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## Theoretical framework

In this chapter, a very brief introduction to the Standard Model of the particle physics is presented. It starts with the description of the four fundamental interactions and their relative Lagrangian. Then the violation of the CP symmetry is presented in order to introduce its formalism in the Standard Model. This first part of this chapter ends with a short list of the main observed issues of the Standard Model. It allows to introduce the New Physics research. First is described how the Standard Model can be expressed as an eﬀective theory and the way the New Physics eﬀect can be introduced. Then, some of the main laboratories to search for New Physics in rare decays are presented and a particular attention is given to the radiative decays. Observables related to these decays, as the photon polarisation, are discussed. This chapter ends with a status on the latest constraints on some parameters of the Standard Model expressed as an eﬀective theory for which the radiative decays are very sensitive.

Standard Model and CP violation

The Standard Model of particle physics

The Standard Model (SM ) of Particle Physics is the theoretical framework which describes three of the four fundamental forces: the strong interaction and the electroweak interaction (the unification of the electromagnetic and weak forces into a common formalism). The SM is a gauge theory based on the symmetry group: SU(3)c SU(2)L U(1)Y (2.1) where SU(3)c is the gauge group describing the strong interaction (the c subscript corresponds to the color charge) and SU(2)L U(1)Y is the gauge group describing the electroweak force. SU(2) refers to the weak isospin (the L subscript means it involves left handed states only) and the U(1)Y refers to the weak hypercharge.

Electromagnetic interaction

The electromagnetic force is mediated by the photon and its dynamic is described by the following Lagrangian: L = (i @ m) q 1 F F (2.2) where the first part is the free fermion Lagrangian and the middle term describes the interaction between the vector field and the fermion current with a strength measured by q, the electric charge of the fermion. A is the photon field. The last term is the kinetic Lagrangian term for the vector field where F is the electromagnetic field strength tensor and is defined as: F =@A @A (2.3)

Strong interaction

The strong force is mediated by gluons, and is acting on quarks. This force is responsible of the quark confinement into the hadrons. The six quarks are separated into three families in which there is two flavours of quarks usually referred as Up and Down quarks with reference to the first family quarks. There is another quantum number related to the strong interaction: the color charge of the quarks, which can be red, blue or green. The Lagrangian which describes the dynamic of the strong force for a particular flavour of the quark of one family is similar to the Lagrangian of the electromagnetic force: L = [i @ m ] gs( a 1 (2.4) )Ga + ga ga where, as well as for the electromagnetic Lagrangian, the first term is the free quark Lagrangian for a given flavor. The quark field is made of three components, one per color charge: r = @ bA ; = ( r b g) (2.5) g
The second term in Eq. 2.4 describes the interaction of the quark current ja = a with the gluon field Ga , where a is the color index and a are the Gell-Mann matrices. There are 8 Gell-Mann matrices, corresponding to the 8 gluons (a=1,…,8). gs is the strong force coupling constant, representing the strength of the interaction. The last term represents the kinetic term of the gluon field and ga is the strong field strength tensor: g = @ G @ G gsfabcG G (2.6) where fabc is the structure constant of the SU(3)c group and is defined as: 2a ; 2b = ifabc 2c (2.7)
Since the SU(3)c group is non abelian, this last term introduces couplings between three or four gluons.

Electroweak interaction

With the electroweak force, fermions can interact throughout the charged currents j , mediated by W gauge bosons or neutral currents jY , mediated by Z and gauge bosons.
The charged currents of the e interaction with a e can be written as:
j = L eL (2.8)
j+ = eL L (2.9)
Only left handed fermions can interact via these charged currents. It is possible to write the quarks and leptons fields qL and lL as left handed doublets, the quark family structure is the same as for strong interaction with an Up and a Down quark type per doublet: qL = dL ; lL = eL (2.10)
Fermions can also interact through neutral currents which, in the case of left handed leptons, can be written as: j3 = 1 lLlL 1 lL lL (2.11)
Whereas, only left handed fermions interact through charged current, right handed fermions also interact through neutral current. Only massive quarks and leptons have right handed component. Therefore, in addition to the two left handed fermions doublets qL and lL, three singlets are added: uR, dR and eR, corresponding to right handed up, right handed down and right handed electron singlets, respectively. It is necessary to introduce the weak hypercharge Y defined via the Gell-Mann–Nishijima formula which links the electric charge Q of a particle to its weak isospin I3 and Y [1] [2]: Q=I3+ Y (2.12)
Therefore, the weak hypercharge jY is written as: jY = 2jEM 2j3; (2.13) where, jEM is the electromagnetic current which for the leptons is written as: jEM = lL lL lR lR; (2.14) where lR is the right handed leptons field. Then, the weak hypercharge current is: jY = 2lR lR lL lL lL lL (2.15)
The neutral current mediators are a mixture of the neutral W 3 field of the weak SU(2)L and the weak hypercharge field B of the U(1)Y . These physical bosons are:
Z = W3 cos W B sin W (2.16)
A = W3 sin W + B cos W
where, W is the weak mixing angle. A can be identified as the photon while Z is the second neutral boson often referred as the Z. The kinetic term of the Lagrangian for the electroweak field is: Lkinetic = B B where Wi (i = 1; 2; 3) and B are the field strength tensors for the weak isospin and weak hypercharge fields, defined as follow: Wi = @ Wi @ Wi g ijkWj Wk B =@B @ B
where ijk are the structure constants of the SU(2) group, and defined from the Pauli matrices commutation relations: h i = i ijk i ; j k (2.19)

### Electroweak symmetry breaking

At this point, all the masses of the boson fields and fermions vanish since the corresponding Lagrangian terms are not invariant under SU(2) U(1). If the photon is indeed massless, others bosons as well as fermions are massive. Therefore, the SU(2)L U(1)Y symmetry is broken by introducing a complex scalar field with a non zero vacuum expectation ( p ), referred as [3]. The relative Lagrangian is: L = @ y@ V ( ) (2.20) where the potential V ( ) contains either mass and interaction terms: V ( ) = 2 y + ( y )2
As for the left handed fermions, a scalar SU(2) doublet is introduced: (+) = (0)
The choice of the charge is led by the fact that, to be invariant under SU(2)L the hypercharge must be 1. The Lagrangian for these scalar fields is: LH = (D )y(D ) 2 y ( y )2 with > 0 and 2 < 0. D is the covariant derivative: D = [@ + igW + ig0y B ] (2.24) Where g = e and g0 = e are the coupling constants of the electroweak sin W cos W interaction and y is defined as: y = Q T3 = 1 where Q is the electromagnetic charge and T3 is the proper value of the operator 3=2 for the (0). According to the choice of the unitarity gauge, the charged field (+) can be set to 0 and (0) is real. Therefore, the scalar field becomes: vp+2 (2.25) = 0 H H is a real field corresponding to the Higgs boson. Including this expression into the Lagrangian density given in Eq. 2.23, the following couplings to the gauge bosons are obtained: 1 g2 1 (2.26) LH = @H@H+ (v + H)2[W yW + Z Z ] 2 4 2cos2 W
This term introduces masses for the electroweak bosons and these masses are related: MW = 1 vg = MZ cos W (2.27)
It is not allowed to introduce a fermionic mass term such as: Lm = m since it breaks the gauge symmetry. However, it is possible to write a lagrangian corresponding to the coupling of the fermions doublets with the scalar doublet introduced above. The associated coupling constant can be connected to the mass of the fermions. In the unitary gauge, this Yukawa lagrangian becomes:
LY = (1+ H (2.28) v )[mqd qdqd + mqu ququ + mlll]
The fermion masses are free parameters of the theory and have to be measured experimentally.

The CP symmetry violation

Symmetries in physics, which could be discrete or continuous, are seen as the invariance of a system under a certain type of transformation. The invariance under a symmetry can be translated into a conservation law. For instance, invariance under time translation results in the conservation of the energy while rotational invariance results in the conservation of the angular momentum.
In the quantum field theory, three discrete symmetries are introduced, the symmetry of charges (C), of parity (P) and of time (T). The first one is the transformation which relates a particle with its anti-particle. The P symmetry transforms the quadri-momentum of a particle as follows: P :(! ) ( t; ! x ) (2.29)
As a direct consequence of the P symmetry, the momentum vector is also changed: P : p 7! p . However, the angular momentum is conserved: P : L 7!L . Then, the T symmetry is the transformation which reverses the time of a system: T :(! ) ( ! ) (2.30) t; x 7! t; x
These symmetries are conserved by the strong and the electromagnetic interaction. Schwinger (1951) [4], Lüders (1954) [5] and Pauli (1956) [6] have shown that, on the most general assumptions (causality, locality, Lorentz invariance), any quantum field theory is invariant under the combined operations of C, P and T. This is known as the CPT theorem and, as a direct consequence, the mass and the lifetime of a particle are exactly the same as for antiparticle. The most sensitive test of the CPT theorem to date is the K0– K0 masses diﬀerence [7]: jmK 0mK0 j < 4 10 19: (2.31)
The parity violation in the weak interactions has been predicted in 1956 by Lee and Yang [8] and observed experimentally by Wu et al. the same year [9]. The violation of the C symmetry in the weak interactions have been demonstrated by the observation of  large diﬀerences between right handed and left handed neutrinos by Goldhaber, Grodzins and Sunyar in 1958 [10]. The CP symmetry was considered to be non violated until Christenson, Cronin, Fitch and Turlay demonstrated the CP violation in the decays of neutral kaons: KL0 ! + [11].
Considering a generic neutral meson M0 produced in strong interactions. eigenstates: jM i and jM0i are not CP eigenstates: CP jM i = jM0i (2.32)
The relative CP eigenstates are constructed as follows: M+ = p (jM i + jM i) (2.33) M = p (jM i j M i)
Therefore, the flavor eigenstates can be written in terms of the CP eigenstates (the p is for normalisation): 0 1 (jM+i + jM i) M = p (2.34)
The time evolution of the flavor eigenstates is described by the following equation: d M0 i = H ; = M0 (2.35) dt where, H is the Hamiltonian. It is written as the combination of two hermitian matrices
M and which represent the mass and the decay width of the neutral meson, respectively: H = M i (2.36)
As the CPT invariance requires the masses and the decay rates of the particle and the anti-particles to be equal, the diagonal elements of the H matrix are equal: Hi 11 = H22.
The fact that the M and matrices are hermitian implies that H21 = M12 12. The eigenstates of this Hamiltonian are the physical states: heavy one (H) and light one (L).
These eigenstates can be written as: i (2.37) HjMH;Li = (mH;L 2 H;L)jMH;Li
Both MH and ML states have distinct masses: mH =6 mL and distinct decay widths: H =6 L. The mass and decay width diﬀerences as well as the average decay width are defined as: m mH mL; LH ; 1 (H+L) (2.38)
It is now possible to write the physical eigenstates in terms of the flavour eigenstates as:
jMH i = pjqj21+[pj2)
jMLi = pjqj21+[pj2)
(pjM i + qjM (2.39)
From these equations it can be deduced that if the CP symmetry is not violated ( pq = 1), the physical states coincide with the CP eigenstates. Three types of CP violation can be observed:
The CP violation can occur in the mixing of the neutral meson. This is due to the fact that jqj =6 jpj and so physical states do not correspond to flavour eigenstates.
The second type of CP violation occurs if the decay amplitude of a process and its CP conjugate are not equal: M0 ! f) 6= M0 ! f) =) 0 ! 6= 1 (2.40)
where are the decay widths and A the related amplitudes. This is often called “direct” CP violation and is the only one available for charged particles which cannot mix. The CP violation measurement presented in this thesis is a “direct” CP violation.
The last type of CP violation occurs in interference of mixing and decay, when the meson and its anti-meson can decay to a common final state configuration. M 0 0 ! f) 6= 0 ! M 0 ! f) (2.41)

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#### The CKM matrix

In 1964, the CP violation was observed and the known quarks were: u, d and s. The theory of Cabibbo described the interaction between the u and the d or s quarks by introducing the Cabibbo angle [12]. In 1973, Kobayashi and Maskawa introduced the Cabibbo-Kobayashi-Maskawa (CKM) matrix which describes the interaction between the Up quarks and the Down quarks [13]. Since the CPT symmetry is not violated while the CP one is, the T symmetry should be violated too. Therefore, the CKM matrix shall be complex. The CKM matrix is a unitary matrix: V yV = 1. Assuming N is the number of quark families, N is also the dimension of the CKM matrix. For N = 2 families, the number of parameters for a complex unitary matrix is only one: the Cabibbo angle. But, for N = 3 families, there are four parameters, correspond-ings to three Cabibbo angles, referred as 12;23;13, and a phase : the CP violation is possible.
This prediction of three quarks (and leptons) families has been experimentally confirmed by the discovery of the c quark in 1974 [14], the in 1975 [15], the b quark in 1977 [16] and the t quark in 1995 [17] [18].
The CKM matrix can be parametrized as follows:
VCKM = 0 Vcd Vcs Vcb 1 = 0 s12c23 c12s23s13ei c12c23 s12s23s13ei s23c13 1
Vud Vus Vub @ s12s23 c12c13 s12c13 s13e i A
@Vtd Vts Vtb A c12c23s13ei c12s23 s12c23s13ei c23c13
where, cij = cos ij and sij = sin ij and ij are the the mixing angles of the three quark families: the Cabibbo angles. The elements of the CKM matrix are measured by experiments:
jVudj: study of decays (n! pe e, p! ne+ e)
jVusj: semileptonic decays of kaons: K ! ( )l l and ! K decay. Also accessible through jVus=Vudj measurement (K ! = ! and ! K = ! )
jVubj: semileptonic decays: B ! Xul
jVcdj: semileptonic decays of D mesons, c particles production by neutrino anti-neutrino interactions
jVcsj: semileptonic decays of D mesons, leptonic decays of Ds+
jVcbj: semileptonic decays of B mesons in c particles
jVtdj and jVtsj: neutral B mesons oscillations
jVtbj: measurement of the ratio of branching fractions: B(t!W b) B(t!W q)
The values of all the Vij elements of the CKM matrix are extracted using all the experimental data. The most recent results [7] are:
0 jVudj = 0:97427 0:00014 jVusj = 0:22536 0:00061 jVubj = 0:00355 0:00015 1
VCKM = j V : 0:00061 j V csj = 0:97343 0:00015 j V cbj = 0:0414 0:0012
cdj = 0 22522 +0:00033 : +0:0011 : 0:00005
tdj = 0 00886 0:00032 tsj = 0 0405 0:0012 tbj = 0 99914 (2.43)
It shall be noticed that a clear hierarchy is visible between the elements of the CKM matrix. The diagonal elements are close to 1 while a symmetry is observed between the lateral elements. The elements corresponding to the transition between the first and the second quark families are bigger than those corresponding to the transition of the second and the third families which are bigger than those corresponding to the transition of the first and the third families. The hierarchy of the elements of the CKM matrix is sketched in the Fig. 2.1.

1 Introduction
2 Theoretical framework
2.1 Standard Model and CP violation
2.1.1 The Standard Model of particle physics
2.1.2 The CP symmetry violation
2.1.3 The CKM matrix
2.1.4 The Standard Model issues
2.2 Testing the New Physics
2.2.1 An effective theory
2.2.2 New Physics laboratories
2.2.4 The photon polarisation
2.2.5 Constraints on the C7 and C0
3 Experimental setup
3.1.1 The collider, the particles and the detectors
3.1.2 Accelerating protons
3.1.4 Flavor physics at LHC
3.2 The LHCb spectrometer
3.2.1 Overview of the detector
3.2.2 The vertexing and the tracking system
3.2.3 The magnet
3.2.4 The charged particle identification system
3.2.5 The calorimeters
3.2.6 The muon chambers
3.3 The LHCb trigger system and software environment
3.3.1 The trigger
3.3.2 The hardware level 0 trigger
3.3.3 The software trigger
3.3.4 Triggered on signal, or not
3.3.5 The particle identification algorithm
3.3.6 Efficiency of the particle identification requirements
3.3.7 The converted photons reconstruction algorithm
4 Signal selection
4.1 Samples and simulation
4.1.1 Data samples
4.1.2 Simulation
4.2 Signal preselection
4.2.1 Stripping
4.2.2 True signal sample definition
4.2.3 Dielectron track type samples
4.2.4 Trigger selection
4.2.5 High
4.2.6 Converted photon selection
4.2.7 Mass windows
4.2.8 Events with multiple candidates
4.3 Background rejection
4.3.1 Contamination from misidentified events
4.3.2 Contamination from V e+e􀀀 decays
4.3.3 Contamination from partially reconstructed decays
4.3.4 Combinatorial background
4.4 Selection efficiencies, background contaminations
4.4.1 Selection summary
4.4.2 Selection efficiencies
4.4.3 Background contamination
4.5 Conclusion
5 Model, fits and measurements
5.1 Modelisation and fitting procedure
5.1.1 Signal and background models
5.1.2 Fit to the data
5.2 Validation of the fit
5.2.1 Data and simulation comparison
5.2.2 Fraction of events in each sample
5.2.3 Pseudo data samples generation
5.3 Systematic uncertainties
5.3.1 Fraction of events, data and simulation differences
5.3.2 L0-Trigger
5.3.3 Fit parameters
5.3.4 Double misidentified B0! K0 events
5.3.5 Data and simulation differences
5.3.6 Magnet polarity asymmetry
5.3.7 Systematic uncertainties summary
5.4 Results
5.4.1 Measurement of the ratio of branching fractions
5.4.2 Measurement of the CP asymmetry of B0! K0
5.5 Limit on the branching fraction of the B0 decay
5.5.1 The CLS method
5.5.2 Result
6 Conclusion and outlook
A Correlation tables from reference fit
B Selection variable comparison
C L0-trigger distribution comparison
D Systematics from fixed parameters

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