# Neutrino mixing and the charged lepton sector

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## Theoretical overview and motivation

This first chapter provides an overview of our current theoretical understanding of particle physics and outlines the main motivation for the experimental search described in chapter 4.
I begin by presenting the Standard Model (SM) of particle physics in section 1.1 before focusing on the results from the latest Lepton Universality (LU) tests and their implications on charged Lepton Flavour Violation (cLFV) in section 1.2.

The Standard Model of particle physics

The Standard Model of particle physics is a quantum field theory (QFT) that aims at describing the fundamental interactions and elementary particles. With the notable exception of the gravitational force, which is currently unaccounted for in the model, it provides a full depiction of our present understanding of fundamental physics.
The whole model is predicated upon the principle of local gauge invariance with regards to the symmetry group SU(3)C SU(2)L U(1)Y where the SU(3)C factor is responsible for the strong interaction while the SU(2)L U(1)Y one governs the unified electroweak interaction.
The field content of the theory can be split in three categories. Spinor fields (spin 12 ) are associated with the elementary particles composing the matter. Vector fields (spin 1) represent the gauge bosons encoding the fundamental interactions. Lastly, a scalar field (spin 0) is responsible for the masses of the massive particles.

Matter content

The building blocks of matter are the fermions (spin 12 particles). The ones that participate in the strong interaction are called quarks, while the rest are referred to as leptons. Fermions are split into three families (or generations) of increasingly growing mass. For each family, there are two quarks (of electric charge Q = +23 and Q = 13 , respectively) and two leptons (of electric charge Q = 1 and Q = 0). These are called, for the first family, the up and down quarks, the electron and the electron neutrino. Finally, the spinor fields identified to the fermions have a left-handed component and a right-handed one. Only the former takes part in the weak interaction (hence the L in SU(2)L).
All these particles are depicted in figure 1.1 along the bosons described in the following two sections. It is important to note that each quark carries a specific flavour quantum number and that, for each generation, a single further such number is shared by both leptons.
The properties of the fermions are encoded in the representation of the gauge group in which they transform. So, by indexing the representations of SU(3)C and SU(2)L by their dimension and specifying the weak hypercharge Y (eigenvalue of the generator of U(1)Y ), the full fermionic content of the SM can be expressed as depicted in table 1.1.
It should be pointed out that the right-handed neutrinos described here can not participate in any interaction. They are added to the model in order to explain the experimental observation of neutrino masses and oscillations. These particles have not been observed yet.

Gauge bosons and interactions

The dynamics of fermions and gauge bosons is encoded in the Standard Model la-grangian. This lagrangian includes terms for the kinetic energy and self-interactions of gauge bosons (Lg), terms for the kinetic energy of fermions and their interac-tions with gauge bosons (Lf ), as well as terms related to the Higgs boson sector and the masses of particles (Lh). LSM = Lg + Lf + Lh
This subsection focuses on the first two classes of terms through the analysis of the symmetry groups governing the strong and electroweak interactions.
The algebra of the SU(3)C group has eight generators, composing its dimension-8 adjoint representation. These can be associated to eight massless vector particles, noted Ga=1;:::;8, called gluons. Beside these octuplets and the triplets associated to each quark discussed in the previous section, all SM particles are singlets un-der SU(3)C . As a consequence, only quarks and gluons participate in the strong interaction.
Things procede in a similar fashion for the electroweak interaction, embodied by the SU(2)L U(1)Y group. Here, the three vector fields linked to the generators of the SU(2)L algebra, W b=1;:::;3, mix with the one from the U(1)Y group, B , to form the physical W , Z and gauge bosons.
The self-interaction term of the lagrangian is a direct product of the algebrae just discussed and can be written as Lg = 1 (B B + W b Wb + Ga Ga ) with the diﬀerent field strenght tensors obtained by the general relation Aa = @ Aa @ Aa + gfabcAb Ac where, for any given group, the Aa stand for the generators’ vector fields, g is the coupling constant of the interaction and fabc is the structure constant of the group. The last term, hence, trivially vanishes for B but not for the other tensors.
Of course, the lagrangian term accounting for the interactions between fermions and vector bosons is also tightly linked to the symmetry group of the SM. This term is given by Y a a b b Lf = f (i@ g1 B g2 W g3 G )f whith an implicit sum over all fermions f (f being the adjoint spinor) and where the gi are the coupling constants of the three interactions, the a are the Pauli matrices of SU(2) and the b are the Gell-Man matrices of SU(3). The term in parenthesis is the gauge covariant derivative of the SM (times i). It is paramount to point out that aW a and bGb are a (2 2) and a (3 3) matrix respectively. Thus, they will intervene only with the fermionic fields of the appropriate dimension. The aforementioned statements that right-handed particles do not participate in the weak interaction or that only quarks couple to gluons follow directly from this.

### The Higgs mechanism and the CKM matrix

The lagrangian considered up to this moment describes the dynamics of massless particles. This is due to the fact that the gauge bosons transform under the SM symmetry group in a way that would not allow for a typical mass term to be invariant. Moreover, it is impossible to construct a mass term for fermions that is a singlet under SU(3)C SU(2)L U(1)Y (the right-handed neutrino case being an exception in this regard). Nevertheless, the massiveness of fermions and of the W and Z bosons is a well established experimental fact.
The Higgs mechanism provides an elegant solution to this problem by intro-ducing a complex scalar field whose ground state breaks the SU(2)L symmetry. As a consequence of this spontaneous symmetry breaking, this scalar boson gen-erates dynamically the masses of the fermions and of the weak interaction gauge bosons. It transforms as (1; 2; +12 ) under the SM group. The Higgs sector of the lagrangian is then Lh = (D )y(D ) m2( y ) + n( y )2 + Ly with the first term describing the interaction of the gauge bosons with the Higgs field, the last term responsible for the fermion masses and the rest depicting the Higgs self-interactions and mass (the aforementioned non vanishing vacuum expectation value of the field being insured by the choice m2; n > 0).
The D in the formula is the gauge covariant derivative of the previous subsec-tion. Of course, since the Higgs field is an SU(3)C singlet, the bGb factor does not apply here. Expanding this term of the lagrangian, one can show that mass terms and interaction terms with the Higgs field arise for the weak interaction bosons by identifying does not pick up a mass nor does it interact with the Higgs boson, since the ground state of the scalar field is such that U(1)Q is not spontaneously broken.
It is however clear that this complex scalar doublet also allows for fermionic mass terms in the lagrangian of the form Ly = yeEL eR y EL e;R ydQL dR yuQL uR where the yi are the Yukawa couplings of the diﬀerent fermions, EL and QL are the first generation fermion SU(2)L doublets and further generations as well as hermitian conjugation are implicit.
If this mechanism has no problems providing a mass to the weak interaction bosons and to the fermions, it comes short of explaining the experimentally mea-sured values for these masses. Another unexplained feature of the mass sector of the SM is the fact that the weak eigenstates of the quarks do not coincide with their mass eigenstates. The passage from one basis to the other is encoded in the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix, noted VCKM.
0s01 = 0Vcd Vcs Vcb 1 0s1
B d0 C B Vud Vus Vub d
b0 weak Vtd Vts Vtb C BbC mass
A rich phenomenology derives from this quark mixing in charged weak currents, with neutral mesons’ oscillations and the CP violation (i.e. the breaking of the invariance under space inversion and conjugation of all charges) in hadronic weak decays being prime examples.
It is interesting to note that, contrary to the W currents, the neutral weak currents (the ones mediated by the Z boson) have a flavour diagonal structure and, as such, conserve flavour. Finally, the VCKM structure is also found in the neutrino sector, as it will be discussed in the next section.

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#### Charged Lepton Flavour Violation

Lepton Universality and Lepton Flavour Conservation (LFC) are two accidental ingredients of the Standard Model, in the sense that they are not prescribed by any fundamental symmetry of the theory nor they are implied by any fundamental principle.
In the case of LFC, in fact, it even turns out that the symmetry is not there at all. Indeed, neutrino mixing conclusively proves that lepton flavour is not conserved and even implies cLFV, although at unmeasureably low rates.
Nonetheless, the link between the universality of lepton interactions and the conservation of lepton flavour is deep, and the recent experimental tensions with regards to LU could be pointing to a cLF violating New Physics (NP) within the experimental reach and potentially able to shed light upon some of the dark spots of the SM.

Neutrino mixing and the charged lepton sector

The observation of neutrino oscillations [1] [2] [3] [4] made great waves in particle physics, proving that neutrinos were not massless as previously thought and challenging the particle content and the renormalizability of the theory. The origin of neutrino masses is one of the biggest mysteries in particle physics today, even bringing into question our comprehension of the nature itself of these particles.
An obvious consequence of this mixing, is that lepton flavour is not a conserved quantum number. This is also true for charged lepton processes, where the LFV proceeds through neutrino oscillations in loops, as shown in figure 1.2. However, such transitions are suppressed by factors proportional to ( mij )4, mij being the appropriate neutrino mass diﬀerence and MW the mass of the W boson. As a result, their branching ratios are extremely small within the SM. For instance, all diagrams involving cLFV in the second and third generations are suppressed by a factor on top of any other relevant term, as the VtbVts CKM suppression for the specific case of the B0 ! K 0 decay.
Since cLFV is far beyond experimental reach according to the Standard Model, any observation of such a process is an unambiguous sign of NP. Moreover, it is clear that the charged lepton sector is intimately tied to the neutrino one. Thus, any input from NP in the former should translate into a better understanding of the latter.
Finally, it is interesting to remark that all fermions, with the exception of the charged leptons, directly mix. The VCKM structure discussed in section 1.1 is replicated in the neutrino sector in the PMNS unitary matrix
0 1 = 0U 1 U 2 U 3 1 021
B e C B Ue1 Ue2 Ue3 C B 1 C
named after Pontecorvo, Maki, Nakagawa and Sakata. The diﬀerences with the quark sector lies in the fact that for the neutrinos only the flavour basis is exper-imentally accessible and that the quark transition are mediated by gauge bosons. Moreover, the hierarchic structures of these matrices are not equal and, contrary to what is true for the quarks, the charged leptons’ mass eigenbasis coincides with their weak interaction one. There is, however, no reason for this last happenstance to hold beyond the Standard Model.

Implications of the Lepton Universality tests

Lepton Universality is not a fundamental building block of the theory. However, it is a well documented experimental fact, most notably tested in the decays of the weak interaction vector bosons [5] [6], of the lepton [7] and of the kaon [8].
The same is true for the conservation of lepton flavour [9] [10] [11], that has also been verified in the decays of the muon [12]. These results are important constraints for the model builder tackling the LF sector.
Nevertheless, several tensions with regards to the Standard Model emerged from B meson decays, hinting at Lepton Non Universality (LNU). It is the case of the measurement of the ratio of branching fractions, for diﬀerent lepton species, in b! c‘ transitions. A prime example is [13] [14] [15] RD = B(B0 ! D + ) = 0:306 0:016 (stat) 0:022 (syst); B(B0!D + ) lying about 2 standard deviations away from its Standard Model prediction. RD [14] [15] has also being measured and exhibits similar tensions.

Introduction
1 Theoretical overview and motivation
1.1 The Standard Model of particle physics
1.1.1 Matter content
1.1.2 Gauge bosons and interactions
1.1.3 The Higgs mechanism and the CKM matrix
1.2 Charged Lepton Flavour Violation
1.2.1 Neutrino mixing and the charged lepton sector
1.2.2 Implications of the Lepton Universality tests
2 The LHCb experiment at the LHC
2.2 The LHCb detector
2.2.1 The Vertex Locator
2.2.2 The magnet and the tracking system
2.2.3 The Cherenkov detectors
2.2.4 The calorimeter system
2.2.5 The muon chambers
2.2.6 The trigger
2.2.7 Real time alignment and calibration
3 Tracking with the SciFi subdetector in the LHCb upgrade
3.1.2 Particle identification subdetectors
3.1.3 Tracking subdetectors
3.2 Tracking strategy
3.2.1 Track types
3.2.2 Tracking sequence
3.3 The PrHybridSeeding
3.3.1 Overview of the Hybrid Seeding
3.3.2 The x-z projection step
3.3.3 Stereo step and full track selection
3.4.1 Simulated geometries
3.4.3 Impact of the layer number on the Seeding
3.4.4 Profiling of the Hybrid Seeding performance
3.5 Alternative seeding algorithms
3.5.1 The projective approach
3.5.2 Layer inefficiencies in the Progressive Seeding
3.5.3 Progressive Seeding refinement and variants
3.5.4 Combined Seeding
4 The B0! K0 analysis
4.1 Analysis strategy
4.2 Dataset and simulated samples
4.2.1 Dataset description
4.2.2 Monte Carlo samples
4.3 B0 mass reconstruction
4.4 Event selection
4.4.1 Trigger selection
4.4.2 Stripping selection
4.4.3 Fiducial region
4.4.4 Multivariate selection against the combinatorial background (BDTAC)
4.4.5 Multivariate selection for candidates (BDTTAU)
4.4.6 Particle identification selection
4.4.7 Daughters mass cuts
4.4.8 Fisher discriminant on isolation variables and flight distance
4.4.9 Vetos
4.4.10 Multiple candidates
4.5 Efficiencies
4.5.1 PID efficiencies
4.6 Background studies
4.6.1 Background yields estimate: the ABCD method
4.6.2 Monte Carlo background checks
4.7 Control channel
4.7.1 Anti combinatorial BDT (BDTAC)
4.7.2 Overall control channel selection
4.8 Systematic uncertainties
4.8.1 Efficiencies
4.8.2 Normalization channel fit
4.8.3 Background estimate
4.9 Limit setting
Conclusions
A Tracking algorithms for the LHCb upgrade
A.1 PrPixelTracking
A.2 PrVeloUT
A.3 PrForwardTracking
A.4 PrHybridSeeding
A.5 PrLongLivedTracking
A.6 PrMatchNN
Bibliography

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