Charmless hadronic three-body decays of neutral B mesons with a Kos in the final state in the LHCb experiment : branching fractions and an amplitude analysis

HAL Id: tel-01599261

https://tel.archives-ouvertes.fr/tel-01599261

Submitted on 2 Oct 2017

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Charmless hadronic three-body decays of neutral B

mesons with a Kos in the final state in the LHCb

experiment : branching fractions and an amplitude

analysis

Louis Henry

To cite this version:

Louis Henry. Charmless hadronic three-body decays of neutral B mesons with a Kos in the final state

in the LHCb experiment : branching fractions and an amplitude analysis. High Energy Physics

-Theory [hep-th]. Université Pierre et Marie Curie - Paris VI, 2016. English. �NNT : 2016PA066428�.

�tel-01599261�

E ole do torale des S ien es de laTerre etde l'environnement etPhysique de l'Univers,

Paris- ED N.560

Laboratoire de Physique Nu léaire etdes Hautes Energies - UMR 7585

Charmless hadroni three-body de ays of neutral

B

mesons with

a

K

0

S

in the nal state in the LHCb experiment: bran hing

fra tions and an amplitude analysis

Désintégrations hadroniques à trois orps sans harme de mésons beaux ave un

K

0

S

dans l'état nal dans l'expérien e LHCb : mesure de rapports d'embran hement etune

analyse en amplitude

par

Louis Henry

Jury omposé de:

Sébastien Des otes-Genon Rapporteur

Dire teur de re her he (LPT)

Maurizio Pierini Rapporteur

Resear her (CERN)

Tim Gershon Membre du jury

Professor (University of Warwi k)

Sandrine Lapla e Membre du jury

Dire teur de re her he (LPNHE)

François Le Diberder Membre du jury

Professeur(Université Paris-Diderot)

Guy Wilkinson Membre du jury

Professor (Oxford University)

Jean-Bernard Zuber Membre du jury

Professeur (Université Pierre etMarie Curie)

Eli Ben Haim Dire teur de thèse

Les dernières orre tions se mettent en pla e, la soutenan e paraît déjà loin, et j'appose

es derniers mots à e manus rit. Curieusement 'est eux-là, auxquels je pense depuisle

début, qui meprennent leplus de temps. C'est qu'il s'agit de n'oublierpersonne.

Tout d'abord, je souhaite remer ier une fois de plus mon dire teur de thèse, Eli Ben

Haim,quim'auraa ompagnédeprèstoutaulongde estroisannées. Grâ eàsa apa ité

à analiserdes pensées pas toujourstrès ordonnées, j'ai pu profondément évoluer en tant

que personne et en tant que her heur. J'ajoute par ailleurs une mention spé iale à sa

patien esanslimitespour orrigeraumotprèstoutesmesprodu tionsé rites, orre tions

qui donnaient l'impression qu'il aurait été plus rapide de repeindre ma page entière en

rouge. J'espère sin èrementpouvoir ontinuer àproterdesa justessed'analyseetde ses

qualités humaines tout aulong de ma arrière.

Jetiens àexprimer toute magratitude l'ensembledu groupe LHCb du LPNHE pour

leur gentillesse et leur disponibilité. Mer i aussi à mon parrain, José O ariz, qui m'a

a - ompagné et supporté mes râleries (surtout en troisièmeannée), toujours ave lesourire.

Je remer ie l'en adrement du laboratoire, et notamment sa grande ouverture aux

étudi-ants, qu'ils viennent de la li en e ou de l'extérieur. Ce n'est pas un hasard que mon

premier onta t ave lemonde de la physique des parti ules après mon é ole se soit fait

dans es lo aux. Enn, mer i à tous mes ollègues do torants: nos dis ussions

inter-minablessur tous lessujets au ours de pauses afé àrallonge memanquent déjà. Petite

penséespé ialepourMathilde, quiauramarqué profondémenttout ledébutde mathèse,

et pour l'ensemble de mes préde esseurs. Je leur souhaite bonne han e pour le futur,

dans la re her he eten-dehors.

Je remer ie l'ensemble des membres de mon jury, et notamment mes deux

rappor-teurs, SébastienDes otes-Genon etMaurizioPierini,qui ontpu relire et ommentermon

manus rit en moins d'un mois. Le présent do ument doit beau oup à l'ensemble des

re-marques du jury et à leurs questions in isives. Je remer ie Jean-Bernard Zuber d'avoir

a epté de présider e jury, et de m'avoir onseillé dans mes hoix de arrière en M1.

Mer i aussi à l'ensemble du groupe KShh, et plus parti ulièrement à Tom, Stéphane et

Rafael. Je remer ie sin èrement Diego Milanes, pour son aide, sa onstante sympathie,

et nos nombreuses soirées et ex ursions. Ses onseils, prodigués autour d'un afé, d'une

bière, oudans un stade de base-ball,m'ontété extrêmement pré ieux.

Jepensebienentenduàmafamille,quim'aforgéetatentédepuis27ans de analiser

labouled'énergie dont ilsavaienthérité. J'essaie toujoursde me montreràlahauteur de

et Thibaut, qui m'a ompagnent depuis tant d'années. On peut di ilement trouver

hemins plus diérents, maisvotre présen e m'est toujoursaussi hère dans es moments

importants.

Enn, toute ma gratitude et ma tendresse à Camille, qui aura eu la mal han e de

roiser un thésard audébut de son voyage et la grâ e de lesupporter. Son soutien et sa

Cemanus ritprésenteplusieursétudesdesdésintégrationsde mésons

B

0

et

B

0

s

entrois orps non- harmés, dont un méson

K

0

S

. Ces études portent sur les données enregistrées

parl'expérien eLHCbpendantleRunIdu LHC, orrespondantàuneluminositéintégrée

de

R L = 3

fb

−1

.

Unepremièreanalyse onsisteenunemesuredesrapportsd'embran hementdesmodes

B

0

d,s

→ K

S

0

h

+

_h

′

₋

, où

h

(

′

₎

désigne un kaon ouun pion. Lespré édentes mesures par LHCb

desrapportsd'embran hementsde esmodesdedésintégration,rapportésà eluidumode

B

0

_{→ K}

0

S

π

+

_π

−

,sontmisàjour.Deplus,lebutprin ipalde etteanalyseestdere her her

le mode

B

0

s

→ K

S

0

K

+

_K

−

, pas en ore observé par les analyses pré édentes. Les rapports

d'embran hement relatifssont mesurés :

B (B

0

s

→ K

S

0

π

+

_π

−

₎

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.26

± 0.04(stat.) ± 0.02(syst.) ± 0.01(f

s

/f

d

),

B (B

0

_{→ K}

0

S

K

±

π

∓

)

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.17

± 0.02(stat.) ± 0.00(syst.),

B (B

0

s

→ K

S

0

K

±

π

∓

)

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 1.84

± 0.07(stat.) ± 0.02(syst.) ± 0.04(f

s

/f

d

),

B (B

0

_{→ K}

0

S

K

+

_K

−

₎

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.59

± 0.02(stat.) ± 0.01(syst.),

(1)

Unepremière observation de

B

0

s

→ K

S

0

K

+

_K

−

est rapportée, ave une signi an eglobale

de

3.7 σ

.

Uneanalysenon-étiquetéedesaveur etindépendantedu tempsdu plandeDalitzde la

désintégration

B

0

_{→ K}

0

S

K

+

_K

−

estprésentée, enutilisantl'appro heisobare.Lesrapports

d'embran hement quasi-deux- orps des désintégrations

B

0

_{→ K}

0

S

φ

0

,

B

0

_{→ K}

0

S

f

′

2

(1525)

,

B

0

_{→ K}

0

S

f

0

(1710)

, et

B

0

_{→ K}

0

S

χ

c0

sont mesurés. Ils sont ompatibles ave les mesures pré édentes de BaBar, àl'ex eption de

B

0

_{→ K}

0

This dissertation presents several studies of the de ays of both

B

0

and

B

0

s

mesons to harmless three-body nal states in ludinga

K

0

S

meson. They use the data re orded by

the LHCb experimentduring RunI ofLHC , orrespondingtoanintegrated luminosityof

R L = 3

fb

−1

.

A rst analysis onsists of the measurement of the bran hing fra tions of

B

0

d,s

→

K

0

S

h

+

_h

′

₋

de ays, where

h

(

′

₎

designates a kaon ora pion. Pre eding LHCb measurements

of bran hing fra tions for all de ay hannels, relative to that of

B

0

_{→ K}

0

S

π

+

_π

−

, are

updated. Furthermore, the primary goal of this analysis is to sear h for the, as yet,

unobserved de ay

B

0

s

→ K

S

0

K

+

_K

−

. The relativebran hing fra tionsare measured tobe:

B (B

0

s

→ K

S

0

π

+

_π

−

₎

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.26

± 0.04(stat.) ± 0.02(syst.) ± 0.01(f

s

/f

d

),

B (B

0

_{→ K}

0

S

K

±

π

∓

)

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.17

_{± 0.02(stat.) ± 0.00(syst.),}

B (B

0

s

→ K

S

0

K

±

π

∓

)

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 1.84

_{± 0.07(stat.) ± 0.02(syst.) ± 0.04(f}

s

/f

d

),

B (B

0

_{→ K}

0

S

K

+

_K

−

₎

B (B

0

_{→ K}

0

S

π

+

π

−

)

= 0.59

_{± 0.02(stat.) ± 0.01(syst.),}

(2) A rst observation of

B

0

s

→ K

S

0

K

+

_K

−

is reportedwith aglobal signi an e of

3.7 σ

. A avour-untagged, time-independent Dalitz-plot analysis of

B

0

_{→ K}

0

S

K

+

_K

−

is

pre-sented,usingtheisobarapproa h. Thequasi-two-bodybran hingfra tionsof

B

0

_{→ K}

0

S

φ

0

,

B

0

_{→ K}

0

S

f

′

2

(1525)

,

B

0

_{→ K}

0

S

f

0

(1710)

, and

B

0

_{→ K}

0

S

χ

c0

are measured. They are ompati-blewith previous measurements from BaBar, ex ept for

B

0

_{→ K}

0

1 Theory 2

1.1 Introdu tion . . . 2

1.2 Violation of the

CP

symmetry . . . 3

1.2.1 Introdu tiontosymmetries . . . 4

1.2.2 The

C

,

P

,and

T

symmetries . . . 5

1.2.3 Neutral mesons mixingand

CP

violation . . . 6

1.2.4 The CKM matrix and the KM me hanism . . . 10

1.2.5 The unitarity triangles . . . 12

1.2.6

B

0

os illationsand the

β

angle . . . 14

1.3 Amplitudeanalyses on epts . . . 16

1.3.1 Three-body parti le de ays and the Dalitzplot. . . 16

1.3.2 The square Dalitz plot . . . 18

1.3.3 Quasi-twobodyde ays . . . 20

1.3.4 The isobar model . . . 20

1.3.5 Resonan e dynami s . . . 22 1.3.6 Nonresonant amplitude . . . 24 1.4 The study of

B

0

d,s

→ K

S

0

h

+

_h

−

de ays . . . 25 1.4.1

B

0

d,s

→ K

S

0

h

+

_h

′

₋

de ay amplitudes . . . 25 1.4.2 Previous studies of

B

0

d,s

→ K

S

0

h

+

_h

′

₋

and

B

0

d,s

→ K

S

0

K

+

_K

−

de ays . 26 2 Des ription of the LHCb experiment 28 2.1 The Large Hadron Collider . . . 28

2.2 The LHCb dete tor . . . 30

2.2.1 Beam onditions atthe LHCb intera tion point . . . 33

2.2.2 The magnet . . . 33

2.2.3 The tra king system . . . 34

2.2.4 The RICH1 and RICH2 . . . 39

2.2.5 Calorimeters. . . 41

2.2.6 Muon hamber . . . 43

2.3 Parti le identi ationinLHCb . . . 44

2.4 Trigger system in the LHCb experiment . . . 45

2.4.1 The hardware trigger(L0 trigger) . . . 45

2.5 Monte-CarlosimulationsinLHCb . . . 47

2.5.1 The Gaussframework . . . 47

2.5.2 The DaVin i framework. . . 48

2.5.3 Data/MC dis repan ies. . . 49

3 Fast Monte-Carlo method for ba kground studies 51 3.1 Strategy of the fast MCmethod . . . 51

3.2 Study of a

B

0

_{→ (K}

∗0

_{→ K}

0

S

π

0

_)(ρ

0

_{→ π}

+

_π

−

₎

sample . . . 53

3.3 Study of the resolution model appliedto other hannels . . . 56

3.4 Study of generator-levelre onstru tion ee ts . . . 56

3.4.1

K

0

S

re onstru tion mode . . . 57

3.4.2 Reprodu tionof the generator-leveldistributions. . . 59

3.5 Complete fast Monte-Carlotest on

B

0

_{→ K}

∗0

_ρ

0

. . . 60

3.6 Con lusion . . . 62

4 Measurement of the bran hing fra tions of the

B

0

d,s

→ K

S

0

h

+

_h

′

₋

modes 65 4.1 Analysisstrategy . . . 65

4.2 Toolsand formalism of the

B

-meson invariantmass t . . . 69

4.2.1 The unbinned maximum extended likelihoodt . . . 69

4.2.2 Gaussian onstraints . . . 70

4.2.3 The

s

Plots

method . . . 70

4.3 The

B

-meson invariantmass tmodel . . . 71

4.3.1

B

0

and

B

0

s

signal . . . 71 4.3.2 Charmed ontributions . . . 73 4.3.3

Λ

ba kground . . . 73

4.3.4 Beauty baryons ba kgrounds . . . 73

4.3.5 Combinatorialba kgrounds . . . 75

4.3.6 Cross-feeds . . . 76

4.3.7 Partiallyre onstru ted ba kgrounds. . . 76

4.4 Results of the mass t . . . 78

4.5 Validationof the mass t model . . . 84

4.6 Estimationof systemati un ertainties . . . 89

4.6.1 Totalun ertainties on yields . . . 94

4.7 Modelling the signal distribution overthe Dalitz plot using

s

Plots

. . . 101

4.8 Measurement of the bran hing fra tions. . . 102

4.8.1 Internal onsisten y . . . 102

4.8.2 Combinationof bran hing fra tions . . . 103

4.8.3

B

0

s

→ K

S

0

K

+

_K

−

observation signi an e . . . 104

4.8.4 Comparison with previous measurements . . . 105

5 Dalitz-plot analysis of

B

0

_{→ K}

0

S

K

+

_K

−

108

5.1 Analysis ontext and strategy . . . 108

5.2 Reoptimizationof the BDT sele tion . . . 111

5.2.1 Strategy of the reoptimization . . . 111

5.2.2 Results of the reoptimization . . . 113

5.3 Yieldsof the signaland ba kground spe ies. . . 115

5.4 Ba kground distributions . . . 116

5.4.1 Combinatorialba kground modelling . . . 116

5.4.2 Cross-feeds modelling. . . 118

5.5 E ien y variationsa ross the Dalitz plot . . . 118

5.5.1 Un ertainty estimation pro edure . . . 119

5.5.2 A eptan e of the generator-level ut . . . 119

5.5.3 Sele tion e ien y . . . 120 5.5.4 PID e ien y . . . 122 5.5.5 Totale ien ies . . . 123 5.6 Data-t model. . . 133 5.6.1 Baseline model . . . 133 5.6.2 Fit results . . . 134 5.7 Fit validation . . . 139

5.8 Evaluationof systemati un ertainties. . . 142

5.8.1 Fit-biasestimation . . . 142

5.8.2 General methodto evaluate systemati un ertainties . . . 142

5.8.3 E ien ies. . . 144

5.8.4 Signal and ba kground yieldsestimations . . . 144

5.8.5 Ba kgroundshapes . . . 145

5.8.6 Totalexperimentalsystemati un ertainties . . . 146

5.8.7 Resonan e shape parameters . . . 146

5.8.8 Fixed isobar parameters . . . 147

5.8.9 Modelun ertainties . . . 148

5.9 Con lusion . . . 149

A Corre ting sWeights in the presen e of xed yields 154 A.1

s

Plots

with xed yields. . . 154

A.2 RooStats implementationof the

s

Plots

method . . . 156

A.3 Proposed method and test . . . 156

A.4 Con lusion . . . 164

B Goodness-of-t riteria 165 C Fast MC method for ba kground studies - other hannels 169 C.1

B

0

_{→ K}

0

S

(η

→ π

+

_π

−

_π

0

₎

. . . 169 C.2

B

0

_{→ K}

0

S

π

+

_π

−

_γ

. . . 169 C.3

B

0

_{→ (K}

∗0

_{→ K}

0

S

π

+

_)π

+

_π

−

. . . 170

C.4

B

0

_{→ (K}

∗0

_{→ K}

0

S

π

0

_)(φ

_{→ K}

+

_K

−

₎

. . . 172 C.5

B

+

_{→ (K}

∗0

_{→ K}

0

S

π

+

_)(φ

_{→ K}

+

_K

−

₎

. . . 173 C.6

B

0

s

→ (K

∗0

→ K

S

0

π

0

_)(φ

_{→ K}

+

_K

−

₎

. . . 175 Referen es . . . 177

The study of of

b

-hadronde ays to hadroni nal states with no harmedparti les allow for a ri h array of studies. A few examplesare the measurementsof bran hing fra tions,

CP

asymmetries, weak and strong phases; they probe the dynami s of weak and strong intera tions. The typi al bran hing fra tions of these modes are below

10

−5

and thus

their analyses are feasible only with large data samples and the use of powerful tools to

reje t ba kground. The LHCb experimentatthe CERNLarge Hadron Collider(LHC) is

anadequateexperimentalenvironmentfortheseanalyses,oeringthe possibilitytostudy

de ays of light

B

mesons,

B

s

mesons and

b

baryons. This dissertation des ribes two analyses of

B

0

d,s

→ K

S

0

h

+

_h

′

₋

de ays, where

h

(

′

₎

rep-resents a kaon or a pion, that were performed with the 3fb

−1

dataset olle ted by the

LHCb experiment during the years 2011 and 2012, at entre-of-mass energies of 7 and

8

TeV

,respe tively. The de ays understudy are dominated by looptransitions,that may have ontributions fromparti lesbeyond the standard model. The measured observables

are therefore probes for new physi s. A rst analysis onsists inthe measurement of the

six bran hing fra tions of these modes, relative to that of

B

0

_{→ K}

0

S

π

+

_π

−

. This in ludes

a sear h for the mode

B

0

s

→ K

S

0

K

+

_K

−

, that has never been observed before. A se ond

study isthe rstamplitudeanalysis(orDalitz-plotanalysis)ofthemode

B

0

_{→ K}

0

S

K

+

_K

−

fromLHCb . It ontainsameasurementthebran hingfra tionsofintermediatestatesthat

intervene in the de ay, using the isobar approximation. This is the rst su h study of

this mode inLHCb ; itwillbepursued in stepsof in reasing omplexitywith the growing

dataset, and willbe ome more and more sensitiveto new physi sobservables.

This dissertation is organized as follows. Se tion 1 shortly reviews the theoreti al

framework, as well as on epts related to the amplitude analysis. It also gives a short

overviewofexistingresults. Se tion2thendes ribestheLHCbexperimentandtherelated

on epts that are useful for the understanding of the analysis work. The presentation

of my work is then separated into three parts. Firstly, Se . 3 presents an alternative

pro edure to simulate ba kground events. This pro edure is used in the measurement

of

B

0

d,s

→ K

S

0

h

+

_h

′

₋

bran hing fra tions. Se ondly, Se . 4 des ribes the measurement of

the bran hing fra tions of

B

0

d,s

→ K

S

0

h

+

_h

′

₋

modes, along with the sear h for the missing

B

0

s

→ K

S

0

K

+

_K

−

mode. Finally, Se . 5 presents the untagged, time-independent

Dalitz-plot analysis of the

B

0

_{→ K}

0

S

K

+

_K

−

Theory

1.1 Introdu tion

The Standard Model (SM) of parti le physi s des ribes the intera tion of fundamental

parti les through the strong and ele troweak intera tions [13℄. It is an outstandingly

su essful theory that predi ts nearly all the measurements ever performed with great

pre ision. There are however some hints that point ata larger theory, the SM being an

ee tive model of that theory at lower energies:

•

the SM doesnot explain the numberof fermiongenerations nor their highly hierar- hi al stru ture in termsof mass. Instead, masses of parti lesformthe bulkof free

parameters of the SM (13 out of 18);

•

the SM does not in lude gravity. In fa t, general relativity is even mathemati ally in ompatiblewithquantum eldtheory (QFT).The SMhasthen tobeanee tive

theory that annot be validat the Plan k energy s ale;

•

the SM does not provide a andidate for old dark matter, whose ontribution to the mass ontent of the Universe is found to be about ves times larger than that

of ordinary matter[4℄;

•

there is no me hanism in the SM that explains the smallness of the mass of the Higgs boson. Indeed, quantum ontributions to the Higgs boson mass from Grand

Uni ation or Plan k-s ale parti les would make the mass huge, unless there is a

ne-tuning an ellation between the radiative orre tions and the bare mass [5℄.

This problemmay besolved by the presen e of physi sbeyond the SMatlowmass

s ale (1

TeV

), whi hwould provide amore natural an ellation;

•

the SM fails to a ount for the matter-antimatterasymmetry observed in the Uni-verse.

Theseissues motivate the sear h for new physi s (NP),and alsoprovidesome hints that

Dire t sear hes look for the produ tion of on-shell parti les beyond the SM, su h as

supersymmetri parti les (squarks, gluinos) [4℄. Indire t sear hes fo us on deviations of

measurements of observables from a theoreti al SM predi tion due to the ee t of

o-shell NP parti les. Thesesear hes requireboth a lean theoreti alpredi tion anda lean

experimentalmeasurement so that possible deviations an beattributed to the ee ts of

NP; they are better performed on de ays where a ontributionfrom NP is expe ted. In

general,dire t sear hes need ana urate des riptionof theba kground,whereas features

of the ba kground an beusually inferred from data inindire tsear hes.

The violationof the

CP

symmetry, des ribed in Se . 1.2, isa feature of the Standard Model whi h is strongly related to the matter-antimatter asymmetry in the Universe.

1

It depends onfew parameters of the Standard Model, thus its predi tive power is rather

high. The study of the violation of this symmetry in

B

0

d,s

→ K

S

0

h

+

_h

′

₋

de ays provides

opportunities to perform indire t sear hes for NP. Indeed, de ays of the type

B

0

d,s

→

K

0

S

h

+

_h

′

₋

, where

h

(

′

₎

are kaons or pions, are dominated by so- alled penguin diagrams

that in lude a loop of virtual parti les. Parti les of NP ould ontribute inside of that

loop and ause a deviation of some observables from the SM predi tion. Additionally,

thesede aysalsoprovidearelatively leanexperimental ontextintheLHCb experiment,

wheresample purities largerthan 90% an bea hieved.

Se tion 1.2 details the Standard Model des ription of the

CP

violation, and Se . 1.3

presents some general on epts of amplitude analysis. Finally, Se tion 1.4 presents an

overview of the motivations and experimental ontext of the study of

B

0

d,s

→ K

S

0

h

+

_h

′

₋

de ays.

1.2 Violation of the

CP

symmetry

The violation of the

CP

symmetry, des ribed in Se . 1.2.2 is a key fa tor to understand thematter-antimatterasymmetryoftheUniverse. Indeed,therequired onditionssothat

amodel ouldallowforamatter-antimatterasymmetry, denotedSakharov onditions [7℄,

are

•

the existen e of an intera tion that doesnot onserve the baryon number;

•

the existen e of an intera tion that violates both the

C

and

CP

symmetries;

•

non-thermal equilibrium.

The baryonnumber isnot onserved in somenon-perturbativeele troweakpro esses, for

instan ethe pro esses alledsphalerons[8℄. Theexisten eofsu hpro essesrelieshowever

on the existen e of a

CP

violationat the perturbative s ale. 1

As des ribed in thefollowing,

CP

violationis akeyingredientto explainthis asymmetry,but this

CP

violationis toosmall by9orders ofmagnitude to explainthe matter-antimatterasymmetryof the Universe[6℄.

Symmetries play a fundamental role in modern physi s, as they onstitute the building

blo ks of any Lagrangian theory. They an be ontinuous or dis rete. Continuous

sym-metries are familiesof symmetries that depend ona ontinuous parameter. For instan e,

U(1)

is a group of the ontinuous, global symmetries that des ribe rotations in a plane. It an bedened as

{t

α

∈ U(1); y → y × e

iα

},

(1.1)

wherey is a omplex numberand

α

isa real number.

Continuous, global symmetries an be extended into gauge symmetries, where the

parameter is itselfa fun tionof the positionin spa eand time. For instan e, the gauged

version of the global

U(1)

symmetry would be

{t

α

∈ U(1); y(x) → y(x) × e

iα(x)

},

(1.2) wherexisapositioninspa e-time,

y(x)

isa omplexoperator,and

α(x)

isarealfun tion. The Standard Model is a gauge theory of the

SU(3)

C

⊗ SU(2)

L

⊗ U(1)

Y

group. This underlying stru ture onstrains the parti le ontent of the theory and the intera tions

between these parti les.

Thestrong intera tion isdes ribed by the underlying

SU(3)

C

symmetry, wherethe

C

standsfor olour hargeoftheintera tion. Propertiesofthatsymmetrygroupnaturally

yieldthegluonself-intera tion,whi histheunderlying auseforthe onnementofquarks

into olourless hadrons.

Theele tromagneti andweakintera tions are des ribed by the underlying

SU(2)

L

⊗

U(1)

Y

symmetry, where the

L

stands for left-handed and the

Y

for the hyper harge. The left-handed aspe t of the

SU(2)

L

symmetry is what explains the nonexisten e of right-handed neutrinos, and thus the violation of parity (see Se . 1.2.2) by the weak

intera tions. The

SU(2)

L

⊗ U(1)

Y

symmetry is spontaneously broken at the urrent Universe energydensity, leavingonlythe residual

U(1)

Q

symmetry thatisresponsiblefor ele tromagneti intera tionandwhosemediatoristhemasslessphoton

γ

. Theme hanism ofthatsymmetrybreaking,wheretheva uumexpe tationvalueofoneofthe s alarelds

of the theory is nonzero, is known as the Higgs me hanism. This me hanism gives rise

to the masses of fermions and of the gauge bosons of the weak intera tion,

W

±

and

Z

0

, and has been onrmed by the dis overy of the Higgs boson by the ATLAS and

CMS experiments in 2011 [9,10℄. The weak intera tion is the only one known to ouple

dierent avours. In the quark se tor, it ouples up-type quarks (

u

,

c

,

t

) and down-type quarks (

d

,

s

,

b

).

Dis retesymmetriesdonotdepend ona ontinuousparameter,and annotbegauged.

The onservationofthesequantumnumbersinapro essgovernedbyanintera tionthatis

invariantunderthe orrespondingsymmetryallowstobuildsele tionrules. Thefollowing

se tiondes ribesthreeofthese dis retesymmetries,

C

,

P

,

T

,aswellasthe

CP

and

CP T

produ ts.

1.2.2 The

C

,

P

, and

T

symmetries

The harge- onjugation operator

C

The harge- onjugationoperator

C

transformsaparti letothe orrespondingantiparti le. This antiparti le shares all the properties of the original parti le, ex ept for reversed

ele tri , avour, and olour harges. The Lagrangians of the ele tromagneti and strong

intera tion are invariantunder

C

, unlikethe Lagrangian of the weak intera tion.

The parity operator

P

The parity operator is dened as the reversal of allthe spatial oordinates of a 4-ve tor,

whilethe time omponentis onserved. It onserves allthe harges ofthe parti leand its

spin. The angular momentum

L

is onserved, whi h means that the sign of the heli ity of the parti le, dened as

H =

L.p

|p|

,

(1.3)

isreversed. Hen e

P

transformsleft-handed(

H =

−1

)parti lesintoright-handed(

H = 1

) parti les, and inversely. The heli ity is strongly related to the hirality of the parti le,

whi h denes its transformationunder

P

.

2

In ontrary tothe heli ity, however, hirality

does not depend onthe referen eframein the ase of massive parti les.

Following the observation that parity is onserved by the ele tromagneti and strong

intera tions, weak intera tion was initiallythoughtto onserve that symmetry. However,

Lee and Yang [11℄ raised on ern that the weak intera tion ould be sensitive to the

hirality of parti les ( hiral intera tion). This was onrmed by the observation that

β

de ays only emit left-handed neutrinos [12℄. More generally,only left-handed parti les (and right-handed antiparti les) intera t via the weak intera tion.

The

T

operator

Thetime-reversaloperator

T

is omplementarytotheparityoperator

P

,asittransforms

(t, x)

into

(

−t, x)

. Itis onserved by theele tromagneti andthe strongintera tions. The rst dire t observation of the violation of the

T

symmetry by the weak intera tion has been performed inthe study of the

B

0

system [13℄.

2

The

CP

and

CP T

operators

The previous results on

C

and

P

operators ould mean that the produ t of the

C

and

P

operator, denoted

CP

, is onserved by weak intera tions as this operator transforms left-handed neutrinosintoright-handed antineutrinos[14℄. Therst demonstrationof

CP

violationin naturehas been obtained through the study of the mixingof neutral mesons

su h asthe

K

0

[15℄ and the

B

0

[16℄

The

CP T

theorem states that the Lagrangian of the SM must be invariant under the

CP T

produ t. This is related to Lorentz invarian e and lo ality. Sear hes for

CP T

violationhave for nownot found any signi antviolation.

Under the assumption of the

CP T

theorem, any observation of violation of the

T

or the

CP

symmetry results in the violation of

CP

or

T

, respe tively. This has led to the rstobservationoftime-reversalsymmetryviolationintheneutralkaonsystem,underthe

assumption of

CP T

[17℄. Additionally,measurementsof the

T

-violationhave onstrained the violation of

CP

by the strong intera tion tosmallerthan

10

−10

[18℄. This onstitutes

the strong

CP

problem, as the strong intera tion ould in prin iple violate

CP

. We onsider inthe rest of this dissertation that the strong intera tion if

CP

- onserving.

Se tion1.2.3des ribesthe me hanismof

CP

violationinthemixingofneutralmesons, along with the dierent types of

CP

violation in the Standard Model.

1.2.3 Neutral mesons mixing and

CP

violation

We onsider a neutral meson

|P

0

_i

su h that

|P

0

_{i 6= |P}

0

_i

, de aying to a nal state

f

. Thereare threedierent bases that an be used todes ribethe

|P

0

_i

-

|P

0

_i

system:

• |P

0

_i

and

|P

0

_i

(avour eigenstates);

•

_√

1

2

(

|P

0

_{i + |P}

0

_i)

and

1

√

2

(

|P

0

_{i − |P}

0

_i)

(

CP

-eigenstates);

• |P

L

i

and

|P

H

i

(eigenstates of the Hamiltonian).

In the two eigenstates of the Hamiltonian, L and H stand for light and heavy,

re-spe tively. The weak Hamiltonian onserves

CP

if and only if the eigenstates of the Hamiltonianare also eigenstates of

CP

.

The ee tive Hamiltonian

H

,des ribing the evolution of an initialstate ontaining a mixture of

|P

0

_i

and

|P

0

_i

(and ignoringnal states), an be writtenas

H = M

−

i

2

Γ,

(1.4)

where

M

and

Γ

are hermitianmatri es dened as

M =

m

11

m

12

m

∗

12

m

22



, Γ =

Γ

11

Γ

12

Γ

∗

12

Γ

22



.

(1.5)

The

CP T

invarian e requires that the diagonal terms of these matri es are equal. The introdu tionofthematrix

Γ

intheHamiltonianremovesitspropertyofhermiti ity,whi h is linked to the onservation of probability. This allows to introdu e the lifetimeof the

states des ribed by this Hamiltonian, as the square of the wave-fun tion that des ribes

them is de reasing exponentially with time.

The S hrödingerequation that governs the time-evolutionof a wave-fun tionis

i

d

|Ψ(t)i

dt

= H

|Ψ(t)i.

(1.6)

The integration of this equation appliedto the

|P

L,H

i

states yields

|P

L,H

(t)

i = |P

L,H

ie

−i

(

M

L,H

−

i

2

Γ

L,H

)

t

(1.7)

where

(M

L,H

−

i

2

Γ

L,H

)

are the orresponding eigenvalues of the Hamiltonian. The terms

p

and

q

are dened as the (nonvanishing) oe ients that allowto hange the basis

|P

L

i = p|P

0

i + q|P

0

i,

|P

H

i = p|P

0

i − q|P

0

i,

(1.8)

where

|p|

2

₊

_|q|

2

_{= 1}

. Conversely, these oe ients an be used towrite

|P

0

i =

1

2p

(

|P

L

i + |P

H

i),

|P

0

_{i =}

1

2q

(

|P

L

i − |P

H

i).

(1.9) We remark that if

p = q =

1

√

2

,

|P

L

i

and

|P

H

i

are exa tly equal to

1

√

2

(

|P

0

_{i + |P}

0

_i)

and

1

√

2

(

|P

0

_{i − |P}

0

_i)

, and

CP

is onserved.

Finally, ombining Eq. 1.9 and 1.7, the time-evolutionof

|P

0

_i

and

|P

0

_i

states writes

|P

0

_{i(t) = f}

+

(t)

|P

0

i +

q

p

f

−

(t)

|P

0

_i,

|P

0

_{i(t) = f}

+

(t)

|P

0

i +

p

q

f

−

(t)

|P

0

_i,

(1.10)

Table1.1 Experimental average for

∆m

and

∆Γ

indierent neutral-meson systemsfrom [19℄.

B

0

mixing parameters

∆m

d

( ps

−1

)

0.510

± 0.003

∆Γ

d

/

Γ

d

0.001

± 0.010

|q/p|

1.0009

_{± 0.0013}

B

0

s

mixing parameters

∆m

s

( ps

−1

)

17.757

± 0.020 ± 0.007

∆Γ

s

/

Γ

s

0.124

± 0.009

|q/p|

1.0038

_{± 0.0021}

where

f

_±

(t) =

1

2



e

−i

(

M

L

−

₂

i

Γ

L

)

t

_{± e}

−i

(

M

H

−

i

₂

Γ

H

)

t



_.

(1.11)

We dene the quantities

∆m = m

H

− m

L

, ∆Γ = Γ

L

− Γ

H

,

(1.12) and obtain

f

_±

(t) =

1

2



e

−im

L

t

_e

−

1

₂

Γ

L

t

h

₁

_{± e}

−i∆mt

_e

−

1

₂

∆Γt

i

_.

(1.13)

This fun tiongoverns the mixinginthe

|P

0

_i



|P

0

_i

system.

The

∆m

and

∆Γ

parameters an be predi ted from SM al ulations, and experimen-tallymeasured. Table1.1summarizesthe urrentworldaveragesforthe

B

0

and

B

0

s

meson systems [19℄.

We onsiderthe de ay of the

|P

0

_i

mesontoa nalstate

f

, asso iatedwith the ampli-tude

A

f

. 3 The parameter

λ

f

=

q

p

A

f

A

f

(1.14)

ontains the informationabout

CP

violation inthat de ay. Indeed, if the modulus of

λ

f

is not 1, or if its imaginary part is not vanishing,

CP

violation in the

|P

0

_{i → f}

de ay

o urs. Dening the threeobservables

3

Inthefollowing,the onjugatede ayof

|P

0

_i

to

f

isasso iatedwiththeamplitude

A

C

f

=

1

_{− λ}

2

f

1 + λ

2

f

,

(1.15)

S

f

=

2

_ℑ(λ

f

)

1 + λ

2

f

,

(1.16)

A

∆Γ

f

=

−

2R(λ

f

)

1 + λ

2

f

,

(1.17) (1.18)

the de ay rate of

|P

0

_i

as afun tion of time writes

Γ(t)

_∝

e

−Γ

|P 0i

t

2τ



cosh

 ∆Γt

2



+

_A

∆Γ

_f

sinh

 ∆Γt

2



+ (C

f

cos (∆mt)

− S

f

sin (∆mt))



,

(1.19) where

τ = (

Γ

L

+Γ

H

2

)

−1

,

Γ

|P

0

_i

=

Γ

L

+Γ

H

2

, and

∆Γ = Γ

L

− Γ

H

. It is ne essary to perform a time-dependent analysis of a de ay in order to measure all the

CP

-violation ee ts with pre ision, aswellastodeterminethe avourof theneutralmesonthatde ays(tagging).

In the ase where several hannels ontribute to the total amplitude, the amplitudes

A

and

A

of the total de ay an be written

A =

X

i

A

i

e

i(φ

i

−δ

i

)

,

A =

X

i

A

i

e

i(φ

i

+δ

i

)

,

(1.20) wherethe sum runsoverthe hannels ontributingtothe amplitudeand

A

i

isthe magni-tudeof the ontributionof ea h hannel. Thephases

φ

i

and

δ

i

are the

CP

- onservingand

CP

-violating omponents of the phase orrespondingthe ea h hannel,respe tively. The ee t of the

CP

symmetry an onlyindu e a dieren ein phase, not magnitude,in ea h hannel taken separately. However, inthe presen e of two ormore ontributing hannels,

the dieren e inthe patternof interferen e indu ed bythe

CP

-violatingphase an result in

CP

violation inthe de ay.

Three types of

CP

violation sour es an be distinguished, with dierent physi al in-terpretations.

CP

violation in de ays

In presen e of several ontributions to the amplitude that both have a relative

CP

- onserving phase and dierent

CP

-violatingphases, the rate ofade ay and its onjugate may be dierent. Indeed, in the ase of two ontributingdiagrams,

wheretherelative

CP

- onservingphasebetweendiagrams1and2is

2φ

,and

δ

1,2

isthe

CP

-violatingphase between these diagrams. Ifboth the

CP

- onserving andthe

CP

-violating phases are not 0,the de ay rates related to

A

and

A

are dierent.

Thisis theonly possibletype of

CP

violation inde ays of hargedmesonsorbaryons.

CP

violation through mixing

As underlined before,

CP

violation an be indu ed by the mixing of neutral mesons. Considering for instan e Eq. 1.8,

CP

is violated in the mixing of neutral mesons if and only if

|

p

q

| 6= 1

. As shown inTab. 1.1, this ratio is onsistent with1 inthe ase ofthe

B

0

and

B

0

s

mesons.

CP

violation in interferen e between mixing and de ay

Anothertype of

CP

violation is asso iated to the interferen e between mixingand de ay pro esses of neutralmesons tothe same

CP

-eigenstate. Contrary tothe

CP

violationin de ay, it does not require several hannels to ontribute to the amplitude, as the

inter-feren e happens between the mixed and unmixed amplitudes. This type of

CP

violation o urs in ase that the imaginary part of

λ

takes a nonzero value. The parameter that outlines this measurement is ontained inthe term

S

f

.

Dire t and indire t

CP

violation

CP

violation anbealternatively lassiedintodire torindire t

CP

violation. Dire t

CP

violation orrespondsto

CP

violationthrough de ay, whereasindire t

CP

violationrefers to

CP

violation through mixing or through the interferen e between mixing and de ay. AsshowninTab. 1.1,the

CP

violationinthe mixingofthe

B

0

meson an benegle tedin

most ases, and thus indire t

CP

violation often refers to interferen e between mixing and de ay when onsidering de ays of the

B

0

meson.

1.2.4 The CKM matrix and the KM me hanism

As des ribed in Se . 1.1, avour eigenstates are eigenstates of the ele troweak

intera -tion. They are however not ne essarily eigenstates of the strong intera tion, or of the

Hamiltonian. This se tion des ribes how the hange of basis between eigenstates of the

ele troweakintera tionandoftheHamiltonianintrodu esanirredu iblephaseintheSM,

and thus to

CP

violation, whenthree or morequark generations exist.

We onsider the hange of basis between the quark eigenstates of avour and of the

Hamiltonianby the matri es

U

f

L

and

U

f

R

, dened su h as

M

mass

=



U

_L

f



†

M

flavour

U

R

f

,

(1.22)

where

M

mass

and

M

flavour

are the matri esthat des ribe quark urrents inthe mass basis and the avour basis, respe tively. The idea of dierent bases to des ribe the mass

and the weak eigenstates was rst proposed by Cabibbo [20℄. The motivation was to

explain the suppression of the de ay of strange parti les, and thus the long lifetime of

these parti les. The GIM me hanism is an extension of this on ept that requires the

existen e of a se ond-generationup-type quark, the

c

quark [21℄. It allows toforbid any avour- hanging neutral urrent at tree-level inthe Standard Model.

A

2

× 2

unitary matrix

V

an bedes ribed by a single real parameter. Startingfrom the original

2

× 4

real parameters (e.g. magnitudesand phases), unitarity relationsstate that

∀(i, j),

X

k

V

ik

V

jk

∗

= δ

ij

,

∀(i, j),

X

k

V

ki

V

kj

∗

= δ

ij

,

(1.23)

whi hremoves4parameters. Finally,phasesbetweenquark urrentsare physi ally

mean-ingless, thus removing

2N

− 1 = 3

parameters, leaving only one real parameter. The omparisonwith realorthogonal matri esleadstodening thisparameter asanangle

θ

C

, and so

V =

 cos θ

C

sin θ

C

− sin θ

C

cos θ

C



.

(1.24)

This idea has rst been proposed with the two lightest quark generations, this angle

θ

C

being named the Cabibbo angle. Kobayashi and Maskawa have proposed to extend this idea to three quark generations and showed how this resulted in the introdu tion

of a physi al phase in the SM, responsible for

CP

violation [22℄.

4

Indeed, an extension

of the dis ussion above shows that a

3

× 3

unitary matrix an be des ribed by 4 real parameters, one of whi h being an irredu ible phase. The

3

× 3

basis- hanging matrix in the ase of three quark generations is alled the Cabibbo-Kobayashi-Maskawa (CKM)

matrix. The 2008 Nobel prize of physi s was awarded to Kobayashi and Maskawa after

pre ise measurements of

CP

violation showed that it was indeed onsistent with their des ription.

The CKM matrix is writtenas

V

CKM

= (U

L

u

)

†

U

R

d

=





V

ud

V

us

V

ub

V

cd

V

cs

V

cb

V

td

V

ts

V

tb





.

(1.25)

Itisimportanttonotethat,duetothefa tthatavour- hangingneutral urrents(FCNC)

are forbidden at tree-level in the SM, up-type quarks are only paired with down-type

quarks, and inversely. Following the dis ussion on the number of degrees of freedom,

this matrix an beparameterized by three real parametersand one imaginaryparameter.

These three angles are dened as

θ

12

(= θ

C

)

,

θ

13

, and

θ

23

. For ea h angle

θ

ij

, its osine and sine are noted

c

ij

and

s

ij

, respe tively, and the CKM matrix may be writtenas

4

V

CKM

=





c

12

c

13

s

12

c

13

s

13

e

−iδ

−s

12

c

23

− c

12

s

23

s

13

e

iδ

c

12

c

23

− s

12

s

23

s

13

e

iδ

s

23

c

13

s

12

s

23

− c

12

c

23

s

13

e

iδ

−c

12

s

23

− s

12

c

23

s

13

e

iδ

c

23

c

13





,

(1.26)

where

δ

is the irredu ible phase. Sin e the term

s

12

is small, this form of the CKM matrix an be written as anexpansion of

λ = s

12

≈ 0.22

, and three parameters that are lose to unity:

A =

s

23

λ

2

,

ρ =

s

13

λs

23

cos δ

, and

η =

s

13

λs

23

sin δ

. This yields the Wolfenstein

parameterization [23℄

V

CKM

=





1

_{− λ}

2

_/2

_λ

_Aλ

3

_(ρ

_{− iη)}

−λ

1

_{− λ}

2

_/2

_Aλ

2

Aλ

3

₍₁

_{− ρ − iη)}

_−Aλ

2

₁





+

O(λ

4

).

(1.27) Finally,the

ρ = ρ



1

₋

λ

2

2



, η

= η



1

₋

λ

2

2



(1.28)

terms an be dened toyield the Burasparameterization [24℄ whi his is validat

O(λ

5

₎

V

CKM

=





1

− λ

2

_/2

_{− λ}

4

_/8

_{λ +}

_O(λ

7

₎

_Aλ

3

_(ρ

_{− iη)}

−λ + A

2

_λ

5

_[1

_{− 2(ρ + iη)] /2}

₁

_{− λ}

2

_/2

_{− λ}

4

_{(1 + 4A}

2

_{) /8}

_Aλ

2

₊

_O(λ

8

₎

Aλ

3

₍₁

_{− ρ − iη)}

_−Aλ

2

_{+ Aλ}

4

_[1

_{− 2(ρ + iη)] /2 1 − A}

2

_λ

4

_/2





(1.29)

1.2.5 The unitarity triangles

The unitarity of the CKM matrix an beformulatedas

L

∗

_i

L

j

=

X

i

V

_ik

∗

V

jk

= δ

ij

,

C

_i

∗

C

j

=

X

i

V

_ki

∗

V

kj

= δ

ij

,

(1.30)

where

L

i(j)

and

C

i(j)

are the

i

th

(

j

th

) line and olumn, respe tively. These unitarity

onstraints yield9 equations,among whi h six involve dierent lines or olumns

5

:

5

V

_ud

∗

V

us

+ V

cd

∗

V

cs

+ V

td

∗

V

ts

= 0,

(1.31)

V

_ud

∗

V

ub

+ V

cd

∗

V

cb

+ V

td

∗

V

tb

= 0,

(1.32)

V

us

∗

V

ub

+ V

cs

∗

V

cb

+ V

ts

∗

V

tb

= 0,

(1.33)

V

_cd

∗

V

ud

+ V

cs

∗

V

us

+ V

cb

∗

V

ub

= 0,

(1.34)

V

_td

∗

V

ud

+ V

ts

∗

V

us

+ V

tb

∗

V

ub

= 0,

(1.35)

V

_td

∗

V

cd

+ V

ts

∗

V

cs

+ V

tb

∗

V

cb

= 0.

(1.36) These onstraints an be represented by trianglesin the omplex plane, denoted by

uni-taritytriangles. Mostofthemin ludetermsofdierentordersin

λ

,thus orrespondingto at triangles. Equation 1.32 and 1.35,however, onlyin lude termsthat are proportional

to

λ

3

.

Thetriangle dened by Eq. 1.32is often alledthe unitaritytriangle,as ithas been

the fo us of many measurements. Indeed, the three sides of this triangleare all of order

λ

3

, ompared to other triangles that are atter. Alternatively, it is referred to as the

B

0

unitarity triangle. Its sides are normalized by

V

∗

cd

V

cb

, and itsinternalangles are thus dened as:

α = arg



−

V

tb

∗

V

td

V

∗

ub

V

ud



= arg



−

1

− ρ − iη

ρ + iη



+

O(λ

2

_),

(1.37)

β = arg



−

V

cb

∗

V

cd

V

∗

tb

V

td



= arg



1

1

_{− ρ − iη}



+

O(λ

4

_),

(1.38)

γ = arg



−

V

_V

ub

∗

_∗

V

ud

cb

V

cd



= arg (ρ + iη) +

_O(λ

2

).

(1.39)

Figure 1.1 shows a sket h of this unitarity triangle spe ifying the angles and the

expres-sions of the lengthsof its sides [19℄.

Theangles and thesides of the triangle an bemeasured experimentally,to onstrain

thelo ationofitsapex.

6

Thesedierent onstraintssetbythemeasurementsmustoverlap

in atleast one regionof spa e sothat the unitarity of the CKM matrix isrespe ted.

Fig-ure 1.2 shows the status of the onstraintsonthis unitaritytriangle,fromthe CKMtter

ollaboration [19℄. These onstraints arise from the measurement of physi s observables

by several experiments. They in lude

•

themeasurementof

ε

K

and

ε

′

K

(

CP

-violatingparametersoftheneutralkaonsystem) [26℄;

•

the onstraint on

∆m

d

,measured rst by the UA1 [27℄and ARGUS [28℄ ollabora-tions; urrent world average is dominated by

B

-fa tories and LHCb ;

6

Thefreedomtosettheoriginofthereferentialanditsorientation anbeusedtosettwoofthetree apexesof thetriangleto0and1,leavingonlyoneapextobedetermined.

•

the onstrainton

∆m

s

, rstly measured by CDF [29℄;LHCb [30℄is dominatingthe urrent world average

7

;

•

the measurement of

β

performed in

b

→ ccs

modes by BaBar [31℄, Belle [32℄, and LHCb [33℄;

•

the measurement of the angle

α

, measured in time-dependent analyses of

b

→ uud

de ays su has

B

→ ππ

,

B

→ ρρ

, and

B

→ ρπ

;

•

the onstraint on

γ

, set with the best pre ision in harmed

B

tree de ays, and measuredbyCDF,BaBar,Belle,andLHCb . Itisoneoftheleastknown parameters

of the

B

0

unitarity triangle.

The mixingphase between the

B

0

s

and the

B

0

s

is noted

φ

s

,and isequal to

φ

s

=

−2β

s

= arg



−

V

ts

V

tb

∗

V

cs

V

cb

∗



,

(1.40)

where

β

s

is one of the anglesof the

B

0

s

unitarity triangledened by Eq.1.33. The LHCb experimentdisposes of alarge sample of

B

0

s

mesons that allows ittoimprove onstraints on this triangle.

1.2.6

B

0

os illations and the

β

angle Asdis ussed in Se . 1.2.3,avoured neutralmesons(

K

0

,

D

0

,

B

0

, and

B

0

s

)os illate when they propagate. The short-range terms related to these os illations an be des ribed at

rst order by box diagramslikethose shown in Fig.1.3. Long-range terms and upper

or-7

The ratio

∆m

d

/

∆m

s

is leaner thanthe individualobservables,asit an elssomehadroni un er-tainties.

CKMtter ollaboration [25℄

Figure 1.3  Se ond-order weak intera tion Feynman diagrams that give rise to the mixing of the

B

0

meson. The virtualloopinboth diagramsis dominatedbythetop-quark.

ders arenegle ted. The ontributionfromvirtualquarksinsideoftheloopare dominated

by the top-quark. It is then a very good approximation to onsider the amplitude tobe

proportionalto

V

tb

V

∗

td

/V

tb

∗

V

td

,whosephaseisequalto

−2β

at

O(λ

4

₎

. Thisexpression also

yieldsthat

|q/p| = 1 + O(λ

4

₎

,thusstrongly suppressing

CP

violation inthe mixingof

B

0

The angle

β

an be extra ted from various hannels that allowto measure the inter-feren e between the mixing and the de ay of

B

0

mesons. Considering a

B

0

_{→ f}

de ay,

where

f

is a

CP

eigenstate and only one pro ess ontributes to the amplitude, no dire t

CP

violationis possible and

S

f

= sin (arg (λ

f

)) = sin



arg

 q

p

A

f

A

f



= η

f

sin 2β,

(1.41)

where

η

f

=

±1

is the eigenvalue of the

f

nal state. The observable

S

f

an be extra ted from ananalysis that measures

Γ(t)

(time-dependent analysis).

De aysoftheform

B

0

_{→ K}

0

S

(K

0

L

)(cc)

aredominatedbythetree-leveltransition

b

→ ccs

and thus allow for a lean measurement of the angle

β

by means of a time-dependent analysis. This allows toextra t a lean measurement of

β

inmodes where nosigni ant ontribution from NP pro esses is expe ted. This value an then be ompared to the

value of

β

frommodes that in ludea virtual loop. Charmless

B

0

de ays involve an underlying

b

→ qqs

transition. They are strongly suppressed at tree level as the only tree-level ontribution involves a

b

→ u

transition, that is suppressed by a fa tor

λ

2

in bran hing fra tions ompared to a

b

→ c

transition. Figure 1.4 shows a ompilation of the CKM angle

β

and of

β

eff

as of 2014 [19℄, in the

b

_{→ ccs}

and the

b

→ qqs

transitions, respe tively. Thesetwoaverages are ompatible,but most of the

b

→ qqs

measurements are smallerthan measurementsin

b

→ ccs

modes.

1.3 Amplitude analyses on epts

1.3.1 Three-body parti le de ays and the Dalitz plot

The dierential ross-se tion asso iated with the de ay of a parti le of mass

M

and mo-mentum

P

into

n

parti lesof momenta

p

i

and energies

E

i

is

dΓ =

(2π)

4

2M

|M|

2

dΦ

n

(P ; p

1

...p

n

),

(1.42) where

dΦ

n

(P ; p

1

, ...p

n

) = δ

4

(P

−

n

X

i=1

p

i

)

n

Y

i=1

d

3

_p

i

(2π)

3

_2E

i

(1.43)

is thephase-spa e elementof volume,and thes attering matrix

M

ontains allthe infor-mation relatedto underlyingdynami s (su has resonan es or hadroni fa tors).

Conser-vation of momentum is ensured by the Dira fun tion

δ

.

Inthe ase of three-body de ays, the previous equationbe omes

dΓ =

1

(2π)

5

1

16M

2

|M|

2

dE

1

dE

3

dαd(cos β)dγ

(1.44)

Figure 1.4  World average of

β

from [19℄, extra ted from

b

→ ccs

de ays (left) and

b

→ qqs

de ays (right). The worldaverage from

b

→ ccd

isalso indi atedin theright hand-sidegure.

where

E

1

and

E

3

aretheenergyofparti les1and3intherestframeofthemotherparti le. The angles

α

,

β

, and

γ

are the Euler anglesthat denethe plane where momentaof the daughters are ontained. Here, the initialtwelve degrees of freedoms are redu ed to ve

when the onservation of momentum and the masses of the three nal-state parti les is

taken into a ount.

Inthe aseofthede ayofa(pseudo-)s alarparti leintothree(pseudo-)s alarparti les,

the pro essisisotropi . This meansthatthe dependen yonangles an beintegrated out,

furtherredu ingthenumberofdegreesoffreedomfromvetotwo. Equation1.44be omes

dΓ =

1

(2π)

3

1

8M

|M|

2

dE

1

dE

3

.

(1.45)

This equation an berewritten as

dΓ =

1

(2π)

3

1

32M

3

|M|

2

dm

2

₁₂

dm

2

₁₃

,

(1.46)

wherethe

m

ij

massesarethe invariantmassesof theparti lepair

ij

. This amplitudeonly depends on two variables, whi h allows to represent the whole phase-spa e on a single

plane. A graphi representation of this plane is alleda Dalitz plot [34℄.

The onservationofmomentumandthe massofthemotherparti leset onstraintson

(

m

₂₃

)

_max

0

1

2

3

4

5

0

2

4

6

8

10

m

₁₂

(

Ge

V

2

₎

m

23

(

Ge

V

2

)

(m

₁

+m

₂

)

2

(M−m

₃

)

2

(M−m

₁

)

2

(m

₂

+m

₃

)

2

(

m

₂₃

2

)

_m

_in

2

2

2

℄

M

℄

m

′

_θ

′

m

′

_{= 1}

π

arccos 2

m

ij

−m

m

ij

in

m

max

i

j

−m

m

ij

in

−1 ,

θ

′

_{= 1}

π

θ

ij

,

m

max(m

_ij

in)

ij

m

max

ij

=M −m

k

m

m

in

ij

=(m

i

+m

j

),

plot. [36℄

where

M

is the mass of the mother parti le and

m

i,j,k

is the mass of the daughter

i

,

j

, or

k

. The angle

θ

ij

is the heli ity angle of a given

ij

system, whi h is dened between the momentaof the parti les

k

and

i

in the

ij

rest frame. These oordinates are dened between 0 and 1,and the hange of oordinates between the regular Dalitzplot and the

square Dalitzplot is dened as

dm

2

_ij

dm

2

_jk

_{→ | det J|d m}

′

dθ

′

,

(1.51)

|det J| = 4





p

∗

_ij





|p

∗

_k

|

δm

ij

δ m

′

δ cos θ

ij

δθ

′

,

(1.52)

J

being the Ja obian of the transformation. The momenta

p

∗

ij

=

q

E

2

ij

− m

2

ij

and

p

∗

k

=

pE

2

k

− m

2

k

are dened in the

ij

rest frame. Figure 1.6 shows the distribution of this Ja obianover the square Dalitz plot.

Thisrepresentation is espe ially useful in harmless

B

de ays, as they populateareas oftheDalitzplot losetoitsboundaries,duetothesmallmassofintermediateresonan es

ompared tothe mass of the

B

meson. Additionally, froma te hni al point of view, the square shape of this plot allows to bin the plane more easily.

A major dieren e between the usual Dalitz plot and the square Dalitz plot is that

the squareDalitzplot areaisnot proportionaltothe elementofphase spa e. This means

that stru tures over the square Dalitz plot are not ne essarily related to any dynami s,

unlike in the usual Dalitz plane. This is illustrated by Fig. 1.6, as the Ja obian an be