Development of same side ﬂavour tagging algorithms for ...cds.cern.ch/record/2015250/files/CERN-THESIS-2015-040.pdf · di loro, così da ottenere un unico algoritmo di “Same Side

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Università degli Studi di Milano Bicocca

FACOLTÁ DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Tesi di Laurea Magistrale in Fisica

Development of "same side" flavour tagging algorithms

for measurements of flavour oscillations and

CP violation in the B0 mesons system

Relatore: Prof.ssa Marta Calvi

Correlatore: Dott. Basem Khanji

Candidato: Davide Fazzini

Matricola n◦: 727161

Anno Accademico 2013 - 2014

I’ll hold on to the world tight someday.

I’ve got one finger on it now; that’s a beginning.

Ray Bradbury, Farenheit 451

I

Abstract

In this thesis new developments of Flavour Tagging algorithms for the LHCb experiment

are presented. The Flavour Tagging is a very usefull tool which allows to determine the

flavour of the reconstructed particles, such as the B0 mesons. A correctly identification of

the flavour is fundamental in certain measurements such as time-dependent CP violation

asymmetries or the B0 ↔ B0 oscillations. Both these type of measurements are exploited by

LHCb experiment in the research of new physics beyond the Standard Model.

The new developments achieved in this work concern an optimization of the Same Side

Tagger algorithms, using protons and pions correlated in charge with the signal B0 to infer

its initial flavour. Then two combinations are implemented: the first is a combination of

the SS Pion Tagger (SSπ) and the SS Proton Tagger (SSp) in a unique Same Side (SS) tagging

algorithm; the second one is the final combination (SS +OS) of this new SS Tagger with the

Opposite Side (OS) Tagger combination already implemented.

To unfold the signal from the background events it has taken advantage of the sPlot

technique, which allows to calculate a per-event sWeight exploiting a discriminant variable,

i.e. the invariant mass of the B0 meson. The new SS taggers are implemented by means a

multivariate analysis based on a Boost Decision Tree (BDT) algorithm. The goal of this BDT

is to optimize the separation between “signal” (right charge correlated) and “background”

(wrong charge correlated) particles and to identify the most probable tagger candidate. The

input variables used to train the BDT include both kinematic and geometric variables. Then

the sample is divided in categories according to the BDT output value and for each one a

mistag probability is estimated by means an unbinned fit on the asymmetry oscillations.

This procedure allows to predict the mistag value (η) of a certain event directly from the

BDT response.

This analysis is performed on the data sample collected by the LHCb experiment in 2012,

corresponding to the B0 −→ D−(→ K+π−π−)π+ decay channel. Then the data samples

collected in the 2011, corresponding to the same decay channel and using two different

II

event selections, are used to obtain a validation of the flavour oscillation calibration. In

order to verify the goodness of the calibration both samples are divided in categories and

the true mistag (ω) is calculated through an unbinned fit. Thus a plot η vs ω can be used

to check the corrected calibration. An additional validation is performed on a different data

sample corresponding to the B0 −→ K−π+ decay mode. As last step the systematic effects

are studied to check the dependence of the tagging response on the event properties.

The new SSπ provides a tagging effective efficiency εe f f = 1.64± 0.07%, showing an

improvement of the performance by about 20% with respect to previous tuning. On the

other hand the new SSp yields a tagging power compatible to the result achieved with the

previous tuning (i.e. εe f f = 0.47± 0.04%). The two combinations SS and SS + OS provide

a tagging effective efficiency εe f f = 1.97± 0.10% and εe f f = 5.09± 0.15% respectively. The

algorithms developed in this thesis will be available as new taggers for the next CP violation

measurements at the LHCb experiment.

III

Sintesi

In questo progetto di tesi sono presentati nuovi sviluppi riguardanti gli algoritmi di “etichet-

tatura del sapore” (Flavour Tagging) per l’esperimento LHCb. Al centro degli studi condotti

a LHCb vi sono l’osservazione di decadimenti rari dei quark b e c e le misurazioni di vi-

olazione di CP, che potrebbero rivelare un nuovo tipo di fisica non spiegabile tramite il

Modello Standard. Per raggiungere l’elevata precisione richiesta da queste misure, è di fon-

damentale importanza ottenere una corretta identificazione del “sapore” degli adroni pe-

santi ricostruiti, come ad esempio i mesoni B0. A tale scopo la tecnica di Flavour Tagging si è

dimostrata essere un metodo molto efficace.

In particolare gli sviluppi ottenuti riguardano un’ottimizzazione degli algoritmi di “Same

Side Tagging” che, sfruttando la correlazione di carica presente tra il mesone di segnale e

il pione (SSπ), o protone (SSp), generato dalla sua frammentazione, cercano di determi-

narne il sapore. Successivamente le risposte di questi due tagger sono state combinate tra

di loro, così da ottenere un unico algoritmo di “Same Side Tagging” (SS); in un’ultima fase

si è proseguito con la sua combinazione con un “Opposite Side Tagger” (OS) generale, già

implementato.

Per separare i contributi di segnale e fondo presenti nelle n-tuple utilizzate, è stata imp-

iegata la tecnica degli sPlot la quale, sfruttando la massa invariante del mesone di segnale

come “variabile discriminante”, permette di attribuire a ciascun evento un peso (sWeight).

L’implementazione degli algoritmi è stata sviluppata attraverso un’analisi multi-variata

basata su un “Albero di Decisione Potenziato” (Boost Decision Tree, BDT), il cui obiet-

tivo è quello di ottimizzare la separazione delle tracce di segnale (correttamente correlate

in carica) da quelle di fondo (con la correlazione di carica errata) e di identificare il miglior

candidato per il tagging. Per migliorare l’efficacia di questa identificazione la BDT viene

allenata sia con variabili cinematiche che geometriche. In seguito il campione analizzato

viene diviso in categorie secondo la risposta fornita dalla BDT stessa e per ognuna di esse

viene effettuato un fit non binnato alle oscillazioni di sapore per determinare la probabilità

IV

di errata etichettatura (mistag). Tramite questo procedimento è possibile predire la mistag

(η) associata ad ogni evento direttamente dal valore fornito dalla BDT.

L’analisi è stata effettuata su un campione di dati corrispondente al canale di decadi-

mento B0 −→ D−(→ K+π−π−)π+, raccolto da LHCb nel corso del 2012. Per eseguire

dei controlli sulla stabilità degli algoritmi implementati, sono stati utilizzati due campioni

contenenti eventi raccolti durante il 2011 nello stesso canale di decadimento. La differenza

nei due casi risiede nel diverso contributo della componente di fondo, dovuta ad una se-

lezione degli eventi di segnale più larga in uno dei due casi. Per verificare la bontà della

calibrazione entrambi i campioni sono stati suddivisi in categorie, in modo da poterne cal-

colare la reale mistag (ω) attraverso un fit non binnato delle oscillazioni di sapore. É stato

quindi possibile controllare la calibrazione attraverso un grafico che mettesse in relazione

ω con η. È stato effettuato successivamente anche un controllo su un differente canale di

decadimento utilizzando un campione di eventi B0 −→ K−π+. Infine alcune sistematiche

sono state studiate in modo tale da poter valutare la presenza di eventuali dipendenze dalle

proprietà degli eventi.

I risultati finali ottenuti con il SSπ mostrano un’efficienza efficace di tagging εe f f =

1.64± 0.07% con un miglioramento delle prestazioni del 20% rispetto al tagger precedente-

mente sviluppato. L’efficienza del SSp è compatibile con quella raggiunta dal tagger attuale,

εe f f = 0.47± 0.04%. Le due combinazioni, SS e SS + OS invece forniscono rispettivamente

un’efficienza efficace di εe f f = 1.97± 0.10% e εe f f = 5.09± 0.15%. Gli algoritmi sviluppati

in questa tesi saranno disponibili come nuovi tagger per le prossime misure di violazione

di CP eseguite a LHCb.

V

Table of Contents

Frontispiece I

Abstract II

Sintesi IV

Table of Contents VIII

List of Figure XI

List of Table XV

Introduction 1

1 Theoretical Introduction 3

1.1 The Standard Model 3

1.2 Fermions and Bosons 5

1.3 The interactions in the SM 6

1.3.1 The electroweak interaction 6

1.3.2 The strong interaction 7

1.3.3 The Higgs mechanism 8

1.4 The CKM formalism 9

1.5 CP violation 13

1.5.1 Mixing of neutral pseudoscalar mesons 14

1.5.2 Types of CP Violation 16

2 The LHCb experiment 20

2.1 The Large Hadron Collider 20

2.2 b production at LHCb 22

VI

2.3 The LHCb detector 23

2.3.1 The beam pipe 24

2.3.2 The VErtex Locater 24

2.3.3 The Tracking System 25

2.3.4 The Magnet 28

2.3.5 The Ring Imaging Cherenkov 30

2.3.6 The calorimeter system 32

2.3.7 The Muon Stations 34

2.3.8 Trigger 35

3 Same Side Pion Tagger 37

3.1 The Flavour Tagging 37

3.1.1 Definitions 38

3.1.2 Same Side Taggers 40

3.2 Same Side tagger 41

3.3 SSπ tagger development using 2012 data sample 42

3.3.1 sWeights estimation 42

3.3.2 Training of the SS pion tagger 45

3.3.3 Performance and calibration 49

3.4 Validation on the 2011 data sample 55

3.5 Validation on the B0 → K+π− 2012 data sample 58

4 Same Side Proton Tagger 65

4.1 SSp tagger development using the 2012 data sample 65

4.1.1 SS proton training 65




5 Tagger combination 76

5.1 Combination of taggers 76

5.2 SSp and SSπ combination 77

5.2.1 Combination of the SS taggers on the B0 → D−π+ 2012 data sample 78

5.2.2 Combination on B0 → D−π+ 2011 data sample 80

5.2.3 Combination on the B0 → K+π− 2012 data sample 82

VII

5.3 SS and OS combination 83

5.3.1 Combination of SS taggers with the OS tagger on the B0 → D−π+

2012 data sample 84

5.3.2 Combination on the B0 −→ D−π+2011 data sample 87


5.4 Measurement of ∆md 90

6 Systematics 93

6.1 Systematic uncertainties 93

6.2 Dependence of the SS tagging on pT of the signal B 94

6.3 Dependence of the SS tagging on the magnet polarity 98

6.4 Dependence of the SS tagging performances on the B flavour 99

6.4.1 Dependence on the B flavour at decay 99

6.4.2 Dependence on the B flavour at production 101

7 Conclusion 104

A sPlots technique 106

A.1 sPlot properties 108

A.2 sPlot application 108

B Boost Decision Tree classifier 110

B.1 Boosting method 112

C Monte-Carlo analyses 113

D Validation on a different cuts selection 116

D.1 Validation for the SS Pion Tagger 119

D.2 Validation for the Proton Tagger 119

D.3 Validation for the SS Tagger combination 120

D.4 Validation for the SS+OS Tagger combination 121

Bibliografia 124

Ringraziamenti 125

VIII

List of Figures

1.1 B0d and B0

s Unitary Triangles 11

1.2 Combined fit results of the B0d Unitarity Triangle 14

1.3 Box diagrams of B-mixing 18

1.4 Diagrams of CPV in interference 18

2.1 A schematic representation of the LHC collider 21

2.2 Feymann diagrams for the bb production 22

2.3 The LHCb acceptance 23

2.4 A y-z section of the LHCb detector 24

2.5 Layout of TT detection layers 26

2.6 Layout of IT detectors 26

2.7 A section of OT station 27

2.8 Track classification 28

2.9 Tracking system in the magnet 29

2.10 Dominant component of the magnetic field 29

2.11 Perspective view of the LHCb dipole magnet 30

2.12 Schematic view of RICH detectors 31

2.13 Cherenkov angle vs particle momentum for RICH radiators 31

2.14 Kaon(left) and proton(right) identification and pion misidentification as a

function of the track momentum measured on 2011 data. Plots for two differ-

ent ∆ log L are shown. 32

2.15 Side view of the LHCb muon system 34

3.1 B tagging sketch 38

3.2 Feymann diagrams for the B0 hadronization 41

3.3 Mass fit for the B0 → D−(Kππ)π+ 2012 data sample 44

IX

3.4 Output of the SSπ BDT 48

3.5 Distribution of the input variables used in the SSπ BDT 49

3.6 Time distribution of the events in 2012 data sample 50

3.7 Mixing asymmetry for signal events 52

3.8 Calibration plots for the B0 → D−π+ 2012 data sample 54

3.9 Mass fit for the B0 → D−(Kππ)π+ 2011 data sample 56

3.10 Time distribution of the events in 2011 data sample 56

3.11 Calibration for the B0 → D−π+ 2011 data sample 57

3.12 Calibration for the B0 → D−π+ 2011+2012 data sample 58

3.13 Mass fit for the B0 → K+π− 2012 data sample 60

3.14 Time distribution of the events in B0 → K+π− 2012 data sample 62

3.15 Calibration for the B0 → K+π− 2012 data sample 63

3.16 Comparison of the Bpt distributions for different decay channels 64

4.1 Output of the SSp BDT 67

4.2 Distribution of the input variables used in the SSp BDT 68

4.3 Mixing asymmetry for signal events 70

4.4 Calibration plots for the B0 → D−π+ 2012 data sample 71


4.6 Calibration for the B0 → D−π+ 2011+2012 data sample 73


5.1 Mixing asymmetry for signal events for the B0 → D−π+ 2012 data sample 79




5.5 OS calibration for the B0 → D−π+ 2012 data sample 84

5.6 Mixing asymmetry for the B0 → D−π+ 2011 data sample 85

5.7 OS calibration for the B0 → D−π+ 2012 data sample 86


5.9 SS+OS calibration for the B0 → K+π− 2012 data sample 90

5.10 Mixing asymmetry plots to estimate ∆md using the B0 → D−π+ 2011 data

sample 91

6.1 BpT distribution and the splitting in three bins 95

X

6.2 Calibration plots for the SS pion in BpT bins 96

6.3 Calibration plots for the SS proton in BpT bins 97

6.4 Calibration plots for the SS pion according to the magnet polarity 98

6.5 Calibration plots for the SS proton according to the magnet polarity 99

6.6 Calibration plots for the SS pion according to the Bid defined from the final

state 100

6.7 Calibration plots for the SS proton according to the Bid by the final state 101

6.8 Calibration plots for the SS pion according to the Bid from the tagger charge 102

6.9 Calibration plots for the SS proton according to the Bid from the tagger charge 103

A.1 Distributions of signal and background for some variables 109

B.1 Sketch of a decision tree 111

D.1 Mass fit for the B0 → D−(Kππ)π+ 2011 data sample 118

D.2 Time distribution of the events in 2011 data sample 118

D.3 Calibration for the B0 → D−π+ 2011 data sample 119



D.6 SS+OS calibration for the B0 → D−π+ 2011 data sample 122

XI

List of Tables

1.1 Fermions list in the Standard Model 5

1.2 Boson list in Standard Model 6

1.3 Weak flavor quantum numbers of leptons and quarks 8

1.4 Angles of B0d triangle from UTfit 12

1.5 Value of Wolfenstein parameters from UTfit 13

1.6 Experimental values of VCKM parameters 13

2.1 Integrated luminosity delivered to LHCb 22

3.1 Selection cuts for the decay channel B0 → D−π+ 43

3.2 Results of the fit to the mass distribution 2012 data sample 44

3.3 Preselection cuts applied for SS pion tagging algorithm 46

3.4 Input variables used to train SSπ 47

3.5 Input variables ranking 48

3.6 Acceptance parameters for the 2012 data sample 50

3.7 Performances for the BDT categories determined from the asymmetry fit 51

3.8 Parameters of the 3rd polynomial for the B0 → D−π+ 53

3.9 Calibration parameters and tagging performances for the B0 → D−π+ 2012

data sample 53

3.10 Performance of the current SSπ tagger on the B0 → D−π+ 54

3.11 Results of the fit to the mass distribution 2011 data sample 55

3.12 Acceptance parameters for the 2011 data sample 55


data sample 57

3.14 Calibration parameters and tagging performances for the B0 → D−π+ 2011+2012

data sample 57

XII

3.15 Selection cuts for the decay channel B0 −→ K+π− 59

3.16 Results of the fit to the mass distribution B0 → K+π− 2012 data sample 61

3.17 Acceptance parameters for the B0 → K+π− 2012 data sample 62

3.18 Calibration parameters and tagging performances for the B0 → K+π− 2012

data sample 62

3.19 Comparison between the tagging powers for B0 → K+π− data sample 64

4.1 Preselection cuts applied for SS proton tagging algorithm 66

4.2 Input variables used to train SSp 67

4.3 Input variables ranking 68

4.4 Performances for the BDT categories determined from the asymmetry fit 69

4.5 Parameters of the 3rd polynomial for the B0 → D−π+ 71


data sample 71

4.7 Performance of the current SSp tagger on the B0 → D−π+ 72


data sample 72

4.9 Calibration parameters and tagging performances for the B0 → D−π+ 2011+2012

data sample 73

4.10 Calibration parameters and tagging performances for the B0 → K+π− 2012

data sample 74

4.11 Comparison between the tagging powers for B0 → K+π− data sample 75

5.1 Performances of the sub-sample with both taggers for the B0 → D−π+ 2012

data sample 78

5.2 Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2012 data sample 79

5.3 Performances of the SS combination on the 2012 data sample with 79

5.4 Performance of the current SS tagger on the B0 → D−π+ 80

5.5 Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample 80

5.6 Performances of the SS combination on the 2011 data sample 81

5.7 Performances of the SS combination on the B0 → D−π+ 2011+2012 data sample 82

5.8 Calibration parameters of the SS combination for the B0 → K+π− 2012 data

sample 82

5.9 Performances of the SS combination on the B0 → K+π− 2012 data sample 83

XIII

5.10 OS calibration parameters and tagging performances for the B0 → D−π+

2012 data sample 84

5.11 SS+OS calibration parameters for the B0 −→ D−(→ Kππ)π+ 2012 data sample 85

5.12 SS+OS performances of the sub-sample with both taggers for the B0 → D−π+

2012 data sample 86

5.13 Performances of the OS combination on the 2012 data sample 87

5.14 Performance of the current SS + OS tagger on the B0 → D−π+ 87

5.15 OS calibration parameters and tagging performances for the B0 → D−π+

2011 data sample 87

5.16 SS+OS calibration parameters for the B0 −→ D−π+ 2011 data sample 87

5.17 Performances of the SS+OS combination on the 2011 data sample 88

5.18 Performances of the SS+OS combination on the B0 → D−π+ 2011+2012 data

sample 88

5.19 OS calibration parameters and tagging performances for the B0 → K+π−

2012 data sample 89

5.20 SS+OS calibration parameters for the B0 → K+π− 2012 data sample 89

5.21 SS+OS tagging performances for the B0 −→ K+π− 2012 data sample 89

5.22 Fit results to estimate ∆md 92

5.23 Comparison of the mixing frequency ∆md results 92

6.1 Calibration parameters for the SS pion in BpT bins 95

6.2 Calibration parameters for the SS proton in BpT bins 97

6.3 Calibration parameters for the SS pion according to the magnet polarity 98

6.4 Calibration parameters for the SS proton according to the magnet polarity 99

6.5 Calibration parameters for the SS pion according to the Bid defined by the

final state 100

6.6 Calibration parameters for the SS proton according to the Bid defined by the

final state 101

6.7 Calibration parameters for the SS pion according to the Bid defined by the

tagger charge 102

6.8 Calibration parameters for the SS proton according to the Bid defined by the

tagger charge 103

C.1 Origin of the right charged correlated pions 113

C.2 Origin of the wrong charged correlated pions 114

XIV

C.3 Comparison of the results provide with MC and data 114

D.1 Results of the fit to the mass distribution 2011 data sample 116

D.2 Selection cuts for the decay channel B0 −→ D−(Kππ)π+ 117

D.3 Acceptance parameters for the 2011 data sample 118

D.4 Calibration parameters and tagging performances for the B0 → D−π+ 2011

data sample 119

D.5 Calibration parameters and tagging performances for the B0 → D−π+ 2011

data sample 120

D.6 Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample 120

D.7 Performances of the SS combination on the 2011 data sample with the second

event selection 121

D.8 OS calibration parameters and tagging performances for the B0 → D−π+

2011 data sample 121

D.9 SS+OS calibration parameters for the B0 → D−π+ 2011 data sample 122

D.10 SS+OS performances of the SS combination on the 2011 data sample with the

second cuts selection 122

XV

Introduction

The Standard Model (SM) is a relativistic quantum field theory which describes successfully

the interactions of fundamental particles up to the actual energies except the gravity. In this

theory when a physical system is invariant under a particular transformation a symmetry

arises and, as the Noether’s theorem stands, it is related to a conservation law by a one-

to-one correspondence. There are two types of symmetry: continuous, as rotations in space

or translation in time, and discrete, like charge conjugation (C) or parity (P). As it was

discovered in 1957 and in 1964, the weak interactions don’t conserve both P and C and

neither their product (CP).

The LHCb experiment is devoted to to the study of the b and c hadrons decay, and

in particular in the measurements of CP violation. To provide these measurements the

“Flavour Tagging” technique is exploited in order to know the initial flavour of the meson

reconstructed, such as a B0d. This procedure is performed by means of several algorithms,

using the informations from the b quark fragmentation which originates the signal B me-

son (Same Side tagging - SS) or the informations from the decay chain of the opposite B

(Opposite Side tagging - OS).

The aim of this thesis is to present some new developments of Flavour Tagging algo-

rithms for the LHCb experiment. In particular this thesis provides an optimization of the SS

taggers which use a pion or a proton created during the hadronization process of the sig-

nal B meson to infer its initial flavour. This new algorithms exploit a multivariate classifier

based on a “Boost Decision Tree” to choose the best tagger candidate providing a proba-

bility of the tagging decision to be correct. In order to improve this selection the BDT uses

both geometrical and kinematic variables related to the tracks, to the B meson or to the

event itself.

1

LIST OF TABLES

The data sample used to develop the two algorithms corresponds to B0 −→ D−(→

K+π−π−)π+ decay mode collected during the 2012 by LHCb. Then the 2011 data samples

corresponding to the same decay mode, but with different background contribution, are

used to test the performances and the calibration of the estimated tagging probability.

Then the results obtained with the SSπ and the SSp taggers are combined, in a first

time, in a unique SS tagger to achieve better performances and in a second time this new

SS tagger is combined with a general OS tagger, already implemented.

In Chapter 1 a summary of the theoretical background of the physics studied by LHCb

is reported. In Chapter 2 the LHCb spectrometer is described. In Chapter 3 the details of

the SSπ tagger implementation,its performances and calibration are reported. In Chapter 4

the development of the SSp tagger is described and its performances are shown. In Chapter

5 the combination of a unique SS tagger and the final combination of a SS + OS tagger

are described, showing their performances and calibration. In Chapter 6 some systematic

effects are analyzed for the SSπ and the SSp taggers. In Chapter 7 a summary of all the

results is reported.

2

1

Theoretical Introduction

Contents

1.1 The Standard Model 3

1.2 Fermions and Bosons 5

1.3 The interactions in the SM 6

1.3.1 The electroweak interaction 6

1.3.2 The strong interaction 7

1.3.3 The Higgs mechanism 8

1.4 The CKM formalism 9

1.5 CP violation 13

1.5.1 Mixing of neutral pseudoscalar mesons 14

1.5.2 Types of CP Violation 16

In this chapter the Standard Model (SM) of particle physics is shortly described. The

subatomic particles and the fundamental interactions are described in section 1.2 and in

section 1.3. The SM describes particles as fermion fields and their interactions are mediated

by the exchange of boson fields. Then a brief description of CP violation and the flavour

tagging technique are given in section 1.5 and 3.1.

1.1 The Standard Model

The Standard Model (SM) of particle physics is a quantum field theory which, combin-

ing the special relativity with quantum mechanics, describes three of the four fundamental

forces of nature (the Electromagnetic Force, the Weak Force and the Strong force).

Its Lagrangian is symmetric under the transformations of the SU(3) × SU(2) × U(1)

gauge group. SU(3) group describes the strong coupling while the SU(2)×U(1) describes

3

1 - Theoretical Introduction

the electroweak interaction (the unification of the weak and the electromagnetic forces pre-

dict by the model).

The Gravitational Force is not accounted in the Standard Model, though some theories

to unify the gravity with the Strong and Electroweak Forces are in development phase.

One hypothesis predicts that the gravitational force is mediated by a single boson, named

graviton, which posses an intrinsic spin of two unit.

The Standard Model consists of two type of elementary particles: fermions, which have

a odd-half integral spin, and bosons, which have an integer spin. The fermions (leptons

and quarks) are the building block of the matter and anti-matter. A description of a group

of fermion must be antisymmetric under interchange of two particles, it means that they

obey to the Fermi-Dirac statistic:

|1, , 2, ..., i, ..., j, ...N〉 = −|1, , 2, ..., j, ..., i, ...N〉

On the other hand the bosons are mediators of the interactions between the particles. A

description of a group of bosons must be symmetric under interchange of two particles and

thus they obey to the Bose-Einstein statistic:

|1, , 2, ..., i, ..., j, ...N〉 = |1, , 2, ..., j, ..., i, ...N〉

The last particle observed, foreseen by the Standard Model, is the Higgs Boson. It is a

massive, chargeless, boson with zero spin (thus, it’s a scalar boson). The Higgs field, trough

the mechanism of “Spontaneous Symmetry Breaking”, is the reason why some fundamen-

tal particles are massive, even though the symmetries controlling their interactions require

them to be massless. It also answers several other long-standing puzzles in physics, such as

the reason the weak force has a much shorter range than the electromagnetic force.

A relativistic quantum field theory based on a hermitian Lagrangian, invariant under

Lorentz transformations, is also invariant under the product of the tree operators C,P,T (CPT

theorem) but not under one transformation separately. C denotes the charge conjugation

transformation, turning particles in their respective anti-particles; the parity transformation

P inverts the space coordinates of the field while the time reversal operator change the sign

of time coordinate. According to CPT theorem particles and anti-particles must have equal

masses and decay times.

4


1st generation 2nd generation 3rd generation

Leptonsνe < 2 eV νµ < 2 eV ντ < 2 eV

e 511 KeV µ 105.7 MeV τ 1.78 GeV

Quarksu 2 MeV c 1.27 GeV t 173 GeV

d 5 MeV s 95 MeV b 4.18 GeV

Table 1.1: Fermions described in the Standard Model. The respective masses are given in parenthesis

1.2 Fermions and Bosons

The fermions are the matter constituents and can be divided in leptons and quarks. Accord-

ing to the Standard Model there are three generations of fermions organized by increasing

mass. Each generation consisting of a lepton, a neutrino, a positively charged quark, a neg-

ative charged quark and their respective anti-particles.

The leptons described in the theory are the electron (e−), the muon (µ−), the tau (τ−)

and their associated neutrinos (νe, νµ and ντ). Each lepton posses an anti-particle which is

exactly the same in all the observable ways except the charge, that is opposite in sign. They

are denoted as e+, µ+, τ+, νe, νµ, ντ1.

Six quarks exist: up (u), down (d), charm (c), strange (s), top (t), bottom (b). The posi-

tively charged quarks (u,c,t) carry 23 of the fundamental unit charge, while the negatively

charged quarks (d,s,b) carry −13 of the fundamental unit charge. Their relatively anti-particles

are denoted as u, d, c, s, t and b. Leptons and quarks are listed in Table 1.1 [1]. The quarks

posses an additional property, called color charge. There are three varieties of color: red,

green and blue and the respective anti-colors. The color charge is related to the strong force

just like the electric charge is related to the electromagnetic force.

Because of the Color Confinement (see section 1.3) quarks cannot be isolated singularly,

and therefore cannot be directly observed. Instead they clump together to form hadrons.

There are two types of hadrons:

• “mesons”: consist of bound states quark anti-quark pairs (qq) and have integral spin;

1The relationship between neutrinos and their anti-neutrinos, both chargeless, is more complicated than

charge conjugation.

5


interaction bosons mass relative strength

Electromagnetic γ 0 αem ∼ O(10−2)

WeakW± 80.4 GeV

αW ∼ O(10−6)Z0 91.2 GeV

Strong g (g1, . . . , g8) 0 αs ∼ O(1)

- H0 125.9 GeV -

Table 1.2: Bosons described in the Standard Model with their mass and relative strength of the in-

teraction.

• “baryons”: consist of bound states of three quarks (or anti-quarks) and have odd-half

integral spin;

Even though bound states of five quarks, called penta-quarks, are expected they have yet

to be observed.

Because the strong force doesn’t interact with hadrons, each quark bound state must be

colorless, just like the leptons. For the mesons this condition can happen combining a color

with its anti-color (e.g. red plus anti-red), while the baryons must posses a quark for each

color (thus, red plus green plus blue is equal to colorless).

1.3 The interactions in the SM

The bosons are the force-carries (γ, W±, Z0), which mediate the interaction between the

fermions and the Higgs boson (H0) that gives mass to the particles. These particles are

listed in Table 1.2 [1]. The interactions are introduced in the theory requiring that the the

SM Lagrangian is invariant under local gauge transformations of the SU(3)× SU(2)×U(1)

group.

1.3.1 The electroweak interaction

The photon (γ) is a massless, chargeless particle with a unit spin. It is the gauge boson

associated to the electromagnetic force. This interaction acts on all the charged particles

and it is easily visible at a macroscopic level. This force is described through the Quantum

Electrodynamics (QED) and is associated to the symmetry generated by the U(1) gauge

group.

6


The weak force is mediated by the Z0 and the W± gauge bosons.This interaction acts

only on the left-handed particles2, this is an effect of the V-A form (Vector minus Axial)

form of the Lagrangian, which contains terms that project out the left-handed component

of the state. For this reason the left-handed particles are represented as a isospin doublet,

while the right-handed particles are considered as isospin singlet. The symmetry associated

to this interaction is generated by the SU(2) gauge group.

The Standard Model predicts the unification of these two interactions into the SU(2)×

U(1) gauge symmetry, associating the massless gauge bosons ~Wµ = (W1µ, W2

µ, W3µ) and Bµ.

However this symmetry is broken by the Higgs mechanism, which decouples the weak

and the electromagnetic forces originating the photon (gauge field Aµ) and the three weak

bosons (gauge field Wµ and Zµ).

W±µ =W1

µ ± iW2µ√

2

Aµ = − sin θWW3µ + cos θW Bµ

Zµ = cos θWW3µ + sin θW Bµ

(1.1)

In these formulas θW is the Weinberg angle defined as θW = arctan (g′/g), where g and

g′ are the U(1)Y and SU(2)L couplings constant.

The weak Lagrangian that describes the coupling of the charged gauge bosons to the

fermions is:

Lew = − g√2

W+µ (νγµ(1− γ5)l + quγµ(1− γ5)qd + h.c.) (1.2)

The weak flavour quantum numbers related to the SM fermions are reported in Table

1.3.

1.3.2 The strong interaction

The strong interaction is described trough the Quantum Chromodynamics (QCD), a non-

Abelian gauge theory, introducing a new quantum number, named color. The color struc-

ture can be represented by the SU(3) gauge group. This force is mediated by 8 massless,2The handedness of a particles refers to its chirality determined by whether the particle transforms in a right-

or left-handed representation of the Poincaré group. However the Dirac spinors representation have both the

components, thus it is possible to define the projection operators which project out the component of the state.

The form of these operators is: (1± γ5)/2. If the particle is massless the chirality is the same as helicity, defined

as the projection of its spin relative to the direction of its momentum

7


generations I I3 Y Q

νeL

eL

νµL

µL

ντL

τL

1/2+1/2 -1/2 0

-1/2 -1/2 -1

eR µR τR 0 0 -1 -1uL

d′L

cL

s′L

tL

b′L

1/2+1/2 +1/6 +2/3

-1/2 +1/6 -1/3

uR cR tR 0 0 +2/3 +2/3

d′R s′R b′R 0 0 -1/3 -1/3

Table 1.3: The symbol I denotes the weak isospin and I3 is its third component, Q is the electric

charge (given in unity of the elementary charge e) and Y = Q − I3 is the weak hyper-

charge. The weak eigenstates (d’, s’, b’) are related to the mass eigenstates (d, s, b) trough

the CKM matrix, described in section 1.4

chargeless gauge bosons, called gluons, that posses two color varieties and can self-interact.

The gluons and the quarks are the only particles that carry non-vanishing color charge and

thus participate in strong interactions.

Unlike the other interactions, the strong force doesn’t diminish in strength with increas-

ing distance, the effect is the “color confinement”. Because of this phenomenon, when two

quarks become separated, at some point it is more energetically favorable for a new quark-

antiquark pair to spontaneously appear. As a result, instead of seeing the individual quarks

in detectors, many "jets" of color-neutral particles (mesons and baryons) are observed. This

process of particles production is called “hadronization” or “fragmentation”.

On the other hand, at short distance the QCD coupling decreases and thus the quarks

behave as free particles (in terms of strong interaction). This phenomenon is known as

“asymptotic freedom”.

1.3.3 The Higgs mechanism

The Higgs mechanism is a mathematical model that explains why and how gauge bosons

could still be massive despite their governing symmetry. The “Higgs field” breaks the sym-

metry laws of the electroweak interaction and the weak bosons are able to have mass. The

introduction of a scalar field, with a potential V(Φ) = −µ2|Φ|2 + λ2|Φ|4, breaks the sym-

8


metry of the theory choosing spontaneously one of the degenerate ground states as the true

ground, resulting in the appearance of massless Goldstone bosons. The Higgs field can be

represented as a doublet of complex scalar fields:

Φ(x) ≡

Φ+(x)

Φ0(x)

(1.3)

The minimum of the potential is chosen as:

Φ(x) =1√2

( 0√−µ2

λ + h(x)

)(1.4)

with expectation value on vacuum state equal to: |〈0|Φ0(x)|0〉| ≡ v√2, where v = −µ√

λ. The

Higgs field is responsible also for the mass of the fermions through the extension of the

Higgs mechanics to Yukawa’s interaction. For each fermion’s generation the Yukawa’s La-

grangian can be written as:

LY = − 1√2(v + H)(cddd + cuuu + cl ll + cννν) (1.5)

where u = type-up quark, d = type-down quark and l = lepton and ν = neutrino. The fermion

masses are calculated as:

Mi = civ√2

(1.6)

where i = u, d, l, ν.

1.4 The CKM formalism

The fermions and the bosons should be massless in order for the electroweak interaction

to be invariant under local gauge transformations. However due to the coupling to the

Higgs field they acquire mass trough the Spontaneous Symmetry Breaking mechanism.

The resulting mass eigenstates are not the same as the eigenstates of the weak interaction

but a their linear combination, as Cabibbo suggests in 1963 [2].

The Cabibbo-Kobayashi-Maskawa (CKM) matrix (VCKM) is the quark mixing matrix in-

troduced to describe the transformation needed to switch from the quark mass eigenstates

to the weak ones and vice versa. This transformation consist in a rotation of the down-type

quarks:

d′

s′

b′

= VCKM

d

s

b

(1.7)

9


where VCKM is a unitary matrix3 defined as:

VCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

(1.8)

In general a n × n complex matrix possess a priori 2n2 real parameters, however the

unitary conditions introduce 2 · 12 · n · (n − 1) = n2 − n constraints for the off-diagonal

elements and n for the diagonal elements. Through a redefinition of the quark fields 2n− 1

phases can be removed.

The final number of the parameters is:

Npar = 2n2 − n− (n2 − n)− (2n− 1) = n2 − 2n + 1 = (n− 1)2 (1.9)

In the quark case n = 3 thus the matrix can be completely defined by 4 parameters: three

real rotation angles and one phase δ responsible for all CP-violating (CPV) phenomena in

flavor-changing processes. These parameters are free in Standard Model so they should be

determined experimentally.

The unitary constraints are:

1)VusV∗ub + VcsV∗cb + VtsVtb = 0

2)VudV∗ub + VcdV∗cb + VtdVtb = 0

3)VudV∗us + VcdV∗cs + VtdV∗ts = 0

4)VudV∗td + VusV∗ts + VubV∗tb = 0

5)VcdV∗tb + VcsV∗ts + VcbV∗tb = 0

6)VudV∗cd + VusV∗cs + VubV∗cb = 0

(1.10)

These conditions can be resumed as follows:

3

∑k=1

Vki ·Vkj = δij (1.11)

Each condition can be geometrically represented as a triangle in the complex plane,

called “Unitary Triangle”. In total there are six equivalent triangles in the Standard Model

and their area4 value is a direct measure for the predicted amount of CP-violation. The

3It means that VCKMV†CKM = 1

4The area of the Unitary Triangles is equal to half of the Jarlskog invariant defined as: Im[VijVklV∗ij V∗kl ] =

J ∑m,n εikmε jln

10


conditions 2) and 4) in equation 1.10 are very important for the CPV studies because of the

similar lengths of their three sides and amplitudes of their internal angles. They are shown

in Figure 1.1. These characteristics provide tests of the CKM matrix because they can lead

to large CP violating asymmetries between the matrix elements. The other triangles posses

a very short side, thus they are very close to degenerate in a line.

(0, 0)(1, 0)

(ρ, η)

Re

Im

∣∣∣VtsV∗usVcdV∗cb

∣∣∣

∣∣∣VudV∗ubVcdV∗cb

∣∣∣∣∣∣ VtdV∗tb

VcdV∗cb

∣∣∣γ β

α

(0, 0) (1− λ2

2 + ρλ2, ηλ2)

(ρ, η)

Re

Im

∣∣∣VtsV∗usVcdV∗cb

∣∣∣∣∣∣VtbV∗ub

VcdV∗cb

∣∣∣∣∣∣VtdV∗ud

VcdV∗cb

∣∣∣γ′

β′α′

βs

Figure 1.1: The two main important Unitary Triangles. On the left the triangle from 2) and on the

right the triangle from 4).The sides are scaled of a factor |VcdV∗cb| = Aλ3, while the ver-

tices are calculated using the Wolfenstein parameterization, explained at the of this sec-

tion

The triangle 2) is known also as “B0d triangle” because its angles and sides can be mea-

sured through the B0d decays. The values of the angles are given by:

α = arctanVtdV∗tbVudV∗ub

β = arctanVcdV∗cbVtdV∗tb

γ = arctanVudV∗ubVcdV∗cb

(1.12)

where α+ β+ γ = π. The angles of the triangle 4) are related to the α, β and γ in agreement

to the following relations:

α′ = α β′ = β− βs γ′ = γ + βs (1.13)

where βs is the angle between the real axis and the lower side of the triangle. Experimentally

βs is found close to 1◦ (βs = 0.0182± 0.0009 [3]).

The constraints on these angles can be obtained from measurements of many processes

and, through a global fit, the values extrapolated can provide a test for the Standard Model

accuracy. Values different from the expected ones would be a confirmation of new physics

as many extensions to the Standard Model predict.

The global fit performed by UTfit group achieved the results reported in Table 1.4.

11


Parameter Final value

α 88.6± 3.3

β 22.03± 0.86

γ 69.2± 3.4

Table 1.4: Estimated values of the angles of B0d Unitary Triangle trough a global fit performed by

UTfit group[4].

A parameterization of the CKM matrix is the “Chau-Keung parameterization”, where

VCKM = R23 × R13 × R12.

R12 =

c12 s12 0

−s12 c12 0

0 0 1

R23 =

1 0 0

0 c23 s23

0 −s23 c23

R13 =

c13 0 s13e−iδ

0 1 0

−s13eiδ 0 c13

(1.14)

VCKM =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

(1.15)

where sij = sin θij, cij = cos θij and i, j = 1, 2, 3 are the generations5. The angles θij must

be chosen in the first quarter, while δ is the only one parameter which can introduce effects

of CP violation and must be: 0 < δ < 2π.

Another parameterization, known as “Wolfenstein parameterization”, is obtained ex-

panding as a power series of the parameter λ = |Vus|:

VCKM =

1− λ2

2 λ Aλ3(ρ− iη)

−λ 1− λ2

2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

+ O(λ4) (1.16)

where

λ =|Vus|√

|Vud|2 + |Vus|2= sin θc

Aλ2 = λ∣∣∣Vcb

Vus|

Aλ3(ρ + iη) = V∗ub

(1.17)

5The angle θ12 is known also as Cabibbo angle (θc).

12


Parameter Final value

A 0.821± 0.012

λ 0.22534± 0.00065

ρ 0.132± 0.023

η 0.352± 0.014

Table 1.5: Estimated values of the Wolfenstein parameters trough a global fit performed by UTfit

group [4].

Element Value Measurement Channel

|Vud| 0.97425± 0.00018 Nuclear beta decays

|Vus| 0.22543± 0.00077 Semileptonic kaon decays

|Vcd| 0.22529± 0.00077 ν scattering from valence d quarks

|Vcs| 0.97342+0.00021−0.00019 Semileptonic D meson decays

|Vcb| 0.04128+0.00058−0.00129 Semileptonic B meson decays

|Vub| 0.00354+0.00016−0.00014 Semileptonic B meson decays

|Vtd| 0.00858+0.00030−0.00034 B0 mixing assuming |Vtb| = 1

|Vts| 0.04054+0.00057−0.00129 B0

s mixing assuming |Vtb| = 1

|Vtb| 0.99914+0.00005−0.00003 Single-top-quark production

Table 1.6: The current values of VCKM matrix elements [5]

The constraints on this parameters estimated by means of a global fit performed by UTfit

and CKMfitter groups are shown in Figure 1.2, while the experimentally values obtained

by UTfit are reported in Table 1.5, where ρ = ρ

(1− λ2

2

)and η = η

(1− λ2

2

).

Input to these global fits are the measurements of the angles and sides of the Unitary

Triangles from meson decays and mixing as reported in Table 1.6. .

1.5 CP violation

The electromagnetic and strong forces are invariant under parity symmetry and charge

conjugation, on the other hand the weak interaction violates both in a maximal way. The

13


ρ-1 -0.5 0 0.5 1

η

-1

-0.5

0

0.5

1γ

β

α

)γ+βsin(2

sm∆dm∆

dm∆

Kε

cbVubV

)ντ→BR(B

Summer14

SM fitγ

γ

α

α

dm∆

Kε

Kε

sm∆ & dm∆

SLubV

ν τubV

βsin 2

(excl. at CL > 0.95)

< 0βsol. w/ cos 2

exc

luded a

t CL >

0.9

5

α

βγ

ρ

1.0 0.5 0.0 0.5 1.0 1.5 2.0

η

1.5

1.0

0.5

0.0

0.5

1.0

1.5

excluded area has CL > 0.95

Winter 14

CKMf i t t e r

Figure 1.2: Combined fit results of the B0d Unitarity Triangle performed by UTfit [3] (left) and CKMfit

[4] (right) groups.

parity violation (PV) was observed for the first time in 1956 with an experiment of the β-

decay (Wu at al. 1957 [6]). The eigenvalues of the parity operator that a particle can assume

are +1 (“particle right-handed”) and -1 (“particle left-handed”).

The weak interaction violates also the product CP as proved in 1964 with the experiment

of the neutral kaons realized by James Cronin and Val Fitch [7]6.

1.5.1 Mixing of neutral pseudoscalar mesons

The neutral particle oscillation is the transmutation of a neutral particle into another neutral

particle due to a change of a non-zero internal quantum number via an interaction which

does not conserve that quantum number.

A flavor eigenstate P0 transforms under CP symmetry as:

CP|P0〉 = −|P0〉 (1.18)

where P0 can be any neutral mesons (e.g. B0, K0, D0).

In the same way the eigenstate P0 transforms like:

CP|P0〉 = −|P0〉 (1.19)

6From the CPT theorem and the CP violation (CPV) follows that also the time reversal symmetry can be

violated.

14


These two transformations lead to write the CP eigenstates as linear combination of the

flavor eigenstates:

|PCP evens 〉 = 1√

2(|Ps〉 − |Ps〉)

|PCP odds 〉 = 1√

2(|Ps〉+ |Ps〉)

(1.20)

The time evolution of the flavor eigenstates is described by the Schrödinger equation:

iδ

δtΨ(t) = HΨ(t) (1.21)

where H = Hweak because only the weak interactions can induce transitions with flavour

change. Ψ(t) =

a(t)

b(t)

is the state function assuming as initial state:

Ψ(0) = a(0)|P〉+ b(0)|P〉 (1.22)

The Hamiltonian operator can be written as:

H = M− i2

Γ =

M11 − i2 Γ11 M12 − i

2 Γ12

M21 − i2 Γ21 M22 − i

2 Γ22

(1.23)

where the mass matrix M and the decay matrix Γ are hermitian 2× 2 matrices, thus M12 =

M∗21 and Γ12 = Γ∗21. M12 is the dispersive part of the transition amplitude, while Γ12 is the

absorptive part.

Diagonalizing the Hamiltonian lead to the following mass eigenstates:

|PL〉 = p|P0〉+ q|P0〉

|PH〉 = p|P0〉 − q|P0〉(1.24)

where |q|2 + |p|2 = 1. If q = p = 1/√

2 the mass and CP eigenstates are equal.

Using the eigenvalues ωL and ωR, the mass and width differences can be calculated as:

∆m ≡ mH −mL = Re(ωH −ωL)

∆Γ ≡ ΓH − ΓL = −2Im(ωH −ωL)(1.25)

where the index H or L is related to the “heavy” and the “light” mass eigenstate.

Solving the eigenvalues problem the relation of the ratio q/p with the off-diagonal ele-

ments, M12 and Γ12, it is found:

qp=

√M∗12 −

i2 Γ∗12

M12 − i2 Γ12

(1.26)

15


where δm = M11 −M22 and δΓ = Γ11 − Γ22.

The flavor eigenstates evolve according to the following expressions:

|P0(t)〉 = g+(t)|P0〉 − qp

g−(t)|P0〉

|P0(t)〉 = g+(t)|P

0〉 − qp

g−(t)|P0〉(1.27)

where

g±(t) =12

(e−imH t− 1

2 ΓH t ± e−imLt− 12 ΓLt)

(1.28)

represent the time dependent probabilities of the state remaining unchanged (+) or oscillat-

ing into its charge conjugate state (-).

Introducing the the average values of mass and lifetime as:

m =mH + ML

2Γ =

ΓH + ΓL

2

it follows that:

g+(t) = e−imteΓt/2[

cosh(

∆Γ4

t)

cos(

∆m2

t)− sinh

(∆Γ4

t)

sin(

∆m2

t)]

g−(t) = e−imteΓt/2[− sinh

(∆Γ4

t)

cos(

∆m2

t)+ i cosh

(∆Γ4

t)

sin(

∆m2

t)] (1.29)

|g±(t)|2 =e−Γt

2

[cosh

(∆Γ2

t)± cos (∆m t)

]g∗+(t)g−(t) =

e−Γt

2

(− sinh

(∆Γt

2

)+ i sin (∆mt)

) (1.30)

These equations demonstrate that the probability of P0 to become a P0, or vice versa,

oscillates as a function of time and depends on the mass difference ∆m and on lifetime

difference ∆Γ.

1.5.2 Types of CP Violation

The CP Violation indicate a difference between a process and its CP conjugate. Considering

a neutral meson decay in a certain final state “f”, or in its conjugate, the decay rate of the

process Γ is calculated using the equation 1.31.

Γ f (t) ≡ Γ(P0(t)→ f ) =∣∣〈 f |H|P0(t)〉

∣∣2Γ f (t) ≡ Γ(P0

(t)→ f ) =∣∣〈 f |H|P0

(t)〉∣∣2 (1.31)

The CPV arises if Γ f 6= Γ f

It can happen in three different ways in neutral meson decay:

16


• CP violation in the Decay

• CP violation in Mixing

• CP violation in the Interference of Mixing and Decay

CP Violation in the Decay

Direct CP violation takes place when the rate of a process and of its conjugate are different.

In order that this condition occurs it is necessary that the decay amplitude consists at least of

two elements. The term of interference must contain a “weak” phase (φ), which change sign

under CP transformation, and a “strong” phase (δ), that preserves CP. Direct CP violation

is the only type of CP violation possible for charged mesons.

The decay amplitudes A f and A f are defined as:

A f = 〈 f |H|P0(t)〉 A f = 〈 f |H|P0(t)〉 (1.32)

where f is a flavor-specific final state.

The time-independent CP asymmetry is written as:

ACP =Γ(P→ f )− Γ(P→ f )Γ(P→ f )− Γ(P→ f )

=1−

∣∣∣ A fA f

∣∣∣21 +

∣∣∣ A fA f

∣∣∣2 (1.33)

Values of A fA f6= 1 would indicate effects of CPV.

CP violation in Mixing

This kind of violation takes place in neutral mesons mixing and is possible to the difference

between mass and CP eigenstates. The evolution of the physical mass state are described in

equation 1.27.

The p and q coefficients denote the relative proportions of B and B states making up the

mass eigenstates and play an important role in Indirect CP violation. A values of the ratio

q/p different from 1 demonstrates that CP symmetry is violated.

∣∣∣∣ qp

∣∣∣∣ 6= 1 =⇒ Prob(P0 → P0) 6= Prob(P0 → P0) (1.34)

The time-dependent asymmetry can be written as:

ACP(t) =Γ(|P0(t)〉 → f )− Γ(|P0

(t)〉 → f )

Γ(|P0(t)〉 → f )− Γ(|P0(t)〉 → f )

=1−

∣∣ qp

∣∣41 +

∣∣ qp

∣∣4 (1.35)

17


B0s,d

s, d

s, d

t, c, u

W

b

W

b t, c, u

B0

s,d B0s,d

s, d

s, d

t, c, u

W−

b

b

t, c, u

W+

B0

s,d

Figure 1.3: Example of leading order box diagrams involved in B0d- B0

d mixing.

where f is a flavour−specific final state (e.g. the semileptonic decay). The Feymann dia-

grams for the leading order box interactions involved in B0d-B0

d mixing are shown in Figure

1.3.

CP Violation from the interference

If the final state studied is accessible to both P0 and P0 the effects of CP Violation can still

occur even if there is no CPV neither in the decay nor in the mixing individually. The time-

dependent decay rate contains a term λ f defined as:

λ f =qp

A f

A f(1.36)

According to this definition λ f is invariant under arbitrary re-phasing of the initial and

final states, thus it is a potential observable in neutral mesons decays.

CPV effects take place if λ f 6= ±1 and this condition can be satisfied even if |q/p| = 1 and

|A f /A f | if Im(λ f ) 6= 0.

B0d

d

d

c

b

W

t, c, u

W

t, c, u b

W−

c

s }K0

}J/Ψ

B0d

d

c

s

W+ c

b

}J/Ψ

}K0

Figure 1.4: The CP Violation can be caused from the interference of these two diagrams.

18


The time-dependent asymmetry, if ∆Γ = 0, can be written as:

ACP(t) =Γ(|P0(t)〉 → fCP)− Γ(|P0

(t)〉 → fCP)

Γ(|P0(t)〉 → fCP)− Γ(|P0(t)〉 → fCP)

=

= S fCP sin (∆mt) + C fCP cos (∆mt)

(1.37)

where the coefficients S fCP and C fCP are equal to:

S fCP =2Im(λ fCP

1 + |λ fCP |2C fCP =

1− |λ fCP |2

1 + |λ fCP |2(1.38)

In case of absence of CP violation in mixing and in decay, that occurs when |λ fCP | = 1,

CP Violation in the interference can take place from the sine term: S fCP = Im(λ fCP). For the

B0d system the CPV can be caused from the interference of the diagrams shown in Figure

1.4.

19

2

The LHCb experiment

Contents

2.1 The Large Hadron Collider 20

2.2 b production at LHCb 22

2.3 The LHCb detector 23

2.3.1 The beam pipe 24

2.3.2 The VErtex Locater 24

2.3.3 The Tracking System 25

2.3.4 The Magnet 28

2.3.5 The Ring Imaging Cherenkov 30

2.3.6 The calorimeter system 32

2.3.7 The Muon Stations 34

2.3.8 Trigger 35

LHCb (Large Hadron Collider Beauty) experiment is one of the four main experiments

at the LHC (Large Hadron Collider). It is specialized in b-physics and its goal is to search

for physics beyond the Standard Model in the CP violation and rare decays sectors. These

searches can shed a light on the matter-antimatter asymmetry puzzle in the Universe.

2.1 The Large Hadron Collider

The Large Hadron Collider (LHC), represented schematically in Figure 2.1, is the world

largest particle accelerator. In The LHC consists of a 27Km ring of superconducting magnets

with a number of accelerating structures to boost the energy of the particles along the way.

20

2 - The LHCb experiment

Inside the accelerator, two high-energy proton beams travel at a speed close to the speed of

light before they are made to collide.

The collisions take place in four interaction points corresponding to the main experi-

ments : ATLAS, CMS, LHCb, ALICE. The first two are general purpose experiments while

LHCb is dedicated to heavy flavour and rare decays physics and ALICE is dedicated to

lead-ion collisions.

Figure 2.1: A schematic representation of the LHC collider

The collider is designed to operate at an energy of√

s = 14 TeV and a design luminosity

L = 1034 cm−2 s−1 in the final configuration [8]. The beams are structured in 2808 bunches

containing each ∼ 1011 protons 25 ns spaced, the interaction frequency is then 40 MHz.

The beams travel in opposite directions in separate beam pipes and are guided around the

accelerator ring by a strong magnetic field (8.33 T) maintained by dipolar super-conducting

electromagnets.

The instantaneous luminosity delivered by LHC at the IP-8 (Interaction point 8, where

LHCb is located) is lower with respect to the design luminosity of LHC in order to limit

the number of interactions per bunch crossing. The technique by which the instantaneous

luminosity is lowered is called luminosity leveling and consists of adjusting the transversal

beam overlap. During the Run I period (2010-2012) LHCb took data with different beam and

luminosity conditions; the number of collision per bunch (µ) and the integrated luminosity

(Lint) along with the peak luminosity (Lpeak) and the center of mass energy for the different

21


data taking are reported in Table 2.1.

Year Lint√

s µ Lpeak

2010 37 pb−1 7 TeV 1− 2.5 1.6 · 1032 cm−2 s−1

2011 1.0 fb−1 7 TeV 1.5− 2.5 4.0 · 1032 cm−2 s−1

2012 2.2 fb−1 8 TeV ' 1.8 4.0 · 1032 cm−2 s−1

Table 2.1: the data taking condition at LHCb during the Run I period (2010-2012).

2.2 b production at LHCb

At the LHC inelastic pp interactions occur in beam pipe. From these collisions couple bb are

produced in several ways, but the dominant contribution come from sea gluons and quarks

through the “gluon-gluon” and “quark-quark” fusion. The feyman graph of both mode are

shown in Figure 3.2.

g1

g2

b

b

q1

q2

b

b

Figure 2.2: Feynman diagrams for the production of a pair bb quarks at LHCb.[9]

The two b quarks are produced inside a narrow cone, which can be oriented in the

forward direction or in the backward direction. Both the cones pointing to the interaction

vertex, as shown in Figure 2.3.

The number of events which produce couples bb can be calculated as:

Nbb = σ(pp→ bbX) · Lint (2.1)

where σ is the cross section and Lint is the luminosity.

The measured cross section at√

s = 7 TeV is σ(pp → bbX) = (284± 20± 49)µb [10],

thus the number of bb couple is 3 · 1011. About a forth of the total number is produced in

the forward direction as shown in Figure 2.3 [11].

22


0/4π

/2π/4π3

π

0

/4π

/2π

/4π3

π [rad]1θ

[rad]2θ

1θ

2θ

b

b

z

LHCb MC = 7 TeVs

Figure 2.3: Azimuthal angle distribution of the bb quark pairs. The red part of the distribution is the

LHCb acceptance.

2.3 The LHCb detector

The LHCb detector is a single arm spectrometer optimized to maximize the detection ef-

ficiency of b-hadrons produced at LHC. The layout of the detector is shown in Figure 2.4

[12]. The coordinate system is chosen such that the z axis corresponds to the beam pipe axis,

the y axis is the vertical (non-bending plane) one and x is horizontal (bending plane). The

acceptance in the x-z plane is 10− 300 mrad and 10− 250 mrad in the y-z plane. It consists

of several sub detectors:

• Vertex Locator (VELO)

• Tracking system

• Dipolar magnet

• Two Ring Imaging Cherenkov detectors (RICH1 and RICH2)

• Electromagnetic Calorimeter (ECAL)

• Hadronic Calorimeter (HCAL)

23


• Muon detector

In the following section, each of these sub-detectors will be described along with the

trigger system.

Figure 2.4: A y-z section of the LHCb detector

2.3.1 The beam pipe

The proton beams circulate in an Ultra High Vacuum pipe, called “beam pipe”. The beam

pipe consists of four sections, three of them are made of beryllium while the fourth sec-

tion is made of stainless steel. Beryllium is chosen in order to minimize the probability of

the particles produced in the interaction point to create secondary particles. In the VELO

(described in section 2.3.2) region it is made of high strength aluminum alloys.

2.3.2 The VErtex Locater

The VErtex Locator (VELO) is the part of the LHCb spectrometer closest to the collision

region, inside the LHC vacuum pipe. It allows to observe and reconstruct the decays of

B-mesons, which have displaced decay vertex ( ∼ 0.5 cm) because of their relatively long

lifetimes. The precise measurement of primary and secondary vertexes (PV and SV) of

the decays is fundamental for CP violation time dependent measurements and also to re-

duce the combinatorial background. For B-mesons the resolution of the PV depends on the

number of tracks in the event. On average it’s 60 µm in the z direction and 10 µm in the

perpendicular direction. The sub-detector consists of two rows of half-moon-shaped silicon

stations, each 0.3 mm thick. A small cutout in the center of stations allows the main LHC

24


beam to pass through freely. The stations are made by two type of sensors: r sensors mea-

sure the radial distance of the particle tracks from beam axis while the φ sensors measure

their polar angle. The first two stations are used for the L0 trigger level. The tracks can be

reconstructed with polar angles between 15 mrad and 390 mrad. The VELO is also impor-

tant for the impact parameter measurement, the resolution is ∼ 15 µm at high transverse

momentum (∼ 10 GeV) and ∼ 300 µm at low transverse momentum (∼ 0.3 GeV)

2.3.3 The Tracking System

The tracking system enables the trajectory of each particle passing through the detector

and their momentum to be recorded and is absolutely crucial for reconstructing B-particle

decays.

It comprises four large rectangular stations, each covering an area of about 40 m2 : one

station (TT) is located between RICH-1 and the dipole magnet, while the other three stations

(T1-T3) are located over 3 meters between the magnet and RICH-2.

Two detector technologies are employed:

1. The Silicon Tracker: uses silicon microstrip detectors It comprises the entire TT station

and a cross-shaped area (the Inner Tracker) around the beam pipe in stations T1-T3.

Its total sensitive surface is approximately 11 m2 .

2. The Outer Tracker: uses straw-tube drift chambers with 5 mm cell diameter and covers

the largest fraction of the detector sensitive area in stations T1-T3. The total sensitive

area is 80.6 m2.

The Silicon Tracker

The Silicon Tracker (ST), which is placed close to the beam pipe, uses silicon microstrip de-

tectors with a strip pitch of approximately 200 µm. Each of the four Silicon Tracker stations

consists of four detection layers. The vertical layers are called x-layers, while the u and v-

layers are rotated by an angle of 5◦ and −5◦ The first two x-u layers are separated by the

others two v-x layers by 27 cm along the beam axis.

The TT detection layers are shown in Figure 2.5. The ST comprise two detectors: the Tracker

Turicensis (TT) and the Inner Tracker (IT).

The TT is a 150 cm wide and 130 cm high planar tracking station that is placed upstream

of the LHCb dipole magnet and covers the full acceptance of the experiment. The strips are

500 µm thick with a pitch of 183 µm and the single hit resolution of the TT is about 50 µm.

25


(a) v layer

132 .

4 cm

7.74 cm

138.6 cm

7.4

cm

(b) x layer

150.2 cm

131.1

cm

127.1 cm

132.

8 cm

(c) u layer

Figure 2.5: Layout of TT detection layers [13]

The IT is a 120 cm wide and 40 cm high cross-shaped made by four detector boxes. It’s

placed at the center of three large planar tracking stations downstream of the magnet. The

strip sensors are 320 µm thick for boxes above and below the beam line, and 410 µm thick

for the other two. The pitch between the sensors is 200 µm and the single hit resolution is

50 µm. The Inner Track x and u layers in cross-shaped configuration are reported in Figure

2.6.

Figure 2.6: Layout of Inner Tracker x and u layers in the cross-shaped configuration

26


The Outer Tracker

The Outer Tracker (OT) is located in the three tracking stations covering the area outside the

IT acceptance. The three stations are of equal size with the outer boundary corresponding

to an acceptance of 300 mrad in the horizontal plane and 250 mrad in the vertical one.

The design of the three OT stations is modular. Each is built from 72 separate modules

supported on four independently moving aluminum frames (18 modules per frame). A

module consists of two panels and two sidewalls, which form a mechanically stable and

gas-tight box, and contain up to 256 straw tubes filled with a mixture of argon (70%), carbon

dioxide (28.5%) and oxygen (1.5%). A section of the OT station is shown in Figure 2.7.

Figure 2.7: A section of OT station

The straw tubes are wound from two layers of foil material. An inner layer of carbon-

doped Kapton (Kapton XC) acts as a cathode for the collection of the positive ions. The

outer layer, made of a polyimide-aluminum laminate, provides shielding and together with

the anode wire forms a transmission line for the effective transport of the high-frequency

signals. The inner diameter of the straws is 5.0 mm and the pitch between them is 5.25 mm.

The spatial resolution of the single straw tube is 200 µm.

Track reconstruction

The hits on each tracking detector are combined in order to form particle trajectories. Given

the detectors used to build the tracks they are classified as:

• VELO tracks: tracks that contain only hits of the VELO detector. They allow a precise

determination of the primary vertex as they have typically a large polar angle.

27


• Upstream tracks: tracks that are reconstructed with hits on the VELO and TT stations.

They are low momentum tracks that are bent out of the acceptance in the dipole mag-

net. Although their momentum resolution is reduced, they can be used in some B

decay analyses.

• Downstream tracks: these tracks are reconstructed with the TT and the T stations. They

are useful to reconstruct long lived particles that decay outside the VELO acceptance,

like K0S.

• T tracks: tracks that are only reconstructed in the T stations.

• Long tracks: tracks that contain hits both in the VELO and in all the tracking detectors.

They are the most important for B physics measurements because they are the best

quality physics tracks of LHCb.

The different track types are shown in Figure 2.8.

Figure 2.8: Track classification

The relative resolution of long tracks is between δp/p = 0.35% for low momentum

tracks (∼ 10 GeV/c) and δp/p = 0.55% for high momentum tracks (∼ 140 GeV/c).

2.3.4 The Magnet

A dipole magnet is used together with the tracking stations to determine the momentum

of charged particles. Charged particle trajectories are bent when traversing a magnetic field

and the curvature enables their momentum and charge to be determined. The bending of

28


Figure 2.9: Principal idea of the tracking system and the “momentum kick” method

the magnet can to first order be approximated as a single kick at the center of the magnet, the

track as the combination of two straight lines. The momentum of the particle is inversely

proportional to the difference of the track slope in the Velo and the track slope in the T-

Stations (“momentum kick” method), as shown in Figure 2.9.

The magnet consists of two coils of conical shape placed symmetrically one above and

one below the beam pipe each 7.5 m long, 4.6 m wide and 2.5 m high. A perspective view

of the dipole magnet is shown in Figure 2.11

The dipole field has a free aperture of±300 mrad horizontally and±250 mrad vertically.

The magnetic field is along the y axis and its integrated value is 4 T·m.

Figure 2.10: Dominant component of the magnetic field

The magnetic field can be inverted to minimize systematic errors due to the detector

29


Figure 2.11: Perspective view of the LHCb dipole magnet. The interaction point is located behind

the magnet

asymmetries that can limit the precision of charge asymmetry measurements. In Figure

2.10 the trend of the magnetic field along the z-axis is shown.

2.3.5 The Ring Imaging Cherenkov

The experiment’s two Ring Imaging Cherenkov (RICH) detectors are built for particle iden-

tification (PID), fundamental for LHCb measurements [12]. A schematic view of the RICH

is shown in Figure 2.12.

They are responsible for identifying different particles that result from the decay of B

mesons, including pions, kaons and protons. PID is crucial to reduce background in selected

final states.

RICH detectors work by measuring emissions of Cherenkov radiation that consists of

photons. This phenomenon occurs when a charged particle traverse a medium with a speed

higher than a threshold speed vt = c/n, where c is the speed of light and n is the refractive

index of the medium itself. This radiation is emitted at a specific angle θc = arccos (1/nβ),

where β is the ratio between the particle’s speed and the speed of light. The Figure 2.13

shows the Cherenkov angle as a function of the momentum for different particles.

The PID hypothesis is made associating the Cherenkov ring image to a track. At first all

particles are identify as a pion, then for each particle hypothesis a likelihood is calculated

30


250 mrad

Track

Beam pipe

Photon

Detectors

Aerogel

VELOexit window

Spherical

Mirror

Plane

Mirror

C4F10

0 100 200 z (cm)

Magnetic

Shield

Carbon Fiber

Exit Window

(a) Side view schematic of RICH-1

120mrad

Flat mirror

Spherical mirror

Central tube

Quartz plane

Magnetic shieldingHPD

enclosure

2.4 m

300mrad

CF4

(b) Top view schematic of RICH-2

Figure 2.12: Schematic view of RICH detectors

θC

(mra

d)

250

200

150

100

50

0

1 10 100

Momentum (GeV/c)

Aerogel

C4F10 gas

CF4 gas

eµ

p

K

π

242 mrad

53 mrad

32 mrad

θC max

Kπ

Figure 2.13: Cherenkov angle vs particle momentum for RICH radiators

31


(a) kaon-pion identification (b) proton-pion identification

Figure 2.14: Kaon(left) and proton(right) identification and pion misidentification as a function of

the track momentum measured on 2011 data. Plots for two different ∆ log L are shown.

using the informations from the RICH, calorimeters and the muon system (described in

section 2.3.6 and 2.3.7). In the end a discriminating variable is calculated as the logarithm of

the difference between the likelihood of the track and the likelihood of the pion hypothesis.

As can be seen from Figure 2.14 [14], choosing ∆ log LK−π > 0 (kaon hypothesis better

than pion hypothesis) the average kaon efficiency identification over the momentum spec-

trum is∼ 95% while the pion misidentification is ∼ 10%. Choosing ∆logLK−π > 5 result in

a pion misidentification of ∼ 3%. Also for the proton hypothesis, the choice ∆logLp−π > 5

reduce the pion misidentification.

2.3.6 The calorimeter system

The calorimeter system is designed to stop particles(photon, electrons and hadrons) as they

pass through the detector, measuring their energy and position. These measurements are

also used by the trigger system to select events interesting for the LHCb experiment (de-

scribed in section 1.3.8). The calorimeter system consists of several layers:

• the Scintillating Pad Detector (SPD)

• the Pre-Shower Detector (PS)

• the Electromagnetic Calorimeter (ECAL)

• the Hadron Calorimeter (HCAL)

The calorimeters are segmented in the x-y plane such that the channel density is higher

towards the beam pipe where the particle density is higher. These detectors use scintillating

32


materials to detect the shower of photons, electrons and positrons produced when particles

pass through them. The angular acceptance is between 300 mrad and 30 mrad horizontally

and 250 mrad vertically.

The Scintillating Pad Detector and the Pre-Shower

The SPD and PS indicate the electromagnetic character of the particles hitting the calorime-

ter system, i.e. they determine if the particles are charged or neutral. They are used at the

trigger level (L0) in association with the ECAL to indicate the presence of electrons, pho-

tons, and neutral pions.

The SPD and PS consist of scintillating pads with a thickness of 15 mm, inter spaced

with a 2.5X0 lead converter. They are placed right after the first muon station and collecting

the light using wavelength-shifting (WLS) fibers.

The Electromagnetic Calorimeter

The showers initiate in PS are detected by the ECAL. It employs “shashlik” technology of

alternating scintillating tiles (2 mm thick) and lead plates (4 mm thick). The cells’ surface

is 4 cm × 4 cm , 6 cm × 6 cm and 12 cm × 12 cm in the inner, middle and outer parts of

the detector, respectively. The overall detector’s dimensions are 7.76 m × 6.30 m, covering

an acceptance of 25 mrad < θx < 300 mrad in the horizontal plane and 25 mrad < θy <

250 mrad in the vertical one. Light is collected by WaveLength-Shifting (WLS) fibers and

PhotoMultiplier Tubes (PMTs).

The ECAL energy resolution is given by :

σ(E)E

=10%√

E⊕ 1.5%

where E is the energy and ⊕mean the sum in quadrature.

The Hadronic Calorimeter

The HCAL is positioned outside the ECAL and detects particles originated in hadronic

showers. Its internal structure consists of thin iron plates absorber inter spaced with scintil-

lating tiles arranged parallel to the beam pipe. Like ECAL, the inner and outer parts of the

calorimeter have different cell dimensions: 13 cm × 13 cm and 26 cm × 26 cm,respectively.

In the lateral direction tiles are inter spaced with 1 cm of iron matching with the hadron

radiation length in iron (X0); while in the longitudinal direction the length of tiles and iron

33


spacers corresponds to the hadron interaction length (λI) in iron. Light is collected with the

same principle used in ECAL.

The HCAL energy resolution is given by :

σ(E)E

=80%√

E⊕ 10%

2.3.7 The Muon Stations

The muon is the only detectable particle that is able to pass through the calorimeters with-

out losing completely its energy. Muons are present in the final states of many CP-sensitive

B decays and play a major role in CP asymmetry and oscillation measurements. LHCb ex-

periment uses five muon stations (M1-M5), gradually increasing in size, to identify and

reconstruct muons. The system covers an acceptance between±300− 20 mrad horizontally

and ±258 − 16 mrad vertically. Each station is divided into four regions, R1 to R4, with

increasing distance from the beam axis The stations M2-M4 are interleaved by 80 cm thick

iron wall to absorb hadronic particles. Average muon identification efficiencies of 98% can

be obtained with a level of pion and kaon misidentification below 1%. The hadron misiden-

tification probabilities are below 0.6%. A side view of the muon stations is shown in Figure

2.15.

Figure 2.15: Side view of the LHCb muon system, showing the position of the five stations. The first

station is placed before the calorimeters and the other four after them, interleaved with

the muon shield. [12]

34


2.3.8 Trigger

The LHC bunch crossing frequency is 40 MHz, of which the LHCb detector at nominal

luminosity (2 · 1032 cm−2s−1 [8]) will see events with at least one visible interaction 1 at a

rate of 10 MHz. The purpose of the LHCb trigger system is to reduce the event rate from 40

MHz to 2 KHz. The LHCb trigger consists of two stages:

• Level-0 (L0) trigger synchronous with the bunch crossing frequency, is implemented

in hardware and reduces the event rate to less than 1.1 MHz

• High Level Trigger (HLT), is a C++ software trigger and runs on a dedicated Event

Filter Farm (EFF);it reduces the event rate to 2 KHz

Level-0 Trigger

Using informations from VELO, L0 trigger estimates the number of primary interactions

in each bunch crossing and also provides the possibility to veto events with multiple PV

(pile-up trigger).

Because B mesons have a large mass, they often decay producing particles with large

transverse momentum (pT) and energy (ET). The L0 trigger attempts to reconstruct:

• the highest ET hadron, electron and photon using the calorimeters’ informations. The

event is triggered if the ET is greater than a certain threshold.

• the two highest pT muons combining the informations of the five muon stations. The

event is triggered if the pT is greater than a certain threshold (single-muon) or if the

sum of the two highest momenta is greater than a certain threshold (di-muon).

The total latency of the L0 trigger is 4 µs, including time-of-flight of the particles, cable

delays, delays in the front-end electronics and the time necessary to process the data and to

derive a decision.

High Level Trigger

The High Level Trigger (HLT) analyses the events that are selected by the L0 trigger. HLT

consists of two level named HLT-1 and HLT-2.1An interaction is defined to be visible if it produces at least two charged particles with sufficient hits in the

VELO and T1âT3 to allow them to be reconstructible

35


• HLT-1: its purpose is to reduce the rate to 30 kHz allowing a full pattern recogni-

tion on the remaining events. Information from a tracking sub detector and applying

requirements on the pT and the impact parameter (IP) with respect to the primary

vertex (PV) are added. This reduces the CPU time needed for decoding and pattern

recognition algorithms.

• HLT-2: It performs a full event reconstruction using the Kalman Filter algorithm to fit

the tracks. After the track fit, the HLT-2 stage applies cuts either on invariant mass,

or on pointing of the B momentum towards the primary vertex. The resulting ex-

clusive and inclusive selections, called topological lines, aim to reduce the rate to 2

KHz,which is the final output rate of LHCb trigger system.

In the topological lines a multi body candidate is built, starting from two particles to

make a two-body object.

36

3

Same Side Pion Tagger

Contents

3.1 The Flavour Tagging 37

3.1.1 Definitions 38

3.1.2 Same Side Taggers 40

3.2 Same Side tagger 41

3.3 SSπ tagger development using 2012 data sample 42

3.3.1 sWeights estimation 42

3.3.2 Training of the SS pion tagger 45




In this chapter the development of a new Same Side Pion tagger is described. The al-

gorithm implementation uses a sample with the B0 −→ D−π+ decay channel, collected

by LHCb in 2012 corresponding to 2 f b−1 of pp collisions at√

s = 8TeV. It is described in

section 6.4.2. A first validity check is performed using the data of 2011 of the same channel

corresponding to 1 f b−1 taken at√

s = 7TeV, it is described in section D.1, then another

check is performed using a different event selection as described in section ??.

3.1 The Flavour Tagging

The Flavour Tagging (FT) at LHCb is a fundamental tool to study the b-hadron decays and

the CP violation. These measurements require the knowledge of the quark content in the B

37

3 - Same Side Pion Tagger

3/20 Ulrich Eitschberger | Updates on Flavour Tagging | 72nd LHCb week | June 19th, 2014

Flavour Tagging: Determine B production flavours SS Pion SS Kaon Signal Decay

Same Side

Opposite Side

OS Vertex Charge OS Muon OS Electron

OS Kaon

PV

Figure 3.1: A sketch of an event generated by a bb pair. In green the signal B, in red the OS taggers

and in violet the SS taggers.

meson, i.e. the hadron flavour. The identification of this flavour is further complicated by

the oscillation of the neutral B mesons: B0d ↔ B0

d and B0s ↔ B0

s .

In LHCb, B mesons are produced as bb pairs which are correlated in charge. One of

them is reconstructed to be the signal decay, while the other, called tagging B or opposite B,

is used to tag the initial flavour of the first one.

At LHCb the Flavour Tagging is performed using several algorithms, called “taggers”,

exploiting the informations from the fragmentation of the b quark in the signal B and there-

fore known as Same Side (“SS”) taggers, or using the informations from the decay chain of

the opposite B, known Opposite Side (“OS”) taggers 3.1. The data ntuples studied in the

analysis developed in this thesis are written within the LHCb analysis framework called

DaVinci. In particular the DaVinci version 35r0 has been used.

3.1.1 Definitions

For each tagger a decision (d) is assigned to the signal B particle by looking at the charge of

the tagger:

• d = 1 =⇒ the B meson contains a b quark

• d = −1 =⇒ the B meson contains a b quark

38


• d = 0 =⇒ no particle is available to identify the meson

Each event is classified as:

• correctly tagged (R), if the tag decision is equal to the B flavour at the production

• incorrectly tagged (W), if the tag decision is different to the B flavour at the production

• untagged (U), if the tag decision is equal to 0

The ratio of tagged events is represented by the tagging efficiency (εtag) and can be

calculated as:

εtag =R + W

R + W + U(3.1)

In the same way is possible to determine the fraction of wrong tagged events, called

“mistag fraction” (ω), like:

ω =W

W + R(3.2)

The mistag fraction can be calculated directly only using flavour specific decays (i.e the

charge of the decay products is correlated with the flavour of the B particle) for charged

mesons , such as Bu; in these cases the mistag estimation is straight forward because the

wrong and the right tagged events can be identified comparing the tagger decision to the

charge of the Bu particle. For the Bd and Bs decays there is an additional complication:

due to flavour oscillation only the flavour of the Bd (or Bs) particle at decay can be known

from decay products. In this particular case the mistag must be extracted from a fit to the

time-dependent asymmetry of the flavour oscillations. For the B decays in CP state, such

as Bs −→ J/ΨΦ, there is no way to determine the mistag directly in this channel hence

the mistag for these decays has to come from an external flavour-specific decays; those are

named the calibration channels.

Assuming that the “mistag fraction” and the “tagging efficiency” are not depending

from the initial flavour,the decay rate (Γ) and the CP asymmetry are written as:

Γm(B(t)→ f ) = εtag[(1−ω)Γ(B(t)→ f ) + ωΓ(B(t)→ f )]

Γm(B(t)→ f ) = εtag[(1−ω)Γ(B(t)→ f ) + ωΓ(B(t)→ f )](3.3)

where Γm and Γm are the measured decay rate of B meson to a final state (f) and its CP

conjugate.

39


Am =Γm(B(t)→ f )− Γm(B(t)→ f )Γm(B(t)→ f ) + Γm(B(t)→ f )

= (1− 2ω)A(t) = DA(t) (3.4)

where

• Am is the measured asymmetry and A is the true asymmetry

• D = 1− 2ω is the dilution term, which has the effect to decrease the amplitude of the

oscillation

The real asymmetry and its statistical error is given by:

A =Am

1− 2ωσA =

σAm

1− 2ω(3.5)

where the error on ω has been neglected.

Using the error propagation on 3.4 and knowing that

1− A2m =

4ΓmΓm

(Γm + Γm)2(3.6)

the error on the measured asymmetry is:

σAm =

√4ΓmΓm

(Γm + Γm)3=

√1− A2

m

Γm + Γm=

√1− A2

mNtag

=

√1− A2

mεtagN

(3.7)

where Ntag is the number of events in which the initial flavour is known and N is the total

number of events. Thus the error on the true asymmetry is evaluated as:

σA =σAm

1− 2ω=

√1− A2

m√εtagN(1− 2ω)

(3.8)

From the equation 3.8 it follows that to improve the statistical error on the asymmetry it

is necessary to maximize the “effective tagging efficiency” or “tagging power”(εe f f ) [15],

defined as:

εe f f = εtagD2 = εtag(1− 2ω)2 (3.9)

3.1.2 Same Side Taggers

Same Side tagger algorithms exploit the charge correlation in the fragmentation chain of the

signal meson to define its flavour. In this thesis the signal channel B0 −→ D−π+ is studied.

In the case of B0 (bd) a d quark is available to produce particles like: π+, π0, p, Λ0. If the

additional particle is charged, it can be used to identify the flavour of the B meson. The

particles which are studied in this thesis are:

40


• positive pion π+ (du), described in Chapter 3

• anti-proton p (d, u, u), described in Chapter 4

In Figure 3.2 two possible diagrams for the B0 hadronization are shown. [16].

u

d

bd

u

}π+

}B0

ud̄d

bd

u

d }p

}B0

d

Figure 3.2: Feynman diagrams for the hadronization of B0 with π+ (left) or p (right) production.

3.2 Same Side tagger

The idea to develop a Same Side (SS) tagging algorithm arises in the context of the research

of excited b-hadron states1 (B∗∗, Λ∗∗b , Σ∗∗b ) [17]. These states can decay via strong interaction

to the ground state b-hadron and an additional particle, such as π or p. If the additional

particle is charged, then it can be used to identify the initial flavour of the signal B meson.

In order to perform this identification, these studies exploit the invariant mass distribution

as well as the kinematic and geometric correlations between the tagger candidate and the

b-hadron. Actually some SS taggers, which exploit a multivariate analysis based on a BDT

to identify the flavour of the signal B, have been already developed during the 2012 by

Antonio Falabella from the Università degli studi di Ferrara [18]. In particular he studied

the SS proton and then the SS pion taggers. Thus the analysis presented in this chapter and

in the following one aspire to find an improvement in the results achieved by these taggers

already implemented.

In this chapter and in the following one, the implementation of the SS pion and SS

proton tagging algorithms are described. To develop these taggers a multivariate classifier

based on a “Boost Decision Tree” (BDT) is used to select the particles which can be usefull

for the tagging (the BDT method is described in Appendix B while its application in the SS

tagging is described in section 6.4.2).

1These excited b-hadron states, which decaying strongly, can be reconstructed selecting a B meson originated

from the primary vertex and an additional track from the same vertex, given the negligible lifetime.

41


The sample chosen to tune these BDT corresponds to the B0 −→ D−(→ K+π−π−)π+

sample collected by the LHCb experiment during the 2012 data taking. The reason of this

choise lies in the nature itself of this decay channel: as explained in the section 3.1, the

flavour tagging technique can be applied only on a flavour specific channel, like the B0 −→

D−π+. In this channel indeed the additional particle (π or p) produced by the b-fragmentation

allows to tag the flavour of the signal B at the production, not at the decay because of the

flavour time-dependent oscillations.

The BDT training can occur in two possible ways: the first consists in the use of a data

sample while the second in the use of a Monte-Carlo (MC) simulation. In a first time the

MC sample was chosen because of the possibility to know the MC-truth about the event

properties. In particular, some analysis about the origin of the many pions created in the

events was performed in order to possibly increase the separation ability of the BDT. They

are reported in Appendix C. However in a second time the train on a data sample was

preferred, because the MC did not reproduce correctly the data, thus it was not possible to

fully trust its response. The analysis and the results reported in the following chapters don’t

take in account the training performed on the MC sample.

The main difference between a MC sample and a data sample is that the first one con-

tains only the signal events while the second one is polluted with background events. In

this thesis the sPlot technique has been exploited in order to get rid the background con-

tribution. According to this method, described in Appendix A, known the distributions of

the invariant mass of the signal B,it is possible to obtain a per-event sWeight which can be

used to weight the data distribution and thus unfold the signal from the background. This

procedure is described in detail in the following section.

3.3 SSπ tagger development using 2012 data sample

3.3.1 sWeights estimation

In this section the analysis of B0 −→ D−(→ K+π−π−)π+ 2012 data sample is described.

To select the B0 candidate and to reduce as much as possible the background events in the

sample the cuts, reported in Table 3.1 [18], are applied. Using the sPlot technique, briefly

illustrated in Appendix A, a per-event weight is calculated allowing to unfold the contribu-

tions of background and signal.

42


Variable Description Cut

cuts for the B0 candidate

D mass Invariant mass of the D 1845 < mD < 1895

D time decay time of the D > 0

FDχ2(D) Fly distance significance of the D > 1

IPχ2(D) Impact parameter significance of the D wrt PV > 4

IPχ2(B) Impact parameter significance of the B wrt PV < 16

B(pointing) cosine of the angle between B momentum and its

direction

> 0.9999

cuts for the particles in the final state

π+ isMuon Identification of the positive pion as muon = 1

IPχ2(π+) Impact parameter significance of the π+ wrt PV > 36

IPχ2(π−) Impact parameter significance of the π− wrt PV > 9

IPχ2(π−) Impact parameter significance of the π− wrt PV > 9

IPχ2(K+) Impact parameter significance of the K+ wrt PV > 9

PIDK (π+) ∆(log LK − log Lπ) of the π+ < 2

PIDK (π−) ∆(log LK − log Lπ) of the π− < 5

PIDK (π−) ∆(log LK − log Lπ) of the π− < 5

PIDK (K+) ∆(log LK − log Lπ) of the K+ > 0

mass vetoes

Ds − veto Mis-Id: D− → (K+π−π−)↔ D− → (K+π−K−) |m−mDs | > 30

Λc − veto Mis-Id: D− → (K+π−π−)↔ D− → (K+π−p) |m−mΛc | > 30

Table 3.1: Selection cuts for the B0 candidate, for the particles in the final state and mass vetoes for

the decay channel B0 → D−(Kππ)π+ [19].

This separation is achieved through a fit to the B-candidate mass distribution (consid-

ered the “discriminating variable”), using the following probability density function (PDF):

P = (1− fB)S + fBB (3.10)

where

• fB is the fraction of background in the sample

43


• S is signal component described by two Gaussian functions with common mean MB:

S = S(m) = fm · G(m; MB; σm,1) + (1− fm) · G(m; MB; σm,2) (3.11)

• B is the background component described by a decreasing exponential:

B = B(m) = exp(α ·m) (3.12)

Parameter Description Value

MB [MeV/c2] Mean B mass value 5283.90 ± 0.04

σm,1 [MeV/c2] σ of the first Gaussian 16.33 ± 0.14

σm,2 [MeV/c2] σ of the second Gaussian 26.48 ± 0.40

fm fraction of the first Gaussian 0.644 ± 0.018

α [MeV−1] slope of the exponential function -5.30 ± 0.20

Nsig Number of signal events 328120 ± 536

Nbkg Number of background events 21855 ± 771

S/B Signal over background ration 15.013 ± 0.530

Table 3.2: Results of the fit to the mass distribution 2012 data sample

)2^+) (GeV/cπm(D^- 5.2 5.25 5.3 5.35 5.4 5.45 5.5

)2E

vent

s / (

0.0

03 G

eV/c

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000TotalSignalBackground

5.2 5.25 5.3 5.35 5.4 5.45 5.5

Pul

l

-5

0

5

Figure 3.3: Mass fit for the B0 → D−(Kππ)π+ 2012 data sample. The blue curve represents the pdf

written in equation 3.10. It is composed by two components: the signal component (red)

and the background component (green). Below the plot the normalized residuals (pulls)

are shown.

44


In Figure 3.3 the plot of the fit to the mass distribution on the 2012 data sample is shown

while in Table 3.2 the results for the PDF parameters are reported. In the following analysis

the background contribution will be removed from the data samples weighting each event

with its sWeight estimated by the mass fit described above.

3.3.2 Training of the SS pion tagger

To determine the flavour of the B signal meson is necessary to develop a tagging algorithm

to identify the particle which is rightly correlated in charge. To perform this aim as better as

possible a multivariate technique is implemented using kinematic and geometric variables

to select the best candidate to be used for tagging. The algorithm implemented is based on a

“Boost Decision Tree” (BDT) classifier [20] using the “MisClassificatorError” as separation

criterion and the “AdaBoost” as boosting method (more details in Appendix B). A BDT

algorithm has been preferred respect a naive cut-based approach because, as demonstrated

in other similar analysis, it provides better performance [18]. To develop the BDT algorithm

the sample has been divided in three sub-samples: the first sub-sample is used to perform

the BDT training, the second sub-sample uses the BDT training results to choose the right

tagger candidate and in the third sub-sample a calibration of the BDT is performed. Then

an efficiency value of the tagging method is calculated using the second sub-sample. The

events has been divided according to the following relation:

Ns = Neventmod3 (3.13)

where Nevent is the number associated to the event and Ns is the number of events in sub-

sample. The track charge is used to classify the tracks with the right or wrong charge corre-

lation with respect the B signal meson. For the Same Side pion such correlation is defined

as:

Right −→ B0π+ or B0π−

Wrong −→ B0π− or B0π+

However using a neutral channel calls for some caution because the B0 mesons can

undergo flavour oscillation reversing the charge correlation. To reduce the contribution of

flavour-oscillated events during the BDT training, a cut on the decay time has been applied

to t < 2.2 ps. This cut allows to remove the greater part of the oscillated events without

reducing drastically the statistics of the sample. To perform the BDT training the first sample

is divided further in two subsamples: the first is used for the BDT training phase (“training

45


sample”) while the second is used as a statistical independent sample to check possible

overtraining effects, as described in [20].

In order to improve the BDT performance some preselection cuts are applied removing

a priori some events as illustrated in [18]. The preselection cuts applied are shown in Table

3.3.


Selection cuts on the tagging particle

pT Tranverse momentum > 400 [MeV/c2]

IP/σIP Impact parameter significance wrt PV < 4

Ghost prob Probability that a track is a random combination of hits < 0.5

PIDp ∆(log Lp − log Lπ) < 5

PIDK ∆(log LK − log Lπ) < 5

χ2track/nd f Quality of track fit < 5

IPPU/σIPPU Impact parameter significance wrt Pile-Up > 3

Selection cuts on the B+tagging system


∆Q m(B + track)−m(B)−m(track) < 1200 [MeV/c2]

∆η Difference between signal B and tagging track

pseudorapidity

< 1.2

∆φ Difference between signal B and tagging track φ angle < 1.1

χ2vtx Quality of B vertex fit < 100

Table 3.3: Preselection cuts applied for SS pion tagging algorithm. The first group contains cuts

applied to the tagging particle while the cuts in the second group are related to the

“B+tagging” system.

The list of the variable used in the training is reported in Table 3.4 while in Table 3.5

their importance ranking in the BDT decision is reported. These variables should show

great differences between right and wrong charge correlated tracks [18]. For the variables

whose slope is very irregular a logarithm is applied to improve the BDT performance.

46


Variable Description

Track related variables

log p Momentum

log pT Tranverse momentum

log IP/σIP Impact parameter significance wrt PV

Ghost prob Probability that a track is a random combination of hits

PIDK ∆(log LK − log Lπ)

Signal B variables


B+track variables


∆Q m(B + track)−m(B)−m(track)

∆R√

∆φ2 + ∆η2

log ∆φ Difference between signal B and tagging track polar angle

log ∆η Difference between signal B and tagging track pseudorapidity

Event related variables

Ntracks in PV Number of degrees of freedom in the fit of Primary Vertex

Table 3.4: Input variables used to train the BDT for the SS pion tagging algorithm. The first group

contains the variables related to the tagging track, the variables in the second group are

related to the signal B meson, the variables in the third group are related to “B+tagging

track” system while in fourth group contains the event related variables.

The distributions for the input variables are reported in Figure 3.5 The results of the

BDT for the training and test samples are shown in Figure 3.4. Since the two distributions

are slightly shifted the BDT output can be used to distinguish the right correlated charge

tracks from the wrong ones.

47


Rank Variable Variable Importance

1 log(Ptrackt ) 1.522e-01

2 ∆Q 1.495e-01

3 ∆R 1.181e-01

4 log(PVndo f ) 1.174e-01

5 log(∆φ) 8.398e-02

6 log(Ptott ) 7.564e-02

7 log(P) 7.368e-02

8 log(BPt) 5.395e-02

9 log(∆η) 4.417e-02

10 gprob 3.719e-02

11 log(ipsig) 3.481e-02

12 PIDK 1.453e-02

Table 3.5: Ranking of the input variables according to their importance in the BDT response.

ssPion response

0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

dx

/ (1

/N)

dN

0

0.5

1

1.5

2

2.5 Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

KolmogorovSmirnov test: signal (background) probability = 0.000195 (0.00584)

U/O

flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

TMVA overtraining check for classifier: ssPion

Figure 3.4: Output of the SS pion BDT. The red distribution is related to the right charge correlated

tracks while the blue distribution corresponds to the wrong charge correlated tracks.

Both the distributions are normalized to their own number of entries.

48


(a)

(b)

Figure 3.5: Distribution of the input variables considered for the BDT training. The red curve rep-

resents the right charge correlated tracks while the blue curve corresponds to the wrong

ones.

3.3.3 Performance and calibration

The shift of the BDT output distributions allows to identify the tracks usefull for the B

flavour tagging. This track selection is used in the second sample to select the best pion

tagger candidate, in case of multiple pion tagger candidates the pion with the highest BDT

value is picked. In order to calculate the tagging power for the SS pion tagger an estimation

of the mistag fraction probability (ω) is needed. The flavour oscillation can not be calculated

as shown in 3.2, instead it is estimated performing an unbinned Likelihood fit to the mixing

asymmetry of the signal events.

49


The fit is performed using RooFit package2[21]. The function used is:

f (t; q, ω) = e−t

τBd · (1 + q(1− 2ω) · cos(∆mdt)) (3.14)

where the value of Bd lifetime and its mixing frequency value are fixed to τBd = 1.519 ps

and ∆md = 0.510 ps−1 respectively [22]. The time acceptance function used is extracted

previously from the data in a first time and it is described by:

A(t) =(α(t− t0))β

1 + α(t− t0))β· (1 + γt) (3.15)

The acceptance parameters are calculated through a fit on the decay time, fixing τBd to 1.519

ps. The parameter values from the fit are reported in Table 3.6 and the plot is shown in

Figure 3.6.

α β t0 γ

1.97± 0.04 1.00 0.272± 0.002 −0.053± 0.001

Table 3.6: Acceptance parameters calculated with the fit on the B decay time for the 2012 data sam-

ple. The β parameter is fixed to one allowing a better fit convergence.

tau (ps)2 4 6 8 10 12 14 16 18

Eve

nts

/ (

0.18

025

ps

)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

time distribution

2 4 6 8 10 12 14 16 18

Pu

ll

-5

0

5

Figure 3.6: Time distribution of the events in 2012 data sample

This approach allows to calculate an average mistag which, by definition, gives an un-

derestimation of the tagging power respect the value calculated with the per-event mistag.

This effect is due to the quadratic dependence between εe f f and ω expressed by the equa-

2RooFit is a collection of classes built to support B physics most used Pdfs

50


tion:

εtag ·∑i

(1− 2ωi)2

Ntag> εtag ·

(1− 2 ∑

i

ωi

Ntag

)2

(3.16)

where 0 < i < Ntag.

However, this formula takes in account all the events contained in the data sample, to elim-

inate the background contribution is necessary applied the sWeights for each event. The

sWeights addition gives:

εtag ·∑i

(1− 2ωi · si)2

∑i si> εtag ·

(1− 2∑i ωi · si

∑i si

)2

(3.17)

where εtag = ∑i si/ ∑j sj with 0 < i < Ntag and 0 < j < Ntag+untag.

To reduce this underestimate the sample is splitted in BDT output bins (“categories”) and

the mistag is determined for each of them through a simultaneous fit on the asymmetry

oscillations. The mistag values are reported in Table 3.7 while the asymmetry plots for the

categories are shown in Figure 3.7. The asymmetry plots show a dependence between the

amplitude of the oscillation and the BDT response: the amplitude is higher for higher values

of BDT output, which correspond to smaller values of the mistag probability.

BDT category ω [%] εtag [%] εe f f [%]

[−1.0,−0.2] 48.0 ± 0.3 25.03 ± 0.09 0.04 ± 0.01

[−0.2, 0.0] 46.2 ± 0.4 18.53 ± 0.08 0.11 ± 0.02

[0.0, 0.1] 44.8 ± 0.5 8.99 ± 0.06 0.10 ± 0.02

[0.1, 0.2] 43.1 ± 0.6 6.35 ± 0.05 0.12 ± 0.02

[0.2, 0.35] 41.3 ± 0.6 5.56 ± 0.05 0.17 ± 0.03

[0.35, 0.5] 37.5 ± 0.8 3.50 ± 0.04 0.22 ± 0.03

[0.5, 0.7] 32.1 ± 0.7 3.84 ± 0.04 0.49 ± 0.04

[0.7, 1.0] 26.7 ± 1.1 1.48 ± 0.03 0.32 ± 0.03

TOT - 73.31 ± 0.16 1.57 ± 0.07

Table 3.7: Mistag probability, tagging efficiency and tagging power for the seven BDT categories

determined from the asymmetry fit of the test sample.

51


tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat0

(a) -1.0 < BDT < -0.2

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat1

(b) -0.2 < BDT < 0.0

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat2

(c) 0.0 < BDT < 0.1

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat3

(d) 0.1 < BDT < 0.2

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat4

(e) 0.2 < BDT < 0.35

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat5

(f) 0.35 < BDT < 5

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat6

(g) 0.5 < BDT < 0.7

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat7

(h) 0.7 < BDT < 1.0

Figure 3.7: Mixing asymmetry for signal events in the test sub-sample divided in categories.

52


Fitting the dependence of ω vs BDT output a per-event mistag value (η) can be esti-

mated. A 3rd order polynomial function is found to fit well the data:

ηi = p0 + p1 · BDTi + p2 · BDT2i + p3 · BDT3

i (3.18)

where ηi is the average expected mistag in the i− th BDT bin. The η distribution is shown

in Figure 3.8.

Then the BDT calibration is performed on the third sample, statistically independent

from the samples used so far. In this sample the ω is calculated through the asymmetry

plots dividing the data in BDT bins, as done previously. Thus a plot of ω vs η is fitted with

a linear function:

ω = p0 + p1 · (η − 〈η〉) (3.19)

where 〈η〉 is the average predicted mistag on the full sample. If η is correctly calibrated p0

should be equal to 〈η〉 and p1 should be equal to 1.

The parameters estimated from the two fit are reported in Table 3.8 and in Table 3.9,

while the fit plots are shown in Figure 3.8.

In the third sample a per-event mistag is determined through the polynomial and the

linear fits starting from the BDT output. In this way it has been possible to calculate a more

precise tagging power, as it has been described in equation 3.16. The results achieved are

reported in Table 3.9.

p0 p1 p2 p3

0.452± 0.003 −0.12± 0.01 −0.12± 0.04 −0.08± 0.07

Table 3.8: Parameters of the 3rd polynomial for the B0 −→ D−(→ Kππ)π+ 2012 data sample (test

sub-sample).

p0 p1 〈η〉 εtag [%] εe f f [%]

0.441± 0.003 0.982± 0.049 0.444 71.83± 0.22 1.64± 0.10

Table 3.9: Calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+ 2012

data sample (calibration sub-sample).

In this case the events with a per-event mistag larger than 0.5 are discarded and con-

sidered untagged. The results indicate that the calibration is correct within the statistical

53


uncertainties. The performances achieved provide an improvement of the 20% respect to

the current SSπ tagger, whose effective tagging efficiency is reported in Table 3.10.

ssPion-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

ω

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

/ ndf 2χ 0.4072 / 4p0 0.002795± 0.4523 p1 0.01238± -0.1168 p2 0.03679± -0.1222 p3 0.07103± -0.08121

/ ndf 2χ 0.4072 / 4p0 0.002795± 0.4523 p1 0.01238± -0.1168 p2 0.03679± -0.1222 p3 0.07103± -0.08121

(a) Curve η vs BDT output

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.9976 / 6

p0 0.0025± 0.4411 p1 0.0487± 0.9818

> η< 0± 0.4441

/ ndf 2χ 0.9976 / 6p0 0.0025± 0.4411 p1 0.0487± 0.9818

> η< 0± 0.4441

(b) Calibration curve ω vs η

Entries 83769Mean 0.4412RMS 0.04723

η0 0.1 0.2 0.3 0.4 0.5

a.u.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Entries 83769Mean 0.4412RMS 0.04723

eta_histo

(c) η distribution

Figure 3.8: Calibration plots for the B0 → D−π+ 2012 data sample. On the left the polynomial curve

on the test sub-sample and in the middle the linear fit on the calibration sub-sample. On

the right the eta distribution is shown. The magenta area shows the confidence range

within ±1σ.

εtag [%] εe f f [%]

60.38± 0.20 1.44± 0.09

Table 3.10: Performances obtained using the current SSπ tagger on the B0 → D−π+ 2012 data sam-

ple.

54


3.4 Validation on the 2011 data sample

A first validation has been performed on the B0 −→ D−(→ Kππ)π+ events collected

in 2011 corresponding to 1 fb−1 taken at√

s = 7 TeV. The cuts applied to select the B

candidates in this sample are the same as the ones described in Table 3.1. The aim is to

verify that the algorithm developed can be used on a data sample completely different

from the one used for the tuning without losing its tagging ability. The analysis follows the

same path described in the previous section. The sWeights are calculated with the same

parametrization used in the 2012 data. In Figure 3.9 the fit mass distribution is shown and

in the Table 3.11 the fit parameters are reported.










Table 3.11: Results of the fit to the mass distribution 2011 data sample

Then the sample is divided in categories and for each one the value of a “predicted

mistag” (η) is calculated through the parameters of the 3rd polynomial function, calculated

with the 2012 sample, and using the BDT response as independent variable. The true mistag

(ω) is calculated through an unbinned Likelihood fit using the formula reported in equation

3.14 and dividing the sample into categories.

α β t0 γ

2.17± 0.05 1.00 0.27± 0.03 −0.05± 0.01

Table 3.12: Acceptance parameters calculated with the fit on the B decay time for the 2011 data sam-


The acceptance parameters are estimated in this new sample through a fit on the decay

55


time. In Figure 3.10 the time plot is shown while the parameter values are reported in Table

3.12.

)2^+) (GeV/cπm(D^- 5.2 5.25 5.3 5.35 5.4 5.45 5.5

)2E

vent

s / (

0.0

03 G

eV/c

0

1000

2000

3000

4000

5000

6000

7000

8000


5.2 5.25 5.3 5.35 5.4 5.45 5.5

Pul

l

-5

0

5

Figure 3.9: Mass fit for the B0 → D−(Kππ)π+ 2011 data sample. The blue curve represents the pdf



are shown.

tau (ps)2 4 6 8 10 12 14 16 18

Eve

nts

/ (

0.18

025

ps

)

0

1000

2000

3000

4000

5000

6000

7000

time distribution

2 4 6 8 10 12 14 16 18

Pu

ll

-5

0

5

Figure 3.10: Time distribution of the events in 2011 data sample

If the BDT is well calibrated and unaffected by overtraining in each categories η should

be close to ω. The calibration plot is shown in Figure 5.3 while in the Table 3.13 the fit

parameters and the performances are reported. The tagging power is calculated using a

per-event mistag.

56


p0 p1 〈η〉 εtag [%] εe f f [%]

0.440± 0.002 1.08± 0.04 0.444 70.65± 0.13 1.76± 0.07


data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.85 / 6

p0 0.002276± 0.4402 p1 0.04422± 1.075

> η< 0± 0.4444

/ ndf 2χ 1.85 / 6p0 0.002276± 0.4402 p1 0.04422± 1.075

> η< 0± 0.4444

Figure 3.11: Calibration for the B0 → D−π+ 2011 data sample, plot ω vs η

The result for the p0 and p1 are compatible respectively with 〈η〉 and 1 by about 2σ, so

the estimated mistag is calibrated. These results prove that both the calibration parameters

and the performances found with 2012 data are compatible with 2011 data sample.

In Figure 3.12 the calibration plot of the merge between 2011 and 2012 data samples is

shown, and the tagging performances are reported in Table 3.14.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.441± 0.002 1.03± 0.03 0.444 71.19± 0.09 1.70± 0.05

Table 3.14: Calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+

2011+2012 data sample

In this case p0 is compatible with 〈η〉 within 2σ variation while p1 is correct within

the statistical error. Thus the performances on this sample are compatible with both the

2011 and 2012 data sample by about 1σ. Another validation has been perform on the same

channel B0 −→ D−(→ Kππ)π+ but using a different cuts selection. This check allows to

57


study a possible dependence of the tagging performances on the background contamination

of the sample. The details and the results about this selection are reported in Appendix D.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 2.073 / 6

p0 0.001683± 0.4407 p1 0.03275± 1.033

> η< 0± 0.4442

/ ndf 2χ 2.073 / 6p0 0.001683± 0.4407 p1 0.03275± 1.033

> η< 0± 0.4442

Figure 3.12: Calibration for the B0 → D−π+ 2011+2012 data sample, plot of ω vs η. The magenta

area shows the confidence range within ±1σ.

3.5 Validation on the B0 → K+π− 2012 data sample

Finally a last validation is performed on a data sample collected by the LHCb experiment

during the 2012, corresponding to their B0 −→ K+π− decay mode. This validation can

be usefull to identify any possible dependence from the decay channel utilized. As shown

for the other channels analyzed, by means the sPlot technique has been possible to get

rid the background from the data. The cuts selection applied on this channel in order to

improve the estimation of the sWeights are reported in Table 3.15. In this case fitting the B

invariant mass distribution is not straight forward, because in addition to the combinatorial

background other significant contributions, which were negligible in the B0 −→ D−π+

decay mode, affect the fit [?]. The most important contamination coming from the B0s −→

K+π− which, even if characterized by a lower branching ratio than the signal channel, can

alter meaningfully the distribution.

Another background contaminations arises both from the partially reconstructed three-

body decays of B mesons, such as B0 −→ K−ρ+(π+π0) or B+ −→ π+π+π−, where only

two of the daughters are used to reconstruct the B candidate and from the combinato-

rial background. There are also two further background contributions represented by the

misidentification of the K or the π, i.e. B0 −→ π+π− and B0s −→ K+K−, but in this analysis

they are not taken in account because of their negligible effect on the fit.

58



Mother cuts

B0 mass Invariant mass of the B 4.9 < mB0 < 5.6 GeV/c2

IPB Impact parameter of the B wrt PV < 0.06 mm

B0 time Decay time of the B > 0.9 ps

pT(B0) Tranverse momentum of the B > 2200 MeV/c

Combination cuts

DOCA Distance of closest approach between

the two daughters

< 0.08 mm

Daughters cuts

min(ph+T , ph′−

T ) Minimum pT between the two

daughters

> 1100 MeV/c

max(ph+T , ph′−

T ) Maximum pT between the two

daughters

> 2800 MeV/c

min(IPh+ , IPh′−) Minimum IP between the two

daughters

> 0.15 mm

max(IPh+ , IPh′−) Minimum IP between the two

daughters

> 0.30 mm

PID cuts

h = K Identification of a hadron as a kaon PIDK > 3 &

∆(log LK − log LP) > −5

h = π Identification of a hadron as a pion PIDK < −3 & PIDp < 5

Table 3.15: Selection cuts for the the decay channel B0 −→ K+π−. The cuts are applied to the B0

candidate (mother), and to the two hadron h+, h− (daughters).

59


The PDF used to fit the signal B mass distribution consist of the sum of two Gaussians

centered on the same mean value, one to describe the distribution core and the other to rep-

resent the distribution tails, with one “Crystal Ball”. The pollution sources instead are mod-

eled using different PDFs: the B0s contamination is reproduced by the sum of two Gaussians,

centered on the same value, the partially reconstructed three-body events are estimated by

means of an Argus function while the size of the combinatorial background, which a priori

is not described by any analytical function, is evaluated through a negative exponential,

whose expression has been reported in equation 3.12.

)2) (MeV/cπ K+ →m(B4900 5000 5100 5200 5300 5400 5500 5600

)2

Even

ts / (

7 M

eV

/c

0

2000

4000

6000

8000

10000

Kpi→B

Total

B0d → K

+π −

B0s → K

−π +

B→ 3 − body

Comb. bkg

4900 5000 5100 5200 5300 5400 5500 5600

Pu

ll

5

0

5

Figure 3.13: Mass fit for the B0 → K+π− 2012 data sample. In the figure the blue curve represents

the total pdf. The signal component is shown in green, the background coming from

the B0s decay is colored in black, the partially reconstructed contamination is the red

curve and the combinatorial is represented in magenta. Below the plot the normalized

residuals (pulls) are shown.

The Crystal Ball PDF consists of a Gaussian core portion and a power-law low-end tail

and is described by the equation 3.20, while the Argus distribution is reported in formula

3.21 [21].

f (x; α, n, x, σ) = N ·

exp

(− (x−x)2

2σ2

), for x−x

σ > −|α|(n|α|

)ne−

12 α2(

n|α|−|α|−x

)n , for x−xσ ≤ −|α|

(3.20)

f (x; x, c, p) = x(

1−( x

x

)2)p

· exp{

c(−( x

x

)2)}

(3.21)

60



Bd → K+π− PDF

µBd Mean value of Bd mass distribution 5285.16± 0.10

σ1Bd σ of the core Gaussian 18.91± 0.12

σ2Bd σ of the tail Gaussian 43.40± 0.81

f coreBd Fraction of the core Gaussian 0.76± 0.01

αCB,Bd α parameter of the Crystal Ball 1.74± 0.06

σCB,Bd σ parameter of the Crystal Ball 15.05± 9.58

nCB,Bd n Parameter of the Crystal Ball 3.579± 0.934

fCB,Bd Fraction of the Crystal Ball 1.00± 0.07

Bs → K−π+ PDF

µBs Mean value of Bs mass distribution 5373.97± 0.82

σ1Bs σ of the core Gaussian 24.00± 4.02

σ2Bs σ of the tail Gaussian 17.36± 3.32

f coreBs Fraction of the core Gaussian 0.43± 0.54

αCB,Bd 2.288 1.933

Argus PDF

Thmass Mean value of the mass distribution 5161.1± 0.6

argpar Shape parameter −18.48± 0.50

Combinatorial background PDF

aComb Slope of the exponential function −0.001± 0.000

Yields of the distributions

Nsig Yield of the signal distribution 71060± 305

NBs2Kpi Yield of the B0s background 4135± 187

NPartPhysic Yield of the three-body background 19283± 325

Ncomb Yield of the combinatorial background 103236± 484

Table 3.16: Results of the fit to the mass distribution B0 → K+π− 2012 data sample

The final fit on the B mass distribution is shown in Figure 3.13 and the fit parameters are

listed in Table 3.16.

61


Estimated the sWeights the background yield is completely unfold from the signal events.

Then, according to the steps followed so far, the acceptance function is fitted fixing both the

Bd lifetime and the mixing frequency ∆md to the pdg values. The fit on the decay time is

presented in Figure 3.14 and it allows to evaluate the parameters of the acceptance function


α β t0 γ

11.01± 0.12 1.00 0.84± 0.09 −0.068± 0.002

Table 3.17: Acceptance parameters calculated with the fit on the B decay time for the B0 → K+π−

2012 data sample. The β parameter is fixed to one allowing a better fit convergence.

(ps)τ

1 2 3 4 5 6 7 8 9 10

Even

ts / (

0.0

915 p

s )

0

500

1000

1500

2000

2500

3000

3500

4000

4500

time distribution

1 2 3 4 5 6 7 8 9 10

Pu

ll

5

0

5

Figure 3.14: Time distribution of the events in B0 → K+π− 2012 data sample

Obtained these parameters, they are kept fixed and an unbinned Likelihood fit per BDT

categories is performed to determine the true mistag ω for each one of them. These values

are plotted against the predicted mistag η evaluate per categories by means the parameters

found in section 6.4.2 for the 3rd polynomial. The calibration plot is shown in Figure 3.15

and its parameters are related in Table 3.18.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.450± 0.004 1.15± 0.10 0.456 66.50± 0.18 1.08± 0.08

Table 3.18: Calibration parameters and tagging performances for the B0 → K+π− 2012 data sample

62


η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.54 / 3

p0 0.003658± 0.4498 p1 0.1123± 1.146

> η< 0± 0.4564

/ ndf 2χ 5.54 / 3p0 0.003658± 0.4498 p1 0.1123± 1.146

> η< 0± 0.4564

Figure 3.15: Calibration for the B0 → K+π− 2012 data sample, plot of ω vs η. The magenta area

shows the confidence range within ±1σ.

Also in this case the fit parameters results calibrated, indeed the value of p0 is close

within 2σ from the mean value of η and p1 is compatible to 1 within 1σ. However the

tagging power achieved is lower than the one found in each sample of the B0 → D−π+.

This discrepancy is due, at least in a measure, to the different BpT distribution of the two

samples, which can be observed in Figure 3.16. The tagging power dependence on the BpT ,

is analyzed in detail in Chapter 6.

To verify that this tagging power loss is due to the different kinematic, i.e. the transverse

momentum of the signal B, the events are reweighted according to the ratio of the pT distri-

butions of the B0 → D−π+ and B0 → K+π− 2012 data sample and the performances have

been recomputed. This reweighting procedure entails an increasing of the performances,

making them compatible to the ones found in the other channel, and corroborates the hy-

pothesis of a dependence between the tagging power loss and the softer BpT spectra. The

comparison between the new tagging power achieved after the reweighting with the previ-

ous one is reported in Table 3.19.

63


εtag [%] εe f f [%] εtag [%] εe f f [%]

without BpT reweighting with BpT reweighting

65.30± 0.18 1.20± 0.11 67.93± 0.18 2.15± 0.22

Table 3.19: Comparison between the tagging powers obtained in the B0 → K+π− 2012 data sample

before and after the reweighting using the ratio of the transverse momentum of the signal

B.

Pt_sig_Kpi

Entries 5937106

Mean 6.376

RMS 3.229

of signal B [GeV/c]T

p0 5 10 15 20 25 30

events

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Pt_sig_Kpi

Entries 5937106

Mean 6.376

RMS 3.229

B0d → K

+π −

B0d → D

−π +

(a)

Pt_ratio

Entries 41

Mean 6.544

RMS 4.996

of signal B [Gev/c]T

p0 5 10 15 20 25 30

ratio

0

1

2

3

4

5

6

Pt_ratio

Entries 41

Mean 6.544

RMS 4.996

Pt_ratio

(b)

Figure 3.16: Comparison of the pt normalized distributions of the signal B for B0 → K+π− (red) and

the B0 → D−π+ (blue) 2012 data samples, respectively. In the right plot the ratio of the

two distributions is shown.

64

4

Same Side Proton Tagger

Contents

4.1 SSp tagger development using the 2012 data sample 65

4.1.1 SS proton training 65




In this chapter a development of a Same Side Proton tagger is illustrated. The step fol-

lowed are similar to the ones described in the previous chapter for the SS pion. The BDT

training is performed on the B0 −→ D−π+ data sample collected in 2012 and it is described

in 6.4.2, then the validation using the B0 −→ D−π+ 2011 data sample is described in section

4.2 and using the second event selection in section D.2. Because the sWeights are indepen-

dent from the tagger chosen they are not recalculated in this chapter, but each sample uses

the sWeights estimated in the previous chapter.

4.1 SSp tagger development using the 2012 data sample

4.1.1 SS proton training

As shown in Figure 3.2 the fragmentation of the b quark can produce also protons in addic-

tion to the signal B0d. Thus also the charge of the proton, because of the quark correlation,

can be used to infer the flavour of the signal B meson. However in this case the charge cor-

relation is the opposite with respect to the SS pion’s one, thus the tagging decision used to

implement the BDT is defined as:

Right −→ B0 p̄ or B̄0 p

65

4 - Same Side Proton Tagger


Selection cuts on the tagging particle


IP/σIP Impact parameter significance wrt PV < 4

Ghost prob Probability that a track is a random combination of hits < 0.5

PIDp ∆(log Lp − log Lπ) > 5

IPPU/σIPPU Impact parameter significance wrt Pile-Up > 3

Selection cuts on the B+tagging system


∆Q m(B + track)−m(B)−m(track) < 1300 [MeV/c2]

∆η Difference between signal B and tagging track

pseudorapidity

< 1.2

∆φ Difference between signal B and tagging track φ angle < 1.2

χ2vtx vertex goodness fit < 100

Table 4.1: Preselection cuts applied for SS proton tagging algorithm. The cut on PIDp is comple-

mentary to the one used in SSπ and a cut on PIDK has been added.

Wrong −→ B0 p or B̄0 p̄

The sample is divided in sub-sample using the same way used for the SS pion and

also the BDT options chosen are the same, i.e. MisClassificatorError as separation criterion

and AdaBoost as boosting method. The aim of the BDT is to separate the right charged

correlated protons from the wrong charged correlated ones and to identify the best proton

tagger candidate. To improve the separation power of the BDT some preselection cuts are

applied to remove tracks that would be discarded anyway. These cuts are reported in Table

4.1. The cut on the particle identification of the track (PIDp) is complementary to the one

used for the SSπ, so the SSp candidates are a disjoint sample respect to the previous one.

The input variables used in the training are reported in Table 4.2 and in the Table 4.3

they are listed in order of importance in the BDT decision. The input variable distributions

are shown in Figure 4.2 The results of the BDT for the training and test samples are shown

in Figure 4.1. Also in this case the two distributions are slightly shifted, thus the BDT output

can be used to distinguish the right correlated charge tracks from the wrong ones.

66


Variable Description

Track related variables

log p Momentum


log IP/σIP Impact parameter significance wrt PV

B+track variables


∆Q m(B + track)−m(B)−m(track)

∆R√

∆φ2 + ∆η2

log ∆η Difference between signal B and tagging track pseudorapidity

Event related variables

Ntracks in PV Number of degrees of freedom in the fit of Primary Vertex

Table 4.2: Input variables used to train the BDT for the SS proton tagging algorithm. The first group

contains the variables related to the tagging track, the variables in the second group are

related to the signal B meson, the variables in the third group are related to “B+tagging

track” system while in fourth group contains the event related variables.

ssProton response

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

dx

/ (1

/N)

dN

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

KolmogorovSmirnov test: signal (background) probability = 0.00325 (0.00968)

U/O

flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

TMVA overtraining check for classifier: ssProton

Figure 4.1: Output of the SS proton BDT. The red distribution is related to the right charge correlated

tracks while the blue distribution corresponds to the wrong charge correlated tracks.

Both the distributions are normalized to their own number of entries.

67


In case of more than one tagger candidate per event, the one with the highest BDT

output value is chosen.

Rank Variable Variable Importance

1 log(PIDp) 2.656e-01

2 log(Ptrackt ) 1.426e-01

3 log(Ptott ) 1.163e-01

4 log(PVndo f ) 1.010e-01

5 log(∆η) 9.578e-02

6 log(P) 9.459e-02

7 ∆Q 7.975e-02

8 ∆R 6.742e-02

9 log(ipsig) 3.685e-02

Table 4.3: Ranking of the input variables according to their importance in the BDT response.

(a)

(b)

Figure 4.2: Distribution of the input variables considered for the BDT training. The red curve rep-

resents the right charge correlated tracks while the blue curve corresponds to the wrong

ones.

68


4.1.2 Performance and calibration

The performances of SS proton algorithm are evaluated with the same procedure described

in section 6.4.2 for the SS pion. The first step is to use the BDT output to divide the sam-

ple in categories. For each bin an unbinned fit to the mixing asymmetry is performed in

order to determine the mistag value ω. The mistag values are reported in Figure 4.4 and the

asymmetry plots for the categories are shown in Figure 4.3.

BDT category ω [%] εtag [%] εe f f [%]

[−1.0,−0.2] 51.0 ± 0.5 8.15 ± 0.05 0.003 ± 0.001

[−0.2, 0.0] 48.8 ± 0.5 11.75 ± 0.06 0.007 ± 0.005

[0.0, 0.1] 48.5 ± 0.6 6.77 ± 0.05 0.006 ± 0.005

[0.1, 0.2] 46.9 ± 0.6 6.08 ± 0.05 0.023 ± 0.009

[0.2, 0.35] 45.3 ± 0.6 6.76 ± 0.05 0.060 ± 0.016

[0.35, 0.5] 43.5 ± 0.7 4.22 ± 0.04 0.072 ± 0.017

[0.5, 0.7] 38.7 ± 0.9 2.36 ± 0.03 0.120 ± 0.021

[0.7, 1.0] 29.6 ± 1.3 1.16 ± 0.02 0.193 ± 0.025

TOT - 39.10 ± 0.14 0.48 ± 0.04

Table 4.4: Mistag probability, tagging efficiency and tagging power for the seven BDT categories

determined from the asymmetry fit in the test sub-sample

Also in this case the asymmetry plots show a dependence between the amplitude of the

oscillation and the BDT response: the amplitude is higher for higher values of BDT output,

which correspond to smaller values of the mistag. In the first category the evaluated mistag

is greater than 0.5, thus both the related tagging power and tagging efficiency are not taken

in account. The dependence ω vs BDT output is fitted with a 3rd order polynomial function,

thus a per-event mistag value (η) can be estimated. The η distribution is shown in Figure

4.4. In the Table 4.5 the estimated value of the fit parameters are reported and the Figure 4.4

shown the fit plot. Then the BDT calibration is performed on the statistically independent

sample using the linear function in equation 3.18. The results of the calibration and the

linear plot are shown in Table 4.6 and in Figure 4.4.

69


tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat0

(a) -1.0 < BDT < -0.2

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat1

(b) -0.2 < BDT < 0.0

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat2

(c) 0.0 < BDT < 0.1

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat3

(d) 0.1 < BDT < 0.2

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat4

(e) 0.2 < BDT < 0.35

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat5

(f) 0.35 < BDT < 0.5

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat6

(g) 0.5 < BDT < 0.7

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.6

0.4

0.2

0

0.2

0.4

0.6

cat7

(h) 0.7 < BDT < 1.0

Figure 4.3: Mixing asymmetry for signal events. The plots are obtained with the sPlots technique.

70


p0 p1 p2 p3

0.48± 0.003 −0.07± 0.01 −0.05± 0.04 −0.17± 0.06

Table 4.5: Parameters of the 3rd polynomial for the B0 −→ D−(→ Kππ)π+ 2012 data sample (test

sub-sample)

ssProton-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

ω

0.2

0.3

0.4

0.5

0.6

0.7

/ ndf 2χ 1.685 / 4p0 0.003271± 0.4831 p1 0.01261± -0.07304 p2 0.03947± -0.05043 p3 0.06292± -0.1749

/ ndf 2χ 1.685 / 4p0 0.003271± 0.4831 p1 0.01261± -0.07304 p2 0.03947± -0.05043 p3 0.06292± -0.1749

(a) Curve η vs BDT output

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 2.533 / 6

p0 0.003381± 0.4601 p1 0.08044± 0.9071

> η< 0± 0.462

/ ndf 2χ 2.533 / 6p0 0.003381± 0.4601 p1 0.08044± 0.9071

> η< 0± 0.462

(b) Calibration curve ω vs η


η0 0.1 0.2 0.3 0.4 0.5

a.u.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14 Entries 46207Mean 0.4601RMS 0.03685

eta_histo

(c) η distribution

Figure 4.4: Calibration plots for the B0 → D−π+ 2012 data sample. On the left the polynomial curve

on the test sub-sample and on the right the linear fit on the calibration sub-sample. On

the right the eta distribution is shown. The magenta area shows the confidence range

within ±1σ

p0 p1 〈η〉 εtag [%] εe f f [%]

0.460± 0.003 0.91± 0.08 0.462 39.64± 0.15 0.47± 0.04


data sample

71



33.16± 0.14 0.51± 0.03

Table 4.7: Performances obtained using the current SSp tagger on the B0 → D−π+ 2012 data sam-

ple.

As done for the SS pion, the tagging power is calculated using using the per-event

mistag, determined with the polynomial function. The results achieved are reported in Ta-

ble 4.6. The events with a per-event mistag greater than 0.5 are discarded and considered

untagged. In this case p0 is compatible with 〈η〉 within the statistical error, while p1 is com-

patible to 1 within 1σ. In Table 4.7 the performances obtained using the SSp available in the

sample are shown. The tagging power achieved with the two algorithms are compatible

within the statistical error.

4.2 Validation on the 2011 data sample

The same validation on the sample done for SS proton studies using B0 −→ D−(→ Kππ)π+

events collected in 2011 corresponding to 1 fb−1 taken at√

s = 7 TeV center of mass energy.

The calibration plot is shown in Figure 4.5 while in the Table 4.8 the fit parameters and the

performances are reported. The tagging power is calculated using a per-event mistag.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.469± 0.003 1.06± 0.07 0.461 41.11± 0.13 0.48± 0.04


data sample

The result for the p0 and p1 are compatible respectively with 〈η〉 and 1 by about 2σ, so

the estimated mistag is calibrated. These results prove that both the calibration parameters

and the performances found with 2012 data are compatible with 2011 data sample.

In Figure 4.6 the calibration plot of the merge between 2011 and 2012 data samples is

shown, and the tagging performances are reported in Table 4.9.

72


p0 p1 〈η〉 εtag [%] εe f f [%]

0.465± 0.002 0.99± 0.05 0.462 40.45± 0.10 0.47± 0.03

Table 4.9: Calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+

2011+2012 data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.071 / 5

p0 0.003004± 0.469 p1 0.07051± 1.056

> η< 0± 0.4612

/ ndf 2χ 1.071 / 5p0 0.003004± 0.469 p1 0.07051± 1.056

> η< 0± 0.4612

Figure 4.5: Calibration for the B0 → D−π+ 2011 data sample, plot ω vs η

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.382 / 5

p0 0.002246± 0.4651 p1 0.05304± 0.9906

> η< 0± 0.4616

/ ndf 2χ 1.382 / 5p0 0.002246± 0.4651 p1 0.05304± 0.9906

> η< 0± 0.4616

Figure 4.6: Calibration for the B0 → D−π+ 2011+2012 data sample, plot of ω vs η. The magenta area


In this case p0 is compatible with 〈η〉 by about 2σ while p1 is correct within the statistical

error. Thus the performances on this sample are compatible with both the 2011 and 2012

73


data sample within the statistical error.

4.3 Validation on the B0 → K+π− 2012 data sample

The last validation for the SSp is executed on a different decay channel in order to verify

that there is no dependence between the BDT output and the decay mode analyzed. The

B0 → K+π− 2012 data sample has been used and the sWeights evaluated in section 3.5 are

applied to rid the background events. The calibration plot is shown in Figure 4.7 and the

results obtained from the linear fit to η are reported in Table 4.10.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.458± 0.004 0.82± 0.14 0.468 47.33± 0.19 0.43± 0.05

Table 4.10: Calibration parameters and tagging performances for the B0 → K+π− 2012 data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 6.5 / 4

p0 0.004331± 0.458 p1 0.1401± 0.8214

> η< 0± 0.4675

/ ndf 2χ 6.5 / 4p0 0.004331± 0.458 p1 0.1401± 0.8214

> η< 0± 0.4675


shows the confidence range within ±1σ

The calibration of both the fit parameters is found to be correct within the 2σ. Similarly

to the SSπ, also the performances achieved by the SSp are lower than the ones obtained

in the B0 → D−π+ samples. Thus the same reweighting procedure is followed, using the

ratio of the two BpT distributions as weight. The behavior is similar to what was found with

the SSπ: the tagging power increases and becomes compatible with to the results obtained

in the other samples. In Table 4.11 the tagging efficiencies achieved before and after the

reweighting are compared.

74


εtag [%] εe f f [%] εtag [%] εe f f [%]

without BpT reweighting with BpT reweighting

47.33± 0.19 0.43± 0.05 49.75± 0.19 0.53± 0.06

Table 4.11: Comparison between the tagging powers obtained in the B0 → K+π− 2012 data sample

before and after the reweighting using the ratio of the transverse momentum of the signal

B.

75

5

Tagger combination

Contents

5.1 Combination of taggers 76

5.2 SSp and SSπ combination 77

5.2.1 Combination of the SS taggers on the B0 → D−π+ 2012 data sample 78

5.2.2 Combination on B0 → D−π+ 2011 data sample 80


5.3 SS and OS combination 83

5.3.1 Combination of SS taggers with the OS tagger on the B0 → D−π+

2012 data sample 84

5.3.2 Combination on the B0 −→ D−π+2011 data sample 87


5.4 Measurement of ∆md 90

In this chapter the SS tagger combination (SS pion + SS proton) 5.2 and the combination

SS tagger + OS tagger 5.3 are studied. The theoretical formulas used to achieve the com-

bination of the properties of all available taggers are reported in section 5.1. In section 5.4

the tagging decisions of the final tagger combination (SS+OS) are used to estimate ∆md by

means a simultaneous unbinned fit on the categories.

5.1 Combination of taggers

When more than one tagger is available per event the tagging decisions and mistag proba-

bilities provided by each tagger can be combined into a final decision on the initial flavour

76

5 - Tagger combination

of the signal B using the following equations:

p(b) = ∏i

(1 + di

2− di(1− ηi)

)p(b) = ∏

i

(1− di

2+ di(1− ηi)

)(5.1)

where

• p(b) (p(b)) is the probability that the signal B contains a b-quark (b-quark)

• di is the decision taken by the i-th tagger based on its charge using this convention:

di = 1 −→ signal B contains a b-quark, thus is a B0d

di = −1 −→ signal B contains a b-quark, thus is a B0d

• ηi is the predicted mistag probability of the i-th tagger.

Then these probabilities are renormalized as:

P(b) =p(b)

p(b) + p(b)P(b) = 1− P(b) (5.2)

If P(b) > P(b) the combined tagging decision is d = +1 and the final mistag probability

is η = 1− P(b). Otherwise if P(b) > P(b) the combined tagging decision and the mistag

probability are d = −1 and η = 1− P(b) [23].

5.2 SSp and SSπ combination

In the previous chapters the B identification was done only with one type of taggers: the

SS pion or the SS proton. For each tagger the probability of the tag decision to be wrong

was estimated event by event by means of a BDT which combined some input variable

related to the tagger or to the event. The BDT output was calibrated in order to calculate a

per-event mistag. However there is an overlap between the SS proton and SS pion taggers

where the same event is tagged by both taggers. In these cases can be usefull to combine the

tagging decisions and the mistag probabilities of each tagger into a final tagging decision

and mistag probability to infer the initial flavour of the signal B meson with higher accuracy.

The same track can not be used for both taggers as the PIDp cuts applied to the candidate

are mutually exclusive: PIDp < 5 for the SS pion and PIDp > 5 for the SS proton. Thus

there are not correlations among the two types of tagger.

Each sample has been divided in three sub-samples in order to separate events with

only one tagger from the events with both taggers:

77


• events tagged only by a π −→ ω is estimated with the SSπ calibration (Chapter 3)

• events tagged only by a p −→ ω is estimated with the SSp calibration (Chapter 4)

• events tagged both by π and p −→ ω is estimated combining the two taggers (Section

5.1)

While the mistag probability provided by the first two sub-samples is well calibrated as

proved with the previous validations, the calibration of the mistag estimated in the third

sub-sample has to be checked. In order to verify the well calibration of the combination,

the last subsample is divided further in categories and for each one the true mistag ω is

estimated through an unbinned fit on the time oscillations. This time the category division

is based on the value of the combined mistag and for each categories η is calculated as the

weighted average using the sWeights. Thus a linear fit of ω vs η can be performed.

5.2.1 Combination of the SS taggers on the B0 → D−π+ 2012 data sample

The analysis on this sample is made dividing the events in three sub-sample as explained in

the previous section. The events of the third sub-sample are divided in categories in order

to check the calibration of the tagger combination. In Figure 5.1 the plots of the flavour

oscillation asymmetry of each category are shown and in Table 5.1 the performances are

reported. In Figure 5.2 the η distribution obtained in the total sample is shown.

η category ω [%] εtag [%] εe f f [%]

[0.45, 0.50] 48.0 ± 0.5 17.57 ± 0.12 0.03 ± 0.01

[0.40, 0.45] 43.9 ± 0.6 12.35 ± 0.09 0.19 ± 0.04

[0.30, 0.40] 37.8 ± 0.8 6.07 ± 0.07 0.36 ± 0.05

[0.0, 0.30] 25.6 ± 1.5 1.58 ± 0.04 0.38 ± 0.05

TOT - 37.58 ± 0.17 0.96 ± 0.08

Table 5.1: Tagging performances of categories of sub-sample with both taggers for the B0 → D−π+

2012 data sample

The calibration plot for the third sub-sample is reported in Figure 5.2, while in Table 5.2

the fit parameters are listed. The result for the p0 is compatible with 〈η〉 by about 2σ while

p1 is correct within the statistical error.

78


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(d) 0 < η < 0.3

Figure 5.1: Mixing asymmetry for signal events for the B0 → D−π+ 2012 data sample. The plots are

obtained with the sPlots technique.

p0 p1 〈η〉

0.440± 0.004 0.98± 0.06 0.433

Table 5.2: Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2012 data sample

Sub-sample εtag [%] εe f f [%]

SSπ 35.27± 0.14 0.88± 0.07

SSp 8.94± 0.09 0.11± 0.02

SS(π + p) 34.35± 0.14 0.99± 0.07

TOT 78.57± 0.12 1.97± 0.10

Table 5.3: Tagging performances of the SS combination for the B0 −→ D−(→ Kππ)π+ 2012 data

sample

79


η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.171 / 2

p0 0.003512± 0.4402 p1 0.0604± 0.9806

> η< 0± 0.4327

/ ndf 2χ 1.171 / 2p0 0.003512± 0.4402 p1 0.0604± 0.9806

> η< 0± 0.4327

(a) Calibration curve ω vs η


η0 0.1 0.2 0.3 0.4 0.5

a.u.

0

0.02

0.04

0.06

0.08

0.1Entries 91301Mean 0.4404RMS 0.05203

eta_SS_histo

(b) η distribution

Figure 5.2: On the left the calibration plot for the B0 → D−π+ 2012 data sample. On the right the

distribution of the predicted mistag (η) for the total sample is shown. The magenta area

shows the confidence range within ±1σ

The performances obtained in this sample are reported in Table 5.3, where the efficien-

cies reported are calculated using the corrected calibration for the per-event mistag.

The same combination has been developed using the SSπ and the SSp taggers cur-

rently available, whose performances are reported in Table 5.4. The combination realized

by means the SS taggers implemented in this thesis provides an improvement in tagging

performances of 10% respect to those obtained with the current taggers.


68.35± 0.14 1.79± 0.08

Table 5.4: Performances obtained using the current SS tagger on the B0 → D−π+ 2012 data sample.

5.2.2 Combination on B0 → D−π+ 2011 data sample

In this section the same procedure described in the previous section to divide the third

sub-sample has been followed. This analysis represents a validation of the SS tagger combi-

nation developed in this chapter. The calibration plot is reported in Figure 5.2 and in Table

5.2 the fit parameters are shown.

p0 p1 〈η〉

0.439± 0.004 0.92± 0.05 0.432

Table 5.5: Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample

80


In this case both p0 and p1 are compatible respectively with 〈η〉 and 1 by about 2σ.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.1982 / 2

p0 0.00315± 0.4393 p1 0.05066± 0.9242

> η< 0± 0.432

/ ndf 2χ 0.1982 / 2p0 0.00315± 0.4393 p1 0.05066± 0.9242

> η< 0± 0.432

Figure 5.3: Calibration for the B0 → D−π+ 2011 data sample, plot of ω vs η. The magenta area


The final performances obtained in this sample are reported in Table 5.6, where the

efficiencies reported are calculated using the corrected per-event mistag.


SSπ 33.46± 0.13 0.91± 0.06

SSp 7.76± 0.07 0.12± 0.02

SS(π + p) 36.47± 0.13 1.02± 0.07

TOT 77.69± 0.12 2.05± 0.07

Table 5.6: Tagging performances of the SS combination for the B0 −→ D−(→ Kππ)π+ 2011 data

sample

As done in the previous chapters the 2012 and 2011 data samples are merged in a only

one sample to decrease the statistical uncertainties. The performances obtained in this new

sample are reported in Table 5.7. The tagging power found in the 2012,2011 and 2011+2012

data samples are compatible within the statistical errors.

81



SSπ 34.28± 0.10 0.90± 0.05

SSp 7.93± 0.05 0.10± 0.02

SS(π + p) 35.61± 0.10 1.01± 0.06

TOT 77.81± 0.08 2.01± 0.06

Table 5.7: Tagging performances of the SS combination for the B0 −→ D−(→ Kππ)π+ 2011+2012

data sample.

5.2.3 Combination on the B0 → K+π− 2012 data sample

The performances and the calibration have been also cross-checked on the B0 → K+π− 2012

data sample. The calibration parameters extrapolated by the linear fit are listed in Table 5.8

while the plot is shown in Figure 5.8.

p0 p1 〈η〉

0.450± 0.005 0.90± 0.09 0.440

Table 5.8: Calibration parameters of the SS combination for the B0 → K+π− 2012 data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5p0 0.004633± 0.4499

p1 0.08923± 0.9036

> η< 0± 0.4404

p0 0.004633± 0.4499

p1 0.08923± 0.9036

> η< 0± 0.4404



In this calibration the p1 parameter is correct within the statistical error while p0 is com-

patible to the mean value of η within 2σ. The tagging efficiencies for each sub-sample and

82



SSπ 24.97± 0.16 0.55± 0.07

SSp 12.52± 0.12 0.12± 0.03

SS(π + p) 39.21± 0.18 0.78± 0.08

TOT 76.70± 0.25 1.44± 0.11

Table 5.9: Tagging performances of the SS combination for each sub-sample of the B0 −→ K+π−

2012 data sample.

the total ones are reported in Table 5.9

Also in this case, it is observable a tagging power loss, which can be accounted to the

dependence of the SS taggers on the different BpT spectra of the decay modes.

5.3 SS and OS combination

As shown in the previous section, combining two taggers allows to improve the tagging

performances. Thus as a last step of this analysis the combination of the general SS tagger

and Opposite Side tagger (OS) is performed. The general OS tagger represents the combi-

nation of OS electron, OS muon, OS kaon and OS vertex charge which is based on the

inclusive reconstruction of a secondary vertex corresponding to the Opposite B decay.

Using the same procedure used in the previous section, each sample is divided in three

sub-samples in order to separate events with only one tagger from the events with both

taggers:

• events tagged only by a SS tagger −→ ω is estimated with the SS calibration (Section

5.2)

• events tagged only by OS tagger −→ ω is estimated with the OS calibration

• events tagged both by SS and OS taggers −→ ω is estimated combining the two tag-

gers (Section 5.1)

Also in this case the mistag estimated in the third sub-sample has to be check, thus the

sub-sample is divided in categories according to the eta value and for each one the true

mistag ω is estimated through an unbinned fit on the time oscillations.Then a linear fit of ω

vs η can be performed.

83


5.3.1 Combination of SS taggers with the OS tagger on the B0 → D−π+ 2012

data sample

The first step of this analysis is to check the corrected calibration of the OS tagger combi-

nation on the B0 −→ D−(→ Kππ)π+ 2012 data sample. The calibration plot is shown in

Figure 5.5 and in Table 5.10 the fit parameters and the tagging performances obtained using

only the OS tagger contribution are reported, where the efficiencies reported are calculated

using the calibrated per-event mistag. In Figure 5.5 the distribution for the mistag predicted

by OS tagger is shown.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.379± 0.003 0.91± 0.04 0.369 38.01± 0.15 3.32± 0.16

Table 5.10: OS calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+

2012 data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.43 / 4

p0 0.00336± 0.3789 p1 0.0355± 0.905

> η< 0± 0.3692

/ ndf 2χ 5.43 / 4p0 0.00336± 0.3789 p1 0.0355± 0.905

> η< 0± 0.3692



η0 0.1 0.2 0.3 0.4 0.5

a.u.

0

0.005

0.01

0.015

0.02

0.025


eta_OS_histo

(b) η distribution

Figure 5.5: The OS calibration for the B0 → D−π+ 2012 data sample is shown on the left while on

the right the η distribution is reported. The magenta area shows the confidence range

within ±1σ

This calibration is used to calculate the corrected mistag in the sub-sample containing

the events tagged only by the OS tagger. Then the events tagged by both taggers are split-

ted in categories, in order to check the corrected calibration of the new combined mistag.

In Figure 5.6 the asymmetry plot on the categories are shown and in Table 5.12 the perfor-

mances of each category are reported. The distribution of the mistag predicted by the tagger

combination is reported in Figure 5.7.

84


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cat0

(f) 0.20 < η < 0.00

Figure 5.6: Mixing asymmetry plots for signal events for the B0 → D−π+ 2011 data sample.

The calibration plot for the third sub-sample is reported in Figure 5.7 while in Table 5.11

the fit parameters estimated are listed.

p0 p1 〈η〉

0.371± 0.004 1.02± 0.04 0.367

Table 5.11: SS+OS calibration parameters for the B0 −→ D−(→ Kππ)π+ 2012 data sample

85


η category ω [%] εtag [%] εe f f [%]

[0.45, 0.50] 47.9 ± 0.8 6.80 ± 0.08 0.02 ± 0.01

[0.40, 0.45] 42.7 ± 0.8 6.79 ± 0.08 0.15 ± 0.03

[0.35, 0.40] 38.5 ± 0.9 5.84 ± 0.07 0.31 ± 0.05

[0.30, 0.35] 33.5 ± 1.0 4.35 ± 0.06 0.47 ± 0.06

[0.20, 0.30] 25.8 ± 0.8 5.54 ± 0.07 1.30 ± 0.10

[0.0, 0.20] 15.7 ± 1.2 2.00 ± 0.04 0.94 ± 0.07

TOT - 31.33 ± 0.16 3.18 ± 0.15

Table 5.12: SS+OS tagging performances of categories of sub-sample with both taggers for the B0 →

D−π+ 2012 data sample

The results for the p0 and p1 are compatible with 〈η〉 and 1 respectively within the

statistical error.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.193 / 4

p0 0.003712± 0.3705 p1 0.03809± 1.017

> η< 0± 0.3664

/ ndf 2χ 1.193 / 4p0 0.003712± 0.3705 p1 0.03809± 1.017

> η< 0± 0.3664



η0 0.1 0.2 0.3 0.4 0.5

a.u.

0

0.01

0.02

0.03

0.04

0.05

0.06


eta_histo

(b) η distribution

Figure 5.7: On the left the SS + OS calibration for the B0 → D−π+ 2012 data sample is reported

and on the right the predicted mistag distribution is shown. The magenta area shows the

confidence range within ±1σ.

The final performances obtained in this sample after the combination are reported in

Table 5.13. The final combination has been implemented also using the SSπ and the SSp

taggers currently available, whose performances are reported in Table 5.14. The combina-

tion realized by means the SS taggers developed in this thesis with the OS tagger provides

an improvement in tagging performances of about 7% respect to those obtained with the

current taggers.

86



SS 46.83± 0.15 1.19± 0.08

OS 7.53± 0.08 0.75± 0.05

SS + OS 29.67± 0.14 3.15± 0.12

TOT 84.03± 0.11 5.09± 0.15

Table 5.13: Tagging performances of the OS combination for the B0 −→ D−(→ Kππ)π+ 2012 data

sample


79.63± 0.12 4.75± 0.13

Table 5.14: Performances obtained using the current SS + OS tagger on the B0 → D−π+ 2012 data

sample.

5.3.2 Combination on the B0 −→ D−π+2011 data sample

A validation of the performances is performed on the B0 −→ D−(→ Kππ)π+ 2011 data

sample.

The performances found using only the OS tagger and its calibration parameter are

reported in Table 5.15. Also on this case the efficiencies are calculated using the calibrated

per-event mistag.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.373± 0.003 0.93± 0.03 0.365 37.14± 0.13 3.53± 0.14

Table 5.15: OS calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+

2011 data sample

p0 p1 〈η〉

0.367± 0.003 1.04± 0.03 0.360

Table 5.16: SS+OS calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample

In the third sub-sample the calibration plot is reported in Figure 5.8, while in Table 5.16

87


the fit parameters are listed. In this case both p0 and p1 are compatible respectively with 〈η〉

and 1 by about 2σ. The efficiencies obtained on this sample are shown in Table 5.17.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.942 / 4

p0 0.003391± 0.3671 p1 0.03414± 1.039

> η< 0± 0.3597

/ ndf 2χ 5.942 / 4p0 0.003391± 0.3671 p1 0.03414± 1.039

> η< 0± 0.3597

Figure 5.8: Calibration for the B0 → D−π+ 2011 data sample, plot of ω vs η. The magenta area



SS 48.16± 0.14 0.91± 0.08

OS 7.61± 0.07 0.11± 0.05

SS + OS 29.53± 0.13 3.28± 0.14

TOT 85.30± 0.12 5.37± 0.17

Table 5.17: Tagging performances of the SS+OS combination for the B0 −→ D−(→ Kππ)π+ 2011

data sample


SS 47.69± 0.10 1.25± 0.05

OS 7.57± 0.05 0.79± 0.04

SS + OS 28.66± 0.09 3.23± 0.10

TOT 83.92± 0.07 5.27± 0.12

Table 5.18: Tagging performances of the SS+OS combination for the B0 −→ D−(→ Kππ)π+

2011+2012 data sample.

88


The performances obtained merging the 2012 and 2011 data samples in a only one sam-

ple, in order to decrease the statistical uncertainties, are reported in Table 5.18. The effective

tagging efficiency is compatible with the one found in 2012 and 2011 data samples within

the statistical error.

5.3.3 Combination on the B0 → K+π− 2012 data sample

The last validation for the final SS + OS tagger is performed using the B0 −→ K+π− 2012

sample. The performances found using only the OS tagger and its calibration parameter are


p0 p1 〈η〉 εtag [%] εe f f [%]

0.386± 0.003 1.10± 0.04 0.376 34.67± 0.16 3.61± 0.27

Table 5.19: OS calibration parameters and tagging performances for the B0 −→ K+π− 2012 data

sample

The sub-sample containing the events tagged by both OS and SS tagger is splitted in

categories and the calibration results are reported in Table 5.20. The linear fit is shown in

Figure 5.9.

p0 p1 〈η〉

0.375± 0.006 1.12± 0.07 0.366

Table 5.20: SS+OS calibration parameters for the B0 −→ K+π− 2012 data sample

In this case both p0 and p1 are compatible within the statistical error to 〈η〉 and 1, re-

spectively.


SS 49.67± 0.19 0.97± 0.09

OS 6.73± 0.09 0.87± 0.07

SS + OS 23.67± 0.16 3.05± 0.17

TOT 80.09± 0.26 4.89± 0.20

Table 5.21: Tagging performances of the SS+OS combination on the B0 −→ K+π− 2012 data sample.

89


η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5p0 0.005582± 0.3745

p1 0.05092± 1.119

> η< 0± 0.3655

p0 0.005582± 0.3745

p1 0.05092± 1.119

> η< 0± 0.3655

Figure 5.9: SS+OS calibration for the B0 −→ K+π− 2012 data sample, plot of ω vs η. The magenta


The final performances obtained in this sample are reported in Table 5.21. Also for the

final SS +OS tagger the effective tagging efficiency is found lower than the one obtained in

the first decay channel. This loss is accounted only to the SS tagger as the tagging power of

the OS tagger, reported in Table 5.19, is compatible within the statistical error to the values

achieved in the B0 → D−π+ decay channel.

5.4 Measurement of ∆md

The first observation of the B0 ↔ B0 mixing and the measurement of its strength were per-

formed in 1987 [24]. The oscillation frequency in the B0 − B0 system (∆md) is given by the

mass difference between the heavy and light mass eigenstates. The world average measure-

ment is ∆md = 0.510± 0.003 ps−1 [25].

In this analysis the results achieved with the final tagger combination are used to esti-

mate the value of ∆md on the 2011 data sample corresponding to the B0 → D−π+ decay

mode, using the event selection which involves a low background contribution. This sample

is chosen because it contains independent events from the ones used to tune the SS taggers.

To estimate ∆md the sample is divided in BDT categories and then an unbinned simultane-

ous fit on the asymmetry oscillations is performed on each category. In the previous fits the

∆md parameter was fixed to the world average value while here it is floated.

The asymmetry plots are shown in Figure 5.10 and in Table 5.22 the ∆md and the mistag

values are reported.

90


tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

cat5

(a) 0.45 < η < 0.50

tau (ps)2 4 6 8 10 12 14 16 18

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ym

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mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

cat4

(b) 0.40 < η < 0.45

tau (ps)2 4 6 8 10 12 14 16 18

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ym

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mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

cat3

(c) 0.35 < η < 0.40

tau (ps)2 4 6 8 10 12 14 16 18

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ym

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mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

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cat2

(d) 0.30 < η < 0.35

tau (ps)2 4 6 8 10 12 14 16 18

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ym

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try

in

mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

cat1

(e) 0.30 < η < 0.20

tau (ps)2 4 6 8 10 12 14 16 18

As

ym

me

try

in

mix

Sta

te

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

cat0

(f) 0.20 < η < 0.00

Figure 5.10: Mixing asymmetry plots for signal events to estimate ∆md using the B0 → D−π+ 2011

data sample.

The result of the fit found is ∆md = 0.511± 0.006 , which is in good agreement with

world average measurement.

This ∆md value is compared in Table 5.23 to the one estimated using the SS + OS tag-

ger combination implemented with the taggers currently available, by means a unbinned

simultaneous fit on the asymmetry oscillations. The SS + OS algorithm developed in this

thesis reduces the statistical uncertainty of the ∆md value, evaluated in this channel, of the

30%.

91


Parameter ω using ∆md floated ω using ∆md fixed

ω[0.45,0.50] 46.9 ± 0.3 47.5 ± 0.3

ω[0.40,0.45] 43.1 ± 0.4 43.9 ± 0.5

ω[0.35,0.40] 36.7 ± 0.6 37.3 ± 0.6

ω[0.30,0.35] 31.9 ± 0.7 32.7 ± 0.8

ω[0.20,0.30] 25.4 ± 0.6 25.0 ± 0.7

ω[0.0,0.20] 16.8 ± 1.0 17.0 ± 1.1

∆md 0.511± 0.006 0.510

Table 5.22: Parameter values found from the simultaneous fit to estimate ∆md. The values are com-

pared to the one obtained dividing the sample in the same categories and fixing ∆md to

the world average measurement.

∆md world ∆md current ∆md new

0.510± 0.003 0.508± 0.008 0.511± 0.006

Table 5.23: Comparison of the mixing frequency ∆md results: the first one is the world average value,

the second value is the one provided with the current taggers and the last one is the value

estimated using the taggers developed in this thesis.

92

6Systematics

Contents

6.1 Systematic uncertainties 93

6.2 Dependence of the SS tagging on pT of the signal B 94

6.3 Dependence of the SS tagging on the magnet polarity 98

6.4 Dependence of the SS tagging performances on the B flavour 99

6.4.1 Dependence on the B flavour at decay 99

6.4.2 Dependence on the B flavour at production 101

In order to not compromise the statistical precision obtainable on any CPV measurement

at LHCb, a good control of the systematic uncertainties on the tagging parameters must be

achieved. In this chapter some possible sources of systematic uncertainty are discussed.

6.1 Systematic uncertainties

The time-dependent CP violation asymmetries measurements require the reconstruction of

the final state and frequently the determination of the initial state flavour. Same charge and

flavour dependent effects may exist, which can indeed bias the measurement. The most

important ones are:

• Production asymmetries: as the LHC collider is a proton-proton machine, the initial frac-

tion of b and b hadrons is not expected to be the same. This asymmetry at b production

is a function of rapidity and pt, reaching values of few percent, as reported in [26] and

[27].

• Charge dependence: because of the different particle/antiparticle interaction with mat-

ter, the calibration of the mistag probability of different B flavour might be different.

93

6 - Systematics

The data samples collected at LHCb can be splitted according to the magnet polar-

ity in order to check the existence of asymmetries of the detector efficiency or of the

alignment accuracy; indeed these systematics could introduce important differences

in the tagging performance.

• Asymmetries in tagging efficiency: the flavour tagging algorithm developed in this thesis

rely on measuring the charge of the selected tracks. If the particle reconstruction effi-

ciency has a charge dependence, it will result in a difference in the tagging efficiency

for b and b hadrons. These systematic errors can be measured by splitting the sample

into two subsamples according to the signal flavour, determined by the reconstructed

final state, or the tagging decision.

Additional systematic sources are:

• the number of reconstructed primary vertices (PV);

• the track multiplicity;

• sWeights applied to unfold signal from background;

but their expected effect is negligible respect to the previous ones. Indeed the channels

used into the validations, described in the previous chapters, have event properties differ-

ent from the tuning sample. Thus the number of reconstructed PV and the track multiplicity

should not compromise neither the tagging power nor the caibration as the validation re-

sults are compatible within the statistical error to the one found in the tuning sample. For

the same reason also the sWeights should not change significantly the performance because

of different S/B ratio of the data samples used in the analysis.

In the following sections the main systematic effects related to the mistag are analyzed

both for the SS pion and SS proton taggers, using the sample where they were tuned, i.e.

the B0 −→ D−(→ K+π−π−)π+ 2012 data sample.

6.2 Dependence of the SS tagging on pT of the signal B

In previous studies it has been found that the transverse momentum (pT) of the signal B

can have an important influence on the tagging performances [28, 23]. This dependence has

been studied first for the SS pion tagger.

In order to check this dependence the events have been splitted in three sub-sample

according to the pT of the B [0-7.5 GeV/c, 7.5-15 GeV/c and > 15 GeV/c] , as shown in

Figure 6.1.

94

6 - Systematics

htempEntries 116656

Mean 9.795

RMS 5.007

[GeV/c]T

B p0 10 20 30 40 50

even

ts

0

2000

4000

6000

8000

10000

htempEntries 116656

Mean 9.795

RMS 5.007

BPt

Figure 6.1: BpT distribution on the B0 → D−π+ 2012 data sample. The sample has been splitted in

three bins defined by the red dashed line.

To reject correctly the background contribution from the signal events, the sWeights are

evaluated for each sub-sample by means a fit on B mass distribution using the sPlot tech-

nique [29] because the S/B ratio depends on the pT of the reconstructed B0 signal. Then the

sub-samples have been divided further in BDT categories and for each one a value for the

mistag ω is extrapolated using a time-dependent asymmetry fit on the flavour oscillations.

As last step a plot of ω against η, the predicted mistag calculated through the 3rd polyno-

mial parameters found in Section 6.4.2, is performed to guarantee the independence of the

calibration from B transverse momentum.

The calibration plots obtained for the three categories are shown in Figure 6.2 and in

Table 6.1 the comparison between the fit parameters found in the sub-samples to the ones

obtained in Section 6.4.2 is reported.

Sample p0 p1 〈η〉

BpT ≤ 7.5 GeV/c 0.454± 0.004 1.02± 0.21 0.460

7.5 < BpT ≤ 15 GeV/c 0.441± 0.005 0.70± 0.20 0.439

BpT > 15 GeV/c 0.415± 0.006 1.07± 0.09 0.412

total 0.441± 0.003 0.98± 0.05 0.444

Table 6.1: Calibration parameters obtained for SS pion on the B0 → D−π+ 2012 data sample splitted

in BpT bins.

95

6 - Systematics

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 3.648 / 3

p0 0.004305± 0.4539 p1 0.2056± 1.021

> η< 0± 0.4603

/ ndf 2χ 3.648 / 3p0 0.004305± 0.4539 p1 0.2056± 1.021

> η< 0± 0.4603

(a) BpT ≤ 7.5 GeV/c

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 3.272 / 3

p0 0.003449± 0.4341 p1 0.07319± 0.9958

> η< 0± 0.4389

/ ndf 2χ 3.272 / 3p0 0.003449± 0.4341 p1 0.07319± 0.9958

> η< 0± 0.4389

(b) 7.5 < BpT ≤ 15 GeV/c

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 6.513 / 3

p0 0.006385± 0.4147 p1 0.09741± 1.089

> η< 0± 0.4118

/ ndf 2χ 6.513 / 3p0 0.006385± 0.4147 p1 0.09741± 1.089

> η< 0± 0.4118

(c) BpT > 15 GeV/c

Figure 6.2: Calibration plots for SS pion on the B0 → D−π+ 2012 data sample. The sample has been

splitted in three BpT bins. The magenta area shows the confidence range within ±1σ.

The results show a dependence of 〈η〉 on the BpT , in particular to higher values of the

B traverse momentum correspond lower values of mistag. This dependence could be en-

larged by the fact that the BpT is used as input variable to train the BDT. However in each

sub-sample the predicted mistag remains well calibrated, indeed the values of p0 and p1 are

compatible to the expected ones within the statistical error. Because of the small number of

events available in the sub-samples the parameter errors are bigger than the ones found in

the overall sample.

The same procedure is followed to check also the SS proton dependence from the B

pT. The fit parameters achieved are listed in Table 6.2 and compared to the ones reported

in Section 6.4.2. In Figure 6.3 the calibration plots are shown. Also in this case the same

dependence of the 〈η〉 on the BpT can be observed. Also in this case, even if not directly,

the BpT information is used in the BDT training through the pT of the total system. The

statistical errors are again much bigger than the ones obtained using the complete sample.

The fit parameters of each sub-sample are compatible with the predicted value within less

96

6 - Systematics

than 2 σ, so the expected mistag is still well calibrated.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.2191 / 2

p0 0.005178± 0.468 p1 0.1951± 0.8203

> η< 0± 0.4705

/ ndf 2χ 0.2191 / 2p0 0.005178± 0.468 p1 0.1951± 0.8203

> η< 0± 0.4705

(a) BpT ≤ 7.5 GeV/c

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 4.547 / 2

p0 0.004432± 0.4616 p1 0.1105± 0.8749

> η< 0± 0.4586

/ ndf 2χ 4.547 / 2p0 0.004432± 0.4616 p1 0.1105± 0.8749

> η< 0± 0.4586

(b) 7.5 < BpT ≤ 15 GeV/c

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.9315 / 2

p0 0.009982± 0.4211 p1 0.2049± 0.9706

> η< 0± 0.4386

/ ndf 2χ 0.9315 / 2p0 0.009982± 0.4211 p1 0.2049± 0.9706

> η< 0± 0.4386

(c) BpT > 15 GeV/c

Figure 6.3: Calibration plots for SS proton on the B0 → D−π+ 2012 data sample. The sample has

been splitted in three BpT bins. The magenta area shows the confidence range within

±1σ.

Sample p0 p1 〈η〉 εtag

BpT ≤ 7.5 GeV/c 0.468± 0.005 0.82± 0.20 0.471 46.47± 0.23

7.5 < BpT ≤ 15 GeV/c 0.462± 0.004 0.87± 0.11 0.459 44.15± 0.21

BpT > 15 GeV/c 0.421± 0.010 0.97± 0.20 0.439 29.13± 0.38

total 0.460± 0.003 0.91± 0.08 0.462 39.64± 0.15

Table 6.2: Calibration parameters obtained for SS proton on the B0 → D−π+ 2012 data sample

splitted in BpT bins.

97

6 - Systematics

6.3 Dependence of the SS tagging on the magnet polarity

Particles and anti-particles can interact differently with the detector material, generating a

non negligible effect on tagging, both on mistag and on efficiency. This systematic has been

studied comparing samples collected with opposite polarities of the magnetic field.

The steps followed are the same as the ones described in the previous section. The cal-

ibration plots obtained for the SS pion and SS proton taggers are shown in Figures 6.4 and

6.5 respectively. In Tables 6.3 and 6.4 are reported the comparisons of the fit parameters for

the two sub-samples with the values found in the Sections and .


magnet up 0.447± 0.004 0.93± 0.07 0.443

magnet down 0.435± 0.004 1.06± 0.07 0.443

total 0.441± 0.003 0.98± 0.05 0.444


according to the magnet polarity.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 2.348 / 4

p0 0.003614± 0.447 p1 0.07094± 0.9297

> η< 0± 0.4431

/ ndf 2χ 2.348 / 4p0 0.003614± 0.447 p1 0.07094± 0.9297

> η< 0± 0.4431

(a) magnet up sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.672 / 4

p0 0.003559± 0.4345 p1 0.06795± 1.061

> η< 0± 0.4428

/ ndf 2χ 5.672 / 4p0 0.003559± 0.4345 p1 0.06795± 1.061

> η< 0± 0.4428

(b) magnet down sub-sample


splitted in two sub-samples according to the magnet polarity used to collect the data. The

magenta area shows the confidence range within ±1σ.

For both the taggers the fit parameters obtained in all the sub-samples are compatible

within the statistical error to the expected ones, thus the predicted mistag is well calibrated

on the data collected with both the magnet polarities. In this case no dependence effects of

the 〈η〉 on the cut used to split the sample are observed.

98

6 - Systematics


magnet up 0.458± 0.005 0.99± 0.12 0.461

magnet down 0.460± 0.005 0.87± 0.12 0.461

total 0.460± 0.003 0.91± 0.08 0.462


splitted according to the magnet polarity.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.71 / 3

p0 0.004664± 0.4582 p1 0.1205± 0.986

> η< 0± 0.4614

/ ndf 2χ 0.71 / 3p0 0.004664± 0.4582 p1 0.1205± 0.986

> η< 0± 0.4614

(a) magnet up sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 4.572 / 3

p0 0.004587± 0.4596 p1 0.1175± 0.8684

> η< 0± 0.4614

/ ndf 2χ 4.572 / 3p0 0.004587± 0.4596 p1 0.1175± 0.8684

> η< 0± 0.4614

(b) magnet down sub-sample


been splitted in two sub-samples according to the magnet polarity used to collect the

data. The magenta area shows the confidence range within ±1σ.

6.4 Dependence of the SS tagging performances on the B flavour

In this section the flavour dependences in the tagging performance have been studied. The

sample has been divided in two sub-samples according to two variables:

• the flavour of the B at decay 6.4.1;

• the flavour of the B at production as provided by the tagger 6.4.2

In both the cases the procedure used to check these systematics is the same followed in the

previous sections.

6.4.1 Dependence on the B flavour at decay

In this systematic check the sample has been divided according to the electric charge of the

particles created in the final state:

99

6 - Systematics

• D−π+ −→ B0

• D+π− −→ B0

In the Figure 6.6 the calibration plot for the two independent sub-sample are shown for

the SS pion tagger and in Table 6.5 the fit parameters extrapolated are compared to the one

obtained in Chapter


B0 0.441± 0.004 0.94± 0.07 0.443

B0 0.438± 0.004 1.05± 0.07 0.443

total 0.441± 0.003 0.98± 0.05 0.444


according to the flavour of the B at decay defined by the particle charge of the final state.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 3.533 / 4

p0 0.003579± 0.4405 p1 0.07024± 0.9378

> η< 0± 0.4431

/ ndf 2χ 3.533 / 4p0 0.003579± 0.4405 p1 0.07024± 0.9378

> η< 0± 0.4431

(a) B0 sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 0.6063 / 4

p0 0.003619± 0.4382 p1 0.06998± 1.046

> η< 0± 0.4432

/ ndf 2χ 0.6063 / 4p0 0.003619± 0.4382 p1 0.06998± 1.046

> η< 0± 0.4432

(b) B0 sub-sample


splitted in two sub-samples according to the B flavour at decay identified by the electric

charge of the particles in the final state. The magenta area shows the confidence range

within ±1σ.

Looking at the results reported in Table 6.5 no difference on the calibration performances

is found in the two sub-samples. The fit parameters are the corrected ones within the sta-

tistical errors, thus there are no significant dependences of the tagging calibration on the B

flavor, identified from the charged particles observed in the final state.

The results obtained for the SS proton tagger are shown in Figure 6.7 and in Table 6.6

where they are compared to the ones obtained in Section

100

6 - Systematics


B0 0.471± 0.005 0.86± 0.12 0.461

B0 0.449± 0.005 1.01± 0.12 0.462

total 0.460± 0.003 0.91± 0.08 0.462


splitted according to Bid defined by the particle charge of the final state.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 4.323 / 3

p0 0.004607± 0.4706 p1 0.1194± 0.8572

> η< 0± 0.4613

/ ndf 2χ 4.323 / 3p0 0.004607± 0.4706 p1 0.1194± 0.8572

> η< 0± 0.4613

(a) B0 sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.442 / 3

p0 0.004639± 0.4485 p1 0.1192± 1.009

> η< 0± 0.4617

/ ndf 2χ 1.442 / 3p0 0.004639± 0.4485 p1 0.1192± 1.009

> η< 0± 0.4617

(b) B0 sub-sample


been splitted in two sub-samples according to the B flavour identified by the electric

charge of the particles in the final state. The magenta area shows the confidence range

within ±1σ.

There is a small dependence of the calibration on the B flavour at decay shown by the

difference of more than 2σ between the p0 parameters.

6.4.2 Dependence on the B flavour at production

In this second check the sample has been divided according to the electric charge of the

particles chosen as tagger by the BDT. For the SS pion tagger the charge correlation follows

this combination:

• π+ −→ B0

• π− −→ B0

In the Figure 6.8 the calibration plot for the two independent sub-sample are shown

while in Table 6.7 is shown the comparison of the fit parameters to the previous ones, re-

101

6 - Systematics

ported in Section .


B0 0.432± 0.004 1.08± 0.07 0.440

B0 0.447± 0.004 0.94± 0.07 0.440

total 0.441± 0.003 0.98± 0.05 0.444


according to the flavour of the B at production defined by the electric charge of tagger

candidate.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5

/ ndf 2χ 1.069 / 3p0 0.003591± 0.4323 p1 0.07193± 1.076

> η< 0± 0.4404

/ ndf 2χ 1.069 / 3p0 0.003591± 0.4323 p1 0.07193± 1.076

> η< 0± 0.4404

(a) B0 sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.559 / 3

p0 0.003608± 0.4466 p1 0.07232± 0.9401

> η< 0± 0.4401

/ ndf 2χ 1.559 / 3p0 0.003608± 0.4466 p1 0.07232± 0.9401

> η< 0± 0.4401

(b) B0 sub-sample


splitted in two sub-samples according to the B flavour at production identified by the

electric charge of the particle chosen as tagger. The magenta area shows the confidence

range within ±1σ.

Instead the charge correlation for the SS proton tagger is:

• p −→ B0

• p+ −→ B0

The results obtained for the SS proton tagger are reported in Table 6.8 where they are com-

pared to the ones obtained in Section . In Figure 6.9 the two calibration plots are shown.

102

6 - Systematics


B0 0.468± 0.005 0.87± 0.12 0.461

B0 0.456± 0.005 1.18± 0.13 0.462

total 0.460± 0.003 0.91± 0.08 0.462


splitted according to the flavour of the B at production defined by the electric charge of

tagger candidate.

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.592 / 3

p0 0.004601± 0.4676 p1 0.1248± 0.8745

> η< 0± 0.4613

/ ndf 2χ 5.592 / 3p0 0.004601± 0.4676 p1 0.1248± 0.8745

> η< 0± 0.4613

(a) B0 sub-sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 2.382 / 3

p0 0.004646± 0.4563 p1 0.1277± 1.182

> η< 0± 0.4621

/ ndf 2χ 2.382 / 3p0 0.004646± 0.4563 p1 0.1277± 1.182

> η< 0± 0.4621

(b) B0 sub-sample


been splitted in two sub-samples according to the flavour of the B at production iden-

tified by the electric charge of the track chosen as tagger. The magenta area shows the

confidence range within ±1σ.

Also in this systematic control the predicted mistag using both the SS pion and the SS

proton result well calibrated to the true mistag found in each sub-sample using the fit on

the time flavour oscillations. In all the sub-samples the parameters p0 and p1 are compatible

to 〈η〉 and 1 respectively for less than 2σ.

From the results achieved for each systematic check studied in this chapter, the fit pa-

rameters p0 and p1 are well calibrated within the statistical uncertainty. Thus the systematic

error can be neglected respect to the statistical one.

103

7

Conclusion

In this thesis the optimization and the calibration of the Same Side Pion (SSπ) and the

Same Side Proton (SSp) taggers have been presented. These taggers use the pion or the

proton produced in the hadronization of the b quark to the signal B meson to tag its initial

flavour. A multivariate classifier based on “Boost Decision Tree” (BDT) method [20] has

been exploited to select the best tagger candidate, among the track in the event, and to

estimate the probability of the tagging decision to be correct.

All the analyses are developed after the unfolding of the background contribution by

means the sPlot technique [29], using the B invariant mass as “discriminant variable”. The

BDT-based algorithms are trained on the B0 −→ D−(→ K+π−π−)π+ data sample col-

lected by LHCb experiment during the 2012. The training exploits both kinematic and ge-

ometric variables related to the track or to the event to discriminate the right charge corre-

lated particles from the wrong charge correlated ones. The BDT output is used to evaluate

the per-event mistag probability after a calibration procedure.

The new tuning for the SSπ, applied on the B0 −→ D−(→ K+π−π−)π+ 2012 data

sample, provides a effective tagging efficiency of εe f f = 1.64 ± 0.07% which is ∼ 20%

larger than the current result obtained from a similar tagger [18, 23]. The tuning for the SSp

provides an tagging power of εe f f = 0.47± 0.04%, which is compatible within the statistical

errors to the current performance [18].

These results are checked on various independent data samples, for each case the per-

formances and the calibration are consistent within the statistical errors with the ones ob-

tained in the tuning sample. In particular for the SSπ, the tagging power obtained in the

the B0 −→ D−(→ K+π−π−)π+ 2011 data sample is εe f f = 1.76 ± 0.07% and in the

B0 −→ K+π− 2012 data sample is εe f f = 1.08± 0.08%. For the SSp the two tagging powers

obtained in the B0 −→ D−(→ K+π−π−)π+ 2011 data sample and in B0 −→ K+π− 2012

104

7 - Conclusion

data sample are εe f f = 0.48± 0.04% and εe f f = 0.43± 0.05%, respectively. The differences

in the tagging power achieved in the B0 −→ K+π− decay mode both for the SSπ and for

SSp, respect to the B0 −→ D−π+, can be accounted to the different kinematic properties of

the signal B in the two channels.

Some systematics have also been studied in order to observe possible dependences of

the tagging performances on the event properties. The source of possible systematic errors

analyzed in this thesis are the BpT dependence, the dependence on the magnet polarity and

the dependence on the flavour at decay of the B signal meson, identified by the charge of

the particles produced in the final state or the flavour at production identified by the charge

of the tagger candidate (tagging decision). The mistag calibration obtained in the different

cases remain consistent within the statistical errors. A dependence of 〈η〉 on the BpT has

been found, as reported also in some previous analyses.

The combination of the tagger responses entails an improvement on the tagging per-

formances [23]. For this reason the last part of this thesis is focused on the combination

of the SSπ and the SSp taggers in a unique Same Side Tagger (SS) and then on the fi-

nal combination of the SS tagger with the OS general tagger, already implemented. The

SS tagger provides an effective tagging efficiency of εe f f = 1.97 ± 0.10% on the B0 −→

D−(→ K+π−π−)π+ 2012 data sample. This value is consistent to the one found in the

B0 −→ D−(→ K+π−π−)π+ 2011 sample (εe f f = 2.05± 0.07%) and in the B0 −→ K+π−

2012 data sample (εe f f = 1.44 ± 0.11%). The results provides by the final combination

SS + OS are: εe f f = 5.09± 0.15% for the B0 −→ D−(→ K+π−π−)π+ 2012 data sample,

εe f f = 5.27± 0.12% for the validation on the B0 −→ D−(→ K+π−π−)π+ 2011 sample and

εe f f = 4.89± 0.20% for the cross-check on the B0 −→ K+π− 2012 data sample.

This final tagger combination has been used to evaluate the oscillation frequency in

the B0 − B0 system (∆md), given by the mass difference between the heavy and light mass

eigenstates. The value obtained in the B0 −→ D−π+ data sample (1 fb−1) is ∆md = 0.511±

0.006 , which is in good agreement with world average measurement (i.e. ∆md = 0.510±

0.003 ps−1). The statistical error related to this value is reduced of 30% respect to the one

evaluated with the current taggers.

The improvements introduced by these optimized SS algorithms and by the two tagger

combinations will contribute to increase the precision of the measurement based on the

flavour identification in the B0d system, such as the evaluation of sin 2β [24].

105

A

sPlots technique

This technique analyzes the events of a sample assuming that they are characterized by two

set of variables:

• the first one is a set of variables for which the distributions of all the sources of events

are known (“discriminating variables”)

• the second one is a set of variables for which the distributions of some sources of

events are either truly unknown (“control variables”)

The sPlot technique allows to reconstruct the distributions for the control variables with-

out making use of any a priori knowledge on this variable [29]. An essential assumption

is that the control variables are uncorrelated with the discriminating variables. This tech-

nique exploits the maximum Likelihood method using the discriminating variables. The

log-Likelihood used is:

L =N

∑e=1

ln{ Ns

∑i=1

Ni fi(ye)}−

Ns

∑i=1

Ni (A.1)

where

• N is the total number of events considered

• Ns is the number of species of events populating the data sample (signal and back-

ground)

• Ni is the number of events expected on the average for the ith species

• y represents the set of “discriminating variables”, which can be correlated with each

other

• fi(ye) is the value of the pdf of y for the ith species and for the event e

106

A - sPlots technique

The set of “control variables” x doesn’t explicitly appear into the log-Likelihood. The

aim of the sPlot technique is to unfold the true distribution (Mn(x)) of x for events with

nth species, from the knowledge of the pdfs ( fi) of y. If x and y are two set of uncorrelated

variables, the total pdf fi(x, y) can be factorized into products Mn(x) fi(y). An estimate (M̃n)

of x for the nth species can be built as:⟨Nm M̃n(x)

⟩=∫

dydxNs

∑j=1

Nj Mj(x) f j(y)δ(x− x)Pn

= Nm

Ns

∑j=1

Mj(x)Nj

∫dy

fn(y) f j(y)

∑Nsk=1 Nk fk(y)

(A.2)

where Pn1 is the naive weight defined for each event. Using the inverse of the covariance

matrix V−1nj defined as:

V−1nj =

∂2(−L)∂Nn∂Nj

=N

∑e=1

fn(ye) f j(ye)

(∑Nsk=1 Nk fk(ye))2

(A.3)

Introducing the average of covariance matrix:

〈V−1nj 〉 =

∫dy

fn(y) f j(y)

∑Nsk=1 Nk fk(y)

(A.4)

the equation A.2 can be rewritten as:⟨M̃n(x)

⟩=

Ns

∑j=1

Mj(x)Nj〈V−1nj 〉 (A.5)

Inverting this matrix equation, the distribution Mn(x) can be obtained:

Nm Mn(x) =Ns

∑j=1〈Vnj〉

⟨M̃j(x)

⟩(A.6)

Thus the appropriate weight is the covariance-weighted quantity defined by:

sPn(ye) =∑Ns

i=1 Vnj f j(ye)

∑Nsk=1 Nk fk(ye)

(A.7)

Using this sWeight the distribution of x can be obtained from the sPlot histogram:

Nn s M̃n(x)δx ≡ ∑e⊂δx

sPn(ye), (A.8)

which reproduces, on average, the true binned distribution:⟨Nm s M̃n(x)

⟩= Nm Mn(x) (A.9)

Because the covariance matrix enters explicitly in the definition of sWeight, these values can

be positive or negative, and the estimators of the true pdfs are not constrained to be strictly

positive.

1Pn = Pn(ye) =Nn fn(ye)

∑Nsk=1 Nk fk(ye)

is the correct weight if x is totally correlated with y.

107


A.1 sPlot properties

The distribution s M̃n is guaranteed to be normalized to unity and the sum over the species

reproduces the data sample distribution of x. Thus the sPlots technique satisfies other two

properties:

• Each x-distribution is properly normalized. The sum over the x-bins of Nn s M̃δx is

equal to Nn:N

∑e=1

=s Pn(ye) = Nn (A.10)

• In each bin the sum over all species of the expected numbers of events equals to the

number of events actually observed. For any event:

Ns

∑j=1

sPj(ye) = 1 (A.11)

For this reasons, the sPlot return a correct representation of the distribution of each

control variable in the set x.

A.2 sPlot application

The sPlot technique prove itself to be a very usefull tool to unfold the background and

signal contributions. Indeed to eliminate the background is sufficient weighting each event

for the sWeight evaluated by means a fit on the “discriminating variables”. In this work

thesis the variable chosen as the discriminant one is the invariant mass of the B meson.

A simple application of the sPlot technique can be exploited to separate the distribution

of the event and track variables. For some variables the comparisons of the normalized

distributions of signal and background are shown in Figures A.1.

In the case of the decay time τ the signal distribution slope is larger than the background

one, the reason can be found in combinatorial nature of the background itself, described

properly by a simple exponential function, while the signal distribution is determined by

the B0d lifetime and it is represented by a convolution between an exponential function and

a resolution model function.

108


time_sigEntries 349966

Mean 2.107

RMS 1.437

(ps)τ1 2 3 4 5 6 7 8 9 10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

time_sigEntries 349966

Mean 2.107

RMS 1.437

signalsignal

background

(a) τ distribution

P_sigEntries 349966

Mean 136

RMS 79.95

p (MeV/c)0 100 200 300 400 500 600

0

0.01

0.02

0.03

0.04

0.05

P_sigEntries 349966

Mean 136

RMS 79.95

signal

signalbackground

(b) p distribution

Pt_sigEntries 349966Mean 9.788RMS 4.905

(MeV/c)T

p0 5 10 15 20 25 30 35 40 45

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Pt_sigEntries 349966Mean 9.788RMS 4.905

signal

signal

background

(c) pT distribution

Figure A.1: The normalized distributions for the decay time (τ), p, pT and ∆R variables of the signal

(blue) and background (red) components.

109

B

Boost Decision Tree classifier

A decision tree (DT) is a machine-learning technique which creates a powerful multivariate

discriminant combining several weak classifiers. It is created through a binary segmentation

procedure with the task to classify the events in “signal” or “background” using the infor-

mations provide from some input variables. The events are analyzed at the nodes, where a

cut on one of the input variable is applied. According to this cut the events are splitted in

two daughter node as shown in Figure B.1. Consequently, any event that fails to pass a cut

is not thrown away immediately as background, but continue to be analyzed. This proce-

dure is repeated until a stop condition is reached, as a minimal number of events in a node

or a maximal ratio between signal and background events. A node which reached the stop

condition is called “leaf” node and according to its purity value its classified definitively as

signal or background.

At each node an input variable is chosen, then the corresponding cut which maximizes

the separation between signal and background is calculated and executed. In this work the

method used to quantify the separation is called “MisClassificatorError” and it is defined

as:

gain = 1−max(p, 1− p) (B.1)

The purity p is:

p =S

S + B=

∑s ws

∑s ws + ∑b wb(B.2)

where

• wi is the weight assign to each event classified as signal (s) or background (b)

• S (B) is the weighted total number of signal (background) events which landed on the

node

110

B - Boost Decision Tree classifier

RootNode

Xi > c1 Xi < c1

Xj > c2

Xi > c1

Xj < c2 Xj > c3 Xj < c3

B S S

SB

Xk < c4Xk > c4

Figure B.1: Sketch of a decision tree. Starting with a single node (“root”) the decision tree grows

by means a sequence of binary cuts on the discriminant variables. Each node uses the

variable which allows the best separation between signal and background. The terminal

nodes (“leaf”) are identified as “signal” or “background” according to the ratio S/B.

Thus this classifier assume the value 0 if the purity is 1 or 0 (pure signal or pure back-

ground) and is maximized if the purity is 0.5 (maximally mixed sample) The maximum

separation is defined as the maximum change in the gain between the mother node and the

two daughters node:

∆ gain = gainM − fL · gainL − fR · gainR (B.3)

where

• L = left daughter node, R = right daughter node and M = mother node

• fL(R) = is the weighted fraction of events in the daughter node.

The best cut corresponding to the maximum of ∆ gain is calculated and executed for

both the left and right nodes.

In a first step, called “training”, a set of known signal and background events, each with

a weight wi, is used to build a tree structure of cuts node by node. In a second step, called

“test”, the tree structure is used to infer and to separate the signal from the background

events from an unknown set of events. A test event, through the cut conditions, follows the

path along the tree according to pass (right daughter node) or to fail (left daughter node)

111

B - Boost Decision Tree classifier

until it lands on a leaf. The classifier value, i.e. the decision tree result, D(i) for a test event

is equal to the purity of the leaf which it reached.

B.1 Boosting method

The Boosting is a technique to enhance and to increase the stability with respect the sta-

tistical fluctuations in the classification and regression performance. The improvement is

achieved creating several decision trees (100-1000 trees) and combining their results to pro-

vide a final classifier value. A decision tree classifier to which is applied a boost method is

called Boost Decision Tree (BDT).

In this work an adaptive boost, called “AdaBoost”, is chosen. The basic idea of the Ad-

aBoost is to attribute a higher weight wi to the misclassified events during the training of

the following tree. According to the following definitions for the ith event:

• yi the true nature of the event: +1→ signal and −1→ background

• cmi (xi) the response of the mth tree: +1→ signal and −1→ background

where x is the tuple of the input variables, the ith event is misclassified if yi 6= cmi (xi)

Thus the first training starts with the original event weights1 while the subsequent trees

are trained using a modified sample where the weights of previously misclassified events

are multiplied by a common boost weight αm, defined as:

αm =1− errm−1

errm−1(B.4)

where errm−1 is the misclassification rate of the previous tree and is calculated like:

errm = ∑yi 6=cm

i (xi)

wi (B.5)

The weights of the new sample are renormalized such that the sum of weights remains

constant.

The final classifier value is calculated as:

Ci(xi) =1

Ntrees·

Ntrees

∑m=1

ln αm · cmi (xi) (B.6)

A small value for Ci(xi) is a sign of background-like event, while large value indicates a

signal-like event.

1In general the weights are initialized to 1

112

C

Monte-Carlo analyses

Before to choose definitively to use the 2012 data sample to tune the BDT instead of the

Monte-Carlo (MC) sample to develop the SS pion, some studies were performed exploiting

the MC-truth about the event properties. In particular the aim of these studies was to find

a way to purify the most as possible the sample removing all the events which produced

pions both positive and negative. From these events indeed none usefull contribution could

come from because the charge correlation between the pion and the flavour of the signal B

is lost. The idea was to remove these pions from the training to improve the BDT discrim-

ination using the MC-truth about the “mother ID”. In the Table C.1 and C.2 the origins of

the tracks identified as right charged correlated pions and as wrong charge correlated pion

respectively are shown.

ID Ratio [%] Particle Dominant decay

0 21.2 PV -

113 16.2 ρ0 π+π−

213 15.1 ρ+/− π+/−π0

223 13.3 ω π+π−π0

310 4.9 Kshort π+/−

221 3.3 η π+π−

313 2.9 K∗0 K+/−π+/−

323 2.5 K∗+/− K0π+/−

331 1.1 η′ π+π−η

Table C.1: The mother ID provides an usefull information on the true origin of these tracks. In the

table also the fraction and the dominant decay are reported. The origin with less than 1%

of the pions are not listed

113

C - Monte-Carlo analyses

ID Ratio [%] Particle Dominant decay

0 20.6 PV -

113 16.5 ρ0 π+π−

213 15.2 ρ+/− π+/−π0

223 13.6 ω π+π−π0

310 5.0 Kshort π+π−

221 3.3 η π+π−

313 2.9 K∗0 K+/−π+/−

323 2.6 K∗+/− K0π+/−

331 1.1 η′ π+π−η

Table C.2: The mother ID provides an usefull information on the true origin of these tracks. In the

table also the fraction and the dominant decay are reported. The origin with less than 1%

of the pions are not listed

As shown in the Tables the neutral resonances are ρ0, ω, Kshort, η and η′ and in their

dominant decays both positive and negative pions are created; these pions correspond to

about the 40% of total. As explained above for all these pion the charge correlation with the

signal B meson is lost, therefore they are useless in order to improve the BDT separation

between right and wrong correlated pions. Thus a BDT training was performed after had

removed all pions coming from these neutral resonances. However the results provide ap-

plying this tuning on the data sample were found worst than the ones obtained using the

tuning performed directly on the data sample. The results obtained with the two tuning

methods are reported in Table C.3 corresponding to the B0 −→ D−(→ K+π−π−)π+ decay

mode.

tuning on MC [%] tuning on data [%]

εe f f 1.41± 0.08 1.64± 0.07

Table C.3: Effective tagging efficiency found tuning the BDT on a MC sample and on a data sample.

In both the cases the channel analyzed corresponds to the B0 → D−π+ decay mode.

A possible reason of this worsening of the performances could be related to the use of a

“too clean” sample for the tuning: the BDT trained on a sample devoid of the pions coming

from these resonances, is not able to discriminate correctly the right and the wrong pions in

114

C - Monte-Carlo analyses

the data, where these resonances can not be eliminated. Because of this studies have been

proved themselves useless in order to obtain an improvement of the tagging performances,

they have not been execute for the SS proton.

115

D

Validation on a different cuts selection

For all the taggers developed in this thesis, another validation has been perform on the

B0 −→ D−(→ K+π−π−)π+ 2011 data sample but using a different cuts selection. The

cuts applied in order to obtain this selection, reported in Table D.2, are looser than the

ones used in the previous selection, allowing for a large background contribution in the

sample. This check allows to study a possible dependence of the tagging performances on

the background contamination of the sample.

Following the steps reported in section 6.4.2 a fit on the mass distribution is performed

in order to estimate the sWeights for each sample. The parametrization used is the same as

the one reported in the equation 3.10. The B mass distribution is fitted using the pdf reported

in the equation 3.10 in order to calculate the sWeights. To improve the performances of the

fit an additional cut on the B mass is applied: 5200 < mB < 5380 [MeV/c2]. The fit results are

reported in Table D.1 and the plot is shown in Figure D.1.










Table D.1: Results of the fit to the mass distribution 2011 data sample

116

D - Validation on a different cuts selection


cuts for the B0 candidate

PIDK(bachelor2 π) DLLK−π of the π < 0

IPχ2(bachelor π) Impact parameter significance of the π wrt PV > 9

PIDK (K from D) ∆(log LK − log Lπ) of the K from D > 0

D mass Invariant mass of the D 1848 < m < 1890

IPχ2(D) Impact parameter significance of the D wrt PV > 4

IPχ2(B) Impact parameter significance of the B wrt PV < 16

B(pointing) cosine of the angle between B momentum and

its direction

> 0.9999

cuts for the tagging track

IPχ2 Impact parameter significance < 16

pT Tranverse momentum > 400 MeV/c2

χ2track/nd f Quality of track fit < 5

Ghost prob Probability that a track is a random

combination of hits

< 0.5

IPPU Impact parameter with respect to pile up

vertexes

> 9

cuts for the “B + tagging track” system

pT Tranverse momentum > 3000 MeV/c2

cos θ θ is the angle between the B momentum and the

B+track momentum in the B+track rest frame

> -0.5

∆Q m(B + track)−m(B)−m(track) < 2500 MeV/c2

χ2vtx Quality of the vertex fit < 100

Table D.2: Selection cuts for the B0 candidate, tagging track and “B+tagging track” system for the

decay channel B0 −→ D−(Kππ)π+ [18].

The acceptance parameters have been calculated with the same function used in Section

6.4.2. The time fit is shown in Figure D.2 and the parameter values are reported in Table

D.3.

117


)2^+) (GeV/cπm(D^- 5.2 5.22 5.24 5.26 5.28 5.3 5.32 5.34 5.36 5.38

)2E

vent

s / (

0.0

018

GeV

/c

0

2000

4000

6000

8000


5.2 5.22 5.24 5.26 5.28 5.3 5.32 5.34 5.36 5.38

Pul

l

-5

0

5

Figure D.1: Mass fit for the B0 → D−(Kππ)π+ 2011 data sample. The blue curve represents the pdf



are shown.

α β t0 γ

2.00± 0.05 1.00 0.25± 0.1 −0.063± 0.002

Table D.3: Acceptance parameters calculated with the fit on the B decay time for the 2011 data sam-


tau (ps)2 4 6 8 10 12 14

Eve

nts

/ (

0.14

8 p

s )

0

1000

2000

3000

4000

5000

6000

7000

time distribution

2 4 6 8 10 12 14

Pu

ll

-5

0

5

Figure D.2: Time distribution of the events in 2011 data sample

118


D.1 Validation for the SS Pion Tagger

Following the steps reported in section 6.4.2 a calibration is performed using the BDT

trained on the tuning sample collected during the 2012.

The calibration plot is shown in Figure D.3 and the performances are reported in Ta-

ble D.4. In this case p0 and p1 are compatible respectively to 〈η〉 and 1 by about 2σ. The

performances are compatible with the ones reported in the other channels. These results

demonstrate that the BDT output and the following calibration are not dependent on the

background contribution presents in the sample.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.435± 0.002 1.07± 0.04 0.447 70.52± 0.11 1.81± 0.08

Table D.4: Calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+ 2011

data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 4.828 / 6

p0 0.002137± 0.4345 p1 0.0433± 1.073

> η< 0± 0.4467

/ ndf 2χ 4.828 / 6p0 0.002137± 0.4345 p1 0.0433± 1.073

> η< 0± 0.4467

Figure D.3: Calibration for the B0 → D−π+ 2011 data sample, plot of ω vs η. The magenta band


D.2 Validation for the Proton Tagger

The calibration plot for the 2011 data sample is shown in Figure D.4 and the performances

are reported in Table D.5 The tagging power is calculated using a per-event mistag.

In this case p0 is compatible to the expected value within the statistical error, while p1

119


is compatible to 1 within 2.5σ. Also the tagging power is compatible to the values found in

the previous channels within 2σ.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.464± 0.002 1.16± 0.07 0.462 40.20± 0.12 0.58± 0.06

Table D.5: Calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+ 2011

data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 11.5 / 5

p0 0.002772± 0.4641 p1 0.06723± 1.158

> η< 0± 0.4623

/ ndf 2χ 11.5 / 5p0 0.002772± 0.4641 p1 0.06723± 1.158

> η< 0± 0.4623

Figure D.4: Calibration for the B0 → D−π+ 2011 data sample, plot of ω vs η. The magenta area


D.3 Validation for the SS Tagger combination

For this analysis the same steps reported in section 5.2 are followed. The calibration plot

for the third sub-sample is reported in Figure D.5 while in Table D.6 the fit parameters are

listed. In this case p0 is compatible respectively with 〈η〉 by about 3σ while p1 is compati-

ble to 1 within the statistical error. The performances obtained in this sample are reported

in Table D.7, where the efficiencies reported are calculated using a per-event mistag. The

tagging power is compatible with the values calculated in the other samples by about 2σ.

p0 p1 〈η〉

0.437± 0.003 0.96± 0.05 0.426

Table D.6: Calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample

120


η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 1.965 / 2

p0 0.002915± 0.4373 p1 0.04677± 0.9553

> η< 0± 0.4263

/ ndf 2χ 1.965 / 2p0 0.002915± 0.4373 p1 0.04677± 0.9553

> η< 0± 0.4263

Figure D.5: Calibration for the B0 → D−π+ 2011 data sample, plot of ω vs η. The magenta area



SSπ 31.39± 0.11 0.93± 0.07

SSp 8.49± 0.07 0.14± 0.03

SS(π + p) 36.32± 0.11 1.14± 0.08

TOT 76.20± 0.11 2.16± 0.08

Table D.7: Tagging performances of the SS combination for the B0 −→ D−(→ Kππ)π+ 2011 data

sample with the second event selection.

D.4 Validation for the SS+OS Tagger combination

The performances found using only the OS tagger and its calibration parameter are reported

in Table D.8. The sub-sample containing the events tagged by both OS and SS tagger is

splitted in categories, as done in the previous analysis.

p0 p1 〈η〉 εtag [%] εe f f [%]

0.367± 0.003 0.95± 0.03 0.361 37.06± 0.13 3.35± 0.12

Table D.8: OS calibration parameters and tagging performances for the B0 −→ D−(→ Kππ)π+

2011 data sample

121


The calibration plot of this sub-sample and its fit parameters are shown in Figure D.6

and in Table D.9, respectively. In this case p0 is compatible to 〈η〉 within the statistical error

while p1 is compatible to 1 by about 3σ.

p0 p1 〈η〉

0.359± 0.003 1.08± 0.03 0.356

Table D.9: SS+OS calibration parameters for the B0 −→ D−(→ Kππ)π+ 2011 data sample

η0 0.1 0.2 0.3 0.4 0.5

ω

0

0.1

0.2

0.3

0.4

0.5 / ndf 2χ 5.597 / 4

p0 0.003222± 0.3587 p1 0.03048± 1.079

> η< 0± 0.3556

/ ndf 2χ 5.597 / 4p0 0.003222± 0.3587 p1 0.03048± 1.079

> η< 0± 0.3556

Figure D.6: SS+OS calibration for the B0 −→ D−π+ 2011 data sample, plot of ω vs η. The magenta


The final performances obtained in this sample are reported in Table D.10. The tagging

power is compatible with the ones found in the other samples by about 2σ.


SS 46.33± 0.12 1.40± 0.09

OS 8.11± 0.06 0.97± 0.06

SS + OS 27.22± 0.11 3.69± 0.12

TOT 81.66± 0.09 5.46± 0.16

Table D.10: SS+OS tagging performances for the B0 −→ D−(→ Kππ)π+ 2011 data sample with the

second cuts selection

122

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Ringraziamenti

Giunto alla conclusione di questo lavoro, vorrei ringraziare innanzitutto la mia relatrice,

Prof.ssa Marta Calvi, per l’opportunità concessami di lavorare su una tematica interessante

e stimolante, facendomi apprezzare il mondo della ricerca universitaria.

In ugual modo ringrazio anche il mio correlatore, Dott. Basem Khanji, per essersi di-

mostrato sempre disponibile a fornirmi utili consigli e spiegazioni per superare i problemi

che man mano ho incontrato nello sviluppo del presente lavoro.

Vorrei ringraziare anche la la Dott.ssa Stefania Vecchi dell’Università di Ferrara per il

tempestivo supporto fornito nella preparazione dei set di dati necessari alle analisi svolte

in questa tesi.

Ringrazio i miei compagni di corso Fra, Mannu, Fra e tutti gli altri per tutti i consigli

dati in questi anni e sopratutto in questo periodo.

Ringrazio allo stesso modo i miei amici Gian, Riki, Mary, Nicco, Marco, Vale e Carlo per

il loro sostegno e per essere stati sempre presenti. Grazie per tutti i bei momenti passati

assieme.

Ringrazio infine tutta la mia famiglia per essermi sempre stata vicino, per avermi aiutato

in questo periodo universitario ed avermi sempre spinto ad andare avanti, permettendomi

così di arrivare dove sono ora.

Tutti voi, chi più chi meno, avete contribuito a rendermi la persona che sono in questo

momento.

Grazie!

126

Documents

Development of same side ﬂavour tagging algorithms for ...cds.cern.ch/record/2015250/files/CERN-THESIS-2015-040.pdf · di loro, così da ottenere un unico algoritmo di “Same Side