheunitarityrelationsistodisplaytheminthe omplex

plane, hen e as a triangle.Fig. 1.1 proposes su h 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

∗



1.1

CP

violation in the SM 7

Figure1.1: The unitaritytrianglewithsides of the same

λ

order with

α

,

β

and

γ

angles asso iated.

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

Inquantumme hani s,the

CP

transformation ombines harge onjugation

C

withparity

P

transformations. The parity operator,

P

, inverts the algebrai sign of all spa e oordinates

usedinthedes riptionofaphysi alpro ess. Asexample,if theparityoperatorisperformed

on a s alar wavefun tion

ψ(x, y, z, t)

, the latter will transform it to

ψ(−x, −y, −z, t)

. The

parity onservationor

P

-symmetryimpliesthat anyphysi alpro ess willpro eedidenti ally

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

physi al pro ess would onserve parity. However, a number of experiments were performed

(

e.g.

Wu experiment [23℄) and showed that, for pro esses involving weak intera tion, the

P

-symmetry violated.

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

intrinsi additivequantum numbers,astheele tri harge,thebaryonquantumnumber,the

lepton quantum number, the strangeness, et . The

C

-symmetry,as the

P

-symmetry, means

thesymmetry ofphysi allawsunderthe harge onjugation transformation. Thissymmetry

is onserved by ele tromagnetism, gravity and strong intera tion, but violated in the weak

intera tions [24℄.

Thus, ombiningthetwooperators

P

and

C

,the

CP

operatorwilltransform,forinstan e,

a left-handed ele tron

e

−

L

into a right-handed positron

e

+

R

1

. Therefore, if

CP

were an exa t

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

of this symmetryissubtle and has been di ulttoexplore. However, Croninand Fit h[25℄

performeda beam experimentin1964 inwhi h they measuredthe de ayof neutralkaons in

1

W

_W

d

¯b

b

¯

d

t, c, u

¯

t,

¯

c,

u

¯

B

0

B

¯

0

Figure1.2: One of the two box diagrams des ribing 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 intera tion, inthe neutral kaon mixing.

Toillustratethemanifestationof

CP

violationwithweakintera tionintheSM,let'sapply

theoperator

CP

tothersttermoftheLagrangiandensityshowninEq.(1.4)(

L

(1)

W

CP

−→ L

(1)′

W

)

L

(1)

W

= i

g

1

√

2

( ¯

U

Li

γ

µ

U

ik

u

U

d†

kj

D

Lj

W

µ

+

) ,

(1.10)

L

(1)′

W

= i

g

1

√

2

( ¯

D

Li

γ

µ

U

ik

d

U

u†

kj

U

Lj

W

µ

−

_{) .}

(1.11)

Therefore if the matrix element

U

d

ik

U

u†

kj

is omplex wewillhave

L

(1)

W

6= L

(1)′

W

, whi himplies a

CP

violation. Then the

δ

phase introdu ed in the CKM matrix is a sour e of

CP

violation

inthe weak intera tion.

1.1.5

CP

violation in neutral

B

se tor

Despitealargenumberof attemptstoobserve

CP

violationphenomena,ittookalmostforty

years to rea h a se ond observation of it. Before addressing the

CP

violation in neutral

B

mesons, a brief overview is given in the following subse tion dis ussing the quantum

me hani sof neutral

B

mesons.

1.1.5.1 The quantum me hani s of neutral

B

meson mixing

The neutral

B

mesons are pseudo-s alarmesons whi h an have twoavor states,

B

0

made

of

d

-quarkand

¯b

-quark,and

B

0

s

made of

s

-quarkand

¯b

-quark. They anea hmixwith their

respe tive antiparti le, 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 des ribed in term of two physi al states ombination of the

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 oe ients satisfying the relation

|p|

2

_{+ |q|}

2

_{= 1}

.

The states

|B

L

i

and

|B

H

i

are the lighter and heavier mass eigenstates, respe tively. The

time-dependent S hroedingerequation for these states reads

i

∂

∂t

 p

q



= H

eff

 p

_q



,

(1.13)

where

H

eff

isthe ee tive Hamiltoniandes ribing the neutralmesons mixingasfollows

H

eff

= M − i

Γ

2

=

 M

11

M

12

M

21

M

22



−

₂

i

 Γ

_Γ

11

Γ

12

21

Γ

22



,

=

 ω

L

0

0

ω

H



.

(1.14)

M

and

Γ

are

2 × 2

Hermitian matri es des ribing the mass and de ay rate omponent of

H

eff

, respe tively. 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

, respe tively, and

Γ

L

and

Γ

H

their de ay rate ounterpart. The 2-parti le system

{B

0

_{, ¯}

_B

0

_}

is hara terized by

5 physi al observables (named also mixingobservables): the mass and de ay rate averages,

the dieren es in mass and de ay rate, and its " ompositionfra tion"

|q/p|

. The mass and

de ay rate averages are

m =

m

H

+ m

L

2

,

Γ =

Γ

H

+ Γ

L

2

.

(1.16)

The dieren es inmass and de ay 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 whi h mass eigenstate

has the longer lifetime. The sign of

∆Γ

is predi ted,by the SM,tobe negative, but has not

yet been established, while is well established in

B

0

s

-

B

0

s

mixing (

∆Γ

s

= (0.091 ± 0.008) ×

10

12

_s

[20℄). Thevaluesfound fortheworldaverageof themassdieren emeasurements[20℄,

are

∆m

B

0

= (3.337 ± 0.033) × 10

−10

_MeV

and

∆m

B

0

s

= (1.1691 ± 0.00014) × 10

−8

_MeV

. As

mentioned above, the de ay rate dieren e has on the ontrary not yet been observed and

we onsider itnegligiblein the following study.

 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 o urring inthe a tion of

CP

operator onthe state

|B

0

_i

(

|B

0

_i

)whi htransforms 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 de ay rate of a

|B

0

_i

meson produ ed at time

t = 0

to a nal state

f

at time

t

is

1.1

CP

violation in the SM 11

dΓ

B

0

_{→f ( ¯}

_{f )}

(t)

dt

= |hf( ¯

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 de ay rates of the initially produ ed 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

de ay 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

= hf|T |B

0

_i

and

A

¯

¯

f

= h ¯

f |T |B

0

i

are the de ay amplitudes for

|B

0

_i

and

|B

0

_i

de aying to the nal state

|fi

and

| ¯

f i

, respe tively, 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

= hf|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,dis ussed indetails inthe following se tion. 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 de ay rates

Γ(B

0

_{→ f)}

and

Γ(CP (B

0

_{→ f))}

,where

CP (B

0

_{→ f)}

isthe pro ess

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

noti eabledieren es amongpro essesand their orresponding

CP

- onjugates. The

observa-tionof

CP

isrelatedtotheinterferen e betweendierentamplitudesthat ontributetothese

Figure 1.3: Diagrams showing the three type of

CP

violation: (A)

CP

violation in de ay, (B)

CP

violation in mixing and(C)

CP

violation between de ays withand without mixing.

detailsare giveninSe tion1.1.5.3. The possiblemanifestation of

CP

violation an be

lassi-ed in three ategories: (A)

CP

violation inde ay, (B)

CP

violationin mixingand (C)

CP

violation between de ays with and without mixing(Mixing-indu ed

CP

violation). Fig. 1.3

illustrates ea h manifestation type of

CP

violation. In ea h ase there is a orresponding

observable of

CP

violation. All

CP

violation observables in the pro esses of

B

0

/

B

0

de ay-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 de ay

This type of

CP

violationis adire t

CP

violation,whi hrequires aavour-tagging

informa-tion on the initialstate in the neutral

B

de ays,

i.e.

a distin tion between the de ays of

B

0

and

B

0

to a nalstate

f

and

f

¯

,respe tively, where

CP |fi = e

2iθ(f )

| ¯

f i .

θ(f )

here is is an arbitrary phase. The manifestation of

CP

violation in this ase o urs 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 pro ess 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

1.1

CP

violation in the SM 13

hen e, the

CP

violation in de ay o urswhen

| ¯

A

_f

¯

|

|A

f

|

6= 1

=⇒

CP violation.

(1.34)

If several amplitudes

j

ontribute to the de ay

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 be omes

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 pro esses that ontributes to the nal state

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

CP

violation,

whi h the amplitude

A

f

should have at least two ontributing omplex amplitudes with

dierent weak and strongphase, the reason forthat omes fromthe fa t that

CP

- onjugate

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

CP

violation

in de ay is most thoroughly studied in

b

-hadron de ays 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 indire t

CP

violation, whi h implies that the os illation

from

B

0

to

B

0

is dierent from the os illation

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

,

=





hB

0

|H

_eff

|B

0

i





−





hB

0

|H

_eff

|B

0

i









hB

0

|H

_eff

|B

0

i





+





hB

0

|H

_eff

|B

0

i





.

(1.38)

To he kthatthedieren ebetween





hB

0

|H

_eff

|B

0

i





and





hB

0

|H

_eff

|B

0

i





isasign ofmixing

hB

0

|H

eff

|B

0

i

CP

−−→ hB

0

|(CP )

†

(CP )H

eff

(CP )

†

(CP )|B

0

i

=

_hB

0

_{|(CP )}

†

_H

CP

_eff

_{(CP )|B}

0

_i

=

e

−4iθ(B

0

)

_hB

0

_|H

CP

_eff

_|B

0

_{i ,}

(1.39)

hB

0

|H

eff

|B

0

i

=

e

4iθ(B

0

₎

hB

0

|H

CP

eff

|B

0

i ,

(1.40) where

H

CP

eff

= (CP )H

eff

(CP )

†

and

θ(B

0

₎

is the arbitrary unphysi al phase introdu ed in

Eq. (1.20). So, if

CP

is a symmetry of

H

eff

then

[H

eff

, CP ] = 0

, whi h implies

H

eff

= H

CP

eff

=⇒





hB

0

|H

_eff

|B

0

i





=





hB

0

|H

_eff

|B

0

i





.

(1.41)

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

|B

L

i

,

|B

H

i

}, the

CP

asymmetrybe omes

A

CP

=







p

q





 −







q

p













p

q





 +







q

p







.

(1.42)

Therefore,

CP

violation inmixingo urs if









p

q









6= 1

=⇒

CP violation in mixing.

(1.43)

as wasintrodu edearlier 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

0

s

-

B

0

s

mixings is expe ted 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-indu ed

CP

violation

CP

violation in the interferen e between de ays with and without mixing o urs for the

de ays of

B

0

and

B

0

to anal state

f

whi h is a

CP

-eigenstate

B

0

_{(→ B}

0

_{) → f ← (B}

0

_←)B

0

,

_{CP |fi = η}

CP

|fi ,

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 interferen e of mixing and de ay amplitudes

A(B

0

_{→ B}

0

_{→ f}

CP

)

and

A(B

0

_{→ f}

CP

)

,respe tively.

The time-dependentmixing-indu ed

CP

asymmetry reads

A

CP

(t) =

dΓ

_B0→f

(t)

dt

−

dΓ

_B0→f

(t)

dt

dΓ

_B0→f

(t)

dt

+

dΓ

_B0→f

(t)

dt

.

(1.44)

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

CP

asymmetry reads

where

S ≡ S

f

CP

= η

f

CP

2Im(λ

f

CP

)

1 + |λ

f

CP

|

2

,

C ≡ C

f

CP

= η

CP

1 − |λ

f

CP

|

2

1 + |λ

f

CP

|

2

.

(1.46)

Therefore, mixing-indu ed

CP

violationo ursif

S 6= 0

.

1.2 Constraints on CKM matrix elements

1.2.1 Magnitudes of the matrix elements

The nine CKM matrix elements

V

jk

are, in prin iple, a essible experimentally thanks to

the oupling

W

±

_q

j

_q

_¯

k

. The aimof this se tion isto re apitulate the numeri al values of the

CKM matrix elements urrently measured:

ˆ

|V

ud

|

: this matrix element an be measured by means of three dierent methods: the

nu lear

β

de ay, the neutron lifetime and the pion

β

de ay

π

+

_{→ π}

0

_e

+

_ν

. Currently,

the world best average [30℄ reads:

|V

ud

| = 0.97425 ± 0.00022

.

ˆ

|V

us

|

: this matrix element is mainlydeterminedfrom the measurement of the

semilep-toni kaon de ays. The urrent average value from the PDG [20℄ is

|V

us

| = 0.2253 ±

0.0008

.

ˆ

|V

cd

|

: the magnitude ofthis matrix element an beevaluatedfromsemileptoni harm

de ays,

e.g. D → πlν

. The other possibilityto measure this parameter isvia neutrino

and antineutrino intera tions[31℄. The average value given by the PDG [20℄ is

|V

cd

| =

0.225 ± 0.008

.

ˆ

|V

cs

|

: the magnitude of this matrix element an be determined dire tly by means of

the semileptoni de ays of

D

orleptoni de ays of

D

s

. It isalsopossibletouse avour

tagged Wde ays [32℄. The world average value [20℄ reads:

|V

cd

| = 0.986 ± 0.016

.

ˆ

|V

cb

|

: this parameter an be determined from ex lusive and in lusive semileptoni

de ays of

B

mesons into harm, its average value given by the PDG [20℄ reads:

|V

cb

| = (41.1 ± 1.3) × 10

−3

.

ˆ

|V

ub

|

: the determinationof this parameter an be performed from the in lusive de ay

B → X

u

l¯

ν

, where

X

u

is harmless hadroni nal state. In addition, ex lusive

deter-mination of

|V

ub

|

ome from the study of

B → πl¯ν

l

. The average value given by the

PDG [20℄ is

|V

ub

| = (4.13 ± 0.49) × 10

−3

.

ˆ

|V

td

|

and

|V

ts

|

: Thesetwoparameters an bemeasured in theos illationof

B

0

-

B

0

and

B

0

s

-

B

0

s

, where top quark appears in box diagrams, or in rare de ays where top quark

an be found in loop diagrams in the SM. The average value given by the PDG [20℄

for the ratio of the magnitudes of these matrix elements is:

|V

td

/V

ts

| = 0.21 ± 0.04

.

ˆ

|V

tb

|

: the determination of this element matrix is made using the ratio of bran hing

fra tions[33℄

R = B(t → W b)/(t → W q) = |V

tb

|

2

_/(

P

q

|V

tq

|

2

) = |V

tb

|

2

,where

q = b, s, d

.

Another possible determination of

|V

tb

|

makes used the single top-quark-produ tion

ross se tion. The average given by the PDG [20℄ for the magnitude of this matrix

1.2.2 CKM angles

The measurements of the angles of the unitarity CKM triangle Fig. 1.1 are important for

determining the degree of

CP

violation in the standard model. To onstraint these three

angles

α

,

β

and

γ

,several

CP

-violatingobservables an be used:

ˆ

α

angle: thetime-dependent

CP

asymmetryin

b → u¯ud

de aydominatedmodes,allows

to measure

sin 2α

. Pra ti ally, the measurements are taken on the de ays

B → ππ

,

B → ρπ

and

B → ρρ

.

ˆ

β

angle: it represents the mixing angle of the

B

0

mesons in the SM and is

mea-sured through time-dependent

CP

asymmetry to a nal state

CP

-eigenstate ( f:

Se -tion 1.3.1).

ˆ

γ

angle: it an be measured in tree-level

B

de ays. For example, the interferen e of

B

−

_{→ D}

0

_K

−

and

B

−

_{→ ¯}

_D

0

_K

−

givesa ess to the

γ

angle.

Fig. 1.4 superimpose all re ent CKM onstraints determined under the SM hypothesis

inthe plane (

ρ, ¯

¯

η

)(see Ref.[3℄).

1.3 Charmless three-body neutral

B

de ays

The

B

0

and

B

0

s

mesons oer a relevant environment for studying

CP

violation as the

non-squashed unitarity triangles involve quark transitions with the

b

-quark. Several parameters

ofthe unitarityCKMtriangle anbea essibleby

B

mesonsphysi s,su hasthe magnitude

of the matrix elements

|V

cb

|

,

|V

ub

|

,

|V

td

|

and

|V

ts

|

as well as the three angles given in the

previous se tion

α

,

β

and

γ

.

Among the dierent types of neutral

B

de ays, the three-body harmless neutral

B

de- aysisthepro ess studiedinthis thesiswork. This typeofde aysprovidesari hlaboratory

for studying dierent aspe ts as

CP

violation, strong intera tion, onstraints in the CKM

triangle et .. In addition,the spe i harmless hadroni nal state, in ludinga

K

0

S

meson, namely

B

0

d,s

→ K

S

0

π

+

_π

−

,

B

0

d,s

→ K

S

0

K

±

_π

∓

and

B

0

d,s

→ K

S

0

K

+

_K

−

, has a variety of physi s

interpretations. The nal states

B

0

_{→ K}

0

S

π

+

_π

−

and

B

0

_{→ K}

0

S

K

+

_K

−

allow for the

measure-mentof the weak phaseof

B

0

-

B

0

mixingin

b → q¯qs

transitions, whi h an be obtained, for

example,byatime-dependenttaggedanalysisofthethree-bodyDalitzplot. The omparison

of the weak phase determination in

b → q¯qs

and

b → c¯cs

transitions an be a measure of

New Physi s (NP) ontributions (see Se tion 1.3.1),under the assumption that the

b → c¯cs

transition is dominated by SM pro esses. Studying the de ay

B

0

s

→ K

S

0

π

+

_π

−

is a

ne es-sary ingredient for a lean extra tion of

γ

in harmless de ays (Ref. [35℄) by means of an

analysis of the ratio of the amplitude of the isospin-related mode

B

0

s

→ K

−

π

+

π

0

and its

harge onjugate, where a dire t dependen e on the weak phase (

β

s

+ γ

) is exhibited. In

that ase, ananalysis of the

B

0

s

→ K

S

0

π

+

_π

−

Dalitzplot willbe required[36℄. Eventuallythe

de ay

B

0

s

→ K

S

0

K

±

_π

∓

an allowtomeasure theweak phaseof

B

0

s

-

B

0

s

mixing,analogouslyto

B

0

_{→ K}

S

0

π

+

_π

−

. However, the signi ant lifetimedieren e between the lightand heavy

B