Search for heavy neutrinos in B decays at LHCb

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Search for heavy neutrinos in

B decays at LHCb

by Niels Ruijter

Student nr. 11292296

Report Bachelor Project Physics and Astronomy, size 15 EC

Conducted between 30-3-2020 and 10-7-2020

Submitted on 10-7-2020

Faculty of Science, University of Amsterdam

bfys group, Nikhef

Supervisor:

dr. Wouter Hulsbergen

Daily Supervisor:

Valeriia Lukashenko MSc.

Second Examiner:

prof. dr. ir. Paul de Jong

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Summary

The neutrino masses are much smaller than the masses of other Standard Model particles. The seesaw mechanism offers a possible explanation for the smallness of neutrino masses by introducing a heavy neutrino N next to the very light neutrino we know. In this project a preparatory study into the decay B− → µ−e+_π−_{, mediated by such a heavy neutrino, and its detection at LHCb} was conducted. For the heavy neutrino a mass and lifetime hypothesis of 2500 MeV and 100 ps respectively was used. In the project both Monte Carlo simulated data and LHCb 2016 data from Run 2 with√s = 13 TeV were used. The effects of using a B mass constraint in combination with a PV constraint and using downstream tracks in the LHCb detector next to the standard long tracks were studied. The B constraint improved the neutrino mass resolution, while adding downstream tracks to the long tracks increased sensitivity to longer lived neutrinos. A limit on the B branching fraction for the particular model used was set at B (B± → µ±_{N ) = 1.96 × 10}−9 _{(95% C.L.).}

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Popular summary (Dutch)

Het Standaardmodel van de deeltjesfysica beschrijft de allerkleinste bouwstenen van het univer-sum. Alles om je heen, of het nu je fiets is, de piek op de boom tijdens kerstmis of een ster aan de nachtelijke hemel, op zijn meest fundamentele niveau is het opgebouwd uit de deeltjes in het Standaardmodel. Alleen de zwaartekracht wordt niet beschreven door het Standaardmodel. Er zijn twee soorten deeltjes in het Standaardmodel; fermionen en bosonen. Fermionen zijn de materiedeeltjes, terwijl bosonen de deeltjes zijn die krachten overbrengen. Er zijn 12 soorten (of ’flavours’) fermionen, die zijn onderverdeeld in drie generaties van vier. In elke generatie zitten twee soorten quarks, en twee soorten leptonen. De leptonen in elke generatie bestaan uit een elek-trisch geladen lepton en een neutrino. De fermionen in elke generatie hebben ongeveer dezelfde massa, met uitzondering van de neutrino’s. De massa’s van de neutrino’s zijn vele malen kleiner dan de massa’s van de andere fermionen. Het is niet bekend waarom dit zo is. Het ’seesaw mechanism’ biedt een mogelijke verklaring voor de kleine massa van het neutrino. In deze theorie bestaat er naast het lichte neutrino ook een zwaar neutrino. Een consequentie van deze theorie is dat in sommige situaties het neutrino gelijk is aan zijn eigen antideeltje. Elk fermion in het Standaardmodel heeft zijn eigen antideeltje met een tegengestelde elektrische lading, maar verder dezelfde eigenschappen. Neutrinos hebben geen elektrische lading, zodat het mogelijk is dat het neutrino en het antineutrino gelijk aan elkaar zijn.

In dit project is een verkennend onderzoek gedaan naar zware neutrinos in een bepaald verval van het B meson. Bij dit verval worden een muon, elektron en een pion gevormd. Het muon en elektron zijn geladen leptonen, het pion is een ander soort meson. Een meson is een deeltje bestaande uit twee soorten quarks. Het B meson bevat een bottom quark; deze komen alleen voor bij zeer hoge energi¨en, zoals gehaald worden in de LHC deeltjesversneller op het CERN in Gen`eve. De LHCb detector bij de LHC is speciaal ontworpen om deeltjes die een bottom quark bevatten te detecteren. In het onderzoek is een hypothese gebruikt voor een zwaar neutrino met massa van 2500 MeV (4.46 × 10−27 kg) en een vervaltijd van 100 ps (10−10 s). In het onderzoek zijn zowel een simulatie van het B meson verval en de detectie ervan in de LHCb detecor als echte data van de LHCb uit 2016 gebruikt. De effectiviteit van verschillende technieken om het verval in de LHCb detector te reconstrueren zijn bekeken in het onderzoek. Ook zijn verschillende selecties gemaakt in de data om het signaal van het verval met het neutrino in de gesimuleerde data te isoleren en tegelijkertijd zoveel mogelijk achtergrond uit de echte LHCb data weg te gooien. Uiteindelijk is er een limiet gezet op de ’branching fraction’ van het B meson met het zware neutrino. De branching fraction is de fractie van alle mogelijke vervallen van het B meson die via het zware neutrino vervallen. Deze branching fraction geldt alleen voor het specifieke model gebruikt in dit onderzoek, dus een verval naar een muon, elektron en pion en de massa- en levensduur-hypothese van het zware neutrino. Het blijkt dat deze branching fraction niet hoger kan zijn dan 1.96 × 10−9.

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Acknowledgements

Firstly I would like to thank my supervisor Wouter Hulsbergen for giving me the opportunity to conduct my bachelor project at the the Nikhef bfys group about this fascinating subject. During the project I greatly developed both my experimental and theoretical skills and learnt many new things.

Many thanks to my daily supervisor Valeriia (Lera) Lukashenko for helping me out during the project on a daily basis and always being available for answering my questions.

Thanks also to Paul de Jong for acting as second examiner for this project and for asking some tricky questions during my talk about the project.

I want to thank my fellow final year physics bachelor Madelon Geurts for proofreading this thesis from the perspective of a bachelor student and pointing out things that were unclear.

Finally thanks to my parents for proofreading the final version of this thesis (despite having no physics background at all) and filtering out the last spelling and grammar mistakes.

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1 Introduction

In the Standard Model of particle physics it was assumed that neutrinos are massless particles. However, since the discovery of neutrino oscillations in 1998, it is known that neutrinos must have a mass. The neutrino masses are, however, very small compared to the masses of the other Standard Model particles. The smallness of the neutrino mass could be explained by the seesaw mechanism [1]. In the seesaw mechanism a new heavy neutrino N is hypothesized, that exists next to the light neutrino ν that we know. Because the seesaw mechanism would explain the smallness of the neutrino mass, heavy neutrinos are an interesting object of study.

If a Majorana mass term is added to the Dirac mass term in the mass term of the neutrino Langrangian, the seesaw mechanism can be derived. The Majorana mass term provides a direct coupling between right-handed neutrinos and left-handed antineutrinos. If the Majorana mass term exists, decays that violate lepton number conservation are possible.

Research has been conducted by the LHCb collaboration into lepton number violating B decays mediated by a heavy neutrino [2], [3]. These studies looked into the decay B− → µ−_{N with} N → µ−π+, where the two muons have the same sign. To have two same sign muons in the final state the heavy neutrino must behave as a Majorana particle. No signal was found in the studies and upper limits were set on the B decay branching fraction for heavy neutrinos in a mass range between 250 and 5000 MeV and lifetimes ranging from 0 to 1000 ps. The branching fractions were of the order of 10−9 _{for lifetimes smaller than 1 ps, of the order of 10}−8 _{for lifetimes around 100} ps and of the order of 10−7 for lifetimes up to 1000 ps. At a mass of 5000 MeV the upper limits were an order of magnitude higher. Upper limits were set on the coupling of heavy neutrinos to muons as well. Together with research into heavy neutrinos conducted by other collaborations, the limit on the coupling between heavy neutrinos and muons as a function of neutrino mass is displayed in Fig. 1 [4]. The regions that are excluded by previous studies are in grey.

Figure 1: Limits on the coupling between heavy neutrinos and muons as a function of neutrino mass, from [4]. The regions that are excluded by previous studies are in grey.

This project also looks into a lepton number violating B decay mediated by a heavy neutrino. The complete decay is B− → µ−_e+_π−_{; the corresponding Feynman diagram is shown in Fig. 2.} Simultaneously also the decay with opposite charges is studied. The advantage of this decay mode is that, contrary to the studies previously conducted at LHCb, the heavy neutrino does not have to behave as a Majorana particle to have an opposite sign muon and electron in the final state. The neutrino only has to mix between flavours, which it does because it has a mass. Electrons are, however, harder to detect than muons, because they loose energy through bremsstrahlung and they can fly out of the detector acceptance. But looking for heavy neutrinos in a decay with two opposite sign muons in the final state is difficult because there is a large amount of background coming from other Standard Model decay modes with two opposite sign muons in the final state.

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2 THEORETICAL BACKGROUND 6

With an opposite sign muon and electron in the final state, there is no background coming from other Standard Model decay modes.

Figure 2: Feynman diagram for the decay B−→ µ−

e+π−mediated by a heavy neutrino, which is studied in this project.

A preparatory study into the decay in Fig. 2 and its detection at LHCb is performed in this project. In the study both Monte Carlo simulated data and LHCb 2016 data from Run 2 are used. For the heavy neutrino a hypothesis of mass mN = 2500 MeV and lifetime τN = 100 ps is used. This mass is not yet excluded by previous studies, as can be seen in Fig. 1. The neutrino is assumed to be relatively long lived, based on its small coupling constant and helicity suppression because of its right-handedness.

Because the neutrino is expected to be long lived, more than just the standard long tracks in the LHCb detector are used in this study. Tracks are the trajectories of charged particles through the detector. Next to the long tracks also downstream tracks are used. Downstream tracks are sensitive to longer lived signals.

The effect of using a B mass constraint is investigated. The goal of using this constraint is to improve the neutrino mass resolution. As the electron looses energy through bremsstrahlung, information on the kinematics of the decay is lost. This negatively impacts the mass resolution of the neutrino, because the mass is determined from the kinematics of the decay products. With the B mass constraint the momenta of the decay products are adjusted such that B invariant mass is what it is supposed to be for the B; 5279 MeV [5]. With the adjusted momenta, the neutrino mass resolution could be improved. The B mass constraint is used in combination with a PV constraint. The PV constraint requires the B to point to the primary vertex, which is where a proton-proton interaction happened. To accomplish this the PV constraint further adjusts the particle momenta.

2 Theoretical background

2.1 The Standard Model

The Standard Model of particle physics describes the universe at its most fundamental level. That is, the elementary particles and the interactions between them [1]. Only gravity is not part of the Standard Model.

The elementary particles consist of spin 1₂ fermions and spin 1 bosons. Fermions make up the matter content of the universe and come in two groups: quarks and leptons. For both quarks and leptons there are six different types or flavours, as shown in Fig. 3. Each fermion has a corresponding antiparticle that has the same mass, but opposite charge. The fermions are divided into three different flavour generations. Apart from neutrinos, the fermions in each different generation have masses of roughly the same order of magnitude. The neutrino masses are much smaller than the masses of the other fermions. Currently it is not understood why the mass hierarchy between different generations exists and how the small neutrino masses fit into it.

Bosons are the force carrying particles in the Standard Model. There are four fundamental forces in nature, of which three are described by the Standard Model. These are the electromag-netic force, the strong force and the weak force. The fourth fundamental force is the gravitational force. The weak force is the only force that can change particle flavour. Finally there is the spin

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zero Higgs boson, which was introduced to make the other Standard Model particles obtain their mass.

Figure 3: The elementary particles in the Standard Model of particle physics. In the public domain, parameters from [5].

2.2 Neutrino oscillations

In 1998 it was discovered in the Super-Kamiokande experiment that neutrinos can change flavour when travelling over large distances [6]. Further evidence was provided by the SNO experiment in 2001 [7]. This neutrino flavour changes are neutrino oscillations. They occur because the mass eigenstates of the neutrino are not the same as the weak flavour eigenstates [1]. The mass eigenstates ν1,2,3 are the eigenstates of the free particle Hamiltonian. The weak eigenstates νe,µ,τ are the eigenstates that participate in the weak interaction. The different types of eigenstates are related by the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix:

  νe νµ ντ  =   Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ 1 Uτ 2 Uτ 3     ν1 ν2 ν3  . (1)

When a neutrino state |ψi is formed in a weak interaction, the state is a mixture of the three different mass eigenstates:

|ψi = Ue1∗ |ν1i + Ue2∗ |ν2i + Ue3∗ |ν3i . (2) The neutrino state propagates as a linear superposition of the mass eigenstates until it participates in another weak interaction, where the wavefunction collapses into a weak eigenstate. In this in-teraction a charged lepton of particular flavour is produced which can be observed in experiment. If the mass eigenstates are not the same, phase differences arise between the wavefunction com-ponents leading to the neutrino oscillations. Therefore, to explain neutrino oscillations, neutrinos are required to have a mass. This meant a modification to the Standard Model, in which neutrinos were assumed to be massless.

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2.3 Chirality and handedness

Chirality is a concept in the Standard Model in which particle states are decomposed in left-handed (denoted with subscript L) and right-handed (denoted with subscript R) chiral components [1]. The chirality operator is the γ5 _{matrix, which is defined as}

γ5= iγ0γ1γ2γ3, (3)

where the γµ _{are the Dirac matrices.}

In the ultrarelativistic limit, in which E >> m, the eigenstates of the chirality operator are equal to the helicity eigenstates. The helicity h is the normalized spin component along the direction of the momentum of the particle:

h = S · p

|p| . (4)

Here S is the spin vector of the particle and p the momentum vector of the particle. When helicity is positive, the momentum and spin are in the same direction and the particle is right-handed. When helicity is negative, the momentum and spin are in the opposite direction and the particle is left-handed.

Chiral projection operators are used to decompose particle states in left- and right-handed chiral components. The chiral projection operators PL and PR are defined as

PL= 1 2 14×4− γ 5_, ₍₅₎ PR= 1 2 14×4+ γ 5_. ₍₆₎

The total particle state ψ is the sum of the left- and right-handed chiral components:

ψ = ψL+ ψR= PLψ + PRψ. (7)

For antiparticle states PL projects out the right-handed chiral component and PR projects out the left-handed chiral component. The weak interaction vertex factor contains a factor PL, which means only left-handed particle states and right-handed antiparticle states participate in the weak interaction. Since neutrinos only interact through the weak interaction only left-handed neutrinos and right-handed antineutrinos should be observable, which is confirmed by experiment [8].

2.4 Majorana particles

All Standard Model fermions obey the Dirac equation:

(iγµ∂µ− m) ψ = 0. (8)

The solutions are four dimensional complex Dirac spinors [1].

Particle states can be replaced by their corresponding antiparticle states by the combination of the charge conjugation operator ˆC and parity operator ˆP . This CP conjugation flips the charge and changes left-handedness to right-handedness. CP conjugation is defined as

ψC= ˆC ˆP ψ = iγ2γ0ψ∗, (9)

where ψC is the CP conjugated or antiparticle spinor and γ2 and γ0are Dirac matrices. For Dirac particles the particle and antiparticle states are not the same, i.e.

ψC6= ψ. (10)

The solutions to (8) are real if all non-zero components of the γµ_{matrices are purely imaginary} [9]. This is the case in the Majorana representation of the γµ _{matrices. The real solutions to (8)}

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are Majorana particles. For Majorana particles the particle and antiparticle states are the same, i.e.

ψC= ψ. (11)

In the seesaw mechanism the neutrino can behave as a Majorana particle. If that is the case, decays that violate lepton number conservation are possible. The different generations of leptons in the Standard Model are assigned a corresponding lepton number. There are Le, Lµ and Lτ lepton numbers, which equal one for particles and minus one for antiparticles. In Standard Model interactions the lepton numbers are conserved. The lepton numbers would not be conserved when the neutrino behaves as a Majorana particle, because when that is the case the neutrino and antineutrino can couple to each other.

2.5 The seesaw mechanism

From neutrino oscillations it is known that neutrinos have mass, which means there must be a mass term in the neutrino Langrangian [1]. This can be done by assuming the neutrino is a Dirac particle, the Dirac mass term LD is:

LD= −mD ψLψR+ ψRψL . (12)

Here mD is the Dirac mass, ψ a neutrino particle spinor and ψ the adjoint spinor (ψ = ψ†γ0, where γ0 _{is a Dirac matrix). If this is the origin of the neutrino mass, right-handed neutrinos and} left-handed antineutrinos exist. Because the neutrino mass is so small though, the neutrino mass might be of different origin.

There is a freedom to include additional terms in the Langrangian mass term containing right-handed neutrino spinors and left-right-handed antineutrino spinors. Therefore a Majorana mass term LM containing these spinors can be added:

LM = − 1 2M ψ c RψR+ ψRψ c R . (13)

Here M is the Majorana mass. The superscript c denotes CP conjugation so ψc

Ris the antiparticle spinor of the right-handed neutrino spinor, which is the left-handed antineutrino spinor. The factor 1₂ is to avoid double counting. The Majorana mass term provides a direct coupling between the neutrino and antineutrino. Therefore it is only possible to add a Majorana mass term to the Langrangian mass term for neutrinos, because they have charge zero. If neutrinos would have had charge, charge conservation would be violated in the Majorana mass term. The difference between the Dirac and Majorana mass terms is shown in Fig. 4.

Figure 4: The difference between the Dirac and Majorana Mass terms, from [1]. In the Dirac mass term there are both left- and right-handed neutrinos (and antineutrinos). In the Majorana mass term there are only right-handed neutrinos and left-handed anti-neutrinos (denoted ν here), with a direct coupling between them.

By using that ψLψR is equal to ψRcψcLthe Dirac mass term can be written as LD= − 1 2mD ψLψR+ ψ c Rψ c L + h.c. (14)

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3 THE LHCB DETECTOR 10

Here h.c. denotes the Hermitian conjugated term.

The Dirac and Majorana mass terms can be combined to obtain LDM = − 1 2 mDψLψR+ mDψ c Rψ c L+ M ψcRψR + h.c. (15)

This expression can be rewritten in matrix form: LDM = − 1 2 ψL ψ c R 0 mD mD M ψc L ψR + h.c. (16)

The matrix can be diagonalized to obtain two mass eigenvalues:

m±=

M ± Mp1 + 4mD2/M2

2 . (17)

In the seesaw mechanism it is assumed that M >> mD, which results in a very light neutrino ν with

|mν| ≈ mD2

M , (18)

and a very heavy neutrino N with

mN ≈ M. (19)

The light neutrino would be almost completely left-handed and the heavy neutrino would be almost completely right-handed. Therefore the heavy neutrino would be almost completely transparent to the weak interaction, explaining why only the light neutrinos are observed.

It should be stressed that the introduction of the Majorana mass term in the seesaw mechanism does not imply that the neutrino is a Majorana particle. The neutrino only behaves as a Majorana particle in the sense that there is a direct coupling between the right-handed neutrino and the left-handed antineutrino.

3 The LHCb detector

3.1 Detector configuration

The LHCb experiment at the LHC at CERN in Geneva is dedicated to the study of b-hadrons, which are hadrons containing a bottom quark. The goals of the LHCb experiment are to find new physics in CP violation and look for rare and exotic decays [10]. The LHCb detector is built in the forward direction, covering an angle from approximately 10 mrad to 250 mrad in the vertical plane and 300 mrad in the horizontal plane. This angular configuration is chosen because at LHC energies b-hadrons are predominantly produced in a forward and backward cone of the same angles.

The LHCb detector consists of multiple subdetectors as well as a magnet. The design of the detector is shown in Fig. 5. The different parts of the detector are the following:

• Vertex Locator (VELO), built around the point where the LHC proton beams, coming from the ± z directions, meet. It is the first part of the charged particles tracking system and its purpose is to locate vertices. A vertex is a point where a particle interaction happens. • RICH1 Cherenkov detector, the first part of the particle identification system. It covers

charged particles with low momentum in the range of ∼1-60 GeV.

• TT tracking station, the second part of the tracking system. Charged particles leave hits in the tracking stations from which tracks can be reconstructed. It is placed before the magnet to provide extra hits in this region.

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3 THE LHCB DETECTOR 11

• Magnet, used together with the tracking system to measure the momentum of charged particles. To account for possible detector asymmetries, half the data at LHCb is taken with magnet up (MU) configuration, and half with magnet down (MD) configuration.

• T1-T3 tracking stations, the final parts of the charged particles tracking system. • RICH2 Cherenkov detector, second Cherenkov detector in the particle identification

system. It covers particles with high momentum in the range of ∼15-100 GeV and beyond. • Calorimeter system (consisting of SPD/PS, ECAL and HCAL), part of the particle identification system and used to identify hadrons, electrons and photons and to measure their energies and positions.

• M1-M5 muon stations, the muon stations are part of the particle identification system and are used to identify muons. M1 is placed before the calorimeter system, M2-M5 behind it.

Figure 5: The design of the LHCb detector at the LHC, from [10].

3.2 Data processing

Too much data is produced at LHCb for all of it to be stored. Therefore, LHCb data is processed in a specific data flow that only stores promising events. An event is a crossing of the proton beams through the detector. The data flow can be divided in three steps [11]:

1. A trigger system makes a first selection of the data coming from the detector.

2. Tracks are reconstructed from the detector hits in the raw triggered data and particle iden-tifications are made.

3. Candidates are formed from the tracks and particle identifications. A candidate is a recon-structed particle. In the data used in this study, candidates are formed in the following way:

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4 ANALYSIS STRATEGY 12

(b) Electrons and pions are combined to form N candidates. At this point cuts are applied on certain variables of the candidates; the stripping cuts.

(c) The N candidates are combined with muons to form B candidates, stripping cuts are applied here as well.

The stripping cuts are described by a stripping line. A stripping selection describes similar events and is defined by the stripping line.

In Monte Carlo simulation at LHCb the collisions of protons, particle decays and the interaction of particles with the detector are simulated. The simulated detector hits are then converted into signals that mimic the real detector. This way the simulated data can be passed through the same data flow as the real data.

Charged particles that travel through the detector leave hits in the various components of the LHCb tracking stations. From these hits the tracks are reconstructed. Different types of tracks are possible, which are shown in Fig. 6 [12]. The different tracks are listed below:

• VELO track, hits only in the VELO. • Upstream track, hits in the VELO and TT.

• Long track, hits in VELO and T1-T3 (also possibly TT). • Downstream track, hits in TT and T1-T3.

• T track, hits in T1-T3.

It is possible to have more than one track for the same particle. If that is the case, the track containing the most information (hits in more parts of the tracking system) is stored. Also fake tracks can be reconstructed, which is caused by mismatching of hits before and after the magnet. These fake tracks do not represent a real particle.

In this study long tracks and downstream tracks are used. The neutrino candidate is formed from either two long tracks or two downstream tracks. The muon is always a long track.

Figure 6: Different types of tracks in the LHCb detector, from [12].

4 Analysis strategy

Various aspects of the heavy neutrino mediated decay B−→ µ−_e+_π− _{and its detection at LHCb} are analyzed in this project. This is done using a Monte Carlo data sample and a LHCb 2016 data sample from Run 2. In the analyses the Monte Carlo data represents the signal decay, which is the heavy neutrino mediated B decay. The data samples are described in Section 5.

In Section 6 the stripping lines are analyzed. This is done by applying the cuts from the stripping lines on the Monte Carlo data, in which no stripping is performed. By doing this

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5 DATA SAMPLES 13

stripping efficiencies for both long tracks and downstream tracks can be computed. When the stripping cuts are applied on the Monte Carlo data, the data can be compared to the LHCb 2016 data. This is done by plotting the B and N invariant mass distributions. The N invariant mass distributions are also plotted using the B mass constraint, to see its effect.

Further selection cuts can be made to separate signal-like candidates from the background. The process of determining selection cuts is described in Section 7. Different methods are used for determining the selection cuts. First cuts are made for the B and N decay vertex χ2 _{and the B} impact parameter χ2_{, based on plots of those variables from both the Monte Carlo data and 2016} data. Cuts on more variables are made using ROC curves and significance. The rejecting power of a variable is determined using a ROC curve. The optimal value to cut is then determined by looking at a plot of the significance. The selection efficiencies of the determined cuts are computed in the Monte Carlo data, again both for long tracks and downstream tracks. With the selected data new invariant mass distributions are plotted, to visualize the effect of the selections.

In Section 8 the signal efficiency is determined as a function of neutrino lifetime. This is done using the stripped and selected Monte Carlo data. The signal efficiency is a measure for how effective the detector is in measuring the decay signal. The efficiency is determined for long and downstream tracks separately as well as for the different tracktypes combined, to see the differences.

With the selected data an upper limit can be set on the branching fraction of B±→ µ±_{N for} the particular model used in this study. That is for mN=2500 MeV, τN=100 ps and an electron and a pion in the final state. The method used in computing the limit and the result of the computation of the limit are described in Section 9.

5 Data samples

5.1 Monte Carlo data

The Monte Carlo data sample used in this study is generated using the hypothesis of mN=2500 MeV and τN=100 ps. Information on the Monte Carlo data sample is shown in Tab. 1. In the table n (long tr.) and n (downstream tr.) are the number of candidates for long tracks and downstream tracks respectively, n (accepted) is the number of generated events that are left after the generator level cuts and recthe reconstruction efficiency. The generator level cuts require the decay products to fall within the angular acceptance of the LHCb detector, defined in the cuts as 10-400 mrad. The reconstruction efficiency is defined as the number of accepted events that are reconstructed into candidates:

rec=

n (candidates)

n (accepted) . (20)

Here n (candidates) is the number of candidates that are reconstructed, which in this case is defined as n (candidates) = n (long tr.) + n (downstream tr.).

n (long tr.) n (downstream tr.) n (accepted) rec

MU 26257 15711 288317 0.14556(66)

MD 24230 14657 289157 0.13448(63)

Total 50487 30368 577474 0.14001(46)

Table 1: Information on the Monte Carlo data sample used in the study

5.2 LHCb 2016 data

A LHCb 2016 data sample from Run 2 is used, for which the center of mass energy √s = 13 TeV. Information on the 2016 data sample is shown in Tab. 2. In the table n (long tr.) and n

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6 STRIPPING CUTS 14

(downstream tr.) are the number of candidates for long tracks and downstream tracks respec-tively and L is the luminosity. The luminosity is a measure for the number of proton-proton interactions. The stripping lines applied in the data sample are StrippingBu2MajoMuLLLine and StrippingBu2MajoMuDDLine for long tracks and downstream tracks respectively. The same stripping cuts are applied for both tracktypes, but they use different input tracks. A table with a description of the stripping cuts in the stripping lines is found in Appendix A.

n (long tr.) n (downstream tr.) L [pb−1_]

MU 2625424 1799404 683.75

MD 2704855 1853481 697.17

Total 5330279 3652885 1380.92

Table 2: Information on the 2016 data sample used in the study

6 Stripping cuts

6.1 Stripping efficiency

The efficiencies of the stripping cuts from the stripping lines are determined by applying them on the Monte Carlo data. This is done for long tracks and downstream tracks separately. The stripping efficiency stripis defined as

strip=

n (stripping)

n (total) . (21)

Here n (stripping) is the number of candidates left after the cut is applied and n (total) is the total number of candidates before the cut. The value of n (total) is equal to the values for n (long tr.) for long tracks and n (downstream tr.) for downstream tracks in Tab. 1. The results of the computation are shown in Tab. 3. The efficiencies vary between 0.8 and 1.0 except for the BPVDIRA cut which is significantly lower then the others, especially for downstream tracks. With the stripping cuts applied there are 28309 candidates left for long tracks and 6971 for downstream tracks.

Table 3: Stripping efficiencies in the Monte Carlo data

Particle Cut strip

(long tr.) strip (downstream tr.) π TRGHOSTPROB<0.5 1.0 1.0 TRCHI2DOF<3.0 0.99439(33) 0.99872(21) PT>300.0∗MeV 1.0 1.0 MIPCHI2DV(PRIMARY)>9.0 0.99851(17) 0.99559(38) e TRGHOSTPROB<0.5 1.0 1.0 TRCHI2DOF<3.0 0.98776(49) 0.99460(42) PT>300.0∗MeV 1.0 1.0 MIPCHI2DV(PRIMARY)>9.0 0.99758(22) 0.99460(42) µ TRGHOSTPROB<0.5 1.0 1.0 TRCHI2DOF<3.0 0.99840(18) 0.99845(23) PT>500.0∗MeV 1.0 1.0 MIPCHI2DV(PRIMARY)>16.0 0.8659(15) 0.8874(18)

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6 STRIPPING CUTS 15

Particle Cut strip

(long tr.) strip (downstream tr.) N P>5000.0∗MeV 1.0 1.0 PT>500.0∗MeV 0.97456(70) 0.99299(48) VFASPF(VCHI2)<16.0 0.99750(22) 0.99483(41) VFASPF(VMINVDDV(PRIMARY))>1.0 0.99935(11) 1.0 BPVVDCHI2>25.0 0.99972(7) 0.99937(14) ADWM(KS0,WM(’pi+’,’pi-’))>35.0 1.0 0.99993(5) ADWM(Lambda0,WM(’p+’,’pi-’))>35.0 1.0 0.99997(3) ADWM(Lambda0,WM(’pi+’,’p-’))>35.0 1.0 1.0 B P>5000.0∗MeV 1.0 1.0 VFASPF(VCHI2)<10.0 0.9262(11) 0.9519(12) BPVDIRA>0.9999 0.7208(20) 0.2737(26) MIPCHI2DV(PRIMARY)<16.0 0.8917(14) 0.9151(16) VFASPF(VMINVDDV(PRIMARY))>1.0 0.9183(12) 0.9279(15) BPVVDCHI2>30.0 0.8959(14) 0.8448(21) (5279-500.0)∗MeV<M<(5279+700.0)∗MeV 0.97215(73) 0.9584(11) Total 0.5607(22) 0.2296(24)

6.2 Invariant mass distributions after stripping

With the stripping cuts applied in the Monte Carlo data the B and N invariant mass distributions from the Monte Carlo data and the 2016 data can be plotted. This is done for long tracks and for downstream tracks separately. The N invariant mass distributions are plotted with and without the B constraint.

In Fig. 7 plots of the B invariant mass distributions from the stripped Monte Carlo data are shown. The distributions are spread around the B mass.

In Fig. 8 plots of the N invariant mass distributions from the stripped Monte Carlo data are shown, with and without the B constraint. Both distributions peak around the N mass of 2500 MeV. Using the B constraint the peak becomes sharper, so the mass resolution is improved.

(a) Long tracks (b) Downstream tracks

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6 STRIPPING CUTS 16

Figure 8: N invariant mass distributions, obtained from the stripped Monte Carlo data. In the red distributions the B constraint is applied, in the blue distributions it is not.

In Fig. 9 plots of the B invariant mass distributions from the 2016 data are shown. For long tracks a bump is visible just below the B mass, while for Downstream tracks there is only background.

In Fig. 10 plots of the N invariant mass distributions from the 2016 data are shown, with and without the B constraint. Various peaks are visible in the distributions. In the long track distribution there is a peak at 1750 MeV. This is the decay D0_{→ K}±_π±_{, where the D}0 _{is formed} in the decay B±→ D0_π±_{. A smaller peak is visible at 3100 MeV, this is the decay J/ψ → e}+_e−_. The J/ψ is formed in the decay B± → J/ψK±_{. These decay modes cause the bump in the long} track B invariant mass distribution in Fig. 9. The bump is shifted to the left of the B mass because the mass is calculated with the wrong hypothesis of a muon, electron and pion in the final state.

In the downstream track distribution in Fig. 10 a sharp peak is visible at 450 Mev, this is the decay K_S0 → π+_π−_.

In both of the distributions in Fig. 10 there is a dip on the low edge of the distribution. This is caused by the veto cuts in the stripping lines.

For both tracktypes the effect of the B constraint is that the background peaks are spread out.

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7 SELECTION CUTS 17

Figure 10: N invariant mass distributions, obtained from the 2016 data. In the red distributions the B constraint is applied, in the blue distributions it is not.

7 Selection cuts

7.1 Cuts on B and N χ

2

variables

In the selection further cuts are made to separate signal-like candidates from the background. First cuts are applied on the B decay vertex χ2_{, N decay vertex χ}2 _{and the B impact parameter} χ2_{. The impact parameter is a measure for the distance between the primary vertex and the track.} To determine at which values to cut, the distributions of the variables are plotted. This is done both for the stripped Monte Carlo data and the 2016 data, by comparing the distributions the cutvalue is chosen. The plots of the distributions can be found in Appendix B.

The distributions have an exponential shape from which the tail can be cut. The decay vertices have one degree of freedom. For the B and N decay vertex χ2 _{a cutvalue of 4.0 is chosen, which} corresponds to two standard deviations. The impact parameter has two degrees of freedom, therefore a less tight cutvalue of 10.0 is chosen for the B impact parameter χ2_.

The efficiencies of the selection cuts are computed by applying the selection cuts on the stripped Monte Carlo data. The efficiencies are computed for long tracks and downstream tracks separately. The selection efficiency sel is defined as

sel=

n (selection)

n (total) . (22)

Here n (selection) is the number of candidates left after the cut is applied and n (total) the number of candidates before the cut, which is the number of long track or downstream track candidates in the stripped Monte Carlo data. The results of the computation are shown in Tab. 4. With the selection cuts there are 22925 long track candidates and 5956 downstream track candidates left in the Monte Carlo data. In the 2016 data there are 1731897 long track candidates and 964465 downstream track candidates left.

Cut sel (long tr.) sel (downstream tr.) B decay vertex χ2 _{< 4.0} _0.8863(19) _0.9188(33) N decay vertex χ2_{< 4.0} _0.9414(14) _0.9366(29) B impact parameter χ2 _{< 10.0} _0.9643(11) _0.9834(15)

Total 0.8098(23) 0.8544(42)

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7 SELECTION CUTS 18

7.2 ROC curves and significance

For other variables selection cuts are made based on ROC curves and significance. A ROC curve is a plot of the background rejection versus the signal efficiency of a cut on a certain variable. The background rejection is one minus the background efficiency. The background efficiency bkg is defined as

bkg=

nbkg (after cut) nbkg(before cut)

, (23)

where nbkg(before cut) and nbkg(after cut) are the number of background candidates before and after the cut is applied. Similarly the signal efficiency sig is defined as

sig=

nsig (after cut) nsig(before cut)

, (24)

where nsig (before cut) and nsig (after cut) are the number of signal candidates before and after the cut. Each point on the ROC curve represents a value of the cut. The larger the area under the ROC curve, the better the variable is in rejecting background while keeping signal.

The exact value to cut is then determined using a plot of the significance versus the value of the cut. The significance S is computed as

S = nsig (after cut) p1 + nbkg(after cut)

. (25)

The ideal value to cut is in the region where the significance peaks. The cutvalue is chosen as the most efficient cut that is within ten percent of the maximum of the significance. However for particularly powerful variables a lower value for the cut is chosen, because in that case a lower cutvalue keeps more signal while rejecting almost the same amount of background.

In this study the Monte Carlo data is used as entirely signal, while the 2016 data is assumed to be entirely background. In the Monte Carlo data the stripping cuts are applied, the B and N χ2 cuts are applied in both Monte Carlo and 2016 data. The variables that are used in the selection are the following:

• µ, e and π impact parameter χ2_{, selecting a high impact parameter χ}2_{for these particles} makes their tracks less correlated with the primary vertex. This means the particles are defined better, because they are not created at the primary vertex.

• PID variables, used to filter out background decays. PID gives the logarithmic likelihood of a particle to be either a particle of type x or a pion:

PIDx= log Lx− log Lπ. (26)

Here Lx is the likelihood of the particle to be of type x and Lπthe likelihood of the particle to be a pion. A positive PID means the particle is more likely to be of type x, a negative PID means the particle is more likely to be a pion. First the decay D0_{→ K}±_π± _{is rejected} by requiring the PIDe of the electron to be positive, this ensures an electron in the final state. Then the second electron in J/ψ → e+e− is rejected by requiring the PIDe of the pion to be negative. Finally the third particle is required to be a muon by taking a positive PIDµ for the muon.

• N lifetime, this study looks for long lived neutrinos, therefore neutrinos with a short lifetime are rejected.

The ROC and significance plots of the above variables are made for long tracks and downstream tracks and can be found in Appendix C. The chosen cutvalues for long tracks are shown in Tab. 5. The chosen cutvalues for downstream tracks are showm in Tab. 6. In the tables the corresponding

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7 SELECTION CUTS 19

selection efficiencies of the cuts are shown as well. The selection efficiency sel is computed as in (22), where in this case n (total) is the number of long track or downstream track candidates in the Monte Carlo data with both the stripping cuts and the B and N χ2 selection cuts applied. With all selection cuts applied there are 9073 long track candidates and 1774 downstream track candidates left in the Monte Carlo data. In the 2016 data there are 54 long track candidates and 289 downstream track candidates left.

Cut sel µ impact parameter χ2 _{> 50.0} _0.8869(21) e impact parameter χ2> 2500.0 0.7441(29) π impact parameter χ2> 2500.0 0.7936(27) e PIDe> 5.0 0.8500(24) π PIDe< -2.0 0.8659(23) µ PIDµ > 2.0 0.9556(14) N lifetime > 7.5 ps 0.7492(29) Total 0.3941(32)

Table 5: Long track selection efficiencies for cuts based on ROC curves and significance.

Cut sel µ impact parameter χ2 > 100.0 0.8811(42) e impact parameter χ2 > 300.0 0.7834(53) π impact parameter χ2 _{> 300.0} _0.7535(56) e PIDe> 5.0 0.8360(48) π PIDe< -2.0 0.7740(54) µ PIDµ > 4.0 0.8506(46) N lifetime > 20.0 ps 0.9718(21) Total 0.2975(59)

Table 6: Downstream track selection efficiencies for cuts based on ROC curves and significance.

7.3 Invariant mass distributions after selection

To visualize the effects of the selections, the B and N invariant mass distributions with the selection cuts applied are plotted. To better visualize the correlations between the distributions, the B and N invariant mass distributions are plotted against each other in a two dimensional scatterplot. The scatterplots are made separately for long tracks and downstream tracks, and they are made for the Monte Carlo and the 2016 data.

In Fig. 11 the scatterplots for the Monte Carlo data are shown with and without the B constraint applied in the N mass. There is a clear concentration of candidates in the region where the B mass and the 2500 MeV N mass intersect. However, without the B constraint there is a linear correlation between the B and N mass. For a lower B mass the N mass is also lower, this is because the electron has lost energy through bremsstrahlung. With the B constraint this loss of energy is compensated by adjusting the momenta of the decay products, which improves the N mass resolution.

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7 SELECTION CUTS 20

(a) Long tracks, without B constraint (b) Downstream tracks, without B constraint

(c) Long tracks, with B constraint (d) Downstream tracks, with B constraint

Figure 11: Scatterplots of the B and N invariant mass distributions with and without the B constraint applied in the N mass. The distributions are obtained from the Monte Carlo data with the stripping cuts and selection cuts applied.

In Fig. 12 the scatterplots for the 2016 data are shown, the B constraint is applied in the N mass. In the downstream track plot it is clearly visible that most of the N mass values are smaller than 2500 MeV.

Figure 12: Scatterplots of the B and N invariant mass distributions from the 2016 data with the selection cuts applied. The B constraint is applied in the N mass.

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8 SIGNAL EFFICIENCY AS A FUNCTION OF N LIFETIME 21

8 Signal efficiency as a function of N lifetime

The signal efficiency as a function of the neutrino lifetime τ is determined from the stripped and selected Monte Carlo data. The efficiency is determined for long tracks and downstream tracks separately as well as for combined tracks. The first step in determining the efficiency is to plot the distribution of τ from the stripped Monte Carlo data. The distribution is rebinned to have approximately the same number of entries per bin. The bins are then normalized, the exponential decay function of the neutrino is used for this:

n(t) = n(0)e−t/τN_. ₍₂₇₎

Here n(t) is the number of particles left after time t, n(0) is the number of particles at t = 0, which is equal to the total number of accepted events in Tab. 1, and τN is the neutrino lifetime of 100 ps. Over each bin the integral of the exponential decay function is computed. For each bin the bin content is then divided by the corresponding integral.

The signal efficiency for long tracks and downstream tracks separately as well as for combined tracks is shown in Fig. 13. For long tracks the efficiency is highest for short lifetimes, while it goes to zero for longer lifetimes. For downstream tracks the efficiency peaks around 50 ps and is slightly higher at 100 ps than for the long tracks. The downstream track efficiency goes to zero for short lifetimes because short lived neutrinos are reconstructed as long tracks. The effect of combining the long and downstream tracks is that the efficiency falls away less rapidly for longer lifetimes.

(c) Combined tracks

Figure 13: Signal efficiency as function of neutrino lifetime for long tracks and downstream tracks sepa-rately as well as for combined tracks.

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9 BRANCHING FRACTION LIMIT 22

9 Branching fraction limit

An upper limit can be set on the branching fraction of B±→ µ±_{N for the particular model used} in this study, so for mN=2500 MeV. τN=100 ps and with an electron and a pion in the final state. To compute the branching fraction, the following formula is used:

B B± → µ±N = 1

nsig nB

. (28)

Here B is the branching fraction, is the total efficiency in the Monte Carlo data, nsig is an upper limit on the number of signal candidates in the 2016 data and nB is the total number of B mesons produced in the 2016 data sample. The B constraint is used for the neutrino, because in Section 6 and Section 7 it was shown that it improves the mass resolution. In Section 8 it was shown that when combining long tracks and downstream tracks the efficiency is slightly improved for longer neutrino lifetimes. Therefore, when computing the limit, the long tracks and downstream tracks are combined. Because the limit is set for the particular mass of 2500 MeV, first a cut on the N mass is made. The cut is defined as 2350 MeV < mN < 2750 MeV.

To compute nsig, a signalbox is defined as 5100 MeV < mB < 5400 MeV in the B mass range. To represent the background, two sideboxes are defined as 4800 MeV < mB< 5100 MeV and 5400 MeV < mB < 5700 MeV. The ratio between the size of the sideboxes and the signalbox is 2.0. In both the signalbox and the sideboxes the number of candidates from the selected 2016 data are counted. In the signalbox there are 3 long track candidates and 0 downstream track candidates. In the sideboxes there are 2 long track candidates and 6 downstream track candidates. Then nsig is computed using the method described in [13]. This method is implemented in the ROOT TRolke class [14]. The TRolke class takes the total number of candidates in the signalbox, the total number of candidates in the sideboxes and the ratio between the sizes of the boxes as input. With this input TRolke computes nsigwith a desired confidence level (C.L.). The above procedure resulted in nsig= 4.28 (95% C.L.).

Then nB is computed as

nB= L σB, (29)

where L is the total luminosity of the 2016 data sample from Tab. 2 and σB the B cross section. For σB the reported value in [15] at

√

s = 13 TeV is used; σB= 86.6 ± 6.4 µb.

In [15], the cross section is given for a rapidity range of 2.0 < y < 4.5, which is where LHCb is sensitive. The rapidity y of a particle is defined as

y =1 2log

E + pz E − pz

, (30)

where E is the energy of the particle and pz the momentum component along the beam direction [5]. The fact that the cross section is only known in the rapidity range 2.0 < y < 4.5, means that (29) only gives the number of produced B mesons in this range. This has to be taken into account in the computation of from the Monte Carlo data. First it needs to be checked what the rapidity is of the B mesons within the N mass cut and the B signalbox mass cut. In Fig. 14 the B rapidity distribution from the stripped and selected Monte Carlo data with the B and the N mass cuts is shown. In the plot the long track and downstream candidates are combined. It can be seen that the candidates fall inside the rapidity range of 2.0 < y < 4.5.

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9 BRANCHING FRACTION LIMIT 23

Figure 14: The rapidity distribution of the Monte Carlo candidates in the signalbox.

Knowing that the B mesons from the stripped and selected Monte Carlo data within the B and the N mass cuts fall inside the desired rapidity range, can be computed as

= n (signalbox)

n (accepted). (31)

Here n (signalbox) is the number of candidates in the stripped and selected Monte Carlo data that fall within the ranges 5100 MeV < mB < 5400 MeV and 2350 MeV < mN < 2750 MeV, and n (accepted) the total number of accepted events in Tab. 1. This expression for is equivalent to

= rec× strip× sel× sig, (32)

where rec would be the total reconstruction efficiency from Tab. 1, strip the total stripping efficiency for combined tracks, sel the total selection efficiency for combined tracks and sig the efficiency of the B and the N mass cuts in the stripped and selected Monte Carlo data.

The number of candidates at each stage of the signal taking is shown in Tab. 7, where n (long track) are long track candidates, n (downstream track) downstream track candidates and n (total) candidates for combined tracks. For comparison the numbers are shown for the 2016 data as well. With the numbers in Tab. 7 the Monte Carlo efficiency can be computed. In Tab. 8 the efficiency of each step of the signal taking as well as the total efficiency is shown.

Combining the terms in (28) resulted in an upper limit on the B±→ µ±_{N branching fraction} of B (B±→ µ±_{N ) = 1.96 × 10}−9 _{(95% C.L.).}

Monte Carlo data 2016 data

n (long tr.) n (downstream tr.) n (long tr.) n (downstream tr.)

Accepted events 577474 (total) -

-Reconstructed candidates 50487 30368 -

-Candidates after stripping 28309 6971 5330279 8983164

Candidates after selection 9073 1774 54 389

Candidates after B and N

mass cuts 8834 1707 3 0

Table 7: Number of candidates at each stage of the signal taking in the Monte Carlo data adn in the 2016 data for comparison.

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10 DISCUSSION 24 rec 0.14001(46) strip 0.4362(17) sel 0.3075(25) sig 0.9718(16) (total) 0.01825(18)

Table 8: Efficiencies in the Monte Carlo data at the different stages of the signal taking as well as the total Monte Carlo efficiency.

10 Discussion

The current upper limit on the B branching fraction for the specific final state used in this study is set at 6.3 × 10−3 (90% C.L.) [16]. The study in which this limit was set looked for B and D decays to all possible charge conserving final states with two charged leptons and either π± or K±. In the study no specific masses or lifetimes were hypothesized for the mediating particles. However, the branching fraction limit set in the study performed in this project is six orders of magnitude smaller.

In the search for heavy neutrinos in B decays with two same sign muons and a pion in the final state previously performed at LHCb, the limit on B (B±→ µ±_{N ) for a neutrino with a} mass of 2500 MeV and a lifetime of 100 ps was of the order of 10−8. This means the limit on B (B±_{→ µ}±_{N ) with the same neutrino mass and lifetime hypothesis but with N → e}+_π− _{set in} this study is one order of magnitude smaller.

Improvement of the signal taking is possible at the level of the selection cuts, which could be optimized by using multivariate analysis. Instead of optimizing cuts on single variables separately, in multivariate analysis cuts on a set of multiple variables are optimized together. Optimizing the selection cuts would maximize the sensitivity to the branching fraction and result in a tighter limit. It was also found that signal is lost because of the too tight BPVDIRA stripping cut. Signal taking could be improved by rerunning the stripping with this cut loosened.

Normally at LHCb a an upper limit is set on the branching fraction using a B decay mode X with a known branching fraction as control mode. This removes dependence on the luminosity and B cross section and leads to a more precise measurement. The branching fraction of B± _{→ µ}±_N is then computed in the following way:

B B±→ µ±N = (B ±_{→ X)} (B±_{→ µ}±_{N )} nsig nX B B±→ X . (33)

Here (B±_{→ X) is the total Monte Carlo efficiency of the control mode, (B}±_{→ µ}±_{N ) the total} Monte Carlo efficiency of the signal decay, nsigthe upper limit on the number of signal candidates and nX the number of candidates of the control mode. To determine (B± → X) a Monte Carlo data sample of the control mode should be generated. In this data sample the control mode should then be reconstructed.

In a continuation of this study, the decays that showed up in the 2016 data could be used as control mode. This are B±→ D0_π± _{and B}± _{→ J/ψK}±_{. Since these decays show up in the data,} the same stripping lines could be used. To determine nXthe size of the control mode peaks should be estimated from the data.

Only a small part of the LHCb data was analyzed. By analyzing a larger amount of data, statistical errors become smaller. This results in a better determination of the upper limit on the number of signal candidates. Also, a larger amount of data means a possible signal can be distinguished from the background more easily.

In this study a search was performed for heavy neutrinos with only one specific neutrino mass and lifetime. The study could be extended by looking for different masses and lifetimes, as was done in the search at LHCb for heavy neutrinos in the decay B− _{→ µ}−_µ−_π+_{. The branching} fraction limit can then be determined as a function of neutrino mass and lifetime. It was however found that using the B mass constraint spreads out the background peaks of D0_{and J/ψ. This is}

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11 CONCLUSION 25

no problem for the neutrino mass used in this study, but when masses closer to the D0 _{and J/ψ} masses are used the background peaks can shift under the signal peak.

11 Conclusion

In this project a preparatory study into the heavy neutrino mediated decay B− → µ−_e+_π−_{and its} detection at LHCb was performed. For the heavy neutrino a hypothesis of mass mN = 2500 MeV and lifetime τN = 100 ps was used. An upper limit was set on the branching fraction of B− → µ−N for the particular model at B (B±→ µ±_{N ) = 1.96 × 10}−9 _{(95% C.L.). This limit is six orders} of magnitudes smaller than the current branching fraction limit on the decay B− → µ−_e+_π−_. However, this limit was set without a hypothesis for the mass of lifetime of the mediating particle. The limit set in this study is also one order of magnitude smaller than the limit on B (B±→ µ±_{N )} set in the previous search for heavy neutrinos at LHCb. In that study the same mass and lifetime hypothesis was investigated, but a different final state with two same sign muons was used.

The effect of using a B mass constraint in combination with a PV constraint was studied, using the B constraint improves the mass resolution of the neutrino. Downstream tracks were used in addition to the standard LHCb long tracks. The effect of adding the downstream tracks is to increase sensitivity to longer lived neutrinos.

The signal taking could be improved, both at a selection and stripping level. The selection could be improved by using multivariate analysis, while the stripping could be improved by using a less tight BPVDIRA cut.

Normally at LHCb a decay mode with a known branching fraction is used as a control mode, which removes the dependence on the luminosity and cross section and makes for a more precise measurement. In a continuation of this study the decay modes B± → D0_π± _{or B}± _{→ J/ψK}± could be used as control mode.

The study could be extended by looking for neutrinos with different masses and lifetimes, this way the branching fraction limit can be determined as a function of neutrino mass and lifetime. However, the B constraint spreads out background peaks, which is a problem if the neutrino mass is closer to the background peaks.

Finally more data should be analyzed to be more effective in finding possible signal and oth-erwise set a tighter limit.

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A DESCRIPTION OF STRIPPING CUTS 26

Appendix A

Description of stripping cuts

Particle Cut Description

π

TRGHOSTPROB<0.5 probability of the track being fake TRCHI2DOF<3.0 track fit χ2 _{per degree of freedom}

PT>300.0∗_MeV _{transverse momentum}

MIPCHI2DV(PRIMARY)>9.0 minimum χ

2 _{distance of the track} to the primary vertex

e

PT>300.0∗MeV transverse momentum

µ

N

P>5000.0∗MeV momentum

VFASPF(VCHI2)<16.0 χ2 _{of the decay vertex}

VFASPF(VMINVDDV(PRIMARY))>1.0

minimum distance between the decay vertex and the primary

vertex

BPVVDCHI2>25.0 χ

2 _{distance from the best} primary vertex

ADWM(KS0,WM(’pi+’,’pi-’))>35.0 veto cut for background decay ADWM(Lambda0,WM(’p+’,’pi-’))>35.0 veto cut for background decay ADWM(Lambda0,WM(’pi+’,’p-’))>35.0 veto cut for background decay

B

P>5000.0∗MeV momentum

VFASPF(VCHI2)<10.0 χ2 _{of the decay vertex}

BPVDIRA>0.9999

cosine of the angle between the momentum and the line from the best primary vertex to the decay

vertex

MIPCHI2DV(PRIMARY)<16.0 minimum χ

2 _{distance of the track} to the primary vertex VFASPF(VMINVDDV(PRIMARY))>1.0

minimum distance between the decay vertex and the primary

vertex

BPVVDCHI2>30.0 χ

2 _{distance from the best} primary vertex

M>(5279-500.0)∗MeV mass

M<(5279+700.0)∗MeV mass

Table 9: Description of the stripping cuts in the used stripping lines StrippingBu2MajoMuLLLine and StrippingBu2MajoMuDDLine.

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B PLOTS OF B AND N χ2 _VARIABLES ₂₇

Appendix B

Plots of B and N χ

2

variables

(a) Monte Carlo data, long track (b) Monte Carlo data, downstream track

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B PLOTS OF B AND N χ2 _VARIABLES ₂₈

(30)

B PLOTS OF B AND N χ2 _VARIABLES ₂₉

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C ROC AND SIGNIFICANCE PLOTS 30

Appendix C

ROC and significance plots

(a) ROC curve (b) Significance

Figure 18: Long track ROC and significance plots for the µ impact parameter χ2 cut.

Figure 19: Downstream track ROC and significance plots for the µ impact parameter χ2cut.

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Figure 21: Dpwnstream track ROC and significance plots for the e impact parameter χ2 _cut.

Figure 22: Long track ROC and significance plots for the π impact parameter χ2 _cut.

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Figure 24: Long track ROC and significance plots for the e PIDe cut.

Figure 25: Downstream track ROC and significance plots for the e PIDecut.

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Figure 27: Downstream track ROC and significance plots for the π PIDecut.

Figure 28: Long track ROC and significance plots for the µ PIDµcut.

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Figure 30: Long track ROC and significance plots for the N lifetime cut.

(a) ROC curve _{(b) Significance}

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References

[1] M. Thomson, Modern Particle Physics (Cambridge University Press, Cambridge, 2013) [2] R. Aaij et al. (The LHCb collaboration), Searches for Majorana neutrinos in B− _{decays, Phys.}

Rev. Lett. D85, 112004 (2012)

[3] R. Aaij et al. (The LHCb collaboration), Search for Majorana neutrinos in B− → π+_µ−_µ− decays, Phys. Rev. Lett. 112, 131802 (2014)

[4] G. Cvetiˇc, A. Das, J. Zamora-Sa´a, Probing heavy neutrino oscillations in rare W boson decays, J. Phys. G 46, 075002 (2019)

[5] P. A. Zyla et al. (Particle Data Group), Review of particle physics, to be published in Prog. Theor. Exp. Phys. 2020, 083C01 (2020)

[6] F. Yukawa et al. (The Super-Kamiokande collaboration), Evidence for oscillation of atmo-spheric neutrinos, Phys. Rev. Lett. 81, 1562-1567 (1998)

[7] Q. R. Ahmad et al. (The SNO collaboration), Measurement of the rate of νe+ d → p + p + e− interactions produced by8B solar neutrinos at the Sudbury Neutrino Observatory, Phys. Rev. Lett. 87, 071301 (2001)

[8] M. Goldhaber, L. Grodzins, A. W. Sunyar, Helicity of Neutrinos, Phys. Rev. 109, 1015-1017 (1958)

[9] P. B. Pal, Dirac, Majorana and Weyl fermions, Am. J. Phys. 79, 485-498 (2011)

[10] A. Augusto Alves Jr. et al. (The LHCb collaboration), The LHCb Detector at the LHC, JINST 3, S08005 (2008)

[11] LHCb starterkit lessons, The LHCb dataflow (2015). Online: https://lhcb.github.io/ starterkit-lessons/first-analysis-steps/dataflow.html, accessed April 3th 2020 [12] V. Lukashenko, Master Thesis (Nikhef, 2019)

[13] W. A. Rolke, A. M. Lopez, J. Conrad, Limits and Confidence Intervals in the Presence of Nuisance Parameters, Nucl. Instrum. Meth. A551, 493-503 (2005)

[14] TRolke Class, ROOT version 6.20/02 (2020)

[15] R. Aaij et al. (The LHCb collaboration), Measurement of the B±production cross-section in pp collisions at√s = 7 and 13 TeV, JHEP 12, 026 (2017)

[16] A. J. Weir, S. R. Klein, G. Abrams et al., Upper Limits on D± and B± Decays to Two Leptons Plus π± or K±, Phys. Rev. D 41, 1384-1388 (1990)

Search for heavy neutrinos in B decays at LHCb