Search for rare four-body charm decays with electrons in the final state and long track reconstruction for the LHCb trigger by Alessandro SCARABOTTO x 4.3.1 Phase-space model correction . . . . .

The rare four-body charm decays D 0 → π + π -e + e -and D 0 → K + K -e + e -receive contributions from flavour-changing neutral current c →u ℓ + ℓ -transitions. Measurements of such decays are unique probes of beyond-the-standard-model effects as new heavy particles can contribute in electroweak quantum loops. The search for these rare charm decays is presented using a sample of proton-proton collisions data collected during the Run 2 of LHC (2015-2018), corresponding to an integrated luminosity of 6 fb -1 and recorded with a center-of-mass energy of 13 TeV at the LHCb experiment. The goal of the study is the first observation of these decays and their branching fraction measurement. The analysis is limited by the statistics and the low hardware-trigger efficiency on the electrons. The upgrade of the LHCb detector, which was recently completed and is now being commissioned, will record data with a five times greater instantaneous luminosity and with a new trigger system, and this analysis is therefore expected to greatly benefit from it.

The trigger system was completely redesigned for the upgrade by removing the first-level hardware trigger stage and selecting events with a fully software-based trigger at the 30 MHz collision rate of the LHC. This demanding task is performed by the first step of the high-level trigger, optimised for Graphic Processing Units (GPUs). In order to select interesting events, the trigger must reconstruct the tracks traversing the whole LHCb detector and leaving hits in the three tracking detectors: the vertex locator, the upstream tracker and the scintillating fiber tracker. However, at the beginning of the Run 3 of LHC (2022), the upstream tracker was not yet installed, leading to the necessity of modifying the long track reconstruction strategy. A new algorithm presented in this thesis allows the reconstruction of tracks traversing the whole LHCb detector without using the upstream tracker information. This reconstruction method was successfully tested and commissioned throughout the 2022 data taking and currently used in production in 2023.

SORBONNE UNIVERSITÉ

Resumé

Docteur en Philosophie

Recherche de désintégrations rares à quatre corps de mesons charmes avec des electrons dans l'état final et reconstruction de longues traces pour le système de déclenchement de LHCb

par Alessandro SCARABOTTO Les désintégrations rares à quatre corps de mésons charmés D 0 → π + π -e + e - et D 0 → K + K -e + e -reçoivent des contributions par les transitions à courant neutre avec changement de saveur de type c →u ℓ + ℓ -. Les mesures de ces désintégrations sont fondamentales pour la recherche de physique au-delà du modèle standard parce que des particules plus massives peuvent contribuer dans les boucles quantiques electrofaibles. La recherche de désintégrations charmées D 0 → π + π -e + e -et D 0 → K + K -e + e -est présentée sur ce manuscrit en utilisant des données de collisions proton-proton du Run 2 du LHC (2015-2018), qui correspondent à une luminosité intégrée de 6 fb -1 obtenues avec une énergie au centre de masse de 13 TeV dans l'expérience LHCb. Le but de l'analyse est la première observation de ces désintégrations et la mesure de leur rapports d'embranchement. L'analyse est toutefois limitée par la statistique et la faible efficacité de selection du système de déclenchement de premier niveau. L'upgrade du détecteur LHCb, qui a été recemment achevé et se trouve maintenant dans une phase de preparation, va prendre des données avec une luminosité instantanée cinq fois plus grande qu'au Run 2 avec un tout nouveau système de déclenchement, dont l'analyse presentée ici va beneficier.

Le système de déclenchement a été complètement changé en enlevant le premier niveau du système qui était développé sur un système électronique pur. Le nouveau système doit donc sélectionner les événements avec un système purement logiciel, directement au taux de collisions de 30 MHz du LHC. Ceci a été développé et optimisé sur des processeurs graphiques (GPUs). Pour sélectionner les événements intéressants et réduire le taux en entrée, le système de déclenchement doit reconstruire les traces dites longues, qui traversent tout le détecteur LHCb et laissant du signal sur les trois trajectographes de LHCb: le vertex locator, le upstream tracker et le scintillating fiber tracker. Cependant, pendant la prise de données de 2022, le upstream tracker n'a pas été installé, ce qui veut dire que la stratégie de reconstruction des traces longues devait être modifiée. Un nouvel algorithme de reconstruction présenté dans ce manuscrit permet la reconstruction des traces longues sans utiliser l'information de l'upstream tracker. Cette nouvelle methode a été testée et mise en place en grandeur nature dans LHCb pendant la prise de données de 2022 et encore utilisé couramment en 2023. This has been for me really three wonderful years of my life, through more happy and more stressful times. I would like here to acknowledge all the people who contributed and were close to me in this period.

First, I would like to thank Vava, my supervisor, for accepting me into this journey and being an amazing supervisor. From the outside, sometimes he seems a bit strict, but he really always treated me in such a kind way and helped me every time I was in need. He is the kind of supervisor who manages to let you do your thing when you need more freedom or pushes towards the goal when it's the time to conclude a project. I hope I learned as much I could from you and that we can continue to work together in the coming years.

I would like to thank the jury members for accepting to attend my PhD defense, taking the time to read this document and for the useful comments and questions.

I want to thank Marianna and Dominik, who followed me in every step of my PhD, and are really more than colleagues/supervisors to me, I would say more like friends. They were really always there for me, for any stupid questions. The work that I achieved is in a big part thanks to them.

I would like to thank the PhD students and people in the LPNHE lab: Giorgos, Romain, Sara, Julianna, Ariel, Ream, Lukas, Louis, Tommaso, Cri, Lucile, Luis, Michelangelo, Mykola, Renaud, Vlada, Aurelien, ... We really had a rough time the first year with covid but were always together, "stuck" in the lab. Probably, also because of this, we grew up as a strong group and even went on vacation all together, remember Cannes! My happiness of the first year is for sure due to all of you guys.

My second and third year, I was at CERN, where I met many amazing people and went through many adventures. I would like to thank the people from the CERN volleyball team (Giulia, Gaia, Luca, Sara, Marzia, ... and many others). I spent with them so much time of my days at CERN, both in the gym playing volleyball or outside having fun. From the period at CERN, I cherish the moments in the control room (even if they were very tough times) with the HLT1 people (Dorothea, Daniel, Mari, Lukas, ...) when we were staying at work so late to achieve small goals every day. Work can be also fun and makes you meet many amazing people.

I would like to thank my friends from Fano and Ferrara. I will not mention all their names here, but you might see them after my defense around in Paris with the 2006 Italy jersey joking around about everything as usual. They have always been my place to come back in Italy (both when I go to Fano and Ferrara) and where I could just relax and forget about everything. They are really just amazing. I also want to thank you my friends around the world, always present in my life, even if sometimes we don't have the chance to see each other all the time.

I would like to thank my uncles, aunts, cousins, grandparents all in Fano, with a special thought for Nonno Gigio. I am really lucky to have such a family who loves me so much.

Finally, I conclude by saying thank you to my parents and Caro, even if probably one thank you is not even enough. You are really there for me every moment, when I am happy, sad, angry, stressed, scared, ... I managed to go through and finish this PhD mainly thanks to you who supported me in every decision, even in the difficult ones. I hope I am there for you as much as you are there for me, even if most of the time I am far away. Thank you so much for everything. Ale

Chapter 1

Introduction

Particle physics aims to study the most basic building blocks of the Universe surrounding us by describing the laws regulating their interactions. The culmination of the description of the nature in terms of particles takes form in the Standard Model (SM). The theory of the SM was formulated during the first half of the twentieth century and aims to describe the fundamental particles and their interactions in the frame of relativistic quantum field theory. Over the years and through many experimental results, the SM has been confirmed as a well-tested theory, providing precise predictions for a wide range of physics processes. However, it is widely recognized that some phenomena are not accounted for in the SM, pointing to the idea that a more fundamental theory must exist. The SM does not fully describe the matter-antimatter asymmetry, the observed neutrino oscillations, the existence of dark matter and dark energy, the hierarchy problem and does not incorporate the theory of gravitation. For these reasons, currently, the physics community is focused on finding hints of beyond-the-standard-model (BSM) effects.

Experimentally, BSM searches can be performed directly, looking for new particles not described by the theory, or indirectly, probing properties of the SM decays, with the advantage of testing even higher energy scales. The Conseil Européen pour la Recherche Nucléaire (CERN) hosts several experiments able to test the SM theory. These experiments study the high energy proton-proton collisions provided by the Large Hadron Collider (LHC), reaching a center-of-mass energy up to 13.6 TeV, the highest collision energy ever achieved in a laboratory. With the study of such high energy interactions, the experiments at the LHC follow both direct and indirect approaches to look for BSM physics. For example, a large part of the ATLAS and CMS collaborations physics program is dedicated to the search for heavy new particles in direct production from the collisions. Instead, the LHCb collaboration mainly focuses its attention on the study of flavoured heavy hadrons, looking for BSM effects indirectly. Indirect searches can be performed via the investigation of rare or forbidden processes in the SM. New particles can contribute in the quantum loops, modifying the decay rates or the angular distributions of the particles involved in the transitions.

In this respect, flavour-changing neutral current (FCNC) rare decays can only occur via electroweak quantum loops and thus are very sensitive to potential enhancements of the decay rates. In the past years, many studies and measurements on FCNC decays were performed in the K and B sectors, while the complementary charm transitions are still a relatively unexplored area. Measurements of FCNC charm decays are unique up-type quark probes for these processes.

The D 0 → h + h -ℓ + ℓ -(h = π, K and ℓ = e, µ) are among the most promising rare charm channels. These decays can proceed via the FCNC c → uℓ + ℓ -transitions with a branching fraction SM prediction of O (10 -9 ). However, contributions due to tree-level decays via an intermediate resonance can increase the branching fraction up to O (10 -6 ). Experimentally, access to the FCNC component can be gained by measuring the branching fractions in regions of the dilepton mass away from the resonances or performing studies of angular and charge-parity (CP) asymmetries. The comparison with the theoretical predictions of these observables can reveal hints of BSM physics. Additionally, BSM effects can be studied comparing the D 0 → h + h -µ + µ -muon modes with the D 0 → h + h -e + e -electron modes with the so-called lepton flavour universality ratios. The SM predicts these ratios to be one with an uncertainty of O(1 %), as most of the hadronic uncertainties coming from the resonant component cancel out in the ratio. New physics particles could couple differently to the lepton species and modify the expected values of such ratios.

Thanks to the huge cc cross-section available at the LHC, the LHCb collaboration has invested a major effort in observing and measuring SM decay properties of the D 0 → h + h -µ + µ -decays. Instead, the D 0 → h + h -e + e -rare decays have not been observed yet, since experimentally they are more challenging compared to the equivalent muon modes. Electrons interact more with the material of the detector, making the evaluation of the properties of the decays less precise at LHCb. Furthermore, the studies of decays with electrons in the final state are affected by a lower trigger efficiency compared to decays involving muons. This thesis presents the search for D 0 → h + h -e + e -decays at LHCb and sets the first steps to look for BSM effects in the rare charm four-body electron modes. The analysis is statistically dominated and a larger data sample would open the possibility in the future to measure CP and angular observables of these processes and to compare their differential decay rates to the muon mode ones.

The upgrade of the LHCb detector aims to collect more data by increasing the instantaneous luminosity by a factor five compared to previous data taking. In particular, one of the major limitations of the LHCb detector was the hardware firstlevel trigger. The hardware trigger signal yield saturates for decays with hadrons and electrons in the final state when increasing the luminosity. For this reason, the hardware trigger was removed and a fully software-based trigger system was implemented. In this respect, one of the main challenges is reading out the detector and performing the event reconstruction at the LHC 30 MHz pp collision rate.

Thirty million events per second are reconstructed by the first high level trigger optimised for Graphic Processing Units (GPUs). GPUs are excellent for highly parallelized tasks, such as tracking, and the LHCb collaboration implemented an optimised framework, Allen, to fully exploit them. In order to select interesting events, the trigger must reconstruct so-called long tracks coming from charged particles traversing the full LHCb detector, leaving hits in the three tracking subdetectors: the VErtex Locator (VELO) and the Upstream Tracker (UT) before the dipole magnet, and the Scintillating Fiber tracker (SciFi) after the magnet. These tracks are crucial to select events based on high momentum and displaced signatures in the detector. The UT is fundamental when extrapolating tracks upstream of the magnet as it provides a first momentum estimate and charge measurement of each track, thanks to the residual magnetic field outside the region where the magnet is located. However, during the 2022 data taking, it was not yet installed. The second part of this thesis focuses on describing a GPU-based algorithm able to reconstruct long tracks without using the UT information. The algorithm was commissioned and deployed for the 2022 data taking, and it is currently in use for the 2023 one.

The thesis first introduces the theoretical background needed to study the rare charm decays in chapter 2. This chapter presents the SM predictions of these decays and the observables which are interesting to measure to search for physics BSM. The LHCb experimental environment is presented in chapter 3, focusing on the aspects and performance of the detector relevant to study rare charm decays with electrons in the final states. Then, chapter 4 discusses the experimental methods used to search for the D 0 → h + h -e + e -decays, from the data selection to the expected sensitivity. The upgrade of the LHCb detector is introduced in chapter 5 with a particular focus on the upgrade of the trigger system and the reconstruction algorithms used at the first-level trigger. Finally, the long track reconstruction is presented in chapter 6 describing in details the relevant algorithms and showing the performance evaluated on simulated samples.

Chapter 2

Rare charm physics within the standard model of particle physics

The standard model of particle physics, the theory describing how elementary particles interact with each other, is presented in this chapter. The rare charm c → uℓ + ℓ - transitions are introduced and discussed via an effective field theory which describes them. The study of such processes is crucial when searching for beyond-thestandard-model effects and various experimental ways to probe them are presented. Finally, the phenomenology of D 0 → h + h -e + e -decays (h = π, K) is discussed in order to introduce their experimental search, which is one of the main focuses of this thesis.

The Standard Model

The Standard Model (SM) of particle physics was formulated from mid-1950s and developed over the years by many scientists around the world [START_REF] Sheldon | Partial-symmetries of weak interactions[END_REF]2,[START_REF] Salam | Weak and Electromagnetic Interactions[END_REF]. The SM describes three out of the four fundamental forces as physical manifestations of local symmetries in a gauge theory, built as the product of three groups:

SU(3) C ⊗ SU(2) L ⊗ U(1) Y , (2.1)
where the SU(3) C group is associated to the strong interaction while SU(2) L ⊗ U(1) Y is responsible for the electroweak sector, i.e. the combined electromagnetic and weak interactions.

The fourth fundamental force, gravity, is not described by the SM.

Once the gauge group is specified, the SM is totally determined by assigning how elementary particles are charged under it. In particular, by distinguishing among the different force carriers, i.e. the gauge bosons, the SM describes:

• the electromagnetic force: mediated by the photon γ and coupled to particles that carry electromagnetic charge;

• the weak force : mediated by the W and Z bosons and coupled to particles that carry isospin;

• the strong interaction: mediated by eight gluons and coupled to particles that carry colour charge.

Elementary particles are bosons and fermions, namely particles with respectively integer and half-integer spin. The fermions are classified in quarks and leptons and separated in three generations. Each of the generations contains:

• an up-type quark with charge + 2 3 : (u,c,t);

• a down-type quark with charge -1 3 : (d,s,b);

• an electrically charged lepton (charge -1): (e -,µ -,τ -);

• a neutrino: (ν e ,ν µ ,ν τ ).

All electric charges are in units of the charge of the proton e ∼ 1.6 × 10 -19 C [START_REF] Zyla | Review of particle physics[END_REF]. The strong force is described by the theory of quantum chromodynamics (QCD) and particles that interact with the strong force carry colour charge, denoted as red, green and blue, or anti-colours, anti-red, anti-green and anti-blue. The force mediators, the gluons, couple to colour charged particles and carry a unit of colour and unit of anti-colour resulting in eight possible combinations. Gluons are also coloured, leading to gluon-gluon interactions.

The coupling strength of the strong force, α s , is evaluated through perturbation theory for different momentum scales. The perturbation approach relies on expansions in powers of α s and the Feynman diagrams [START_REF] Riddell | The Number of Feynman Diagrams[END_REF] constitute a pictorial way of representing the calculations. The values of α s decrease as the relevant momentum scale grows [START_REF] Olive | Review of Particle Physics[END_REF], often referred as the asymptotic freedom property of QCD. This means that, for example, within a proton, quarks behave as nearly free. On the other hand, the perturbative method breaks down at the Λ QCD confinement scale or long distances, as the coupling gets too large: this phenomenon is known as confinement. Quarks and gluons are not observed as free particles, because by increasing distance between them, the production of a new quark-antiquark pair is instead energetically preferred, leading to the formation of bound states of quarks. These bound states, called hadrons, are colourless, hence can be composed of quark-antiquark pairs, the mesons, or combination of three quarks, the baryons. More complicated systems composed of four or five quarks are possible and have been recently confirmed experimentally [START_REF] Aaij | Observation of the resonant character of the Z(4430)state[END_REF][START_REF] Aaij | Observation of J/ψp resonances consistent with pentaquark states in Λ 0 b → J/ψpKdecays[END_REF]. At energies below the confinement scale Λ QCD , non-perturbative methods have to be employed to describe the QCD interactions, such as the lattice QCD theory. In charm physics, since the mass of the c-quark is only slightly above Λ QCD , QCD effects lead to large uncertainties on theoretical predictions when evaluated with perturbation theory.

The combination of the electromagnetic and weak interactions, SU(2) L ⊗ U(1) Y , can be understood in the context of the spontaneous symmetry breaking given by the Higgs mechanism in the SM [START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF][START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF][START_REF] Guralnik | Global Conservation Laws and Massless Particles[END_REF]. The Higgs scalar field leads to the mixing of the three SU(2) L gauge bosons W 1 , W 2 , W 3 and the single U(1) Y gauge boson B into the two electrically charged W + , W -and two neutral Z 0 ,γ bosons. As previously discussed, the W ± and Z 0 become the mediators of the weak force while the photon γ becomes mediator of the electromagnetic interaction. Furthermore, the Higgs mechanism gives rise to the mass of massive particles and predicts the existence of a scalar boson, known as the Higgs particle. The existence of this particle was confirmed in 2012 by both the ATLAS [START_REF] Aad | Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF] and CMS [START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF] collaborations.

The particle content of the SM is summarised in Figure 2.1, including all fermions and bosons cited in the section.

Flavour-changing processes

Within the SM, flavour-changing processes can only occur via weak interactions, through the exchange of the W boson. The charged-current Lagrangian L CC can be written as:

L CC ∼ ūL γ µ V CKM d L + h.c., (2.2)
Figure 2.1: The elementary particles composing the Standard Model grouped by family, species, generation and interaction types [START_REF] Zyla | Review of particle physics[END_REF].

with V CKM the Cabibbo-Kobayashi-Maskawa (CKM) matrix [START_REF] Cabibbo | Unitary Symmetry and Leptonic Decays[END_REF][START_REF] Kobayashi | CP-violation in the renormalizable theory of weak interaction[END_REF], u and d respectively the up and down fields and γ µ the Dirac matrices. The CKM matrix is a complex and unitary 3 × 3 matrix with four degrees of freedom, which can be defined by three angles and a CP-violating phase. Each matrix element (V CKM ) ij is related to the probability of a quark i decaying into a quark with flavour j. Within the SM, the CKM matrix is the dominant source of CP violation, where the CP transformations include a simultaneous charge conjugation (C) and parity (P) variation. Experimentally, CP violation was first observed in kaon system in the decays of K 0 L mesons [START_REF] Christenson | Evidence for the 2π Decay of the K 0 2 Meson[END_REF] and more recently in the band c-sectors [START_REF] Aubert | Measurement of CP-Violating Asymmetries in B 0 Decays to CP Eigenstates[END_REF][START_REF] Abe | Observation of Large CP Violation in the Neutral B Meson System[END_REF][START_REF] Aaij | Observation of CP violation in charm decays[END_REF]. The CKM matrix can be parametrised in terms of the Wolfenstein parameters (λ, A, ρ, η) [20]:

V CKM =   V ud V us V ub V cd V cs V cb V td V ts V tb   =   1 -λ 2 /2 λ Aλ 3 (1 -ρ -iη) -λ 1 -λ 2 /2 Aλ 2 Aλ 3 (1 -ρ -iη) Aλ 2 1   + O(λ 4 ), (2.3
) with λ ∼ 0.23, A ∼ 0.83, ρ ∼ 0.16 and η ∼ 0.35.

The diagonal elements of the matrix describe transitions within the same generation and are O [START_REF] Sheldon | Partial-symmetries of weak interactions[END_REF]. These transitions are referred as Cabibbo-favored and are described by tree-level Feynman diagrams. Transitions between two different generations are suppressed by powers of λ, known as Cabibbo or doubly-Cabibbo suppressed and described respectively by off-diagonal or two off-diagonal elements.

The unitary condition of the matrix V CKM V † CKM = 1 imposes:

∑ i V ij V * ik = δ jk ; ∑ j V ij V * kj = δ ik , (2.4)
hence the orthogonality among rows and columns of the matrix. These six constraints can be represented as triangles in the complex plane (ρ-η), known as the unitarity triangles. The areas of all unitarity triangles are equal and related to the measure of CP violation [START_REF]The CKM matrix and the unitarity triangle[END_REF]. Most of the triangles are very elongated, making the CP violation effects harder to precisely measure. However, the triangle describing the transitions of band d-quarks is predicted to have sides approximately of the same length to each other O(λ 3 ) and described by the following condition:

V ud V * ub + V cd V * cb + V td V * tb = 0. (2.5)
Due to the importance of this triangle, it is usually known as the unitarity triangle.

It is then normalised to V cd V * cb , obtaining as vertices (0,0), (0,1) and (ρ,η). Most of the CKM matrix elements are determined through direct measurements, looking at tree-level processes, in order to directly extract the values of |V ij | [START_REF] Workman | Review of Particle Physics[END_REF]. However, some elements have low precision, such as |V tb | and |V cs |, or are too suppressed to be directly measured, i.e. |V td | and |V ts |. In these cases, indirect measurements are performed looking at higher-order processes. A representation of the current experimental results is shown in the (ρ-η) plane, see Figure 2.2. The determination of the apex (ρ,η) of the triangle acts as closure test of the SM. In fact, any inconsistencies in the (ρ-η) plane can give hints about the physics beyond the SM [START_REF]The CKM matrix and the unitarity triangle[END_REF].

Processes that change the generation of a quark without changing its charge are known as flavour-changing neutral currents (FCNC). The structure of the Lagrangian and the unitarity of the CKM matrix set all the couplings between quarks of the same charge of different generations to zero. Hence, the FCNC processes cannot proceed at tree level under the SM but only at loop-level via the exchange of an opposite type quark. In addition to the CKM matrix terms, FCNC decays are further suppressed by the Glashow-Iliopoulos-Maiani (GIM) mechanism [START_REF] Glashow | Weak Interactions with Lepton-Hadron Symmetry[END_REF]. The GIM cancellation is proportional to the mass squared of the quark i in the internal loop, following the relation m 2 W /m 2 i , with m W the mass of the W boson. Typical orders of magnitude for the branching fractions of FCNC processes are below 10 -6 , setting these decays in the so-called regime of rare decays.

Examples of FCNC transitions are the b →s ℓ + ℓ -and b →d ℓ + ℓ -processes with two oppositely charged leptons ℓ ± in the B sector, s →d ℓ + ℓ -in kaon decays and c → uℓ + ℓ -in charm decays.

A schematic of c →u ℓ + ℓ -transitions is shown in Figure 2.3, illustrating loop processes with at least two flavour-changing weak interactions, the so-called penguin and box diagrams. The GIM cancellation in c →u ℓ + ℓ -transitions is dominated by the b-quark contribution in the loop ∼ m 2 W /m 2 b which is much stronger than in b → sℓ + ℓ -decays where the t-quark can enter as intermediate particle in the loop, leading to a GIM cancellation proportional to m 2 W /m 2 t . Thus, rare charm decays are approximately three orders of magnitude more suppressed than the equivalent processes in beauty sector.

Effective field theory for rare charm decays

The role of charm hadron decays is special when studying FCNC processes, as they are the only bound systems testing up-type quark dynamics.

At the charm scale, the high-energy dynamics involved in the loop diagrams, i.e. the W boson and b-quark (see Figure 2.3), cannot be resolved. Model independent effective field theories (EFT) provide a solution to this problem. An effective Lagrangian is constructed as a sum of all possible local four-fermion operators O i which may contribute to the decay, with the so-called operator product expansion (OPE). The effective Lagrangian is defined as:

L e f f ∼ ∑ i C i O i , (2.6)
with C i the corresponding coupling strengths, known as Wilson coefficients [START_REF] Kenneth | Non-Lagrangian Models of Current Algebra[END_REF]. These coefficients are calculated by matching the EFT to the full SM, evaluating them at a high energy scale μ, typically the W mass. In this way, all the heavy degrees of freedom with m > μ are integrated out and absorbed in the Wilson coefficients. The degrees of freedom with m < μ, generally from the light quark fields, contribute to the operators and are considered massless. The renormalization group equation [START_REF] Collins | The Problem of scales: Renormalization and all that[END_REF] is then used to obtain the Wilson coefficients values at the relevant energy scale, in charm physics, at the mass of the charm quark m c . The effective Lagrangian can be written at the charm scale m c for c →u ℓ + ℓ - decays as:

L e f f ∼ ∑ q=d,s V * cq V uq (C 1 (m c )O (q) 1 + C 2 (m c )O (q) 2 + 10 ∑ i=3 C i (m c )O i ), (2.7)
with the full set of operators O 1-10 defined as [START_REF] Chetyrkin | Weak radiative B meson decay beyond leading logarithms[END_REF][START_REF] Bobeth | Photonic penguins at two loops and m t dependence of BR[B → X s l + l -][END_REF][START_REF] Gambino | Anomalous dimension matrix for radiative and rare semileptonic B decays up to three loops[END_REF][START_REF] De Boer | Higher-order Wilson coefficients for c → u transitions in the standard model[END_REF]:

O (q)
1 ∼ ( ūL γ µ T a q L )( qL γ µ T a c L )

O 3 ∼ ( ūL γ µ c L ) ∑ q=u,d,s,c
( qγ µ q)

O 5 ∼ ( ūL γ µ γ ν γ α c L ) ∑ q=u,d,s,c

( qγ µ γ ν γ α q) O 7 ∼ m c ( ūL σ µν c R )F µν O 9 ∼ ( ūL γ µ c L )( lγ µ ℓ) O (q) 2 ∼ ( ūL γ µ q L )( qL γ µ c L ) O 4 ∼ ( ūL γ µ T a c L ) ∑ q=u,d,s,c
( qγ µ T a q)

O 6 ∼ ( ūL γ µ γ ν γ α T a c L ) ∑ q=u,d,s,c

( qγ µ γ ν γ α T a q)

O 8 ∼ m c ( ūL σ µν T a c R )G a µν O 10 ∼ ( ūL γ µ c L )( lγ µ γ 5 ℓ)

where q = d, s, b quarks, γ µ the Dirac matrices, F µν the electromagnetic field strength tensor and T a the Gell-Mann matrices, defined in the study of strong interaction [START_REF] Workman | Review of Particle Physics[END_REF]. The fields q L,R are defined from the chirality operator P L,R as q L,R = P L,R q = 1 2 (1 ∓ γ 5 )q.

At the scale μ = m W , the C 3-9 coefficients are zero as the d-, sand b-quarks are treated as massless and the GIM mechanism is fully effective. Only C 1 and C 2 receive non-zero contributions from the matching procedure at μ = m W . Possible non-zero values of the C 3-9 coefficients can arise from the mixing of the O 3-9 operators with the four-quarks operators O 1 and O 2 below the b-quark mass scale m b . The mixing leads to very small values of C 3-9 at the charm scale [START_REF] De Boer | Higher-order Wilson coefficients for c → u transitions in the standard model[END_REF].

The axial-vector operator O 10 does not mix at any scale, leading to a zero value of C 10 . At the charm scale, all the heavier degrees of freedom are integrated out of the Lagrangian. This means that the Z boson does cannot contribute in the loop diagrams of c →u ℓ + ℓ -transitions. Instead, diagrams with a photon γ in the loop are still allowed as the photon is massless. The C 10 Wilson coefficient receives no contributions from these diagrams as the photon is a vector boson.

This has a crucial impact on the phenomenology of charm, making it unique among the other heavy quark systems. For instance, the angular asymmetries of the rare charm decays are expected to be zero in the SM with a null uncertainty [START_REF] De Boer | Higher-order Wilson coefficients for c → u transitions in the standard model[END_REF].

Effective field theories and the values of the Wilson coefficients are usually used to compare measurements with theoretical predictions. Contributions from new physics can modify the c →u ℓ + ℓ -processes at loop level or even resulting in BSM first order tree-level transitions, leading to variations of Wilson coefficient values compared to the SM expectations.

Beyond the Standard Model

The Standard Model is an extremely successful theory which has been confirmed by many experimental measurements. However, some remaining questions are still unanswered which motivates the particle physics community to look for physics beyond the SM.

First, the SM only includes three out of the four fundamental forces in nature. Gravity has not been quantised yet, and no unified theory can explain all interactions at the same time.

Astronomical observations of galaxy cluster rotation curves [START_REF] Sofue | Rotation curves of spiral galaxies[END_REF] or gravitational lensing [START_REF] Massey | The dark matter of gravitational lensing[END_REF] imply the existence of matter and energy which is not visible. The SM only accounts for 5 % of the world. The rest seems to not interact with the SM particles and is called dark matter, representing 27 %. The remaining 68 %, the socalled dark energy, is responsible for the accelerating expansion of the universe.

The observation of neutrino oscillations [START_REF] Ahmad | Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory[END_REF] indicates that neutrinos are massive particles. The oscillations can be described by introducing the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) rotation matrix [START_REF] Maki | Remarks on the unified model of elementary particles[END_REF][START_REF] Pontecorvo | Inverse beta processes and nonconservation of lepton charge[END_REF], similarly to the CKM matrix in quark sector. However, the origin of the neutrino masses remains unclear.

Another unsolved puzzle is the difference between the amount of matter and anti-matter in the Universe. Assuming that the initial state of the Universe was symmetric between the two terms, a process which favors matter over antimatter must exist. This process must respect the Sakharov conditions [START_REF] Sakharov | Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe[END_REF] allowing for: the violation of the baryon number, interactions out of thermal equilibrium and the presence of CP and C violation. However, the amount of CP violation present in the SM [START_REF] Glennys | Baryon asymmetry of the universe in the standard electroweak theory[END_REF][START_REF] Huet | Electroweak baryogenesis and standard model CP violation[END_REF][START_REF] Gavela | Standard model CP violation and baryon asymmetry. Part 1: Zero temperature[END_REF] can only account for a very small fraction of the matter-antimatter asymmetry seen.

Finally, in the SM, no symmetry protects the mass of the Higgs boson which can receive corrections from all scales present in nature, up to the Planck scale ∼ 10 19 GeV/c 2 . The observed Higgs mass of 125 GeV/c 2 [START_REF] Aad | Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF][START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF] can be only explained if these high energy corrections cancel out. One possibility is to extend the SM gauge symmetry including additional particles. For example, in supersymmetric models [START_REF] Martin | A Supersymmetry primer[END_REF], a supersymmetric boson partner is added for each of the SM fermion and vice versa. Alternative theories are represented by little Higgs models [START_REF] Schmaltz | Little Higgs review[END_REF], where the Higgs particle is a pseudo-Nambu-Goldstone boson from a spontaneously broken higher global symmetry. The space-time dimensionality can also be enlarged, as suggested by Randall-Sundrum models [START_REF] Randall | A Large mass hierarchy from a small extra dimension[END_REF][START_REF] Randall | An Alternative to compactification[END_REF], postulating warped extra dimension and solving the hierarchy problem by lowering the Planck scale. Furthermore, new particle species, so-called leptoquarks [START_REF] Doršner | Physics of leptoquarks in precision experiments and at particle colliders[END_REF][START_REF] Buchmüller | Leptoquarks in lepton-quark collisions[END_REF], can be included in the SM trying to define a more fundamental grand unified theory (GUT). Leptoquarks can lead transitions from quarks to leptons carrying both lepton and baryon numbers and acting as a unification of the fermions.

Experimentally, beyond the Standard Model effects are tested with both direct and indirect searches. In direct studies, signatures of new particles not predicted by the SM are investigated. At the Large Hadron Collider, the ATLAS and CMS experiments look for the production of heavy new particles in high energetic proton-proton collisions. The main limitation here is the available center-of-mass energy provided by the accelerator. Furthermore, direct searches try to probe the interactions of dark matter particles with the visible matter. This is the goal of large volume shielded detectors such as XENON [START_REF] Aprile | The XENON1T Dark Matter Search Experiment[END_REF] exploiting the nuclei of a liquid xenon target to observe a possible matter-dark matter interaction.

Indirect searches are based on precise measurements of the SM decay properties, with the advantage of probing even higher energy scales compared to direct studies. Considering an effective field theory valid up to a cut-off scale Λ, defined to be larger than the electroweak scale v ∼ 246 GeV, the decay amplitude of a flavour-changing transition of a fermion ψ i to another ψ j is:

A(ψ i → ψ j + X) = A 0 ( c SM v 2 + c NP Λ 2 ), (2.8)
with c SM and c NP multiplicative constants related respectively to the standard model and new physics contributions to the corresponding decay. When studying decays with c SM ≪ 1, new physics effects can have a large impact even if Λ ≪ v, probing a high energy scale. This means that indirect searches can test any energy scale, but are only limited by the experimental and theoretical uncertainties on the measurements.

In flavour physics, often c SM ≪ 1 as it is proportional to the off-diagonal small entries of the CKM matrix or the studied processes are forbidden at tree level, such as the FCNC transitions. As an example, in the case of the B-B mixing,

√ c SM ∼ |V td | ∼ 10 -3
, therefore this observable can probe new physics up to the 10 3 TeV scale in models with c NP ∼ 1. Flavour-changing neutral current transitions are also perfect in order to probe higher energy scales as they are forbidden at tree level. In such decays, NP effects could be easily seen compared to a small SM signal.

The LHCb collaboration has invested a major effort in precisely measuring FCNC decay properties in order to constrain NP models. Generally three types of analyses are performed, as illustrated by the following examples in the b →s ℓ + ℓ -sector:

1. Branching fraction measurements: the direct evaluation of the decay rates.

One of the major LHCb results is the measurement of the branching fraction of the B 0 s →µ + µ -decay with a value of (3.09 +0.46+0.15 -0.43-0.11 ) × 10 -9 [START_REF] Aaij | Measurement of the B 0 s → µ + µdecay properties and search for the B 0 → µ + µand B 0 s → µ + µγ decays[END_REF]. The result is compatible with the SM predicted value and constrains many NP models.

2. Angular analysis: the study of the parameters which describe the distributions of the angles between the particles involved in the decays, known as angular observables. They can be defined such that the form factors cancel out to first order, reducing the hadronic uncertainties. Therefore, these observables have more precise theoretical predictions compared to the decay rates. Experimentally, the LHCb collaboration performed an angular analysis in the B 0 →K * µ + µ -and B + →K * + µ + µ -decays [START_REF] Aaij | Measurement of CP-averaged observables in the B 0 → K * 0 µ + µdecay[END_REF][START_REF] Aaij | Angular analysis of the B + → K * + µ + µdecay[END_REF] which are currently in tension of 2-3 standard deviations with the SM and in the B 0 s → ϕµ + µ -decay [51] which is compatible with both SM predictions and the other angular experimental results.

3. LFU tests: measurement of the branching fraction ratios R H in a q 2 range, [q 2 min , q 2 max ], defining the so-called Lepton Flavour Universality (LFU) ratios:

R H = q 2 max q 2 min dB(H b →Hℓ + 1 ℓ - 1 ) dq 2 q 2 max q 2 min dB(H b →Hℓ + 2 ℓ - 2 ) dq 2
, with H b any hadron with valence quark b, H the final state particle and ℓ 1,2 different leptons between e, µ and τ particles. The q 2 is equivalent to the squared dilepton invariant mass. The advantage of defining such observables is the cancellation of hadronic effects in the ratio, achieving very precise theoretical predictions, especially for the light lepton modes. Corrections to the expected unity value of the ratio are only due to quantum electrodynamics (QED) effects [START_REF] Bordone | On the Standard Model predictions for R K and R K *[END_REF][START_REF] Bordone | The LFU ratio R π in the Standard Model and beyond[END_REF]. The LHCb collaboration recently measured simultaneously the LFU ratios in the B + →K + ℓ + ℓ -(R K ) and B 0 →K * ℓ + ℓ -(R K * ) modes obtaining results compatible with SM predictions [START_REF] Aaij | Test of lepton universality in b → sℓ + ℓdecays[END_REF][START_REF] Aaij | Measurement of lepton universality parameters in B + → K + ℓ + ℓand B 0 → K * 0 ℓ + ℓdecays[END_REF].

Similar measurements have been performed in the charm sector, studying FCNC c →u ℓ + ℓ -. Charm decays are further suppressed by the GIM mechanism compared to b →s ℓ + ℓ -transitions by approximately three orders of magnitude [START_REF] Paul | On D → X u l + lwithin the Standard Model and Frameworks like the Littlest Higgs Model with T Parity[END_REF], making their measurements much more challenging. Furthermore, charm decays are dominated by hadronic contributions at low energy which are evaluated with non-perturbative methods. These methods lead to SM predictions with larger uncertainties compared to the equivalent decays in the B sector [START_REF] De | Null tests from angular distributions in D → P 1 P 2 l + l -, l = e, µ decays on and off peak[END_REF].

Theoretically and experimentally, the cleanest rare charm decay is the two-body decay D 0 →µ + µ -with a purely leptonic final state, as theoretical hadronic uncertainties are minimal due to the precise lattice calculation of the D decay constant. Due to the additional helicity suppression, the expected SM branching fraction is O(10 -18 ) [START_REF] Burdman | Rare charm decays in the standard model and beyond[END_REF]. The FCNC component dominated by the long distance contribution via an intermediate two-photon state with an expected branching fraction of O(10 -13 ) [START_REF] Burdman | Rare charm decays in the standard model and beyond[END_REF][START_REF] Wang | Decays D + (s) → π(K) + ℓ + ℓand D 0 → ℓ + ℓin the MSSM with and without R-parity[END_REF]. A recent LHCb measurement sets an upper limit on the BF to be < 3.1 × 10 -9 at 90 % confidence level [START_REF] Aaij | Search for rare decays of D 0 mesons into two muons[END_REF], representing the most stringent available limit on these decays.

When looking at rare multibody c →u ℓ + ℓ -decays, such as D +(0) → h + (h -)µ + µ - (h = K, π), the helicity suppression is much lower but the non-perturbative hadronic contributions become dominant. Long distance (LD) processes through intermediate resonant states in the form of D +(0) → h + (h -)V(→ µ + µ -) can enhance the branching fractions up to O(10 -6 ) [START_REF] Cappiello | Standard Model prediction and new physics tests for D 0 → h + hℓ + ℓ -(h = π, K : ℓ = e, µ)[END_REF][START_REF] De | Flavor and new physics opportunities with rare charm decays into leptons[END_REF]. Sensitivity to the short distance (SD) FCNC component can be gained by studying:

• BF as a function of q 2 : probing q 2 regions away from the resonances;

• angular observables: defining them such as the influence of the hadronic uncertainties is reduced in order to gain sensitivity in the SD physics;

• CP violation: testing the decay rates of D 0 -D 0 or D + -D -.

Such studies are performed by LHCb in the D 0 → h + h -µ + µ -channels obtaining results compatible with the SM predictions within the uncertainties [START_REF] Aaij | Measurement of angular and CP asymmetries in D 0 → π + πµ + µand D 0 → K + Kµ + µdecays[END_REF][START_REF] Aaij | Angular analysis of D 0 → π + πµ + µand D 0 → K + Kµ + µdecays and search for CP violation[END_REF].

As in the B sector, the LFU ratios in charm have well controlled SM predictions. The four-body BF ratio R D P 1 ,P 2 is defined in a q 2 range [q 2 min , q 2 max ] as:

R D P 1 ,P 2 = q 2 max q 2 min dB(D→P 1 P 2 µ + µ -) dq 2 q 2 max q 2 min dB(D→P 1 P 2 e + e -) dq 2 , ( 2.9)
for a decay of a charmed D meson into two hadrons, P 1 and P 2 , and two leptons, in the formula muons and electrons. Phase-space and electromagnetic corrections to the ratios are highly suppressed leading to expected SM values of unity with uncertainties at the percent level [START_REF] Bordone | On the Standard Model predictions for R K and R K *[END_REF][START_REF] Huber | Electromagnetic logarithms in B → X s l + l[END_REF][START_REF] Bobeth | CP Asymmetries in bar B → K * (→ Kπ) lℓ and Untagged Bs , B s → ϕ(→ K + K -) lℓ Decays at NLO[END_REF]. BSM effects can be seen in both R D ππ and R D KK ratios in different q 2 ranges [START_REF] De | Null tests from angular distributions in D → P 1 P 2 l + l -, l = e, µ decays on and off peak[END_REF]. Experimentally, these observables are still limited statistically as the D 0 → π + π -e + e -and D 0 → K + K -e + e -decays have not been yet observed. This thesis focuses on presenting the first LHCb study on D 0 → h + h -e + e -rare charm decays, foreseeing a future LFU test in the charm sector.

Rare charm decays: D

0 → h + h -e + e -
Rare semi-leptonic four-body charm decays of the form D 0 → h + h -e + e -can either proceed via long-distance (LD) interactions through decays of intermediate mesonic vector or scalar resonances D 0 → h + h -V(→ e + e -), or at short distances (SD) via FCNC transitions. Here, h + h -are two oppositely charged pairs of hadrons, either pions π + π -or kaons K + K -.

The Feynman diagrams for the LD and SD contributions of D 0 → π + π -e + e - and D 0 → K + K -e + e -decays are shown in Figure 2.4. The LD diagram (left) shows a generic neutral resonance V decaying into a dielectron pair. On the right, the FCNC process is described by one of the possible short distance transitions, the electroweak penguin diagram. Figure 2.4: Feynman diagrams of D 0 → π + π -e + e -and D 0 → K + K -e + e - processes. On the left, the LD diagram decaying via an intermediate resonance. On the right, a higher-order electroweak penguin transition is shown [START_REF] Dominik | Search for new physics in rare four-body charm decays at LHCb[END_REF].

The phase space of these decays is quite complex as the dihadron pairs can also stem from intermediate resonances. The two pions in the D 0 → π + π -e + e -decay are expected to arise from intermediate ρ and ω resonances, with the dominant contribution coming from the ρ resonance, as the ω resonance weakly couples to pions. The dielectron system is dominated by the ρ resonance, ω and ϕ vector mesons decays. Contributions from the pseudoscalar η ( ′ ) resonances are subleading [START_REF] De | Null tests from angular distributions in D → P 1 P 2 l + l -, l = e, µ decays on and off peak[END_REF]. The expected differential branching fraction as a function of the squared dielectron mass m 2 (e + e -) is shown in Figure 2.5 (left). The total expected branching fraction has been estimated to be [START_REF] Cappiello | Standard Model prediction and new physics tests for D 0 → h + hℓ + ℓ -(h = π, K : ℓ = e, µ)[END_REF]:

B(D 0 → π + π -e + e -) expected ∼ 1.3 × 10 -6 .
The FCNC non-resonant contribution is estimated to be around 10 -9 -10 -10 . At low-dielectron masses, bremsstrahlung contributions, where a photon is radiated from the weak interaction vertex converting into a dielectron pair, can enhance the branching fraction to ∼ 5.3 × 10 -7 [START_REF] Cappiello | Standard Model prediction and new physics tests for D 0 → h + hℓ + ℓ -(h = π, K : ℓ = e, µ)[END_REF].

Due to the higher mass of the kaons compared to pions, D 0 → K + K -e + e -decays have a substantially smaller phase space. The two kaons mainly proceed via the ϕ intermediate resonance while the ρ and ω resonances dominate the dielectron spectrum, as shown in Figure 2.5 (right). The total expected branching fraction has been estimated to be [START_REF] Cappiello | Standard Model prediction and new physics tests for D 0 → h + hℓ + ℓ -(h = π, K : ℓ = e, µ)[END_REF]:

B(D 0 → K + K -e + e -) expected ∼ 1.1 × 10 -7 .
The non-resonant contributions is estimated to be O(10 -10 ), while the bremsstrahlung effect ∼ 5.4 × 10 -7 .

Experimentally, these decays have not been observed yet. Currently, the world's most stringent limits on the BF of D 0 → h + h -e + e -decays have been set by BESIII [START_REF] Ablikim | Search for the rare decays D → h(h ( ′ ) )e + e[END_REF] in the order O(10 -5 ), as shown in Table 2.1.

Channel

BF upper limit D 0 → π + π -e + e -< 0.7 × 10 -5 at 90 % CL D 0 → K + K -e + e -< 1.1 × 10 -5 at 90 % CL Table 2.1: Experimental status of rare charm four-body decays with electrons in the final states [START_REF] Ablikim | Search for the rare decays D → h(h ( ′ ) )e + e[END_REF]. Upper limits are stated with 90% confidence level.

The D 0 → h + h -µ + µ -decays have already been observed by LHCb [START_REF] Aaij | Observation of D 0 meson decays to π + πµ + µand K + Kµ + µfinal states[END_REF]. The results of the BF measurements are reported in Table 2.2 in both integrated and differential regions of the dimuon mass. Assuming the LFU to be conserved, this is also a benchmark for what it is expected for

D 0 → h + h -e + e -decays. Channel Total low mass η ρ/ω ϕ high mass [×10 -8 ] [×10 -8 ] [×10 -8 ] [×10 -8 ] [×10 -8 ] [×10 -8 ] D 0 → π + π -µ + µ -96.4 ± 12 7.8 ± 2.1 < 2.4 at 90 % CL 40.6 ± 5.7 45.4 ± 5.9 < 2.8 at 90 % CL D 0 → K + K -µ + µ -15.4 ± 3.2 2.6 ± 1.3 < 0.7 at 90 % CL 12.0 ± 2.7 Table 2.2: Measured BF for D 0 → h + h -µ + µ -
decays in single dimuon bins and integrated [START_REF] Aaij | Observation of D 0 meson decays to π + πµ + µand K + Kµ + µfinal states[END_REF]. Statistical and systematic uncertainties are added in quadrature. Upper limits are stated with 90% confidence level.

The Cabibbo favored mode D 0 → K -π + e + e -is expected to be dominated by SM amplitudes and not sensitive to NP effects. Recently, the BaBar collaboration published the first observation of the D 0 → K -π + [e + e -] ρ 0 /ω decay in the ρ/ω dilepton mass range 675-875 MeV/c 2 . The measured branching fraction is B(D 0 → Kπ + [e + e -] ρ 0 /ω ) = (4.0 ± 0.5) × 10 -6 and compatible with the SM expectations [START_REF] Cappiello | Standard Model prediction and new physics tests for D 0 → h + hℓ + ℓ -(h = π, K : ℓ = e, µ)[END_REF].

Assuming a SM rate, the available statistics accumulated by BaBar and BESIII is expected to be insufficient to make further observations in the four-body dielectron modes. Due to the huge charm production cross-section at the LHC, LHCb has a unique physics reach in studying these decays.

Chapter 3

The LHCb experiment

The chapter discusses the LHCb experiment, starting from a brief description of the LHC accelerator complex. The detectors composing the LHCb experiment and their performances are introduced and presented. The reconstruction of electrons is described, as well as the challenges involved when performing an analysis which includes electrons, such as the search for D 0 → h + h -e + e -decays.

The Large Hadron Collider at CERN

The Large Hadron Collider (LHC) is the largest and most powerful particle collider in the world. It was designed and built by the Conseil Européen pour la Recherche Nucléaire (CERN) between 1998 and 2008. The CERN organisation includes more than 10000 scientists from hundreds of universities and laboratories from all over the world [START_REF] Evans | LHC Machine[END_REF]. The accelerator is located in a tunnel 26.7 km long, 3.8 m wide and with a depth ranging from 50 to 175 m under the France-Switzerland border near Geneva. The tunnel hosts two adjacent rings with counter-rotating beams in order to be able to collide protons in four crossing points where the major experiments are located:

• Two general purpose detectors designed to study collisions producing high transverse momentum particles: ATLAS (A Toroidal LHC ApparatuS) and CMS (Compact Muon Solenoid);

• LHCb (Large Hadron Collider beauty), a general purpose detector in the forward region, optimised for the study of b and c hadron decays;

• ALICE (A Large Ion Collider Experiment), a detector designed to study the quark-gluon plasma with heavy ion collisions.

Inside the tunnel, a total of 1232 superconductive dipole magnets are used to keep the beams in a circular trajectory, while 392 quadrupole magnets are necessary to keep the beam focused. Stronger quadrupoles are placed near the colliding points to steer and further squeeze the beams to achieve maximal luminosity at the interaction point.

The particles are accelerated in different steps, see Figure 3.1. They are first accelerated in the linear machine LINAC4 reaching a beam energy of 50 MeV. The protons get then injected to the first circular accelerator, the Proton Synchrotron Booster (PSB), then to the Proton Synchrotron (PS) reaching 26 GeV. Finally, the Super Proton Synchrotron (SPS) prepares and boosts the energy of the proton beams to 450 GeV to be ready to get injected into the LHC ring. Proton bunches circulating in the LHC ring are composed of 1.15 × 10 11 protons separated from each other by 25 ns×c, with a collision rate of 40 MHz.

The LHC started its operation in 2010 delivering beams at 3.5 TeV of energy and reaching an instantaneous luminosity peak of 4 × 10 32 cm -2 s -1 , during the Run1 period (2010-2012). After a first long shutdown of 2 years, LS1, LHC increased the beam energy to 6.5 TeV (13 TeV total center-of-mass energy) during the period named Run2 (2015-2018). After a second long shutdown of 3 years, LS2, the accelerator reached the record center of mass energy of 13.6 TeV in July 2022 with the start of Run3. The LHC will run for close to four years at this energy, providing greater precision and discovery potential than ever before.

The LHCb detector

LHCb is a single-arm spectrometer with a forward angular acceptance which covers the range 10-300 (10-250) mrad in the horizontal (vertical) plane [START_REF] Alves | The LHCb detector at the LHC[END_REF]. The geometry of the detector is optimised to study heavy-flavour hadron decays which, at high energy, are produced in a small cone around the beamline. In fact, at the LHC pp inelastic collisions produce bb pairs through the gluon-gluon fusion and quarkantiquark annihilation mechanisms, shown in Figure 3 The number of events with a bb pair can be calculated as:

N bb = σ(pp → bbX) × L int , (3.1)
where σ is the cross-section and L int is the integrated luminosity. The cross-section σ(pp → bbX) for 13 TeV collisions in the LHCb pseudorapidity range is 154 ± 14µb over the full range of b-hadron transverse momentum [START_REF] Aaij | Measurement of the b-Quark Production Cross Section in 7 and 13 TeV pp Collisions[END_REF]. During Run1 and Run2 data takings, LHCb ran at an instantaneous luminosity of 2-4 ×10 32 cm -2 s -1 , one order of magnitude lower than ATLAS and CMS, see Figure 3.4. LHCb levels the luminosity at a constant value during a full LHC fill by adjusting the beam overlap. This maximises the integrated luminosity collected during the fill while maintaining the same trigger configuration and reduces systematic uncertainties. Furthermore, for most of the LHCb physics programme, collecting data at the same luminosity of ATLAS and CMS is not beneficial, as some trigger selection rates do not scale with luminosity, see the LHCb upgrade discussion in section 5.1. The integrated luminosity collected by LHCb during Run 1 and 2 is about 9 fb -1 . The charm cross-section for 13 TeV collisions in the LHCb pseudorapidity range is σ(pp → ccX) = 2940 ± 240 in a range of c-hadron transverse momentum below 8 GeV/c [START_REF] Aaij | Measurements of prompt charm production cross-sections in pp collisions at √ s =13 TeV[END_REF]. It is almost 20 times larger than bb one, making the LHCb experiment a perfect environment to study charm decays. In particular, the rare charm decays discussed in the thesis particularly benefit from it. Furthermore, the flexibility of the trigger and the excellent particle identification capabilities of the LHCb detector are key elements when studying such rare decays, as explained in the next sections. O,x,y,z) is centered at the collision point, with the z axis along the beam line and the y-axis along the vertical. Tracks produced in the LHCb acceptance are bent by a dipole magnet with magnetic field lines along the y-direction. This defines the bending plane as (x-z) and non-bending one as (y-z). The tracking system is composed of several sub-detectors:

• the VErtex LOcator (VELO), a silicon-strip vertex detector surrounding the beam interacting region;

• the Tracker Turincensis (TT), a large silicon-strip detector located upstream of the magnet;

• the T-stations, three stations of detectors placed downstream of the magnet and each composed of two separate subdetectors located in different regions: the Inner Tracker (IT), a silicon micro-strips detector, and the Outer Tracker (OT), a drift-tube gas detector.

Thanks to this complex tracking system, the momentum of charged particles is measured with a relative uncertainty of less than 1 %. Furthermore, the impact parameter, the minimum distance of a track to a pp collision, resolution of (15 ± 29/p T ) µm [START_REF] Steinkamp | on behalf of the[END_REF] achieved by the VELO is fundamental in the identification of displaced band cdecay vertices. Charged hadron identification is performed by two Ring Imaging Cherenkov Detectors (RICH). The first detector (RICH1) is located upstream of the magnet and designed for low momentum particles, from 2 to 60 GeV/c, while RICH2 is located downstream of the magnet and is optimised for momenta from 15 to 100 GeV/c of the particles. Both detectors use gas radiators and are designed to mainly separate pions, kaons and protons, but are used also for electron and muon identification. The particle identification ability of the LHCb detector is a key feature for flavour physics studies. A calorimeter system is placed downstream of RICH2 and provides information on the energy deposited by electrons, photons, charged and neutral hadrons as well as their spatial coordinates. Lastly, five muon stations in most downstream region are used to detect muons produced in the collisions. Part of the information from the subdetectors is quickly processed by a low-level hardware trigger (L0) for a first selection of interesting events, then a high-level software trigger (HLT) looks for more complex signatures.

The tracking system

At the LHCb experiment, the tracking subdetectors are positioned both downstream and upstream of the magnet in order to measure the momenta of the charged particles. LHCb uses a dipole magnet composed of two symmetric coils arranged above and below the beam axis [START_REF]LHCb magnet: Technical Design Report[END_REF]. The magnetic bending power of 4 T•m and peak strength of 1.1 T (see Figure 3.6) allow measuring with an excellent precision the momentum of the particles. In fact, a charged particle is bent in the (x-z) plane when passing through the magnetic field and the curvature of the trajectory is proportional to the momentum of the particle. The polarity of the magnetic field is reversed frequently during the data taking in order to keep under control systematic effects on the tracking and charge asymmetries in CP-violation measurements.

Figure 3.6: Distribution of the magnetic field strength across the full length of the detector. The different types of tracks are displayed [START_REF] Alves | The LHCb detector at the LHC[END_REF].

The LHCb tracking system is designed to reconstruct different types of tracks, as shown in Figure 3.6. The most relevant for physics analysis are the so-called long tracks which originate from a vertex in the VELO and traverse the whole tracking system. Downstream tracks are very important for long-lived particle studies, such as the K 0 S or Λ particles. They are produced from a vertex downstream of the VELO and leave hits in the TT and T-stations. Furthermore, the LHCb detector is able to reconstruct upstream tracks, leaving hits only in the VELO and TT, and T-tracks with signatures only in the T-stations.

VErtex LOcator

The VELO detector surrounds the beam interaction point. Its main goal is to locate primary vertices (PV) from pp collisions and secondary vertices (SV) from long-lived beauty and charm hadrons decays to allow for measurements of their decay times and impact parameters (IP). This information is crucial to discriminate signatures of heavy-flavour decays from background. An excellent vertex resolution is required for CP violation measurements, e.g. to study fast B 0 s -B 0 s mesons oscillations. Therefore, a good PV resolution as a function of number of tracks (N tracks ) and IP resolution as a function of p T are strong indicators of the excellent performance of the detector. The distribution of primary vertex and impact parameter resolutions are shown in Figure 3.7. The IP resolution linear dependence on 1/p T is a consequence of multiple scattering and the geometry of the vertex detector. At high-p T , the IP resolution is dominated by the PV resolution rather than track reconstruction one, as high-p T particles undergo little multiple scattering while the PV reconstruction is mainly dependent on the number of tracks associated to it. The VELO covers a length of around 1 meter and consists of 42 double-sided modules [START_REF]Technical Design Report[END_REF]. Each module is composed of two layers of silicon micro-strip sensors, one measuring the radial distance from the beam axis (R-sensor) and the other measuring the azimuthal angle (Φ-sensor), as shown in Figure 3.8. The detector was installed inside the LHC vacuum vessel and the sensors are separated only by a 300 µm thin aluminum foil from the LHC vacuum inside a secondary vacuum. This is done in order to protect the sensors from the radio-frequency waves produced by the beam and at the same time to minimise the material traversed by charged particles before they cross the sensors. During the injection of the LHC beams, the VELO modules are moved by 29 mm in the horizontal direction opposite to the beam pipe and subsequently moved back once the beam is declared stable, to reduce the amount of radiation interacting with the sensors. Once the VELO is closed, the closest active strip is at 8.2 mm from the beam. A scheme of the detector layout is shown in Figure 3.9.

The Tracker Turincensis

The TT upstream tracker detector consists of four planar layers of silicon microstrips, shown in Figure 3.10, separated by 27 cm along the z direction. A total of 143 thousand strips are present, with a length of 10-40 cm and pitch of 183 µm, achieving an excellent position resolution of around 50 µm in the bending plane [START_REF]LHCb inner tracker: Technical Design Report[END_REF]. The strips are placed vertically to exploit the fact that the tracks are bent in the horizontal plane by the dipole magnet, covering a total active area of 8 m 2 . Shorter strips are employed in the inner region, where the particle density is the highest, while longer strips are used in the outer region. The four detection layers are arranged in the stereo layout x-u-v-x, with vertical strips in the x layers, while strips are rotated by a stereo angle of -5 • and +5 • for the u and v layers respectively. Thanks to this, the combination of the u and v measurements allows to extract the y position of the track and provides the 3D information needed for the full reconstruction. The TT is crucial when reconstructing downstream tracks and helps the long track reconstruction by providing hit measurements before the magnet.

The T-stations

Three tracking stations (T1-T3) are located downstream of the magnet. As the particle occupancy is much higher in the central region compared to the outer one, two Charged particles require a minimum momentum of 2 GeV to reach the T-stations, forming a long track. These tracks are very important for physics analysis and often used to measure performance of the LHCb tracking system. The reconstruction efficiency for these type of tracks as a function of momentum is shown in Figure 3.12 reaching an efficiency above 94 %. Furthermore, the figure shows the momentum resolution distribution as a function of momenta, with values below 1 % for a wide range of momenta.

The particle identification system

Particle identification (PID) at the LHCb experiment is ensured by three detectors: the Ring Imaging Cherenkov detectors (RICH), a calorimeter system and the muon stations. PID is a key element for flavour physics studies and its performance is discussed in this section.

Ring Imaging Cherenkov detectors

The RICH detectors are particularly important to distinguish between different types of hadrons, i.e pions, kaons and protons. They rely on the Cherenkov effect: a particle propagating through a material, called radiator, of refractive index n, with a velocity v larger than the speed of light in this medium, c/n, emits photons in a cone The mass of the particle m is then evaluated from the momentum estimate provided by the tracking system, as m = p/v, with the velocity v retrieved by θ measurement from the RICH detector. Two RICH detectors are responsible for charged hadron identification in the momentum range 2-100 GeV/c, see Figure 3.13 [START_REF]LHCb RICH: Technical Design Report[END_REF]. The first detector, RICH1, is located upstream of the magnet between the VELO and the TT and uses C 4 F 10 gas as radiator with refractive index n = 1.0014. It is designed to identify lower momentum particles from 2 to 60 GeV/c covering the full LHCb acceptance. The downstream detector, RICH2, located between the T-stations and the muon system, covers the 15-200 GeV/c momentum range with a limited angular acceptance ranging from 15 to 120 mrad horizontally and 100 mrad vertically, covering approximately a quarter of the solid-angular acceptance of the spectrometer. In fact, only higher momentum particles reach RICH2 as softer ones are bent out of the acceptance by the magnet. Around 70 % of long tracks with p > 15 GeV/c are in the RICH2 acceptance, while more than 95 % above 60 GeV/c. The RICH2 detector uses CF 4 gas as radiator with a refractive index n = 1.0005.

Both RICH detectors use a system of plane and spherical mirrors to reflect and focus the rings of Cherenkov photons to detection planes where Hybrid Photon Detectors (HPDs) are installed. The HPDs are position-sensitive sensors designed to detect photon rings and measure their radius, thus the θ Cherenkov angle. The distribution of the Cherenkov angle as function of the momenta of the particles is shown in Figure 3.14.

The calorimeter system

The calorimeter system measures the energies and the positions of clusters deposited by electrons, photons and hadrons. This information is used to select candidates with high deposited transverse energy for the L0 trigger and for particle identification. Several subdetectors compose the calorimeter: Scintillating Pad Detector (SPD), PreShower (PS), the electromagnetic calorimeter (ECAL) and the hadronic The ECAL is a heterogeneous lead/scintillator sampling calorimeter located after RICH2, designed to measure the energy of photons and electrons. The separation between photons and electrons is made by the SPD and PS located before the ECAL. A thickness of lead layer between the SPD and PS is chosen in a way that photons start the electromagnetic shower in it. In addition electrons leave hits also in SPD which allows their distinction from photons. Once this separation is achieved, the energies of these particles are determined by the ECAL with an energy resolution of [START_REF] Beteta | Calibration and performance of the LHCb calorimeters in Run 1 and 2 at the LHC[END_REF]:

σ E E = 10 % √ E ⊕ 1 %, (3.3)
with a stochastic term of 10 %, due to fluctuations related to the physical development of the shower, and a 1 % constant term, corresponding to mis-calibrations, non-linearities or leakages of the detector. Neutral and charged hadrons deposit their remaining energy in the HCAL. The detector is composed of alternating layers of scintillating tiles and iron absorbers. The measured energy resolution is [START_REF] Beteta | Calibration and performance of the LHCb calorimeters in Run 1 and 2 at the LHC[END_REF]:

σ E E = 70 % √ E ⊕ 10 %, (3.4)
with a stochastic term of 70 % and constant term of 10 %. Different segmentation schemes were designed in the calorimeter system due to the different occupancy of the detector between the inner and outer regions. The SPD, PS and ECAL detectors are divided into three regions, while the HCAL only in two, see Figure 3.16. Figure 3.16: Regions of different granularity for the ECAL (left) and HCAL (right) [START_REF]LHCb calorimeters: Technical Design Report[END_REF].

The muon detector

The LHCb muon system consists of five stations M1 to M5 with the goal of identifying and triggering on muons, see Figure 3.17 [START_REF]LHCb muon system: Technical Design Report[END_REF]. The five stations are made of Multi Wire Proportional Chambers (MWPC) except for the inner part of M1, where the particle flow is higher, which is made of triple Gas Electron Multiplier (GEM). The first station is located before the calorimeter in order to improve the transverse momentum resolution of the muon trigger. The M2 to M5 stations are located after the HCAL and each interleaved by 80 cm of thick iron absorbers, therefore only muons with a minimum momentum of 6 GeV/c can cross the whole system. Each of the muon stations is divided in four regions (R1-R4) of different granularity, having a finer segmentation closer to the beam pipe and coarser in the outer regions.

Particle identification performance and variables

Information from the tracking system, the RICH detectors, the calorimeter system and the muon chambers are gathered and combined to provide a particle identification likelihood for each track. The hadron identification relies on the π/K/p separation evaluated by the two RICH detectors. As mentioned in section 3.2.2, electrons and photons are identified and distinguished using the SPD, the PS and the ECAL detectors. The muon chambers isolate muons from all the previously mentioned particles. The RICH information is also used to help identify electrons and muons. Figure 3.17: Side view of the muon system (left). Layout of a single muon station displaying the four regions of granularity (right) [START_REF]LHCb muon system: Technical Design Report[END_REF].

A PID likelihood is evaluated for each particle hypothesis using the information from the described detectors. For example, the total pion π likelihood is defined as:

L(π) = L RICH (π) × L CALO (!e) × L MUON (!µ), (3.5)
with L RICH/CALO/MUON , respectively, the RICH, the calorimeter or the muon likelihood values for the enquired particle to be identified, correspondingly, as a pion, or not identified as an electron e or a muon µ.

In order to define a global PID variable, all particle hypotheses are compared to the pion one, the most abundant charged particle at LHCb. A log-likelihood difference ∆LL for any particle h = π, K, µ, e is evaluated as follows:

PID(h) = ln L(h) -ln L(π) = ln L(h) L(π) . (3.6)
The kaon and electron PID performances in terms of efficiency of correct identification and mis-identification are shown in Figure 3.18 as a function of momentum. In order to improve the PID performance, LHCb has introduced a neural-networkbased PID variable, called ProbNN [START_REF] Derkach | Machine-Learning-based global particle-identification algorithms at the LHCb experiment[END_REF]. The information from the tracking system, the calorimeter, the RICH detectors and the muon chambers is combined in an algorithm trained on simulation samples to identify each type of particle. The output of the neural network is used to identify a given track as a K, π, e, p or µ particle. The advantage compared to the standard ∆LL PID variables is that the ProbNN variables take into account correlations among the various subdetectors and various ∆LL, resulting in better performances as shown in Figure 3

Data flow at LHCb

In order to process and store data from the proton-proton collisions at the 40 MHz bunch-crossing rate of the LHC, a data flow is in place to reduce the full rate by three orders of magnitude and store the relevant data to disk. The LHCb trigger system performs the data filtering and selection down to 12 kHz of output rate. It exploits typical bor c-hadron signatures to select interesting events. Several offline applications are further used to provide data to the analyst.

In order to keep the data-simulation differences as small as possible, simulated samples undergo the same processing chain as real data.

The trigger system

The LHCb trigger system consists of a hardware L0 and software high-level trigger HLT stages. The hardware trigger applies very basic selections in order to filter the 40 MHz bunch-crossing rate of the LHC, down to about 1 MHz from which the LHCb subdetectors can be read out. The high-level trigger runs on the Event Filter Farm (EFF) and it is split into two further steps: HLT1 and HLT2. The first stage, HLT1, performs a partial event reconstruction and applies selection constraints on the reconstructed tracks. The second stage, HLT2, performs the full event reconstruction, looking for more complex event signatures. The various trigger classifications, so-called trigger lines, are implemented with different set of requirements and adjustable rates. These settings are stored in a Trigger Configuration Key (TCK) that is unique for each configuration. During Run1 the p T threshold, below which tracks are not reconstructed in HLT1, was tighter due to throughput constraints. This reduced the efficiency of selecting charm decays which have generally particles with lower momenta compared to bdecays. Furthermore, this meant the data had to be re-reconstructed offline to be able to reach the final precision, resulting in differences between the online and offline selections. The high-level trigger was redesigned and optimised for Run2. In Run2, momentum thresholds could be lowered and a new line selecting two-track vertices was introduced, increasing the efficiency for band c-hadrons with low transverse momentum. This means events could be reconstructed with offline-like quality at HLT2 level. The real-time alignment and calibration greatly contributed in this sense. It is performed run-by-run on the events selected by HLT1 and stored to a buffer before being reconstructed by HLT2.

The trigger scheme of the LHCb experiment is illustrated in Figure 3.20. The final goal of the trigger system is to reduce the incoming data rate down to about 12.5 kHz, around 0.6 GB/s of data to be saved in storage.

L0 hardware trigger

The first stage L0 trigger is fully implemented in hardware and designed to operate synchronously with the interaction rate. Its goal is to select events with high transverse energy (E T ) clusters in the calorimeter or high momentum muon tracks. In fact, events including heavy-flavour decays are characterised by high transverse energy and momentum signatures compared to QCD backgrounds.

The calorimeter system searches for high E T clusters of 2 × 2 cells in the ECAL and HCAL. The E T of the candidate cluster is compared to a threshold, see Table 3.1, and if the event contains at least one candidate above the threshold, it is accepted.

The calorimeter information is used to distinguish hadrons, electrons and photons, firing the corresponding L0Hadron, L0Electron and L0Photon lines. Limited information is available to separate electrons and photons within the L0 system. The identification relies on the amount of electromagnetic showering in the PS detector, before photons and electrons reach the ECAL, and on possible hits in the SPD detector. The L0Photon trigger line working point is set to achieve high purity but relatively low efficiency. Many real photons are selected by the L0Electron selection which instead has a high efficiency with lower purity.

The L0 muon trigger selects events with muon tracks having a minimum p T . The transverse momentum of each track is estimated assuming the particle associated with it originates at the interaction point and crosses the five muon stations. For each event, the two muon candidates with highest p T are used to form a trigger decision based on the highest one (L0Muon) or both (L0DiMuon). In order to boost the efficiency of finding B 0 s →µ + µ -decays, very loose occupancy cuts are used for the L0DiMuon decision, in particular on the number of SPD hits. Muon p T requirements are shown in Table 3.1 for different years of data taking.

In all hardware triggers, except the L0DiMuon line, busy events with the number of SPD hits > 450 are rejected in order to avoid saturation in the HLT processing farms in Run2 [START_REF] Callot | Using the SPD multiplicity in the Level-0 trigger[END_REF].

year L0Muon [ GeV/c ] L0Electron [ GeV ] L0Hadron [ GeV ] 2015 p T > 2.80 E T > 2.70 E T > 3.60 2016 p T > 1.80 E T > 2.40 E T > 3.70 2017 p T > 1.35 E T > 2.11 E T > 3.46 2018 p T > 1.75 E T > 2.38 E T > 3.75
Table 3.1: L0 trigger lines thresholds for the Run2 data taking years.

High Level Trigger

Events selected by the L0 trigger are further processed by the two HLT stages. The HLT trigger lines are composed of a set of reconstruction and selection steps with access to the full detector information, allowing for more complex decisions compared to the L0 trigger. This high computational effort is performed by the Event Filter Farm (EFF) consisting of 1700 nodes. The first software trigger stage, HLT1, reduces the L0 input data rate by a factor 20 [START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF] for Run1 (Run2) selecting events after performing a partial reconstruction of the event. In fact, at this stage it is enough to reconstruct high p T and high p displaced tracks, typical signatures of B and D meson decays. A full three-dimensional pattern recognition algorithm is run to reconstruct tracks in the VELO, which can be extrapolated to the beamline to find primary vertices (PV). Selected VELO tracks are further extrapolated to the T-stations forming long tracks and their momentum is measured. With this information, the following decisions are mainly used for analysis:

• HLT1TrackMVA: requiring a single-track with high p T displaced from all PVs;

• HLT1TwoTrackMVA: selecting two-track vertices with large p T and displacement from any PV. This selection line was added during the Run2 data taking to improve the heavy-flavoured decays selection performance. Event passing the HLT1 trigger are streamed to the second high level trigger stage, HLT2, where a full reconstruction is performed. Here, the whole detector information is used, including data coming from calorimeter and RICH detectors. Exclusive and inclusive decay modes are selected and stored by the HLT2 trigger lines reaching a 12 kHz data output rate in Run2. Inclusive selections rely on the presence of heavily displaced two-, three-, four-particles vertices, while exclusive ones aim to select decay channels with requirements on the PID of the final state particles. As shown in Figure 3.20, between HLT1 and HLT2, events are buffered to disk while performing an online detector calibration and alignment. This is critical to achieve the best possible reconstruction at HLT2 level with updated detector information. Two types of HLT output, so-called streams, are used:

• Full stream, where the entire raw event is saved. This allows for the data to be reprocessed at a later stage, improving the reconstruction or selecting new decay channels. These datasets are usually very large and a set of preselection requirements, so-called Stripping, is applied to reduce their size. For a given analysis, decay channels are usually extracted by the analyst from the output of the Stripping.

• Turbo stream, by saving only partially the raw detector and event information directly from the HLT2 output. Several configurations are available, for example, saving only the signal candidates or only pre-defined interesting parts of the event. The Turbo stream is mainly composed by exclusive lines selecting heavy-flavour decays. Furthermore, this format was used to collect dedicated calibration samples to be used for tracking and PID data-simulation corrections.

The offline software framework

The full LHCb software, both online and offline, is executed within the GAUDI framework [START_REF] Barrand | GAUDI -The software architecture and framework for building LHCb data processing applications[END_REF]. It is used for generation and data processing ensuring the availability of all LHCb tools and algorithms across the experiment by enforcing a global consistency of all LHCb analyses. The main components of the data processing are visualised in Figure 3.22:

• Generation and simulation: The Gauss package [START_REF] Clemencic | The LHCb simulation application, Gauss: Design, evolution and experience[END_REF] is responsible for the generation of simulated Monte Carlo (MC) events. Primary proton-proton interactions are simulated using PYTHIA [START_REF] Belyaev | Handling of the generation of primary events in Gauss, the LHCb simulation framework[END_REF], while decays of band c-hadrons are generated with the EvtGen package [START_REF] Lange | The EvtGen particle decay simulation package[END_REF]. The Geant4 [START_REF] Allison | Geant4 developments and applications[END_REF] toolkit is further used to simulate the interaction of particles with the detector;

• Digitization: The Boole application [97] emulates the LHCb detector response of the simulated event. The digitised output is the MC-equivalent of the real data detector response;

• Trigger: Both simulated and real data are processed through the three stages of the trigger (L0, HLT1 and HLT2) as explained in section 3.2.3, by the Moore framework [98];

• Reconstruction: Signal hits in the detector are clustered, and the full reconstruction is performed by the Brunel application offline [99]. Particle properties (i.e. momentum, PID probability variables, ...) are evaluated and stored in the so-called Data Storage Tape (DST) files;

• Analysis: The DaVinci software package [100] uses the information saved in the DST files to evaluate several kinematical and topological variables while performing the event selection. Determination of quantities like the invariant masses of the decayed particles, their distance of flight or the decay times is all done in this step. The application allows the production of ROOT Ntuple output data files [START_REF] Brun | ROOT -An object oriented data analysis framework[END_REF] on which are performed physics analyses. This data processing chain is used to produce simulated samples of the decay channels D 0 → π + π -e + e -, D 0 → K -π + e + e -and D 0 → K + K -e + e -which are used in the rare charm analysis presented in the next chapter. Their decay chain is produced with the EvtGen package, where a sum of resonant and non-resonant contributions is assumed, see Table 3.2. Simulated samples are used in the analysis to estimate the reconstruction and selection efficiencies and determine the mass shapes used when fitting the data, as further explained in chapter 4.

Channel

Component Fraction D 0 → K -π + e + e -K * (→ K -π + )ρ(→ e + e -) 0.16

K * (→ K -π + )ω(→ e + e -) 0.16 K * (→ K -π + )e + e - 0.16 K -π + ρ(→ e + e -) 0.16 K -π + ω(→ e + e -) 0.16 K -π + e + e - 0.16 D 0 → K + K -e + e -ϕ(→ K + K -)ρ(→ e + e -) 0.25 ϕ(→ K + K -) → e + e - 0.25 K + K -ρ(→ e + e -) 0.25 K + K -→ e + e - 0.25 D 0 → π + π -e + e -ρ(→ π + π -)ρ(→ e + e -) 0.125 ρ(→ π + π -)ω(→ e + e -) 0.125 ρ(→ π + π -)ϕ(→ e + e -) 0.125 ρ(→ π + π -)e + e - 0.125 π + π -ρ(→ e + e -) 0.125 π + π -ω(→ e + e -) 0.125 π + π -ϕ(→ e + e -) 0.125 π + π -e + e - 0.125

Electron reconstruction at LHCb

Electrons behave in a very different way with respect to hadrons or muons which makes their study experimentally more challenging. A good understanding of their behaviour is crucial for the work presented in this thesis. Three main factors affect the reconstruction and identification of electrons:

• Trigger: the L0 trigger selects candidates with electrons depositing high transverse in the calorimeter. These energy thresholds are higher for example compared to the muon momentum requirements at the L0 stage, as reported in section 3.2.3. This is mainly due to the higher occupancy of the ECAL detector compared to the muon chambers.

• Resolution: due to their low mass, electrons produce a large amount of bremsstrahlung photons when interacting with the material of the detector, which results in a worse momentum and energy resolution.

• PID: the poorer momentum estimate of the electrons, caused by the bremsstrahlung, makes the particle identification information less reliable.

Calorimeter response and electron identification

The electron identification is mainly controlled by the RICH detector and the calorimeter system. As mentioned in section 3.2.2, electrons are separated from photons and hadrons in the calorimeter thanks to their different signatures of the produced showers. In short, electrons deposit hits in SPD detector, which makes it possible to separate them from photons, and do not leave any hits in the HCAL, which helps distinguish them from hadrons. Furthermore, the energy deposited by a particle in the ECAL detector over the track momenta (E/p), is a powerful discriminating variable against charged hadrons. Indeed, as shown in Figure 3.23, the E/p distribution for electrons is very close to unity, with some resolution effects, while hadrons deposit only a little fraction of their energy in the ECAL detector, shifting the E/p distribution towards low values.

Figure 3.23: Energy deposited on the ECAL over track momenta (E/p) distribution for electrons (red) and hadrons (blue) [START_REF] Aaij | LHCb Detector Performance[END_REF].

Bremsstrahlung

Because of their very light mass, the probability of an electron to emit bremsstrahlung radiation when interacting with the material of the detector is higher compared to muons and hadrons, which is almost negligible in comparison. The photons are emitted preferentially collinearly to the direction of the electron track reducing the original energy of the electron. Bremsstrahlung effects complicate the reconstruction and momentum measurements of these particles. If the radiation is emitted before the magnet, for example when the electron interacts with the material of the VELO, RICH1 or TT, the photon trajectory follows the VELO or TT track direction of flight, while the electron is bent by the magnetic field. Therefore, the electron and the photon clusters in the ECAL are detected in different positions, as illustrated in Figure 3.24. Both the measured momentum and energy deposited in the ECAL of the electron are lower than the original ones, which means the E/p distribution mentioned in the previous section is not affected. Nevertheless, this effect is seen on the invariant mass resolution of decays involving electrons, which is deteriorated.

In order to improve this last aspect, a dedicated bremsstrahlung recovery algorithm was implemented within the LHCb software: the BremAdder algorithm. It extrapolates both the VELO and TT track associated with the electron to the ECAL detector, in order to open a search window and look for clusters compatibles with a Figure 3.24: Schematic representation of the bremsstrahlung photon emission by an electron before and after the magnet. The BremAdder algorithm searches for photon signatures in the ECAL in a search window defined by the VELO and TT track extrapolations to the calorimeter. photon hypothesis, see Figure 3.24. In this way, the bremsstrahlung photon candidate energy can be added back to the four-momentum of the electron.

However, given the strict selection thresholds of the algorithm (p T photon > 75 MeV/c) which is mainly due to the high occupancy of the ECAL detector, the correct bremsstrahlung photon is not always found. Furthermore, a random wrong photon can be associated with the electron candidate adding its energy to the fourmomentum of the particle. Even if the correct photon is found, its energy estimate is limited by the ECAL resolution. All these effects significantly degrade the invariant mass resolution of decays with electrons in the final state, which makes them complicated objects to analyse. The mass distributions of simulated D 0 → K -π + e + e - and D 0 → K -π + [µ + µ -] ρ 0 /ω decays are compared in Figure 3.25a. The D 0 mass resolution for the electron mode is more than three times larger than for to the muon mode.

Roughly 50 % of the reconstructible photons (p T > 75 MeV/c) lie inside the calorimeter geometrical acceptance and do not share cells with other particles. The BremAdder algorithm achieves a 95 % purity of reconstructed photons within this ∼ 50 %, see Figure 3.26. If the radiation is produced after the magnet, the photon has a collinear direction to the electron track and thus reaches the same ECAL cells. Additionally, in this situation, the momentum of the electron is measured before its bremsstrahlung emission and is therefore not affected by it. Assigning bremsstrahlung photons to a track is a powerful electron identification signature.

The mass distributions of D 0 → K -π + e + e -decays is shown in Figure 3.25b comparing candidates with or without bremsstrahlung photons attached to any of the final state electrons. Not reconstructing the radiated photon creates a left-hand tail in the distribution due to the energy loss. If the correct bremsstrahlung photon is reconstructed, the mass distribution is re-centered around the D 0 mass peak. However, if not all the radiated photons are reconstructed, the mass distribution can feature a left-hand tail and furthermore, due to the wrong photon association and the limited calorimeter resolution, a right-hand tail is also present.

Chapter 4

Search for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays

The search for rare charm four-body decays with electrons in the final state is presented in this section. The author is the main responsible for all aspects of the data analysis, starting from the signal candidates selection, the accurate evaluation of the selection efficiencies from simulated samples and determination of the yields from a fit to the collision data, and finally extracting unbiased central values as well as accurate statistical and systematic uncertainties on the physical quantities of interest.

Analysis strategy

The goal of the analysis is to perform a search for rare charm D 0 → π + π -e + e -and D 0 → K + K -e + e -decays, as they have not been observed yet. The analysis focuses on determining their branching fractions or, if no significant signal is found, setting an upper limit on them.

In order to determine a branching fraction experimentally, both the yield of the total number of produced D 0 mesons and the fraction of them decaying into the signal final state are needed. This is quite challenging as the D 0 production rate must be measured with high precision, evaluating both the luminosity and the D 0 cross-section. An alternative approach consists in performing the branching fraction measurement relative to a normalisation channel. The normalisation channel is generally chosen to have very similar topology to the signal decay in order to profit from the cancellation of systematic uncertainties in the ratio. In this analysis, the D 0 → K -π + e + e -decay is chosen as normalisation channel. It has very similar topology compared to the D 0 → h + h -e + e -signal decays (h = π, K) and is expected to be dominated by SM amplitudes and not be sensitive to new physics, as discussed in section 2.3. The D 0 → K -π + e + e -decay has been observed by the BaBar collaboration and its branching fraction measured to be B(D

0 → K -π + [e + e -] ρ 0 /ω ) = (4.0 ± 0.5) × 10 -6 in the ρ 0 /ω region of dielectron mass 675 < m(e + e -) < 875 MeV/c 2 [103].
The following formula applies when measuring the signal decay rate

B (D 0 → h + h -e + e -) relative to the normalisation channel B (D 0 → K -π + [e + e -] ρ 0 /ω ): B(D 0 → h + h -e + e -) B(D 0 → K -π + [e + e -] ρ 0 /ω ) = N(D 0 → h + h -e + e -) N(D 0 → K -π + [e + e -] ρ 0 /ω ) ϵ(D 0 → K -π + [e + e -] ρ 0 /ω ) ϵ(D 0 → h + h -e + e -) , (4.1
) where N are the observed yields and ϵ are the selection efficiencies. The analysis uses the full Run 2 data set corresponding to 5.7 fb -1 , including the 2015-16 data taking years with 1.9 fb -1 and the 2017-18 ones with 3.8 fb -1 , recorded at a centre-of-mass energy of 13 TeV. The Run 1 data set is not used for the analysis as the exclusive trigger lines selecting the signal decays were not present.

In order to increase the sensitivity to beyond-the-standard-model effects, as explained in section 2.2, the measurement is performed in regions of dielectron mass. The dilepton mass bins are defined in Table 4.1 and correspond to the ones used in the search for D 0 → h + h -µ + µ -decays [START_REF] Aaij | Observation of D 0 meson decays to π + πµ + µand K + Kµ + µfinal states[END_REF]. Three regions are expected to be dominated by intermediate resonances (η, ρ/ω and ϕ) while the low and high mass bins are defined such that the resonance influence is minimal. The 'very low' mass region (< 211.32 MeV/c 2 ) is kinematically accessible only for the dielectron modes and it is not considered for the analysis to keep consistency with the muonic measurement. Due to the limited phase-space, the ϕ and 'high' mass regions are not accessible for

D 0 → K + K -e + e -decays.
D 0 → π + π -e + e - [ ] D 0 → K + K -e + e - [ ]
Table 4.1: Dilepton mass bins considered for the analysis. The last two bins are not available kinematically for D 0 → K + K -e + e -. Due to the smaller electron mass, an additional dilepton mass region ('very low') is kinematically allowed for dielectron modes, which is, however, not considered in this analysis to keep consistency with the muon modes [START_REF] Aaij | Observation of D 0 meson decays to π + πµ + µand K + Kµ + µfinal states[END_REF].

The branching fractions are measured in each dielectron mass bin, however, if no significant signal is found, an upper limit on the BF is set. The total branching fraction integrated over the dielectron-mass bins is computed at the end combining the single measurements.

The analysis is blind, meaning that it is performed keeping the signal yields and all mass spectra in the data samples unknown. Blinding is a common procedure in experimental high-energy physics in order to avoid any potential experimenter's bias on the measurement. The D 0 → K -π + e + e -data instead is not blinded, as it was already observed by the BaBar collaboration, and it is used to validate various parts of the analysis.

In order to perform the measurement of Equation 4.1, both ratios of efficiencies and yields must be evaluated. The selection efficiencies are determined applying the full signal selection defined in section 4.2 on simulated samples. Differences between simulation and collision data are accounted for with a correction procedure explained in section 4.3. Data control samples are used to determine/validate parts of the efficiencies which are known to be poorly reproduced in simulation.

The yields are determined through maximum likelihood fits to the D 0 mass candidates distributions after the full selection, as defined in section 4.4. Due to different mass shapes and background levels, a simultaneous fit is performed separating candidates without or with bremsstrahlung photons attached to the electrons during the reconstruction, denoted as brem0 and brem1 categories hereafter. The brem1 category includes all events with at least one bremsstrahlung photon attached to either of the electrons. Furthermore, to account for possible signal bin migration from one dielectron-mass bin to another due to reconstruction effects, the fit is made simultaneously between the dilepton-mass regions.

Dominant systematic uncertainties are related to yield and efficiency determination. The uncertainties are evaluated directly on the ratio of yields and efficiencies to profit from cancellations in Equation 4.1. Details on the studies of systematic uncertainties can be found in section 4.5.

The results are reported in section 4.6 evaluating the expected limits on each of the dielectron mass regions.

Signal selection

The signal and normalisation D 0 decays are required to arise from a D * + (2010) →D 0 π + decay chain, with the D * + meson produced directly in a pp interaction. The decay chain is illustrated in Figure 4.1. Selecting D 0 candidates produced from D * + decays is very beneficial to reduce background contaminations. The prompt D 0 cross-section from pp collisions is ∼ 4 larger than the production via D * decay. However, the expected background is ∼ 40 more abundant in prompt production. The measurement is expected to be dominated by the background with low signal yields which means that 40 times more background would significantly reduce the precision of the measurement.

The D * + charged meson decays via the strong interaction instantaneously at the PV into a D 0 meson and a charged pion. Due to the small mass difference between the D * (2010.25 ± 0.05 MeV/c 2 ) and D 0 (1864.84 ± 0.05 MeV/c 2 ) [START_REF] Zyla | Review of particle physics[END_REF], the pion has very small momentum, and is therefore often referred to as a soft pion π + s . The soft pion has on average a momentum of ∼ 7 GeV/c compared to the D 0 candidates with an average of ∼ 90 GeV/c. The D 0 meson decays via the weak interaction into the signal final states with a lifetime of 0.41 ps [START_REF] Zyla | Review of particle physics[END_REF], which means it flies on average 12 mm in the laboratory frame before decaying. This distance is sufficient to reconstruct the secondary vertex produced by the D 0 meson decay and distinguish it from the PV.

Expected backgrounds

All the relevant backgrounds must be understood before defining an efficient signal selection, such that the background rejection can also be optimised. The background contributions which can mimic the D 0 → h + h -e + e -signal can be separated in two categories: random combinations of tracks which accidentally pass the selection, known as combinatorial background, and physical backgrounds, where some final state particles are either not reconstructed or misidentified. Inside the physical background category, the most dominant contribution is given by the hadronic four-body decays misidentified as signal. These two main backgrounds are here presented in more detail:

• Combinatorial background: candidates composed of randomly associated tracks.

The complex decay topology imposes tight constraints on the kinematics, resulting in a strong combinatorial background suppression. However, this background cannot be reduced to a negligible level and must be considered in the next steps of the analysis. A dedicated multivariate analysis is performed to suppress it.

• Hadronic four-body decays: decays in the form of

D * + →D 0 (→ h + h -π + π -)
π + with a D 0 decaying into four hadrons and the pion pair misidentified as two oppositely charged electrons. The double pion-to-electron misidentification probability is low (O(10 -4 )), however, these decays have branching fractions several orders of magnitude larger than the signal, as reported in the first column of Table 4.4. In order to reduce the misidentified backgrounds, PID variable requirements can be applied to the electron candidates. An optimisation of such selection requirements is performed in the analysis. Furthermore, the bremsstrahlung acts as a particle identification variable for these misidentified decays. Pions rarely emit bremsstrahlung radiation, which means that if a photon is reconstructed and associated to a particle, this one must be a real electron. Random photon association can happen and is estimated to occur with a rate of O(5 %).

Hadronic four-body decays with three mis-identifications can also contribute as backgrounds, but are much more suppressed compared to the double mis-idenfication ones. The mis-identification of kaons to electrons is a factor 10 more rare than pion to electron. However, in order to suppress these backgrounds, particle identification requirements are added on the hadrons h, as shown in the analysis data selection. The hadron identification requirement helps also reducing the so-called crossfeed backgrounds, such as

D 0 → K -π + e + e -, D 0 → π + π -e + e -or D 0 → K + K -e + e -
decays mis-identified as another signal decay. The selection requirements suppress these backgrounds to an almost negliglible level, however a systematic uncertainty is assigned due to a small residual contribution expected (see systematic uncertainties section 4.5).

Partially reconstructed decays of type D * + →D 0 (→ h + h -η) π + with an intermediate resonance η →e + e -γ with a non-reconstructed photon γ, are also potential backgrounds in the search for D 0 → h + h -e + e -decays. However, these decays are mainly present in the low mass dilepton mass region, as the dielectron invariant mass can only be lower than the η mass (∼ 548 MeV/c 2 ).

Furthermore, D * + mesons not arising from pp collisions but from a secondary decay of a B hadron, for example B 0 →D * + µ -ν µ , known as secondary decays, are considered as background as they could increase the number of fitted signal events. Due to the relatively long lifetime of the B hadrons, the D * + production vertex is likely to be separated from the PV and these backgrounds suppressed by the decay chain reconstruction. Moreover, the rate of secondary decays is very similar for the normalisation D 0 → K -π + [e + e -] ρ 0 /ω and the signal D 0 → h + h -e + e -decays, therefore cancelling out in the ratio of signal yields. The effect is estimated to be below 0.3 % and is therefore neglected in the analysis.

A similar reasoning is considered for D 0 mesons arising directly from the pp collisions, referred as untagged decays. The rate of untagged decays of the D 0 meson is very similar for the normalisation and signal channels, with an effect below 0.1 % on the efficiency ratio evaluated on simulated samples.

All the neglected physical backgrounds not mentioned in this section have a small impact in the analysis and are considered in a dedicated systematic uncertainty study discussed in section 4.5.3.

Trigger and reconstruction

At LHCb, all events recorded to disk are filtered through three trigger stages. However, not all saved events are considered for the analysis. The trigger strategy is based on a limited number of trigger lines, which optimise the signal retention. Furthermore, this facilitates the trigger efficiency data-simulation corrections which can be measured with higher precision.

At L0, events are required to fall into one of the following two exclusive categories:

• L0Electron or L0Muon triggered independently of the signal (TIS), in the following referred to as 'L0 independent', LOI;

• if not satisfied by the previous requirement, triggered by L0Electron at least on one of the two electrons in the final state particles (TOS), called 'L0 lepton' LOL.

The signal candidates are split between the LOI and LOL categories and the separation is very useful when correcting for the L0 response difference between data and MC, see section 4.3.6. At HLT1, signal events are selected by a positive line response of Hlt1TrackMVA or Hlt1TwoTrackMVA on the signal D 0 candidates. These lines select events with highp T displaced two-tracks vertices or single tracks.

Finally, at HLT2 level, candidates are selected by an exclusive line optimised for the signal decay topology. The HLT2 selection performs the full reconstruction combining final state particles to build the decay chain

D * + →D 0 π + s with D 0 → h + h -e + e -.
First, only events with at least one reconstructed PV are considered. Four charged particles satisfying some loose track quality criteria (χ 2 /dof < 4) and not coming from a PV (IP χ 2 > 3) are selected to form a D 0 candidate. The IP χ 2 of the particle is defined as the difference between the χ 2 obtained when fitting the PV with and without the considered particle. Furthermore, some loose requirements on the momentum and transverse momentum of the final state tracks, p > 3 GeV/c and p T > 300 MeV/c, are imposed to get rid of a large amount of combinatorial background, generally tending to have particles with lower momenta. The four tracks must all be consistent with a common secondary vertex (SV) with a maximal distance of closest approach (DOCA) smaller than 0.3 mm from it.

The D 0 vertex is further fitted, with a requirement on the vertex χ 2 /dof < 15, and it must be displaced from the PV (FD χ 2 > 49). The flight distance (FD) is defined as the length of the distance between the PV and the SV. The FD χ 2 variable relates the information of the FD to the uncertainties of the vertex fits, such that a small FD is more significant for well measured vertices than for those having a large uncertainty. Furthermore, the cosine of the direction angle (DIRA) of the combined D 0 candidate is required to be > 0.9999. The DIRA is defined as the angle between the momentum vector ⃗ p of the reconstructed intermediate particle with respect to the vector connecting the PV and SV ( ⃗ FD). Signal D 0 candidates are required to have an impact parameter IP χ 2 lower than 25.

The D 0 candidate mass is computed from the sum of the four-momenta p µ = (E, p x , p y , p z ) of the final state particles. The energy is evaluated using the momentum of the particle and a given mass hypothesis E = m 2 PDG + p 2 with m PDG the mass value from the PDG [START_REF] Zyla | Review of particle physics[END_REF]. The mass computation heavily relies on the tracking system precisely measuring the momentum. The D 0 candidates mass is required to be in the range 1700 -2050 MeV/c 2 .

In the final step of the reconstruction, a low momentum pion with p > 120 MeV/c and track quality χ 2 /dof < 5 is combined with the D 0 meson to form a D * + candidate. The D * + vertex is fitted and required to have vertex fit quality of χ 2 /dof < 25.

After the trigger selection, the data is processed by the so-called Stripping, rerunning the reconstruction to add some further selection requirements and compute useful variables for the data analysis. The selection requirements used by the HLT2 trigger and stripping are very often similar, see for comparison Table 4.2. Some simple ∆LL particle identification requirements are added for each of the final state particles (pions, kaons and electrons) in the stripping line. Furthermore, the mass difference between the D * + and D 0 mesons, ∆m, is evaluated and required to be in the range 129.4-181.4 MeV/c 2 . This variable is a powerful tool to reject the combinatorial background and it is used in the analysis selection explained in the next section.

The HLT2 and stripping requirements were changed from the first part of the Run2 data taking (2015-16), reported in appendix A.3.1, to the second part (2017-18), shown in Table 4.2. Some kinematics and topological selections were loosened (e.g. IP) to increase the signal statistics or tightened (e.g. DOCA) to reject combinatorial background with almost no loss in signal efficiency. Furthermore, the D 0 mass range was enlarged from 1800-1950 MeV/c 2 to 1700-2050 MeV/c 2 in 2017 in order to have a better control on the electron mode decays, due to their worse mass resolution compared to the muon modes caused by bremsstrahlung effects. For these reasons, as well as different efficiencies among the periods, the subsamples 2015-16 and 2017-18 are kept separate when performing the signal fits.

After performing the full reconstruction, it is possible that more than one signal candidate is found in a single event, usually referred as multiple candidates. It is very likely that the multiple candidates are due to an artifact of the reconstruction algorithms rather than being true signal decays. If, after the full selection, more candidates are found in the same event, one is chosen randomly. This requirement removes less than one percent of the final selected candidates.

Particle Variable HLT2 line Stripping

K, π, e p > 3 GeV/c 3 GeV/c p T > 300 MeV/c 300 MeV/c Track χ 2 /dof < 4 3 IP χ 2 > 3 3 K ∆LL K > - -5 e ∆LL e > - -2 D 0 m(D 0 ) > 1700 MeV/c 2 1700 MeV/c < 2050 MeV/c 2 2050 MeV/c Vertex χ 2 /do f < 15 20 FD χ 2 > 49 16 DIRA > 0.9999 0.9999 max DOCA > 0.3 0.3 IP χ 2 < 25 36 p > - 3 GeV/c p T > - 2 GeV/c π s p T > 120 MeV/c 120 MeV/c Track χ 2 /dof < 5 3 D * + Vertex χ 2 /do f < 25 20 ∆m = m(D * + ) -m(D 0 ) > - 129.4 MeV/c 2 < - 181.4 MeV/c 2

Preselection

Candidates fulfilling the trigger and stripping selections defined in the previous section are further filtered with a set of preselection requirements aiming to reject the maximum amount of background with little signal loss. The preselection requirements are summarised in Table 4.3. In order to reduce the amount of fake tracks, all final state tracks are required to have a ghost probability smaller than 0.3.

particle Variable D 0 → h + h -e + e - K, π, e, π s ghost probability < 0.3 π ProbNNpi > 0.7 K ProbNNK > 0.7 D 0 IPχ 2 < 9 D * + A pT > -0.4 D * + DTF χ 2 > 0 D * + DTF ∆m > 144 MeV/c 2 < 147 MeV/c 2
Tight neural network PID variable (ProbNN) cuts are applied on the D daughter hadrons (ProbNNh > 0.7) to suppress cross-feed of D 0 → hh (′) e + e -decays due to hadron mis-identification, for example D 0 → K -π + e + e -decays mis-id as the signal decay D 0 → K + K -e + e -. This background is shifted in D 0 mass due the kaon mass assigned to mis-identified pions, but still forming a broad peak in ∆m = m(D * )m(D 0 ). By plotting ∆m in the high D 0 -mass region above 1900 MeV/c 2 before applying a hadron PID cut, a peak appears as shown in Figure 4.2 at 144-147 MeV/c 2 , which disappears after the PID cuts. Residual contributions of this background are estimated and considered as a source of systematic uncertainty in section 4.5.3.

Requiring that the D 0 IP χ 2 is smaller than 9 and that the asymmetry between the p T of the D * candidate and p T of all particles reconstructed within a cone around the D * A pT is greater than -0.4, helps to remove part of the combinatorial background with very little loss of signal. A cut around the ∆m = m(D * + )m(D 0 ) peak of the distribution (∆m = 145.426 ± 0.002 MeV/c 2 [START_REF] Zyla | Review of particle physics[END_REF]), is performed after refitting the full decay chain constraining the D * decay vertex to coincide with the PV using DecayTreeFitter (DTF) [START_REF] Wouter | Decay chain fitting with a Kalman filter[END_REF]. The DTF simultaneously fits the full decay chain using a Kalman filter under the PV contraint, improving the momentum and mass resolutions. The DTF is required to converge and output a valid covariance matrix. A cut around the DTF ∆m, 144 < ∆m < 147 MeV/c 2 , which rejects 80% of the combinatorial background with an efficiency of 80% on the signal is applied (see Fig. 4.3). The choice of the ∆m cut is explained in nex section, optimising the signal over background selection by maximising a figure of merit. On the left, the ∆m spectrum of D 0 → K -π + e + e -decays after trigger and preselection cuts using 2017-18 data, where the red lines show the ∆m cut applied in the preselection. On the right, D 0 mass distribution before (black) and after (red) the ∆m requirement.

Multivariate selection

The combinatorial background is further suppressed using a multivariate analysis. The multivariate method exploits as training samples signal and background proxies. This information is combined into a single variable, the classifier output, used in a classification phase to distinguish signal and background in a sample of unknown composition.

A Boosted Decision Tree (BDT) [START_REF] Roe | Boosted decision trees, an alternative to artificial neural networks[END_REF] classifier method is used in the analysis. A decision tree is a binary-structured classifier based on repeated yes/no decisions taken on single variables. In this way, the phase space is split into many regions classified as signal or background, depending on the majority of training events that end up in each leaf node. The boosting of the classifier extends this concept from one tree to several, forming a so-called forest. The output of the forest is finally combined into a single classifier, as a weighted average of all the individual decisions of the trees. Boosting stabilises the decisions' response with respect to fluctuations in the training sample and is able to considerably enhance the performance of the classifier.

Separate BDTs are implemented for the two signal channels D 0 → π + π -e + e - and D 0 → K + K -e + e -to discriminate signal from combinatorial background. The classifier is implemented in the Toolkit for Multivariate Analysis (TMVA) [START_REF] Hoecker | TMVA 4 -Toolkit for Multivariate Data Analysis with ROOT[END_REF]. It is trained using simulated signal samples as signal proxy filtered by trigger strategy and preselection, defined in the previous sections. Data samples in the brem0 category and selecting candidates with D 0 mass > 1900 MeV/c 2 (also filtered by trigger and preselection) are used as the background proxy. In the brem0 category, only combinatorial background is expected in the high-mass sideband (> 1900 MeV/c 2 ), as this region is sufficiently separated from the signal decays.

The following variables are found to show separation in the signal and background distributions and are used in the training (see Figure 4.5), avoiding variables correlated with the D 0 mass:

• the logarithm of the cosine of the D 0 direction angle, log(DIRA);

• the D 0 flight distance (FD) with respect to its PV, FD(D 0 );

• the largest distance of closest approach of the D 0 daughters, max(DOCA);

• the D 0 vertex χ 2 , χ 2 vertex ;
• the χ 2 of the decay tree fitter, DTF χ 2 ;

• the momentum of the soft pion, p(π s );

• the transverse momentum of the soft pion, p T (π s );

• the impact parameter of the soft pion with respect to its primary vertex, IP(π s ).

The BDT classifier distributions are shown in Figure 4.4. For the signal, the BDT output distribution is flat while on a background sample lower values are favored. A BDT selection requirement can be chosen to reject a large quantity of combinatorial background enhancing the signal. A dedicated optimisation is studied in the next section.

In order to avoid optimising the BDT on a specific training sample, two disjoint training subsamples are tested, selecting them by even or odd event numbers. The BDT is trained on one subsample, then applied to the other and vice versa.

→ π + π -e + e -and D 0 → K + K -e + e -BDT.
The plots show the comparison between the signal proxy in black (simulation) and background proxy in red (data mass sideband) distributions for each variable.

trained respectively on the subsamples with even or odd event numbers. No difference is seen.

D 0 → π + π -µ + µ -and D 0 → K + K -µ + µ -BDTs
trained on the even and odd (event number) subsamples. This crosscheck validates the classifier results on independent training subsamples.

The BDT trained on the 2015-16 samples provides lower performance and stability compared to the classifiers trained on the 2017-18 samples due to limited sample statistics. Therefore, the BDTs trained using 2017-18 samples are also applied to 2015-16 data. To crosscheck no significant differences is seen in the distributions among the years, a classifier is trained using as training samples simulation from 2015-16 (as "signal") and 2017-2018 (as "background"). As shown in Figures 4.7, the resulting receiver operating characteristic (ROC) curve appears as a straight line. The ROC curve shows the signal efficiency versus the background for a given BDT output requirement. If it appears as a straight line, it indicates that there are no significant differences between the two samples used, justifying the merge of the years. The BDT output is plotted as a function of the D 0 meson, dilepton or dihadron masses on a background sample to check for any possible correlation which must be avoided. The distributions are shown in Figure 4.8 for D 0 → K -π + e -e + decays. To get a pure background sample, the ∆m requirement is inverted and a cut on D 0 vertex χ 2 > 40 is added. In addition, correlations are searched for in signal samples as shown in Figure 4.9 removing the blinded signal by inverting the ∆m requirement and selecting m(D 0 ) > 1900 MeV/c 2 . None of the BDT profile distributions show large correlations.

Selection optimisation

To optimise the signal selection and reduce the two major backgrounds (combinatorial and fully-hadronic misidentified decays), a figure of merit is evaluated for different requirements on the BDT classifer output and the ProbNNe PID selections on the two electrons. A two-dimensional scan is performed computing the Punzi figure of merit (FoM) [START_REF] Punzi | Sensitivity of searches for new signals and its optimization[END_REF]:

FoM = ϵ sig a/2 + N bkg , (4.2)
where ϵ sig is the signal efficiency, evaluated from simulated signal samples, and N bkg the total expected background yield in the signal region. The parameter a is set to a = 5 to optimise the selection for a statistical signal significance of 5 σ. The Punzi figure of merit is particularly useful for searches as no assumption on the signal yield is required and the signal efficiency can be evaluated from simulation. The signal region is defined as 1700-1900 MeV/c 2 for brem0 and 1700-2050 MeV/c 2 for brem1, see distributions of the D 0 mass in Figure 3.25b for both brem categories. The optimisation is performed separately for D 0 → π + π -e + e -and D 0 → K + K -e + e - and for brem0 and brem1 categories.

The background yield is evaluated as N bkg = N comb + N mis-id . The combinatorial component N comb is estimated by fitting the data in a ∆m sideband of the same width as the signal window defined by 150 < ∆m < 153 MeV/c 2 and m(D 0 ) > 1900(1950) MeV/c 2 for brem0 (brem1) and extrapolating the measured yield to the signal region. In order to consider the non-uniformity of ∆m spectrum, in the extrapolation, a correction of 5% (15%) is applied in brem0 (brem1) category. The correction factors are obtained from the ratio of background yields measured in D 0 → Kπ + ee + in the two ranges 150 < ∆m < 153 MeV/c 2 and 144 < ∆m < 147 MeV/c 2 .

A crosscheck is performed removing this 5% (15%) correction and no change in the optimal selection point is observed. The mis-id background yield N mis-id is evaluated from a fit to the normalisation mode in the brem0 category with a fit model explained in section 4.4, and using the formula:

N mis-id,brem0 (D 0 → h + h -π + π -) = N mis-id • R B • R ϵ , (4.3)
with N mis-id the yield of the brem0 mis-id evaluated from the fit of real data in

D 0 → K -π + [e + e -] ρ 0 /ω , R B = B(D 0 → h + h -π + π -)/B(D 0 → K -π + π + π -)
the ratio of branching fractions from PDG (see values in Table 4.4) and

R ϵ = ϵ(D 0 → h + h -π + π -)/ϵ(D 0 → K -π + π + π -
) evaluated from the fully hadronic MC after applying the selection. The misID yield in brem1 is evaluated as:

N mis-id,brem1 (D 0 → h + h -π + π -) = N mis-id,brem0 (D 0 → h + h -π + π -)R brem1/brem0 , (4.4
) with R brem1/brem0 the ratio of selected candidates in brem1 and brem0 evaluated MC (see Table 4.4).

Decay B [×10 -3 ] R B R brem1/brem0 [%] D 0 → K -π + π + π -82.2 ± 1.4 5.4 ± 3.0 D 0 → π + π -π + π -7.56 ± 0.20 0.092 ± 0.003 5.2 ± 2.4 D 0 → K + K -π + π -2.47 ± 0.11 0.030 ± 0.001 7.4 ± 3.1 Table 4.4: Fully hadronic 4-body D 0 decays branching fractions from PDG. The R B = B(D 0 → h + h -π + π -)/B(D 0 → K -π + π + π -) and R brem1/brem0
values are shown in the last two columns. The R brem1/brem0 ratio is evaluated from simulation and used to estimate the misID background yield in brem1 category from the brem0 fit.

Figure 4.10 shows the results for both D 0 → π + π -e + e -and D 0 → K + K -e + e - signal decays and both brem categories for 2017-18 data. A common optimal point between brem0 and brem1 is chosen as the final simultaneous fit constrains the brem1 mis-id yield to the one fitted in the brem0 category. The optimal cut is chosen from the brem1 category, as it is expected to be the most significant one for the overall analysis:

• BDT > 0.5 and ProbNNe > 0.4 for

D 0 → π + π -e + e - • BDT > 0.4 and ProbNNe > 0.4 for D 0 → K + K -e + e -.
The 2015-16 selection is aligned to the 2017-18 optimal point. All the values of efficiencies and yields used to evaluate the figure of merit are reported in the appendix A.1.2.

The optimisation method hereby described is used to perform a study testing selections with alternative ∆m ranges. The full method is applied multiple times varying the ∆m requirements reported in Table 4.5. The table shows the values of the maximal figure-of-merit for each configuration. The range 144-147 MeV/c 2 is kept since the observed background yields allow for a more stable fit in the low statistics dilepton-mass bins.

∆m cut [ MeV/c 2 ] D 0 → π + π -e + e -D 0 → K + K -e + e -

Efficiency evaluation

The total selection and reconstruction efficiency ratio

ϵ(D 0 → h + h -e + e -)/ϵ(D 0 → K -π + e + e -
) is evaluated to determine the BFs with equation 4.1. In order to correct parts of the efficiency with data-driven methods, the total efficiency is written as product of the individual terms as follows:

ϵ = ϵ acc • ϵ reco-strip • ϵ selection • ϵ trig , (4.5)
defined as: with N the yields before and after the selection step indicated in the subscript. Each partial efficiency is defined as a conditional efficiency, meaning under the condition of passing all previous steps. The following sections explain how each individual efficiency is determined for the signal and normalisation channels in regions of dielectron mass. The generated resonance model and reconstructed dilepton mass are corrected in simulation with data-driven methods to account for data-MC differences.

ϵ acc = N in
The efficiency in each dilepton-mass bin is defined as following:

ϵ m(ee) bin = N gen&reco in bin N gen in bin , (4.6)
with N gen in bin the number of events generated in the dilepton-mass bin and N gen&reco in bin the number of events generated and reconstructed in the same dileptonmass bin. By definition, the efficiency evaluated in this way excludes all "leaking in/out" events with candidates migrating from one dielectron mass region to another. This effect is due to the poor momentum resolution of the electron candidates which does not allow a perfect evaluation of the dielectron invariant mass. When performing the fit on data in a specific dilepton-mass region, the "leaking in" events are fitted as a separate component and not considered in the signal yield estimation.

Phase-space model correction

In the signal simulated samples, an incoherent sum of resonant and non-resonant components is generated since the true amplitude composition is unknown. The fractions of generated contributions are listed in Table 3.2.

The simulated dilepton and dihadron mass spectra are corrected using the corresponding spectra as measured in four-body muon modes from Ref. [START_REF] Aaij | Angular analysis of D 0 → π + πµ + µand D 0 → K + Kµ + µdecays and search for CP violation[END_REF]. Backgroundsubtracted and efficiency-corrected dimuon and dihadron mass spectra for 4.11. The mass distributions are normalised and a non-uniform binning is chosen in order to avoid negative weights in the low statistics tails. The dilepton mass of D 0 → π + π -µ + µ -decays is plotted until 1100 MeV/c 2 as no signal was observed in the high-mass dilepton bin and the tail of distribution is kept blind. Dihadron mass distribution are plotted until 1200 MeV/c 2 due to a preselection requirement in the muon-mode analysis.

D 0 → π + π -µ + µ -, D 0 → K -π + [µ + µ -] ρ 0 /ω and D 0 → K + K -µ + µ -are shown in Figure
The resolution difference between the reconstructed and generated dimuon and dihadron mass in

D 0 → h + h -µ + µ -decays is below 1 %, as shown in appendix A.2.
Based on this assumption, the dielectron and dihadron mass distributions at generator level in the D 0 → h + h -e + e -simulated samples are corrected according to the evaluated correction weights in the muon modes. For D 0 → π + π -e + e -decays, uniform correction weights are assumed above 1100 MeV/c 2 , extrapolating from the last histogram bin. Analogously the correction is applied to candidates with dihadron

→ π + π -µ + µ -, D 0 → K -π + [µ + µ -] ρ 0 /ω and D 0 → K + K -µ + µ -decays.
The distributions are taken from Ref. [START_REF] Aaij | Angular analysis of D 0 → π + πµ + µand D 0 → K + Kµ + µdecays and search for CP violation[END_REF] in the supplementary material. mass greater than 1200 MeV/c 2 . The reconstructed mass spectra before and after correction are reported in Figure 4.12 for both dielectron and dihadron masses.

The reweighted simulated samples are used to evaluate the final efficiency ratio and perform the branching fraction measurement. The correction is also important to precisely evaluate the signal leakages from one dilepton mass bin to another since it depends on the underlying decay model.

The effect of the correction is O(10%) and visualised in Figure 4.13, where the ratio of efficiencies (signal over normalisation) is evaluated with and without the resonance-model correction. The relative difference per brem category considered in the analysis is shown. Further possible simulation-data discrepancies are considered as source of systematic uncertainties.

Dielectron mass resolution and bin migration

The efficiencies are evaluated in regions of the reconstructed m(e + e -) in simulation, as defined at beginning of the section. Thus, resolution effects can affect the efficiency evaluation. The m(e + e -) resolution is generally better in simulated samples when compared to data. The differences in resolution mainly arise from two sources:

• The simulated and real material budget of the detector is not perfectly aligned, which impacts the bremsstrahlung effects, and therefore the momentum resolution.

• The imperfect simulation of occupancy and calorimeter responses, which affects the bremsstrahlung recovery and leads to a change in the energy associated with each electron.

To take into account these effects, a mass smearing is applied to the simulated samples based on fits of the J/ψ resonance in m(e + e -) using B + → K + J/ψ(→ e + e -) control samples in data and MC. The following formula applies (method adapted from Ref. [START_REF] Aaij | Measurement of lepton universality parameters in B + → K + ℓ + ℓand B 0 → K * 0 ℓ + ℓdecays[END_REF]):

m smeared = m true + s σ • (m reco -m true ) + (µ f it data -µ f it MC ) + (1 -s σ ) • (µ f it MC -m(J/ψ) PDG ), (4.7
) with m true the true (generated) dilepton mass, m reco the reconstructed dilepton mass in simulation, s σ the σ data σ MC ratio parameterising the m(e + e -) resolution difference determined from the fits, µ f it data and µ f it MC the mass value determined from fits to data and MC and m(J/ψ) PDG the PDG value of the J/ψ mass. The parameters are evaluated for different years of data taking and bremsstrahlung categories, considering brem0, brem1 and brem2 respectively with 0, 1 or ≥ 2 photons attached to the electrons, see Table 4.6. Unless specified otherwise, from now on the reconstructed dilepton mass in simulation refers to the corrected m smeared for the purposes of the efficiency ratio evaluation. MC . The values of the parameters are reported for different years and bremsstrahlung categories respectively with 0, 1 or ≥ 2 photons attached to the electrons.

brem0 brem1 brem2 sσ ∆µ [ MeV/c 2 ] µ MC [ MeV/c 2 ] sσ ∆µ [ MeV/c 2 ] µ MC [ MeV/c 2 ] sσ ∆µ [ MeV/c 2 ] µ MC [ MeV/c 2 ]
To understand the effect of the correction, the reconstruction-stripping efficiency is evaluated using the uncorrected and corrected dilepton mass and the relative difference is shown separately for the two brem categories considered in the analysis. An effect of O(10%) is found and shown in Figure 4.14 for the 2017-18 data taking conditions. Since the dilepton mass resolution is worse in data compared to MC, the dilepton-mass corrected efficiency is smaller than the uncorrected one. The comparison of the corrected and uncorrected dilepton mass spectra can be found in Figure 4. [START_REF] Kobayashi | CP-violation in the renormalizable theory of weak interaction[END_REF].

An alternative method using D + s → π + ϕ(→ e + e -) decays as control sample is tested and used as a cross-check (see section 4.5.2).

Acceptance

The LHCb detector only covers a limited region of the solid angle in the forward region, meaning only a fraction of the produced signal decays falls within acceptance. The efficiency of the LHCb acceptance is determined from simulated samples, selecting events with all charged final state particles falling within a polar angle range between 10 and 400 mrad. Large simulated samples of signal and normalisation decays are generated and the fraction of candidates which is found within acceptance are used to determine the efficiencies in bins of dilepton mass. The ratio of acceptances for signal and normalisation are shown in the first column of Figure 4.22.

Reconstruction and Stripping

Simulated candidates falling within the acceptance are required to be correctly reconstructed and passing the stripping line selection. After the offline reconstruction, simulated samples can contain a small contamination of incorrectly reconstructed backgrounds. These are usually in the order of a few percent. A truth-matching method is applied to remove these backgrounds.

The stripping and reconstruction efficiency is evaluated from truth-matched simulation requiring candidates to be correctly reconstructed and passing the stripping line requirements, defined in Table 4.2.

A data-driven method is used to correct for tracking efficiency differences between simulation and real data, implemented in the TrackCalib LHCb tool [START_REF] Aaij | Measurement of the track reconstruction efficiency at LHCb[END_REF]. This tool provides data-simulation correction maps in bins of momentum and pseudorapidity of the particles and number of tracks in an event. The maps are evaluated using a tag and probe method with J/ψ → µ + µ -decays. The tagging track is a well-reconstructed muon, while the probe is a partially reconstructed track omitting information from one of the tracking subdetectors (VELO, TT or T-stations). This allows to evaluate the reconstruction efficiency for the missing subdetector. The single efficiencies can then be combined to evaluate the total tracking efficiency in each kinematic bin. The ratio of efficiencies obtained with the tag and probe method in data and simulation determines the correction factors which are used in this analysis.

Correction factors obtained for muon tracks can be used for hadron tracks by taking into account differences in their material interaction. Approximately 11% of the kaons and about 14% of the pions cannot be reconstructed due to hadronic interactions that occur before the last T-station. The total material budget in the simulation has an uncertainty of 10%, so a systematic uncertainty of 1.1% for kaons and 1.4% for pions is considered when applying the correction [START_REF] Aaij | Measurement of the track reconstruction efficiency at LHCb[END_REF].

A similar method is used to correct electron tracking efficiencies [START_REF] Aaij | Performance of the LHCb trigger and full real-time reconstruction in Run 2 of the LHC[END_REF]. Here, the tag and probe method is performed on B + → J/ψ(→ e + e -)K + by also tagging the kaon as a correctly reconstructed particle. The correction maps are provided in bins of ϕ, η and p T of the electron.

The data-simulation corrections weights are applied for each electron track in the decays. The final efficiencies are then evaluated per brem category and data taking year. The ratio of reconstruction-stripping efficiencies (signal over normalisation) are shown in the second column of Figure 4.22.

The results are shown for each dilepton mass bin. Single efficiency values are reported in appendix A.3.1. The data-simulation corrections lead to a 2-3% relative change on the single channel reconstruction-stripping efficiencies, while the effect on the efficiency ratio ϵ(D 0 → h + h -e + e -)/ϵ(D 0 → K -π + e + e -) is below 0.5 %. This study is reported in appendix A.5.1.

Offline selection

The offline selection efficiency is evaluated from simulation and includes the preselection and optimised BDT and PID requirements explained in the previous sections.

To correct for simulation-data differences in the PID response, a data-driven resampling method (PIDGen) is used [START_REF] Aaij | Selection and processing of calibration samples to measure the particle identification performance of the LHCb experiment in Run 2[END_REF]. The technique used exploits PID distributions from calibration data as PDFs to generate a resampled PID variable for different kinematic regions. The resampling is performed in bins of p, p T and nTracks. When correcting the PID variables of the electrons, B + → J/ψ(→ e + e -)K + decays are used as a calibration sample while when correcting the ones of the hadrons D * + → D 0 (→ K -π + )π + decays are used. The PIDGen tool transforms the PID variable of a simulation sample in such a way that its distribution matches the one seen in the calibration sample, following the kinematics of each particle. The advantage of this method is that the correlation of the PID variable present in simulation with other variables, for example related to the kinematics of the particle, is preserved.

The method is validated by comparing the background-subtracted (using the sPlot technique [START_REF] Pivk | sPlot: A statistical tool to unfold data distributions[END_REF]) to the PIDGen resampled distributions of D 0 → K -π + e + e - decays in Figure 4. [START_REF] Christenson | Evidence for the 2π Decay of the K 0 2 Meson[END_REF]. The sPlot method allows to unfold the signal and background contributions in a given data sample. Typically, the invariant mass is used as a discriminating variable and is described by a model which includes the two components, signal and background. When performing a fit to the data sample with such a model, event-by-event weights, known as s-weights, are extracted from the model. The weights are used on other control variables to separate the contribution to each category. The method assumes that the control variable is uncorrelated with the discriminating variable. In the analysis, the sPlot is often used to compare the distributions of the variables in simulation and background subtracted data, to assess the data-simulation differences. No relevant differences are observed in both brem categories between new resampled PID distributions in simulation and the s-weighted ones in data, as shown in the pulls distributions of Figure 4. [START_REF] Christenson | Evidence for the 2π Decay of the K 0 2 Meson[END_REF].

The signal selection efficiency can then be evaluated using resampled PID variables. Values are reported in appendix A.3.1. The ratio of efficiencies (signal over normalisation) are shown in Figure 4.22 (third column).

Trigger

The trigger response in simulation is known to not perfectly reproduce the one in collision data. Simulated samples must be corrected for data-simulation differences. However, evaluating the trigger efficiency in data is not a trivial task. In data, only triggered events are saved to disk, which means it is impossible to evaluate the absolute trigger efficiency as the number of triggered events over the total.

The TISTOS method proposes a solution to evaluate the trigger efficiency with a data-driven approach [START_REF] Tolk | Data driven trigger efficiency determination at LHCb[END_REF]. It is based on selecting a subsample of data with a certain trigger decision, the tag sample, which contains also the events selected by trigger decision of interest. The main assumption consists in the fact that the trigger efficiency in the defined subsample is representative of the whole sample. The efficiency is evaluated in bins of kinematic variables to avoid an incorrect evaluation of it. In fact, the chosen tag sample could bias the efficiency evaluation if the TISTOS method is performed integrating over the relevant kinematic variables. For example, in order to evaluate the efficiency of a TOS trigger decision, one can define a subsample selected by a TIS requirement. In this way, instead of evaluating the efficiency over the total number of events ϵ TOS = N TOS N total , it is estimated as ϵ TOS = N TOS&TIS N TIS , with N TOS&TIS the number of events passing both TIS and TOS decisions and N TOS(TIS) the yield of events passing only the TOS (TIS) decision.

In the analysis, the TISTOS method is applied to a calibration sample in both data and simulation to evaluate the trigger efficiency in kinematic bins. The data/simulation efficiency ratio is used to evaluate correction weights to be further applied to the simulation. The method is used to correct both L0 and HLT1 trigger responses. The calibration samples are composed of B 0 → K * (→ K -π + )J/ψ(→ e + e -) decays selected with simple cuts to obtain a pure signal, including mass cuts around B 0 and J/ψ masses (± 50 MeV/c 2 ). Correction weights are evaluated in bins of particle kinematics per year and magnet polarity.

L0 correction

For the sake of simulation-data efficiency corrections, two L0 trigger categories, L0I and L0L, are defined as follows:

• L0I: (L0Muon TIS or L0Electron TIS)

• L0L: (L0Electron TOS (l0) or L0Electron TOS (l1)) and !L0I with l0 and l1, the two final states electrons. This strategy facilitates the L0 efficiency correction as the two categories are by definition exclusive.

Correction weights can be derived from data-MC efficiency ratios as follows:

w L0I = ϵ data L0I ϵ MC L0I , w L0L = ϵ data L0L ϵ MC L0L , with ϵ data L0I = N TISTOS N TOS data and ϵ MC L0I = N TISTOS N TOS MC , ϵ data L0L = N TISTOS N TIS data and ϵ MC L0L = N TISTOS N TIS MC , (4.8)
where N TISTOS denotes the fraction of candidates triggered by the TIS and TOS categories, while N TIS (N TOS ) is the fraction of candidates triggered by the TIS (TOS) category.

The L0L efficiency correction is evaluated in bins of the transverse energy deposited by the electrons in the electromagnetic calorimeter (ECAL), E T , since this is the quantity responsible for a positive decision of L0Electron. As the granularity of the ECAL is different in its three regions (see Figure 4.17), the deposited energy resolution is also expected to vary between them. Thus, the L0L efficiency correction is evaluated in bins of E T and ECAL regions.

The TISTOS method is performed using as TIS tag: (L0Muon TIS or L0Electron TIS). The L0L weights as a function of the transverse energy E T of the electron are shown in Figure 4.18 for different ECAL regions. The distributions here reported correspond to the 2018 data taking year. The 2015, 2016 and 2017 distributions are reported in appendix A.3.2. The L0L category efficiencies are evaluated from the logical or condition of L0Electron TOS for each electron, which are not mutually exclusive. Considering that P(A ∪ B) = P(A) + P(B) -P(A ∩ B), the L0L efficiencies for both electrons can be expressed as:

ϵ data,MC L0L = ϵ data,MC L0L (e + ) + ϵ data,MC L0L (e -) -ϵ data,MC L0L (e + ) • ϵ data,MC L0L (e -). (4.9)
In this way, the final weight expression is:

w L0L = ϵ data L0L ϵ MC L0L = ϵ data,MC L0L (e + ) + ϵ data L0L (e -) -ϵ data L0L (e + ) • ϵ data L0L (e -) ϵ data L0L (e + ) + ϵ MC L0L (e -) -ϵ MC L0L (e + ) • ϵ MC L0L (e -) . ( 4

.10)

The L0I trigger category contains mainly events triggered by the particle resulting from the hadronisation of the opposite heavy flavoured quark in c c or b b production. To account for kinematic correlations, the efficiency can be evaluated in bins of the momentum of the signal candidate, i.e. p T (B 0 ) in the control sample, and total number of tracks in the event. The TISTOS method is performed using an (L0Hadron TOS or L0Electron TOS) subsample as tag, triggered on either the hadrons or electrons decay final states. The L0I weights as a function of p T (B 0 ) are shown in Figure 4. [START_REF] Aaij | Observation of CP violation in charm decays[END_REF]

HLT1 correction

The HLT1 trigger correction is performed similarly to the L0 category one. The corrections are binned in six regions of nTracks. The weights are evaluated as:

Total trigger efficiency

The L0 and HLT1 corrections are applied as per-event weights to simulated candidates. For each event, either the L0L or L0I weight is applied, as the two categories are mutually exclusive. The values of the single efficiencies are reported in appendix A.3.1. The ratios of efficiencies (signal over normalisation) are shown on the fourth column in Figure 4.22.

Total efficiency ratio

The efficiency ratios ϵ(D 0 → h + h -e + e -)/ϵ(D 0 → K -π + e + e -), from equation 4.1, are evaluated in regions of dilepton mass as reported in Figure 4.21 for the data taking years 2017-18, while the 2015-16 plots are reported in appendix A.3.1.

The uncertainty is evaluated using a bootstrapping approach. For each event 100 different Poisson distributed weights are assigned, with a mean value of 1. The random generation of the weights is done setting a unique seed for each event. With this method, efficiency is re-evaluated 100 times, each time using the i th bootstrapped weight for each event.

The mean and spread of the 100 ratio of efficiencies distribution are finally used to determine the nominal ratio of efficiencies values and their statistical uncertainty.

The efficiency ratios for the intermediate steps discussed in the previous sections can be found in

Fit model and yield determination

Yields of signal and normalisation channels are obtained through extended a maximumlikelihood fit to the mass distributions of D 0 candidates. The branching fractions are directly evaluated in the fit as a function of the yields and efficiencies with Equation 4.1. Fits are performed simultaneously for the two brem categories and the 2015-2016 and 2017-2018 data-taking periods, where the signal BF is treated as shared parameter between the categories. Furthermore, the fit is performed simultaneously in all dilepton mass bins to constrain the bin-by-bin signal leakages.

After the selection, the data D 0 mass distribution is assumed to be composed of:

• the signal:

D 0 → π + π -e + e -, D 0 → K + K -e + e -and D 0 → K -π + [e + e -] ρ 0 /ω ;
• the misID peaking background, four-body fully hadronic decays D 0 → h + h ( ′ )-π + π -with two pions mis-identified as electrons;

• combinatorial background (random combination of tracks forming a candidate passing the selection);

• only relevant for low dilepton mass regions, backgrounds from D 0 → h + h (′)-η(→ e + e -γ) decays;

• signal leakage between neighbouring dilepton mass bins (only when relevant).

The probability density functions (PDF) describing these components are determined from simulation or data sidebands, separately for brem categories. Candidates passing the preselection are fitted in each dilepton mass bin separately (including truth-matching for the MC samples). Due to a lack of statistics in the smaller dielectron mass bins, the BDT and PID cuts optimised in the selection are not applied when fitting the simulation samples to determine the PDFs. Furthermore, the models retrieved from the 2017-18 samples are also used in the final fit for the 2015-16 data taking years. In the final fit to collision data, all parameters of the PDFs are fixed to the values found in simulation letting only the signal and background yields free to float. All effects related to chosen fit models are considered in the systematic uncertainty subsection 4.5.3.

Signal

A Crystal Ball distribution is used to describe the brem0 component of the signal:

f (x; α, n, x, σ) = N • exp (-(x-x) 2 2σ 2 ), for x- x σ > α A • (B -x- x σ ) -n , for x- x σ ≤ α (4.12) where A = ( n |α| ) n • exp (-|α| 2
2 ), B = n |α| -|α| and with n, α, σ and x as free parameters. This PDF consists of a core Gaussian with a long low-end tail which allows the description of candidates whose bremsstrahlung photons are not recovered.

The signal brem1 component is described by a combination of a Crystal Ball and Gaussian distributions with common mean. This signal shape is different from the brem0 component due to the finite calorimeter resolution which widens the distribution, and candidates with incorrectly added photons, which lead to a tail towards higher D 0 masses. Figures 4. [START_REF] Charles | CP violation and the CKM matrix: Assessing the impact of the asymmetric B factories[END_REF] show the simulation distributions fitted in the integrated dielectron-mass spectrum. The two signal PDFs are also fitted to simulated data in the individual dielectronmass bins. To avoid taking the bin-by-bin leakage into account in the shape, the candidates are required to be generated and reconstructed in the same dielectron mass bin. The leakage contribution is discussed in the next section.

All fit parameters are fixed when fitting the rare decay modes and possible shape variations are considered as sources of systematic uncertainties. The single dilepton mass bin distributions and values of the PDFs parameters are reported in appendix A.4.

Dilepton mass bin migration

A study is performed to assess the impact of the signal leakage between dilepton mass bins. The leakage fraction is evaluated from simulation, requiring candidates to be generated in one dielectron mass bin and reconstructed in another. The expected yield estimation is then performed assuming the muon mode branching fractions (Table 2.2), the D 0 → K -π + e + e -signal yields (Table 4.8) and the ratio of efficiencies (subsection 4.3.7). The results are reported in Table 4.7, showing the estimated leakage yields, together with the fraction (in %) of reconstructed candidates over the generated ones in each dilepton mass bin.

The largest effects are seen for D 0 → π + π -e + e -candidates generated in the ϕ dilepton mass bin and leaking down to the (ρω) bin in both brem categories. The upwards leakage in the brem1 category of the (ρω) bin to the ϕ, and ϕ migrating to the high mass dielectron mass bin are also taken into account in the simultaneous fit. All other signal bin leakages are estimated to be < 1 event and thus not considered in the analysis.

A Bukin PDF [START_REF] Bukin | Fitting function for asymmetric peaks[END_REF] is used to define models for the aforementioned leakages fitting simulation samples. It is a 5-parameter distribution, convolving a Gaussian and an exponential distribution, which allows to describe the asymmetric shapes of the bin-by-bin leakages. The simulated samples are selected requiring candidates to be generated in ϕ dilepton mass bin but reconstructed in the:

• (ρω) mass bin in the brem0 category and brem1 category ;

• high mass mass bin in the brem1 category; and generated in the (ρω) mass bin then reconstructed:

• ϕ mass bin in the brem1 category.

All distributions overlaid by the corresponding models are shown in Figure 4.24. As shown in the figures, the distributions are shifted with respect to the nominal signal PDFs.

The yields of these components are constrained in the simultaneous fit from the corresponding dilepton mass bin signal yield using fractions evaluated from MC:

• (29.6 ± 1.0) % for ϕ to (ρω) leakage in the brem0 category;

• (14.4 ± 0.6) % for ϕ to (ρω) in the brem1 category;

• (4.3 ± 0.3)% for ϕ to high mass leakage in the brem1 category;

• (3.4 ± 0.2) % for (ρω) to ϕ mass bin leakage in the brem1 category;

The other dielectron mass leakages reported in Table 4.7, both for D 0 → π + π -e + e - and D 0 → K + K -e + e -, are not considered due to the very low yield expectation and no dedicated PDFs are added to the final fit.

Bin migration effects are considered and estimated also when performing the fit to the normalisation channel D 0 → K -π + e + e -. The 'leaking in' effect inside the 675-875 MeV/c 2 dilepton mass bin is estimated from MC to be 5.5 % for the brem0 and 8.2 % for the brem1 category. To account for these effects, when evaluating the signal branching fraction with the formula Equation 4.1, the normalisation yield is scaled by a correction factor of 0.945 ± 0.04 for the brem0 and 0.918 ± 0.04 for the brem1 category.

(d) D 0 → K + K -e + e -brem1
Table 4.7: Estimated dilepton mass bin leakage yields for D 0 → π + π -e + e -and D 0 → K + K -e + e -separately for brem categories. The estimation is done assuming the muon modes BFs in Table 2.2, D 0 → K -π + e + e -yield and efficiencies. In parenthesis, the percentage of reconstructed candidates from the overall generated in each dielectron mass bin. The generated dilepton mass bin are shown horizontally while the reconstructed one vertically.

Mis-identified hadronic background

Simulated fully hadronic four-body decays of the form D 0 → h + h (′)-π + π -with double π → e mis-identification, filtered by the preselection, are used to determine the D 0 mass distribution of the double mis-id peaking background. The candidates are reconstructed and selected as signal decays, including the possibility to attach (predominantly random) bremsstrahlung photons to the misidentified pions. This allows to determine PDFs of the misID background for both brem categories. Since the mass shapes of signal and mis-id background in the brem1 category show no large discriminating power (compare the right-hand columns of Figure 4.23 and Figure 4.25), the fraction is fixed in the final fit from the values evaluated on simulation (see Table 4.4). In this way, the mis-id brem0 yield is fitted directly on data while the brem1 one is constrained and self-calibrating from the brem0 one. A Bukin PDF [START_REF] Bukin | Fitting function for asymmetric peaks[END_REF] is used to describe both brem components of the misID.

Combinatorial background

To ensure a stable fit, the combinatorial PDF shape is determined from a fit on ∆m sideband data (> 150 MeV/c 2 ) and m(D 0 ) > 1900 MeV/c 2 . The D 0 mass requirement is used to avoid promptly produced D 0 decays with randomly associated tagging pion in the high ∆m region. A first order polynomial is used to describe the combinatorial background. Figure 4.26 shows the fits for the integrated dielectron-mass range. All parameters are fixed in the final fit to data. Alternative shapes are used to evaluate a systematic uncertainty associated with this choice, see section 4.5.3.

Partially reconstructed η background

Simulated samples of D 0 → h + h (′)-η(→ e + e -γ) decays reconstructed as signal decays are used to determine the D 0 mass shapes of partially reconstructed backgrounds with a non-reconstructed γ. These decays are mainly present in the lowmass region of dielectron mass.

A Bukin PDF [START_REF] Bukin | Fitting function for asymmetric peaks[END_REF] is used to describe both brem components of the D 0 → h + h (′)-η(→ e + e -γ) backgrounds (see Figure 4.27).

The fit model is validated using D 0 → K -π + e + e -data in the low mass dilepton region 211-525 MeV/c 2 , adding the η background component (see Figure 4.28). The quality of the fit is greatly improved adding the η component.

→ K -π + π -π + (middle) D 0 → π + π -π + π -and (bottom) K + K -π + π -
with double π to e mis-identification. The fit are performed separately for brem categories integrated over the dielecton-mass region of interest.

Fit results

The fit to the normalisation mode is shown in Figure 4. 29 (2015-16) and Figure 4.30 (2017-18), separately for π + π -e + e -and K + K -e + e -optimised selections. The yields found are reported in Table 4.8. The fits are performed for the two brem categories fixing all parameters of the PDFs, leaving only the yields free to float. The mis-id background yield in the brem1 category is fixed to be 5 % of the brem0 mis-id yield, determined from simulations as described earlier.

The fit to signal data samples is performed blinding the D 0 mass distributions and shifting the branching fraction results by a random value. It is performed simultaneously in all dilepton mass bins in order to constrain the bin-by-bin leakages. All parameters of the models used are fixed and only the background and signal yields are free to float. The background yield values are reported in Table 4.9 for both D 0 → K + K -e + e -and D 0 → π + π -e + e -. The yields are shown for the data taking years 2015-16 and 2017-18 and for the brem categories considered in the analysis. The absolute statistical uncertainty on the branching fraction measurements are reported in Table 4.10 for both D 0 → π + π -e + e -and D 0 → K + K -e + e -decays in regions of dielectron mass. Table 4.10: Absolute uncertainty on the branching fraction values for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays retrieved from a simultaneous fit on all dilepton mass regions considered in the analysis.

D 0 → K -π + [
D 0 → π + π -µ + µ -(top) or D 0 → K + K -µ + µ -(bottom).
D 0 → π + π -µ + µ -(top) or D 0 → K + K -µ + µ -(bottom).

Pseudoexperiments studies

Pseudoexperiments are run to test and validate the signal fit strategy. Mass distributions are generated for each dilepton mass bin, brem category and year of data taking considered in the analysis. The number of generated events for each component of the model (signal and backgrounds) is randomised following a Poisson distribution with mean value:

• values reported in Table 4.9 for the D 0 → π + π -e + e -and D 0 → K + K -e + e - backgrounds;

• assuming the branching fractions reported in Table 2.2 to generate the signal yields (the upper limit values are used for the dilepton mass bins with no observation).

Once the mass distributions are generated, they are fitted using the nominal model with a simultaneous fit. Negative yields are allowed for both signal and background to avoid any possible bias and ensure a proper estimation of the uncertainties.

An asymmetrical definition of the pulls is used for a better interpretation of the fit uncertainties in the low statistics dilepton mass bins:

p =      X f it -X true σ + X f it , if X f it > X true X f it -X true σ - X f it , otherwise (4.13)
where X f it is the fitted value, X true the expected true value and σ + X f it (σ - X f it ) the high (low) asymmetric uncertainty values of a generic variable X.

The pull distributions of the branching fractions per dilepton mass bin are shown in Figure 4.31 and their mean and sigma in Table 4.11. The values are obtained by running a set of 1000 pseudoexperiments with 99 % convergence rate.

The pull values in the low-statistics dilepton mass bin are biased towards negative values. This is caused by the non-gaussian regime reached by the fit due to low values of signal and background yields. In this situation, the fit does not properly estimate the uncertainty on the branching fraction measurement which could bias the pull value. The uncertainty here needs to be evaluated with alternative methods, such as the Feldman-Cousins method [START_REF] Gary | A Unified approach to the classical statistical analysis of small signals[END_REF].

Furthermore, the fit is validated testing different signal branching fraction values, in particular assuming for all dilepton mass bins BF = 10 -6 , 10

Systematic uncertainties

The systematic uncertainties affecting the final BF measurements are discussed in the following sections. They are mainly divided in three categories related to:

• normalisation mode BF: the branching fraction of D 0 → K -π + e + e -is taken as external input from the BaBar measurement and used in the formula Equation 4.1. The uncertainty related to it is added as systematic uncertainty;

• Efficiency ratio: the effiencies are evaluated from simulation and corrected step-by-step with data driven methods. The uncertainty of each correction step is taken as systematic, as well as the statistical uncertainty on the final ratio of efficiencies evaluation;

• Yields evaluation: different fit models are tested to evaluate a systematic uncertainty on the yields evaluation. Furthermore, neglected backgrounds are also considered as source of systematic, as well as the statistical uncertainty on the normalisation yield evaluation.

Normalisation mode BF

The normalisation mode branching fraction [START_REF] Lees | Observation of the Decay D 0 → Kπ + e + e[END_REF] is measured to be:

B(D 0 → K -π + e + e -) = (4.0 ± 0.5(stat) ± 0.2(syst) ± 0.1(norm BF)) × 10 -6 , (4.14)
which includes the statistical, systematic and the uncertainty related to D 0 → K -π + π + π -normalisation BF used in the measurement. The total relative uncertainty ∆B norm B norm is 13.7 %, adding these in quadrature.

This uncertainty, just as in the D 0 → K -π + e + e -BF value, is kept separate in the final measurement. Indeed, if a more precise D 0 → K -π + e + e -BF measurement will be available in the future, the value in Equation 4.14 can be replaced and the signal BF recomputed.

Efficiency ratio

This section investigates the systematic uncertainties related to the efficiency evaluation, in particular focusing on limited statistics of the simulated samples and on data-simulation corrections presented in section 4.3. Uncertainties are evaluated directly on the efficiency ratio to profit from cancellations due to similarities between the normalisation and signal modes.

In the following sections, the systematic effects are reported for each dilepton mass bin combining the 2015-16 and 2017-18 efficiencies by their corresponding integrated luminosities. The following formula is used:

ϵ = L 15-
∆R ϵ R ϵ = (R alternative ϵ -R nominal ϵ ) R nominal ϵ , (4.17)
with R nominal ϵ and R alternative ϵ the ratio of efficiencies for the nominal and alternative corrections tested to evaluate the systematic uncertainties.

Limited statistics of the simulated samples

The limited simulated sample size leads to a statistical uncertainty on the ratio of efficiencies which is propagated as a systematic uncertainty. Since the samples have been reweighted, the statistical uncertainty is evaluated through a bootstrapping method defining the efficiencies as in Equation 4.15. The ratio of efficiencies (signal over normalisation) is evaluated for each of the 100 bootstrapped samples and the mean and RMS of the ratio of efficiencies distributions are used to determine the systematic effect as a relative uncertainty. The values are reported for each dilepton mass bin in 4.12: Systematic uncertainty on the efficiency ratio due the limited MC statistics. Each efficiency is evaluated 100 times with a bootstrap method and the spread of their distribution is taken as statistical uncertainty.

Tracking efficiency correction

Tracking data/simulation corrections are presented in section 4.3.4 and aim to correct hadrons and electron reconstruction efficiencies in simulation. Muon correction maps are used to correct kaon and pion track efficiencies. This introduces a 1.1 % and 1.4 % systematic respectively for kaons and pions, as explained in 4.3.4 and in [START_REF] Aaij | Measurement of the track reconstruction efficiency at LHCb[END_REF].

When considering the ϵ(D 0 →π + π -e + e -) ϵ(D 0 →Kπ + e + e -) and

ϵ(D 0 →K + K -e + e -)
ϵ(D 0 →Kπ + e + e -) ratios, two electrons and one hadron coincide between the signal and normalisation channels. Their uncertainty is expected to be 100 % correlated, therefore cancelling in the ratio. A relative systematic uncertainty is assigned for the pion/kaon difference in the ratio, specifically 1.1 % for the kaon and 1.4 % for the pion.

The total relative systematic uncertainty on the efficiency ratio due to the tracking data/simulation correction is given by the squared sum ∆R Track

ϵ /R ϵ = 1.1 % ⊕ 1.4 % = 1.8 %.

Particle identification efficiency correction

The particle identification data-simulation correction is performed by resampling the PID response with the PIDGen package [START_REF] Poluektov | Correction of simulated particle identification response in LHCb using kernel density estimation[END_REF]. Two systematic effects are considered on this correction:

• Uncertainties arising from the limited statistics in the simulated calibration samples. Alternative PID templates obtained from five different bootstrapped samples are used to test and assign a systematic;

• Uncertainties arising from the choice of a specific kernel width when parameterising the PID control samples. Two extra PID templates with different kernel width are used for testing.

In the PIDGen framework [START_REF] Poluektov | Correction of simulated particle identification response in LHCb using kernel density estimation[END_REF], the PID response, p T , η, nTracks variables are first binned in the calibration sample (with 100 bins each), then 4D smeared with a certain kernel width size. Changing the kernel width size is therefore equivalent to change the binning width.

The efficiency ratio is evaluated with the nominal and alternative PID corrections and the largest relative difference is assigned as systematic uncertainty. Effects are summarised in Table 4. [START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF]

Trigger efficiency correction

The major systematic effect for the data-simulation trigger efficiency correction is due to the choice of the kinematics binning scheme. As a systematic study, finer and coarser binning schemes (double and half size binning from the nominal) are tested and the largest deviations on the efficiency ratio are reported in Effects are caused by the choice of binning when evaluating the correction weights for L0L, L0I and HLT1 trigger requirements.

Kinematic and multiplicity reweighting

The simulated event multiplicity and detector occupancy are known to show deviations from data. These have an effect on variables like the number of hits in the SPD detector (nSPDhits) or other occupancy proxies such as the number of unique tracks (nTracks). Data-simulation disagreements differ in each subdetectors. However, the mis-match between data and simulation is expected to be very similar for signal and normalisation channels, and therefore to mostly cancel out in the ratio. No correction is applied by default and a systematic uncertainty is assigned as follows.

Data-simulation correction weights can be evaluated plotting nSPDhits and nTracks for both simulated samples and D 0 → K -π + e + e -background-subtracted data (with the sPlot method). The weights are defined dividing the two distributions separated by brem category (see Figure 4.32). The relative difference between the nominal and reweighted ratio of efficiencies is considered as systematic uncertainty. An averaged systematic value over the dilepton mass bins of ∆R kin ϵ /R ϵ = 1.8 % for D 0 → π + π -e + e -and 1.5 % for D 0 → K + K -e + e -decays is obtained.

Dielectron-mass resolution correction

An alternative m(e + e -) smearing method is tested to assign a systematic uncertainty on the dielectron mass resolution data-MC correction. Data and simulation D + s → π + ϕ(→ e + e -) decay samples are used to determine an alternative m(e + e -) smeared compared to the nominal one, which uses B + → K + J/ψ(→ e + e -) decays.

The following formula applies:

m smeared = m true + s σ • (m reco -m true ) + (µ f it data -µ f it MC ) + (1 -s σ ) • (µ f it MC -m(ϕ) PDG ), (4.18
) with m true the true (generated) dilepton mass, m reco the reconstructed dilepton mass in simulation, s σ the σ data σ MC ratio parameterising the m(e + e -) resolution difference determined from the fits, µ f it data and µ f it MC the mass value determined from fits to data and MC and m(ϕ) PDG the PDG value of the ϕ mass.

To order to select D + s decays (PDG mass 1968.37 MeV/c 2 [START_REF] Zyla | Review of particle physics[END_REF]). The D + s mass requirement is tighter on the lower mass side (> 1920 MeV/c 2 ) to maximally reduce the leakage of D + → π + ϕ(→ e + e -) decays into the mass window and in the brem0 category also to further reduce the leakage from the left side of hadronic decays D + s → π + π -π + which are not considered in the simple fit.

The fit model is composed of a Bukin PDF for the signal and an exponential distribution for the combinatorial background. The fits are performed separately for different brem categories, considering candidates with 0, 1 and 2 or more bremsstrahlung photons attached to the electrons. Separate parameters are evaluated for each category and applied to the signal MC.

Fits to the dilepton mass distributions in simulation and data from 2017-2018 data taking years are shown in Figure 4.33. The dielectron mass range is chosen around [850-1200] MeV/c 2 in order to fully fit candidates coming from the ϕ resonance (m(ϕ) = 1019.46 MeV/c 2 [START_REF] Zyla | Review of particle physics[END_REF]).

The relative difference of the ratio of efficiencies computed with the alternative and nominal methods are used as systematic uncertainties reported in Table 4.15. 4.15: Systematic uncertainty on the efficiency ratio due the dilepton mass correction. The alternative method using as calibration sample D + s → π + ϕ(→ e + e -) is compared to the nominal correction with B meson decays. The relative difference is when evaluating the efficiencies is reported in the table for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays in different dilepton mass bins.

Correction of the decay model

Systematic uncertainties related to the decay-model correction are evaluated varying within uncertainties the single dilepton and dihadron correction weights. In this way, one hundred different correction maps are generated and applied to the simulated samples. The spread of their effiency ratio is taken as the uncertainty. The values of the systematic are shown in Table 4.16 for the different dilepton-mass bins and signal decays. The same method is applied for both acceptance and full selection (for decays within the acceptance) efficiencies.

Summary of systematic uncertainties on efficiency ratio

A summary of all systematic uncertainties related to the efficiency ratio is shown in

Yield estimation

In order to evaluate the systematic uncertainties associated with the invariant mass fit, pseudoexperiments are generated assuming alternative models or injecting neglected background contributions. The pseudoexperiments are then fitted both with the nominal and alternative models and the systematic uncertainty on the yields ratio R N defined:

∆R N R N = (BF nominal -BF alternative ) BF nominal , (4.19)
with BF nominal/alternative the branching fraction results from the fit using the nominal and alternative models respectively. When testing the systematic uncertainty due to the neglected backgrounds, the fitted BF is compared to the one used to generate the signal yield. Furthermore, a systematic uncertainty is assigned due to the statistical uncertainty on the evaluation of the normalisation yield.

Fit model

For the determination of the signal and normalisation yields, a study is performed varying the different components of the fit PDFs, motivated by the limited accuracy of the simulation to reproduce collision data and the inability to select the background samples with the full signal selection. The following variations from the nominal fit models presented in section 4.4 are considered for both normalisation D 0 → K -π + e + e -and signal decays D 0 → π + π -e + e --D 0 → K + K -e + e -:

• Signal PDF:

testing a Bukin PDF as an alternative to the Crystal Ball nominal model; -variations α and n parameters of the Crystal Ball nominal PDF according

to their uncertainty to study the effect of the radiative tails modeling;

retrieving the model by fitting the 'corrected' D 0 mass distribution in simulation, re-computed from the m(e + e -) smeared dielectron mass.

• MisID PDF:

variations of the shape with alternative selections, as no PID and BDT requirements are applied in the nominal model due to the low statistics of the simulated samples;

variation within uncertainty of the mis-id brem1/brem0 fraction (values in Table 4.4) used to compute the misid brem1 yield.

• Combinatorial PDF:

shapes determined in different ranges of ∆m (150-155, 155-160, > 160 MeV/c 2 ) from the nominal model with ∆m > 150 MeV/c 2 ;

• η background PDF:

variation within uncertainties of the parameters of the PDFs (only relevant for the 'low mass' dilepton mass bin in D 0 → π + π -e + e -and D 0 → K + K -e + e -).

• Leakage fractions:

variation within uncertainties of the leakage fractions defined (only relevant for D 0 → π + π -e + e -decays in the ρ/ω, ϕ and high mass dielectron mass regions).

Systematic uncertainties are also assigned testing the 2015-16 and 2017-18 differences in the model and by applying the full selection when retrieving the model from MC.

Pseudoexperiments are generated in order to evaluate systematic uncertanties on the ratio of yields determination. The mass distributions are generated using the alternative PDFs and fitting them twice, once with the nominal and once with the alternative model. The alternative model is also used to fit the D 0 → K -π + e + e -data in order to retrieve N(D 0 → K -π + e + e -) alternative , which is then used as input when fitting the pseudoexperiments with the alternative model. The relative difference between the two branching fractions results, nominal and alternative, is taken as systematic uncertainty using Equation 4. [START_REF] Aaij | Observation of CP violation in charm decays[END_REF].

The pseudoexperiments are produced assuming the muon modes BFs in Table 2.2 to generate the signal yield, while the background yield values are taken from the signal fit results. The value of the yields is randomised with a Poisson distribution for each of the pseudoexperiments.

The systematic uncertainties evaluated with this method are reported in Table 4.18 and Table 4.19 respectively for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays.

Model variations

∆R model N /R model N D 0 → π + π -e + e - 211
D 0 → π + π -e + e -fits.

Neglected backgrounds

Contributions of possible background sources which have been neglected in the final fit are estimated and considered as systematic uncertainties. The following mis-id and/or partially reconstructed decays are considered:

• the signal decays D 0 → π + π -e + e -, D 0 → K -π + e + e -and D 0 → K + K -e + e - mis-identified as another signal channel with one or two hadron wrong identification, also called signal crossfeed backgrounds;

• fully hadronic

D 0 → π + π -π + π -, D 0 → K -π + π + π -, D 0 → K + K -π + π -and D 0 →K -K + K -π +
decays with three or four mis-identifications. The double mis-id with two pions as two electrons is already considered in the fit;

• double muon to electron mis-identified backgrounds 4.19: Systematic effects on the ratio of yields evaluation by varying the model shape PDFs for D 0 → K + K -e + e -fits.

D 0 → π + π -µ + µ -, D 0 → K -π + [µ + µ -] ρ 0 /ω and D 0 → K + K -µ + µ -; Model variations ∆R model N /R model N D 0 → K + K -e + e
• π 0 partially reconstructed (missing energy from the π 0 ) and mis-identified backgrounds (two pions as electrons):

D 0 → π + π -π + π -π 0 , D 0 → K -π + π + π -π 0 , D 0 → K + K -π + π -π 0 ;
• D 0 →K -π + π -e + ν e decays with pion to electron mis-id and missing energy from the neutrino;

• baryonic decays Σ 0 c (2455) →Λ + c (→ p K π) π and Λ + c (2595) →Λ + c (→ p K π) π π with proton mis-identification and two pions incorrectly identified as two electrons.

As a first check, all these decays are generated with the RapidSim framework [START_REF] Cowan | RapidSim: an application for the fast simulation of heavy-quark hadron decays[END_REF] to understand if their hypothetical D 0 mass distributions are included in the [1700-2050] MeV/c 2 signal range after evaluating their assumed four-body D 0 mass with the mis-identification hypotheses. If the D 0 mass distributions are fully outside the signal range, they can be excluded from the study.

The RapidSim application is used for fast simulation of phase space decays, allowing for quick studies of the properties of particle decays in physics analysis. It includes simpler versions of the reconstruction or detector resolution effects.

All the listed decays are reconstructed as the signal channels D 0 → π + π -e + e -, D 0 → K -π + e + e -and D 0 → K + K -e + e -. The plots in Figure 4.34 show the D 0 mass distributions of the aforementioned backgrounds reconstructed as D 0 → π + π -e + e - decay using RapidSim. The distributions are not normalised to the expected branching fractions, as the goal is just to visualise their mass ranges. Similar plots for D 0 → K -π + e + e -and D 0 → K + K -e + e -decays are reported in appendix A. [START_REF] Aaij | Observation of the resonant character of the Z(4430)state[END_REF].

The backgrounds out of the signal range are reported in Table 4.20 as not-inrange (nr) and thus not considered for this analysis. The table shows the branching fraction values of each background decay [START_REF] Zyla | Review of particle physics[END_REF], multiplied by the mis-identification probabilities reported in Table 4.21.

To simplify the study, the mis-identification efficiencies are taken using the PIDCalib LHCb application [START_REF] Anderlini | The PIDCalib package[END_REF] by applying the ProbNN cuts used in the analysis. The PIDCalib package exploits the use of calibration samples to evaluate the particle identification efficiencies for different particles and PID variable selections. The values in Table 4.21 are used to get a rough estimate of each background contribution and exclude the ones not relevant for the analysis.

Backgrounds BF D 0 → π + π -e + e -D 0 → K -π + e + e -D 0 → K + K -e + e - from PDG 10 -6 4 × 10 -6 10 -7 D 0 → π + π -e + e - 10 -6 signal 2 × 10 -8 5 × 10 -9 D 0 → K -π + e + e - 4 × 10 -6 4 × 10 -7 signal 10 -7 D 0 → K + K -e + e - 10 -7 nr 10 -8 signal D 0 → π + π -π + π - 7 × 10 -3 mis-id 10 -9 10 -9 D 0 → K -π + π + π - 8 × 10 -2 nr mis-id 2 × 10 -8 D 0 → K + K -π + π - 2 × 10 -3 nr nr mis-id D 0 →K -K + K -π + 4 × 10 -6 nr nr nr D 0 → π + π -µ + µ - 10 -6 4 × 10 -10 10 -11 10 -12 D 0 → K -π + [µ + µ -] ρ 0 /ω 4 × 10 -6
3 × 10 -10 10 -11 10 -10

D 0 → K + K -µ + µ - 10 -7 nr 10 -11 10 -9 D 0 → π + π -π + π -π 0 4 × 10 -3 nr 4 × 10 -9 10 -10 D 0 → K -π + π + π -π 0 4 × 10 -2 nr nr 4 × 10 -9 D 0 → K + K -π + π -π 0 3 × 10 -3 nr nr nr D 0 →K -π + π -e + ν e 3 × 10 -4 nr 8 × 10 -7 2 × 10 -8 Σ 0 c (2455) →Λ + c π - 6 × 10 -4 10 -9 5 × 10 -9 4 × 10 -9 Λ + c (2595) →Λ + c π + π - 2 × 10 -6 4 × 10 -10 2 × 10 -9 10 -9
Table 4.20: Background decays BF estimates when mis-identified as the signal channels D 0 → π + π -e + e -, D 0 → K -π + e + e -and D 0 → K + K -e + e -. Their measured branching fractions from PDG are reported in the second column. The Σ 0 c and Λ + c decays assume an extra 0.01 effect due to the difference to their productions rates compared to the D 0 meson. If the backgrounds when reconstructed as the signal channels are out of the D 0 signal region [1700-2050] MeV/c 2 , not-in-range (nr), they are not considered for the analysis. The BF values reported in the main section of the table include the mis-id efficiencies reported in Table 4.21.

The D 0 → K -π + e + e -branching fraction (4 × 10 -6 ) corresponds in data to around 700-800 signal events after the full selection, see Table 4.8. This means that any BF reported in Table 4.20 below 5 × 10 -9 has negligible contribution in our final fit ( < 1 event). These backgrounds are excluded in this study while the ones with a value above 5 × 10 -9 are highlighted in bold in the table.

One of the remaining backgrounds is D 0 →K -π + π -e + ν e which is estimated to contribute with a rate of 8 × 10 -7 when mis-identified as D 0 → K -π + e + e -and 2 × 10 -8 as the D 0 → K + K -e + e -signal decay. A simulated sample of D 0 →K -π + π -e + ν e decays is generated and reconstructed as the signal. The reconstruction efficiency evaluated in such way is O(10 -6 ), while the equivalent efficiency on simulated samples of the signal is O(10 -2 ). Assuming a ratio of efficiencies ϵ (D 0 →K -π + π -e + ν e )/ϵ (D 0 → h + h -e + e -) of 10 -4 , the semileptonic background decay contributes with a rate of 8 × 10 -11 and can be excluded from the study.

The contribution of the remaining backgrounds is also evaluated from simulated events:

• D 0 → K -π + e + e -as D 0 → π + π -e + e -, with K to π mis-identification;

• D 0 → K -π + e + e -as D 0 → K + K -e + e -, with π to K mis-identification;

• D 0 → K -π + π + π -as D 0 → K + K -e + e -
, with K to π and two pions as two electrons mis-identification;

• D 0 → π + π -e + e -as D 0 → K -π + e + e -, with π to K mis-identification;

• D 0 → K + K -e + e -as D 0 → K -π + e + e -, with K to π mis-identification.

Mis-id ProbNN cut efficiency

π → e ProbNNe > 0.4 4 calibration samples integrating over momentum, η and number of tracks. The values are used to assess the mis-id probability on the backgrounds mentioned in the section.

× 10 -3 K → e ProbNNe > 0.4 4 × 10 -4 π → K ProbNNK > 0.7 10 -2 p → K ProbNNK > 0.7 9 × 10 -2 K → π ProbNNpi > 0.7 5 × 10 -2 p → π ProbNNpi > 0.7 3 × 10 -2
Each background decay is reconstructed as the corresponding signal mode and the full selection requirements are applied to evaluate the efficiency ϵ(bkg as signal). To estimate the background yield contributions of these decays, the following formula is used:

N bkg = N D 0 →K -π + e + e -× BF bkg BF D 0 →K -π + e + e - × ϵ(bkg as signal) ϵ(D 0 → K -π + e + e -) , (4.20)
with N D 0 →K -π + e + e -the D 0 → K -π + e + e -yields from the fit to real data, ϵ(D 0 → K -π + e + e -) the selection efficiency of the normalisation mode, the branching fraction BF D 0 →K -π + e + e -= 4 × 10 -6 , BF bkg the BF of the background considered and ϵ(bkg as signal) the selection efficiency of the background selected as signal. The equation is used for both brem0 category with N D 0 →K -π + e + e -= 280 and ϵ(D 0 → K -π + e + e -) = 0.7 × 10 -4 and brem1 category with N D 0 →K -π + e + e -= 550 and ϵ(D 0 → K -π + e + e -) = 1.3 × 10 -4 .

As an example, for the first item on the list, the D 0 → K -π + e + e -decays are assumed as mis-identified and reconstructed as D 0 → π + π -e + e -decays. Here the BF bkg = 4 × 10 -6 while the selection efficiencies are evaluated on the D 0 → K -π + e + e -simulated samples reconstructed and selected as D 0 → π + π -e + e - decays. As expected the main contribution is in the ρ/ω dilepton mass bin 565-950 MeV/c 2 , with 1 event expected in the brem0 and 5 events in the brem1 category.

Similarly, for the D 0 → K -π + e + e -decays mis-identified as D 0 → K + K -e + e - decays, the main contribution appears in the ρ/ω dilepton mass region with 1 event expected in the brem0 and 1 event in the brem1 category.

Pseudoexperiments are generated injecting these background contributions following the estimated mass shape from the simulated samples in order to quantify the effect of neglecting them in the nominal fit model. The systematic uncertainty ∆R bkg N /R bkg N is estimated to be 3.1 % for D 0 → π + π -e + e -and 1.1 % for D 0 → K + K -e + e -decays in the ρ/ω dilepton mass bin.

When performing the same study, for the three remaining decays, both D 0 → π + π -e + e -and D 0 → K + K -e + e -decays reconstructed and mis-identified as D 0 → K -π + e + e -decay, contributions of < 0.1 events are evaluated. For this estimation the branching fractions BF(D 0 → π + π -e + e -) = 10 -6 and BF(D 0 → K + K -e + e -) = 10 -7 are assumed, taken from the muon modes BFs.

The selection efficiency on the D 0 → K -π + π + π -decay mis-identified as D 0 → K + K -e + e -is below 10 -9 which means < 1 event contributes as background.

RapidSim simulation ee backgrounds

π π → D0 π KKK → D0 π π π K → D0 π π π π → D0 π π KK → D0 1 1.5 2

Normalisation mode signal yield

Finally, the statistical uncertainties of the normalisation mode signal yields, reported in table 4.8 for the two optimised selections, are considered as systematic uncertainties. Relative uncertainties in % for the single fits per year and brem categories, and the total one are shown in table 4

Summary systematic uncertainty on ratio of yields

In summary, the individual contributions to the systematic uncertainty on the ratio of yields are added in quadrature to evaluate the final systematic uncertainties reported in Table 4.23 for D 0 → π + π -e + e -and D 0 → K + K -e + e -in different dilepton mass bins.

Channel m(e

+ e -) ∆R model N /R model N ∆R bkg N /R bkg N ∆N stat norm /N stat norm ∆R N R N [ MeV/c 2 ] [%] [%] [%] [%] D 0 → π + π -e + e -

Total systematic uncertainty

The systematic uncertainties related to the yields and efficiencies evaluation are summarised and added in quadrature in 4.24. The values are shown for each dilepton mass bin separately and do not include the external normalisation BF uncertainty. The latter provides the same uncertainty contribution in both channels and all dilepton bins. The systematic uncertainty on the ratio of efficiencies evaluation is dominant in almost all the dilepton mass regions. The dominant contributions are due to the limited statistics of the simulated samples and on the correction of the phase-space model. The best approximation of the real decay model is retrieved from the muon modes dataset, which is mainly statistically limited.

In the ρ/ω dilepton mass bins for both D 0 → π + π -e + e -and D 0 → K + K -e + e - decays, the systematic uncertainty on the yield evaluation is slightly larger compared to one on the efficiency determination. In this dilepton mass region, the uncertainty relative to the crossfeed D 0 → K -π + e + e -background reconstructed as signal increases the total systematic effect on the yield determination. Furthermore, the systematic uncertainty on the decay-model correction is smaller compared to other dilepton mass regions, as the correction is retrieved from the D 0 → h + h -µ + µ - dataset in the ρ/ω dimuon region, which is one of the must abundant of signal thus with lower statistical uncertainty.

Channel m(e

+ e -) [ MeV/c 2 ] ∆R N R N [%] ∆R ϵ R ϵ [%] ∆B B [%] D 0 → π + π -e + e -

Branching fraction determination and limit setting

A fit is performed in all dilepton mass bins simultaneously as explained in section 4.4 in order to measure the branching fraction in each of them. The simultaneous fit includes the two data taking periods (2015-16 and 2017-18) and brem categories (brem0 and brem1) for each dilepton mass bin i with the signal branching fraction B i as a shared parameter:

B i = N i (D 0 → h + h -e + e -) years brem N(D 0 → K -π + e + e -) years brem ϵ(D 0 → K -π + e + e -) years brem ϵ i (D 0 → h + h -e + e -)
years brem B(D 0 → K -π + e + e -) (4.21) with N i (D 0 → h + h -e + e -) the signal yield, N(D 0 → K -π + e + e -) the normalisation yield, the ratio of efficiencies ϵ(D 0 → K -π + e + e -)/ϵ i (D 0 → h + h -e + e -) and B(D 0 → K -π + e + e -) the normalisation mode branching fraction [START_REF] Lees | Observation of the Decay D 0 → Kπ + e + e[END_REF]. The suband over-scripts years and brem are respectively the two data taking periods (2015-16 and 2017-18) and brem categories (brem0 and brem1). This means a single branching fraction B i in the dilepton mass bin i is retrieved from four fits to different data subsamples selected by the two data taking periods and the two brem categories. The dilepton mass integrated branching fraction is computed by summing the individual ones in each dielectron mass bin.

The statistical significance of the signal in each dielectron mass bin is evaluated from the likelihood ratio between the signal-plus-background and only-background hypothesis with Wilks' theorem [START_REF] Wilks | The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses[END_REF]. The likelihood ratio is expected to behave as a χ 2 distribution and thus the p-value can be computed. Moreover, the p-values are translated into the equivalent Gaussian standard deviations σ, with p = ∞ σ 1 √ 2π exp(-x 2 /2)dx. In case the signal significance is below 3σ with respect to the background-only hypothesis, an upper limit on the signal branching fraction is quoted using a frequentist approach, otherwise only the branching fraction value is reported.

More specifically, the CLs method [START_REF] Read | Presentation of search results: The CL s technique[END_REF] is used as implemented in the GammaCombo framework [START_REF] Kenzie | GammaCombo: A statistical analysis framework for combining measurements, fitting datasets and producing confidence intervals[END_REF]. The test statistic q is defined as the negative logarithm of the likelihood ratio:

q = -2 ln L s+b L b , (4.22)
with L s+b and L b the likelihood functions of the fit under respectively the signalplus-background (s+b) and only-background (b) assumptions. The test statistic is constructed in such a way to help the analyst distinguish which hypothesis better describes the data. The value observed in data, q obs , tends to smaller values if the data is better described by s+b hypothesis. To avoid being subject to statistical fluctuations, the distributions of f(q|s+b) and f(q|b) respectively under s+b and b hypothesis are computed by generating pseudoexperiments. Examples are shown in Figure 4. [START_REF] Maki | Remarks on the unified model of elementary particles[END_REF] showing the q-distributions with both assumptions and the value observed on data q obs . One can then define the p-value for the signal-plus-background hypothesis as the probability of finding q greater than q obs p s+b = P(q

≥ q obs |s + b) = ∞ q obs f (q|s + b)dq, (4.23)
and similarly for the background-only hypothesis as the probability of finding q less than q obs p b = P(q ≤ q obs |b) =

q obs -∞ f (q|b)dq. (4.24)
Example p-values are shown in Figure 4.35 with the yellow and green areas respectively, for s+b and b assumptions. One can reject the signal hypothesis if p s+b < α, with for example a confidence level of 95 % taking α = 0.05. This is known as the CL s+b procedure. This method is generally problematic for very small signal sensitivity, as shown in the right plot in Figure 4.35, when f(q|s+b) and f(q|b) are very similar. This corresponds to the case where the expected number of signal events is much smaller than the background ones. For downward fluctuations of the background, the signal-plus-background hypothesis can be rejected where there is in fact no sensitivity. The exclusion probability in this case approaches α = 5 % which is not desirable.

To avoid this problematic behaviour, a modified frequentist approach is defined, the CL s method, normalizing p s+b by the only-background hypothesis:

CL s = p s+b 1 -p b . (4.25)
The signal hypothesis is rejected with a confidence level of 95 % if CL s < α with α = 0.05. In the case of no sensitivity to the signal, CL s has a value close to one and no exclusion is made. If f(q|s+b) and f(q|b) are completely separated, the CL s method is equivalent to CL s+b one.

To apply the CL s method to the analysis, two PDFs are defined: one containing the description of the signal and background components, while the other corresponding to an only-background distribution. Fits to only-background pseudoexperiments are performed to evaluate the likelihood ratio, testing a range of signal branching fractions. The systematics presented in Table 4.24 on the ratio of efficiencies and yields, together with the uncertainty on the normalisation BF, are treated as nuisance parameters and propagated as Gaussian constraints to the likelihoods.

All signal branching fractions where a CL s < 5 %(10 %) is observed are excluded with 95 % (90 %) confidence level. The smallest BF fulfilling this requirement is set as upper limits. The analysis is still blind and currently under review of the collaboration. For this reason, only the expected limits per dilepton mass bin are reported in the following Table 4.25.

The CL s distributions as a function of the signal branching fraction are shown in Figure 4.36 for each dilepton mass bin. The one (two) sigma confidence intervals are shown in light (dark) blue. In these plots 500 pseudoexperiments are run for each BF assumption.

Channel

m(e + e -) [ MeV/c 2 ] expected limits [×10

KKee ) →

Summary

The search for the rare charm decays D 0 → π + π -e + e -and D 0 → K + K -e + e -is performed using proton-proton collision data collected by the LHCb detector during Run 2 (from 2015 to 2018) at a centre-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 6 fb -1 . The expect limits are reported in Table 4.25 for the different dilepton mass regions considered in the analysis. The precision reached by the measurement is O(10 -7 ) on the signal BF. It is expected to observe for the first time the D 0 → π + π -e + e -decay and either observe or set an upper limit on D 0 → K + K -e + e -decay branching fraction. The absolute statistical uncertainty, together with the relative uncertanties on systematic effects and the normalisation channel branching fraction are reported in Table 4.26. One can determine the expected relative statistical uncertainty with respect to the D 0 → h + h -µ + µ -branching fraction values (Table 2.2), under the assumption of lepton flavour universality. One can note that the measurement is still statistically dominated, however, excellent prospects are foreseen for the next data taking runs of the LHCb experiment. Moreover, the dominant systematic uncertainties of this analysis are related to the size of the simulated samples and the knowledge of the phase-space model, the latter being limited by the statistics of the D 0 → h + h -µ + µ -dataset. Hence, these systematic uncertainties can be reduced by generating a higher number of simulated events and collecting more data in the future.

In conclusion, the LHCb upgrades will allow more sensitive studies of the D 0 → h + h -e + e -decays, with the possibility of measuring additional interesting observables as the CP and angular asymmetries. Moreover, the upgraded LHCb trigger system and the implementation of selections dedicated to dielectron modes opens up the possibility to probe new physics through the measurements of branching fraction ratios between dimuon and dielectron modes.

Channel

m(e + e -) Chapter 5

σ stat σ syst /B σ norm /B [ MeV/c 2 ] [×10 -8 ] [%] [%] D 0 → π + π -e + e -

The LHCb upgrade

The upgrade of the LHCb experiment aims to increase the precision of flavour physics measurements by collecting data with an increased instantaneous luminosity. Studies of rare decays, such as the charm D 0 → h + h -e + e -channels, will widely benefit from the data collected during the next years as their measurements are still statistically dominated. One of the major changes of the upgrade consists on a new trigger system which is fully software-based. In this respect, the event reconstruction performed at the first stage of the trigger had to be completely redesigned to be able to cope with the increased luminosity. This chapter discusses the upgraded LHCb detector and the reconstruction sequence of its first level trigger.

The author is the main developer of one of track reconstruction algorithms in this sequence, currently in use for data taking. It is presented in the next chapter.

The Run3 upgrade

After successfully operating from 2010 to 2018, LHCb stopped taking data for three years, during long shutdown 2 (LS2), in order to upgrade its detector. The precision of most of the key flavour physics observables studied and measured by LHCb remains statistically limited [START_REF] Aaij | Implications of LHCb measurements and future prospects[END_REF]. Larger datasets are required to probe the standard model at the level of precision of theoretical calculations and get the sensitivity needed to look for possible new physics effects. LHCb is expecting to collect 50 fb -1 of data over the LHC Run 3 (2022-2025) and Run 4 (2029-2032) operating at an instantaneous luminosity of 2 × 10 33 cm -2 s -1 , five times more compared to Run1-2.

Many of LHCb subdetectors were upgraded in order to cope with the increase of luminosity. The major limitation of the Run1-2 detector was the LHCb hardware trigger whose signal yield saturated for fully hadronic final state decays when increasing the luminosity. As shown in Figure 5.1, the hadronic selection rate was already saturating at the Run1-2 luminosity of 4 × 10 32 cm -2 s -1 . In order to profit of the upgraded conditions and collect significantly more data, it was decided to remove the L0 hardware trigger and implement a fully-software HLT1 stage able to record data at the full LHC rate. Thus signal channels could be selected after a first event reconstruction using both momentum and displacement information, improving the trigger efficiency on the hadronic modes by a factor 2 to 4 [START_REF]LHCb Trigger and Online Upgrade Technical Design Report[END_REF]. Similar selection improvements are expected on electron modes which were also limited by the hardware trigger selections. In this respect, the rare charm D 0 → h + h -e + e - decays will greatly benefit from the trigger upgrade.

The higher luminosity, the higher number of pp interactions, and the new trigger strategy required the full replacement of the readout electronics and substitution of some LHCb subdetectors. The new readout electronics is able to read events at the LHC 30 MHz collision rate. The data processing and computing strategy was also completely revised to deal with the expected increase of data input. The layout of the upgraded LHCb detector is shown in Figure 5.2.

Tracking system upgrade

The entire tracking system was substituted during LS2. The VELO sensors were replaced by new hybrid pixel detectors reaching a record distance of 5.1 mm from the beam pipe. The TT was replaced by four layers of silicon sensors forming the new Upstream Tracker (UT). The UT is covering a larger acceptance and provides a higher granularity compared to the TT. The Scintillating Fibre Tracker (SciFi) replaces the Inner and Outer tracker upstream of the magnet and it is composed of long scintillating fibers read out by silicon photo-multipliers. The goal of the new tracking system upgrade is to achieve a high tracking performance and at the same time to be read out at the LHC 30 MHz collision rate.

VELO

The VELO sensors have been redesigned to cope with the increase of luminosity of the upgrade and its electronics replaced to be able to read out the 30 MHz data input rate [START_REF]LHCb VELO Upgrade Technical Design Report[END_REF]. The sensors exploit the pixellated hybrid silicon detector technology in order to improve the Run2 physics performance of the Velo both in terms of tracking efficiency and impact parameter resolution [START_REF] Eklund | Physics benchmarks of the VELO upgrade[END_REF]. The 52 L-shaped modules are composed of sensors with 55 µm × 55 µm squared pixels. They are located, as shown in

UT

The UT detector is composed of four planes of silicon detectors [START_REF]LHCb Tracker Upgrade Technical Design Report[END_REF] as shown on Figure 5.4. The detector exploits the x-u-v-x geometry, as in the Run1-2 TT detector, where the x-layers are vertical and the stereo u/v-layers are inclined by ±5 • . The silicon strips composing the layers have different width and length, smaller close to the beam pipe and larger in the outer region, matching the expected occupancy. Following

SciFi

The scintillating fiber tracker replaces the IT and OT upstream of the magnet during the LHCb upgrade [START_REF]LHCb Tracker Upgrade Technical Design Report[END_REF]. The SciFi detector is composed of 250 µm diameter plastic scintillating fibres arranged in multi-layered fibre mats. It covers an acceptance of ± 3.2 m horizontally and ± 2.4 m vertically from the edge of the beam pipe. The SciFi tracker is composed of 12 detection planes arranged in 3 stations (T1, T2, T3) with 4 layers each following x-u-v-x configuration, as in the Run 1-2 downstream tracking system. A scheme of the full detector is shown in Figure 5.5. Charged particles propagating through a module of the SciFi detector traverse several fibers of a fibre mat. Photons are produced by the excitation of the polymer core and propagated internally by reflection to the SiPMs located at the end of the fibers outside the LHCb acceptance. The signal photons generated by the same particle have to be clustered to obtain a precise estimation of the hit position in the fibers. The clustering is performed on the front-end electronics and achieves a hit resolution below 100 µm [START_REF]LHCb Tracker Upgrade Technical Design Report[END_REF].

PID detectors upgrade

An excellent particle identification is crucial when taking data in the upgraded conditions to maintain a high final state particle identification of flavour physics decays. The information from the calorimeters and muon system is used at HLT1 level to improve the trigger decisions. Information from the RICH detector is employed to make final trigger selections at HLT2-level before storing data to disk.

The overall layout and concept of the RICH detectors remain unchanged with respect to the Run 1-2 LHCb, although critical modifications were needed to be able to read out the detectors at 30 MHz rate [START_REF]LHCb PID Upgrade Technical Design Report[END_REF]. The full photon detection chain was replaced in RICH1 and RICH2, substituting the former HPDs with multi-anode photomultiplier tubes (MaPMT) equipped with new electronics. Furthermore, due to higher critical level of occupancy in RICH1 caused by the luminosity increase, the spherical mirrors were replaced by new ones with larger radius of curvature leading to an increased focal length.

The increase of the instantaneous luminosity does not require a complete rebuild of the calorimeter and muon systems. The Run 1-2 detectors were kept while the full read-out electronics was changed to cope with LHC input rate. During the LS2, the PS and SPD detectors were removed as they were mainly employed for L0 trigger decisions on electrons, photons and hadrons. The photon and electron identification separation is provided by the ECAL, exploiting its energy resolution. In the muon system, the M1 station was removed during the upgrade phase as its main purpose was to improve the momentum resolution of the hardware muon trigger.

The Run3 dataflow

The upgraded dataflow scheme is presented in Figure 5.6. In average, at most 30 MHz of the input rate corresponds to the non-empty proton-proton collisions and the trigger must be able to reconstruct events at this rate.

The 4 TB/s of input data is handled by the event builder (EB) CPU farm which combines the data from each subdetector and transfers it to the high level trigger. The first stage of the trigger, HLT1, is fully software-based and optimised for graphic processing units (GPU). HLT1 performs a partial event reconstruction in order to select interesting events and reduce the data rate by a factor 30, reaching 1 MHz of output rate. The selected events are then stored in a buffer, while a real-time alignment and calibration of the detector is performed, before being handled by the second level trigger stage, HLT2. Here, a full event reconstruction is performed exploiting the most accurate real-time detector parameters. HLT2 selects events using information from the whole LHCb detector and its optimised reconstruction algorithms are run on a CPU farm. Finally, the events selected by HLT2 are saved to tape and further processed offline by the LHCb software to create the final data samples to be analysed.

Figure 5.6: LHCb upgrade dataflow, from the LHC bunch crossing of 40 MHz down to the offline processing and analysis production [START_REF]RTA and DPA dataflow diagrams for Run 1, Run 2, and the upgraded LHCb detector[END_REF].

The Online system

The LHCb data acquisition is controlled by the Online system, presented in Figure 5.7. It consists of 162 EB servers hosting the back-end receiver FPGA boards (TELL40) and GPUs running the first high level trigger stage. Each TELL40 board receives data from the different front-end electronics of the subdetectors, forming the so-called Multi-Fragment Packets (MFP). The goal of the EB servers is to collect the MFPs of all the subdetectors and group all the information coming from the same packet of events, creating the so-called Multi Event Packets (MEP) containing 30000 events. The MEPs are then transferred internally in the EB server to the GPUs where a first event reconstruction and selection is performed. This internal connection avoids any CPU-CPU transferring, greatly speeding up the data processing.

HLT1

The goal of the first high level trigger, HLT1, is to reduce the non-empty pp collision rate of 30 MHz down to around 1 MHz. HLT1 uses information from the tracking detectors, electromagnetic calorimeter and muon chambers to perform a partial event reconstruction. Minimum momentum requirements are applied in the tracking algorithms in order respect the HLT1 throughput constrains. Furthermore, no information from the RICH is employed to help the particle identification (only used at HLT2 level). The reconstruction includes the primary and secondary vertex finding, tracking and particle identification algorithms using the ECAL and muon stations information. A new sequence of algorithms was developed to handle the event reconstruction in a higher multiplicity environment optimising them for GPUs. The HLT1 reconstruction sequence is presented in section 5.2. The events are finally selected by the so-called trigger lines covering different physics cases. Figure 5.7: Scheme of the upgraded online system. The data received from each subdetector is combined in the event builder farm before being processed by the trigger system [START_REF] Aaij | The LHCb Upgrade I[END_REF].

The partial event reconstruction and selection is performed real-time on the input data thanks to the Allen framework [START_REF]LHCb Upgrade GPU High Level Trigger Technical Design Report[END_REF] using 162 NVIDIA RTX A5000 GPU cards, one per EB node, during the 2022 data taking. GPUs are ideal for HLT1 as each physics event is independent and can be parallelised within the threads of the graphic card maintaining a high throughput. The Allen algorithms exploit the parallelisation power of the GPUs also when reconstructing individual tracks inside an event. GPUs are characterised by a high number of floating point operations per second (TFLOPS) which perfectly fits the huge compute load of HLT1. Thousands of events, corresponding to ∼ 100 kB of raw data each, can be processed at the same time by the GPU fitting into its O(10) GB of memory.

The choice of the GPU technology has also great advantages for the coming years of data taking and future upgrades. In fact, each EB server has three slots where the graphic cards can be placed. A second GPU per server has been purchased and is currently in use for the 2023 data taking. In the coming years, a third GPU per server could be added in order to improve HLT1 performance.

The extra performance can be used to extend the physics reach of HLT1 selections. For example, downstream track reconstruction could be included in the HLT1 sequence to improve the selection of long-lived particles. Furthermore, RICH information could be exploited at HLT1 level which could allow the selection of exclusive decays at very early stages of the LHCb dataflow. The tracking efficiency could be also further improved exploiting the extra GPUs computational power: a new algorithm is presented in chapter 6.

As shown in Figure 5.8, the Allen throughput scales linearly with the TFLOPS, which increases when improving the GPU technology used. This means that in the future it will be sufficient to replace the current GPUs with more performant ones to handle higher multiplicities environments, i.e. increasing the luminosity, as no saturation in throughput is expected. Figure 5.8: Allen throughput for different GPU cards as a function of theoretical 32 bit peak TFLOPS perfomance [START_REF]LHCb Upgrade GPU High Level Trigger Technical Design Report[END_REF]. The throughput performance scales linearly with improved GPU technologies.

HLT2

Events selected by HLT1 trigger selections are passed to an intermediate buffer system before being processed by the second stage of the high level trigger, HLT2. The 40 PB buffer provides enough storage time to calculate the calibration and alignment constants which are crucial for an optimal selection of events. With a sampling process, events are loaded from storage and used to evaluate the necessary constants within minutes to update the HLT1 parameters as soon as possible. Later HLT2 consistently uses the same parameters.

The second high level trigger performs the full event reconstruction and allows inclusive and exclusive selections reducing the rate to around 100 kHz, with a writing bandwidth limit of 10 GB/s to tape. More than a thousand HLT2 lines select interesting events using the information from the reconstruction, tuned for each particular signal topology and/or physics analysis. Unlike HLT1, where a limited number of inclusive lines selecting a large amount of interesting events is sufficient, HLT2 applies both inclusive and exclusive selections. Both rectangular-cut-based and multivariate/neural network-based selections can be deployed.

The number of recorded events can be increased by decreasing the volume of information recorded for each event, the so-called Turbo method [START_REF] Aaij | Tesla: an application for real-time data analysis in High Energy Physics[END_REF][START_REF] Aaij | A comprehensive real-time analysis model at the LHCb experiment[END_REF][START_REF]Computing Model of the Upgrade LHCb experiment[END_REF]. The Turbo mechanism allows full flexibility on the amount of event information stored, referred as selective persistency. Depending on the specific physics channel under study, events information is saved, from the bare minimum of two tracks and their vertex up to a full event.

Offline data processing

The offline data processing consists of a "skimming" and "slimming" of the data, called sprucing, in order to persist selectively the amount of information stored. The sprucing can apply further selection on events in the FULL stream (skimming) which includes about 30 % of total output rate from HLT2. The remaining 70 % of the data is processed through the Turbo mechanism which reduces the data size persisting only the information related to the particle signal candidates (slimming). The data saved in the FULL stream can also be reprocessed in yearly re-sprucing campaigns, while this cannot be done on data in the Turbo stream. A typical example of events saved in the FULL stream are the ones selected by inclusive topological lines, selecting e.g. two or three body decays with requirements on the vertex quality and momenta of the particles. On the contrary, events selected by exclusive lines, requesting specific decay topologies and final state particles, are usually processed in the Turbo model.

Both FULL and Turbo data are then distributed into optimised physics streams to allow analysts to access reduced data sets categorised by the physics topic. The data is further processed by the Analysis Productions framework which produces the final output files ready to be analysed. Analysis Productions is a centralised and automated framework handling the monitoring and archiving of the final output data files.

The offline processing scheme is presented in Figure 5.9. It includes also calibration events processed by the sprucing model. Figure 5.9: Scheme of the offline data flow. Events selected by HLT2 are processed through the sprucing model and the Analysis Productions framework prepares the output files ready to be analysed.

Particle reconstruction at HLT1

The first level trigger of the LHCb experiment fully relies on the upfront charged particle reconstruction to select interesting events. The HLT1 sequence is summarised in Figure 5.10 and includes in steps, the data handling, reconstruction and selection algorithms.

First, the raw data from the detector is decoded by the HLT1 framework, transforming the raw information from the tracking detectors to (x,y,z) values for each hit or energy cluster in the calorimeter. As a second step, the so-called global event cut (GEC), selects all events in which the total sum of UT and SciFi hit multiplicities is less than 10000, rejecting roughly 10 % of the busiest events. All the reconstruction steps presented in Figure 5.10 are further explained in next sections.

During the 2023 data taking, the removal of the GEC was considered as studies of electroweak bosons, such as Z →µ + µ -and W →µ + ν µ decays, could greatly benefit from it by avoiding limiting the systematic uncertainties on the efficiency of this criterion. The removal of the GEC decreases the event processing speed of HLT1 by 1-2 % with an improvement on selection rates of electroweak boson decays of more than 4 %. Tracking algorithms are implemented following three distinct stages. First, the pattern recognition uses hits from the tracking subdetectors to recognise the trajectories of charged particles. Once the first set of hits is found, they can be fitted in a second stage by a model, to determine how likely the hits are to be associated to the same particle. In the final step, the quality of the track fit is used to filter the candidates and reject wrong hit associations.

When implementing a tracking algorithm, the developer needs to optimise a triangular set of fully correlated indicators:

• efficiency: how many reconstructible particles the algorithm achieves to correctly reconstruct;

• ghost rate: the rate of reconstructed tracks which are fake/random combination of hits;

• throughput: how many events per second the algorithm is able to process, normally measured in Hz.

At LHCb, the efficiency and ghost rate are defined as: efficiency = N reconstructed and reconstructible N reconstructible ;

ghost rate = N fake tracks N reconstructed , (

where at least 70 % of the hits in a track must be from the real simulated particle to be defined as reconstructed, otherwise it will be assigned as fake. Different criteria per tracking subdetector at LHCb are set to define a particle reconstructible [START_REF] Li | Tracking Definitions and Conventions for Run 3 and Beyond[END_REF]:

• Velo: the particle generates at least three hits in the Velo layers;

• UT: the particle produces at least one hit in an x layer and one hit in an u/v layer;

• SciFi: the particles generates at least six hits, with at least one hit on one x layer of each station, and at least one hit on each station u/v layers.

Three further indicators are often used when evaluating the performance of an algorithm:

• The clone fraction referring to the tracks associated to the same simulated particle as another reconstructed track:

Clone rate = N clone tracks N reconstructed tracks ; (5.2)
• The hit purity which refers to the fraction of hits that belong to the true particle:

Hit purity = N tracks hits in true particle N track hits ;

(

• The hit efficiency from the number of hits corresponding to the true particle hits in a track:

Hit efficiency = N tracks hits in true particle N true particle hits .

(5.4)

Velo tracking

The LHCb reconstruction starts by looking for tracks leaving hits in the detector closest to pp collisions: the Velo. In this region, particles are expected to travel as straight lines because the effect of the magnetic field is negligible. Velo tracks are crucial to reconstruct primary and secondary vertices and evaluate variables for physics data analysis, such as the impact parameter. Furthermore, their reconstruction efficiency impacts the performance of all subsequent stages in the HLT1 sequence.

The "Search by triplet" algorithm [START_REF] Hugo | A Fast Local Algorithm for Track Reconstruction on Parallel Architectures[END_REF][START_REF] Hugo | Search by triplet: An efficient local track reconstruction algorithm for parallel architectures[END_REF] achieves the necessary physics efficiency together with an optimal computational performance. It exploits a double parallelisation process, optimised for GPUs, at both the event-and track-level. The algorithm uses a typical "local track following" method to reconstruct tracks. Tracks are first seeded from three hit segments in a region where the hit density is lower and the signal easier to distinguish. Then, these tracklets are extrapolated, "followed", to the higher hit density regions of the detector to perform the full reconstruction. At LHCb, this usually corresponds to the zone closer to the pp collisions.

Proton-proton collisions tend to occur close to the x-y plane origin. Hence, the particles originated by them travel in a straight line with a constant phase ϕ in polar coordinates (angle in x-y plane). By consequence, the Velo hits are first sorted by ϕ and z before starting the track reconstruction.

The stages of the "Search by triplet" algorithm are illustrated in Figure 5.11, which are combined in an iterative way:

1. Seeding: search for triplets of hits in consecutive modules opening search windows in the ϕ variable. Every used hit is flagged to avoid including clone tracks;

2. Following: all found seeds are extrapolated as a straight line to the next modules to attach hits to them. To account for the inefficiency of the sensors, tracks are allowed to miss a module. If two consecutive modules are missed, the set of hits is stored as tracklet candidate;

3. Iteration: the process of seeding and following is repeated on all the unflagged hits.

A least mean square straight line fit is performed on all tracklet candidates built by the algorithm. A candidate is accepted and promoted to reconstructed track under a certain fit χ 2 threshold. The "Search by triplet" algorithm achieves a tracking efficiency above 99 % in all ranges of momentum and transverse momentum, as shown in Figure 5.12. The measured ghost rate is around 0.6 %, therefore essentially negligible for the HLT1 purposes.

Primary Vertex finding

As many of the HLT1 selections rely on displaced signatures, the primary vertices (PVs) must be reconstructed with a high efficiency and low fake rate. This is crucial when separating displaced tracks, coming from secondary vertices (SVs), from the prompt ones originated from the PVs. With the increase of luminosity in Run3, the number of PVs is expected to be around 5 per event.

The PV reconstruction fully relies on the tracks found by the Velo reconstruction in the previous section. PVs are usually gaussian distributed around ±20µm in x and y, and ±6cm in z around the center of the interaction region, defined inside the Velo region as shown in Figure 5.13. The HLT1 PV reconstruction algorithm [START_REF] Reiss | Excerpts from the LHCb cookbook -RECEPTS for testing lepton universality and reconstructing primary vertices[END_REF] requires a good knowledge of the LHC beamline position. During the Velo closing procedure, this information can be obtained fitting as a straight line the detected collision points. A PV is reconstructible when it produces at least four reconstructible Velo tracks. The PV reconstruction efficiency is shown in Figure 5.15 as a function of the number of associated tracks of the simulated PV. A primary vertex is considered reconstructed if its simulated position is within the uncertainty of the reconstructed one.

Velo-UT tracking

The Velo tracks reconstructed in subsection 5.2.1 are extrapolated to the upstream tracker to form the so-called VeloUT tracks. These tracks are useful to reconstruct both low momentum particles bent out of the LHCb acceptance and higher momentum ones, reaching the SciFi after the magnet. In fact, VeloUT tracks are input to the forward tracking algorithm, which reconstructs long tracks at HLT1 level.

The "Compass" algorithm [START_REF] Fernandez | A Parallel-Computing Algorithm for High-Energy Physics Particle Tracking and Decoding Using GPU Architectures[END_REF] performs the extrapolation and reconstruction of VeloUT tracks in first high level trigger. As previously mentioned in this chapter, the UT is composed of 4 layers arranged in the x-u-v-x layout. A small fringe magnetic Once the first two hits are found, tolerance windows can be reduced to add the third and fourth hits. A track candidate is confirmed if it is composed of at least three hits. If more track candidates are found for the same input Velo track, a track fit is performed to each of them favoring the one with lowest χ 2 fit value.

Furthermore, the p T -kick method, described in the next chapter, is used to provide a momentum estimate and charge measurement of the track. The momentum resolution achieved is ∼ 15 %.

The reconstruction efficiency as a function of momentum and transverse momentum is shown in Figure 5.17.

Long track reconstruction

The reconstruction of long tracks, traversing all tracking subdetectors, is critical for LHCb as many trigger selections rely on them. It exploits the presence of the magnetic field between the UT and the SciFi to provide a percent-level momentum estimate for the reconstructed particles.

At HLT1 level, VeloUT tracks are extrapolated upstream of the magnet to the SciFi to form long tracks. The "Looking Forward" algorithm [START_REF] Scarabotto | Tracking on GPU at LHCb's fully software trigger[END_REF] exploits all the information from the VeloUT tracks, in particular their momentum and charge, to open search windows and look for hits in the three SciFi stations. The algorithm and its performance are presented in the next chapter.

Kalman fit

The Kalman filter is one of the most accurate methods to estimate track parameters and the corresponding covariance matrix [START_REF] Billoir | Track fitting with multiple scattering: A new method[END_REF]. It consists of a sequence of alternating predicting and updating of a particle state x using detectors measurements. Particles that propagate through the detector at certain z position along the beamline can be described by a 5-element state:

x = {x, y, t x , t y , q p }, (5.5) where (x,y) is the position of the particle in the x-y plane, (t x , t y ) the slopes in x-z and y-z planes, q is the charge and p the momentum of the particle. The error associated with the state is defined by a 5×5 covariance matrix, storing the correlations between all elements. Transport and noise matrices are finally used to transform both the particle state and its covariance matrix, by predicting and updating their elements at subsequent locations in the detector. These steps can be applied as many times as needed, to achieve the best possible estimate of the track parameters.

Due to the tight throughput requirements of HLT1, a simplified version of the Kalman filter is used, which trades off precision for speed. The calculations previously described can be replaced by a parametrization. As the main goal of the HLT1 Kalman fit is to improve the impact parameter (IP) resolution and χ 2 of the tracks, it is performed only in the Velo region. The default HLT1 Kalman filter performs a backward Velo track fit using the momentum estimate from the forward tracking. The IP resolution in both x direction and 3-dimension as a function of 1/p T and the IP χ 2 distribution are shown in Figure 5.18. The plots show the aforementioned distributions using two extra alternative implementations of the Kalman method:

• Simple Kalman: a parametrised Kalman fit applied on the full region of the LHCb detector;

• Full Kalman: full Kalman filter predicting and updating the particle state iteratively using transport and noise matrices.

As shown in Figure 5.18, the default "Velo-only" method achieves similar IP resolution performance compared to the two more complex methods just introduced. However, the full Kalman filter would also allow improved estimates of the momentum of the tracks and better ghost rejection.

Muon identification

Long tracks can be identified as muons by looking at corresponding hits in the four muon stations. Any contribution from the magnetic field can be neglected in the muon chambers region, expecting straight line trajectories. Long tracks with momentum greater than 3 GeV/c are extrapolated from the last layer of the SciFi to the muon stations as straight lines, in order to associate hits to them. Different requirements are set to perform the muon identification, the so-called isMuon, depending on the momentum of the particle:

• 3 < p < 6 GeV/c: minimum 2 hits matched to the long track;

• 6 < p < 10 GeV/c: at least 3 out of 4 stations with a hit associated;

• p > 10 GeV/c: all four stations with a matching hit.

The momentum intervals are evaluated from the muon penetration power in the iron absorbers between the muon stations [START_REF] Lanfranchi | The Muon Identification Procedure of the LHCb Experiment for the First Data[END_REF].

The muon reconstruction efficiency is defined as the number of long tracks correctly identified as muons over the number of reconstructible muons. It is shown as function of momentum in Figure 5.19.

Calorimeter reconstruction

Information from the electromagnetic calorimeter is used at HLT1 level to identify electrons and correct their momentum due to bremsstrahlung effects. The electron reconstruction is a novelty at HLT1 level and is extremely beneficial for physics analyses involving decays with electrons in the final state, as the study of the D 0 → h + h -e + e -channels. Long tracks are extrapolated as straight lines from the last SciFi layer to the calorimeter looking for matching energy deposits. The energy from all the matching cells is summed together and used to measure the energy over momentum value of the particle, E p . As explained in subsection 3.2.4, a particle is identified as an electron if E p > 0.7, meaning it deposited its full energy in the electromagnetic calorimeter. This requirement defines the so-called isElectron identification variable.

Furthermore, the information from the calorimeter is used to reconstruct bremsstrahlung photons and correct the momentum estimate of the electrons. The Velo track associated to a particle satisfying the isElectron requirement, is extrapolated as a straight line looking for energy deposits in the calorimeter. This corresponds to bremsstrahlung radiation emission from the material of the detector before the magnet. If matching cells are found, the energy associated with them is added to the momentum of the particle, updating it accordingly.

Secondary vertex finding

For the LHCb physics purposes and selection, it is important to be able to find and reconstruct decays of particles displaced from the pp collisions. Long tracks are matched together looking for displaced vertices in the Velo region, the so-called secondary vertices.

The secondary vertex reconstruction proceeds in the following steps:

1. Filter tracks: the reconstructed long tracks with momentum greater than 0.2 GeV/c are required to have a χ 2 track < 2.5 from the track fit and a trajectory not directed to the primary vertex, with a condition on the impact parameter evaluated from the Kalman filter χ 2 IP > 4;

2. Two-track vertex finding: the prefiltered tracks originated from the same primary vertex are combined in pairs and their point of closest approach (POCA) is evaluated. The POCA information is used to exclude parallel and noncrossing pair of tracks;

3. Vertex fit: a least mean squared fit is performed on each of the vertices found evaluating the χ 2 vertex . Each two-track vertex fit is required to have a covariance matrix positive definite and a requirement on the quality of the fit χ 2 vertex .

The composite particle objects build by the secondary vertex reconstruction are crucial when selecting events at HLT1 level, as explained in the next section.

HLT1 selections

A collection of trigger lines defines the sequence of reconstruction steps needed to select events and results in trigger decisions. In practice, many HLT1 lines share large portions of the reconstruction.

The HLT1 selections are composed of around a hundred lines mainly divided into four categories:

• Primary inclusive: constitutes the bulk of the LHCb physics program and select events based on their kinematics and topology;

• Calibration: selections for the large calibration samples essential for the datadriven evaluation of the performance of the detector;

• Exclusive: specific physics signatures not covered by the inclusive lines;

• Technical: trigger selections for luminosity determination, monitoring, calibration and alignment.

The primary inclusive lines corresponds to more than 90 % of the LHCb HLT1 trigger rate and are listed here:

• Two-track vertex: requires large p T of two tracks matching in highly displaced vertex from the primary vertices. Many non-muonic decays are selected by this line;

• Displaced single-track: requires large p T of a single track with a significant displacement from the PVs.

• Displaced single-muon: similar to the single-track line, but requiring the track to be identified as a muon;

• Displaced dimuon: requires two tracks identified as muons and forming a vertex in the Velo region;

• High-mass dimuon: selects two muon tracks forming a vertex without any displacement requirement but with a dimuon invariant mass above 2900 MeV/c 2 . It is particularly important for the study of charmonia, both produced promptly and in beauty decays;

• High-p T muon: requires a high p T muon track to be reconstructed within the event. It is used for the study of electroweak physics and searches for exotic signatures;

• Displaced single-electron: requiring a high-p T displaced track identified as an electron;

• Displaced dielectron: selects two electron track forming a displaced vertex.

The calibration selections mainly focus on decays of charm hadrons, such as D 0 →K -π + , which are critical for the data-driven evaluation of particle identification performance. The exclusive lines select decays from the final states measuring the invariant mass of the decaying particle, as D 0 →K + K -, D 0 →π + π -, K 0 S →π + π -, ...

Chapter 6

Long track reconstruction at HLT1

The reconstruction of long tracks traversing the whole LHCb detector is crucial in order to improve the momentum resolution of HLT1 selections from the 15 % achieved by the Velo-UT reconstruction down to values below 1 %.

This chapter introduces the forward tracking reconstruction which allows the extrapolation of Velo-UT tracks to the SciFi detector located after the magnet. Sequential algorithms have been developed in the past years to perform this task. These must be adapted to fully exploit the GPUs parallelisation power at HLT1 level. The "Looking Forward" algorithm, optimised for GPUs, achieves this task and it is presented in this chapter.

During 2022 data taking, the UT was not installed, leading to an immediate necessity of a tracking strategy modification. The author of this thesis is the main developer of a forward tracking algorithm which does not use any information from the UT detector. Details of the implementation of the algorithm are presented in this chapter. The new algorithm has been deployed in production since the start of LHCb commissioning in 2022 and is currently still in use during 2023 data taking.

Forward tracking

The forward tracking problem consists in extending tracks before the magnet, reconstructed in the Velo and UT subdetectors, to the SciFi tracker after the magnet. Particles are deviated by the magnetic field present in their path to the SciFi detector. The equation of motion of a charged particle of momentum ⃗ p, charge q and velocity ⃗ v in a magnetic field ⃗ B is:

d⃗ p dt = q⃗ v × ⃗ B. (6.1)
A mathematical simplification can be achieved by expressing the trajectory of the particle as two linear paths with a kick in the magnet center. The magnetic field can be seen as a lens kicking the flight path of particles from a direction into another at a specific position in space, similarly to an optical model. However, this simple model does not take into account the peripheral conditions of the LHCb magnetic field. The p T -kick method solves this problem by considering a non-constant position of the magnet bending plane and further correction factors [START_REF] Bowen | VeloUT tracking for the LHCb Upgrade[END_REF][START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF]. This method is used to estimate the momentum of a particle given a track state before and after the magnet. Inversely, it can be exploited, given an initial momentum and track state, to predict the track path in the SciFi detector region.

The p T -kick method

The effect of the magnetic field between two detectors is parametrised as an instantaneous kick to the momentum vector at the center of the magnet, z mag , as illustrated in Figure 6.1. The momentum kick, ∆⃗ p is defined as:

∆⃗ p = q • d ⃗ l × ⃗ B,
with an integral of ⃗ B the magnetic field along the path followed by the track.

Since the y and z component of the magnetic field can be neglected, the formula can be simplified as:

∆p x = p final x -p initial x = p( t x, f 1 + t 2 x, f + t 2 y, f - t x,i 1 + t 2 x,i + t 2 y,i ) = q • |d ⃗ l × ⃗ B| x ,
with t x,i (t x, f ) and t y,i (t y, f ) respectively the x and y initial (final) track slopes. The equation can be rearranged as:

q p = 1 q • |d ⃗ l × ⃗ B| x • ( t x, f 1 + t 2 x, f + t 2 y, f - t x,i 1 + t 2 x,i + t 2 y,i
).

(6.2)

In this way, it is sufficient to know the initial and final slopes of the track to have an estimate of the momentum of the particle. The values of

1 q• |d ⃗ l× ⃗ B| x
are stored in a lookup table and used by the algorithms for fast processing. Equation 6.2 can also be used in a reverse way to measure the final slope of the track from a momentum estimate. Figure 6.1: Illustration of the p T -kick method [START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF]. The method is used for both momentum estimations and measure search windows tolerances. The scheme shows the p T -kick between Velo and SciFi tracks, but can be applied to both Velo and UT or VeloUT and SciFi tracks.

From sequential to parallel

The sequential LHCb forward reconstruction is characterised by a histogramming method, the Hough transform [START_REF] Benayoun | The forward tracking, an optical model method[END_REF][START_REF] Callot | The Forward Tracking: Algorithm and Performance Studies[END_REF][START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF]. When extrapolating the Velo-UT tracks to the SciFi detector, by knowing the magnetic field between the two regions, a single hit in the SciFi allows to predict the momentum value and the expect path of the track in whole SciFi. Therefore, groups of hits sharing similar projected positions in x on a reference plane allow to determine the SciFi track segments associated with the Velo-UT input track. The projection of the hits can be achieved thanks to the p T -kick method explained in the previous section. The accumulations of hits on the reference plane are found through a peak-finding search on binned histograms, as shown in Figure 6.2. The group of hits matched to the input track is often referred as hough cluster. In summary, the algorithm iterates sequentially over the input tracks to consider all possible hits in the SciFi detector and build the hough clusters, forming long tracks. Figure 6.2: Hough cluster histogramming method performed by accumulating extrapolated measurements to a reference plane [START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF].

The Hough transform method is optimal for sequential algorithms as it iterates over the input tracks, however, this iteration cannot be parallelised.

An algorithm which can be parellalised over both input tracks and hits combinations is required. In order to do so, the input tracks should be treated in parallel by defining the regions of interest in SciFi detector where to look for hits. This can be done based on their momentum estimate using the p T -kick method. The hits found in such regions of the SciFi layers are combined in parallel to form track seeds. The hit combinations assume a track model in the SciFi region, which is the key element to build correct seeds and obtain the desired tracking performance. The seeds are the matched to the input track and filtered by a quality factor.

The "Looking Forward" algorithm exploits this parallelisation method.

Track model in the SciFi region

Parameterizing analytically tracks is crucial for high-throughput algorithms. At LHCb, the particles trajectories are deviated by the dipole magnet, as a function of their electric charge and momentum. Equation 6.1 leads to the following equations in the three dimensions considering the momentum components p x ,p y and p z :

dp x dz = q(t y B z -B y ), dp y dz = q(B x -t x B z ), dp z dz = q(t x B y -t y B x ), (6.3)
with t x = p x /p z = dx dz and t y = p y /p z = dy dz , the track slopes.

The two differential equations for the tracks slopes in the x-z and y-z planes respectively are: At LHCb, most of the tracks are highly boosted along the z axis and located in the central region where the dominant component of the magnetic field is B y . Following these assumptions, we can consider B x ∼ 0, B z ∼ 0 and approximate the equations for small |t x | and |t y | by keeping only the first order terms, leading to:

dt x dz = q p 1 + t 2 x +
d 2 x dz 2 = t x z ∼ - q p B y , d 2 y dz 2 = t y z ∼ 0. (6.5)
The y-z equation results in a simple linear model as:

y(z) = y 0 + t y (z -z 0 ), (6.6)
where y 0 is the y coordinate at the reference position z 0 . For the x-z track projection, we can consider the B y dependence on z at first order which is valid in the SciFi region:

B y (z) ∼ B 0 + B 1 (z -z 0 ). (6.7)
When we solve the x-z equation we have:

x(z) = x 0 + t x (z -z 0 ) + q p ( B 0 2 (z -z 0 ) 2 + B 1 6 (z -z 0 ) 3 )), = x 0 + t x (z -z 0 ) + q 2p B 0 (z -z 0 ) 2 (1 + dRatio(z -z 0 )), (6.8)
where x 0 is the coordinate at the reference position z 0 . The quantity dRatio = B 1 /3B 0 is roughly constant in the central region of the SciFi detector. In summary, the track fit model used in the long track reconstruction algorithms depends linearly on five adjustable parameters: two related to the y-z projection, y 0 and t y , while three concerning the x-z projection, x 0 , t x and B 0 q p . To write the final track model, we define a x = x 0 , a y = y 0 , c x = q 2p B 0 and dz = zz 0 leading to the so-called dRatio track model [START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF]:

x track (z) = a x + t x • dz + c x • dz 2 • (1 + dRatio • dz), y track (z) = a y + t y • dz.
(6.9)

The track fit is crucial to measure the track quality in any reconstruction algorithm and reject wrong combinations of hits. The model presented in Equation 6.9 is used to measure a χ 2 of each track as:

χ 2 = N hits ∑ i=1 ( x i -x(z i ) σ i ) 2 , ( 6.10)
where x i is the x-position of the hit i, x(z i ) the predicted x-value from the assumed track model at z i and σ i the hit resolution of the hit i. In HLT1, generally, a constant value of σ is assumed as the hit resolution is not stored to save memory.

Long track reconstruction algorithms

The algorithms allowing the long track reconstruction optimised for GPUs are presented in this section. The "Looking Forward" algorithm was the first long track reconstruction method developed for GPUs, paving the path for new algorithms to be developed.

A new algorithm reconstructing tracks with the forward method without using the UT information is presented [START_REF] Scarabotto | Tracking on GPU at LHCb's fully software trigger[END_REF]. Furthermore, an algorithm that first reconstructs track seeds in the SciFi detector and then matches them to Velo segments is developed and tested on GPUs, referred as "Seeding & Matching" [START_REF] Calefice | Standalone track reconstruction on GPUs in the first stage of the upgraded LHCb trigger system and Preparations for measurements with strange hadrons in Run 3[END_REF]. The forward and seeding&matching methods can be combined in a sequence in order to enclose the advantages of both approaches. An algorithm which performs this combination is presented in this section.

Looking Forward

The "Looking Forward" algorithm [START_REF] Hugo | Optimization of high-throughput real-time processes in physics reconstruction[END_REF] is composed of four main steps:

1. Search windows opening: search windows tolerances are evaluated in each SciFi layer extrapolating the input VeloUT tracks;

2. Triplet seeding: triplets of hits are combined using the x layers information;

3. Create tracks: triplets are extended to the other layers and a track fit is performed;

4. Quality filter: candidates are filtered and selected thanks to a track quality factor based on the fit χ 2 .

Search windows opening

VeloUT tracks are extrapolated to the SciFi stations to evaluate the search windows size. The track slope in the non-bending plane is assumed to be constant, t Velo y = t SciFi y . The t y information is first used to understand if the particle is traveling in the upper or lower half of the SciFi, as both detector halves are read out separately. As the bending of the tracks is in the x-z plane, a crossing of both halves is very unlikely. This allows the algorithm to only consider half of the SciFi hits for each track.

In the x-z plane, the p T -kick method, see subsection 6.1.1, is used to estimate the slope of the track in the SciFi region, t SciFi x The estimated momentum from the VeloUT tracking is therefore used.

Each VeloUT track is extrapolated from the z-position of the center of the magnet, z magnet , and then, with the p T -kick method, extrapolated to every SciFi layer using t SciFi x and t SciFi y . In this way, the extrapolated center of the search windows is defined in every layer i as, (x i extrap ,y i extrap ). The size of the search windows in the x layers is defined as:

x min = x extrapx tolx left asym ,

x max = x extrap + x tol + x right asym ,
with [x min , x max ] the search window region in x, x tol the window size or tolerance factor, and x left/right asym a magnetic field asymmetric factor. The x tol factor defines the search window size as:

x tol = 150mm + 2000GeV/c p ,
as a function of the momentum p of the particle. The last term, x left/right asym , takes into account the fact that the magnetic field is not a pure dipole, but is affected by fringe effects mainly away from its center. It is defined as:

x left asym = 100mm, if q = +1 & MagUp OR q = -1 & MagDown 0mm, otherwise , x right asym = 100mm, if q = -1 & MagUp OR q = +1 & MagDown 0mm, otherwise , (6.11)
depending on the charge of the particle q and the polarity of the magnet MagUp or MagDown.

The center of the search window in the u/v layers is defined by extrapolating the track as a straight line from the neighbouring x layer and correcting by the ± sin 5 • factor, respectively of the u and v layer. A tolerance of ±800mm around the center is used to look for hits in the u/v layers.

A hard limit of 32 hits for both x and u/v layers is set for each VeloUT track in order to constrain the number of combinations in the next steps of the algorithm. These are chosen symmetrically around the center of each window.

Triplet seeding

The hits found in the x layers search windows are combined to form tracklet candidates. Triplets of hits (h 0 ,h 1 ,h 2 ) are searched for, either in the first x layer of each T-station, or on the last one. For each VeloUT track, the hits in the first and last layers of each station are tested combining all possible hit doublets (h 0 ,h 2 ). Two conditions are used to accept or reject doublet candidates:

• the bending of the track from the Velo to the UT is in the same direction as the one from the Velo to the SciFi doublet. This condition is used if the input track has momentum < 5 GeV/c, otherwise the bending in the VeloUT region is not sufficient for an accurate charge determination;

• the straight line extrapolation to the center of the magnet z mag , x-value difference:

∆x at z mag = x VeloUT mag -x SciFi mag < O mag (p VeloUT , t doublet x ),
with O mag the x-opening difference condition at z mag dependent on the momentum p VeloUT of the VeloUT track and on the doublet straight line slope t doublet

x . Its value ranges from 8 to 40 mm depeding on the momentum.

These two conditions are used to reject wrong doublet candidates and exclude them from further extrapolations.

Once the doublets are prefiltered, the hit in the second T-station h 1 is added considering a straight line extrapolation between h 0 and h 2 with a sagitta-like correction:

x expected 1 = z 1 • t (h 0 -h 2 ) x + (x 0 -t (h 0 -h 2 ) x • z 0 ) • K x 1 , with x expected 1
the extrapolated x-position of the h 1 hit, z i and x i the z-and x-positions of the respective hits, t (h 0 -h 2 ) x the slope in the bending plane between the h 0 and h 1 hits and K x 1 a sagitta-like correction factor. The value of K x 1 is different for the two cases considered, as it is dependent on the magnetic field: K x 1 = 1 + 1.77513 • 10 -3 when searching for triplets in the first x layers of the stations and K x 1 = 1 + 1.42634 • 10 -3 when searching in the last x layers of each station.

The hit with x-value closest to x ) 2 < T triplet , with T triplet the tolerance maximum value reported in Table 6.1. Triplets passing this condition are validated as tracklet candidates. The maximum number of selected triplets is set to 12 per VeloUT track. If the maximum number is exceeded, the 12 candidates with lower χ 2 triplet are selected.

Create tracks

The triplet candidates formed are extended by looking for hits in the remaining layers. The extrapolation is performed using the model presented in Equation 6.9 in order to look for matching hits. First, tracklets are extended to the three missing x layers, computing the expected x-position of the hit in layer i:

x expected i (z) = a x + t x • dz i + c x • dz 2 i • (1 + dRatio • dz i ), (6.12)
where the values of a x ,t x ,c x and dRatio are evaluated using the position information from the triplet hits and a parametrization of the magnetic field valid for dz i = z iz ref , the difference between the z-position of the i layer and a reference plane at z ref = 8520mm [START_REF] Quagliani | Study of double charm B decays with the LHCb experiment at CERN and track reconstruction for the LHCb upgrade[END_REF]. For each x-layer i, the hit with x-value closest to x expected is added to the triplet if χ 2

x i = (x hit i -x expected i
) 2 < T x with T x a tolerance factor, reported in Table 6.1.

Each tracklet candidate is then extended looking for hits in the u/v layers. Similarly, using Equation 6.12, the x expected is evaluated for each u/v layer correcting by the ± sin 5 • factor.

A hit in the u/v layer j is added to the candidate if χ 2 uv j = (x hit jx expected j

) ) a tolerance parameter dependent on VeloUT track t x and t y , see Table 6.1. The dependence of the parameter on the track slopes allows higher χ 2 tolerances for more peripheral tracks in both x an y.

A tracklet is finally accepted as a track candidate if it is composed of a minimum of 9 hits, counting both x and u/v layers, with at least one hit in the u/v layers per T-station.

Quality filter

The final step of the algorithm filters the track candidates based on their quality factors. First, a linear mean square fit in the y-direction is performed on the candidates evaluating the t expected y slope and χ 2 value χ 2 y i = (y hit iy expected i

) 2 based on the expected and measured y-positions of the u/v hits. The track slope in the nonbending plane is used to reject candidates with ∆t y = t candidate y t expected y > T t y , with T t y a tolerance factor on the y-z slope, with its value reported in The χ 2 tolerances relative to the x-positions of the hits when forming the triplets T triplet and when adding the x-layers T x and uv-layers hits T uv are shown. The tolerance relative to the y-position T y has much larger value compared to the x-position due to the fact that the information only comes from the uv layers and have a poorer resolution. T t y represents the y-z slope difference tolerance when performing the mean squared fit in the y direction.

The total x and y quality factors are determined as:

Q x = N x hits ∑ i=0 χ 2 x i T x + N uv hits ∑ j=0 χ 2 uv j T uv , Q y = N uv hits ∑ i=0 χ 2 y i T y , (6.13)
with χ 2 x i and χ 2 uv j the χ 2 values evaluated in the previous section for each added x-hit i and uv-hit j normalized by their tolerances T x and T uv and χ 2 y-fit the y-fit χ 2 normalized to its tolerance T y . As shown in Table 6.1, the y tolerance values are higher compared to the ones in x, due to the fact that fewer hits are used for the x-y plane fit (only uv hits) and their resolution is poorer compared to the x layers hits.

The total quality factor is evaluated for each track candidate as:

Q = ( Q x nDoF x + Q y nDoF y ) • C(n hits ), (6.14)
with nDoF x-y the fit degrees of freedom in the x and y direction and C(n hits ) a multiplicative parameter dependent on the number of hits which compose the track candidate. The values of C(n hits ) are reported in Table 6.2 and favour tracks composed of more hits. The number of degrees of freedom nDoF x when performing the fit in the x-z plane is nDoF x = n total hits -3 as only information from three x-layers is used. In the y-z plane, nDoF x = n uv hits -2 as only two uv-hits are used for the linear fit. Track candidates are accepted as reconstructed if their quality factor is lower than the tolerance T Q (value reported in Table 6.1). If more than one candidate per single VeloUT track is found, the one with lowest value of Q is kept.

Forward tracking without UT

The upstream tracker was not installed during 2022 data taking leading a necessity of modification of the forward algorithm. The "Forward no UT" algorithm extrapolates Velo tracks directly to the SciFi looking for matching hits [START_REF] Scarabotto | Tracking on GPU at LHCb's fully software trigger[END_REF]. The UT information is used to have a first momentum estimate of the tracks downstream of the magnet. This really helps to reduce the number of hit combinations needed n hits C(n hits ) value 9 5.0 10 1.0 11 0.8 12 0.5 to perform the long track reconstruction, as the search windows size can be greatly reduced. Furthermore, the charge measurement provided by the VeloUT tracking indicates the bending direction in the x-z plane. Without this information, the search windows opening step needs to be completely revisited, as both charge assumptions must be considered. In order to limit the search windows size and limit the number of combinations, the algorithm assumes a minimum momentum of the track of 5 GeV/c or p T of 1 GeV/c. This limits the efficiency of the algorithm to a subset of higher momentum tracks.

The "Forward no UT" follows the same sequence of steps as the "Looking Forward" algorithm with modifications needed to account for the missing UT information. The main variation is applied when defining the search windows tolerances. Smaller modifications are performed in the remaining steps of the algorithm.

Double-sided search windows

The Velo input tracks are extrapolated as a straight line directly to the various SciFi layers to find the x extrap value. As the track momenta are unknown, they are assumed to be the maximum value between p Velo = 5 GeV/c or p Velo = p(p T = 1 GeV/c, t Velo x ,t Velo y ), calculating the momentum using the p T = 1 GeV/c requirement and the (t Velo x ,t Velo y ) slopes values of the Velo track. The following formula applies to calculate the momentum from the transverse momentum:

p(p T , t x , t y ) = p T cos(arctan( 1 √ t 2 x +t 2 y
)) .

The Velo track t y slope is still used to determine which half of the SciFi, lower or upper, can be dismissed when extending the track to the various layers. As no charge measurement is provided by the Velo tracking, both positive and negative charge assumptions are taken into consideration. Double-sided search windows in the x direction are determined, left and right from the center of the extrapolation x extrap . A scheme of the search window strategy is presented in Figure 6.3.

The tolerance of the search window is evaluated with the following a polynomial approximation [START_REF] Hasse | Alternative approaches in the event reconstruction of LHCb[END_REF] assuming the momentum of the Velo track p Velo defined before: (6.15) with c i the coefficients defined in Table 6.3 and t x and t y respectively the slopes in x-z and y-z plane of the Velo input track. The polynomial approximation is used to determine a curvature correction assuming a minimum momentum. It is applied Figure 6.3: Scheme of the search window strategy for the "Forward no UT" algorithm. The Velo track is extrapolated as a straight line to the SciFi layers and double-sided search windows are opened taking into account both charges assumption. The size of the windows is determined by the minimum momentum assumed for each Velo track, p > 5 GeV/c or p T > 1 GeV/c. A small overlap of 1 cm is allowed between the two windows to account for very high momentum tracks, expected to travel in a straight line. instead of the more common Runge-Kutta method [START_REF] Kutta | Beitrag zur näherungsweisen Integration totaler Differentialgleichungen[END_REF], an iterative procedure to approximate solutions of ordinary differential equations, which provides a higher precision result but making it also more computationally demanding. 6.3: Polynomial coefficients tuned on simulation approximating the curvature trajectory of the Velo input track, assuming a minimum momentum p [START_REF] Hasse | Alternative approaches in the event reconstruction of LHCb[END_REF]. They are used in Equation 6.15 to evaluate the size of the search windows.

x tol = 1 p Velo [c 0 + t x (c 1 + c 2 t x ) + t 2 y (c 3 + t x (c 4 + c 5 t x )) + 1 p Velo (c 6 t x + c 7 1 p Velo )],
The double-sided LEFT-RIGHT search windows sizes in the x-layers of the SciFi are defined as: (6.16) with x extrap the Velo track straight line extrapolation x-position in the layer, x tol the window tolerance defined in Equation 6.15 and x overlap = 5 mm an overlap factor between the two windows. As the track extension is performed separately for the two charge assumption, the overlap factor avoids central hits to be considered on only one of the windows. This avoids missing hits for very high momentum tracks which are expected to travel in a straight line around the x extrap value.

x LEFT min = x extrap -x tol , x LEFT max = x extrap + x overlap , x RIGHT min = x extrap -x overlap , x RIGHT max = x extrap + x tol ,
The uv layer search windows size follow a similar approach as Equation 6.16, assuming a x tol = 1200 mm. Similarly, x extrap is evaluated for the u/v layers by extrapolating as a straight line the track using the neighbouring x-layer information and correcting by the ± sin 5 • factor, respectively of the u and v layer.

The limit of hits for each search window is set to 64 hits chosen symmetrically at the center of them. Compared to the default forward algorithm, this limit is doubled as no momentum information is available at this stage and search windows tolerances are higher. In order to compensate for the greater number of hits processed in All the SciFi seeds reconstructed in this way are then combined by a matching algorithm to the Velo tracks:

1. Matching in y-z plane: both Velo and SciFi tracks are extrapolated as a straight line to the matching position z match y = 10 m just behind the last SciFi station. Requirements on the ∆t y , difference of the y-z slopes, and ∆y, difference of y-positions at z match y , are defined to reject wrong combinations.

2. Matching in x-z plane: Velo and SciFi segments are extrapolated as straight lines to the z mag value, similarly to p T -kick method subsection 6.1.1. The ∆t x , difference of the x-z slopes, and ∆x, difference of x-positions at z mag , are measured and used to define a χ 2 match , together with y-z plane variables ∆t y and ∆y. The matching χ 2 match variable is evaluated for each Velo-SciFi tracks combinations. For each SciFi seed, the best χ 2 match value combination with a Velo track is kept, if below a certain χ 2 tolerance.

The seeding&matching reconstruction achieves higher efficiencies at low p or p T compared to the forward tracking algorithm, which is better at high p or p T .

The "Best long track combiner" algorithm combines forward and seeding&matching reconstructions to achieve the best possible tracking efficiency across all momenta of the particles. One of the two algorithms is first run on the full dataset and all hits composing reconstructed tracks, from Velo, UT and SciFi, are flagged and removed from it. The second algorithm is then run on all the unflagged hits. Finally, all reconstructed tracks are used to measure the performances of the sequence. Both combinations are allowed, first running the forward reconstruction and then on the unflagged hits the seeding&matching, or vice versa. They are referred in the performance section as Forward then matching or Matching then forward.

Reconstruction performances

The physics performance of the "Looking Forward", "Forward no UT" and "Best long track combiner" algorithms are evaluated on simulation and discussed in this section. Track categories can be defined using the truth/generated information of the particles from MC, like their generated momenta or origin particle:

• Long: tracks with at least three hits in the Velo, three hits in the UT and ten hits in the SciFi detector;

• Long p > 5 GeV/c & p T > 1 GeV/c: long tracks with momentum greater than 5 GeV/c and transverse momentum greater than 1 GeV/c. The forward method which does not use the UT information is designed and optimised to obtain the best performances in this subrange of momenta;

• from B: tracks required to be originated from a b-hadron decay;

• from strange: tracks originated from a s-hadron decay;

• electrons: electrons tracks, matched with the generated particle ID.

The list of indicators to assess the performance of an algorithm are defined in section 6.1. The two main indicators of interest are the track reconstruction efficiency and ghost rate as a function of p, p T and η. The performances are obtained by running the reconstruction over 5000 simulated events of B 0 s → ϕ ϕ decays. In order to test the electron tracking efficiency, a simulated sample of B 0 →K * e + e -is used. Furthermore, the momentum resolution of long tracks is defined as: dp p = p recop true p true , with p reco the momentum evaluated after the track reconstruction, while p true the momentum from simulation of the associated particle. The momentum resolution as a function of momentum distributions are presented in the next sections for the various algorithms. The throughput of the HLT1 sequence is evaluated by running over simulated minimum bias events with different reconstruction sequences. The sequences are tested for different GPU and CPU cards.

The clone rate, hit efficiency and hit purity as defined in section 5.2 are evaluated for the three algorithms presented in this chapter. Their values are reported in Table 6.4 for both "long" and "long from B" categories, with or without momentum requirements. As shown in the table, the clone rate is always below 1 %. A clone killing procedure which compare the hits of each reconstructed track could be implemented to bring this value to zero. The hit purity, representing the fractions of hits of the reconstructed track in common with the matched track, is above 99 % for all algorithms. The hit efficiency is above 96 % for the three algorithms. The highest hit efficiency is reached by the "Best long track combiner" method as the algorithm achieves a better performance at low momentum where it is more likely to miss a hit.

clone rate [%] hit efficiency [%] hit purity [

Forward performance

The tracking efficiencies are evaluated on a sample of 5000 B 0 s → ϕ ϕ events to test the long from B track category as a function of p, p T and η. The distributions are shown respectively in Figure 6.4 and Figure 6.5 achieving a tracking efficiency above 90 % for tracks with p T > 1 GeV/c. The inefficiency at lower momentum is due to two main factors:

• the VeloUT reconstruction inefficiency below 1 GeV/c of transverse momentum, as shown in the previous chapter in Figure 5.17. Moreover, VeloUT tracks are required to have a minimum p T of 0.4 GeV/c to be extrapolated to the SciFi detector.

• the limited number of hits considered in each tolerance window considered in the SciFi layers. This constraint has to be applied to limit the number of hit combinations performed and though, the throughput.

The algorithm is characterised by a ghost rate around 3 % across a large a range of momenta, as shown in Figure 6.6. The ghost rate increases at low momentum due to similar reasons stated for the tracking efficiency. As particles at low momentum are bending more, it is more likely to mis-model their trajectory and form tracks made of wrong hits. Furthermore, the ghost rate slightly increases at high momentum, as particles tend to travel in a straight line and it is more likely for the algorithm to select wrong hits around the trajectory.

A momentum resolution < 1 % is needed for the LHCb physics purposes selections. Its distribution as a function of momentum is shown in Figure 6. [START_REF] Aaij | Observation of the resonant character of the Z(4430)state[END_REF] Furthermore, the throughput of the full HLT1 sequence is tested on different GPUs and CPUs. A throughput of around 170 kHz is shown in Figure 6.8 for Invidia RTX A5000 cards, default for the HLT1 trigger. This means around 175-180 GPU cards are needed to take care of the full 30 MHz LHC input data rate, matching perfectly the current requirements.

In Figure 6.9, the electron and non-electron tracking efficiencies are compared using B 0 →K * e + e -and B 0 s → ϕ ϕ decay samples. The electron efficiency is around 10-20 % lower compared to non-electron one. Electrons interact more with the material of the detector compared to other particles, changing direction and losing energy due to bremsstrahlung radiation, thus making their reconstruction more difficult.

Forward no UT performance

The goal of the "Forward no UT" algorithm is to achieve similar tracking performances as the nominal forward algorithm for tracks with momentum p > 5 GeV/c and p T > 1 GeV/c, respecting the tight throughput requirements. The efficiencies as a function of p, p T and η are reported in Figure 6.10 for the track category long from B with p > 5 GeV/c & p T > 1 GeV/c and compared to with UT version of the algorithm. The reconstruction efficiency is above 90 % for tracks with p T above 1 GeV/c for both algorithms. The forward method which does not use the UT information is optimised to reconstruct high momentum tracks which tends to have a trajectory close to a straight line. This is why at high momentum and transverse momentum, for example above 50 GeV in momentum or 3 GeV in p T , the algorithm which does not use the UT information performs better than the one using the UT information.

However, due to larger amount of hit combinations computed in the forward algorithm without UT, its ghost rate is larger than in the forward reconstruction with UT, see Figure 6.11. The ghost rate increases as function of momentum as tracks tend to have trajectories similar to straight lines and wrong combinations of hits are easier to be selected and pass the quality filter requirement. The momentum resolution is shown in Figure 6.12 and is comparable between the with UT and no UT algorithms. In fact, for tracks with transverse momentum greater than 1 GeV, the information added by the upstream tracker does not improve the track modeling but mostly helps to reject ghost tracks. It must be considered that in the HLT1 reconstruction no Kalman fit is used on long tracks. In HLT2, the Kalman method improves the momentum resolution of long tracks which include the UT hits by around 20 % compared to the ones reconstructed only by hits in the Velo and SciFi subdetectors.

The throughput of the HLT1 sequence is compared for the with UT and no UT configurations in Figure 6.13 testing different GPUs and CPUs. The no UT sequence excludes all the algorithms using the UT information, simulating a real data taking condition. No UT raw data decoding or VeloUT tracking (subsection 5.2.3) is included. The throughput performance of both sequences is very similar (∼ 170 kHz).

Best long track combiner performance

The "Best long track combiner" algorithm combines the "Looking Forward" and "Seeding & Matching" reconstructions to obtain the best possible physics performance. The plots shown in this section always present three distributions: the performance of the forward algorithm, the seeding&matching and the best long track combiner. Both configurations presented in subsection 6.2.3, are shown: Forward then matching and Matching then forward.

The tracking efficiency distributions as a function of p, p T and η are shown in the Forward then matching Figure 6.14 and Matching then forward Figure 6.15 configurations. The best long track combiner algorithm manages to gain the advantage of the seeding&matching reconstruction at low momentum together with better high momentum efficiency of the forward. The Forward then matching configuration seems to perform generally better, as the forward method has lower ghost rate. The ghost rate as a function of p T is shown in Figure 6. [START_REF] Christenson | Evidence for the 2π Decay of the K 0 2 Meson[END_REF].

The throughput is tested and reported in Figure 6.17. The HLT1 sequences using the best long track combiner algorithm are around 30 % slower than the nominal sequence. s → ϕ ϕ samples, testing the long from B track category. The plots show also the same distributions for the "Looking Forward" algorithm (in black) and "Seeding&Matching" algorithm (in red) for comparison. Figure 6.16: Ghost rate of the "Best long track combiner" algorithm (in blue) as a function of p T . The ghost rate is evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B track category. The plots show also the same distributions for the "Looking Forward" algorithm (in black) and "Seeding&Matching" algorithm (in red) for comparison. Figure 6.17: Throughput of the full HLT1 sequence using the "Looking Forward" or "Best long track combiner" algorithm to reconstruct long tracks for comparison. The values are reported for different GPU and CPU cards. Both configuration of best long track combiner algorithm are tested: Forward then matching and Matching then forward.

Chapter 7 Conclusion

This thesis presents the first LHCb study of the yet unobserved D 0 → π + π -e + e - and D 0 → K + K -e + e -rare charm decays. As a first step, a selection is implemented in order to optimise the background rejection and favor the signal. A multivariate selection based on a boosted decision tree algorithm is used to suppress background from random combinations of charged particles. A second source of background is due to fully hadronic four-body D 0 decays with two hadrons mis-identified as electrons which can be reduced by a multivariate electron identification discriminant. A fit model is implemented to statistically separate the signal from the backgrounds. The fit is performed simultaneously in all dilepton mass regions to enhance the sensitivity of the search. The efficiencies are evaluated from the simulated samples and corrected for data-simulation differences. The branching fractions are measured relative to a normalisation channel, the Cabibbo-favored D 0 → K -π + e + e -decay. Expected upper limits are estimated for both D 0 → π + π -e + e -and D 0 → K + K -e + e - decays, leading to an analysis sensitivity of O(10 -7 ). The dominant systematic uncertainty is relative to the decay model used to generate the simulated samples, which is currently unknown. However, the results are statistically dominated, opening the possibility to be improved with a larger data sample. The analysis is currently under review and a subsequent paper is in preparation.

The LHCb upgrade allows to collect more data by increasing the instantaneous luminosity by a factor five. One of the main limitations of the Run 1-2 detector is the first-level hardware trigger which is removed in the upgrade, such that selections are exclusively based on the event reconstruction. As a preliminary measurement, the trigger efficiency evaluated on simulated samples of D 0 → K -π + e + e -decays is improved by more than a factor three comparing the Run 1-2 and Run 3 firstlevel trigger selections. Furthermore, the new fully-software trigger system can help to reduce the trigger efficiency difference between muons and electrons which is beneficial when studying LFU ratios.

The event reconstruction at the 30 MHz LHC proton-proton collision rate is performed on GPUs at the LHCb first high level trigger. As one of the detectors (upstream tracker) used in tracking algorithms was not yet installed during the 2022 data taking, this required the development of a new method to reconstruct the tracks without relying on the missing information. The tracks reconstructed in the Velo are extrapolated directly after the magnet looking for matching hits in the layers of the SciFi detector. The Velo reconstruction does not provide any charge assumption or momentum estimate to the reconstructed particles, as no magnetic field is present in the Velo region. When extrapolating the Velo tracks to the SciFi detector, a minimum momentum is considered to reduce the size of the tolerance ranges. The algorithm achieves a 90 % tracking efficiency for long tracks with minimum momentum of 5 GeV and transverse momentum of 1 GeV, while respecting the HLT1 throughput

A.2 Supplemental material on the resonance model correction

The resonance model correction presented in subsection 4.3.1 exploits the results of the D 0 → h + h -µ + µ -analysis from Ref. [START_REF] Aaij | Angular analysis of D 0 → π + πµ + µand D 0 → K + Kµ + µdecays and search for CP violation[END_REF] to correct the electron modes resonant model generated in simulation. When applying this correction, the reconstruction resolution effects are not taken into account for the dimuon and dihadron mass. This is a fairly good assumption for hadrons and muons as shown in electron PID 18 16 14 12 10 9.00 8.00 6.00 4.00 2.00 10 9.00 8.00 7.00 6.00 5.00 4.00 4.00 3.00 2.00 7.00 6.00 5.00 5.00 4.00 3.00 3.00 3.00 2.00 1.00 4.00 4.00 3.00 3.00 3.00 2.00 2.00 2.00 1.00 1.00 3.00 2.00 2.00 2.00 2.00 1.00 1.00 1.00 1.00 1.00 2.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 5.00 5.00 6.00 7.00 9.00 10 7.00 5.00 3.00 2.00 4.00 4.00 4.00 4.00 5.00 5.00 3.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 2.00 2.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 0.00 0.00 3.00 4.00 4.00 5.00 5.00 5.00 4.00 3.00 2.00 2.00 2.00 2.00 3.00 3.00 3.00 3.00 2.00 2.00 2.00 1.00 1.00 2.00 2.00 2.00 3.00 3.00 2.00 2.00 1.00 1.00

1.00 1.00 1.00 2.00 2.00 2.00 2.00 2.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 Brem1+ category

A.3.2 Trigger TISTOS method

The appendix shows correction weights distribution evaluated on B 0 → K * (→ K -π + )J/ψ(→ e + e -) data and simulation calibration samples. The weights are evaluated by the TITOS method as explained in subsection 4. Table A.8 shows the effect of the correction on the efficiency ratio ϵ(D 0 → K -π + [e + e -] ρ 0 /ω )/ϵ(D 0 → h + h -e + e -). The effect here is always below 0.5 %.

Table A.8: Effect of the tracking correction procedure on the efficiency ratio for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays in integrated and single dilepton bins.

A.6 Supplemental material on toys studies

Pseudoexperiments are generated as explained in subsection 4.4.6 to validate the fit strategy defined in section 4.4. In subsection 4.4.6, toys are run assuming as signal branching fraction the muon modes values reported in Table 2.2. Furthermore, the fit strategy is tested assuming different signal BFs, in specific 10 -6 , 10 -7 , 10 -8 , 10 -9 for each of the dilepton mass bin. The results for both D 0 → π + π -e + e -and D 0 → K + K -e + e -are reported respectively in Figure A.13 and Figure A.14. The pull distributions are consistent with a normal distribution pulls down to BF = 10 -8 , except for the low statistics η mass bins. When reaching a BF = 10 -9 , the lower statistics dilepton mass bins (signal yield expected reported in the legends), reach a non-gaussian regime needing a different strategy to correctly evaluate the uncertainty.

A.7 Supplemental material on background studies

Different types of backgrounds are studied in section 4.5.3, including mis-identified and partially reconstructed. As a first study, the decays listed in the section are generated with RapidSim and the D 0 mass distributions evaluated assuming the signal final states mass hypotheses. The plots are shown in

Figure 2 . 2 :

 22 Figure 2.2: Unitary triangle in the ρ-η plane with a global fit to the experimental constraints superimposed [23].

Figure 2 . 3 :

 23 Figure 2.3: FCNC c →u ℓ + ℓ -transitions, the electroweak penguin (left) and box (right) diagrams are illustrated [25].

Figure 2 . 5 :

 25 Figure 2.5: Expected differential branching fraction distributions as a function of the squared dielectron mass m 2 (e + e -) (modified from [61]). The plots show the distributions for D 0 → π + π -e + e -(left) and D 0 → K + K -e + e -(right) decays.

Figure 3 . 1 :

 31 Figure 3.1: The CERN accelerator complex.

. 2 .

 2 Their peculiarity is the strong asymmetry of the incoming parton momenta in the laboratory frame which causes the production to happen mainly in a forward or backward cone around the beam pipe. The azymuthal angle distribution of the bb quarks pair is shown in Figure 3.3a with the red zone representing the LHCb detector acceptance. The detector covers a pseudorapidity η range of 2 < η < 5, which corresponds to about a fourth of the total number of bb pairs produced. The comparison of the geometrical acceptance compared to ATLAS and CMS is shown in Figure 3.3b.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Feynman diagrams for the production of a pair of bb quarks in a pp collision at the LHC.

Figure 3 . 4 :

 34 Figure 3.4: Instantaneous luminosity for ATLAS, CMS and LHCb during a long LHC fill in Run 1[START_REF] Aaij | LHCb Detector Performance[END_REF]. The luminosity at the LHCb experiment kept stable in a range of 5 % once reaching the nominal value of 4 × 10 32 cm -2 s -1 by adjusting the beam overlap.

Figure 3 . 5 :

 35 Figure 3.5: Side view of the LHCb experiment during the LHC Run1-2.

Figure 3 . 7 :

 37 Figure 3.7: (A) Primary vertex resolution along x in Run1 and Run2 data taking period as a function of the number of tracks composing the vertex. (B) Impact parameter resolution distribution along the x axis as a function of the inverse transverse momentum measured on Run1 and Run2 data [77].

Figure 3 . 8 :

 38 Figure 3.8: (A) Picture of the assembly of the VELO silicon modules [79]. (B) Schematic of the R and Φ sensors for a VELO module [79].

Figure 3 . 9 :

 39 Figure 3.9: Overview of the VELO detector layout. The modules are also illustrated in both open and closed positions. R and Φ sensors are shown in red and blue respectively [79].

Figure 3 . 10 :

 310 Figure 3.10: The Tracker Turincensis detector design. It is composed of 4 layers in x-u-v-x composition.

Figure 3 .Figure 3 .

 33 Figure 3.11: (A) View of the Outer Tracker. It consists of three T-station (T1,T2,T3) each composed of four layers in the x-u-v-x configuration [81]. (B)Layout of the x-layer (top) and stereo-layer (bottom) for the Inner Tracker [80].

Figure 3 .

 3 Figure 3.13: (A) Side view of RICH1, it uses C 4 F 10 gas as radiator. (B) Side view of RICH2, it uses CF 4 gas as radiator [83].

Figure 3 . 14 :

 314 Figure 3.14: Reconstructed Cherenkov angle θ as a function of momentum for tracks crossing RICH1 C 4 F 10 radiator, for different charged particles in Run1 data [84].

Figure 3 . 15 :

 315 Figure 3.15: Identification of photons, electrons and hadrons with the LHCb calorimeter system.

Figure 3 .

 3 Figure 3.18: (A) Kaon PID performances during 2015 data taking. Kaon efficiency (K → K) and misidentification (π → K) for two different PIDK cuts: > 0 and > 5 (B) Electron PID performances during 2015 data taking. Electron efficiency (e → e) and mis-identification (π → e) for the two different PIDe cuts [88].

. 19 .

 19 The figure shows performances of both ProbNNµ and ProbNNp compared to their respective ∆LL variables.

Figure 3 . 19 :

 319 Figure 3.19: Background rejection rates as function of muon (left) and proton (right) identification efficiency. The variables ∆L and ProbNN are plotted respectively in black and red for comparison, using 2012 data taking samples [89].

Figure 3 . 20 :

 320 Figure 3.20: LHCb trigger processing diagram for Run2 [90].

  These lines are optimised for selecting decaying cand b-hadrons, whose decays are typically displaced from any PV. The efficiency of selecting various charm decays of the single-track line, two-track line and a combination of both is shown in Figure 3.21. A fast muon identification is performed as well, matching hits in the muon chambers with tracks reconstructed in the VELO. The muon-ID information is used to select displaced muons with large p T and dimuon displaced vertices.

Figure 3 . 21 :

 321 Figure 3.21: Efficiency of the HLT1 single-track line (left), two-track line (middle) and a combination of both (right) as a function of the c-hadron p T [77].

Figure 3 . 22 :

 322 Figure 3.22: The Run2 LHCb data flow with the associated applications for both real data and simulation.

Figure 3 .

 3 Figure 3.25: (A) Mass distributions of simulated D 0 → K -π + e + e -and D 0→ K -π + [µ + µ -] ρ 0 /ω decays.The mass resolution of the electron mode is more than three times larger than the muon mode due to bremsstrahlung effects. (B) Mass distributions of D 0 → K -π + e + e -decays from simulated samples, without bremsstrahlung photons attached to the electrons (red -brem0 category) and with one or more photons attached (blue -brem1). The brem0 distribution shows a large tail on the left-hand side of the peak due to the brem radiation energy loss. In the brem1 category, the tail is less pronounced thanks to the brem recovery algorithm which centers the distribution around the peak. The right-hand tail is caused by wrong photon to electron associations and also by the limited resolution of the LHCb calorimeter.

Figure 3 . 26 :

 326 Figure 3.26: Bremsstrahlung radiation z-position origin in the LHCb detector. Regions of VELO (z < 1.5m), TT (2 < z < 3m) and T-stations (7 < z < 10m) are shown. The total number of bremsstrahlung photons are illustrated in blue, while in red all reconstructible ones (p T > 75 GeV/c) and in green the correctly reconstructed ones [102].

  + e -) [ MeV/c 2 ] < 211.32 211.32 -525 525 -565 565 -950 950 -1100 > 1100

Figure 4 . 1 :

 41 Figure 4.1: Illustration of the signal decay chain topology. The D 0 signal candidates arise from D * + →D 0 π + decay produced directly from a primary vertex (PV) pp collision.

Figure 4 . 2 :

 42 Figure 4.2: DecayTreeFitter (DTF) ∆m spectrum of D 0 → K + K -e + e -decays using a data sample in the D 0 high mass region above 1900 MeV/c 2 . The plot shows distributions before and after the hadron particle identification ProbNN selection.

Figure 4 . 3 :

 43 Figure 4.3: On the left, the ∆m spectrum of D 0 → K -π + e + e -decays after trigger and preselection cuts using 2017-18 data, where the red lines show the ∆m cut applied in the preselection. On the right, D 0 mass distribution before (black) and after (red) the ∆m requirement.

Figure 4 . 4 :Figure 4 . 5 :

 4445 Figure 4.4: BDT output distributions for D 0 → π + π -e + e -and D 0 → K + K -e + e - samples. The distributions are shown for the signal using simulation samples and for the combinatorial background using data samples in the D 0 meson high mass region (sideband).

Figure 4 . 6 :

 46 Figure 4.6: Background rejection as a function of signal efficiency forD 0 → π + π -µ + µ -and D 0 → K + K -µ + µ -BDTstrained on the even and odd (event number) subsamples. This crosscheck validates the classifier results on independent training subsamples.

18 Figure 4 . 7 :

 1847 Figure 4.7: Background rejection as a function of signal efficiency forD 0 → π + π -e + e -and D 0 → K + K -e + e -BDTs trained using as proxies the 2015-16 and 2017-18 simulation signal samples. The curve appears as straight line, indicating the absence of differences between the two samples.

Figure 4 . 8 :

 48 Figure 4.8: Averaged profile BDT output as a function of D 0 meson (top), dilepton (middle) and dihadron (bottom) mass for combinatorial background candidates of the normalisation mode D 0 → K -π + e + e -. Both BDTs trained on D 0 → π + π -e + e - (left) and D 0 → K + K -e + e -(right) decays are tested. No significant correlations are found.

Figure 4 . 9 :

 49 Figure 4.9: Averaged profile BDT output as a function of D 0 meson (top), dilepton (middle) and dihadron (bottom) masses for pure combinatorial signal data samples of D 0 → π + π -e + e -(left) and D 0 → K + K -e + e -(right) decays. No significant correlations are found.

9 BDTD 0

 90 .0550.0530.0480.0420.0360.0330.0270.0190.011 0.0420.0590.0570.0520.0440.0400.0360.0290.0210.012 0.0420.0600.0590.0530.0440.0370.0340.0290.0200.012 0.0400.0580.0570.0530.0450.0370.0340.0260.0190.011 0.0390.0570.0580.0540.0460.0390.0360.0300.0210.011 0.0360.0520.0520.0510.0440.0390.0360.0310.0200.011 0.0330.0500.0520.0480.0400.0350.0310.0250.0160.009 0.0280.0400.0390.0370.0310.0250.0240.0190.0130.010 Brem0 category 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0..0430.0460.0480.0500.0500.0450.0390.0270.018 0.0290.0440.0470.0490.0530.0490.0440.0380.0270.018 0.0290.0440.0470.0510.0540.0500.0470.0430.0330.022 0.0290.0420.0450.0490.0540.0480.0450.0430.0390.020 0.0280.0400.0430.0470.0510.0470.0420.0420.0360.019 0.0270.0370.0390.0430.0470.0430.0380.0370.0330.017 0.0250.0350.0350.0370.0390.0360.0320.0310.0270.015 0.0210.0290.0280.0300.0370.0340.0380.0290.0280.KKee FoM values

Figure 4 . 10 :

 410 Figure 4.10: Figure of merit two-dimensional scan over BDT and PID cuts for D 0 → π + π -e + e -(top) and D 0 → K + K -e + e -(bottom) for 2017-18 years. The scan is performed individually for the two brem categories. A common optimal point is found. The values of the figure of merit is shown in each of the bins.

  acceptance /N produced , ϵ reco-strip = N reconstructed and stripped /N in acceptance , ϵ selection = N full offline selection /N reconstructed and stripped , ϵ trig = N passing trigger requirements /N full offline selection ,

Figure 4 .

 4 Figure 4.11: Dimuon (left) and dihadron (right) mass distributions of the background-subtracted and efficiency corrected data of D 0→ π + π -µ + µ -, D 0 → K -π + [µ + µ -] ρ 0 /ω and D 0 → K + K -µ + µ -decays.The distributions are taken from Ref.[START_REF] Aaij | Angular analysis of D 0 → π + πµ + µand D 0 → K + Kµ + µdecays and search for CP violation[END_REF] in the supplementary material.

Figure 4 . 12 :

 412 Figure 4.12: Reconstructed dielectron (top) and dihadron (bottom) mass distributions from simulation of D 0 → π + π -e + e -(left), D 0 → K -π + e + e -(middle) and D 0 → K + K -e + e -(right) decays. The plots show the distributions before and after the resonance model correction from the muon modes data.

Figure 4 . 13 :

 413 Figure 4.13: Effect of the resonance model correction evaluated on the signal over normalisation ratio of efficiencies for each of the dielectron mass bins considered. The effect is evaluated separately for the two brem categories in 2017-18 years of data taking the relative efficiency difference before and after the resonance model correction.

Figure 4 . 14 :

 414 Figure 4.14: Effect of the dilepton mass correction on the reconstruction-stripping efficiency for each of the dielectron mass bins in which the analysis is performed. The effect is evaluated separately for the two brem categories for 2017-18 years of data taking. The relative difference between the efficiency with uncorrected and corrected dilepton mass is shown.

Figure 4 . 15 :

 415 Figure 4.15: Uncorrected and corrected dilepton mass distributions of the D 0 → K -π + e + e -, D 0 → π + π -e + e -and D 0 → K + K -e + e -simulation samples. The correction aims to reduce the differences in mass resolution between data and MC.

Figure 4 . 16 :

 416 Figure 4.16: Comparison of PID distributions of D 0 daughter particles in the decay D 0 → K -π + e + e -from MC (red), PIDGen resampled variable (green) and background-subtracted data (blue). The plots show the 4 daughter particles respective PID distributions in brem0 category (top) and brem1 (bottom).

Figure 4 . 17 :

 417 Figure 4.17: Projections in the x-y plane of the energy deposits in the ECAL. The three different regions are highligthened in different colors.

Figure 4 . 18 :Figure 4 . 19 :

 418419 Figure 4.18: L0L data-MC weights evaluated with the TISTOS method on a calibration sample of B 0 →K * (→K -π + ) J/ψ (→e + e -) decays. The correction weights are evaluated in bins of real L0CaloTool E T

Figure 4 . 20 :

 420 Figure 4.20: HLT1 data-MC weights evaluated with the TISTOS method on a calibration sample of B 0 →K * (→K -π + ) J/ψ (→e + e -) decays. The correction weights are evaluated in bins of nTracks.

Figure 4 .

 4 22 for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays. The values are shown for data taking years 2017-18 and split in brem categories, while 2015-16 plots can be found in appendix A.3.1.

Figure 4 . 21 :

 421 Figure 4.21: Total efficiency ratio as a function of dilepton mass for D 0 → π + π -e + e -and D 0 → K + K -e + e -for the 2017-18 data taking years. The distributions are reported for the different data taking years and brem categories considered in the analysis.

Figure 4 . 22 :

 422 Figure 4.22: Efficiency ratios for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays for different brem categories and split for each intermediate steps, shown from left to right: acceptance, stripping+reconstruction, selection and trigger.

Figure 4 . 23 :

 423 Figure 4.23: Simulated D 0 meson mass distributions with fit projection overlaid for D 0 → K -π + [e + e -] ρ 0 /ω (top), D 0 → π + π -e + e -(mid) and D 0 → K + K -e + e - (bottom). The fit are performed separately for brem categories integrated over the dielectron-mass region of interest.

Figure 4 . 24 :

 424 Figure 4.24: Simulated D 0 meson mass distribution of D 0 → π + π -e + e -candidates generated in the phi 950-1000 MeV/c 2 mass bin and reconstructed in the(ρω) 565-950 MeV/c 2 mass bin for both brem0 and brem1 categories (top) and high mass > 1100 MeV/c 2 bin for only the brem1 category. The (ρω) 565-950 mass upwards leakage towards the ϕ 950-1000 MeV/c 2 in the brem1 category is also included. The fit models projections are overlaid.

Figures 4 .

 4 [START_REF] Dominik | Search for new physics in rare four-body charm decays at LHCb[END_REF] show the fits for the integrated dielectron-mass range. All parameters are fixed in the final fit to data. The fits for the single dielectron-mass bins are shown in appendix A.[START_REF] Zyla | Review of particle physics[END_REF]

Figure 4 . 25 :

 425 Figure 4.25: Simulated D 0 meson mass distribution with fit projection overlaid for(top) D 0 → K -π + π -π + (middle) D 0 → π + π -π + π -and (bottom) K + K -π + π -with double π to e mis-identification. The fit are performed separately for brem categories integrated over the dielecton-mass region of interest.

Figure 4 . 26 :

 426 Figure 4.26: D 0 meson data mass distributions on ∆m sideband data (> 150 MeV/c 2 ) and m(D 0 ) > 1900 MeV/c 2 with fit projection overlaid forD 0 → K -π + e + e -(top) D 0 → π + π -e + e -(mid) and D 0 → K + K -e + e -(bottom).The fit are performed separately for brem categories integrated over the dielecton-mass region of interest.

Figure 4 . 27 :Figure 4 . 28 :

 427428 Figure 4.27: Simulated D 0 meson mass distribution with fit projection overlaid for the background decay D 0→ K -π + η(→ e + e -γ) (top), D 0 → π + π -η(→ e + e -γ) (middle) and D 0 → K + K -η(→ e + e -γ) (bottom) reconstructed and selected as signal decays. The fit are performed separately for brem categories in the low mass dilepton bin 211-525 MeV/c 2 .

  ± 0.5 D 0 → K + K -e + e -

Figure 4 . 29 :

 429 Figure 4.29: Data distribution and fit projection for D 0 → K -π + [e + e -] ρ 0 /ω candidates in the 2015-16 samples, optimised for the two signal selections:D 0 → π + π -µ + µ -(top) or D 0 → K + K -µ + µ -(bottom).The fits are performed separately for brem categories in the dilepton mass range [675-875] MeV/c 2 . The components are shown in different colors: signal (blue), mis-id (orange) and combinatorial (red). All shape parameters are fixed and only yields are free to float.

Figure 4 . 30 :

 430 Figure 4.29: Data distribution and fit projection for D 0 → K -π + [e + e -] ρ 0 /ω candidates in the 2015-16 samples, optimised for the two signal selections:D 0 → π + π -µ + µ -(top) or D 0 → K + K -µ + µ -(bottom).The fits are performed separately for brem categories in the dilepton mass range [675-875] MeV/c 2 . The components are shown in different colors: signal (blue), mis-id (orange) and combinatorial (red). All shape parameters are fixed and only yields are free to float.

  Figure 4.30: Data distribution and fit projection for D 0 → K -π + [e + e -] ρ 0 /ω candidates in the 2017-18 samples, optimised for the two signal selections:D 0 → π + π -µ + µ -(top) or D 0 → K + K -µ + µ -(bottom).The fits are performed separately for brem categories in the dilepton mass range [675-875] MeV/c 2 . The components are shown in different colors: signal (blue), mis-id (orange) and combinatorial (red). All shape parameters are fixed and only yields are free to float.

Figure 4 . 31 :

 431 Figure 4.31: Pull distributions of the D 0 → π + π -e + e -and D 0 → K + K -e + e - decays branching fraction in different dilepton mass bins by generating pseudoexperiments.

Figure 4 . 32 :

 432 Figure 4.32: D 0 → K -π + e + e -simulated sample and s-weighted data distributions of nTracks and nSPDhits. Correction weights are evaluated from the distributions to reweight the signal simulated samples and estimate a systematic due to possible data-simulation differences.

Figure 4 . 33 :

 433 Figure 4.33: Fit to simulation (left) and data (right) from 2017-2018 data taking years of D + s → π + ϕ(→ e + e -) decays, performed in the different brem categories: 0 (top), 1 (middle) and 2 or more (bottom) brem photons attached to the electrons. Channel m(e + e -) [ MeV/c 2 ]∆R reso m(ee) ϵ /R ϵ [%] D 0 → π + π -e + e -211.32-525 1.5 % 525-565 2.2 % 565-950 1.1 % 950-1100 1.9 % > 1100 1.1 % D 0 → K + K -e + e -211.32-525 1.5 % 525-565 2.2 % > 565 1.1 % Table4.15: Systematic uncertainty on the efficiency ratio due the dilepton mass correction. The alternative method using as calibration sampleD + s → π + ϕ(→ e + e -) is compared to the nominal correction with B meson decays. The relative difference is when evaluating the efficiencies is reported in the table for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays in different dilepton mass bins.

Figure 4 . 34 :

 434 Figure 4.34: Background decays with mis-id and/or missing particles reconstructed as D 0 → π + π -e + e -simulated with the RapidSim application[START_REF] Cowan | RapidSim: an application for the fast simulation of heavy-quark hadron decays[END_REF]. The red lines show the signal range [1700-2050] MeV/c 2 from the stripping and HLT2 lines requirements. The background are not normalised to their expected branching fraction.

Figure 4 . 35 :

 435 Figure 4.35: Examples of distributions of the test statistic variable q under the signal+background (s+b) and only-background (b) hypotheses. The actual data result is shown as a vertical line at the value q obs . The p-values, p s+b and p b , are respectively shown in green and yellow. The figure on the right show an example with very small sensitivity to the signal model [121].

Figure 4 . 36 :

 436 Figure 4.36: CL s distributions as a function of the signal branching fraction for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays. The systematics to the ratio of efficiencies, ratio of yields and uncertainty on the normalisation BF are propagated to the likelihoods as Gaussian constraints. The one and two sigma confidence intervals are shown respectively in light and dark blue.

Figure 5 . 1 :

 51 Figure 5.1: Trigger yields for different hadronic and leptonic final state decays as a function of instantaneous luminosity [124].

Figure 5 . 2 :

 52 Figure 5.2: Layout of the upgraded LHCb detector.

Figure 5 . 3 ,

 53 26 on each side of the beam pipe (left and right) and are able to reach a distance of 5.1 mm from it in closed position, 3.1 mm closer compared to Run1-2.

Figure 5 . 3 :

 53 Figure 5.3: Schematic of the upgraded VELO top view in the x-z plane (left) and the nominal layout of the VELO modules in closed position configuration (right) [125].

Figure 5 . 4 ,

 54 in the green region each silicon strip has a pitch of 190 µm and 10 cm length. The innermost yellow and pink regions have respectively full-length half-pitch and half-length half-pitch sensors.

Figure 5 . 4 :

 54 Figure 5.4: Layout of the upstream tracker showing in different colors, regions with different strip width and length [127]. The green region has 190 µm width and 10 cm length strips, the pink one half-width and half-length strips and the yellow one full-length and half-width ones.

Figure 5 . 5 :

 55 Figure 5.5: Scheme of the scintillating fibers tracker three stations shown between the dipole magnet and the RICH2 [127].

Figure 5 .

 5 Figure 5.10: HLT1 data handling, reconstruction and selection algorithms.

Figure 5 . 11 :

 511 Figure 5.11: HLT1 Velo tracking algorithm method [137]. First, step (a), hit triplet candidates are searched for opening windows in ϕ. They are further extrapolated to the following modules in a second stage (b). The seeding and following (c and d) are repeated iteratively to reconstruct all the tracks in the Velo. At every step, used hits are marked as flagged to avoid clones.

Figure 5 . 12 :

 512 Figure 5.12: Velo tracking efficiency as a function of momentum and transverse momentum (p T ) [138].

Figure 5 . 13 : 2 . 3 . 4 . 5 .

 5132345 Figure 5.13: Schematic view of the Velo detector. The primary vertices region is highlighted around the x-y-z coordinates center [125].

Figure 5 . 14 :

 514 Figure 5.14: Histogram filled with z POCA probability density distribution values for different Velo tracks in the same event [139]. Primary vertices are identified by the blue lines and separated by the red ones. From simulation, the green and yellow smaller lines indicate respectively reconstructible and non-recostructible generated PVs.

Figure 5 . 15 : 2 .

 5152 Figure 5.15: Primary vertex reconstruction efficiency as a function of the number of reconstructible tracks produced in the PV [138].

Figure 5 . 16 :

 516 Figure 5.16: Extrapolation of a Velo track to the UT window ranges. The hits found in the main and two neighbouring windows are considered in the UT tracking [140].

Figure 5 . 17 :

 517 Figure 5.17: VeloUT tracking efficiency as function of momentum and transverse momentum [138].

Figure 5 . 18 :

 518 Figure 5.18: Impact parameter resolution as function of the inverse transverse momentum and IP χ 2 distributions for different implementations of the Kalman filter at HLT1 [131]. The default solution is shown in black and features a parametrised Velo-only Kalman fit taking the momentum estimate from the forward tracking. Alternative methods are plotted for comparison: the simple Kalman method features a full detector parametrization, while the full Kalman method uses transport and noise matrices to fit the tracks.

Figure 5 . 19 :

 519 Figure 5.19: Muon identification efficiency as a function of momentum using a B 0 →K * µ + µ -sample [138].

Figure 5 . 20 :

 520 Figure 5.20: Performance of the HLT1 inclusive selections [138]. The plots show the efficiency of displaced single line and two-track vertex selections as a function of momentum. Different signal decays are tested and reported in the legend above the plot.

Figure 6 . 4 :

 64 Figure 6.4: Reconstruction efficiency of the "Looking Forward" algorithm as a function of p and p T evaluated on a 5000 events B 0 s → ϕ ϕ sample, testing the long from B track category.

Figure 6 . 5 :Figure 6 . 6 :

 6566 Figure 6.5: Reconstruction efficiency of the "Looking Forward" algorithm as a function η. The efficiencies are evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B track category.

Figure 6 . 7 :

 67 Figure 6.7: Momentum resolution of the "Looking Forward" algorithm as a function of p. The distribution is evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B track category.

Figure 6 . 8 :

 68 Figure 6.8: Throughput of the full HLT1 sequence using the "Looking Forward" algorithm to reconstruct long tracks. The values are reported for different GPU and CPU cards.

  Number of events [a.u.]

Figure 6 . 9 :

 69 Figure 6.9: Reconstruction efficiency of the "Looking Forward" algorithm as a function of p, p T and η. The efficiencies are evaluated on a simulated B 0 s → ϕ ϕ and B 0 →K * e + e -samples, testing the long from B track category, for electrons and non-electrons.

Figure 6 . 10 :

 610 Figure 6.10: Reconstruction efficiency of the "Forward no UT" algorithm (in red) as a function of p, p T and η. The efficiencies are evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B with p > 5 GeV/c & p T > 1 GeV/c track category. The plots show also the same distributions for the "Looking Forward" algorithm (in black), denominated as with UT, for comparison.

Figure 6 . 11 :

 611 Figure 6.11: Ghost rate of the "Forward no UT" algorithm (in red) as a function of p T . The ghost rate is evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B with p > 5 GeV/c & p T > 1 GeV/c track category. The plot shows also the same distribution for the "Looking Forward" algorithm (in black), denominated as with UT, for comparison.

Figure 6 . 12 :

 612 Figure 6.12: Momentum resolution of the "Forward no UT" algorithm (in red) as a function of p. The distribution is evaluated on a 5000 events B 0 s → ϕ ϕ samples, testing the long from B with p > 5 GeV/c & p T > 1 GeV/c track category. The plot shows also the same distribution for the "Looking Forward" algorithm (in black), denominated as with UT, for comparison.

Figure 6 . 13 :

 613 Figure 6.13: Throughput of the full HLT1 sequence using the "Looking Forward" or "Forward no UT" algorithm to reconstruct long tracks for comparison. The values are reported for different GPU and CPU cards. The no UT sequence excludes both decoding and reconstruction involving UT information.

Figure 6 . 14 :

 614 Figure 6.14: Reconstruction efficiency of the "Best long track combiner" algorithm in the Forward then matching configuration (in blue) as a function of p, p T and η. The efficiencies are evaluated on a 5000 events B 0s → ϕ ϕ samples, testing the long from B track category. The plots show also the same distributions for the "Looking Forward" algorithm (in black) and "Seeding&Matching" algorithm (in red) for comparison.

Figure 6 . 15 :

 615 Figure 6.15: Reconstruction efficiency of the "Best long track combiner" algorithm in the Matching then forward configuration (in blue) as a function of p, p T and η. The efficiencies are evaluated on a 5000 events B 0s → ϕ ϕ samples, testing the long from B track category. The plots show also the same distributions for the "Looking Forward" algorithm (in black) and "Seeding&Matching" algorithm (in red) for comparison.

  two ingredients needed to evaluate the figure of merit. The background yield is evaluated as the sum of the combinatorial yield N comb and the mis-id yield N mis-id . The values of the signal efficiencies in the PID and BDT scan done to optimize the selection are reported in Figure A.1, while the background yields in Figure A.2 for both the combinatorial and the mis-id. .7420.6540.5650.4790.3940.3090.2270.1490.070 0.7690.6820.6000.5200.4400.3630.2860.2100.1390.065 0.7050.6260.5520.4780.4040.3340.2630.1930.1280.060 0.6370.5670.5010.4330.3670.3040.2400.1770.1160.056 0.5650.5030.4440.3830.3260.2680.2110.1570.1020.049 0.4780.4250.3760.3250.2750.2260.1790.1310.0860.040 0.3760.3340.2950.2550.2150.1760.1380.1000.0640.029 0.2420.2150.1890.1600.1350.1100.0860.0630.0400.

Figure A. 1 :

 1 Figure A.1: Signal selection efficiencies used in the figure of merit evaluation. The values are reported for different PID and BDT cuts for the brem0 (left) and brem1 (right) categories. Both D 0 → π + π -e + e -and D 0 → K + K -e + e -values are reported here.

Figure A. 3 .

 3 The resolution difference between generated and reconstructed dihadron mass is always lower than 1 %.

60 D 0

 600 KKee expected mis-id yield

Figure A. 2 :

 2 Figure A.2: Combinatorial and mis-id yield used in the figure of merit evaluation.The values are reported for different PID and BDT cuts for the brem0 (left) and brem1 (right) categories for both D 0 → π + π -e + e -and D 0 → K + K -e + e -decays.

Figure A. 4 :

 4 Figure A.4: Single efficiency ratios for D 0 → π + π -e + e -and D 0 → K + K -e + e -for 2015-16 data taking years.

Figure A. 5 :

 5 Figure A.5: Total efficiency ratio as a function of dilepton mass for D 0 → π + π -e + e -and D 0 → K + K -e + e -for the 2015-16 data taking years. The distributions are reported for the different data taking years and brem categories considered in the analysis.

  3.6. The HLT1 trigger correction weights are reported in Figure A.6 per data taking year. The L0L and L0I trigger correction weights distributions are shown respectively in Figure A.7 and Figure A.8 per data taking year.

Figure A. 6 :

 6 Figure A.6: HLT1 data-MC weights evaluated with the TISTOS method on a calibration sample of B 0 →K * (→K -π + ) J/ψ (→e + e -) decays. The correction weights are evaluated in bins of nTracks.

Figure A. 7 :Figure A. 8 :

 78 Figure A.7: L0L data-MC weights evaluated with the TISTOS method on a calibration sample of B 0 →K * (→K -π + ) J/ψ (→e + e -) decays. The correction weights are evaluated in bins of E T and three ECAL regions.

Figure A. 10 :Figure A. 11 :Figure A. 12 :

 101112 Figure A.10: Simulation distribution fits for the signal D 0 →π + π -e + e -for single dilepton bins. The fit are performed separately for brem categories

Figure A. 13 :

 13 Figure A.13: BF pull distributions of the D 0 → π + π -e + e -decays running 1000 pseudoexperiments and assuming different signal BFs: 10 -6 , 10 -7 , 10 -8 , 10 -8 .

Figure A. 15 :

 15 Figure A.15: Background decays with mis-id and/or missing particles reconstructed as D 0 → K + K -e + e -. The red lines show the signal range [1700-2050] MeV/c 2 from the stripping and HLT2 lines cut. The background are not normalized to their expected branching fraction.

Figure A. 16 :

 16 Figure A.16: Background decays with mis-id and/or missing particles reconstructed as D 0 → K -π + e + e -. The red lines show the signal range [1700-2050] MeV/c 2 from the stripping and HLT2 lines cut. The background are not normalized to their expected branching fraction.

Table 3 .

 3

	2: Resonant and non-resonant fractions simulated for the normalization and
	signal MC.

Table 4 .

 4

2: Requirements for D 0 candidates of the Hlt2 and stripping selections for signal and normalisation modes in 2017-18 samples.

Table 4 .

 4

3: Preselection requirements.

Table 4 . 5 :

 45 Figure of merit optimisation for different ∆m cuts. The table shows the maximised figure of merit values for both D 0 → π + π -e + e -and D 0 → K + K -e + e - performing the 2D BDT vs PID cuts scan.

		144.5-146.5	0.019	0.051
		144-147	0.019	0.056
		143.5-146.5	0.018	0.052
		143-147	0.016	0.049
		142.5-147.5	0.015	0.045
		0.9		0.9
		0.8		0.8
	electron PID	0.4 0.5 0.6 0.7	electron PID	0.4 0.5 0.6 0.7
		0.3		0.3
		0.2		0.2
		BDT		BDT

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0150.0210.0220.0230.0220.0210.0180.0160.0120.008 0.0160.0230.0250.0250.0240.0230.0200.0170.0140.009 0.0170.0240.0260.0270.0260.0260.0220.0180.0140.009 0.0170.0250.0270.0280.0270.0270.0240.0200.0160.010 0.0160.0250.0270.0280.0280.0280.0240.0200.0160.012 0.0160.0240.0270.0280.0270.0270.0230.0190.0150.010 0.0140.0220.0250.0260.0250.0250.0210.0170.0120.011 0.0120.0180.0210.0210.0210.0200.0170.0150.0090.005 Brem0 category 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0090.0140.0150.0160.0170.0170.0170.0160.0130.011 0.0090.0140.0160.0170.0180.0190.0170.0160.0140.012 0.0090.0140.0160.0180.0190.0190.0180.0170.0140.012 0.0090.0140.0160.0170.0180.0190.0180.0170.0140.011 0.0090.0140.0160.0170.0180.0190.0170.0170.0140.011 0.0080.0130.0150.0160.0170.0180.0170.0160.0130.010 0.0080.0120.0140.0150.0160.0160.0150.0140.0120.009 0.0070.0100.0110.0120.0130.0130.0130.0120.0090.006

  2015 1.16 ± 0.02 -5.69 ± 0.78 3075.28 ± 0.50 1.14 ± 0.02 -8.87 ± 0.86 3072.83 ± 0.49 1.17 ± 0.03 -8.94 ± 1.35 3083.42 ± 0.49 2016 1.11 ± 0.01 -4.68 ± 0.30 3074.55 ± 0.47 1.14 ± 0.01 -11.02 ± 0.33 3070.01 ± 0.45 1.18 ± 0.01 -16.00 ± 0.54 3082.32 ± 0.46 2017 1.09 ± 0.01 -4.37 ± 0.28 3075.18 ± 0.66 1.12 ± 0.01 -7.18 ± 0.

	32	3070.92 ± 0.64 1.09 ± 0.01 -9.57 ± 0.51	3080.65 ± 0.63
	2018 1.10 ± 0.01 -4.59 ± 0.27 3074.16 ± 0.65 1.10 ± 0.01 -6.40 ± 0.29	3070.72 ± 0.65 1.08 ± 0.01 -8.09 ± 0.47	3082.12 ± 0.63

Table 4 .

 4 6: Values of the sigma scale s σ , ∆µ and µ MC from the data and MC B + → K + J/ψ(→ e + e -) fits. ∆µ is defined as the difference between µ

	f it data and µ	f it

  and four different nTracks regions. All data taking year distributions are reported in appendix A.3.2.

Table 4 .

 4 e + e -] ρ 0 /ω years 8: Fitted signal and background yields for D 0 → K -π + [e + e -] ρ 0 /ω decays. The values are reported for different data taking years, brem categories and selections optimised on the two signal channels. The brem1 mis-id yields are fixed to 5 % of the brem mis-id. Note the lower yield in the 2015/16 data set with respect to 2017/18 due to the less efficient trigger in that period aside from the doubled luminosity. + π -e + e -

				brem signal	mis-id	combinatorial
	Selection ππ		2015-16 2017-18	brem0 57 ± 13 brem1 64 ± 9 brem0 280 ± 25 260 ± 24 26 ± 7 70 ± 12 5 ± 5 4 0 ± 8 brem1 549 ± 29 13 164 ± 21
	Selection KK		2015-16 2017-18	brem0 49 ± 11 brem1 51 ± 8 brem0 208 ± 21 123 ± 19 20 ± 6 34 ± 10 0 ± 2 2 0 ± 4 brem1 433 ± 25 6 116 ± 2
		211-525	17.4 ± 5.1	7.8 ± 4.1	38.0 ± 7.1	0.4 ± 0.3	0.0 ± 0.6	0.0 ± 2.9
		525-565	4.1 ± 2.3	0.2 ± 1.8	9.4 ± 3.6	0.0 ± 0.1
	2015-16	565-950	14.4 ± 4.5	38.2 ± 7.0	115.8 ± 11.7 2.0 ± 0.8
		950-1100	0.0 ± 1.0	9.1 ± 3.6	27.6 ± 6.3	0.5 ± 0.3
		> 1100	1.8 ± 1.7	9.4 ± 3.4	34.0 ± 6.2	0.5 ± 0.3
		211-525	78.3 ± 12.2	13.4 ± 7.3	29.5 ± 10.8	0.7 ± 0.5	22.6 ± 7.9 24.0 ± 8.6
		525-565	12.0 ± 4.4	1.6 ± 2.8	5.0 ± 2.9	0.1 ± 0.1
	2017-18	565-950	89.2 ± 12.0	41.2 ± 10.7	36.5 ± 11.

m(ee) [MeV] comb. brem0 mis-id brem0 comb. brem1 mis-id brem1 eta brem0 eta brem1 D 0 → π

Table 4 .

 4 9: Background yields from D 0 → π + π -e + e -and D 0 → K + K -e + e -fits on 2015-16 and 2017-18 data for the different dielectron mass bins considered in the analysis. The combinatorial and mis-id background yields are reported for each mass bin and brem categories. The partially reconstructed η background is only relevant in the lowest dilepton mass bin 211-525 MeV/c 2 .

	1800	1850	1900	1950

Table 4 .

 4 -7 , -8 , 10 -9 . The results are reported in appendix A.6. 11: Mean and sigma pull values evaluated running 1000 pseudoexperiments to validate the signal fit. The results are obtained fitting a Gaussian to the pull distributions as shown in Figure4.31. The signal is generated assuming the muon mode branching fractions reported in Table2.2 (the limit value is used for the mass bins with no observation).

	Channel	m(e + e -) [ MeV/c 2 ] mean	sigma
	D 0 → π + π -e + e -211.32-525	-0.008 ± 0.034 1.022 ± 0.034
		525-565	-0.176 ± 0.039 1.028 ± 0.032
		565-950	-0.020 ± 0.031 0.979 ± 0.023
		950-1100	-0.000 ± 0.031 0.966 ± 0.023
		> 1100	-0.073 ± 0.033 1.018 ± 0.024
	D 0 → K + K -e + e -211.32-525	-0.112 ± 0.036 0.963 ± 0.026
		525-565	-0.132 ± 0.040 0.967 ± 0.029
		> 565	-0.107 ± 0.036 0.998 ± 0.024

Table 4 .

 4 12.

	32-525	6.2 %
	525-565	15.8 %
	565-950	2.8 %
	950-1100	3.5 %
	> 1100	11.9 %
	D 0 → K + K -e + e -211.32-525	3.8 %
	525-565	12.0 %
	> 565	3.9 %

Channel

m(e

+ e -) [ MeV/c 2 ] ∆R stat ϵ /R ϵ [%] D 0 → π + π -e + e -211.

Table

  for each dielectron mass bin.

		calibration sample		∆R PID ϵ	/R ϵ
	D 0 → π + π -e + e -211.32-525	1.2 %	0.7 %	1.4 %
	525-565	1.9 %	0.8 %	2.1 %
	565-950	0.6 %	0.3 %	0.7 %
	950-1100	0.6 %	0.2 %	0.6 %
	> 1100	1.9 %	0.6 %	2.0 %
	D 0 → K + K -e + e -211.32-525	1.1 %	0.8 %	1.4 %
	525-565	1.7 %	0.9 %	1.9 %
	> 565	0.6 %	0.5 %	0.8 %

Channel

m(e + e -) [ MeV/c 2 ] limited sample stats.

[%]

binning

[%]

total

[%]

Table 4 .

 4 [START_REF] Chatrchyan | Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF]: Systematics on the efficiency ratio due to the PID data-simulation correction. Effects are caused by the limited size of the calibration samples used for the PID resampling and to the chosen kinematic binning from which the resampling PDFs are exported.

Table 4

 4

	.14 for each

Table 4 .

 4 14: Systematics on the efficiency ratio due to the trigger data-MC correction.

  MC and data are performed. A selection is applied to reduce the background contamination in data and align it in MC (tight ProbNNe requirements to reject mis-identified hadronic background). A requirement at MeV/c 2 is applied on the D meson mass in

	0.06 0.08 0.1 0.12 0.14 0.16 events normalized		ee brem0 Data Sweighted π K → 0 D MC	0.04 0.06 0.08 0.1 0.12 events normalized	ee brem0 Data Sweighted π K → 0 D MC
		0.04
		0.02					0.02
		0					0
			100	200	300			100	200	300	400
					nTracks						nSpdDigits
	Pulls	5 4 --3 -2 -1 -0 1 2 3 4 5	100	200	300	Pulls	5 4 --3 -2 -1 -0 1 2 3 4 5	100	200	300	400
					nTracks						nSpdDigits
	0.06 0.08 0.1 0.12 0.14 0.16 events normalized		ee brem1 Data Sweighted π K → 0 D MC	0.06 0.08 0.1 0.12 events normalized	ee brem1 Data Sweighted π K → 0 D MC
		0.04					0.04
		0.02					0.02
		0
			100	200	300			100	200	300	400
					nTracks						nSpdDigits
	Pulls	5 4 --3 -2 -1 -0 1 2 3 4 5	100	200	300	Pulls	5 4 --3 -2 -1 -0 1 2 3 4 5	100	200	300	400
					nTracks						nSpdDigits

obtain s σ , µ f it data and µ f it MC , fits to D + s → π + ϕ(→ e + e -)

Table 4 .

 4 [START_REF] Christenson | Evidence for the 2π Decay of the K 0 2 Meson[END_REF]: Systematic uncertainty on the efficiency ratio due the decay model correction. The efficiencies are evaluated fluctuating the model correction within uncertainties and the spread is taken as systematic. Both values for the acceptance and selection efficiencies are reported and summed in quadrature.

Table 4 .

 4 17. In general, the dominant uncertainties are due to the limited MC statistics and the resonance model correction.

	Channel	m(e + e -) [ MeV/c 2 ]	∆R stat ϵ Rϵ	∆R	tracking ϵ Rϵ	∆R PID ϵ Rϵ	∆R Rϵ trigger ϵ	∆R res. model ϵ Rϵ	∆R kin ϵ Rϵ	∆R	reso m(ee) ϵ Rϵ	total ∆Rϵ Rϵ
	D 0 → π + π -e + e -211.32-525 6.2 %	1.8 %	1.4 % 0.8 %	9.2 %	1.8 % 1.5 %	11.4 %
		525-565	15.8 % 1.8 %	2.1 % 1.1 %	9.7 %	1.8 % 2.2 %	19.0 %
		565-950	2.8%	1.8 %	0.7 % 0.8 %	2.3 %	1.8 % 1.1 %	4.7 %
		950-1100	3.5 %	1.8 %	0.6 % 0.7 %	3.3 %	1.8 % 1.9 %	5.8 %
		> 1100	11.9 % 1.8 %	2.0 % 1.2 %	7.8 %	1.8 % 1.1 %	14.7 %
	D 0 → K + K -e + e -211.32-525 3.8 %	1.8%	1.4 % 0.9 %	4.1 %	1.5 % 1.5 %	6.3 %
		525-565	12.0 % 1.8 %	1.9 % 0.9 %	18.9 %	1.5 % 2.2 %	22.7 %
		> 565	3.9 %	1.8 %	0.8 % 0.8 %	2.0 %	1.5 % 1.1 %	5.2 %

Table 4 .

 4 17: Summary of systematic uncertainties on the efficiency ratio per dilepton bin: it includes tracking, PID, trigger, resonances model, kinematic reweighting and bin migration effects.

Table 4 .

 4

18: Systematic uncertainties when varying the model shape PDFs for

Table 4 .

 4 21: Efficiencies of different PID cuts evaluated with the PIDCalib[START_REF] Anderlini | The PIDCalib package[END_REF]

Table 4 .

 4 .22. Selection D 0 → π + π -e + e -2015-16 21.2 % 13.7 % 4.6 % 2017-18 9.5 % 5.7 % Selection D 0 → K + K -e + e -2015-16 14.5 % 10.1 % 4.4 % 2017-18 10.1 % 5.7 % 22: normalisation yield relative uncertainty for selections optimised on the two different signal decay channels.

	years	brem0 brem1 ∆N stat norm /N stat norm

Table 4 .

 4 23: Total systematic uncertainty values on the ratio of yields evaluation.

	211.32-525 4.8 %	\	4.6 %	6.5 %
	525-565	5.1 %	\	4.6 %	5.7 %
	565-950	4.5 %	3.1 %	4.6 %	7.0 %
	950-1100	3.5 %	\	4.6 %	5.6 %
	> 1100	3.7 %	\	4.6 %	5.8 %
	D 0 → K + K -e + e -211.32-525 4.0 %	\	4.4 %	6.1 %
	525-565	4.4 %	\	4.4 %	6.3 %
	> 565	3.8 %	1.1 %	4.4 %	5.7 %

Table 4 .

 4 24: Summary of all systematic uncertainties on the ratio of yields and efficiencies in bins of dilepton mass. The last column shows the total uncertainty contribution.

	211.32-525	6.5 %	11.6 % 13.2 %
	525-565	5.7 %	19.0 % 20.1 %
	565-950	7.0 %	4.7 %	8.4 %
	950-1100	5.6 %	5.8 %	8.1 %
	> 1100	5.8 %	14.7 % 15.8 %
	D 0 → K + K -e + e -211.32-525	6.1 %	6.5 %	8.9 %
	525-565	6.3 %	22.7 % 23.6 %
	> 565	5.7 %	5.2 %	7.7 %

Table 4 .

 4 25: Expected limit values for D 0 → π + π -e + e -and D 0 → K + K -e + e - decays. The results are reported for the dilpeton mass bins considered in the analysis with 90 % and 95 % confidence level.

	-7 ]
	90 % CL (95 %)

Table 4 .

 4 26: Absolute statistical, relative systematic and normalisation channel BF uncertainties on the branching fraction values for D 0 → π + π -e + e -and D 0 → K + K -e + e -decays in the different dilepton mass regions considered in the analysis.

  t 2 y (t x t y B x -(1 + t 2 x )B y + t y B z ), t x t y B y + t x B z ).

	dt y dz	=	q p	1 + t 2 x + t 2 y ((1 + t 2 y )B x

Table 6 .

 6 + t y ) 15 •(t x + t y )

	1.

Table 6 .

 6 1: Values of the parameters used in Forward and Forward no UT algorithms.

Table 6 .

 6 2: Values of the multiplicative parameter C(n hits ) as a function of the number of hits which composes the track candidate. Reconstructible tracks are required to be composed of at minimum 9 hits, with at least one hit in one x-layer and one hit in a uv-layer per T-station.

Table 6 . 4

 64

: Clone rate, hit purity and hit efficiency for the "Forward", "Forward no UT" and "Best long track combiner" algorithms. The values are evaluated on a B 0 s → ϕ ϕ simulated sample.

  electron PID 0.8090.7340.6550.5760.4970.4170.3360.2550.1710.087 0.7590.6890.6160.5430.4680.3930.3170.2400.1610.083 0.7070.6410.5730.5050.4360.3650.2950.2230.1490.076 0.6500.5900.5270.4650.4010.3360.2710.2060.1380.071 0.5840.5300.4730.4170.3600.3020.2430.1850.1240.064 0.5130.4660.4160.3670.3180.2660.2150.1640.1100.056 0.4200.3800.3400.3000.2590.2170.1750.1330.0900.046 0.2880.2610.2320.2050.1780.1500.1210.0910.0620.031

		D 0	ee signal efficiency
					Brem1+ category
			0.8			0.8
			0.6			0.6
			0.4			0.4
			0.2			0.2
		Brem0 category			Brem1+ category
		0.9			0.9
		0.8	0.8		0.8	0.8
	electron PID	0.4 0.5 0.6 0.7	0.2 0.4 0.6	electron PID	0.4 0.5 0.6 0.7	0.2 0.4 0.6
		0.3			0.3
		0.2			0.2
		0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 BDT			0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 BDT

0.8390.7430.6560.5720.4820.3970.3210.2350.1560.076 0.7740.6850.6040.5270.4440.3670.2990.2210.1460.070 0.7060.6260.5510.4790.4030.3350.2750.2020.1320.064 0.6440.5730.5030.4350.3670.3050.2530.1860.1220.060 0.5730.5090.4450.3820.3200.2640.2180.1610.1040.048 0.5000.4430.3860.3300.2770.2280.1890.1390.0900.045 0.4120.3630.3180.2710.2250.1860.1550.1150.0740.037 0.2900.2550.2230.1920.1550.1290.1090.0810.0520.025 0.8130.7370.6600.5790.5040.4220.3370.2560.1710.087 0.7550.6850.6120.5360.4680.3920.3120.2370.1580.081 0.7050.6390.5720.5030.4390.3660.2910.2210.1480.076 0.6490.5900.5260.4640.4040.3380.2680.2030.1360.069 0.5910.5370.4800.4230.3690.3100.2460.1880.1260.065 0.5270.4800.4290.3780.3280.2750.2190.1680.1140.059 0.4450.4060.3630.3180.2750.2310.1830.1410.0960.052 0.3230.2940.2620.2300.1990.1670.1330.1030.0710.037 D 0 KKee signal efficiency

-10 3 -10 2 -10 1 -10

-10 ×

Acknowledgements

this step, tighter χ 2 requirements are defined in the next section.

Triplet seeding, Create Tracks and Quality Filter variations

Triplets of hits (h 0 ,h 1 ,h 2 ) are searched for in the first or last x-layers of the three stations, similarly to the nominal forward algorithm, but only performing combinations either in the LEFT or RIGHT search windows, representing the two charge assumptions. First, all two-hits (h 0 ,h 2 ) combinations are tested using the first and last layers of each station. The condition for a valid doublet is defined as:

with z Velo mag and x doublet mag the x-value straight line extrapolations of the Velo track and the doublet at z mag , while O mag an x opening tolerance at z mag , fixed at 10 mm. The O mag value is set to a quite lower value compared to the nominal forward reconstruction as the algorithm focuses on higher momentum tracks, expecting low-bending trajectories.

Once the valid doublets are found, the x-layer hit from the central station is added with the same procedure explained in section 6.2.1 forming the triplet candidates. The maximum number of selected triplets is increased compared to the forward algorithm which uses the UT information, with a limit of 20.

The last two steps of the algorithm, Create Tracks and Quality Filter, are aligned to what presented for the "Looking Forward" algorithm. The parameters values in Table 6.1 are optimised for the subset of tracks with p > 5 GeV/c or p T > 1 GeV/c. Generally, the χ 2 tolerances are decreased. In fact, higher momentum tracks tend to be more easily modeled compared to low momentum ones, which allows tighter tolerances to be used.

Best long track combiner

An alternative approach to reconstruct long tracks is provided by the "Seeding & Matching" algorithm [START_REF] Calefice | Standalone track reconstruction on GPUs in the first stage of the upgraded LHCb trigger system and Preparations for measurements with strange hadrons in Run 3[END_REF]. The algorithm performs first a standalone track reconstruction in the SciFi, the so-called seeding. It is composed of two main steps:

1. x-z seeding: three-hits combinations are build considering either the first or last x-layers of the three T-stations. For each hit in the x-layer of the first station, a straight-line extrapolation is considered from (x,y,z) = (0,0,0) to look for hits in the other stations (infinite momentum assumption). The search windows tolerances are evaluated assuming a minimum momentum of p = 3 GeV/c. Once the triplet candidates are formed, a parabolic track model in the x-z plane is defined to try to add the remaining three x-layer hits. Tracklets with at least five hits are accepted as good candidates.

2. Adding y-information: a pattern recognition method is performed in the ydirection using hits in the u/v layers. The track model in the x-z plane predicts the x-position of a hit in the first u layer, which thanks to the stereo angle leads to y-position information. Search windows are opened in the other u/v layers assuming a straight line in the y-z plane from the (y,z) = (0,0). For every u/v hit added to the tracklet candidate, the t y value and search window tolerances are updated accordingly. Tracklets are promoted to reconstructed tracks if they are composed by a minimum of 11 hits.

Chapter 7. Conclusion

requirement. The efficiency is comparable to the nominal algorithm using the UT information, in this higher momentum subset of particles. The ghost rate is 4 % higher compared to the nominal reconstruction as the UT information is not used, still a tolerable value to perform physics measurements.

A.1.2 Optimization

This appendix reports the values of the efficiencies and yields used to evaluate the figure of merit in subsection 4.2.5. In specific, as explained in the main section in Equation 4.2, the signal efficiency ϵ sig and the background yield N bkg are the only

The distributions are fitted with a gaussian and the mass resolution is reported in the legend.

A.3 Supplemental material on efficiency ratio determination A.3.1 Efficiency values

The following tables report the values of the single efficiencies per data taking years, brem categories and dilepton mass bins considered in the analysis. The stripping and reconstruction ratio of efficiencies are reported in Table A.2. Offline selection efficiencies in Table A.3 for D 0 → π + π -e + e -and D 0 → K + K -e + e -signal decays and Table A.4 for the normalization D 0 → K -π + e + e -, reported with the two signal selection optimizations.

The ratio of efficiencies for the data taking years 2015-16 are shown in The trigger efficiency is evaluated requiring the full trigger and results are shown in Table A.5 and Table A.6.

A.4 Supplemental material on the fit model

The appendix reports all PDFs taken from simulation to perform the signal fit in bins of dilepton mass. The D 0 → K + K -e + e -and D 0 → π + π -e + e -signal PDFs are shown respectively in