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FACULTY OF SCIENCES
Data driven WZ background
estimation for SUSY searches with
multiple leptons at the energy frontier,
LHC
Author:
Willem Verbeke
Promotor and supervisor:
Prof. Didar Dobur
Co-Supervisor:
Dr. Lesya Shchutska
———————————————————————————–
Academic year 2015-2016
Thesis submitted in partial fulfilment of the requirements for the degree of
Master of Science in Physics and Astronomy
The image on the cover page shows a reconstruction of a proton proton collision
leading to two muons, an electron and missing transverse energy, measured by the
CMS detector. The red lines indicate muon tracks, the orange lines the charged
particle tracks in the tracker, one of which is the electron, for which the ECAL
energy deposit is shown in green. The arrow indicates the missing transverse energy.
Such events, containing three charged leptons and MET form the primary focus of
this thesis.
Acknowledgements
A long journey has come to an end, and at the end of the road there are many people
to whom I owe an expression of gratitude.
First of all I would like to thank you Didar. Your tireless effort to guide me towards
achieving results, helping and correcting me whenever necessary, is what made all
of this possible. Not only that, but you provided me with magnificent opportunities
few thesis students can dream of, such as presenting my results at Fermilab.
Lesya, without your brilliant insights into the analysis, and your endeavor to help
me out whenever I got stuck, I would never have achieved the results I did, and you
have my genuine gratitude for that. And of course I have to thank you for driving
me everywhere during my trip to Fermilab.
Prof. Ryckbosch, if you had not given me the opportunity to do an internship at
CERN during the last summer, I would most likely never have chosen to do a thesis
in experimental particle physics. In retrospect, this turned out to be one of the best
decisions I have made, so I sincerely thank you for giving me this opportunity.
Illia and Tom, you guys helped me debug my analysis code, and answered all of
my undoubtedly annoying questions time and again, and without your help things
would never have progressed the way they did.
Lastly I want to thank my parents. If you guys would ever read this, thank you for
giving me the opportunity to do what I love, and for unconditionally supporting me
throughout all these years!
Contents
1 Introduction
1.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . .
2 The
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Standard Model of particle physics
Standard Model Particles . . . . . . . . . . . . . . . . . . . .
Quantum field theory and the Lagrangian framework . . . . .
Free Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . .
The gauge principle . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Abelian U (1) gauge theory, quantum electrodynamics
2.4.2 Interactions and Feynman diagrams . . . . . . . . . .
2.4.3 Non Abelian gauge theories . . . . . . . . . . . . . . .
2.4.4 Renormalization
and running coupling . . . . . . . . .
N
SU (2)L U (1)Y : electroweak interactions . . . . . . . . . . .
Yukawa couplings . . . . . . . . . . . . . . . . . . . . . . . . .
SU (3): quantum chromodynamics . . . . . . . . . . . . . . .
Standard Model Summary . . . . . . . . . . . . . . . . . . . .
3 Standard Model: the final word?
3.1 A tale of triumph . . . . . . . . . . . .
3.2 Loose ends . . . . . . . . . . . . . . .
3.2.1 Gravity . . . . . . . . . . . . .
3.2.2 Dark matter . . . . . . . . . .
3.2.3 Dark energy . . . . . . . . . . .
3.2.4 Matter-antimatter asymmetry .
3.2.5 Free parameters . . . . . . . .
3.2.6 Hierarchy problem . . . . . . .
3.2.7 Neutrino masses . . . . . . . .
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4 CMS at the LHC
4.1 Hadron colliders: discovery machines . . . .
4.2 LHC: Energy and luminosity frontier . . . .
4.3 The CMS detector . . . . . . . . . . . . . .
4.3.1 CMS coordinate system . . . . . . .
4.3.2 Tracker . . . . . . . . . . . . . . . .
4.3.3 Electromagnetic calorimeter (ECAL)
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6 Software techniques
6.1 Monte Carlo event generation . . . . . . . . . . . . . . . . . . . . . .
6.2 CMS detector simulation . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Search for electroweakinos using a three lepton + MET signature
7.1 Signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Backgrounds and discriminating variables . . . . . . . . . . . . . . .
7.2.1 Backgrounds with three prompt leptons . . . . . . . . . . . .
7.2.2 non promt or fake leptons . . . . . . . . . . . . . . . . . . . .
7.3 Run I electroweakino searches in the three lepton final state . . . . .
7.3.1 Search strategy . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Estimation of the subdominant backgrounds . . . . . . . . .
7.3.3 WZ background estimation by applying data-driven corrections to simulations . . . . . . . . . . . . . . . . . . . . . . .
7.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8 A novel data-driven estimation technique of the WZ background
8.1 Wγ as a proxy to WZ . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Simulation samples used for the proof of principle . . . . . . . . . . .
8.3 Comparison of the kinematic properties of WZ and Wγ in pure simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Object selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Isolation as a background reduction tool . . . . . . . . . . . .
8.4.2 Muon selection . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.4
4.5
4.3.4 Hadronic Calorimeter (HCAL) . . . .
4.3.5 Muon system . . . . . . . . . . . . . .
Trigger system . . . . . . . . . . . . . . . . .
Event reconstruction: particle-flow algorithm
4.5.1 Iterative tracking . . . . . . . . . . . .
4.5.2 Calorimeter clustering . . . . . . . . .
4.5.3 Linking . . . . . . . . . . . . . . . . .
4.5.4 Reconstruction and identification . . .
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5 A supersymmetric solution?
5.1 The principle of Supersymmetry . . . . . . . . . .
5.1.1 chiral/matter supermultiplets . . . . . . . .
5.1.2 gauge/vector supermultiplets . . . . . . . .
5.1.3 other supermultiplets . . . . . . . . . . . .
5.2 Minimal Supersymmetric Standard Model Zoo . .
5.3 Solving the Hierarchy problem and breaking SUSY
5.4 R-parity conservation and dark matter . . . . . . .
5.5 MSSM phenomenology . . . . . . . . . . . . . . . .
5.5.1 mass eigenstates . . . . . . . . . . . . . . .
5.5.2 MSSM sparticle decays . . . . . . . . . . .
5.5.3 Searching for SUSY at hadron colliders . .
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8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.4.3 Electron selection . . . . . . . . . . . . . . . . . . . . .
8.4.4 Photon selection . . . . . . . . . . . . . . . . . . . . .
8.4.5 MET reconstruction . . . . . . . . . . . . . . . . . . .
8.4.6 Jet and HT selection . . . . . . . . . . . . . . . . . . .
8.4.7 Beauty jet tagging . . . . . . . . . . . . . . . . . . . .
Kinematic comparison after detector simulation . . . . . . . .
8.5.1 WZ event selection . . . . . . . . . . . . . . . . . . . .
8.5.2 Wγ event selection . . . . . . . . . . . . . . . . . . . .
8.5.3 Matching reconstructed objects to simulated particles
8.5.4 Kinematic comparison . . . . . . . . . . . . . . . . . .
Reducing FSR and proof of principle . . . . . . . . . . . . . .
8.6.1 Proof of principle for W→ µν . . . . . . . . . . . . . .
8.6.2 Proof of principle for W→ eν . . . . . . . . . . . . . .
Reweighing kinematic variables . . . . . . . . . . . . . . . . .
Statistics and viability . . . . . . . . . . . . . . . . . . . . . .
Resolution comparison . . . . . . . . . . . . . . . . . . . . . .
Backgrounds to Wγ . . . . . . . . . . . . . . . . . . . . . . .
Inclusion of Wjets in the WZ prediction technique . . . . . .
Lepton + photon + MET control sample in data . . . . . . .
8.12.1 Trigger efficiency scale factors . . . . . . . . . . . . . .
8.12.2 Drell-Yan background in the electron channel . . . . .
8.12.3 tt background . . . . . . . . . . . . . . . . . . . . . . .
8.12.4 Data versus MC . . . . . . . . . . . . . . . . . . . . .
8.12.5 Extracting the data driven WZ prediction . . . . . . .
Systematic uncertainties . . . . . . . . . . . . . . . . . . . . .
8.13.1 Lepton PT thresholds in the 3 lepton signal sample . .
8.13.2 Meγ requirement . . . . . . . . . . . . . . . . . . . . .
8.13.3 Simulation uncertainties . . . . . . . . . . . . . . . . .
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9 Conclusions and outlook
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10 Nederlandstalige samenvatting
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Appendices
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A Simulation samples
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B Data samples and luminosity sections
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iii
List of Figures
2.1
2.2
2.3
2.4
2.5
3.1
Figure showing the different fundamental particles present in the
Standard Model. For each particle the mass, electric charge and spin
are listed. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure showing some tree level (i.e. first order perturbation) Feynman
diagrams of QED, all the result of the interaction term obtained by
making the free Dirac Lagrangian invariant under U (1) gauge transformations. (a) shows compton scattering of a photon and an electron,
in (b) the annihilation of a fermion anti-fermion pair into a virtual
photon and the subsequent creation of another fermion pair is shown
and (c) shows the scattering of two fermions by exchanging a virtual
photon. Most Feynman diagrams in this text are made using the tool
[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quartic- and triple gauge couplings between the force carrying bosons,
due to the presence of interaction terms containing three or four gauge
boson fields in the Lagrangian. . . . . . . . . . . . . . . . . . . . . .
Summation of all Feynman diagrams contributing to the measured
electromagnetic coupling between charged fermions. e0 indicates the
bare electric charge, and e(q 2 ) is the electric charge that is finally
measured at a certain momentum scale. The figure has been taken
from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the strong coupling constant’s predicted running to
several measurements. The order of perturbation theory used in the
calculations to extract the coupling constant from the measurements
is indicated. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Figures showing the di-photon and four lepton invariant mass distributions which provided the best signal for CMS’s Higgs discovery.
In both plots the red line represents the expected signal with the
presence of a SM Higgs boson [11]. . . . . . . . . . . . . . . . . . . .
24
iv
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3.2
3.3
3.4
3.5
(a) Comparison of fit results with direct measurements for several
SM parameters in units of the experimental uncertainty. (b) Top
quark mass vs W boson mass with 68%, and 95% confidence limit
contours shown for direct measurements, and fits excluding these two
parameters. The blue contours include the Higgs mass in the fit while
the grey contours do not use it. (c) Plot showing the W boson mass
vs sin2 θef f , with θef f the effective electroweak mixing angle after
higher order perturbative corrections. 68%, and 95% confidence limit
contours are again shown with and without the Higgs mass included
in the fits. All calculations correspond to full fermionic two loop
calculations. [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optic (a) and x-ray (b) images of the bullet cluster, with mass contours shown in green as determined from the weak gravitational lensing effect. [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
One loop self energy corrections to the Higgs mass [25]. . . . . . . .
(a) Feynman diagram showing how an effective mass coupling through
a right handed Majorana neutrino can influence the light neutrino
masses, figure taken from [30]. (b) Neutrinoless double beta decay
induced by introducing a majorana mass term, figure taken from [31].
Visualization of a non-diffractive inelastic proton-proton collision in
which a top quark pair and a Higgs boson are produced. The hard
interaction between the partons is represented by the red dot, the
Higgs and the top quarks by the small red dots. The radiated partons
are shown developing into jets. [32] . . . . . . . . . . . . . . . . . .
4.2 Illustration of the different accelerators and experiments at CERN.
[33] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Cross section of an LHC dipole magnet. [34] . . . . . . . . . . . . .
4.4 Distribution of the number of proton-proton collisions per bunch
crossing at the CMS interaction point during the 8 TeV run (a), and
event display of an 8 TeV bunch crossing with 29 distinct vertices
coming from 29 separate proton-proton collisions in a single bunch
crossing (b) [36], [37]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Sketch of the CMS detector [38]. . . . . . . . . . . . . . . . . . . . .
4.6 CMS coordinate system [38]. . . . . . . . . . . . . . . . . . . . . . .
4.7 Longitudinal cross section of a quarter of CMS’s tracking system.
Solid purple lines represent single-sided silicon strip modules, while
double-sided modules are shown as blue lines. Solid dark blue lines
represent the pixel modules. [41] . . . . . . . . . . . . . . . . . . . .
4.8 Geometric view of one quarter of CMS’s ECAL system. [42] . . . .
4.9 Longitudinal view of a quarter of CMS, on which the positions of the
different HCAL subdetectors are shown, figure from [38]. . . . . . .
4.10 Vertical cross section of CMS, showing the trajectories for several
particle types through the multiple detector layers. [40] . . . . . . .
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4.1
5.1
Triangle diagram, leading to a chiral gauge anomaly [50]. . . . . . .
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5.2
5.3
5.4
6.1
7.1
7.2
7.3
7.4
7.5
7.6
7.7
SM particles, and their supersymmetric partners in the MSSM. Figure
taken from [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Diagram showing a possible proton decay through a virtual strange
squark in the absence of R-parity conservation, figure taken from [28]. 55
Plot showing the expected sparticle pair production cross section at
a proton-proton collider of 8 TeV, and 13-14 TeV. [52]. . . . . . . . 59
Schematic representation of the different steps in the simulation of a
proton-proton collision. [58] . . . . . . . . . . . . . . . . . . . . . .
Pseudo Feynman diagrams depicting the production diagrams of interest to SUSY searches using the three lepton + MET final state.
In diagram (a), the decay of the initial electroweakino pair is mediated by sleptons which can be real or virtual depending on the mass
hierarchy of the SUSY model under consideration. Diagram (b) is
similar to diagram (a), but the chargino decay is now mediated by
a sneutrino which might have a different mass from the sleptons, so
the final state kinematics can be distinctive from diagram (a). In diagram (c) and (d) the electroweakinos decay to the LSP by emitting
electroweak gauge bosons (c) and an electroweak gauge boson and a
Higgs boson (d). Which of the four diagrams depicted above will give
the dominant contribution is model- and mass hierarchy dependent.
Feynman diagram of top quark pair production by gluon fusion, followed by semileptonic decays of the top quaeks, leading to two beauty
quarks, two leptons and two neutrinos. . . . . . . . . . . . . . . . .
Pure simulation of the MT (a), and Mll shapes in WZ events. The
MT is made up of the W decay’s lepton and the MET while the Mll
is calculated using the leptons from the Z decay. . . . . . . . . . . .
Comparison of the recoil components and their resolution in data and
MC. Respectively u1 and its resolution are shown in (a) and (b) while
(c) and (d) show the same distributions for u2 . Every point on these
plots corresponds to values extracted from double Gaussian fits in a
certain Z boson PT bin. [63] . . . . . . . . . . . . . . . . . . . . . . .
=Comparison of the simulated recoil components and their resolution
in Z events as a function of the Z PT and in WZ events as a function
of the WZ system PT . Respectively u1 and its resolution are shown
in (a) and (b) while (c) and (d) show the same distributions for u2 .
Every point on these plots corresponds to values extracted from double Gaussian fits in a certain Z boson, or WZ system PT bin. [63]
[63] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dielectron invariant mass shape in data and MC, before and after
applying lepton energy scale corrections. [63] . . . . . . . . . . . . .
MT versus Mll scatter plot, showing all events with three light lepton
and an OSSF pair. The purple lines mark the different Mll and MT
search regions. [61] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.8
8.1
8.2
8.3
8.4
8.5
8.6
Comparison of the observed yields to those predicted for the backgrounds as a function of MET in the different search region for events
with three light leptons and an OSSF pair. No significant data excess
is observed in any of the search regions. An important thing to take
away from this figure is how large the WZ background is compared
to the others, especially in the onZ region. [61] . . . . . . . . . . . .
Tree level production diagrams of Wγ(left column) and WZ (right
column). Diagrams (a) and (b) depict what is called initial state
radiation (ISR) production of both processes, in which the γ or Z is
radiated by the W boson. Diagrams (c) and (d) go through a virtual
quark propagator, and both the W and the γ or Z are radiated by
the quarks. These diagrams are often called t-channel production.
Note that the upper diagrams are essentially the same for Wγ and
WZ, are forecasted to lead to similar kinematics, though there will
be differences induced by the mass of the Z boson compared to the
massless photon. Unlike the other diagrams, the final diagram (e),
corresponding to final state radiation (FSR) of a photon, is not present
in WZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Production cross section measurements of CMS for several SM processes [64]. The measured cross section of Wγ can be seen to be more
than two orders of magnitude larger than that of WZ. . . . . . . . .
Comparison of several kinematic distributions of Wγ and WZ, both
normalized to unity. In (a) the MT of the lepton coming from the W
decay and the MET is shown, while (b) and (c) show the PT of this
lepton and the MET both of which go directly into the MT calculation. Figure (d) shows the distribution of the angular separation ∆R
between the lepton coming from the decaying W and the Z boson or
the photon. The final bin of every histogram shown is an overflow
bin containing all events falling out of the range of the plot. . . . . .
Comparison of several kinematic distributions in Wγ and WZ events,
normalized to unity. (a) and (b) show the azimuthal angular separation ∆φ between the lepton from the W decay and respectively the
MET vector and the Z boson or photon. (c) depicts the pseudorapidity separation ∆η, between the lepton and the Z boson or photon. In
(d), the PT of the Z boson in WZ and the photon in Wγ is compared.
Validation of the generator matching in simulated WZ events: (a) 2D
plot comparing the PT of the reconstructed leptons to their generator
matches. (b) Angular separation ∆R between the leptons and their
generator matches. Note that there is an overflow bin present in figure
(b), but there are nearly no events populating it. . . . . . . . . . . .
Validation of the generator matching in simulated Wγ events: (a),
(c) 2D plots comparing the PT ’s of leptons, respectively photons to
their generator matches. (b), (d) angular separation ∆R between
respectively the leptons and photons and their generator matches. .
vii
78
81
82
87
88
97
98
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
8.17
8.18
Comparison of the MT shape of Wγ and WZ, reconstructed (a) and
at the MC truth level (b). . . . . . . . . . . . . . . . . . . . . . . . . 100
PT distribution of the lepton from the decaying W compared in Wγ
and WZ, after reconstruction (a), at the MC truth level (b). . . . . . 101
Comparison of respectively the reconstructed (a) and MC truth MET
(b) between Wγ and WZ. . . . . . . . . . . . . . . . . . . . . . . . . 101
The PT of the photon in Wγ events compared to that of the Z, as
reconstructed from its decay products, in WZ events, reconstructed
(a), and at the MC truth level (b). . . . . . . . . . . . . . . . . . . . 102
Comparison of the difference in the azimuthal angle Φ between the
lepton originating from the W decay, and the photon or Z boson, after
reconstruction (a), in MC truth (b). . . . . . . . . . . . . . . . . . . 102
Comparison between WZ and Wγ of the angular separation ∆R between the photon, or Z and the lepton from the decaying W, after
reconstruction (a) and in MC truth (b). . . . . . . . . . . . . . . . . 103
Pseudorapidity difference ∆η between the lepton from the W decay
and the Z or photon, after reconstruction (a), and at the MC truth
level (b), compared in Wγ and WZ events. . . . . . . . . . . . . . . 103
Azimuthal separation ∆Φ between the lepton from the W decay and
the MET, after reconstrution (a), and at the MC truth level (b),
compared in Wγ and WZ events. . . . . . . . . . . . . . . . . . . . . 104
Reconstructed HT and number of jets distributions in Wγ and WZ
events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Plots comparing the generator MET, defined as the sum of all neutrino PT ’s, to the PT of the neutrino coming from the W decay, and
the sum of the PT ’s of all other neutrinos. The left plots were made
for Wγ events while the right plots contain WZ events. The upper
and lower plots show the same distributions, but with different y-ranges.105
Comparison of the MT shapes in Wγ and WZ for muonically decaying
W bosons, plotted on a logarithmic scale, without any additional
cuts (a), after applying the following kinematic requirements on Wγ
events: PT (γ) > 50 GeV (b), ∆Φ(`, γ) > 1 (c) and ∆R(`, γ) > 1 (d).
In every plot, the ratio of the MT curves is shown in the bottom, to
which a constant has been fit by means of the least squares method.
The value of the constant fit, and the goodness of the fit in terms of
χ2 per number of degrees of freedom are listed for every plot. . . . . 107
Comparison of the WZ and Wγ MT shapes in the muon channel
after several kinematic cuts to reduce the FSR contribution in Wγ
have been applied. A constant has been fit to the ratio of the shapes
with the least squares method, and the goodness of fit in terms of χ2
per degree of freedom is indicated together with the fitted constant. 109
viii
8.19 Comparison of the WZ and Wγ MT shapes in the electron channel
after several kinematic cuts to reduce the FSR contribution in Wγ
have been applied. A constant has been fit to the ratio of the shapes
with the least squares method, and the goodness of fit in terms of χ2
per degree of freedom is indicated together with the fitted constant.
8.20 Number of expected events as a function of the MT and the PT of
the Z boson in WZ events (a) and the photon in Wγ events (b).
One can see by eye that there is little correlation, and in fact the
correlation factors between the MT and the PT of the Z, respectively
the photon are calculated to be -0.0265, and -0.0641, indicating that
the correlation is small or non-existent. These plots contain both
muon and electron channel events. . . . . . . . . . . . . . . . . . . .
8.21 Distribution shapes of the MET and PT of the lepton from the W
decay, compared in Wγ and WZ in the muon channel after applying
several kinematic cuts to remove the FSR contribution in Wγ. . . .
8.22 Distribution shapes of the MET and PT of the lepton from the W
decay, compared in Wγ and WZ in the electron channel after applying
several kinematic cuts to remove the FSR contribution in Wγ. . . .
8.23 The lepton PT distribution compared between Wγ and WZ in the
muon channel after applying the reweighing scale factors of (b). The
lepton PT curves now match perfectly by definition since this distribution has been reweighed. . . . . . . . . . . . . . . . . . . . . . . .
8.24 MT shape comparison of Wγ and WZ after reweighing the lepton PT ,
in the electron channel. For every plot a least squares fit is performed,
and the resulting χ2 value is shown. . . . . . . . . . . . . . . . . . .
8.25 MT shape comparison of Wγ and WZ after reweighing the lepton PT ,
in the electron channel. For every plot a least squares fit is performed,
and the resulting χ2 value is shown. . . . . . . . . . . . . . . . . . .
8.26 MET resolution shapes compared in Wγ and WZ in the muon (a)
and electron channel (b). . . . . . . . . . . . . . . . . . . . . . . . . .
8.27 Comparison of the PT (γ) distribution in data and MC in a µ + γ +
MET final state, when using all MC samples out of the box (a) and
after cleaning the overlap between several samples.(b) . . . . . . . .
8.28 MT shape comparison of WZ to Wγ + Wjets, with a least squares fit
to the ratio of the shapes, in the muon channel (a), and the electron
channel (b). The Wγ and Wjets events were given statistical weights
proportional to their expected yields in data, and no reweighing is
applied in these plots. . . . . . . . . . . . . . . . . . . . . . . . . . .
8.29 MT shape comparison of WZ to Wγ + Wjets, with a least squares fit
to the ratio of the shapes, in the muon channel (a), and the electron
channel (b). The Wγ and Wjets events were given statistical weights
proportional to their expected yields in data, and the Wγ + Wjets
events were reweighed using the lepton PT distribution. . . . . . . .
ix
111
112
113
114
114
115
116
118
120
121
121
8.30 Single lepton trigger efficiencies as a function of the lepton’s PT , for
the trigger IsoMu20 (a), and for the triggers Ele23 CaloIdL TrackIdL IsoVL
(MC) and Ele23 WPLoose Gsf (data) (b). . . . . . . . . . . . . . . . 123
8.31 Invariant mass distribution of the electron-photon system, compared
in data and MC. A clear Z boson mass peak can be seen in data and
MC, while both the pixel hit veto, and the conversion safe electron
veto were applied in the photon object selection. From this figure
it becomes extremely clear that electrons and photons are hard to
distinguish. For dramatic effect, the MET and PT (γ) thresholds have
both been lowered to 30 GeV, coming from 50 GeV in our actual
event selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.32 Invariant mass distributions of the lepton-photon system, in the muon
channel with MET and PT (γ) cuts of 30 GeV, and in the electron
channel with both cuts at 50 GeV. . . . . . . . . . . . . . . . . . . . 125
8.33 Number of b-tagged jets in data and MC, in the muon channel (a),
and the electron channel (b). . . . . . . . . . . . . . . . . . . . . . . 126
8.34 Photon PT distribution compared in data and MC, in the muon channel (a), and the electron channel (b). . . . . . . . . . . . . . . . . . . 127
8.35 Lepton PT distribution compared in data and MC, in the muon channel (a), and the electron channel (b). . . . . . . . . . . . . . . . . . . 128
8.36 MET distribution compared in data and MC, in the muon channel
(a), and the electron channel (b). . . . . . . . . . . . . . . . . . . . . 128
8.37 Comparison of the distribution of the Azimuthal angular separation
∆Φ between the lepton and the MET in data and MC, in the muon
channel (a), and the electron channel (b). . . . . . . . . . . . . . . . 129
8.38 Comparison of the distribution of the Azimuthal angular separation
∆Φ between the lepton and the photon in data and MC, in the muon
channel (a), and the electron channel (b). . . . . . . . . . . . . . . . 129
8.39 Comparison of the angular separation ∆R distribution between the
lepton and the photon in data and MC, in the muon channel (a), and
the electron channel (b). . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.40 Comparison of the distribution of the pseudorapidity separation ∆η
between the lepton and the photon in data and MC, in the muon
channel (a), and the electron channel (b). . . . . . . . . . . . . . . . 130
8.41 Distribution of the number of jets, compared in data and MC, in the
electron channel (a), and the muon channel (b). . . . . . . . . . . . . 131
8.42 HT distribution compared in data and MC, in the muon channel (a),
and the electron channel (b). . . . . . . . . . . . . . . . . . . . . . . 131
8.43 MT distribution compared in data and MC, in the muon channel (a),
and the electron channel (b). . . . . . . . . . . . . . . . . . . . . . . 132
8.44 Comparison of the MT distribution in data and MC in the muon
channel on a logarithmic scale. . . . . . . . . . . . . . . . . . . . . . 132
8.45 Comparison of the MT distribution in data and MC in the electron
channel on a logarithmic scale. . . . . . . . . . . . . . . . . . . . . . 133
x
8.46 Comparison of the MT distribution in data and MC in the muon
channel, after requiring the muon to pass a very tight multiisolation
working point, and the tight muon identification criteria as listed
in [70]. The excess in the first bin has been significantly reduced
compared to the plot using looser criteria, indicating that we are
missing a contribution from fake objects in our simulation prediction. 133
8.47 Comparison of the muon channel WZ MT distribution. as determined
from the muon + photon control sample, to the MC prediction. . . 135
8.48 Comparison of the electrons channel WZ MT distribution. as determined from the electron + photon control sample, to the MC prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.49 Comparison of the MT distribution, with and without explicitly requiring the lepton from the W’s decay to have a PT greater than 20
GeV in the muon channel (a), and 23 GeV in the electron channel (b).137
8.50 Influence on the MT shape of vetoing events in which Meγ resides
within the Z-mass window. . . . . . . . . . . . . . . . . . . . . . . . 138
8.51 MT shape in WZ events, as simulated by the Powheg and MC@NLO
matrix element generators, in the muon channel (a), and the electron
channel (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
xi
List of Tables
2.1
Caption for LOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
8.1
Table showing the order in perturbation theory up to which the samples corresponding to the processes of interest were simulated, their
theoretical cross sections and the order up to which this was calculated. The cross section uncertainty was only available for the Wjets
sample. [66] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
xii
Chapter 1
Introduction
Humanity’s deepest grasp of nature is summarized in the Standard Model (SM) of
particle physics, a magnificent theory explaining nigh everything we have ever been
able to observe. In the decades since its development, theoretical and experimental
challenges have risen to the SM, and it is becoming clear that nature has not spoken
its final word. A myriad of experiments is pursuing any signs of new physics in an
attempt to answer some of the outstanding questions in the field, a front-runner
among which is the Compact Muon Solenoid (CMS) experiment at the European
Organization For Nuclear Research’s (CERN) Large Hadron Collider (LHC). At the
time of writing, the LHC is colliding protons at unprecedented energies and luminosities in pursuance of producing unseen particles. This thesis marks a very small
piece in the collaborative effort to analyze the LHC collision data, measured and
collected by CMS.
One of the most prominent of the theories going beyond the SM, is a theory called
Supersymmetry (SUSY), the search for which is one of the primary objectives of the
current LHC operation. The core business of searches for new physics is minimizing
the uncertainties on the estimation of the yields of the SM processes giving the same
signal as the new physics processes of interest, the backgrounds. The smallness of
the background estimation will determine the reach the analysis. The research done
in this thesis consists of exploring a novel technique for estimating one of the primary SM backgrounds for the search for SUSY in its electroweak production mode,
aimed at minimally relying on simulated predictions.
In order to fully appreciate the motivations of searches for new physics, this thesis
begins by briefly illustrating the foundations of the SM as a gauge invariant quantum field theory, some of its recent triumphs, indicating the continuing prowess of
the theory, and some of its greatest shortcomings and challenges. These chapters
are general and non-specific to this thesis, but aim at thoroughly sketching the context in which the research of this thesis was performed. Hereafter a description of
the LHC and CMS machines, what happens in high energy proton-proton collisions,
and how the CMS detector reconstructs all the resulting information, follows. This
part is essential for understanding the later chapters, extensively analyzing data
1
from CMS. Next up is the outlining of SUSY, its phenomenological consequences,
and how to probe for its existence at the LHC. Before we can finally start discussing
new experimental results, a short description of the simulation techniques commonly
employed in high energy collider physics is given. The final chapters are the ones
that are specific to this thesis. Light is shed on a final state topology that seems
very promising for the discovery of electroweak SUSY production, whereafter the
techniques employed, and the results achieved in searches in this final state in data
from the LHC’s previous period of operation are discussed. All of the original results developed during the work on this thesis are found in chapter 8, narrating and
motivating the development of a new data-driven background estimation technique.
1.1
Notations and conventions
Here we list a few conventions used throughout this thesis:
This thesis will extensively use natural units, which mean that two fundamental
dimensional constants, the speed of light c and the reduced Planck constant ~ are
chosen to equal one:
~ = c = 1.
(1.1)
Using natural units greatly simplifies some of the equations appearing in the next
chapters, and allows us to express masses, momenta and energies in the same units,
for which we will use electron volts.
Another convention that is used throughout these thesis is that an implicit summation over relativistic four-vector indices is assumed whenever the same index
appears in a co- and contravariant form.
In the last chapters of this thesis many histograms will be plotted, and every histogram that is shown throughout this thesis will always include overflow binning.
This means that every event which would have fallen outside of the range of the
histogram is placed inside the last bin.
2
Chapter 2
The Standard Model of particle
physics
2.1
Standard Model Particles
When probing ever deeper into the structure of the matter surrounding us, it turns
out that everything consists of just a few elementary building blocks we call particles. Atoms are made up of protons, neutrons and electrons. These atoms are held
together by the strong nuclear and electromagnetic forces, low energy manifestations of respectively quantum chromodynamics (QCD) and quantum electrodynamics (QED). Protons and neutrons on their turn consist of the same two pieces called
the up-quark and the down-quark. The nuclear interactions powering the sun are
made possible by the weak interaction, in these processes yet another particle, the
electron neutrino νe is produced. All the matter we observe is bound in planets,
stars, solar systems and galaxies by the fourth, and by far the weakest of the fundamental forces, gravity. [1] [2]
The up- and down-quarks, the electron and its neutrino make up what is called
the first generation of particles, composing virtually all the matter in our cold, lowenergy universe. When studying particle physics at high energies, like in cosmic rays
or particle colliders, matter particles of a second and third generation are observed.
These particles are more massive copies of the first generation particles, similar in all
other aspects to their lighter siblings. After their production the heavier second and
third generation particles invariably decay to particles of the first generation. The
electron, its heavier partners µ and τ together with their neutrinos are collectively
referred to as leptons. Leptons feel the electromagnetic, weak and gravitational
forces. Another class of particles are the quarks, which are found to be affected
by all four of the fundamental forces and are bound together by the strong force,
forming what we call hadrons. The fundamental forces of the weak, electromagnetic
and strong interactions are mediated by gauge bosons, resulting from the gauge invariance of the SM Lagrangian under certain groups of transformations as will be
described in the next sections. So far Gravity has not been incorporated in the
SM, though many attempts have been made. The particles in nature are known to
3
Figure 2.1: Figure showing the different fundamental particles present in the Standard Model. For each particle the mass, electric charge and spin are listed. [4]
have an internal degree of freedom called spin. The gauge bosons are integer spin
particles, as opposed to the half integer spin matter particles. Integer spin particles
obey Bose-Einstein statistics, and are therefore called bosons, while half-integer spin
particles obey Fermi-Dirac statistics, making them known as fermions. The crucial
difference between Bose-Einstein and Fermi-Dirac statistics is that a fermionic state
must be antisymmetric under the exchange of two particles, whereas a bosonic state
is symmetric. The antisymmetry of fermion states has the direct consequence that
no two fermions can ever be in the same quantum state, which is not forbidden
for bosons. The link between a particle’s spin being integer or half-integer and the
statistics it obeys follows from what is known as the spin-statistics theorem. Aside
from the fundamental fermions and the gauge bosons, a scalar boson, known as the
Higgs, is incorporated into the SM to facilitate gauge invariant mass terms for the
fermions and the weak interaction gauge bosons. In nature matter particles and
some of the gauge bosons are observed to be massive, but writing these mass terms
without a coupling to the Higgs field leads to a disastrous non-renormalizability of
the SM, as shown in [3], directly resulting into an inconsistent theory. The particle
content of the SM is shown in figure 2.1.
4
2.2
Quantum field theory and the Lagrangian framework
The Standard Model of particle physics is formulated as a quantum field theory.
In such a theory the fundamental objects under consideration are operator-valued
fields pervading space and time. Particles are excitations of these quantum fields,
different particles being excitations of distinctive fields. Elementary particle physics
usually goes hand in hand with high energies, so the SM had to be formulated as
a relativistic theory. From this point of view it can be understood relatively easily
that a one-particle approach is not sufficient for the description of the theory due to
the famous E = mc2 , enabling the creation and annihilation of particles. Even at
energies too small for pair production, multi-particle intermediate states appear in
higher order perturbation theory, only existing for short times due to the uncertainty
principle. Another argument for field theory is the presence of causality violating
particle amplitudes in relativistic quantum mechanics, which are canceled between
particles and antiparticles in a quantum field theory as shown in [5].
The field equations governing quantum field theories follow from the action by means
of Hamilton’s principle. This action is expressed as a space-time integral of a Lagrangian density L, which will simply be called Lagrangian from now on. L is a
function of one or more fields ψ(x) and their derivatives ∂µ ψ(x). The field values
always depend on the point in space-time, denoted x, but we will consistently omit
this argument to lighten the notation. The action is given by:
Z
S = L(ψ, ∂µ ψ)d4 x.
(2.1)
The evolution of a system through time has to be along a path of extremal action,
satisfying:
Z ∂L
∂L
∂L
δS =
δψ + ∂µ
d4 x = 0
(2.2)
δψ − ∂µ
δψ
∂ψ
∂µ ψ
∂(∂µ ψ)
The last term in this integral can be rewritten as a surface integral by using Gauss’
divergence theorem, which vanishes since δψ is zero on the borders of the integration
volume. The above equality must hold for arbitrary δψ, implying:
∂L
∂L
− ∂µ
= 0,
∂ψ
∂µ ψ
(2.3)
the famous Euler-Lagrange equation of motion for fields. As a consequence, the dynamics of field theories are completely determined by the form of their Lagrangian.
The Lagrangian point of view eminently facilitates the requirement of Lorentz invariant equations of motion, present in relativistic theories. After all, a Lorentz invariant
Lagrangian will lead to the boosted extremal action being another extremum [3] ,
[5].
5
2.3
Free Dirac field
It can be shown, as done by Dirac that an n × n representation of the Lorentz
Algebra (i.e. the Lorentz group for transformations infinitesimally close to unity)
for spin 21 particles is given by:
i µ ν
[γ , γ ]
(2.4)
4
with the γ matrices defined as a set of n n × n matrices satisfying the following
anticommutation relations:
S µν =
{γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2g µν × 1n×n .
(2.5)
When considering a four dimensional Minkowski space-time, these matrices must at
least be 4 × 4 since there are only three 2 × 2 matrices satisfying the above relation. 3 × 3 matrices are no option either since equation 2.5 implies the matrices are
2
traceless, which is impossible considering the requirement γ µ = 1 or −1 (depending
on the index) implicitly contained in equation 2.5. An object transforming as a
representation of the Lorentz group as generated by the generators in equation 2.4
is called a Dirac spinor, which we will denote as ψ.
The fields corresponding to matter particles, consisting of spin 12 fermions are represented by these spinors. To write a Lorentz invariant Lagrangian for fermions,
Lorentz invariant terms combining these spinors have to be devised. ψ † ψ is not
Lorentz invariant due to the non-unitarity of the Lorentz boost operators. It can be
shown however that ψψ with ψ = ψ † γ 0 is Lorentz invariant since γ 0 anticommutes
with the boost generators S 0i . Similarly ψγ µ ψ can be shown to transform as a
four-vector under Lorentz transformations. Other combinations with clear transformation properties under the Lorentz group, yielding tensors, axial vectors,... can be
made in a similar way. The Lorentz invariant Dirac Lagrangian governing the dynamics of free fermion fields can then be written as: (using Feynman slash notation:
/ = γ µ Oµ )
O
/ − mψψ.
L = iψ ∂ψ
(2.6)
The first term is a kinetic term for free fermions, while the second term determines
the fermion’s mass, together these terms govern the dynamics of free non-interacting
fermions. The famous Dirac equation follows from equations 2.3 and 2.6 and writes:
/ − mψ = 0.
i∂ψ
(2.7)
This equation has both negative and positive energy solutions. The negative energy
solutions propagate backward in time and can be interpreted as positive energy antiparticles traveling forward in time, having opposite charges, but equal masses from
their partner particles. Note that both a particle and its corresponding antiparticle
are linked to the same field. The spinor field operator ψ in equation 2.6 can be interpreted as annihilating particles or creating antiparticles while ψ creates particles
or annihilates antiparticles [3], [5].
6
2.4
The gauge principle
The Lagrangian proposed in equation 2.6 assumes non interacting free fermions,
which is not what we observe in nature. The interactions between particles are introduced in the SM by requiring the Lagrangian to be invariant under particular
local phase transformations called gauge transformations. A P
local phase transformation of a field is a transformation of the form: ψ 0 = exp (i a χa (x)τ a ) ψ where
the τ a ’s are the generators of the gauge group and the χa (x)’s are phase factors
depending on the point in space-time. As opposed to these local phase transformations, global phase transformations have a constant phase factor independent of
the point in space-time. According to Noether’s theorem every global symmetry
in a Lagrangian leads to conserved currents which let us define the charges of the
particles, as shown in [3]. To acquire gauge invariance of the free Dirac Lagrangian,
additional gauge boson fields have to be introduced, coupling to the conserved currents of the corresponding global symmetry, magically leading to all interactions
observed in nature. The easiest way to elucidate this concept is by first considering
the Abelian U (1) gauge transformations leading to QED, the fundamental theory
of electromagnetism and then generalizing this to non-Abelian groups needed for
describing the other fundamental forces.
2.4.1
Abelian U (1) gauge theory, quantum electrodynamics
U (1) is the group of multiplications with a complex phase factor exp (iχ), or equivalently the group of rotations in the complex plane. While the Lagrangian of equation
2.6 is clearly invariant under global U (1) transformations, the Lagrangian will not
be invariant under U (1) gauge transformations due to the presence of the derivative
of the now space-time dependent phase factor. Under a local U (1) transformation
the spinor fields transform as:
ψ 0 = exp (iqχ(x))
(2.8)
ψ 0 = ψ exp (−iqχ(x))
(2.9)
where we explicitly wrote the electric charge q which is some multiple of the fundamental charge e, which could otherwise have been absorbed into the phase factor.
Under these transformations the Lagrangian becomes:
/ − mψψ + iqψ ∂χ(x)
/
L = iψ ∂ψ
ψ
(2.10)
The final term clearly showcases the non gauge invariance of this lagrangian. To
rid ourselves of it we introduce a new field Aµ which corresponds to the electromagnetic four potential. It is well known from electromagnetism that the physical
electric- and magnetic fields corresponding to this potential are invariant under the
transformation:
A0µ = Aµ − ∂µ χ(x)
7
(2.11)
/ into the Lagrangian and require Aµ to transWe can now introduce a term iqψ Aψ
form as like in equation 2.11. This new term transforms as:
/
/ 0 ψ 0 = iqψ Aψ
/ − iqψ ∂χ(x)
iqψ 0 A
ψ
(2.12)
The last terms ofequations 2.10 and 2.12 cancel each other and we end up with a
gauge invariant Lagrangian by having introduced the field Aµ . The introduction of
the new term in the Lagrangian is usually done by replacing the derivative ∂µ in the
original Lagrangian by a covariant derivative Dµ , defined as:
Dµ = ∂µ + iqAµ
(2.13)
This covariant derivative can be interpreted as a derivative taking into account the
phase difference from one point to the next and thus comparing the values of fields
in a more meaningful way than done by ∂µ . The new covariant derivative has the
property:
(Dµ ψ)0 = exp(iqχ(x))Dµ ψ
(2.14)
Which nicely shows the gauge invariance of the new Lagrangian. We can interpret
the field Aµ as the photon field, able to create and annihilate photons. The new
term present in the now gauge invariant Lagrangian represents electromagnetic interactions of the fermions through their coupling with the photon field and we see
that the photons are the force carriers of these electromagnetic interactions. The
strengths of these interactions are proportional to the charge q of the particles involved.
A final term describing the dynamics of the free photon field is needed for completing our picture. This term should be gauge and Lorentz invariant, and only
contain the photon field and its derivatives. This can be facilitated by using the
commutator of the previously introduced covariant derivatives:
[Dµ , Dν ] = iq (∂µ Aν − ∂ν Aµ ) ≡ iqFµν
(2.15)
where Fµν is the electromagnetic field tensor. Using this field tensor we can construct
a term 41 F µν Fµν which is gauge and Lorentz invariant. No mass term for the boson
field has to be added since electromagnetism has massless gauge bosons, explaining
its infinite range. This leaves us with the Lagrangian:
1
/ − mψψ − F µν Fµν
LQED = iψ Dψ
(2.16)
4
The simple requirement of invariance under local phase transformations has given
us the Lagrangian of quantum electrodynamics (QED), providing a full description
of everything linked to electromagnetism. A simple symmetry leads to an understanding of almost everything, from macroscopic phenomena up to scales of about
10−15 m, from life on our planet to the phone in your pocket, a truly magnificent
result [1], [3], [5]!
8
(a)
(b)
(c)
Figure 2.2: Figure showing some tree level (i.e. first order perturbation) Feynman
diagrams of QED, all the result of the interaction term obtained by making the free
Dirac Lagrangian invariant under U (1) gauge transformations. (a) shows compton
scattering of a photon and an electron, in (b) the annihilation of a fermion antifermion pair into a virtual photon and the subsequent creation of another fermion
pair is shown and (c) shows the scattering of two fermions by exchanging a virtual
photon. Most Feynman diagrams in this text are made using the tool [6].
2.4.2
Interactions and Feynman diagrams
/ in
The probability amplitude of interactions resulting from the new term iqψ Aψ
the Lagrangian can be calculated in perturbation theory and each term in this
expansion can be associated with so-called Feynman diagrams. These diagrams,
showing world lines for a certain class of particle paths through space-time, give a
pictorial representation of quantum mechanical transition amplitudes from an initial
to a final state, which are otherwise represented by seemingly arcane equations. With
each part of the diagram, Feynman rules can be associated, making it relatively
straightforward to derive a transition amplitude for the depicted process, as shown
in many books on the subject of particle physics such as [1], [3], [5]. In every vertex
of the diagram particle fields are annihilated, and new ones are created. Each vertex
adds a power of the coupling constant, and virtual particles contribute by means
of their propagator which is momentum dependent. A few examples of Feynman
diagrams are shown in figure 2.2. Besides being used to make perturbation theory
calculations more straightforward, Feynman diagrams can also simply be used to
depict certain processes and interactions between particles. Matter particles are
conventionally represented by straight lines with arrows pointing forward in time
for fermions, and backward in time for antifermions. Electroweak gauge bosons
tend to be represented by wavy lines, gluons by curly lines, and scalars by dashed
lines.
2.4.3
Non Abelian gauge theories
In order to write a Lagrangian invariant under local transformations of non-Abelian
groups, such as SU (N ) groups (except SU (1)), we will have to extend procedure
9
that was developed for the Abelian U (1) group. SU (N ) groups are the groups of
unitary n × n matrices with determinant one. We will again introduce new fields
transforming under the gauge group in order to cancel the terms breaking gauge
invariance in the free Dirac Lagragian, which will again lead to couplings between
fermions and gauge bosons. The bosonic terms in the Lagrangian will however have
to be different from what we wrote down in the Abelian case, as bosons will be
shown to carry the charges of the group and interact among themselves due to the
non-commutation of the generators of non-Abelian groups.
We will start from the Lagrangian:
/
L = iψ ∂ψ
(2.17)
where we have consciously omitted the mass term, the reason for which will become
clear further along the text. The field ψ is now taken to be a multiplet of N spinor
fields:
 
ψ1
 ψ2 
 
 . 

ψ=
(2.18)
 . 
 
 . 
ψN
and transforms as a fundamental representation under SU (N ). So under SU (N )
gauge transformations the multiplet ψ will become (where we implicitly assume a
sum over group indices a if they appear twice):
ψ 0 = exp (igχa (x)τ a ) ψ
(2.19)
ψ 0 = ψ exp (−igχa (x)τ a )
(2.20)
where χa (x) are real and differentiable functions, τ a are the generators of the gauge
group, and g will later be identified as the coupling constant of the interaction, which
could have been absorbed in the phase factors χa (x). In similar fashion to what was
done for QED we can now introduce a covariant derivative:
g
Dµ = ∂µ + i τ a Aaµ .
(2.21)
2
The fields Aaµ are gauge fields corresponding to the transformation group and the
factor 12 is a matter of convention. SU (N ) groups have n2 − 1 generators and
for every generator a gauge field has to be introduced. The fermionic part of the
Lagrangian then becomes:
g
/ − ψτ a A
/ = iψ ∂ψ
/aψ
L = iψ Dψ
2
10
(2.22)
in which we see couplings between the fermions and to n2 −1 new gauge boson fields.
The gauge fields will have to transform differently under the gauge transformations
compared to the Abelian case however, as the generators of the group no longer
commute. To find the transformation properties of the gauge fields we first introduce
the simplified notation:
Aµ = Aaµ τ a
a a
χ = χ(x) τ
(2.23)
(2.24)
We now want to find the transformation law for Aµ under the gauge transformations:
A0µ = Aµ + δAµ .
(2.25)
As shown in the previous section, the covariant derivative must transform as:
g
0
(Dµ ψ) = 1 + i χ(x) Dµ ψ
(2.26)
2
where we considered an infinitesimal SU (N ) transformation this time around. On
the other hand we find from equations 2.19 and 2.25 that:
g
g
g
(Dµ ψ)0 = ∂µ + i Aµ + i δAµ 1 + i χ(x) Dµ ψ
(2.27)
2
2
2
From equations 2.26 and 2.27 we then find:
g
δAµ = −∂µ χ + i [χ, Aµ ] = −∂µ χ − gf abc τ a χb (x)Acµ
(2.28)
2
where f abc are the structure constants of the Lie-group SU (N ). We then find for
the individual gauge fields Aaµ :
δAaµ = −∂µ χa (x) − gf abc χb (x)Acµ .
(2.29)
On account of the last term in the equation above, a free boson term of the same
form as the last term in equation 2.16 will not be gauge invariant. We will need to
add new terms to the original Lagrangian to compensate for this. It can be shown,
as for instance done in the appendix of [3], that the following gauge invariant term
describing gauge boson dynamics can be written in the Lagrangian:
1
Lbosons = − Gaµν Gaµν
4
(2.30)
Gaµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν .
(2.31)
with
The last term in equation 2.31 leads to interactions between the gauge boson fields,
something which was not the case for the photons in QED! These interactions are
depicted in figure 2.3. So everything considered we end up with the following gauge
invariant Lagrangian:
1
g
1
/ − ψτ a A
/ a ψ − Gaµν Gaµν
/ − Gaµν Gaµν = iψ ∂ψ
L = iψ Dψ
4
2
4
11
(2.32)
Figure 2.3: Quartic- and triple gauge couplings between the force carrying bosons,
due to the presence of interaction terms containing three or four gauge boson fields
in the Lagrangian.
2.4.4
Renormalization and running coupling
When measuring particle interactions, we observe interactions with an effective
strength coming from the sum over all Feynman diagrams contributing to this process, up to any order in perturbation theory, as shown in figure 2.4 for QED. The
momenta of virtual particles in loop diagrams are not constrained to a unique value
by the in- and outgoing particle’s momenta. In order to evaluate all possibilities,
one needs to consider integrals over momentum space from zero to infinity, leading
to divergent results. These infinities can be tucked away by absorbing them into
the definition of the coupling constant and the fermion mass. This renormalization
procedure is dependent on q 2 , a measure of the total momentum transfer, and as
such a renormalization scale, usually called µ2 , has to be introduced as the scale at
which the subtractions of the infinities was performed. Higher order contributions
containing fermion propagators cancel each other up to any order in a perturbative
expansion of a gauge invariant theory. These cancellations are known as Ward identities, and if they didn’t hold the effective strength of gauge couplings would depend
on particle masses through their propagators, something which is not observed in
g2
nature [5]. Defining α = 4π
, the scale dependence can be summarized in what is
known as the β-function:
∂α(µ)
(2.33)
∂ ln µ2
β is independent of the renormalization scale and can be calculated through a perturbative expansion. For an Abelian U (1) theory it can be shown to be [5]:
β(α) =
2α2 nf
+ O(α3 )
(2.34)
3π
with nf the number of fermions capable of contributing to the loop diagrams under
consideration. In a more general SU (N ) theory we find:
2α2 2
11
β(α) =
nf − N + O(α3 )
(2.35)
3π 3
3
β(α) =
In Abelian theories, the effective coupling constant is seen to increase with the
momentum transfer scale. In the non-Abelian case however, a negative contribution
12
Figure 2.4: Summation of all Feynman diagrams contributing to the measured
electromagnetic coupling between charged fermions. e0 indicates the bare electric
charge, and e(q 2 ) is the electric charge that is finally measured at a certain momentum scale. The figure has been taken from [1].
due self-interactions of the gauge bosons works the other way. As soon as 2nf <
11N these terms will dominate and lead to a decreasing coupling constant with the
momentum scale. This scenario is referred to as asymptotic freedom [5].
2.5
SU (2)L
N
U (1)Y : electroweak interactions
It has been experimentally established that weak interactions violate parity [7], couple particles of different flavors, and are short-ranged, due to being mediated by
massive gauge bosons. The electromagnetic N
and weak interactions are found to be
described by the unified gauge group SU (2) U (1).
For the SU (2) transformations the left-handed chiral fermion spinors are grouped
into doublets, while the right handed ones are postulated transform as the trivial representation under SU (2), in other words they do not feel this part of the
electroweak interaction. This explains the violation of parity and the conventional
naming of the gauge group as SU (2)L . The left- and right-handed parts of a spinor
are defined as:
1 − γ5
ψ = PL ψ
2
1 + γ5
ψR =
ψ = PR ψ
2
ψL =
(2.36)
(2.37)
with
γ5 = iγ 0 γ 1 γ 2 γ 3
(2.38)
the chirality operator. The operators PL and PR are projection operators, which can
be seen from the property γ52 = 1, and left-handed spinors have a chirality eigenvalue
-1, while right-handed spinors have eigenvalue +1. Since charged weak interactions
can transform leptons into their neutrinos and up-type quarks into down-type quarks
and vice versa, we put the particles that are coupled to each other in this way in
SU (2)L doublets. So we have for respectively lepton and quark doublets:
13
νlL
uL
ΨL =
(2.39)
ΨL =
(2.40)
`L
dL
The elements of the matrices above correspond to the spinor field operators of the
particles. These doublets should transform as fundamental representations under
SU (2) and we can assign a weak isospin to the doublet, in analogy to the spin
assigned to fermions which transform as SU (2) representations under the rotation
group. We follow the convention of assigning a weak isospin of 1/2 to each doublet,
and instead of spin up and down like in the case of the rotation group we now assign
an isospin projection Iw to both components. The upper component is taken to
have Iw = 1/2 while the lower component has Iw = −1/2. Note that the down-type
quarks in the doublets are actually CKM superpositions of the down-type flavor
eigenstates, we will go into more detail on this in the next section. Under SU (2)
gauge transformations the doublets transform as:
σa
Ψ = exp igχ(x)a
Ψ
(2.41)
2
a
with σ a the Pauli spin matrices, σ2 being the generators of SU (2) and g the SU (2)
coupling constant. Under U (1)Y both left- and right-handed spinors are taken to
transform equivalently, as:
Y
χ(x))ψ
(2.42)
2
where Y is the so-called weak hypercharge which will later be related to the electric
charge and g 0 is the U (1)Y coupling constant. Adhering to the procedure described in
the section on non-Abelian gauge theories, we now introduce the covariant derivative:
ψ = exp(ig 0
Dµ = ∂µ + ig
σa a
Y
Wµ + ig 0 Bµ .
2
2
(2.43)
Three gauge fields Wµ1 , Wµ2 and Wµ3 have been introduced for every generator of
SU (2) and one gauge field Bµ for the one generator of U (1), which is the unity
matrix. For (1/2)σ a Wµa we have, using the same representation for the Pauli spin
matrices as in [3]:
σa a 1
Wµ3
Wµ1 − iWµ2
Wµ ≡
W =
(2.44)
−Wµ3
2 µ
2 Wµ1 + iWµ2
/ ΨL , which will appear in the gauge inIt is clear that in terms of the form ΨL W
variant Lagrangian due to equation 2.43, the Wµ3 field will couple particles of the
same flavor as it occupies the diagonal positions of the matrix in equation 2.44.
The linear combinations of Wµ1 and Wµ2 will on the other hand couple the different
spinor fields of the SU (2) doublet. In other words they change a particle’s flavor
and charge. These linear combinations of the gauge fields can be associated with
the physical W + and W − bosons! Since the generator of U (1) is the unity matrix it
is clear that the B µ field, like Wµ3 will preserve a particle’s flavor when coupling to it.
For now the gauge fields we introduced remain massless, and mass terms can not
14
simply be introduced. Mass terms have the form of quadratic couplings of the same
field, which for vector bosons looks like:
m2 µ
A Aµ .
(2.45)
2
For any gauge boson field Aµ this clearly violates gauge symmetry and leads to a
non-renormalizable theory giving nonsensical and infinite predictions in perturbation
theory as shown in [3], [5]. A radically new way to acquire gauge boson masses had
to be concocted. Expanding upon an idea to incorporate gauge boson masses, first
introduced by Brout, Englert and Higgs [8], [9], an SU (2) doublet Φ of complex
scalars with a potential term can be inserted into our Lagrangian [10]:
Lscalar = (Dµ Φ)† (Dµ Φ) − µ2 Φ† Φ − λ(Φ† Φ)2
with
1
Φ= √
2
1
φ + iφ2
,
φ3 + iφ4
(2.46)
(2.47)
λ and µ constants, λ > 0 for the potential to be bounded from below, and µ2 < 0
so that the potential has an infinite set of degenerate minima fulfilling:
v2
µ2
≡ .
(2.48)
2λ
2
Also note that Dµ was used instead of ∂µ to ensure a gauge invariant Lscalar . The
minimum of the potential now corresponds to a nonzero field value, and we can
choose an arbitrary point on the multidimensional sphere of minima as our minimum around which we expand the fields in perturbation theory. For this vacuum
expectation value (VEV) we choose the real component of the bottom scalar:
!
0
h0 |Φ| 0i = √v .
(2.49)
Φ† Φ = −
2
This choice will fix the relations between weak hypercharge, weak isospin and electric
charge as discussed further but is arbitrary and has no physical consequences. Once
the direction of the VEV is chosen, applying infinitesimal SU (2)L transformations
on equation 2.49 shows that the upper component fields of the Φ doublet and the
imaginary part of the lower component are un-physical gauge artifacts. Applying
such an infinitesimal transformation on the VEV yields contributions to all but the
real component of the lower scalar. Every field that can be ”turned on” in this
way can also be ”turned of” with a gauge transformation which has no physical
consequences, making it clear that indeed only the real component of the lower
scalar is a physical field. So we can write:
1
0
(2.50)
Φ= √
2 v+h
It is now unmistakable that by introducing the scalar doublet we get one new scalar
field which is called the Higgs field with the associated boson as an excitation of the
field, the Higgs boson. Writing out the first term in the scalar Lagrangian yields:
15
1
1
(Dµ Φ)† (Dµ Φ) = (∂µ h)† (∂ µ h) + g 2 (Wµ1 + iWµ2 )(W 1µ − iW 2µ )(v + h)2
2
8
1
+ (gWµ3 − g 0 Bµ )(gW 3µ − g 0 B µ )(v + h)2 .
(2.51)
8
Aside from the kinetic term of the Higgs field, one sees that there will be interactions
between the Higgs field and the electroweak gauge fields Wµa and Bµ since terms
coupling their fields are present in equation 2.51. Additionally we now have terms
quadratically coupling the gauge bosons to the VEV:
1
1 2 2
v g (Wµ1 W 1µ + Wµ2 W 2µ ) + v 2 (gWµ3 − g 0 Y Bµ )(gW 3µ − g 0 Y B µ )
(2.52)
8
8
Considering equation 2.45 we see that we effectively get gauge boson mass terms
in our Lagrangian due to their coupling with the VEV. These mass terms are now
gauge invariant since the VEV transforms under gauge transformations too, which
is not the case for a constant mass term introduced ad hoc. For the charged W
bosons we find:
1 2
1
mW (Wµ1 W 1µ + Wµ2 W 2µ ) = v 2 g 2 (Wµ1 W 1µ + Wµ2 W 2µ ),
2
8
(2.53)
so
gv
(2.54)
2
So we have essentially gotten a prediction of the W mass in terms of the SU (2)
coupling constant and the VEV. The mass terms for Wµ3 and Bµ can be rewritten
in matrix form:
mW =
2
1 2
v2
g
3
0
3µ
0
µ
3
Wµ Bµ
v (gWµ − g Y Bµ )(gW − g Y B ) =
−g 0 gY
8
8
−g 0 gY
g 02 Y 2
3µ W
Bµ
(2.55)
We can now choose Y = 1 as the scalar doublet’s hypercharge, this choice is arbitrary,
but once chosen this will fix the weak hypercharges of all other fields. Diagonalizing
the mass matrix we get the eigenvalues 0 and g 2 + g 02 , respectively corresponding to
the eigenvectors:
g 0 Wµ3 + gBµ
p
g 2 + g 02
(2.56)
g 0 Wµ3 − gBµ
p
g 2 + g 02
(2.57)
and
Electromagnetism has an infinite range due to its massless gauge boson, so we associate the first, massless, eigenvector with the photon field. The second eigenvector
16
is massive, and does not change the particle flavor since both Bµ and Wµ3 couple
diagonally in SU (2) space. We associate this field with the physical Z boson and
will call it Zµ . Defining the weak mixing angle θw as:
g0
= tan θw ,
g
(2.58)
Aµ = sin θw Wµ3 + cos θw Bµ
(2.59)
Zµ = cos θw Wµ3 − sin θw Bµ
(2.60)
we can write:
For the Z mass we now find:
mZ =
1p 2
gv
mW
g + g 02 =
=
2
2 cos θw
cosθw
(2.61)
So we find a relation between the W and Z boson masses and the electroweak mixing
angle.1 The bare mass of the Higgs boson itself is given by terms in its potential
term and is:
m2H = 2λv 2
(2.62)
A relationship between weak hypercharge, weak isospin and electric charge, and
between the electroweak coupling constants g and g 0 and QED’s coupling constant
/ ΨL + ig 0 ΨBΨ.
/
e, can now be found by looking at the diagonal terms of igΨL W
Filling in equations 2.59 and 2.60 and requiring that the couplings between the
fermion fields and Aµ are the same as in QED, we find:
q=
Yw
+ Iw
2
(2.63)
and
e = g sin θw = g 0 cos θw .
(2.64)
We have now constructed the Lagrangian of the electroweakN
interactions, and have
shown that through spontaneous breaking of the SU (2)L U (1)Y symmetry to
U (1)QED , by choosing a VEV direction in a scalar doublet with a continuum of
potential minima, we retrieve back QED, but now accompanied by weak interactions
mediated by three massive gauge bosons. The masses of these gauge bosons were
made possible by means of the Higgs mechanism, while retaining renormalizibility.
Since only left handed particles couple to the three Wµa gauge fields the charged weak
interactions will maximally violate parity. The neutral weak interactions, mediated
by the Z boson, will have different couplings to left- and right-handed particles.But
due to the mixing of the Wµ3 field with the Bµ field which couples equally to both
chirality components of the spinors, they are felt by all fermions.
1
Note that all the predicted masses correspond to bare masses which acquire corrections in
perturbation theory.
17
2.6
Yukawa couplings
The matter particles present in nature are massive so we need to have a mass term
of the form mψψ in our Lagrangian. When considering QED this term was written
down without further notice, but after introducing the electroweak interactions of
the previous section we run into problems. The fermion mass term can be rewritten
as:
mψψ = m(ψPL ψ + ψPR ψ) = m(ψPL2 ψ + ψPR2 ψ) = m(ψR ψL + ψL ψR )
(2.65)
using γ52 = 1, γ5† = γ5 and γ5 , γ 0 = 0. In the previous section we let the lefthanded spinors transform as 2D representations of SU (2) while we let the righthanded spinors transform as the trivial representation. The mass term above will
thus no longer be gauge invariant since left- and right spinors transform differently
under the electroweak gauge transformations! Luckily, the scalar Higgs doublet offers
salvation again. Since both left-handed spinors and the scalar doublet transform as
2D SU (2) representations, terms containing ΨL Φ or its hermitian conjugate will
be SU (2) gauge invariant. To make the terms U (1)Y invariant we can then add a
right-handed fermion field. The resulting terms are the so-called Yukawa couplings:
(2.66)
Lyukawa = −gf ΨL ΦψR + ψR Φ† ΨL .
with gf a Yukawa coupling constant which is different for every fermion field. Working this out for a doublet of leptons we find:
v
h
Lyukawa = −gl √ (lL lR + lR lL ) + −gl √ (lL lR + lR lL )
(2.67)
2
2
What we find is gauge invariant mass terms for the charged leptons, acquired by
coupling to the VEV. The size of the mass terms is determined by the Yukawa
coupling constant, and a coupling of the Higgs field directly proportional to the
fermion mass is also obtained. For down type quarks the exact same mass terms
can be written. But how can mass terms for the neutrinos and the up type quarks
be written down after choosing the VEV direction in the lower doublet component?
It can be shown that Φ∗ transforms under an equivalent representation of SU (2) as
Φ, given by:
−1
σ a∗
2
a
iσ exp −iχ (x)
iσ 2
(2.68)
2
implying that we can use
e ≡ iσ 2 Φ∗
Φ
(2.69)
to write gauge invariant mass terms for neutrinos and up type quarks since the VEV
e is located in its upper component. We thus add the terms:
of Φ
e R + ψR Φ
e † ΨL
− gf ΨL Φψ
(2.70)
18
to our Lagrangian. In summary we have introduced gauge invariant mass terms for
the fermions by coupling them to the Higgs doublet, and in doing so we ended up
with couplings of the fermions to the Higgs field proportional to their masses. Opposed to the masses of the W and the Z which are predicted in the Standard Model,
the fermion masses depend entirely on the constant gf which has to be determined
from experiment.
When writing down the fermion mass terms we implicitly assumed the Yukawa
couplings to be diagonal in the particle flavors. There is however no reason why
this should be the case, and when writing the Yukawa couplings in a more general way, one ends up with several matrices for the different particle types that
can not be simultaneously diagonalized. This can be shown to introduce charged
current weak interaction couplings able to couple between particles of different generations. As such a charm quark will for instance be able to decay to a down
quark, or a beauty quark to a charm quark, etc. This flavor violating effect can
be expressed in terms of three angles and one complex phase in the case of three
generations. The matrix relating the mass and flavor eigenstates in the quark
sector is called the Cabibbo–Kobayashi–Maskawa (CKM) matrix, while the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix expresses the relations between
the lepton flavor and mass eigenstates. The presence of a complex phase2 in these
matrices leads to violation charge conjugation + parity symmetry (CP) in the SM.
2.7
SU (3): quantum chromodynamics
Quantum chromodynamics (QCD), or the fundamental theory of the strong interaction is conceptually much easier than the electroweak interactions. But while it is
nigh trivial to write down the QCD Lagrangian using the results of the section on
non-Abelian gauge theories, technical difficulties make this theory extremely hard
to handle in actual calculations. In addition to space- and spin degrees of freedom,
quarks are postulated to have an additional degree of freedom, typically denoted
”color”, which is necessary to make the wave functions of baryons comply with
Fermi-Dirac statistics. We then group the different colored spinors corresponding to
quarks of the same flavor into fundamental 3D representations of SU (3):
 
ψr
ΨSU (3) = ψg 
(2.71)
ψb
where the colours have been denoted as r(red), g(green) and b(blue). These triplets
then transforms under SU (3) gauge transformations as:
λa
)ΨSU (3)
(2.72)
2
The matrices λa are a 3D analogue of the Pauli matrices, commonly known as
the Gell-Man matrices. There are eight such matrices, for the eight generators of
Ψ0SU (3) = exp(iχa (x)
2
Maybe even three in the PMNS matrix if neutrinos turn out to be Majorana fermions.
19
Figure 2.5: Comparison of the strong coupling constant’s predicted running to
several measurements. The order of perturbation theory used in the calculations to
extract the coupling constant from the measurements is indicated. [4]
SU (3). Because the SU (3) transformations operate in color space, the gauge group
of QCD is often called SU (3)c . To formulate a gauge theory one can now replace the
derivative ∂µ in the free particle Lagrangian by the following covariant derivative:
λa a
A
(2.73)
2 µ
, in which 8 new gauge fields, the force carriers of QCD, called gluons, have been
introduced. For these gauge bosons, bosonic terms like in equation 2.30 then have
to be introduced, leading to self interactions between the gluons because of the nonAbelian nature of SU (3). For the gluons no mass terms have to be introduced like
what was done for the electroweak gauge bosons, as they are found to be massless.
Dµ = ∂µ + igs
Considering the section discussing renormalization we see that the condition 2nf <
11N is fulfilled by QCD as there are 6 quark flavors making up the fermion multiplets
taking part in QCD interactions. The resulting running of the coupling constant as
a function of the momentum scale is shown in figure 2.5.
One sees that at low momentum scales the QCD coupling is so large that perturbation theory will no longer be a useful approximation. Because of this two QCD
regimes can be distinguished, based on the momentum transfer scales involved:
• non-perturbative/soft QCD regime:
The strong coupling constant will rapidly increase when going to lower mo20
mentum transfers and longer distances. Perturbation theory can no longer be
applied in this regime and quarks are confined into hadrons.
• perturbative/hard QCD regime:
At adequately high momentum transfers, QCD can be treated in perturbarion
theory. Quarks and gluons can effectively be seen as free, unbound particles
in this regime.
2.8
Standard Model Summary
In this chapter we constructed the Standard Model of particle physics, starting from
a free fermionic Lagrangian that was made invariant under several groups of gauge
transformations. From this simple principle, the presence of electromagnetic interactions mediated by massless photons, weak interactions mediated by three massive
gauge bosons and strong interaction mediated by eight massless gluons were derived.
The masses of both the weak gauge bosons,
N and the matter particles were facilitated
by spontaneously breaking the SU (2)L U (1)Y symmetry to U (1)QED through the
presence of a scalar doublet with a non-vanishing VEV. This scalar doublet leads
to one extra field and a corresponding boson, the Higgs. The hypercharges of all
the particles, and the representation as which they transform under the SM gauge
groups is summarized in table 2.1. One more important
thing
N
N to note is that our
Lagrangian is also invariant under global SU (3)c SU (2)L U (1)Y transformations, leading to a multitude of conserved currents by means of Noether’s theorem.
A direct consequence of this is that in SM interactions the quantities called lepton
and baryon number, ”charges” carried by respectively leptons and baryons, with
antiparticles having the opposite value, will be conserved. One can also intuitively
see this from the form of the couplings of the fermion fields to the gauge bosons, in
which one fermion field is always annihilated while another is created.
4
Right-handed neutrinos are placed in the table, but these particles have never been observed,
which might simply be because they do not feel any of the SM gauge interactions, or because they
simply do not exist. They are needed in the SM if one wants to give neutrinos mass through Yukawa
couplings, but it is not clear if this is in fact the case, more details on this follow in the next chapter.
21
particle multiplet
SU (2)L
U (1)Y
SU (3)c
(νL , lL )
2
-1
1
lR
1
2
1
(νR )
1
0
1
(uL , sL )
2
1/3
3
uR
1
-4/3
3
dR
1
-4/3
3
Higgs doublet
Φ
2
1
1
gluons (8)
Aaµ
1
0
3
W fields (3)
Wµa
2
0
1
B field
Bµ
1
0
1
leptons,
(3 generations)
quarks, (3 generations and colors)
Table 2.1: Summary of the SM gauge transformation properties of all the known
particles4 . For every particle multiplet, the representation as which it transforms
under the different gauge groups is shown, where N denotes the fundamental (N dimensional) representation for SU (N ) and N stands for the adjoint representation
(N 2 −1-dimensional) of SU (N ), and 1 means a trivial representation. For the group
U (1)Y we wrote down the hypercharges instead of the representation since this group
is one-dimensional, so now 0 means a particle does not feel this group or in other
words transforms as the trivial representation.
22
Chapter 3
Standard Model: the final
word?
3.1
A tale of triumph
Originally completed in the 1970’s, the Standard Model has withstood the test of
time and was verified time and again by numerous experiments covering a colossal
energy range. The last of its prediction remaining astray was the Higgs boson, the
experimental discovery of which at the LHC marked the final milestone in the experimental verification of this magnificent theory. While many historical experiments
over the years set the SM in stone, we will show some of the latest results showing
the theory is still remarkably successful.
The Higgs boson can be seen as the linchpin of the SM, its existence manifestly necessary for the theory to work, as its absence would leave the SM non-renormalizable.
The mass of the Higgs boson was constrained to be somewhere below about 1 TeV
by the requirement of quantum mechanical unitarity in the W + W − → W + W −
scattering cross section as shown in [1]. While the LEP and Tevatron colliders only
managed to exclude some mass regions below 1 TeV, the crucial discovery was finally
claimed in 2012 by the CMS [11] and ATLAS experiments [12], almost 50 years after
the prediction in [8] and [9]. The best results were obtained in the decay channels
H → ZZ ∗ → 4` and H → γγ 1 , for which the CMS results are shown in figure 3.1.
Since the discovery of the Higgs boson the electroweak sector of the SM is overconstrained by measurements. This makes it possible to predict certain observables
such as the weak mixing angle and the W boson’s mass to precisions greater than
those attained by direct measurements. Any deviation between the results from the
fits and those from measurements might indicate the presence of new physics, and
as such these fits are crucial tests of the SM. Even new physics at energy scales far
1
The Higgs does not couple to photons directly because they are massless, but can decay to
them through higher order loop diagrams.
23
(a)
(b)
Figure 3.1: Figures showing the di-photon and four lepton invariant mass distributions which provided the best signal for CMS’s Higgs discovery. In both plots the
red line represents the expected signal with the presence of a SM Higgs boson [11].
beyond the reach of modern day experiments can influence these results through
loop corrections. Some of the latest results attained by the Gfitter Group are shown
in figure 3.2, illustrating the tremendous strength of the SM as the SM fits agree
with all current observations [13].
3.2
Loose ends
While fully internally consistent and extremely successful, the SM can not be the
ultimate theory of nature, as it unfortunately leaves us with some gaping questions.
The Standard Model faces challenges from both experimental measurements and
theoretical considerations. A non-exhaustive listing of phenomena that have thus
far remained unexplained by the SM is laid out below.
3.2.1
Gravity
The everyday experience of gravity is perhaps the most obvious argument against
the SM. In the previous chapter we silently ignored gravity, while neither the planet
we live on, nor our sun fueling life would exist if not for gravity. So why is gravity
not present in the SM? The answer is that so far nobody has been able to formulate
a consistent theory of gravity which can be unified with the rest of the SM and is
able to make useful predictions. Many attempts and advances on this topic have
been made, including holographic theories, supergravity theories, etc. [15], [16]. So
far none has been unambiguously shown to do the job.
24
(b)
(c)
(a)
Figure 3.2: (a) Comparison of fit results with direct measurements for several SM
parameters in units of the experimental uncertainty. (b) Top quark mass vs W boson
mass with 68%, and 95% confidence limit contours shown for direct measurements,
and fits excluding these two parameters. The blue contours include the Higgs mass
in the fit while the grey contours do not use it. (c) Plot showing the W boson
mass vs sin2 θef f , with θef f the effective electroweak mixing angle after higher order
perturbative corrections. 68%, and 95% confidence limit contours are again shown
with and without the Higgs mass included in the fits. All calculations correspond
to full fermionic two loop calculations. [13]
25
(a)
(b)
Figure 3.3: Optic (a) and x-ray (b) images of the bullet cluster, with mass contours
shown in green as determined from the weak gravitational lensing effect. [14]
3.2.2
Dark matter
Yet another mystery the SM is unable to cope with is the elusive dark matter,
the existence of which has been presumed to explain a profusion of astronomical
observations made over the past few decades. For example, the magnitude of the
angular velocities of stars in galaxies seem to indicate the presence of far more
mass in these galaxies than we discern. Among many other observations, the most
prominent is perhaps the Bullet Cluster where two galaxy clusters are colliding. In
this collision, the observable matter lags behind the bulk of the mass, as observed
through the weak gravitational lensing effect, making a very strong case for dark
matter’s existence, as opposed to alternatives such as modified dynamics [14]. Many
modern observations show that the largest part of all the mass present in the known
universe must consist of this enigmatic form of matter, which is not accounted for
in the particle spectrum of the SM. [17], [18] According to the latest results of the
Planck collaboration, dark matter contributes about 26% to the energy density in
the universe, while ordinary matter contributes only about 5% assuming the ΛCDM
model to be correct. The other 69% is the subject of the next paragraph [19].
3.2.3
Dark energy
The largest fraction (≈ 69%) of the energy density of our universe is believed to
be in the form of vacuum energy accelerating the expansion of the universe. It is
often called dark energy, a term signifying the fact that we are almost completely
ignorant about its nature. Current observations, such as those from Planck [20] are
consistent with the assumption of a cosmological constant (i.e. constant vacuum
energy density), though slowly varying scalar fields are also a possibility. This
mysterious energy density is not predicted by the SM, and any attempt to calculate
it in the SM leads to cataclysmic result wrong by more than 100 orders of magnitude
[21]!
26
3.2.4
Matter-antimatter asymmetry
One more cosmological objection to SM is that the universe seems to be almost
exclusively made up of matter as opposed to antimatter. Antihelium nuclei have for
instance never been detected in cosmic rays [22], indicating there are no stars made
out of antimatter anywhere near us. As shown in [23], three conditions have to be
met to generate this asymmetry. A process violating baryon number, thermal nonequilibrium, C- and CP-violation, all to be present in the early universe to account
for the observed asymmetry. The SM has no baryon number violating processes,
and the CP-violation present in the weak interactions is not sufficient even if the
other conditions were met. As such, the SM is unable to explain the dominance of
matter in our universe.
3.2.5
Free parameters
The SM has many free parameters, namely the fermion masses, the coupling constants, the CKM and PMNS matrix elements and the constants in the Higgs potential, and the CP-violating phase of QCD2 . The SM provides no explanation for
the values of any of these parameters whatsoever, they can only be deduced by
experiment.
3.2.6
Hierarchy problem
While the discovery of the Higgs boson was a triumph for the SM, with it another
question arose. The observed mass of a particle corresponds to the mass parameter
in the Lagrangian, corrected by all possible higher order loop corrections. More
specifically any quadratic coupling of a field to itself will give contributions to its
mass, and these can go through an infinite amount of loop corrections. These loop
corrections have to be calculated up to a certain cutoff scale at which the theory is
expected to break down. For the SM one usually takes this to be the Planck scale,
where quantum effects of gravity are presumed to become important. The masses
of the gauge bosons and fermions are protected by the gauge- and chiral symmetries
of the SM, and only acquire small contributions which logarithmically depend on
the cutoff energy scale. To give two examples, one can show that all contributions
to the photon mass term cancel, while for instance corrections to the electron mass
integrated up to the Planck scale can be shown to be about a quarter of the bare
electron mass. The scalar Higgs boson’s mass is however unprotected by gauge- or
chiral symmetry and it can be shown that corrections to its mass are proportional
to the square of the cutoff energy scale. Fermion loop corrections for instance, such
as the rightmost diagram in figure 3.4, give the following contributions to the mass
[24]:
δm2h = nf
2
gf2
Λ
1
2
2
2
−Λ
+
6m
log
−
2m
+
O
f
f
8π 2
mf
Λ2
This is a parameter we have not discussed in the previous chapter
27
(3.1)
Figure 3.4: One loop self energy corrections to the Higgs mass [25].
where gf are the earlier introduced Yukawa couplings, Λ is the cutoff energy scale,
Nf is the number of fermions contributing, and mf are their masses. Other contributions, such as the left and middle diagrams in figure 3.4 are similarly dependent
on the cutoff energy scale. Reminiscing the fact the Planck scale was taken to be
the cutoff scale, Λ is about 1019 GeV, making it seem suspicious and that the Higgs
mass was measured to be only about 125 GeV. In order to achieve this number, an
inordinately coincidental cancellation fine tuned up to more than 30 orders of magnitude, between the bare mass and the self energy corrections ought to take place, to
explain the smallness of the observed mass. This can be conceived to be unnatural
and is known as the hierarchy problem. Several solutions exist to this problem, such
as theories with extra dimensions which boast lower cutoff energy scales, or Supersymmetry in which all divergent mass corrections are canceled between bosons and
fermions, as discussed in the next chapter [26], [27], [28].
3.2.7
Neutrino masses
In the previous chapter we introduced Yukawa couplings to the Higgs field in order
to attain fermion masses. It is no problem to write down such terms for neutrinos,
but then one requires the existence of right-handed neutrinos which do not feel any
of the SM gauge groups3 , and as such will be almost impossible to ever detect if
they exist. But multiple limits on neutrino masses, such as those from tritium decay
experiments or from cosmological measurements [4] show that neutrino masses reside
below the eV scale, many orders of magnitudes below the other fermion masses. It
might seem strange that neutrinos are so much lighter than other particles if the
mechanism through which they acquire mass is the same. A popular way to solve this
is by adding a Majorana mass term, coupling right-handed neutrinos to their charge
conjugate field, to the Lagrangian. Such a term does not violate gauge symmetry as
the right-handed neutrinos do not feel any of the SM gauge groups. Large majorana
mass terms can be shown to push down the masses of the left-handed neutrinos
through an effective mass coupling with the right-handed neutrinos, as depicted on
the left of figure 3.5. This mechanism is known as a Seesaw mechanism, and there
are many models expanding upon it. The possible Majorana nature of neutrinos
3
This can be directly derived from the requirement of a gauge invariant Yukawa mass term,
which leads to a weak hypercharge that has to be zero for right-handed neutrinos.
28
(a)
(b)
Figure 3.5: (a) Feynman diagram showing how an effective mass coupling through a
right handed Majorana neutrino can influence the light neutrino masses, figure taken
from [30]. (b) Neutrinoless double beta decay induced by introducing a majorana
mass term, figure taken from [31].
is a hot topic in current research, and can be probed by searching for neutrinoless
double beta decay, shown on the right of figure 3.5. Such a decay is not possible for
SM neutrinos [1], [17], [29].
29
Chapter 4
CMS at the LHC
The analysis presented in this thesis was made using data collected by the Compact
Muon Solenoid (CMS) detector at CERN’s Large Hadron Collider (LHC), currently
the most powerful particle accelerator in the world. In this chapter a brief sketch is
given of of the LHC, and CMS machines, the way particles from the LHC’s collisions
are reconstructed by CMS and why such an hadron collider machine is used.
4.1
Hadron colliders: discovery machines
The LHC is a circular proton-proton collider, built in the tunnel that previously
housed its predecessor, the Large Electron-Positron Collider (LEP) which used to
collide electrons and their antiparticle. For what concerned acceleration in such a
circular machine, hadrons have the distinct advantage that protons are more than
three orders of magnitude more massive than electrons. Accelerated charged particles lose energy by emitting bremsstrahlung when forced to change their direction,
and the cross section for emitting such photons is proportional to the inverse of the
mass of the accelerated particle squared. For this reason the more massive protons
can be accelerated to tremendously higher energies than electrons and positrons. A
proton, not being a fundamental particle, has a very complicated internal structure
however, containing three valence quarks, virtual quark antiquark pairs called sea
quarks and gluons in a very complicated non-perturbative QCD regime. Each of the
constituents of the proton is dubbed a ”parton”. Due to the convoluted structure
of the proton, collisions between protons of the same energy can lead to interactions at different energies and momentum transfers. These interaction energies are
determined by the fraction of the proton’s momentum carried by the interacting
partons. The distribution of the momentum fraction carried by partons is given
by so-called parton-distribution functions (PDF), giving the probability density of
finding a certain type of parton with a specific longitudinal momentum fraction.
So at fixed beam energies, interactions will take place at many different energies,
producing particles of all kinds, masses and energies. This is manifestly not the case
in an electron-positron collider where the leptons annihilate each other at a single
center of mass energy. This means that hadron colliders might be the ideal discovery
machines for novel particles hidden at unknown energy scales. In previous genera30
tions of hadron colliders, such as the Tevatron, proton-antiproton colliosions were
often used, because the valence quarks and antiquarks can then annihilate to form
new particles. The maximum beam intensity, and as a consequence the luminosity
(defined below), would however be much lower as antiprotons are unstable in the
presence of matter and hard to produce. So in the design of the LHC, the option
of proton-proton collisions was chosen instead of proton-antiproton collisions. The
interactions between two protons in a collider like the LHC can be subdivided into
three distinct categories:
• elastic scattering interactions:
In an elastic scattering event, momentum is transferred between the protons,
while both protons remain intact. No quantum numbers are exchanged and
no additional particles are produced in such a process.
• diffractive inelastic scattering interactions:
A diffractive process means that a single (singly-diffractive) or both protons
(doubly-diffractive) break up in the collision. Another possibility is that both
protons remain whole, but create another excited state with neutral quantum
numbers which in turn decays, this is known as central-diffractive process.
• non-diffractive inelastic scattering interactions:
The third category involves at least a single parton of both protons interacting,
with the consequence that both protons shatter. If the interaction between
these partons occurs at high-enough momentum exchange, heavy particles,
such as Z, W, Higgs,... can be produced, and all the rates can be accurately
calculated in perturbation theory since this corresponds to the perturbative
QCD regime. It is clear that non-diffractive inelastic interactions between the
protons in the collider will be the events of prime interest when searching for
the production of new particles.
Another important occurrence in any particle collider is the formation of hadronic
jets. When strongly interacting particles such as the proton’s partons are emitted
from the interaction point, they will radiate more of these particles, slowly losing
their energy. These particles are usually emitted at low angles, and all of them
keep emitting more and more particles until their energy becomes so low they reach
the non-perturbative QCD regime. At this point the remaining particles will be
bound together into hadrons due to the confinement property of QCD. The resulting
structure, of a multitude of hadrons and other particles emitted in a narrow cone is
called an hadronic jet. Such jets are extremely abundant among the events observed
in hadron colliders such as the LHC. A visualization of a non-diffractive protonproton collision leading to several jets is shown in figure 4.1.
4.2
LHC: Energy and luminosity frontier
The LHC is currently marks the high-energy frontier of particle collider experiments,
√
operating at an unprecedented center of mass energy of s = 13 TeV in its second
31
Figure 4.1: Visualization of a non-diffractive inelastic proton-proton collision in
which a top quark pair and a Higgs boson are produced. The hard interaction
between the partons is represented by the red dot, the Higgs and the top quarks by
the small red dots. The radiated partons are shown developing into jets. [32]
32
Figure 4.2: Illustration of the different accelerators and experiments at CERN. [33]
running period, called Run II, after being upgraded from a center of mass energy
of 7 and 8 TeV in its previous run, denoted as Run I. This center of mass energy
defined as:
q
√
(4.1)
s = pµ1 p1µ + pµ2 p2µ
where pµ1 and pµ2 are the four momenta of the colliding protons. The collider is
located near Geneva in Switzerland and has a circumference of 26.7 km, making it
world’s largest particle accelerator. In order to accelerate protons to these titanic
energies, a complex of smaller accelerators is used to boost the protons before finally
injecting them into the LHC. The protons start as hydrogen atoms which are ionized
and subsequently accelerated to 50 MeV by the linear accelerator LINAC 2, after
which they are injected into the Booster, which accelerates them up to about 1.4
GeV. Hereafter they are chained to the Proton Synchrotron and the Super Proton
Synchrotron, circular accelerators of previous generations now used as injectors for
the LHC, where they are accelerated up to respectively 28 GeV and 450 GeV. Finally the protons are injected into two counter rotating beams in the LHC which
accelerates each proton up to 6.5 TeV. The entire complex used to accelerate the
protons is shown in figure 4.2.
The LHC itself accelerates particles by means of an oscillating electric field. Pro-
33
Figure 4.3: Cross section of an LHC dipole magnet. [34]
tons in phase with this electric field are accelerated, while protons out of phase with
the field are decelerated, and as such all protons are automatically grouped into
several bunches. To keep the 6.5 TeV protons in orbit, immense magnetic fields are
needed. The necessary field strengths are are achieved by supercooling magnet coils
built from niobium-titanium cables, a superconductor at low enough temperatures,
to temperatures of 1.9 K, colder than empty space. In order to get two proton beams
traveling in an opposite direction, a two-in-one magnet design is employed. These
so-called twin-bore magnets feature the necessary windings for both beam directions
in a common cryostat and cold mass. Most of the LHC’s magnets, used for keeping
the protons in their circular orbits are dipole-magnets, though several higher-order
magnetic poles are also used however for focusing the proton beams.
At several points across the LHC, its beams are made to cross and collide. Several
experiments are located at these interaction point, one of which is CMS. The amount
of interactions that will effectively take place at these collision points is determined
by the luminosity L of the machine, defined as the number of interactions dN/dt
detected, divided by the cross section σ for such an interaction:
34
1 dN
.
(4.2)
σ dt
Integrating this over the time of operation yields what is known as the integrated
luminosity:
Z
L = Ldt
(4.3)
L=
Once the integrated luminosity is known, the expected number of events of a particular type, with a certain cross section σ, is easily calculated as:
N = σL.
(4.4)
The Luminosity of a collider like the LHC is given by:
N=
f nb Np2
4πσx σy
(4.5)
where f is the frequency at which the protons circulate the accelerator, Np is the
number of protons in a bunch, nb is the number of bunches, and σx and σy are
Gaussian transverse beam profiles. [35]
The LHC in its current incarnation has a peak luminosity of about 1.7· 1034 cm−2 s−1 ,
far higher than previous machines operating at the energy frontier. LEP, for instance, had a luminosity of about 1· 1032 cm−2 s−1 while the Tevatron only reached
5· 1031 cm−2 s−1 , making the LHC a machine both at the energy- and luminosity
frontier. The LHC nevertheless does not have the highest luminosity out of any accelerator, for instance the High Energy Accelerator Research Organisation’s KEKB
has a larger luminosity, but far lower energies.
The LHC’s extremely high luminosity has the distinct advantage that even very
rare processes will occur, and can subsequently be studied. Such a luminosity is not
all sunshine and rainbows though, as it leads to multiple proton-proton collisions
per bunch crossing in an LHC interaction point. This phenomenon is dubbed pileup
and means that interesting hard, non-diffractive events will be accompanied by a
lot of soft activity, and sometimes even other hard interactions. Every interaction
originates from its own vertex, and therefore a good vertex reconstruction is needed
in LHC detectors for distinguishing particles coming from the primary interaction
vertex (usually taken to be the one with the hardest interaction) from those coming
from pileup vertices. The distribution the number of interactions per bunch crossing in the CMS detector is shown in figure 4.4 for the previous 8 TeV run, together
with a recorded event that contained 29 proton-proton collisions in a single bunch
crossing.
35
(a)
(b)
Figure 4.4: Distribution of the number of proton-proton collisions per bunch crossing
at the CMS interaction point during the 8 TeV run (a), and event display of an 8
TeV bunch crossing with 29 distinct vertices coming from 29 separate proton-proton
collisions in a single bunch crossing (b) [36], [37].
4.3
The CMS detector
CMS is one of two multi-purpose detectors specifically designed for the detection of
direct production of new undiscovered particles, the other one being A Toroidal LHC
ApparatuS (ATLAS). The structure is centered around one of the LHC’s collision
points and has a cylindrical shape consisting of a central barrel and two endcaps.
CMS is 21.6 m long, with a diameter of 15 m. It houses the world’s largest and most
powerful superconducting solenoid magnet, boasting a 3.8 T magnetic field. This
magnet is crucial for determining the charge and momentum of high energy particles
by their deflection in its field. After all higher momentum particles will be more
rigid to deflection than low momentum particles, and a particles charge, and by
extension its matter or antimatter nature, determines the direction of its deflection.
CMS consists of various subdetectors which will each be shortly described in the
next sections. The most central part of the detector is the tracker, surrounded
by the electromagnetic calorimeter (ECAL), and the hadronic calorimeter (HCAL).
These subsystems are contained within the magnet’s coil and outside of this the
muon system was placed between the return-yoke layers. A graphical overview of
the CMS detector is given in figure 4.5. In the sections below we will shortly describe
the different aspects of the CMS detector, aside from the forward detector systems,
which were not used either directly of indirectly in this thesis.
4.3.1
CMS coordinate system
Because they will be extensively used in the next sections, it will prove useful to
shortly elucidate the coordinate system used in the CMS experiment. The origin
of the coordinate system is taken to be the interaction point of the LHC’s proton
beams in the center of the CMS detector. The z-axis points along the tangent to the
36
Figure 4.5: Sketch of the CMS detector [38].
37
counter-clockwise rotating proton beam, the y-axis points vertically upwards and
the x-axis points towards the center of the LHC’s ring. The polar angle θ is defined
as the angle measured from the z-axis in the yz-plane, and the azimuthal angle φ
is measured from the x-axis in the yz-plane. The coordinates are sketched in figure
4.6. In practice the pseudorapidity η, defined as:
θ
η = − ln(tan )
(4.6)
2
is often used instead of the polar angle. The reason for this is that the rapidity
difference between two particles is invariant under longitudinal Lorentz boosts along
the beam axis. Rapidity, defined as:
y=
1 E + PL
ln
2 E − PL
(4.7)
with PL the momentum component along the beam axis, can not be conveniently
used to replace the polar angle because it also depends on the energy of the particle.
The pseudorapidity can however be expressed as a function of only the polar angle,
and is equal to a particle’s rapidity in the ultrarelativistic limit1 , which is very often
satisfied for particles produced in the LHC. This means that the pseudorapidity
difference between two particles will be nearly invariant under longitudinal boosts.
Another property of the rapidity is that the particle flux is expected to be constant as
a function of rapidity, meaning this is also approximately true for the pseudorapidity.
The angular separation between two detected particles is usually expressed in terms
of the nearly longitudinal Lorentz boost invariant quantity ∆R defined as:
∆R =
p
∆φ2 + ∆η 2
(4.8)
with ∆φ and ∆η the difference in the η and φ coordinates of the particles. It is self
explanatory that ∆φ is invariant under longitudinal boosts since it is defined in the
plane transverse to the beam direction [38].
4.3.2
Tracker
The first layer of the detector around the interaction point is the tracking system.
It reconstructs trajectories of charged hadrons and leptons with great precision,
information used to measure the momenta and charges of charged particles. Additionally, the tracker is able to reconstruct secondary vertices, necessary for reconstructing with long lived heavy particles such as beauty quarks, and to distinguish
the primary vertex from the usually copious pile-up vertices. In the high pileup
environment of the LHC, an extremely high granularity and a very fast response
time are needed for the tracking system to work appropriately. Other imperative
properties of the tracker, being located at the center of the detector which is utterly
soaked in radiation, are radiation resistance and being lightweight. If the tracker
1
When a particle’s momentum is high enough, the contribution of the mass to the total energy
becomes negligible, and one can assume the particle to be massless when doing calculations without
introducing significant errors.
38
Figure 4.6: CMS coordinate system [38].
Figure 4.7: Longitudinal cross section of a quarter of CMS’s tracking system. Solid
purple lines represent single-sided silicon strip modules, while double-sided modules
are shown as blue lines. Solid dark blue lines represent the pixel modules. [41]
was not lightweight and contained too much material, it would lead to extra photon
conversions, bremsstrahlung, scattering of particles,... all of which are unwelcome
guests. All of the goals outlined above were achieved by a multi-layer detector made
of silicon pixels in the inner layers, needed to cope with the high flux of particles,
and silicon microstrips on the outer layers, together making up the largest silicon detector in the world. When charged particles pass through the silicon of the detector,
they create electron-hole pairs by exciting an electron over silicon’s band-gap. This
electron-hole pair is subsequently drifted towards electrodes by applying an electric
field, after which the signal can be measured. The tracker has diameter of 2.5 m
and measures 5.8 m in length, covering a pseudorapidity range up to |η| < 2.5. A
schematic overview of CMS’s tracker is given in figure 4.7.
39
Figure 4.8: Geometric view of one quarter of CMS’s ECAL system. [42]
4.3.3
Electromagnetic calorimeter (ECAL)
The second layer of the CMS detector is the electromagnetic calorimeter, intended to
measure electron and photon energies with high accuracy by stopping them and measuring the amount of energy they deposit. It consists of PbW O4 , or lead tungstate,
crystals. This is an optically transparent and extremely dense material. On passage of an electron, photon or positron it scintillates, emitting an amount of light
proportional to the energy of the detected particle. The crystals are isolated by a
matrix of carbon fiber, and an avalanche photodiode is placed in the back of each
crystal for reading out the signal. The barrel section of the ECAL stretches over
an |η| range up to |η| < 1.479, whereas the endcap part goes from 1.479 < |η| < 3.
The 1.653 < |η| < 2.6 range is also equipped with a preshower detector with high
granularity, able to distinguish π0 ’s decaying to photons from actual photons, which
is needed because the endcap section of the ECAL has a lower granularity than the
barrel section. A schematic view of the ECAL is shown in figure 4.8.
4.3.4
Hadronic Calorimeter (HCAL)
The hadronic calorimeter’s purpose is to measure the energy of both neutral- and
charged hadrons. When interacting with the detector, hadrons develop a shower by
strongly interacting with nuclei in the material. These showers are relatively slowly
developing compared to electromagnetic showers, and even though the hadrons already interact in the ECAL, they will deposit most of their energy in the HCAL.
The HCAL was designed to put as much absorbant material inside the magnetic coil
and consists of several layers of brass and steel, interwoven with plastic scintilators
reading out the signal with wavelength shifting photodiodes. The HCAL is subdivided into four different detectors. Inside the magnetic coil are the Hadron Barrel
(HB), covering a pseudorapidity range up to |η| < 1.3, and the Hadron Endcap (HE)
which covers the range 1.3 < |η| < 3. The Hadron Outer (HO) detector is layer of
40
Figure 4.9: Longitudinal view of a quarter of CMS, on which the positions of the
different HCAL subdetectors are shown, figure from [38].
scintillators located outside of the magnetic coil. It detects any penetrating hadronic
showers that leaked through the HB, but interacted with the heavy material of the
magnet. The pseudorapidity range 2.9 < η < 5 is covered by the Hadron Forward
(HF) detector which detects particles through the emission of Cherenkov light in its
absorber crystals. The geometry and location of the different HCAL subdetectors
is shown in figure 4.9.
4.3.5
Muon system
Radiating far less bremsstrahlung than electrons due to their far greater mass, and
lacking strong interactions, muons are extremely penetrating, and are generally not
stopped by the HCAL or ECAL. For the purpose of detecting them, the muon
detector system was deployed on the outside edge of the CMS detector. In the barrel
regions, up to |η| < 0.9, where the magnetic field is uniform, drift tube detectors
are used. These drift tubes contain gas which is ionized by passing muons. The
resulting electrons travel to a positively charged wire, and by registring where along
the wire the electrons hit, and calculating the distance between the muon and the
wire, two coordinates of the passing muon are determined. The endcap region from
0.9 < |η| < 2.4 houses cathode strip chambers. They consist of arrays of positively
charged anode wires crossed with negatively charged copper cathode strips. Muons
passing through the chamber create positive ions and free electrons, flocking to the
cathodes and anodes respectively, again yielding two coordinates for the muon. The
cathodes provide a φ measurement, while the anode wires provide an η measurement.
Up to |η| < 2.1 a third type of detector, resistive plate chambers, are used. These
41
Figure 4.10: Vertical cross section of CMS, showing the trajectories for several
particle types through the multiple detector layers. [40]
detectors consist of two gaps made up of parallel plates made from a highly resistivity
plastic, seperated by a volume of gas. One of the plates is positively charged and the
other negative, so if a muon passes through, an electron avalanche is induced, which
is detected by strips placed between the two gaps. While their position resolution is
not up to par with that of the drift tubes or cathode stip chambers, the resistive plate
chambers provide very fast signals, unambiguously matching a muon to a certain
bunch crossing in a high pile-up environment. A summary of all the detection
systems, and the tracks associated with different particle types is shown in figure
4.10.
4.4
Trigger system
At the LHC’s peak performance, proton bunches cross paths in the CMS detector
every 25 nanoseconds or 40 million times per second. Every bunch crossing brings
about multiple proton-proton interactions leading to a total of about one billion
such interactions in the detector every second. As a result of this vast amount of
collisions there is no possibility of measuring and storing the data associated with
all these events. To solve this problem a system of ”triggers” is employed which selects interesting events. Only the events passing these triggers are stored. There are
two levels of triggers, hardware based level 1(L1) triggers which operate extremely
quickly and automatically, looking for simple signs of interesting phenomena, and
the software based level 2 or high level trigger(HLT) which requires the events to be
partially reconstructed.
The signals detected by the different subdetectors are collected in buffers, and seg-
42
ments of the collected information are directly passed to the L1 trigger system,
located close to the detector. A time of 3.2 µs is given for the transition of the
signal to the L1, the calculation of the trigger result and the returning of the decision signal. Custom processors utterly optimized for this task are used to calculate
a decision extremely quickly. Alas, not all subdetectors, for instance the tracker,
provide their information fast enough to be used in the L1 trigger decision, so the
L1 can only rely on the muon system and the calorimeters.
The HLT is made up by dedicated computers, having access to the full event information. For the HLT decisions, particle and event reconstructions are employed.
The HLT trigger algorithms need to execute as quickly as possible, and for this reason they are built up as a series of filters. In order to pass the trigger an event needs
to pass every filter in the corresponding trigger path, and as soon as any filter fails
the trigger path is left and the event can no longer pass the trigger. After passing
the HLT, an event gets a number of identifiers which are then used to sort and store
the event into one of the primary datasets.
4.5
Event reconstruction: particle-flow algorithm
In the last section a brief description of the different subdetectors, and how particles created in the LHC’s collisions yield signals was given. But in order to study
the interactions taking place, one has to figure out what particles were produced,
and determine all their relevant properties for every event. In CMS particle reconstruction is done by the particle-flow algorithm elucidated below with a dissertation
based upon [43].
Particle-flow(PF) aspires to reconstruct all stable and detectable particles in the
event, including muons, electrons, their antiparticles, photons, charged- and neutral
hadrons. This feat is performed by using all CMS subdetector systems in tandem. Neutrinos can not be detected as they only feel weak interactions (which can
be seen from their hypercharge), but conservation of momentum in the transverse
plane makes it possible to define the missing transverse energy (MET)2 variable
which characterizes (confusingly enough) the missing transverse momentum (PT )
in the event. Escaping particles, including neutrinos, and mismeasurements will all
contribute to MET. In order to describe the PF algorithm we will first introduce the
fundamental signal building blocks. These building blocks must be reconstructed
with high efficiency and low fake-rates in very dense environments, to fulfill these
conditions, iterative tracking and calorimeter clustering where developed. The building blocks originating from the same particle are then linked together by a linking
algorithm, after which the PF event reconstruction will take place.
2
It is impossible to determine the missing momentum in the longitudinal direction because the
momenta of the interacting partons are completely unknown. The conservation of momentum in the
transverse plane can be checked by assuming that the interacting partons have negligible transversal
momentum components, leading to a total transversal momentum of zero when summed over all
resulting particles in the event.
43
4.5.1
Iterative tracking
CMS needs to be able to reconstruct particles at very low PT ’s in very busy environments. Even very high PT jets mainly consist of particles with relatively low
momenta (a few GeV/particle in jets with a PT of about a 100 GeV), clearly showcasing that low PT particle reconstruction is essential. When measuring momenta
up to several hundred GeV, but especially at low momenta, the tracker provides a
mightily superior resolution compared to the calorimeters. Conjointly, the tracker
is able to determine the direction of charged particles, pointing to their production
vertex, before the particles trajectories are significantly influenced by the magnetic
field. These properties make the tracker an essential tool for reconstructing all the
different charged particles in an event. We naturally want an efficiency for reconstructing particle tracks as high as possible, in order not to have to rely on the
worse energy and directional resolution of the calorimeters, but we also do not want
to reconstruct any fake tracks. In order to achieve this, tracks meeting very tight
criteria are first determined. These tracks, having almost negligible fake rates, but
only a mediocre efficiency are then removed. Subsequently the track criteria are progressively loosened and more tracks are reconstructed and later removed every time
increasing the overall efficiency, while the fake-rate remains low since the number
of tracks is reduced in every iteration. In the last few iterations, the constraints on
the origin vertex are reduced, leading to the reconstructing of relatively long lived
particles such as τ ’s, kaons, ... All in all, particles with momenta down to 150 MeV,
with as few as three tracker hits and an origin half a meter away from the primary
vertex can be reconstructed with a fake rate below a percent!
4.5.2
Calorimeter clustering
In order to detect and measure the energy of neutral particles, separate them from
charged particle deposits, reconstruct electrons which radiated bremsstrahlung photons, and to improve the energy resolution of charged hadron tracks with high PT ,
or a low track quality, calorimeter clustering is performed. This clustering is independently done in each subdetector. Local maxima are used as seeds, from which
topological clusters are formed by adding cells having at least one side in common
with the seed’s cell. Once all topological clusters are formed, they are subdivided
into as many PF clusters as it contains seeds. The energy of the cells in a topological
cluster is shared between all PF clusters, each PF cluster getting allocated a fraction
proportional to the cell-cluster distance.
4.5.3
Linking
It is obvious that the same particle can give rise to multiple PF elements, such as
a track and a calorimeter deposit for a charged particle. When all PF elements
are determined, the ones corresponding to the same particle have to be linked by a
linking algorithm. This algorithm loops over all possible pairs, and determines the
quality of the link in terms of the distance. The distances are determined as follows
links between several types of elements:
44
• linking charged-particle tracks and calorimeter clusters
Starting from the last hit in the tracker, the charged particle track is extrapolated to the preshower layers, and subsequently to the ECAL to a depth corresponding to the maximum of the longitudinal profile of an electron shower,
and to the HCAL at a depth corresponding to one interaction length. If the
extrapolated cluster is then found to be within the boundaries of a PF cluster, the track and the cluster are matched. If needed cluster boundaries can
be enlarged by one cell in every direction to account for gaps between the
calorimeter cells, the uncertainty on the shower maximum, multiple scattering,... The link distance is then finally defined in terms ∆R as defined in
equation 4.8 between deposits and extrapolated clusters.
• linking bremssstrahlung photons to electrons
In an attempt to reconstruct all energy an electron emitted in the form of
bremsstrahlung photons, tangents to the electron’s track at each layer of the
tracker are extrapolated to the ECAL. If the extrapolated track is within the
boundaries of an ECAL cluster, a link is made.
• linking ECAL, HCAL and preshower clusters
The HCAL is about 25 times less granular than the ECAL, and the ECAL
itself is also coarser than the preshower. If a cluster position in the more
granular detector is found to be within the bounds of a cluster in the coarser
calorimeter a link is established. The cluster in the coarser calorimeters can
again be enlarged by one cell in every direction if necessary. The link distance
is once again defined in terms of ∆R.
• linking a track in the tracker to a track in the muon system
A global fit is made between a muon track and a charged particle track, and if
the χ2 is small enough a link is established. If multiples tracker tracks can be
linked with a muon system track, the match with the smallest χ2 is chosen.
In the end the granularity of the CMS detector leads to blocks of linked particles,
containing only small amounts of elements in most cases. The block size can be
shown to be independent of how busy the environment is, which is rather remarkable.
4.5.4
Reconstruction and identification
After all the blocks are made, the particles will finally be reconstructed. The procedure described below is done for every separate block. Firstly, every global muon,
defined as a link between a charged particle track and a muon system track, brings
about a particle-flow muon if the combined momentum measurement agrees to the
measurement of the tracker within three standard deviations. The muon tracks are
subsequently removed from the block. Additionally an estimate of the energy deposited in the calorimeters by the muon is made, and this deposit is also removed
from the block.
The next particles that are reconstructed are the electrons. Starting from ECAL
45
tracks, the elecron tracks are refit following their trajectory all the way to the ECAL.
The final identification depends on a number of tracker and calorimeter values, and
if the identification is successful, a PF electron is made and the ECAL cluster and
track allocated to the electron are removed from the block.
After the muons and electrons have been removed from a block, the remaining
tracks are only used if the relative uncertainty on their measured PT is smaller than
the calorimetric energy resolution. This resolution is determined by a calibration
procedure discussed in [43]. The remaining tracks are then linked to the calorimeter clusters. Several tracks can be linked to a cluster, and their momenta are then
compared to the calibrated cluster energy. The excess of energy in the calorimeter
deposits compared to the tracks comes from neutral particles. If a track is connected
to multiple clusters, these links are ordered in terms of distance. Then a loop over
this list of links is executed, and as long as the total momenta of the deposits remains lower than the track momentum the links are kept. If the total calorimetric
energy remains smaller than the sum of the energy of the tracks after calibration, a
relaxed search for muons and fake tracks is performed. Any global muons still left in
the block, with a relative momentum uncertainty better than 25% are taken to be
PF muons. Hereafter, the tracks with the largest PT uncertainties are progressively
removed, until the total track momentum is smaller than the calorimetric energy, or
until all tracks with a PT uncertainty of more than 1 GeV have been removed from
the block.
The tracks that still remain become PF charged hadrons. The momentum and
energy of such a charged hadron is taken to be the momentum measured by the
tracker, and the energy deduced from this momentum with the asumption of the
hadron being a charged pion. If the calorimeter and tracker measurements are compatible within uncertainties, the momenta of these charged hadrons are redefined
with a fit between the calorimeter and tracker measurements. If the calibrated
energy of the calorimeter clusters closest to a track are larger than the tracker momentum with a difference greater than the calorimeter resolution, we get additional
PF photons and possibly PF neutral hadrons. If the excess is larger than the total
ECAL energy, this ECAL energy is taken to due to a PF photon while the rest of
the calorimeter energy gives rise to a PF neutral hadron, and if not we only get a
PF photon. Any remaining ECAL and HCAL clusters, not linked to any track are
now assumed to be PF photons and PF neutral hadrons respectively.
After all the particles are reconstructed, the hadrons are clustered into jets by using
the anti-kT algorithm with a cone size of ∆R = 0.4. We will not go into details
about how this exactly works here, as it would lead us too far, but more details can
be found in [44].
46
Chapter 5
A supersymmetric solution?
A particularly attractive extension of the SM is Supersymmetry, or SUSY for short.
SUSY models were originally developed as a tool for making a Grand Unified Theory
(GUT), in which all forces would be described by a single gauge group, because they
have essential property that the coupling constants of the SM gauge groups unify
to a single value at an energy scale around 1016 GeV. Another reason for exploring
SUSY in the past was that it provides a way of getting rid of a vacuum energy,
which is unfortunately known to exist nowadays. Nowadays SUSY is mainly famous
for providing a solution for the Hierarchy problem, all the more contemporary after
the Higgs’ discovery, and providing an excellent dark matter candidate. In order
to provide what some people believe to be a ”natural” solution to the hierarchy
problem, many new particles present in SUSY models are thought to have masses
energetically accessible by the Large Hadron Collider (LHC). Filling up some of
the holes in the SM, and being thought to reside within our reach, SUSY searches
compose one of the primary interests of the current particle physics scene at the LHC.
Below we will briefly describe what Supersymmetry means, and discuss some of its
basic properties and implications without going into field theoretical details. We
will elucidate how SUSY can solve the hierarchy problem and provide a dark matter
candidate. Some details on the phenomenology of the Minimal supersymmetric
Standard Model (MSSM) will be presented, largely based on the last few chapters
of [28]. We will not discuss the many other models such as the Next to Minimal
supersymmetric Standard Model (NMSSM), as this would lead us too far astray
from the actual thesis subject.
5.1
The principle of Supersymmetry
Under a set of very general assumptions the Coleman-Mandula theorem states that
the Lie groups leaving the Lagrangian of a quantum field theory describing particle physics invariant, must be direct products of the Poincaré group (of which the
Lorentz group is a subset) and an internal symmetry group (such as the SM gauge
groups). So the internal and space-time transformations commute and can not mix.
In other words particle types must remain unchanged by the Poincaré transformations while the internal transformation acts only on the particle type. This was
47
clearly the case in the SM in which the gauge groups we discussed clearly didn’t
affect particle momenta or spins, only acting on the particle flavor or color.
Consider having a transformation morphing a bosonic state into a fermionic one
and vice versa, which is a symmetry of the Lagrangian:
Q |bosoni = |f ermioni
(5.1)
Q |f ermioni = |bosoni .
(5.2)
Such a symmetry is called a Supersymmetry. Bringing to mind the Lorentz transformation properties of bosons and fermions discussed in the chapter on the SM,
it is clear that the operator Q must be a spinor. Spinors being inherently complex
objects, one can conclude that Q† must also leave our Lagriangian unchanged if
Q does so. Unlike internal transformation operators, Q and Q† are not Lorentz
scalars and transform as spinors under the Lorentz group, so we are clearly dealing
with a space-time symmetry. The previously discussed Coleman-Mandula theorem
deals with Lie groups, whose structure is determined by the commutation of their
generators. It can be shown that since the structure of the Supersymmetry transformation is determined by the anticommutation relations of the spinor generators,
the Coleman-Mandula theorem can not be applied here. [46] Nonetheless a very
similar, albeit more general, theorem was later formulated by Haag, Lopuszanski
and Sohnius showing that the possible supersymmetries of particle transformations
are also highly restricted. [47] The supersymmetric operators are shown to satisfy
(ignoring the spinor indices for simplicity):
{Q, Q† } = P µ
†
(5.4)
µ
†
(5.5)
{Q, Q} = {Q , Q } = 0
µ
(5.3)
†
[P , Q] = [P , Q ] = 0
where P µ is the four-momentum operator, the generator of space-time translations.
In a supersymmetric theory, the particles will be placed in supermultiplets containing a boson and a fermion. A member of supermultiplet must be given by a linear
combination of Q and Q† acting on the other member. From special relativity we
know Pµ P µ = m2 with m the mass of a certain particle. Since P µ commutes with
both Q and Q† this then implies that particles in the same supermultiplet must have
equal masses! Due to the Coleman-Mandula theorem the supersymmetric generators must also commute with the generators of the gauge groups. This implies that
particles in supermultiplets must transform as the same representation of the gauge
group. In other they feel exactly the same interactions.
Another important property of supermultiplets is that they contain an equal number
of fermionic- and bosonic degrees of freedom. To see this first consider the operator (−1)2s where s is the spin of a particle. Due to the spin-statistics theorem
fermions always have half-integer spin while bosons have integer spin. This means
48
that bosonic states correspond to the eigenvalue 1 of the operators (−1)2s while
fermionic states have eigenvalue −1. Since Q and its hermitian conjugate change
bosons into fermions it then becomes obvious that (−1)2s must anticommute with
them. Now consider a complete set of states |pi, all of which have four momentum P µ . The total number of bosonic degrees of freedom minus the number of
fermionic ones is then given by taking the trace of (−1)2s over such states. Putting
the momentum operator in this trace we then find:
(nB − nF )P µ = P µ T r[(−1)2s ] =
X
hp|(−1)2s P µ |pi
(5.6)
p
via equation 5.3 we find:
=
X
hp|(−1)2s {Q, Q† }|pi
(5.7)
p
using the completeness relation
=
X
P
k
|kihk| = 1 this becomes:
hp|(−1)2s QQ† |pi +
p
XX
hp|(−1)2s Q† |kihk|Q|pi
p
=
(5.8)
k
X
X
hp|(−1)2s QQ† |pi +
hk|Q(−1)2s Q† |ki
p
(5.9)
k
finally employing the anticommutation of (−1)2s we find:
=
X
X
hp|(−1)2s QQ† |pi −
hk|(−1)2s QQ† |ki = 0
p
(5.10)
k
So we have found:
nF = nB
(5.11)
Now that we discussed some basic properties of supermultiplets we will shed light on
the different possible multiplets that can be devised consistent with these properties.
5.1.1
chiral/matter supermultiplets
In order to describe the first kind of supermultiplets we first have to introduce
the concept of a Weyl spinor. Consider the following representation of the gamma
matrices defined in equation 2.5, known as the ”chiral” representation (written in
2 × 2 block form):
0 1
0
σi
0
i
γ =
γ =
(5.12)
1 0
−σ i 0
49
considering equation 2.4 we then have for respectively the boost and rotation generators of the Lorentz group:
i 0 i
−i σ i
0
0i
S = [γ , γ ] =
(5.13)
0 −σ i
4
2
i i j
1 ijk σ k 0
0i
S = [γ , γ ] = (5.14)
0 σk
4
2
The block diagonal form of the equations above clearly suggests that the Dirac
representation of the Lorentz group is in fact reducible. We can thus decompose a
Dirac spinor into two parts transforming under the reduced representations:
ξ
ψ= L
(5.15)
ξR
The two component objects ξL and ξR are called Weyl spinors, and transform as
different representations of the Lorentz group. In terms of the Weyl spinors we can
write the Dirac equation as:
ξL
−m
i(∂0 + σ · ∇)
µ
(iγ ∂µ − m)ψ =
=0
(5.16)
i(∂0 − σ · ∇)
−m
ξR
So we see that the mass of a fermion mixes the two different representations of the
Lorentz group ξL and ξR . It is now clear that these two component Weyl spinors
are the part of the spinor projected out by the projection operators of equation 2.36
since in our representation:
−1 0
(5.17)
γ5 =
0 1
so:
1 − γ5
ξ
ξL
ψL = PL ψ =
(5.18)
= L
0
ξ
2
R
1 + γ5
ξL
0
ψR = PR ψ =
=
(5.19)
ξR
ξR
2
And so the two component Weyl spinors can be associated with left- and righthanded fermion fields [5]. With the newly defined Weyl spinors, which have clearly
defined gauge transformations we can now make a supermultiplet by adding a complex scalar which also has two degrees of freedom. This combination is called a
chiral- or matter supermultiplet. The scalar partners of the fermions are usually
named after their fermionic partner, with ”s” placed in front of the name. And so
they are known as ”sfermions” [28], [48].
50
5.1.2
gauge/vector supermultiplets
Before electroweak symmetry breaking vector bosons are massless, meaning they
contain two helicity (or spin) degrees of freedom in their fields. As such they can
also be combined into a supermultiplet with the Weyl fermions constructed in the
previous section. The left- and right-handed versions of the Weyl fermions partnered
with gauge bosons can not have different gauge transformations since the gauge
bosons transform as the adjoint representation of their gauge group and adjoint
representations have to be their own conjugate. The kind of supermultiplet under
consideration here is called a gauge- or vector supermultiplet. The fermionic partners
of the gauge bosons conventionally carry the same name as the gauge boson, with
”ino” added in the back, so they are known as ”gauginos” [28], [48].
5.1.3
other supermultiplets
Other combinations of fields can be made, satisfying the conditions of having an
equal number of bosonic and fermionic degrees of freedom, but they can always be
shown to be reducible, or to contain non-renormalizable interactions. Supersymmetric theories with more than one SUSY generator Q can also be contrived, but they
can be shown to contain no parity violation, nor chiral fermions in 4D quantum field
theories, making them useless from a phenomenological point of view.
5.2
Minimal Supersymmetric Standard Model Zoo
From the previous section one can conclude that if we want to make the SM supersymmetric, each of the SM particles has to be placed in either a chiral- or a
gauge supermultiplet, and as such has to have a partner with a spin different by
1/2 with the same gauge transformation. The Minimal Supersymmetric Standard
Model (MSSM) is constructed by making the SM supersymmetric, while adding as
few new particles as possible. We will see that new particles will have to be introduced in order to place all SM particles into supermultiplets, and in doing so we
will construct the particle spectrum of the MSSM. New SUSY particles, collectively
called sparticles, will usually be denoted by the same symbol as their SM partners,
but topped with a tilde.
Since left- and right-handed chiral fermions transform differently under the electroweak gauge transformations it is indisputable that they will have to be placed
in chiral supermultiplets, and can not be placed in vector supermultiplets. Every
fermion then needs a complex scalar partner to complete the multiplets, and for this
new particles have to be introduced which we will call ”sfermions”, short for scalar
fermion, or even squarks and sleptons. Note that each SM fermion will have two
scalar partners, one for each of its chirality components. These sfermions, while also
called left and right- sfermions are scalar and transform as trivial representations of
the Lorentz group and as such have no chirality components (or handedness) themselves!
51
Figure 5.1: Triangle diagram, leading to a chiral gauge anomaly [50].
The vector bosons of the SM will have to be placed in vector supermultiplets, and
they need fermionic partners for which the left- and right-handed components transform as the same representation under every gauge group. Such fermions are not
present in the SM and we have to introduce new spin 1/2 gauginos.
Finally, the components of the Higgs doublet have to be placed in supermultiplets.
The doublet components are scalars meaning they have to be a chiral-multiplet.
The additional fermions that have to be introduced as partners get us into trouble
because they will spoil the cancellation of the chiral anomaly present in the SM.
The presence of a gauge anomaly, as for instance induced by the type of triangle
diagram shown in figure 5.1, breaks the gauge symmetry of a theory, leading to nonrenormalizability. For this reason all diagrams of this type have to cancel each other.
This is the case in the SM, and can also be seen as the reason why the hydrogen
atom is neutral because this cancellation requirement relates the hypercharges of
the quarks and leptons. To cancel the new contribution from the fermionic partners
of the Higgs doublet, another scalar SU (2) doublet, with its own fermionic partners
has to be introduced. Each SU(2) doublet leads to two supermultiplets, so we end up
with four supermultiplets, and thus four new weyl fermions we call Higgsinos. In our
description of the SM we noted that we could ”gauge away” multiple components
of the SU (2) doublet and were left with one physical field. We can only choose our
gauge once, so once we choose the unitary gauge for the SM Higgs doublet we will
still be left with four degrees of freedom, or four physical fields, in the second doublet. Long story short: we will have 4 Higgsinos and 5 Higgs bosons in the MSSM.
Two of the Higgsinos can be shown to be neutral, and the two others are charged.
Two of the scalar bosons are charged, and three neutral. One can demonstrate that ,
if not for the anomaly cancellation, two scalar doublets are needed in SUSY models
anyway to give mass to up- and down type quarks. [28] All MSSM particles are
summarized in figure 5.2.
52
Figure 5.2: SM particles, and their supersymmetric partners in the MSSM. Figure
taken from [49].
53
5.3
Solving the Hierarchy problem and breaking SUSY
Adding a fermion loop to a Feynman diagram leads to an extra minus sign in the
amplitude under consideration due to the anticommutation of fermion fields. For
bosons this is not the case, so a Feynman diagram with a fermionic loop will have
the opposite sign of the same Feynman diagram containing a bosonic loop instead.
In the last section we constructed all the supermultiplets present in the MSSM, introducing a new bosonic partner for every SM fermion, and a new fermionic partner
for every SM boson. Another thing that was demonstrated is that particles in the
same supermultiplet must have equal masses. From these considerations, it then
becomes clear that SUSY will elegantly solve the hierarchy problems as every divergent self energy correction to the Higgs mass will be balanced by an equal and
opposite contribution, since every fermionic self energy contribution to the Higgs
mass now gets canceled by an equally large contribution from a bosonic loop and
vice versa!
Introducing a plethora of new, so far unobserved, particles exactly as massive as
the SM particle we all know and love, unfortunately does not work. If these light
SUSY partners existed we would have seen them a long time ago! The solution
to this apparent contradiction is to break SUSY, allowing the SUSY particles to
have different masses from their SM partners. When breaking SUSY, one should
not want to reintroduce the quadratically divergent corrections to the Higgs mass.
This is achieved by what are called ”soft SUSY breaking terms” which leave the
relationships between the dimensionless coupling constants of the SUSY partners,
such as their Yukawa couplings gf intact. The self energy corrections to the Higgs
mass which quadratically depend on the cutoff scale will still cancel in soft SUSY
breaking models, and we end up with logarithmic self energy corrections to the Higgs
mass: [24]
λ
Λ
2
2
δmh ∝ msof t
ln
(5.20)
16π 2 msof t
where msof t are the mass scales of the soft SUSY breaking terms and λ is a representation of all the coupling constants present in the equation, and where we omitted
contributions from higher order loops. The m2sof t largely determine the new masses
of the SUSY partners, so making them too large could lead to a new, albeit less
severe hierarchy problem. For this reason it is believed that many particles in the
MSSM are withing reach of the LHC.
Ideally we would like to get soft SUSY breaking terms in our model by a mechanism
based onN
spontaneous symmetry breaking, making SUSY a hidden symmetry like the
SU (2)L U (1)Y gauge symmetry in the SM. Unfortunately, as of this moment there
is no consensus on how exactly to incorporate this into a consistent model. Multiple
ideas, all extending the MSSM by introducing new particles and interactions, usually at very high mass scales, are being explored. The simplest solution is to just
introduce soft SUSY breaking terms by hand. For the MSSM, soft SUSY breaking
54
Figure 5.3: Diagram showing a possible proton decay through a virtual strange
squark in the absence of R-parity conservation, figure taken from [28].
terms can appear as gaugino mass terms, mass terms and bilinear couplings of the
Higgs bosons, trilinear couplings between sfermions and Higgs bosons and sfermion
mass terms. Considering the discourse of the section on supermultiplets, it is easily
seen that any of these terms will be gauge invariant.
5.4
R-parity conservation and dark matter
Even after SUSY breaking has been introduced, the MSSM has one disturbing property left, namely that it can violate lepton and baryon number which has so far never
been observed in nature. A direct consequence of baryon number violation would
for instance be that protons would decay. An example of proton decay diagram,
possible in our current incarnation of the MSSM, is shown in figure 5.3. To solve
this issue, a new multiplicative quantum number called R-parity, defined as:
PR = (−1)3(B−L)+2s
(5.21)
is taken to be conserved. In this equation B and L indicate the lepton- and baryon
number of a particle whereas s indicates its spin. Supersymmetric particles will
always have R-parity -1, whereas an ordinary SM particle has R-parity 1. This has
two significant conclusions, first of all SUSY particles will always have to be created
in pairs when starting from SM particles, and the lightest supersymmetric particle
(LSP) will be stable since its decay would violate R-parity conservation. In most
SUSY models, the LSP is a weakly interacting neutral particle (the neutralino as
described in the next section), and conserving R-parity it has to be stable. Protecting
lepton- and baryon number has thus provided us with an excellent dark matter
candidate!
55
5.5
5.5.1
MSSM phenomenology
mass eigenstates
The new gauge eigenstates, introduced to form the supermultiplets as discussed
earlier, will mix into several mass eigenstates, which correspond to the physical
particles that should be observed if the theory is valid. The states that will mix
after electroweak symmetry breaking are those with equal charges and color, and
the resulting eigenstates are obtained by diagonalizing the mass terms of the supersymmetric Lagrangian.
The charged electroweak gauginos and the charged higgsinos mix into two ”charginos”,
f
±
±
denoted by χf
1 and χ2 , where a lower index n means a lower mass. The neutral
electroweak gauginos, and the neutral higgsinos will be interwoven to form the four
f0 , χ
f0 , χ
f0 and χ
f0 . Charginos and neutralinos are collectively known as
neutralinos χ
4
1
2
3
electroweakinos, and are of particular interest here since the primary focus of this
thesis was performing a background estimation to be used in the searches for these
f0 ) is usually the LSP,
particles. In conventional models the lighest neutralino (χ
1
also making it very interesting from an astronomical and cosmological point of view.
The neutralino is weakly interacting, neutral, stable, and relatively heavy considering contemporary mass limits like [51], making it an excellent dark matter candidate.
The gluinos are fermions forming a color octet, or in other words transform as
an adjoint representation under SU (3)c . This is a property no other fermion shares,
with the consequence that they can not mix with any other MSSM particle. So one
can conclude that gluinos have to be mass and gauge eigenstates simultaneously.
Any scalar with the same electric charge, R-parity and color can mix. Finding
the mass eigenstates of the squarks and sleptons then comes down to diagonalizing
6 × 6 matrices for the up- and down-type squarks and the charged sleptons, since
there are scalar partners of both the left- and right-handed sfermions. For the sneutrino mass eigenstates we have to diagonalize 3 × 3 matrices, assuming there are no
right-handed SM neutrinos. Note however that if we want to write neutrino masses
in the SM via Yukawa couplings, we would need right-handed neutrinos, and SUSY
partners for it in the MSSM. To avoid flavor changing, and CP violating effects in
the MSSM, the soft SUSY breaking mass terms for the sfermions are usually assumed to be flavor blind. This leads to very small mixing angles in the first- and
second sfermion generations1 , since their Yukawa couplings are almost negligible
compared to the size of the extra mass terms usually introduced. Due to the third
generation’s much larger Yukawa couplings, the resulting masses and mixings can
be quite different from those of the first- and second generation sfermions.
1
The scalar partner of the first- and second generation quarks and leptons, as defined in Chapter
2.
56
5.5.2
MSSM sparticle decays
In this section we shortly describe the decay options each of the SUSY particles
have, assuming R-parity is conserved, and that the lightest neutralino is the LSP.
In these circumstances every SUSY decay chain should eventually end up with an
LSP and a multitude of light SM particles in the final state. The decay chains that
are possible, and those that will dominate heavily depend on the mass hierarchy of
the particular SUSY model in question. Many decays described below might not
be kinematically allowed, meaning that instead of two body decays, decays via a
virtual sparticle have to be considered.
• electroweakino decays:
Being an admixture of the electroweak gauginos and the Higgsinos, the electroweakinos will inherit their couplings. Through these couplings they can
decay into a lepton slepton pair or a quark squark pair, or a lighter electroweakino together with a Z, W or Higgs. In most models the sleptons are
significantly lighter than the squarks, making the lepton + slepton decays
preferable to the quark + squark decays. When none of the above decay channels are kinematically open, decays through virtual sleptons or gauge bosons
will occur.
• slepton decays:
Sleptons will almost surely have two body decays to a lepton and the LSP
available, and additionally to leptons and other electroweakinos depending
on the mass hierarchy of the model. Depending on the gaugino admixture
of the different electroweakinos, left- and right-sleptons might prefer different
decay paths. A left slepton has couplings to the charged gauginos while a
right-slepton only couples to the bino and Higgsino components of the electroweakinos. As such a right-slepton might for instance prefer to decay to the
electroweakino with the largest bino component which is usually the LSP.
• squark decays:
If allowed, two body decays of squarks to a quark and a gluino should dominate
since such a decay goes through QCD vertices. If this channel is kinematically
closed, the squarks will decay to electroweakinos and quarks. Left- and righthanded squarks might pick different decay paths, as in the case of slepton
decays, depending on the mixing of the electroweakinos. The stop and sbottom
squarks have significant Yukawa coupling to the Higgsino components of the
electroweakinos, which might also influence the decay paths these squarks
favor.
• gluino decays:
Gluinos only have QCD couplings, meaning they can only decay via a quark
and a real or virtual squark. The stop and sbottom are usually lighter than
the gluinos, so even if two body decays to the first- and second generation
squarks are forbidden, the third generation channel might be open.
57
5.5.3
Searching for SUSY at hadron colliders
In an hadron collider like the LHC, sparticles can be produced by means of the
following electroweak interactions:
qq
→
f
+f
− f
+f
−
0 f
0
χf
n χm , χn χm , `n `n
(5.22)
ud
→
+f
+
0 g
χf
n χm , `nL νe`
(5.23)
du
→
−f
− e∗
0 g
χf
n χm , `nL ν`
(5.24)
and the following QCD interactions:
gg
→
∗
gege, qf
m qen
(5.25)
qq
→
∗
gege, qf
m qen
(5.26)
geqf
m
(5.27)
qf
m qen ,
(5.28)
gg
qq
→
→
where ∗ indicates an antiparticle. Due to the proton-proton nature of its collisions,
the LHC is expected to mainly produce sparticles through gluon-gluon and quarkgluon fusion, while the other interactions shown above have smaller contributions.
A plot showing the expected production cross section for several types of sparticle
pairs in proton-proton collisions of 8 TeV and 13 TeV is shown in figure 5.4. First
of all it is clear that the 13 TeV production cross sections for all sparticle pairs are
significantly higher than those at 8 TeV, making the current LHC run a quintessential opportunity for the discovery of SUSY. Another thing to note, is that the cross
sections to produce strongly coupling sparticles are much higher than those for sparticles only feeling electroweak interactions. This makes gluinos and squarks ideal
candidates for initial direct production searches, but as these searches continue to
yield null results, their mass limits get pushed higher, making their expected production cross sections smaller and smaller. So if these strongly interacting particles are
very heavy, or even completely out of the LHC’s reach, it might prove advantageous
to search for electroweak production of sparticles.
When SUSY particles are produced, they will decay into a stable weakly interacting LSP in models where R-parity is conserved. This LSP will almost invariable
elude detection, just like neutrinos do in the SM, leading to expectation of a significant amount of MET2 in every event where sparticles are produced. While every
SUSY event should have MET, a profusion of signal topologies for SUSY events can
be thought of, depending on the model, mass hierarchy,... SUSY searches at the
2
Mising transverse momentum, as defined in the previous chapter.
58
Figure 5.4: Plot showing the expected sparticle pair production cross section at a
proton-proton collider of 8 TeV, and 13-14 TeV. [52].
LHC therefore have to cover a multitude of final states, each being sensitive to a
certain SUSY parameter space or model. In order to efficiently search for SUSY or
to exclude it, we have to look everywhere! This thesis is primarily concerned with
electroweakino pair production leading to three leptons and MET, a signal topology
which will be elucidated in much more detail over the next chapters.
59
Chapter 6
Software techniques
When searching for new physics at the LHC, one needs to know what to expect to
find assuming the SM is completely valid. Not only that, the analysis needs to be
optimized, one might for example need to determine the ideal kinematic cuts that
have to be applied to get the maximum signal over background1 ratio, etc. For
these purposes, Monte Carlo (MC) simulations are employed to simulate all the SM
processes that contribute to the background, and sometimes to simulate the signal
processes of interest to interpret the results found in data. Once a physics process
is simulated, it is passed to a simulation of the CMS detector, leading to signals in
this simulated detector which enable one to compare the simulated results to the
measured data in a more effective way.
6.1
Monte Carlo event generation
The first part of the simulation consists of simulating what is known as the hard
process, which is the primary interaction of interest. First of all the colliding particles
are chosen, which in the case of the LHC means two protons. For every proton
collision the interacting partons are then singled out, and their momenta are sampled
from the parton distribution functions of the protons, usually determined from data.
Afterwards, all the Feynman diagrams leading from the initial state of interacting
partons to the process one wants to simulate are calculated by a program known
as a matrix element generator. Depending on the program used this can be done
at the leading order (LO) in perturbation theory or up to next to leading order
(NLO) for increased precision. The matrix element generators used in this thesis
are the LO matrix element generator MadGraph [53] and the NLO generators MC@NLO
[54] and Powheg [55]. After the hard process has been simulated, the simulation is
transferred to a program capable of simulating the higher order effects that go into
the parton showering and hadronization that follow the primary interactions. The
1
The word ”signal” usually refers to the events of interest, which for a lot of analyses means
events originating from some new physics process, though they can also refer to a certain SM process
one wants to do a measurement for. The background on the other hand are the SM processes which
try to make our life miserable by leading to the same final state in the detector as the one we expect
for the signal.
60
Figure 6.1: Schematic representation of the different steps in the simulation of a
proton-proton collision. [58]
initial and final state particles are simulated to radiate gluons, other partons, or
photons and later bind into colorless hadrons due to QCD’s confinement property. If
heavy hadrons are formed they may subsequently decay further, initiating a number
of decay chains, resulting in hadrons, jets and other particles. All the simulated
samples used in this thesis used the program Pythia [56] to carry out these steps.
For simulating the hadronization, peturbation theory can not be used, so Pythia
uses the Lund string model [57], in which the QCD potential between two quarks is
assumed to increase linearly with the distance between them. When two quarks that
were produced, move far enough away from each other, the potential between them
becomes so large, that a new quark-antiquark pair can be formed from the vacuum.
This new pair then screens the color charges of the original from each other. This
process is repeated until the quark energies reach low enough values, corresponding
to hadron masses, at which point hadrons are formed. Once the hadronization and
subsequent decays have been simulated, the simulation of the entire physics process,
and the resulting final state is complete. All the steps in an event simulation are
summarized in figure 6.1.
61
6.2
CMS detector simulation
In order to decently compare any simulated processes to the data taken by the CMS
detector, we need to pass the simulated events through a simulation of the CMS
detector. The samples used for analyses in the CMS collaboration are usually produced by using a full simulation of the CMS detector, based on the GEANT4 toolkit,
which simulates the passage of particles through matter by using MC techniques.
Once all detector signals induced by the simulated particles are calculated, they are
reconstructed by the same software and algorithms that reconstruct the actual data
collected by CMS. These reconstructed simulations can then be used to optimize an
analysis or to compare the SM predictions to the data.
62
Chapter 7
Search for electroweakinos using
a three lepton + MET signature
In general almost any search for physics beyond the SM carried out at the LHC
shares the following steps: Among the gargantuan amount of collisions, the interesting ones are selected by the trigger algorithms discussed in an earlier chapter.
After being reconstructed, the stored data is then statistically analyzed for possible
excesses compared to the background expected from SM processes. To make this
background manageable, it is often reduced by applying kinematic cuts, the optimal
ones usually determined using simulations. When no excess is observed, and all
data is compatible with the SM, limits can be set on the production cross sections of
new particles. These cross sections can be parameterized as a function of the mass
for some models, providing the possibility for setting under limits on the mass of
hypothetical particles.
Most of the LHC’s searches for Supersymmetry are focused on models in which
the production rates are anticipated to be dominated by strongly coupling SUSY
particles, and are usually carried out in final states with excessive hadronic activity.
Reminiscing the sparticle pair production cross sections shown in figure 5.4, one
can conclude that models in which strongly coupling sparticles are within reach,
are indeed better targets for initial searches. But as the mass limits on the sparticles get pushed upwards, their production cross sections are quickly diminishing, in
large part due to the falling of the parton PDFs at very high momentum fractions,
needed to reach energies this high. So if no single strong production search finds any
trace of SUSY, it might become advantageous to start looking for the production
of electroweakinos, which might be far less massive, and are indeed predicted to be
so in many models. The current mass limits on strongly coupling SUSY particles
have at this point already been pushed far higher than those on electroweakinos,
making electroweak SUSY production a well motivated area for conducting a search
for new physics. It is easy to imagine scads of final states which can result from
electroweakino production, but the most interesting ones are those which only a
few, not too common, SM processes can mimic. In other words, we are primarily
interested in final states with low backgrounds from the SM, since these should be
63
by far the easiest channels for making a discovery if sparticles are veritably being
produced.
7.1
Signal model
Consider the pair production of a heavy neutralino and a chargino. Both sparticles
must eventually decay to the LSP, and will typically do so by decaying to leptons
ans sleptons, or by radiating electroweak gauge or Higgs bosons. Such decay paths
can lead to a final state containing multiple leptons, since sleptons can only decay
to the LSP by emitting a lepton, and the SM bosons involved in the decay might
also decay leptonically. Because the LSP, resulting from both electroweakino decay
chains, will almost invariably escape detection, the final state is also expected to
contain a significant amount of MET. This MET can even be further increased
by the potential presence of neutrinos in the signal. The final state topology that
forms the primary interest of this thesis, is that containing three leptons and MET.
The electroweakino pair production diagrams leading to this signal are depicted and
elucidated in figure 7.2.2. Three lepton events hold outstanding potential for the
discovery of new particles, because of the low SM backgrounds in this final state,
with only a few scarce SM processes able to furnish such events.
7.2
Backgrounds and discriminating variables
Although we expect small backgrounds when searching for a three lepton + MET
signal, this does not mean that there won’t be any SM processes mimicking our
signal at all. The SM processes that will mimic our signal can be separated into two
distinct categories. On the one hand there are events that produce three leptons at
the primary interaction vertex, labeled ”prompt leptons”. On the other hand, there
are processes in which one or more of the three leptons originate from a secondary
vertex (non-prompt), or are even faked by another object. Photons and jets can
can be wrongly reconstructed as leptons, and the contribution of such events is
usually reduced by applying additional quality requirements on the reconstructed
leptons. The SM processes contributing to the two background categories are nonexhaustively listed below.
7.2.1
Backgrounds with three prompt leptons
• WZ/Wγ ∗ :
SM production of a W boson together with a Z boson or a virtual photon
(denoted by γ ∗ ) can lead to three leptons and MET if both bosons decay
leptonically. In particular we expect an opposite sign same flavor (OSSF)
lepton pair from the Z or γ decay and and additional lepton and MET from the
W decay, where the MET is provided by the neutrino eluding our detection.
An important thing to note is that one or more of the leptons originating
from the WZ decay might be τ ’s, which decay before detection, leading to an
hadronic signal, or another light lepton and neutrino. The WZ background is
64
(a)
(b)
(c)
(d)
Figure 7.1: Pseudo Feynman diagrams depicting the production diagrams of interest
to SUSY searches using the three lepton + MET final state. In diagram (a), the
decay of the initial electroweakino pair is mediated by sleptons which can be real or
virtual depending on the mass hierarchy of the SUSY model under consideration.
Diagram (b) is similar to diagram (a), but the chargino decay is now mediated
by a sneutrino which might have a different mass from the sleptons, so the final
state kinematics can be distinctive from diagram (a). In diagram (c) and (d) the
electroweakinos decay to the LSP by emitting electroweak gauge bosons (c) and
an electroweak gauge boson and a Higgs boson (d). Which of the four diagrams
depicted above will give the dominant contribution is model- and mass hierarchy
dependent.
65
by far the most prominent background, especially in events where there is a
light lepton OSSF pair with an invariant mass close to that of the Z boson.
This background constitutes the main focus of the research done in this thesis.
• Zγ ∗ :
Consider a leptonically decaying Z boson, joined by a photon. This photon
can be virtual and thus decay, or can undergo a conversion into a lepton pair
when interacting with matter in the detector, leading to a final state with four
leptons. When the photon conversion or decay is asymmetric, in the sense that
one lepton carries away almost all of the photon’s momentum, the other low
momentum lepton might not be detected. So the result is that three leptons
will be reconstructed, and potentially MET furnished by mismeasurements,
such as the escaping lepton. The presence of a significant MET is especially
likely if jets are involved in the event.
• rare SM processes: tt + W/Z, WWW, ZZZ,...:
Many more rare processes with very small cross sections can be thought of
which can lead to three leptons and MET. A top quark pair (denoted tt or TT)
decays via the weak interaction, and almost always leads to two beauty quarks,
due to the small CKM couplings between the third and lower generations of
quarks. When these top decays happen semileptonically, or in other words
when the W emitted during the decay decays to a lepton and a neutrino, we end
up with two leptons and MET. A Feynman diagram showing the production
and semileptonic decay of such a top quark pair is shown in figure 7.2. The
third lepton can then be furnished by the presence of an additional gauge
boson at the original vertex, like a W or a Z. Other processes such as three W’s
decaying to leptons, or two leptonically decaying Z’s where one lepton is lost,
joined by a third Z decaying to neutrinos leading to MET, ... can be thought
of. The cross sections of all these processes are relatively small compared to
the dominant backgrounds, making their contribution rather small.
7.2.2
non promt or fake leptons
• tt:
As mentioned above, top quark pair production can lead to an event with two
leptons. Top quark pair being produced in the absence of any extra electroweak
gauge bosons are far more likely to occur, but they can not lead to three leptons
at the primary interaction vertex. One of the two beauty quarks, coming from
a decaying top, might however decay leptonically as well, leading to a third
lepton. This lepton will generally have a significantly larger impact parameter
(i.e. the distance between its origin and the primary interaction vertex) than
the prompt leptons and might be distinguished in this way as discussed in the
next chapter. The reason for these leptons originating relatively far from the
primary vertex is the longevity of beauty quarks which can again be related
to the very small CKM coupling between the third quark generation and the
others. Nonetheless many top quark pair events will pass all kinematic cuts
66
Figure 7.2: Feynman diagram of top quark pair production by gluon fusion, followed
by semileptonic decays of the top quaeks, leading to two beauty quarks, two leptons
and two neutrinos.
applied in multilepton + MET analyses, and tt is the runner-up in terms of
importance among the other backgrounds, even being as prominent as WZ in
certain parts of the three lepton phase space.
• tW:
A semileptonically decaying top quark produced together with a leptonically
decaying W boson leads to two prompt leptons. The third lepton can again be
supplied by a non prompt lepton from the beauty decay, or from a jet faking
it.
• Z/γ ∗ + jets:
A Z or virtual photon decay, or a photon conversion can lead to two leptons
being produced. A third lepton can be faked by a jet, which can also lead
to a significant MET by means of mismeasurement. An event in which a
leptonically decaying Z or off-shell photon is produced is usually referred to as
a Drell-Yan event, and this term will often be used in the rest of this thesis.
• rare SM processes: WW + jets, ...:
Many other processes with small cross sections leading to two leptons and
MET, and in which a third reconstructed lepton can be faked by jets or photons
can be thought of.
So now how will we search for our SUSY signal among all these backgrounds? As
mentioned in the beginning of the chapter we will hunt for an excess of events
compared to the SM background. To do this we need some variables to define the
phase space regions in which we will look for an excess. One of the most important
variables when searching for electroweakino pair production, leading to three leptons
and MET is the transverse momentum, defined as:
67
MT (1, 2) =
p
2PT (1)PT (2)(1 − cos ∆Φ(1, 2)),
(7.1)
where 1 and 2 indicate the two objects used for calculating MT , and PT and Φ are
respectively the transverse momentum and the azimuthal angle that were defined in
the chapter describing the CMS detector. The transverse mass of two particles can
be seen as the invariant mass1 of these particles in the ultrarelativistic limit2 , after
setting their longitudinal momentum to zero. This transverse mass is calculated using one lepton and the MET. So now what do we expect the MT distribution to look
like in the SM? The primary background to three lepton electroweakino searches is
WZ, and making the MT shape from the W decay’s lepton and the MET from its
neutrino, we can expect a distribution sharply peaked at the W mass. A simulation
of this MT shape in WZ events is shown in figure 7.3. The SUSY processes in figure
are expected to have different MT shapes, with only diagrams (c) and (d) expected
to have a peak at the W mass, but even here the extra MET of the LSP’s is expected
to give significant smearing of the MT shape. So MT is expected to be a powerful
discriminating tool to separate the SM backgrounds from our signal, with SUSY
events expected to show up at very high MT values3 .
The reason behind using the transverse mass instead an invariant mass when looking at W bosons or SUSY particles, is that it is impossible to check conservation
of momentum in the longitudinal direction at CMS. This limits us to only having
an estimate of the missing momentum in the transverse plane (MET), which is due
to all escaping particles and mismeasurements. Alas we have no clue whatsoever
about the longitudinal momentum components of the escaping particles, making it
simply impossible to reconstruct the invariant mass of a W boson, or that of any
other particle containing escaping particles in its decay chain. The conservation of
momentum in the transverse plane can be checked based on the assumption that the
longitudinal momentum components of the initial interacting partons are negligible.
The longitudinal momenta of these partons are however unknown, and hence no
way to check conservation of longitudinal momentum. In a lepton collider on the
other hand one could also check the longitudinal momentum conservation, because
the collision energies are fixed and known.
In all of the diagrams shown in figure 7.2.2, there will be an OSSF pair of leptons coming from the decay of the neutralino, and a third lepton coming from the
chargino decay. The MT will always be calculated using the lepton not being a part
of the OSSF pair and the MET. The invariant mass of the OSSF pair, branded Mll ,
will be used is a second important variable in searches for electroweak SUSY produc1
p
The invariant mass of a particle is defined as Pµ P µ = m, and is by definition invariant under
Lorentz transformations and equal to the particle’s rest mass. When a particle decays, the invariant
mass can be calculated using the energy and momentum of the decay products. Due to energy and
momentum conservation the invariant mass of a system of particles originating from a decay is
equal to the mass of the original particle
2
I.e. ignoring their masses because their contribution to the total energy is negligible at high
enough momenta.
3
Though they can also show up at lower values in some models.
68
√s = 13TeV
CMS Simulation
events /1GeV
events /2GeV
CMS Simulation
40
√s = 13TeV
100
50
20
0
0
50
100
150
200
0
0
50
100
MT(lepton + MET) (GeV)
150
200
Mll (GeV)
(a)
(b)
Figure 7.3: Pure simulation of the MT (a), and Mll shapes in WZ events. The MT
is made up of the W decay’s lepton and the MET while the Mll is calculated using
the leptons from the Z decay.
tion. In a WZ event we expect Mll to be sharply peaked at the Z boson’s mass since
it corresponds to the invariant mass of its decay products, as simulated in figure 7.3.
In the case of the SUSY diagrams shown in figure 7.2.2 only diagram (b) is expected
to have this property, making Mll a potentially powerful discrimination tool for this
search since the WZ background is expected to be significantly less rife outside of
the Z-mass window in Mll . If three leptons of the same flavor are detected, there
will be ambiguity as to which leptons for the OSSF pair. The two leptons , making
up an OSSF pair, with the invariant mass closest to that of the Z boson will be
taken to form the OSSF pair in these cases. This is done because most three lepton
+ MET events observed will come from WZ in which the OSSF pair comes from the
W boson. It is however possible to observe electroweakino production events, and
WZ events in which no OSSF pair is found at all. These events originate from the
short lifetime of τ leptons, which decay to electron, muons or hadrons before their
detection. So if there are leptonically decaying τ ’s present in an event, the possibility of having no OSSF pair arises. Which lepton is chosen for the MT calculation,
and which are used to make up the Mll , is usually determined from a simulation of
Z bosons decaying to τ leptons which in turn decay leptonically. The Mll peak of
the τ decay products is then determined, which is significantly below the Z mass
since the neutrinos from the τ decays escape. The lepton pair closest to this Mll in
terms of invariant mass is then chosen to make up the Mll and the other is used in
the MT . Events containing no OSSF pair will not be considered in the background
estimation developed in the next chapter, and thus are not of vital importance to
this thesis.
69
7.3
Run I electroweakino searches in the three lepton
final state
A search for direct production of electroweakinos and charginos leading to a three
lepton + MET final state was already conducted in the LHC’s Run I. Though we will
not go into exceedingly specific details, such as the exact object selection that was
used, we will shortly describe the general strategy of the search that was performed,
and the results attained in this analysis. While only brief outlines will be given on
the estimation of the subdominant backgrounds, a more detailed explanation of the
WZ background estimation that was performed for this search will be given. It is
important to describe this in detail, so one can understand the procedure and all the
sources of uncertainties. The central point of this thesis is to devise a new method
to do this, and improve upon the old result after all.
7.3.1
Search strategy
The events used in this search were those containing exactly three leptons, with
up to one reconstructed hadronically decaying τ 4 . Because we expect a significant
MET in SUSY events, a MET threshold of 50 GeV was applied, significantly reducing many of the backgrounds discussed above such as Zγ ∗ . The events were then
divided into several exclusive search regions based on their MT , Mll , MET, there
being an OSSF pair in the event, and the presence of an hadronic τ . The amount of
data events observed in every one of these search regions was then compared to the
background predictions in order to find an excess compared to what was expected.
The MET search regions were defined as: 50 GeV < MET < 100 GeV, 100 GeV <
MET < 150 GeV, 150 GeV < MET < 200 GeV, and MET > 200 GeV. The MT bins
used were: MT < 120 GeV , 120 GeV < MT < 160 GeV and MT > 160 GeV. For all
flavor and charge combinations of leptons, the same binning in MET and MT was
used. For events with an OSSF pair the Mll binning that was used was: Mll < 75
GeV, 75 GeV < Mll < 105 GeV (reffered to as onZ) and Mll > 105 GeV. Events
containing an hadronic τ and an OSSF pair were binned separately from those containing only light leptons. but used the same Mll binning. Events in which there
was no OSSF pair were organized into two Mll bins, namely Mll > 100 GeV and
Mll < 100 GeV. These events were further divided into three different categories,
being those without an hadronic τ , those with an hadronic τ and an opposite sign
light lepton pair and those with a same sign light lepton pair and an hadronic τ .
[62]
7.3.2
Estimation of the subdominant backgrounds
Except for the dominant WZ background the major backgrounds in this search are
those in which the third reconstructed lepton is either fake or non-prompt, like in
4
τ ’s are very complicated objects from an experimental point of view, since they can not be
directly detected like electrons or muons, owing to their short lifetime.
70
tt and Z + jets. The background yields from such events were measured from data
by using auxiliary data samples. In a QCD dijet enriched control sample, the probability of for non-prompt leptons to pass a tight relative isolation requirement was
measured. Relative isolation means the energy deposited in the detector in a cone
around the particle divided by the particle’s energy, which is typically small for
prompt-leptons and large for non-prompts as they are surrounded by other activity. All leptons entering a selection are typically required to pass a certain isolation
requirement. More details on isolation are provided in the next chapter. Once the
probability for non-prompts to pass a tight-isolation was known, the amount of three
lepton events in which one of the isolation requirements was inverted was measured.
The two results above were then combined to yield the total amount of background
events originating from non-prompt leptons.
The rare backgrounds leading to three prompt-leptons, such as WWW, WZZ, t
tW, etc, were estimated directly from MC simulations. An extra systematic uncertainty of 50% was assigned to all of their yields to account for the uncertainties in
the theoretical calculations of their NLO cross sections, potential mismodeling and
pileup effects. The overall contribution from these rare SM processes was small, so
no data-driven methods had to be developed to estimate them.
7.3.3
WZ background estimation by applying data-driven corrections to simulations
Because of the WZ background’s dominance in the three lepton + MET final state,
a precise prediction of its yields in every search region is of paramount importance to
the search effort. The Run I searches estimated this background from a simulation,
to which several data-driven corrections were applied to mitigate several sources of
systematic uncertainty. The sources of systematic errors that were corrected for are:
mismodeling of the hadronic recoil, the calibration of the lepton momentum/energy
scale and uncertainties on the event yield normalization due to an insufficient knowledge of the WZ cross section beyond NLO. These sources of uncertainties and the
corrections applied to mend them are individually discussed in more details below.
Hadronic recoil
If the detector effects and the underlying event (all other interactions in same the
proton-proton collision besides the hard interaction, not be confused with pilup) are
mismodeled, this could lead to differences between the MET, and as a consequence
the MT distributions in data and MC. The MET of a WZ event is determined from
the PT of the leptons, and the transverse hadronic recoil of the event, defined as:
X
uT = −MET −
PT (i)
(7.2)
i ∈ leptons
and represents the transverse momentum due all other particles, besides the leptons
that are present in the event. Note that we MET is assumed to be a vector in
this equation. While lepton PT ’s are usually assumed to be well modeled, though
71
as shown in the next section some corrections were still applied to this too, the
hadronic recoil is not as well described. The differences in the hadronic recoil between data and MC were studied in leptonically decaying Z events, after which the
resulting corrections were applied to WZ events. In Z events, the transverse recoil
can be subdivided into a component parallel to the Z bosons direction (u1 ) and a
component orthogonal to its direction (u2 ). One can intuitively expect that u1 is
closely related to the Z bosons PT , as it is essentially the energy balancing it to
conserve momentum since a Z event should have no real MET, owing its MET exclusively to mismeasurements. u2 should on the other hand be largely independent
of the Z PT , and is forecasted to be centered around zero as it is shaped by the
underlying event. To perform the actual measurement of the recoil differences in
data and MC, Z events were binned in terms of their PT . In each PT bin a double
Gaussian likelihood fit was applied to the u1 and u2 distributions, and the mean
value of the fitted double Gaussians were taken to be the recoil components while
their resolution was taken to be the width of these Gaussian fits. For every PT bin,
the recoil components and resolutions as extracted from the Gaussian fits were then
compared between data and MC, and these differences could then be applied as
correction factors depending on the boson PT . This comparison between the recoil
components and their resolution is shown in figure 7.4 for Z events. For WZ events
the correction factors that were applied depended the PT of the WZ system as determined from pure simulations (a W’s PT can not be directly measured since the
neutrino escapes detection) instead of that of the Z. To account for the differences in
the recoil components and their resolutions between WZ and Z events as a function
of the PT of respectively the WZ system or the Z, these distributions were compared
in simulations, shown in figure 7.5. Any discrepancies between the two distributions
were applied as systematic uncertainties.
Lepton energy scale corrections
The MT and Mll distributions directly depend on the energy and energy resolution
of the leptons. So if the reconstructed energy and its resolution are different in data
and MC, this would bring about differences in the MT and Mll distributions. From
the discussion on the PF algorithm in chapter 4, it becomes clear that the momentum measurement for muons is mainly due to the information from the tracker, while
that for electrons also significantly depends on the calorimeter deposits. As such,
the differences between data and MC are smaller when it comes to muon energies
and their resolution, than those for electrons. So energy scale corrections were only
applied for the electrons. For the muons on the other hand systematic uncertainties
of 0.2% and 0.6% on respectively the energy and energy resolution were applied on
the grounds that this was recommended by the muon Physics Object group at the
time the Run-I analysis was done.
The energy scale differences between data and MC for the electrons were determined by looking at the Mll shapes for a Z boson decaying to electrons in data
and MC. This was done in 6 different η bins for the leptons, since the material
distribution of the tracker is η dependent, and the barrel and endcap sections of the
72
(a)
(b)
(c)
(d)
Figure 7.4: Comparison of the recoil components and their resolution in data and
MC. Respectively u1 and its resolution are shown in (a) and (b) while (c) and (d)
show the same distributions for u2 . Every point on these plots corresponds to values
extracted from double Gaussian fits in a certain Z boson PT bin. [63]
73
(a)
(b)
(c)
(d)
Figure 7.5: =Comparison of the simulated recoil components and their resolution
in Z events as a function of the Z PT and in WZ events as a function of the WZ
system PT . Respectively u1 and its resolution are shown in (a) and (b) while (c)
and (d) show the same distributions for u2 . Every point on these plots corresponds
to values extracted from double Gaussian fits in a certain Z boson, or WZ system
PT bin. [63] [63]
74
(a)
(b)
Figure 7.6: Dielectron invariant mass shape in data and MC, before and after applying lepton energy scale corrections. [63]
ECAL will also respond differently to lepton energy deposits. The invariant mass
shape of the Z was measured separately for two electrons coming from every possible
combination of η bins, and to every measurement the mass shape simulated in MC,
convoluted with a Gaussian, was fit. From this fit one could extract the electron
energy scale factors by virtue of the following relationship between the Z invariant
mass in data and MC:
Mll M C
,
(7.3)
αe1 αe2
where αe1 and αe2 are the electron scale factors. The resolution discrepancies between data and MC are determined from the extra Gaussian smearing, with the extra
resolution being the quadratic sum of the electron resolution differences. Through
fitting all the different η combinations, the energy and resolution scale factors for
the electrons could be extracted. Both corrections were later applied to MC, in
order to make it match the uncorrected data. Figure 7.6 shows the comparison of
the Mll distributions for electrons in data and MC, before and after applying the
correction factors to MC. The statistical uncertainties on the fits were propagated
through the entire analysis. To account for possible missing effects, an additional
systematic uncertainty of half the correction factors being applied was postulated.
Mll data =
Event yield normalization
An error in the calculation of the WZ cross section, or the integrated luminosity
could lead to an over or underestimation of the background in every search region.
This problem was dealt with by using a WZ control region in data defined as: 50
GeV < MT < 120 GeV, 75 GeV < Mll < 105 GeV and 50 GeV < MET < 100 GeV.
In this control region, the ratio of the MC to data yields were calculated, and the
75
arising scale factor was used as a nuisance parameter on the WZ prediction. The
scale factors weren’t used as a direct factor for normalizing the WZ yields considering
the existence of some SUSY models predicting a signal in this control region.
WZ prediction uncertainties
There were several sources of systematic uncertainties in the WZ background prediction of the Run I analysis. First of all the statistical uncertainties of the recoil
fits, and the lepton energy scale fits were propagated through the analysis. A 50%
systematic uncertainty was added to the lepton energy scale corrections as earlier
mentioned, and for the recoil corrections, the differences between the recoil in WZ
and Z were taken into account as an additional systematic uncertainty. Another
source of systematic uncertainty was the fact that the top-quark pair background to
the Z signal had to be simulated in MC and subtracted from the data in order to
measure the Z’s Mll shape. The Z mass shape in simulation and the PT distribution
of WZ are prone to theoretical uncertainties which were estimated by comparing LO
and NLO predictions. And then finally, systematic uncertainties on the WZ normalization lead to uncertainties in every search region. In the end the uncertainties
on the WZ background prediction were about 30% in most search regions, slightly
lower in some and up to more than 40% in others. [62]
7.3.4
Results
After all the backgrounds were estimated and all systematic uncertainties were taken
into account, the data yields were compared to the background predictions in every
search region. No significant excess of data events was found in any of the search
regions, and the results were used to put mass limits on electroweakinos in simplified
SUSY models. We will not go into any details on this limit setting here, but the
limits can be found in [61]. All the events with three light leptons and an OSSF
pair, that were observed in data are shown in figure 7.7 on an MT versus Mll scatter
plot, and the comparison of the data and MC yields as a function of MET is shown
in figure 7.8. Similar plots can be found for events without an OSSF pair and or
hadronic taus, but are of less importance to this thesis. The fact that these searches
yielded null results does not mean that we should give up on trying to discover
electroweakino pair production at the LHC. The LHC’s energy has increased, which
can be seen to dramatically increase sparticle pair production cross sections from
figure 5.4. Aside from this, the luminosity has significantly increased, so much more
Run II data will soon be available than there ever was in Run I. The precision of the
WZ background estimation will prove to be one of the crucial pieces in the puzzle
of making a future discovery in this search, and the next chapter is dedicated to a
novel approach for its estimation.
76
Figure 7.7: MT versus Mll scatter plot, showing all events with three light lepton
and an OSSF pair. The purple lines mark the different Mll and MT search regions.
[61]
77
Figure 7.8: Comparison of the observed yields to those predicted for the backgrounds
as a function of MET in the different search region for events with three light leptons
and an OSSF pair. No significant data excess is observed in any of the search regions.
An important thing to take away from this figure is how large the WZ background
is compared to the others, especially in the onZ region. [61]
78
Chapter 8
A novel data-driven estimation
technique of the WZ
background
The prime factor determining the potential reach and sensitivity of an analysis is the
minuteness of the uncertainties on the background estimation that can be achieved.
Determining background yields with a precision as great as possible is therefore the
core business of searches for unknown particles in high energy particle physics experiments. In the last chapter we saw that WZ is the largest background by a significant
margin when looking for electroweakino pair production in three lepton + MET final states. So the total uncertainty on the expected WZ yields entering the search’s
signal selection will be an absolutely crucial and deciding factor for the final grasp
of this analysis. The previous electroweakino searches performed at CMS, estimated
the WZ background by relying on a simulation, to which they applied data driven
corrections as discussed in the previous chapter. Both the systematic and statistical
uncertainties in some of the bins ended up being quite large. To decrease the uncertainty, a new method will be investigated here, attempting to measure the WZ
background directly from data by trying to measure the W transverse mass shape of
WZ in another final state. It is evident that directly measuring the WZ background
itself by looking at three lepton + MET events in data is no option, as we would
be implicitly including any present signal from new particles into our background
estimation. Performing a measurement of the W transverse mass in another final
state topology is possible, but the presence of other particles in the event shall lead
to kinematic differences, influencing the W’s MT . In order to effectively do a measurement from which we can extract WZ’s MT shape we will have to find a way to
get rid of these differences. This chapter starts by motivating and explaining which
final state we intend to use to perform this data-driven WZ estimation. Hereafter
a proof of principle will follow, showing that we can in fact measure the MT from
the W in WZ with another process, and finally the background estimation is performed by using 2.26 fb1 of 13 TeV data collected by CMS in 2015. In this chapter,
all the object and event selection, used at every step of the analysis is extensively
documented.
79
8.1
Wγ as a proxy to WZ
In order to have a precise estimate of the WZ background yields as a function of
the MT of the lepton and the MET originating from the W decay, we intend to
measure the shape of the MT distribution in another, similar process. Measuring
the W’s MT in data, even in another final state, caters the advantage that all poorly
understood and hard to simulate experimental detector effects will be included in
the measurement. This is expected to be especially important for the MET, which
depends on all mismeasurements and particles in the event. The process we intend
to use is SM Wγ production. In the case of a leptonically decaying W, this process
leads to one lepton, MET and a photon. We will attempt to use this photon as a
proxy to the Z boson, which decays to two leptons, present in WZ events. The first
reason for using Wγ is that it has extremely similar production diagrams to WZ at
the tree level. The tree level production diagrams of both processes are depicted
in figure 8.1. Both have similar t-channel and ISR production diagrams, which can
be anticipated to lead to very similar kinematics. If we can in fact extract the WZ
MT shape from data, this might significantly reduce the uncertainty we can attain
on the background estimation. The Z’s mass, compared to the massless photon
will however induce some small kinematic differences between the two processes.
Alas, not all production diagrams are duplicates, as a photon can be radiated by
means of FSR by the lepton, something which is impossible in the case of a Z boson1 .
The presence of such FSR is expected to lead to some kinematic differences between the two processes. One can for instance expect that the photon and the
lepton will be quite close together when the photon is radiated by means of FSR,
leading to a small ∆R between the photon and the lepton. On the other hand this
angular separation can naively be assumed to be quite large in the case of all other
WZ and Wγ production diagrams. If not too many other objects are present in the
event, the photon or Z and the W boson should be emitted almost back to back,
meaning that the ∆φ between them is about π radians, due to the conservation of
momentum in the transverse plane. Another, very important difference is that an
FSR photon will carry away some of the lepton’s momentum, which will be directly
translated into a reduction of the MT in FSR Wγ production compared to WZ and
Wγ’s other production channels. In the next sections we will show comparisons between the kinematic distributions of Wγ and WZ as generated by an MC generator,
and then after reconstruction by a simulated CMS detector. For this we will first
elucidate the MC samples we used and the exact object selection that was employed
during reconstruction.
A further attractive property of Wγ is that its production rates are much higher
than those we can expect for WZ. Its cross section is in fact larger than WZ’s by
more than two orders of magnitude, as shown in figure 8.2. This indicates that we
can expect a large gain in statistics by using Wγ events. Due to the need to remove
1
It was explicitly checked that such events do not occur as explained in section 8.3
80
(a)
(b)
(c)
(d)
(e)
Figure 8.1: Tree level production diagrams of Wγ(left column) and WZ (right column). Diagrams (a) and (b) depict what is called initial state radiation (ISR)
production of both processes, in which the γ or Z is radiated by the W boson. Diagrams (c) and (d) go through a virtual quark propagator, and both the W and
the γ or Z are radiated by the quarks. These diagrams are often called t-channel
production. Note that the upper diagrams are essentially the same for Wγ and WZ,
are forecasted to lead to similar kinematics, though there will be differences induced
by the mass of the Z boson compared to the massless photon. Unlike the other
diagrams, the final diagram (e), corresponding to final state radiation (FSR) of a
photon, is not present in WZ.
81
Production Cross Section, σ [pb]
CMS Preliminary
April 2016
7 TeV CMS measurement (L ≤ 5.0 fb-1)
8 TeV CMS measurement (L ≤ 19.6 fb-1)
13 TeV CMS measurement (L ≤ 1.3 fb-1)
Theory prediction
CMS 95%CL limit
105
≥n jet(s)
104
≥n jet(s)
103
102
=n jet(s)
10
1
10-1
10-2
10-3
W
Z
Wγ
Zγ
WW WZ
All results at: http://cern.ch/go/pNj7
γ γ → EW EW EW
ZZ EW EW
WVγ Zγ γ Wγ γ
qqW qqZ WW Wγ jj ssWW Zγ jj
EW: W→lν , Z→ll, l=e,µ
tt
tt-ch
tW
ts-ch
ttγ
ttW
ttZ
ggH VBF VH ttH
qqH
Th. ∆σH in exp. ∆σ
Figure 8.2: Production cross section measurements of CMS for several SM processes
[64]. The measured cross section of Wγ can be seen to be more than two orders of
magnitude larger than that of WZ.
the FSR contribution from Wγ we will lose some of its events, but the gain in terms
of yields will be shown to be be significant regardless.
Another important reason for choosing a photon as a proxy to the Z boson is the
fact that both the leptons from the decaying Z and a photon are measured with
comparable precision by the CMS detector. These resolutions are important since
the MET resolution depends on the resolution of every object present in an event.
If one of the processes has a significantly worse MET resolution than the other, this
might inflate the high MT tails in this process as it would likely lead to more events
with a very high reconstrcted MET. The presence of the photon in Wγ compared
to two extra leptons in WZ, and potential differences in hadronic activity between
the processes, etc., might all influence the resolution of the MET.
So to summarize, the reasons for using Wγ to measure the WZ background’s MT
shape are:
• Both processes have very similar production diagrams, which we expect to lead
to similar kinematics.
82
process
WZ
Wγ
Wjets
sample accuracy
NLO
NLO
NLO
cross section (pb)
5.26
489
61526.7 ± 2312.7
cross section accuracy
NLO
NLO
NNLO
Table 8.1: Table showing the order in perturbation theory up to which the samples
corresponding to the processes of interest were simulated, their theoretical cross
sections and the order up to which this was calculated. The cross section uncertainty
was only available for the Wjets sample. [66]
• The photon in Wγ and the leptons from the Z decay in WZ are expected to
be reconstructed with a similar resolution, which means the MET resolution
of the processes is likely similar.
• Wγ has a production cross section that is more than two orders of magnitude
larger than that of WZ, so we can expect relatively large yields in data.
8.2
Simulation samples used for the proof of principle
The samples that were used to study the kinematic differences of the Wγ and WZ
processes, and to prove the principle that the MT shapes of the processes can be
made to match, were all produced with the MC@NLO matrix element generator. Every
sample used in this analysis used Pythia8 to handle the hadronization and parton
showering steps of the simulation. Some properties of the samples used for the
different processes, such as the sample cross sections, and the order up to which the
cross section was calculated are listed in table 8.2. The reason we include Wjets will
become clear in a later section, and it can be disregarded for now. An important
thing to mention is that there was a bug present in the WZ sample, as discussed in
[65]. To make sure no bugged events entered the analysis, only events in which M`` >
30 GeV could be used, a cut which fortunately only removes a tiny fraction of events
from the sample. Another WZ sample with NLO precision and without any known
bugs, generated with the Powheg matrix element generator and also interfaced with
Pythia8, was available. There are however two reasons why we opted for using the
earlier mentioned MC@NLO WZ sample. First of all it contains about 10 times more
events, giving us considerably larger statistics to work with, and it included up to
1 jet at the matrix element level. Since WZ produced in tandem with jets will also
enter the event selection of electroweakino searches using a three lepton final state,
it is advantageous to also include this part of the phase space in the data-driven
background estimate we develop. Far more details on the simulation samples used
in this analysis can be found in appendix A.
83
8.3
Comparison of the kinematic properties of WZ and
Wγ in pure simulation
Using the simulations for Wγ and WZ that were discussed in the previous section,
we can compare the ”true” kinematic distributions of Wγ and WZ. The kinematics
simulated by the MC generator are sometimes referred to as MC truth information
because they represent a pure simulation of a particular SM process without experimental effects such as mismeasurements, fake objects, reconstruction inefficiencies,
etc. So looking at the kinematics in a pure simulation should give us a clear image
of the physics that is going on. Several kinematic distributions of Wγ and WZ are
compared in figures 8.3 and 8.4. When kinematic information about the Z boson
is shown below, this actually means the sum of the two lepton vectors coming from
the Z’s decay. At the generator level we could have used the Z information directly,
but we opted for using its decay products since these are also what will be available
when using reconstructed information further on. When the MET is mentioned in
generator level plots, this actually means the sum of the PT ’s of all the neutrinos
in the event, since there are no mismeasuremensts that can cause extra MET in a
pure simulation.
In making these plots we only used events in which there was exactly one generated lepton with Pythia status code 1 coming from a decaying W and two such
leptons coming from a decaying Z, having opposite charges and the same flavor. A
generator particle’s status code being 1 means that it is part of the final state of the
simulation, and these are obviously the particles we are interested in since they are
the ones we will be able to detect. In a pure simulation there is no ambiguity like
there would be in data or a reconstructed simulation about which particle comes
from a Z or W decay etc., since the ”mothers” of all the particles are stored in the
simulation. So we can unambiguously determine which particles come from which
decay in the case at hand. In the case of the photon in Wγ, we are in principle interested in the photon generated at the matrix element level by MC@NLO, as opposed
to photons that are potentially generated in later stages of the simulation by Pythia
as final or initial state radiation. Because there was no obvious way of determining
which photon was exactly the matrix element photon, we used the photon with the
highest PT in the plots below. This is likely the photon of interest since the ”hard”
part of the event is generated by the matrix element generator, while Pythia takes
care of the ”soft” part in these simulations. The new particles introduced in the
parton showering and hadronization steps of a simulated event usually posses relatively low momenta compared to those already generated at the matrix element
level. In a later section we shall return to the subject of the photons generated in
the different steps of the simulation, in a different context.
When looking at the kinematic comparison of Wγ and WZ, there are a few important things to notice:
• The MT distributions do not look similar at all, a fact which can be attributed
to the FSR diagram as anticipated before. For WZ the MT distribution is
84
clearly peaked at the mass of the W boson (about 80 GeV) whereas Wγ has
a displaced second peak at lower MT values. One can intuitively see how the
FSR diagram can lead to this displaced peak as the lepton loses momentum
by radiating the photon, and the MT value is directly proportional to the PT
of the lepton. It is now clear that the FSR contribution in Wγ should be
removed if we hope to use this process to measure the MT shape of WZ.
• Other differences that can be traced back to the FSR can be seen in the
distributions of ∆φ, ∆η and ∆R between the lepton from the W decay and
the photon or the Z boson. In the case of Wγ all these distributions are clearly
peaked around zero which could have been expected since the FSR photons
tend to be close to the lepton radiating them. For WZ on the other hand, the
∆φ distribution is peaked, though not nearly as strongly as the peak at zero
in Wγ, at π which can be understood from conservation of momentum in the
transverse plane. This ∆φ peak at π also translates into ∆R being peaked
around the same value in WZ events.
• The Z boson has a substantially higher PT than the photon on average. The
photon PT distribution seems to decrease after a peak a bit above 10 GeV,
this is not a physical effect but a property of the simulation. Every Wγ event
is required to contain a photon with a PT above 10 GeV, generated at the
matrix element level. Any photon with a lower PT comes from the Pythia.
This means that the photon PT distribution is somewhat distorted in the very
low PT region.
The conclusion that can be drawn from all these plots is that the MT shapes of Wγ
and WZ clearly do not match out of the box, and making them match is the goal of
this thesis! Other kinematic distributions are however seen to be influenced by the
FSR which spoils the MT shape in Wγ as well. By applying cuts on them, such as
cuts on ∆φ and ∆R between the photon and the lepton, and PT cuts on the photon,
we might be able to reduce, or almost completely remove the contribution from FSR
in Wγ.
The reader might wonder at this point why we do not explicitly check how the
Wγ distributions look after removing the FSR instead of just attributing all these
kinematic differences to its presence. After all it is possible to determine the mother
of a particle at the generator level, so one could expect to be able to distinguish the
different diagrams shown in figure 8.1 by asking for the photon’s mother. This way
one could assign an event to the FSR diagram if the photon’s mother is a lepton.
Unfortunately this does not work and the diagrams are impossible to disentangle
because the generator stores many FSR events as a W directly decaying to a photon
a lepton and a neutrino, a coupling non existent in the SM. This makes it that
the photon’s mother is in fact stored as a W boson in many FSR events. Whether
this is a property of the MC generator itself, or if the reason behind this is actual
interference between the Feynman diagrams2 has not been studied. Either way the
2
The different Feynman diagrams contributing to a process are summated, and this sum is then
85
effect remains the same for our purposes, namely that we can not separate the FSR
events from the others by using generator information.
8.4
Object selection
In the previous section we compared the MC truth kinematics of WZ and W γ, but
we actually need to know how these distributions compare after reconstruction by
the detector. If want to know whether the MT shape which we will later measure in
an lepton + photon + MET final state will have the same shape as the one we expect
from WZ, we need to know how the Wγ and WZ MT shapes will compare after all
reconstruction effects are accounted for. Before we can show any reconstructed plots
we need to decide on the selection criteria for selecting different objects. Below we list
all requirements for the reconstruction of all object types used in this analysis. Before
we do this, we will shortly elucidate the concepts of relative isolation, miniisolation
and multiisolation, the latter two of which are used in the selection of muons and
electrons.
8.4.1
Isolation as a background reduction tool
The leptons that are of interest to the analysis at hand come from the hard part of
the interaction, and are in principle not expected overlap with other activity possible
present in the event, such as for instance hadronic jets. We do not wish to select
any fake leptons coming from jets, or non-prompt leptons coming from heavy flavor
meson decays, etc. These backgrounds can be significantly reduced by requiring
leptons to be isolated from other energy deposits or tracks in the detector. The
conventional way of ensuring this employs a variable called relative isolation. This
variable is calculated for every lepton by summing the absolute values of all the PT ’s
measured in the calorimeters and the trackers not belonging to the lepton itself, in
a cone around the lepton. The size of this cone is usually taken to be ∆R < 0.3.
The resulting scalar PT sum is then divided by the PT of the lepton itself to yield
the relative isolation. If this relative isolation variable is large, it means that there
is a lot of activity from other particles around the lepton, meaning it is likely fake,
so upper limits on the relative isolation are then usually applied to reduce the fake
and non-prompt backgrounds.
Accidental overlap between a lepton from the hard interaction and other activity from the event, or from pileup is naturally possible, so isolation cuts will not be
perfectly efficient in letting through the interesting leptons. This inefficiency should
obviously be kept as low as possible, while as few fakes as possible should be able
to pass the isolation requirement. In order to reduce this accidental overlap, we can
reduce the cone size of our isolation, while we still want to retain the ability to catch
non-prompt leptons with our isolation variable. A solution to this problem is the
squared when determining cross sections. The square of this sum is the only physically measurable
property, and it might contain interference terms between the different diagrams.
86
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normalized events /1GeV
normalized events /1.5GeV
CMS Simulation
WZ
0.06
Wγ
0.04
0.02
√s = 13TeV
0.04
WZ
0.03
Wγ
0.02
0.01
0
0
100
200
0
0
300
50
100
150
MT(lepton + MET) (GeV)
(a)
(b)
√s = 13TeV
CMS Simulation
0.04
normalized events /0.035
normalized events /1GeV
CMS Simulation
200
MET (GeV)
WZ
0.03
Wγ
0.02
0.01
√s = 13TeV
WZ
0.08
Wγ
0.06
0.04
0.02
0
0
50
100
150
200
0
0
2
4
PT(lepton) (GeV)
6
∆R(lepton, Z/γ )
(c)
(d)
Figure 8.3: Comparison of several kinematic distributions of Wγ and WZ, both
normalized to unity. In (a) the MT of the lepton coming from the W decay and the
MET is shown, while (b) and (c) show the PT of this lepton and the MET both of
which go directly into the MT calculation. Figure (d) shows the distribution of the
angular separation ∆R between the lepton coming from the decaying W and the
Z boson or the photon. The final bin of every histogram shown is an overflow bin
containing all events falling out of the range of the plot.
87
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normalized events /0.01575
normalized events /0.01575
CMS Simulation
WZ
0.02
Wγ
0.01
0
1
2
0.06
WZ
Wγ
0.04
0.02
0
0
3
√s = 13TeV
1
2
∆Φ(lepton, MET)
∆Φ(lepton, Z/γ )
(a)
(b)
√s = 13TeV
CMS Simulation
normalized events /0.95GeV
normalized events /0.03GeV
CMS Simulation
WZ
0.1
Wγ
0.05
0
0
2
3
4
WZ
0.06
Wγ
0.04
0.02
0
6
√s = 13TeV
50
100
150
200
PT(Z/γ ) (GeV)
∆η(lepton, Z/γ )
(c)
(d)
Figure 8.4: Comparison of several kinematic distributions in Wγ and WZ events,
normalized to unity. (a) and (b) show the azimuthal angular separation ∆φ between
the lepton from the W decay and respectively the MET vector and the Z boson or
photon. (c) depicts the pseudorapidity separation ∆η, between the lepton and the
Z boson or photon. In (d), the PT of the Z boson in WZ and the photon in Wγ is
compared.
88
variable miniisolation, which works in
cone size is now determined by:

0.2

10 GeV
if
∆R =
 PT
0.05
the same way as relative isolation, but the
if PT ≤ 50 GeV
50 GeV < PT < 200 GeV
if PT ≥ 200 GeV
(8.1)
where PT indicates the transverse momentum of the lepton under consideration. As
the lepton PT increases, the objects in the event are more likely have a large boost,
meaning they will automatically be packed closer together. If the cone size remains
fixed, the efficiency for leptons to pass and isolation cut will decrease at very high
PT values, an effect which is far less pronounced in the case of miniisolation. The
dependency on the cone size in the PT region between 50 GeV and 200 GeV comes
from the approximate equality:
∆R ≈
2mmother
PT (mother)
(8.2)
giving the ∆R separation between the decay products in a two body decay of a
boosted mother particle. In this way the miniisolation is optimized to be able to
encapsulate the rest of a beauty quark’s decay products in the cone around a lepton coming from such a decay while it should not include the beauty quark decay
products in the cone around a prompt lepton coming from a top decay. [68] All
things considered, miniisolation is found to perform better than standard relative
isolation in SUSY searches using leptonic final states, so it will also be used for the
multilepton electroweakino search at hand [69]. We will in particular use a version
of miniisolation containing pileup corrections.
Yet another isolation variable combining miniisolation with some extra variables
is what is called multiisolation. Multiisolation combines a miniisolation requirement
with a requirement on the variables PT rel and PT ratio . The PT rel of a lepton is
defined as its momentum component perpendicular to the direction of the closest
jet, from which the lepton is subtracted if it overlaps with the jet. The higher a
lepton’s PT rel , the more unlikely that it has the same origin as the closest jet. A
lepton’s PT ratio on the other hand, is the ratio between its PT and that of the closest
jet, large values for which also tend to indicate the lepton and the jet have different origins. Using these variables, a multiisolation cut requires a lepton to have a
pileup corrected miniisolation smaller than a certain value, and a either a PT rel or
a PT ratio , or both, larger than certain values. So a multiisolation requirement can
be summarized as:
multiiso = miniiso < A and
(PT ratio > B
or
PT rel > C),
(8.3)
where A, B and C are numbers that can be tweaked. Multiisolation is usually
found to better than both relative isolation and miniisolation when used leptonic
SUSY searches, both in terms of fake rate and signal efficieny. For that reason
multiisolation will be employed together with relatively loose miniisolation cuts in
89
the selection for leptons, as recommended by the multilepton SUSY working group
[69].
8.4.2
Muon selection
First of all, the muons in our selection were required to pass the ”medium muon
identification” requirements as defined by CMS’s Muon Physics Object Group of
the CMS collaboration. [70] The requirements needed to pass this identification are
listed below:
• pass ”loose muon identification” defined as:
– object is identified as a muon by the particle flow event reconstruction
which was described earlier in this thesis
– muon is reconstructed as a global muon or a tracker muon. A muon is
global when a muon system track is matched to an inner system track,
with a fit performed by a Kalman filter [72] which updates the paramaters
by iteratively performing a fit to every hit and then proceeding. A tracker
muon on the other hand starts from a tracker track with PT greater
0.5 GeV and a momentum greater than 2.5 GeV. This track is then
extrapolated from the trackers through all detector systems up to the
muon system, accounting for scattering and energy losses. If the track is
matched to a muon system signal the muon is considered a tracker muon.
• The fraction of tracker hits considered valid must be larger than 80%
• one of the following sets of requirements must be fulfilled:
– The global track fit must have a normalized χ2 smaller than 0.3
– The muon must be global.
– The standalone muon track must match the tracker position with a normalized χ2 value smaller than 12.
– A requirement is placed on the absence of kinks in the inner tracker. If
the muons interact with the tracker material, the track will be ”kinked”,
degrading the rest of the muon measurement.
– The different muon segments, namely its track, its HCAL and ECAl
deposits and the muon system track must pass a loose compatibility requirement.
or
– The different muon segments, namely its track, its HCAL and ECAl
deposits and the muon system track must pass a tight compatibility requirement.
Aside from this medium muon id we required muons to pass the following criteria3 :
3
Any kinematic value we cut on refers to a value that has been reconstructed to the primary
interaction vertex, because what happens here is what interests us from a physical point of view.
90
• |η| < 2.4, because the muon system of the CMS detector spans up this pseudorapidity value.
• Pileup corrected miniisolation < 0.4, to reduce the contribution from fake and
non-prompt muons.
• The significance of the 3D impact parameter (i.e. the impact parameter divided by its uncertainty) should be smaller than 4, to reduce contributions
from muons not coming from the primary interaction vertex.
• A loose multiisolation working point has to be passed, again to further reduce
the contribution from non-prompt and fake objects.
8.4.3
Electron selection
For the electrons, a Multivariate analysis (MVA) identification technique developed
by the EGamma Physics Object Group were used [74]. In such an MVA approach,
a single discriminator based on multiple parameters of the measured object is developed by a supervised learning algorithm. Cuts are then applied to this new
discriminator in order to get the best possible signal over background ratio. The
rationale behind using such an MVA is that they can often be shown to have a
significantly better discriminating power between signal and background than the
cut-based identification techniques usually employed, in which cuts are applied separately to a number of variables. There were two version available of this MVA,
, one for application with specific triggers and an other one developed to be used
regardless of the trigger. We used the latter MVA discriminator. The cuts that
were applied are those recommended by the multilepton SUSY working group, in
collaboration with which the research of this thesis was done:
• MVA > 0.87 if |η| < 0.8
• MVA > 0.6 if 0.8 < |η| < 1.479
• MVA > 0.17 if 1.479 < |η| < 2.5
In addition to these MVA cuts, the following requirements on the leptons were
applied:
• |η| > 2.5, since this is pseudorapidity the extent of the tracker. The ECAL
goes further up to |η| > 2.6, but relying solely on the ECAL would significantly
reduce the resolution of the electron momentum measurement.
• Pileup corrected miniisolation < 0.4, to reduce the contribution from fake and
non-prompt electrons.
• The electron has to pass a medium multiisolation working point. The reason
for applying a tighter isolation cut than in the case of muons is that electrons
are easier to fake than muons are, because one can not use information from
the muon system on the outside of the detector for electrons.
91
8.4.4
Photon selection
In a similar way to what was done for electrons, we used an MVA developed by
the EGamma Physics Object Group. In this case we used the MVA cuts that
were recommended by the EGamma Physics Object Group, because there was no
recommendation from the multilepton SUSY working group in this case, as are
normally not used in SUSY searches with multiple leptons. The MVA used for
photons already included isolation, as opposed to the electron MVA, so no further
isolation cuts had to be applied after applying MVA cuts. The cuts that were applied
are [75]:
• MVA > 0.374 if |η| < 1.4442
• MVA > 0.336 if 1.566 < |η| < 2.5
Any photon with an |η| value not in the one of the ranges specified above was
rejected. This gap in the photon selection is related to a gap in the ECAL, which
can be clearly seen on figure 4.8. The MVA threshold values used, are designed
to have a signal efficiency of about 90 % while having minimal fake rates. Two
additional cuts are applied in the selection of the photons to account for possible
fake photons due to electrons:
• Any selected photons has to pass what is called a conversion safe electron
veto. This cut checks if the photon under consideration is matched to any
electron track, which has been found not to come from a photon that underwent
a conversion into an electron-positron pair by interacting with material the
tracker. More details on this cut can be found in [76].
• Any photon with a pixel hit in the tracker matched to its calorimeter cluster
is vetoed. This cut is commonly referred to as a pixel veto, and is generally a
harsher requirement than the conversion safe electron veto.
Conventionally analyses performed in an environment sensitive to fake photons coming from electrons use the pixel veto, while analyses less prone to this fake rate use
the looser conversion safe electron veto. In the analysis that was performed here, the
best results were attained using both cuts, while only slightly reducing the statistics
of the lepton + photon control sample.
8.4.5
MET reconstruction
We defined the MET for the first time in the chapter describing the LHC and the
CMS detector. It was said to correspond to the missing transverse momentum as
induced from the conservation of momentum4 , or equivalently as the negative sum
of the momenta of all the particles in the event 5 :
4
Note that the MET is in principle a vector, but the wording MET is interchangeably used for
both the vector itself and its magnitude. The context should make clear which one is used, for
instance if an angle is calculated with respect to the MET, or MT is determined we mean the vector
whereas when applying cuts on MET or plotting the MET distribution we mean the magnitude of
the vector.
5
We donete vectors by bold symbols.
92
MET =
X
PT (i).
(8.4)
i ∈ particles
The earlier definition of the MET does not entirely correspond to the MET value
that was actually used in the analysis. The MET that was used in this analysis is
the ”type-I” corrected MET [77]. Instead of summing the transverse momenta of
all the particles present in the event, the particles that can be clustered into jets are
clustered before the MET calculation. The resulting jets are then used for the MET
calculation instead of all the separate particles, but only after jet energy corrections
have been applied to these jets. These jet energy corrections are a set of correction
factors that have to be applied to the energy directly measured from a jet deposit
in the detector, to translate this energy into the jet energy at the particle level.
These corrections account for the fact that the calorimeter response to jet deposits
is highly non-linear and not straightforward to interpret. In CMS it is assumed that
all the different effects, such as from pileup, jet flavor, detector response, etc. can
be factorized, and a series of scale factors, which depend on the jet kinematics and
account for all these effects, is applied to the four-momentum of the jets in a certain
fixed order. More details on this can be found in [78] and [79]. In the end what one
has to understand is that a multitude of correction factors have been applied to all
jets present in the event, and the resulting jet transverse momenta are used in the
MET calculation. We can summarize this as:
METtype−I =
X
PT JEC (i) +
i ∈ jets
X
PT (i)
(8.5)
i ∈ other particles
where JEC indicates a jet energy corrected PT and the ”other particles” are those
that are not clustered into jets. Several other types of MET corrections exist, but
it is generally recommended in CMS to use type-I MET for this kind of analysis.
8.4.6
Jet and HT selection
The first selection criteria we apply to jets is a loose jet identification criteria. This
is defined as:
if |η| < 3:
• The fraction the the jet’s energy coming from deposits in the HCAL, allocated
to neutral hadrons by the PF algorithm, has to be less than 99%.
• The jet energy fraction from neutral deposits in the ECAL must be smaller
than 99 %.
• The number of particles clustered into the jet must be larger than one.
• one of the following sets of requirements must be fulfilled:
– |η| < 2.4
– The energy fraction coming from charged hadrons is larger than 0.
93
– At least one charged hadron is clustered into the jet.
– The energy fraction originating from charged hadron deposits in the
ECAL is less than 99%
or
– |η| > 2.4
if |η| > 3:
• The energy deposited by neutral particles in the HCAL is less than 90%.
• More than 10 neutral particles have been clustered into the jet.
The reason a clear distinction is made between |η| < 2.4 and |η| > 2.4 in the criteria
above is that this is the pseudorapidity threshold at which the PF algorithm can still
be reasonably applied, using the tracker, ECAL and HCAL together. For electrons
and photons we could go up to |η| < 2.5 since this range is covered by the tracker,
but for jets this is not the case because jets have a finite cone size. More specifically,
they are clustered within a ∆R cone size of 0.46 by the anti-kT algorithm as specified
in an earlier chapter. This means that we can only associate clusters to tracks, and
specify a certain jet constituent as being charged up to |η| < 2.4. Beyond this range
every jet particle is considered ”neutral”, even though many of them will in reality
be charged.
Aside from the loose jet identification, we apply the following cuts on jets used
in this analysis:
• PT > 30, because the resolution for lower energy jets is significantly degraded,
and the contribution from pileup will be much larger when lowering this threshold.
• |η| > 2.4 in order to only select jets reconstructed by the complete PF algorithm.
• If a photon or a lepton passing the selection criteria outlined in the previous
sections resides within a ∆R cone of 0.4 of the jet, the jet is removed from the
event. This requirement is used to avoid doubly counting objects in an event.
In the kinematic comparison of Wγ and WZ we will also use a variable called HT ,
which is the scalar sum of the transverse momenta of all jets present in the event.
HT can be seen as a measure of the hadronic activity in the event, and a larger
HT can be expected to go hand in hand with a degrading of the MET resolution.
6
One could assume that the |η| threshold should be |η| < 2.3, assuming jet cone size of 0.4,
since the tracker covers a pseudorapidity range up 2.5, but in practice PF still performs well up to
|η| < 2.4.
94
Comparing this value between Wγ and WZ might be useful when investigating if
we can match their transverse mass distributions match in the tails, which strongly
depend on the tails of the MET distribution.
8.4.7
Beauty jet tagging
When looking at a lepton + photon + MET final state to measure Wγ’s MT distribution, a significant background will come from top-quark pair production, a process
described in more details in the previous chapter. The produced top quarks almost
always decay to beauty quarks, which subsequently decay and lead to jets, usually
called a b-jets. By vetoing events containing b-jets we can significantly reduce the
tt background. To apply such a veto we need to be able to identify which jets are bjets. For this an algorithm called the Combined Secondary Vertex (CSV) algorithm
was used. A b-quark is expected to be long lived, leading to a secondary vertex
and tracks with a large impact parameter. The CSV algorithm combines impact
parameter significances, secondary vertex reconstructions and other jet properties,
constructing a likelihood ratio for the jet to be a b-jet or a light-jet. When a jet
passes a certain threshold for this likelihood, we will assume it to come from a bquark, and say it is b-tagged. Here we apply a medium CSV working point which
has an efficiency of about 70% for actual b-jets and a mistag rate around 1% for
light jets. To be specific, for a jet to be b-tagged it had to pass:
• CSV(jet) > 0.679.
8.5
Kinematic comparison after detector simulation
After having specified the selection criteria for all objects we intend to use, we can
start comparing the reconstructed kinematics of the Wγ and WZ processes, and
later use this to determine the ideal kinematic cuts for matching their MT shapes.
When using reconstructed objects we unfortunately do not have information about
the particles mothers anymore so it is not as straightforward to determine which
lepton comes from the W decay anymore, and which two leptons come from the Z
decay in WZ events. Even in Wγ events there is a problem if more than one lepton
is reconstructed. We need to know which lepton to use for the the MT calculation,
in order to decently compare this distribution in the two processes. Another thing to
note is that the events we select in the WZ sample should be only those containing
exactly three leptons as only such events enter the multilepton electroweakino search.
After all we want to do a background estimation for these searches. For these reasons
we apply the following preliminary event selections in WZ and Wγ for the initial
kinematic comparison:
8.5.1
WZ event selection
We use reconstructed WZ events if they pass the following criteria:
• Exactly three light leptons passing the object selection have to be present.
95
• At least one OSSF pair is found in the event.
• The lepton with the highest PT has a PT > 20 GeV.
• The lepton with the lowest PT has a PT > 10 GeV.
• The third lepton has a PT > 15 GeV if it is an electron and a PT > 10 GeV if
it is a muon.
In principle WZ events can also lead to events with three leptons without an OSSF
pair if the Z decays to τ leptons which in turn decay leptonically. Such events
are used in the electroweakino search, but as of this moment the data-driven WZ
background estimation presented here has not been developed for these events and
we will not consider τ leptons anymore in this thesis. Whenever we mention a lepton,
it will be a light lepton. When an event has an OSSF pair and a third lepton of a
different flavor, so an event containing an electron and two muons (eµµ) or a muon
and two electrons (µee), we can unambiguously conclude that the leptons forming
the OSSF pair originate from the Z-decay whereas the third lepton comes from the
decaying W boson. When the event is made up of three leptons of the same flavor,
it is impossible to know exactly which of the possible OSSF pairs originated from
the Z and which lepton came from the W. In these cases the invariant mass of all
possible OSSF pairs is calculated, and the pair yielding the value closest to the Z
mass is chosen as the pair of leptons coming from the Z decay. The third lepton
is then naturally assumed to come from the W and is used in the MT calculation.
The PT cuts applied on the leptons are designed so that only events with leptons on
the efficiency plateaus7 of the dilepton triggers that will later be used in the signal
selection are selected. For now no triggers are used though.
8.5.2
Wγ event selection
For the first kinematic comparison we use Wγ events passing the following selection:
• The event has exactly one lepton passing the object selection with a PT > 20
GeV, needed to be on the plateau of the single lepton triggers that will be
used later.
• At least one photon passing the object selection is found.
In events containing more than one lepton it would no longer be possible to determine
which one originates from the W decay, so only events in which a single lepton passes
the object selection could be used. If multiple photons pass the object selection, it
is ambiguous which one is exactly the photon produced in the primary interaction,
which intend to use as a proxy for the Z boson. We choose the photon with the
maximum PT value for our purposes. This is expected to be the photon generated
at the matrix element level since this is the hard part of the interaction.
7
Triggers used in CMS typically start with very low efficiencies at low PT ’s, which quickly rises
as a function of the PT to a constant value called the plateau.
96
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CMS Simulation
20
150
15
events /5e-05
reconstructed PT(l) (GeV)
CMS Simulation
200
40
30
100
10
20
50
5
10
50
100
150
200
0
0
0
generator P (l) (GeV)
√s = 13TeV
0.001
0.002
0.003
0.004
0.005
∆R(gen lepton, reco lepton)
T
(a)
(b)
Figure 8.5: Validation of the generator matching in simulated WZ events: (a) 2D
plot comparing the PT of the reconstructed leptons to their generator matches. (b)
Angular separation ∆R between the leptons and their generator matches. Note
that there is an overflow bin present in figure (b), but there are nearly no events
populating it.
8.5.3
Matching reconstructed objects to simulated particles
In order to illustrate the effect of the reconstruction, it would be useful to show
the MC truth kinematic distributions for the reconstructed objects that have been
selected. To be able to show the generator versions of reconstructed particles, we
need a way to match reconstructed particles to the generator particles that induced
them. This is done by determining the closest generator particle of the same type
for every reconstructed particle. In order to avoid any ambiguity, in the sense
that multiple particles might have the same match, the generator matching code
written for this thesis used a double loop looking for the generator-reconstructed
particle pair with the closest possible separation. When the particles forming the
best match were determined, they were removed from the lists, and the process was
repeated until all reconstructed particles had a match. In the very rare events were
a reconstructed particle had no match, this event was not used when comparing
generator distributions. The generator matching is validated in figure 8.5 for WZ
events, and in figure 8.6 for Wγ events. The generator matching algorithm seems
to perform well for both photons and leptons, since the reconstructed particle is
almost invariably found to be very close to its generator match, with a very similar
PT . The angular resolution is clearly better for leptons than for photons, which
becomes especially clear when looking at the overflow bins of the ∆R distributions.
The reason for this is most likely tracker has a much better spatial resolution than
the calorimeters, as mentioned in Chapter 4. Only ECAL information is available
for photons, while tracker information can be used for leptons.
97
√s = 13TeV
CMS Simulation
4000
150
4000
100
3000
2000
2000
1000
50
50
100
150
200
0
0
0
0.001
0.002
generator P (l) (GeV)
T
CMS Simulation
0.003
0.004
0.005
∆R(gen lepton, reco lepton)
(a)
(b)
√s = 13TeV
CMS Simulation
200
√s = 13TeV
events /0.0002
reconstructed PT(γ ) (GeV)
√s = 13TeV
events /5e-05
reconstructed PT(l) (GeV)
CMS Simulation
200
8000
150
6000
100
2000
1500
4000 1000
2000
50
50
100
150
200
0
500
0
0
0.005
0.01
generator P (γ ) (GeV)
0.015
0.02
∆R(gen γ , reco γ )
T
(c)
(d)
Figure 8.6: Validation of the generator matching in simulated Wγ events: (a), (c) 2D
plots comparing the PT ’s of leptons, respectively photons to their generator matches.
(b), (d) angular separation ∆R between respectively the leptons and photons and
their generator matches.
98
8.5.4
Kinematic comparison
Once the decision on the preliminary object and event selection has been made,
we can look at the reconstructed kinematic comparison between Wγ and WZ. For
every reconstructed level distribution, the corresponding generator distribution, determined by the generator matching method of the previous section will also be
shown. While many of the reconstructed and MC truth distributions are extremely
similar, we show them as an interesting display of the CMS detector’s resolution for
different objects. The complete kinematic comparison is displayed in figures 8.7 up
to 8.15. One important remark is that ”lepton” always refers to the lepton coming
from the decaying W boson in the plots below. Two additional kinematic distributions that have not been determined at the generator level, namely the number of
selected jets and HT , the scalar sum of all jet transverse momenta, are shown in
figure 8.15.
The first thing that one might notice is that the angular separations ∆R, ∆η and
∆Φ between the photon and the lepton in Wγ are no longer peaked at zero. The
reason for this is simple: isolation requirements were applied in the object selection.
If the angular separation between a lepton and a photon becomes too small, the
lepton will no longer be isolated since the photon deposits energy within the cone
around the lepton used for the calculation of the isolation. So events with a very
small angular separation get vetoed unless the photon energy is very low compared
to that of the electron. When comparing the reconstructed distributions to the generator ones, almost all of the PT and angular distributions seem to match extremely
well, which can be expected considering the fact that photons and leptons are generally well reconstructed in CMS (or in this case in the simulation of the detector).
All variables related to the MET are a different story however. Firstly, the MET
distributions of both Wγ and WZ are significantly smeared out compared to their
generator MET distribution. The reason for this lies in the fact that a plethora of
objects, including jets are present in most events. Jets generally suffer from worse
resolutions, and are much more prone to mismeasurements, than electrons, muons
or photons. A significant number of the events can be seen to contain jets, and the
HT values, which indicated the the amount of hadronic activity in the event can
reach several hundreds of GeV, which might be a reason for the MET smearing.
The relatively bad reconstruction of the MET also has a clear influence on the reconstruction of the azimuthal separation between the MET vector and the lepton,
as shown in figure 8.14. This is the only angular variable with a clear difference
between the reconstructed and MC truth distributions! The widened distribution
of the reconstructed MET, and the relatively poor angular resolution of the MET
vector directly translates into a smeared MT distribution that does not clearly fall
off at the W boson mass anymore in WZ. For W γ events the two separate peaks
have even become indistinguishable after reconstruction.
After reconstruction, the MT shape comparison between Wγ and WZ has not improved, which could also not have been expected since we have not applied any
kinematic cuts to reduce it. All other FSR induced kinematic differences, such as
99
√s = 13TeV
WZ
0.04
Wγ
0.02
0
0
100
200
√s = 13TeV
CMS Simulation
normalized events /3GeV
normalized events /3GeV
CMS Simulation
300
WZ
0.1
Wγ
0.05
0
0
100
200
MT(lepton + MET) (GeV)
(a)
300
MT(lepton + MET) (GeV)
(b)
Figure 8.7: Comparison of the MT shape of Wγ and WZ, reconstructed (a) and at
the MC truth level (b).
peaks at small angular separations between the photon and the lepton in Wγ, are
also still present. Using these differences to remove the FSR is the subject of the
next section. The distributions of the HT and the number of jets can be seen to be
different, with more hadronic activity expected in WZ events. This might influence
the MET resolution, which is studied in one of the next sections.
Another interesting thing to note is that even though the tails of the reconstructed
MET distribution are longer than those of the MC truth distribution, the true MET
can also reach very high values, which can be seen from the size of the overflow bin.
When investigating the high MET values one even finds events with a generator
MET beyond 600 GeV in both Wγ and WZ. So where do these events come from?
We expect the MET to come from the neutrino from the leptonically decaying W in
both processes, but it is worth investigating if this is the only source of true MET in
these high MET events. This was done by comparing the generator MET to the PT
of the neutrino from the W decay, and to the sum of the PT ’s of all other neutrinos
present in the event, as shown figure 8.16. From this figure it immediately becomes
clear that the MET almost exclusively originates from the decaying W, at both low
and high values. So the very high MET values can be concluded to originate from
highly transversely boosted W bosons.
8.6
Reducing FSR and proof of principle
As we anticipated earlier, it should be possible to reduce the contribution of FSR in
Wγ events by applying thresholds on the angular separation between the selected
photon and lepton. Another cut that can be expected to reduce the FSR is applying
a PT threshold on the photon, since a high photon PT threshold makes it unlikely
100
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CMS Simulation
0.1
WZ
Wγ
0.05
0
50
100
150
WZ
Wγ
0.05
0
200
√s = 13TeV
0.1
50
100
PT(lepton) (GeV)
150
200
PT(lepton) (GeV)
(a)
(b)
Figure 8.8: PT distribution of the lepton from the decaying W compared in Wγ and
WZ, after reconstruction (a), at the MC truth level (b).
√s = 13TeV
CMS Simulation
normalized events /2GeV
normalized events /2GeV
CMS Simulation
0.04
WZ
0.03
Wγ
0.02
0.01
0
0
50
100
150
200
√s = 13TeV
WZ
0.06
Wγ
0.04
0.02
0
0
50
100
MET (GeV)
(a)
150
200
MET (GeV)
(b)
Figure 8.9: Comparison of respectively the reconstructed (a) and MC truth MET
(b) between Wγ and WZ.
101
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CMS Simulation
WZ
0.15
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0.1
0.05
0
50
100
150
0.15
WZ
Wγ
0.1
0.05
0
200
√s = 13TeV
50
100
150
PT(Z/γ ) (GeV)
200
PT(Z/γ ) (GeV)
(a)
(b)
Figure 8.10: The PT of the photon in Wγ events compared to that of the Z, as
reconstructed from its decay products, in WZ events, reconstructed (a), and at the
MC truth level (b).
√s = 13TeV
CMS Simulation
normalized events /0.032
normalized events /0.032
CMS Simulation
WZ
0.02
Wγ
0.01
0
0
1
2
WZ
0.02
Wγ
0.01
0
0
3
√s = 13TeV
1
2
∆Φ(lepton, Z/γ )
(a)
3
∆Φ(lepton, Z/γ )
(b)
Figure 8.11: Comparison of the difference in the azimuthal angle Φ between the
lepton originating from the W decay, and the photon or Z boson, after reconstruction
(a), in MC truth (b).
102
√s = 13TeV
0.06
WZ
Wγ
0.04
0.02
0
0
2
√s = 13TeV
CMS Simulation
normalized events /0.07
normalized events /0.07
CMS Simulation
4
0.06
WZ
Wγ
0.04
0.02
0
0
6
2
4
6
∆R(lepton, Z/γ )
∆R(lepton, Z/γ )
(a)
(b)
Figure 8.12: Comparison between WZ and Wγ of the angular separation ∆R between the photon, or Z and the lepton from the decaying W, after reconstruction
(a) and in MC truth (b).
√s = 13TeV
0.06
WZ
Wγ
0.04
0.02
0
0
1
2
3
√s = 13TeV
CMS Simulation
normalized events /0.05GeV
normalized events /0.05GeV
CMS Simulation
4
5
0.06
WZ
Wγ
0.04
0.02
0
0
1
2
∆η(lepton, Z/γ )
(a)
3
4
5
∆η(lepton, Z/γ )
(b)
Figure 8.13: Pseudorapidity difference ∆η between the lepton from the W decay and
the Z or photon, after reconstruction (a), and at the MC truth level (b), compared
in Wγ and WZ events.
103
√s = 13TeV
√s = 13TeV
CMS Simulation
normalized events /0.032
normalized events /0.032
CMS Simulation
WZ
0.02
Wγ
0.015
0.01
WZ
0.03
Wγ
0.02
0.01
0.005
0
0
1
2
0
0
3
1
2
∆Φ(lepton, MET)
3
∆Φ(lepton, MET)
(a)
(b)
Figure 8.14: Azimuthal separation ∆Φ between the lepton from the W decay and
the MET, after reconstrution (a), and at the MC truth level (b), compared in Wγ
and WZ events.
√s = 13TeV
0.15
WZ
Wγ
0.1
0.05
0
200
400
√s = 13TeV
CMS Simulation
normalized events /1GeV
normalized events /5.7GeV
CMS Simulation
600
WZ
0.6
Wγ
0.4
0.2
0
0
2
4
8
10
number of jets
HT (GeV)
(a)
6
(b)
Figure 8.15: Reconstructed HT and number of jets distributions in Wγ and WZ
events.
104
×103
√s = 13TeV
CMS Simulation
generator MET
ν from W decay
300
Σ other ν
events /2GeV
events /2GeV
CMS Simulation
400
ν from W decay
600
Σ other ν
400
100
200
50
100
150
0
0
200
generator MET
800
200
0
0
√s = 13TeV
50
100
150
PT (GeV)
(a)
(b)
√s = 13TeV
CMS Simulation
generator MET
30000
ν from W decay
Σ other ν
20000
events /2GeV
events /2GeV
CMS Simulation
0
√s = 13TeV
generator MET
30
ν from W decay
Σ other ν
20
10
10000
0
200
PT (GeV)
50
100
150
200
0
0
50
100
PT (GeV)
(c)
150
200
PT (GeV)
(d)
Figure 8.16: Plots comparing the generator MET, defined as the sum of all neutrino
PT ’s, to the PT of the neutrino coming from the W decay, and the sum of the PT ’s
of all other neutrinos. The left plots were made for Wγ events while the right plots
contain WZ events. The upper and lower plots show the same distributions, but
with different y-ranges.
105
that a lepton can radiate such a photon. The effects of different kinematic cuts on
the Wγ MT shape, compared to that of WZ is shown in figure 8.17. From these
plots it becomes clear that cuts on the photon’s PT , ∆R(`, γ) and ∆Φ(`, γ) do indeed
make the MT shapes of Wγ and WZ more similar. So our premise that removing
the FSR from Wγ events should leave us with about the same MT distributions in
Wγ and WZ seems to be holding! Out of all the kinematic cuts, applying a PT
threshold on the photon seems to be the most powerful discriminator to eliminate
FSR and match the MT shapes, which can clearly be seen from figure 8.17.
Before we proceed with optimizing all the kinematic cuts that are being applied,
a few remarks have to be made. First of all, we are attempting to provide a WZ
background estimation for electroweakino searches, which means we must estimate
the WZ background for events that will pass the signal selection of these searches.
The Run I search used a MET threshold of 50 GeV, and triggers selecting events with
two leptons, called dilepton triggers. The yet to be performed Run II search will use
these same requirements, so in order to provide a useful background estimation for
these searches the same requirements will have to be applied to the WZ sample here.
The second thing to consider is that we want to measure the Wγ MT distribution
from data, after we found a way match it to that of WZ. In order to do this, we
will also have to apply triggers to our simulation, since only events passing certain
triggers are stored in data. For Wγ events we will use single lepton triggers. The
final thing to note is that electrons are more difficult in terms of reducing the FSR,
and to measure together with photons in data. They are more prone to emitting FSR
photons than muons, and can be faked by photons in the detector, or fake photons
themselves. In order to get a clean Wγ measurement in the electron channel, we will
have to apply some extra cuts compared to the muon channel as will be discussed
below. For this reason, the MT shape matching will have to be done separately for
Wγ events in which the W decays to a muon and those in which the W decays to an
electron. In order to account for any differences in the resolution of electrons and
muons, we will only use muonically decaying W’s to estimate the WZ background
in which which the W decays to a muon, and vice versa for electrons. We will
henceforth refer to events in which the W decays to a muon as the ”muon channel”,
and events in which it decays to electrons as the ”electron channel”.
8.6.1
Proof of principle for W→ µν
After trying multiple combinations of kinematic cuts, the following event selection
was found to give good results:
WZ:
• event selection of section 8.5
• lepton tagged to W decay must be a muon
• MET > 50 GeV
106
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normalized events /6GeV
normalized events /6GeV
CMS Simulation
10−1
WZ
Wγ
10−2
10−3
10
Wγ
10−3
200χ2 / ndf
p0
1165 / 49300
0.9165 ± 0.0057
1
WZ/Wγ
100
30
1
0
0
100
200
300
100
200χ2 / ndf
p0
2
0.5
0
100
200 / 48300
0.9194 ± 0.0164
200
300
MT(lepton + MET) (GeV)
MT(lepton + MET) (GeV)
(a)
(b)
√s = 13TeV
CMS Simulation
normalized events /6GeV
CMS Simulation
WZ
10−2
Wγ
10−3
10−4
√s = 13TeV
WZ
10−2
Wγ
10−3
10−4
0
1.5
100
200χ2 / ndf
p0
293 / 46300
0.9647 ± 0.0072
1
0.5
0
0
WZ/Wγ
WZ/Wγ
0
1.5
0
normalized events /6GeV
WZ
−2
10−4
10−4
WZ/Wγ
√s = 13TeV
0
1.5
100
200χ2 / ndf
p0
228.5 / 46300
0.9684 ± 0.0076
1
0.5
100
200
300
0
0
100
200
300
MT(lepton + MET) (GeV)
MT(lepton + MET) (GeV)
(c)
(d)
Figure 8.17: Comparison of the MT shapes in Wγ and WZ for muonically decaying
W bosons, plotted on a logarithmic scale, without any additional cuts (a), after
applying the following kinematic requirements on Wγ events: PT (γ) > 50 GeV (b),
∆Φ(`, γ) > 1 (c) and ∆R(`, γ) > 1 (d). In every plot, the ratio of the MT curves is
shown in the bottom, to which a constant has been fit by means of the least squares
method. The value of the constant fit, and the goodness of the fit in terms of χ2
per number of degrees of freedom are listed for every plot.
107
• veto on the presence of a b-jet8
• event must pass one of the following dilepton triggers:
– Mu17 TrkIsoVVL Ele12 CaloIdL TrackIdL IsoVL (required a muon with
PT > 17 GeV and an electron with PT > 12 GeV with some additional
identification requirements)
– Mu8 TrkIsoVVL Ele17 CaloIdL TrackIdL IsoVL
– Mu17 TrkIsoVVL Mu8 TrkIsoVVL DZ
Wγ:
• event selection of section 8.5
• lepton must be a muon
• MET > 50 GeV
• PT (γ) > 50 GeV
• ∆Φ(µ, γ) > 1
• veto on the presence of a b-jet9
• event must pass the trigger IsoMu20 (selects events with at least one isolated
muon with PT > 20 GeV)
The MT shape comparison between WZ and Wγ attained after applying these cuts
is shown in figure 8.18. The binning of this plot has been significantly increased
compared to that of the previous plots to be able to see if the distributions match
in every bin after significantly decreasing the Wγ sample’s statistics by applying
several thresholds. From the plot it becomes clear that our method seems to work,
the distributions match well, and the χ2 value of a constant fit in smaller than one.
There does however seem to be a bit of a trend in the ratio around the region of the
W mass, indicating the shapes are not completely similar just yet.
8.6.2
Proof of principle for W→ eν
In this case the following event selection was used:
WZ:
• event selection of section 8.5
• lepton tagged to W decay must be an electron
8
9
This veto will be used in the electroweakino signal selection to reduce the tt background.
Needed to reduce the tt background in data.
108
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normalized events /20GeV
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WZ
10−1
Wγ
10−2
WZ/Wγ
10−3
0
2
100
200χ2 / ndf
p0
12.97 / 14300
0.9787 ± 0.0299
1
0
0
100
200
300
MT(lepton + MET) (GeV)
Figure 8.18: Comparison of the WZ and Wγ MT shapes in the muon channel after
several kinematic cuts to reduce the FSR contribution in Wγ have been applied. A
constant has been fit to the ratio of the shapes with the least squares method, and
the goodness of fit in terms of χ2 per degree of freedom is indicated together with
the fitted constant.
109
• MET > 50
• event must pass one of the following dilepton triggers:
– Mu17 TrkIsoVVL Ele12 CaloIdL TrackIdL IsoVL
– Mu8 TrkIsoVVL Ele17 CaloIdL TrackIdL IsoVL
– Ele17 Ele12 CaloIdL TrackIdL IsoVL DZ
Wγ:
• event selection of section 8.5
• lepton must be an electron
• MET > 50
• PT (γ) > 50 GeV
• ∆Φ(µ, γ) > 1
• event must pass the trigger Ele23 CaloIdL TrackIdL IsoVL
The resulting MT shape comparison is shown in figure 8.19, and it is immediately
obvious that the match between the MT shapes is much worse in the electron channel
than in the muon channel. Especially the trend in the MT ratio near the W mass
that was barely visible in the muon channel has become much more severe, indicating
that it results in leftover effects from FSR which is more prominent in the electron
channel. Many different kinematic cuts have been tried with the goal of improving
the match of the MT shapes, but no significant improvement could be made. There
is no reason to panic however, because we have one trick left up our sleeve, as shown
in the next section.
8.7
Reweighing kinematic variables
In an attempt to improve the matching of the MT shapes in Wγ and WZ, beyond
what can be achieved by applying simple cuts, we can employ reweighing of kinematic variables. Such a reweighing is done by using the ratio of the shapes of a
kinematic distribution in Wγ and WZ, other than the MT , and applying this ratio
as an additional statistical weight to Wγ events in an attempt to compensate some
of the remaining kinematic differences. So which kinematic variable should we use
for this reweighing?
It might at first seem attractive to use the PT of the photon and Z for this reweighing through the argument that the only kinematic difference between Wγ and WZ,
besides the presence of FSR lies in the mass difference of the second gauge boson.
While true, the kinematic differences resulting from the different masses of the Z
110
√ s = 13TeV
normalized events /20GeV
CMS Simulation
WZ
10−1
Wγ
10−2
WZ/Wγ
10−3
0
1.5
100
200χ2 / ndf
p0
49.29 / 14300
0.9044 ± 0.0290
1
0.5
0
0
100
200
300
MT(lepton + MET) (GeV)
Figure 8.19: Comparison of the WZ and Wγ MT shapes in the electron channel after
several kinematic cuts to reduce the FSR contribution in Wγ have been applied. A
constant has been fit to the ratio of the shapes with the least squares method, and
the goodness of fit in terms of χ2 per degree of freedom is indicated together with
the fitted constant.
111
√s = 13TeV
CMS Simulation
√s = 13TeV
200
20
T
0.8
γ P (GeV)
Z PT (GeV)
CMS Simulation
200
150
0.6
100
15
150
10
0.4
100
50
5
0.2
50
100
150
200
0
50
50
100
150
200
MT(lepton + MET) (GeV)
MT(lepton + MET) (GeV)
(a)
(b)
0
Figure 8.20: Number of expected events as a function of the MT and the PT of the
Z boson in WZ events (a) and the photon in Wγ events (b). One can see by eye that
there is little correlation, and in fact the correlation factors between the MT and
the PT of the Z, respectively the photon are calculated to be -0.0265, and -0.0641,
indicating that the correlation is small or non-existent. These plots contain both
muon and electron channel events.
and the photon are of no consequence because the W’s MT is not correlated to the
PT of the second gauge boson. Figure 8.20 shows the number of events as a function
of the PT of the Z or photon and the MT , together with the correlation factors between the two variables. It then becomes clear that they are in fact to a very good
approximation uncorrelated. This can intuitively be seen by considering the fact
that in most events the W boson and the second gauge boson will be emitted nearly
back to back to conserve momentum since the HT distribution is sharply falling,
with a peak at zero. So a higher PT for the photon or Z implies a more boosted
W, which increases the PT of its decay products (the lepton and MET), but also
reduces their angular separation. Looking back at equation 7.1 we can see that both
effects push the MT variable in the opposite direction, and will largely cancel each
other. So we can conclude that the second gauge boson’s PT is not a good variable
for reweighing, as it is uncorrelated from the MT .
The other two obvious candidates for reweighing are the MET and the leptonPT ,
both of which enter the MT calculation directly. This means that their reweighing
will manifestly influence the MT distribution. The MET and lepton PT distributions
after applying all the kinematic cuts of the previous section, are plotted in figure
8.21 for the muon channel and figure 8.22 for the electron channel. The MET shapes
perfectly match in both the muon and electron channel, as one might expect since
the neutrino’s PT should not be influenced by FSR. This means that the MET can
be discarded as a reweighing candidate since its reweighing will change nothing at
all. Even after all the cuts to reduce FSR, significant differences in the lepton PT
distributions of Wγ and WZ remain. These differences are much more pronounced in
112
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0.15
√s = 13TeV
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WZ
Wγ
0.1
WZ
−1
10
Wγ
10−2
350
100
150
200
2
WZ/Wγ
0.05
WZ/Wγ
normalized events /7.5GeV
CMS Simulation
0.2
3
1
1
0
0
50
100
150
200
50
100
50
100
200
150
200
PT(lepton) (GeV)
MET (GeV)
(a)
150
2
(b)
Figure 8.21: Distribution shapes of the MET and PT of the lepton from the W decay,
compared in Wγ and WZ in the muon channel after applying several kinematic cuts
to remove the FSR contribution in Wγ.
the electron channel, indicating that they are in fact FSR artifacts. One recognizes
the same trend in the ratio of the lepton PT shapes as in the MT distributions of
the previous section. So it seems that reweighing the lepton PT might improve the
match of the MT shapes. And indeed, after reweighing all Wγ events with the
scale factors extracted from the lepton PT shape ratio, shown in figure 8.23 as an
illustration, we find an extremely good match between the MT shapes in both the
muon and electron channel as one can see in figures 8.24 and 8.25. Any trend in the
MT shape ratio that remained after applying kinematic thresholds has essentially
been removed, and the shapes match perfectly. We have established the premise
that the WZ MT shape can be extracted from the MT of Wγ!
8.8
Statistics and viability
It was mentioned earlier that the Wγ cross section is more than a hundred times
larger than that of WZ, making it seem like we would immensely gain in statistics
by using a lepton + photon + MET final state for measuring the WZ MT shape.
But everything is not as nice as it might seem at first sight because we are applying
tight kinematic requirements on Wγ events. By applying tight ∆Φ or ∆R cuts, we
will lose about half of our Wγ statistics as seen from figures 8.11 and 8.12 which is
not too problematic. The main issue is however the steeply falling PT distribution of
the photon which is displayed in figure 8.10. From this distribution it becomes clear
that by applying a harsh PT requirement on the photon, which we unfortunately
absolutely need to reduce the FSR, we will lose a very large chunk of our events.
To find the ratio of the amount of Wγ events we expect in data to the number we
expect for WZ we need to compare their cross sections and selection acceptance as
follows:
113
√s = 13TeV
WZ
0.2
Wγ
0.15
√s = 13TeV
CMS Simulation
normalized events /12GeV
normalized events /7.5GeV
CMS Simulation
0.1
WZ
10−1
Wγ
10−2
50
1.5
100
150
WZ/Wγ
WZ/Wγ
0.05
200
1
2
50
100
50
100
150
200
150
200
1
0.5
0
50
100
150
0
200
PT(lepton) (GeV)
MET (GeV)
(a)
(b)
Figure 8.22: Distribution shapes of the MET and PT of the lepton from the W decay,
compared in Wγ and WZ in the electron channel after applying several kinematic
cuts to remove the FSR contribution in Wγ.
√s = 13TeV
10−1
Wγ
10−2
50
100
50
100
150
200
150
200
normalized events /12GeV
CMS Simulation
WZ
WZ/Wγ
normalized events /12GeV
CMS Simulation
√s = 13TeV
3
2
1
1.5
1
0.5
0
50
100
PT(lepton) (GeV)
(a)
150
200
PT(lepton) (GeV)
(b)
Figure 8.23: The lepton PT distribution compared between Wγ and WZ in the muon
channel after applying the reweighing scale factors of (b). The lepton PT curves now
match perfectly by definition since this distribution has been reweighed.
114
√ s = 13TeV
normalized events /20GeV
CMS Simulation
WZ
−1
10
Wγ
10−2
WZ/Wγ
10−3
0
3
100
200χ2 / ndf
p0
6.388 / 14300
1.167 ± 0.048
2
1
0
0
100
200
300
MT(lepton + MET) (GeV)
Figure 8.24: MT shape comparison of Wγ and WZ after reweighing the lepton PT ,
in the electron channel. For every plot a least squares fit is performed, and the
resulting χ2 value is shown.
115
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normalized events /20GeV
CMS Simulation
WZ
10−1
Wγ
10−2
WZ/Wγ
10−3
30
100
200χ2 / ndf
8.533 / 14300
p0
1.041 ± 0.051
2
1
0
0
100
200
300
MT(lepton + MET) (GeV)
Figure 8.25: MT shape comparison of Wγ and WZ after reweighing the lepton PT ,
in the electron channel. For every plot a least squares fit is performed, and the
resulting χ2 value is shown.
116
σWγ · AWγ
NWγ
=
NWZ
σWZ · AWZ
(8.6)
where σ and A indicate the cross section and acceptance. After applying the kinematic cuts previously listed, we find:
NWγ
≈ 6.5
NWZ
(8.7)
in both the muon and electron channel. The ratio of the yields we finally end up
with is not as good as what it would be without any cuts, but the gain in statistics
in the lepton + photon + MET final state compared to the three lepton + MET
final state is still significant.
8.9
Resolution comparison
One of the primary reasons we are going through all this effort to perform a datadriven estimation instead of the MC is that we want to catch all the experimental
effects affecting the high MET tails. Any effect on the MET that is mismodeled
will directly affect the MT distribution, making it of paramount importance to have
a good MET estimate. Considering these arguments it would also be necessary
for the MET resolution in Wγ and WZ to be similar, because if it is completely
different this will obviously translate into differences in the MT distributions which
is exactly what we do not want. We shall define the MET resolution as the generator
MET subtracted from the reconstructed MET, divided by the generator MET. This
resolution is compared in Wγ and WZ in both the muon and electron channel in
figure 8.26. We can see that the distributions seem to match rather well, which
puts the principle of using Wγ to determine the WZ background on even more solid
ground. This result could already be expected as it was shown in the previous
section that the MET shapes of the two processes match perfectly.
8.10
Backgrounds to Wγ
Unfortunately Wγ will have its own backgrounds we shall have to deal with when
measuring the Wγ MT distribution in data. Any process leading to a final state
containing a lepton, a photon and MET will contribute to the background. The
different backgrounds are:
• Wjets:
The dominant background to Wγ are events in which a W boson is created
together with jets. These jets can either fake photons, or contain non-prompt
photons which manage to pass the object selection we apply. Even if the
acceptance for Wjets events can be reduced by several orders of magnitude,
the background will remain large since its production cross section is about a
150 times larger than that of Wγ.
117
√s = 13TeV
CMS Simulation
events /0.4
events /0.4
CMS Simulation
WZ
10−1
Wγ
10−2
√s = 13TeV
WZ
10−1
Wγ
10−2
10−3
10−3
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Figure 8.26: MET resolution shapes compared in Wγ and WZ in the muon (a) and
electron channel (b).
• tt + jets, t + jets:
Events containing a top-quarks and jets provide the second largest background
in the lepton + photon + MET final state. One lepton and MET can be
provided by a leptonically decaying top quark, while the photon is faked by
either a jet or an electron from the other top quark decay. This background
can be significantly reduced by vetoing events with b-tagged jets.
• tt + γ, t + γ:
It is self-evident that a leptonically decaying top, created with a prompt photon
should is able to pass a lepton + photon + MET event selection.
• Z + γ:
A single photon and a lepton can be detected when a leptonically decaying Z
is produced together with a photon, if one of the leptons does not pass the
object selection criteria. The MET has to be provided by mismeasurements,
since there is no ”real” MET in such an event. The background coming from
these events is rather small, but still relevant.
• Drell-Yan + jets:
This is a similar process to Z + γ, with the difference that the photon will now
have to be faked by a jet or a lepton. These events have extremely high cross
sections compared to the signal, and can provide a MET far easier than the Z
+ γ events due to jet mismeasurements. In the muon channel, this background
is still relatively small, but the great potential for electrons to fake photons
makes it very large and problematic in the electron channel.
• γ + jets:
Jets can fake leptons, and lead to significant mismeasurements, making it
possible for events containing only jets and photons to pass our event selection.
118
Because electrons are easier to fake than muons, this background will be much
more important in the electron channel than in the muon channel.
• rare SM processes: ttW, ttZ, ZZγ, WW, ...:
One can think of many rare processes that are able to pass the selection criteria
by means of fake or non-prompt objects. Their production cross sections, and
depending on the process also their acceptances, are very low, making them
almost negligible.
Simulations were used to estimate all of the backgrounds, when extracting Wγ from
data. In comparing data and MC, significant MC excesses were found in almost
every bin of every kinematic distribution when the available MC samples for all
of the backgrounds listed above were used together. The origin of this problem
turned out to lie in overlap between the phase space simulated in several of the
MC samples. To be more specific, samples of processes between which the only
difference is that either a photon or a jet was simulated, like Wjets and Wγ, ttjets
and ttγ, etc. overlap. We will illustrate the overlap by using Wγ and Wjets as an
example, but the principle is exactly the same in the other cases. For every sample,
the hard interaction processes that have to be simulated at the matrix element level,
and the kinematic thresholds above which these objects are simulated are specified
certain files10 . For the Wγ sample the simulated process was proton-proton collisions
leading to a leptonically decaying W boson, a photon which could potentially come
from FSR11 , and up to three jets. The Wjets sample on the other hand simulated
proton-proton collisions resulting in a leptonically decaying W boson and up to four
jets. At the matrix element level these simulations are entirely orthogonal, but as
discussed in the chapter on simulation techniques, the simulation is further processed
by a parton showering program, which is Pythia for all the samples that were used
in this thesis. In the parton showering step, the leptons are able to radiate photons
by means of final state radiation, leading to an overlap between the samples. The
Wγ sample can for instance generate an event at the matrix element level with one
photon and a jet, while a similar event could be simulated in the Wjets sample if
Pythia lets the final lepton radiate a photon. In order to remove the overlap between
Wjets and Wγ, every event in the Wjets sample was removed if it contained any MC
truth photon with a PT greater than 10 GeV, originating from a boson, a quark or
a lepton. The reason for the PT > 10 GeV requirement is that only photons above
this PT threshold were generated at the matrix element level in the Wγ sample.
Even after removing all these events from the Wjets sample, it still contributed to
the background because photons could be faked by jets, or provided by decaying
hadrons such as π0 ’s. A similar overlap removal was applied to all other overlapping
MC samples. The method that was employed to remove this overlap is validated in
figure 8.27, where a comparison between data and MC in the lepton + photon +
MET final state is shown before and after the overlap removal has been performed.
The exact data used, and selection applied in this plot will be mentioned later, but
10
Usually called the run-card and the proc-card.
To be fully specific: up to three QED vertices were allowed for radiating the photon at the
matrix element level.
11
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Figure 8.27: Comparison of the PT (γ) distribution in data and MC in a µ + γ +
MET final state, when using all MC samples out of the box (a) and after cleaning
the overlap between several samples.(b)
for now one just has to take away the fact that data and MC do not match at all
before the overlap removal, while the MC distributions describe the data very well
after the cleaning.
8.11
Inclusion of Wjets in the WZ prediction technique
In order to minimize the reliance on MC in the WZ background estimation, one
can try to incorporate Wjets in the matching of the Wγ and WZ MT shapes. If we
would be able to make the MT shape of WZ match to that of Wγ + Wjets, then
we would not have to rely on the Wjets simulation to subtract this contribution
from the lepton + photon + MET data. Using the exact same event selection and
methodology as outlined above, the MT shape of WZ was compared to that of Wγ
+ Wjets. In the latter distribution, every event was statistically weighed using the
theoretical cross section, so the contributions would be proportional to the amount
of events we expect in data for Wγ and Wjets. The resulting comparison of the MT
shapes is shown respectively with and without reweighing the lepton PT in figures
8.28 and 8.29 for both the electron and muon channel. The match of the shapes, and
the goodness of the constant fit has improved in every case, compared to the results
we achieved by only using Wγ. The contribution we expect from Wjets in the lepton
+ photon + MET final state is relatively small compared to that of Wgamma, and
is found to be about 10% of the Wγ input in both the muon and electron channels.
Albeit small, the extra contribution is another advantage of using Wjets since it will
increase the final amount of data events we have to estimate the WZ background
from.
At first sight it seems unambiguously better to include Wjets into the method, as the
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Figure 8.28: MT shape comparison of WZ to Wγ + Wjets, with a least squares fit to
the ratio of the shapes, in the muon channel (a), and the electron channel (b). The
Wγ and Wjets events were given statistical weights proportional to their expected
yields in data, and no reweighing is applied in these plots.
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Figure 8.29: MT shape comparison of WZ to Wγ + Wjets, with a least squares fit to
the ratio of the shapes, in the muon channel (a), and the electron channel (b). The
Wγ and Wjets events were given statistical weights proportional to their expected
yields in data, and the Wγ + Wjets events were reweighed using the lepton PT
distribution.
121
shapes match better, the simulation reliance is reduced, and the amount of events in
the data sample will increase. There are however a few issues to consider. First of
all, the amount of simulated events in the Wjets sample passing our signal selection
after removing the overlap with Wγ is very small. The amount of simulated events
passing the selection are 48 events in the electron channel, and 47 in the muon
channel, which are very small amounts to draw conclusions from. To make a comparison, we have 126689 simulated WZ events and 2683 Wγ in the electron channel,
and 150400 WZ and 3218 Wγ events in the muon channel. These large statistical
uncertainties make it hard to definitely conclude which is the better method, that
with or without Wjets. The large statistical uncertainties on the Wjets yields make
it technically advantageous to include Wjets in the data-driven prediction for now,
since this spares us from having to subtract the sample from data, saving us from
large statistical fluctuations in the data-driven estimation of the WZ background.
So when performing the data-driven estimate of the WZ background we will include
Wjets into the method.
8.12
Lepton + photon + MET control sample in data
It was shown over the previous sections that a measurement of the Wγ, or Wγ +
Wjets MT shape can be used to estimate the WZ background as a function of MT .
So now we have to actually perform this measurement. The data that was used
for this measurement were the SingleMuon and SingleElectron datasets, collected
during 2015’s LHC operation, at a center of mass energy of 13 TeV with 25ns of
time between bunch crossings. The data had an integrated luminosity of 2.26 fb−1 .
More details on the exact data used in this analysis can be found in appendix B.
Below we will mention a few additional complications that arise when looking at
data events, and hereafter the comparison of data and MC in the lepton + photon
control sample will be shown. We shall then extract the Wγ MT distribution from
this data, and use it to perform a data-driven estimation of the WZ background as a
function of MT . We use the exact same event selection that was discussed in section
8.6 here.
8.12.1
Trigger efficiency scale factors
In the event selection used to proof the principle of the analysis, the triggers IsoMu20
and Ele23 CaloIdL TrackIdL IsoVL were used on Wγ events, depending on the channel. The primary reason for doing this from the start is that we would at some point
need to use triggers to be able to compare data and MC predictions because CMS
datasets consist of events having passed one or more of many different triggers. One
can just apply a trigger to the reconstructed data and a trigger to the simulation samples and hope they will match, but things are not so easy in practice. Every trigger is
supposed to let through events containing one or more objects, starting from certain
PT thresholds, and possibly with some extra quality requirements. These triggers
then have a certain efficiency of letting the objects it selects pass, which depends on
the PT and η of the object attempting to pass. But unfortunately, the trigger simu122
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Figure 8.30: Single lepton trigger efficiencies as a function of the lepton’s PT , for
the trigger IsoMu20 (a), and for the triggers Ele23 CaloIdL TrackIdL IsoVL (MC)
and Ele23 WPLoose Gsf (data) (b).
lations are not perfect, so the trigger efficiency curves as a function of the PT and η
of the lepton can be quite different in data and MC. So ideally one should study the
trigger efficiency curves in data and MC, divide them, and apply the resulting scale
factors to MC events to get a good description of the data. This was done, and the
efficiencies of the triggers that were used are shown as a function of the lepton PT
in data and MC in figure 8.30. The trigger Ele23 CaloIdL TrackIdL IsoVL, used
for electrons, does not even exist in data, but there is an equivalent trigger, namely
Ele23 WPLoose Gsf. One can see that while the muon trigger efficiencies are well
described in simulations, the efficiency curves for electrons are notably different. In
all the plots below, comparing data and MC, the MC events were scaled by the ratio
of the data and MC trigger efficiencies, taken from 2D trigger efficiency maps as a
function of η and PT , provided by Illia Khvastunov. For electrons there was another
complication that was not present for muons. The trigger that was used is devised
to select events with an electron PT greater than 23 GeV, while we applied a PT
threshold of 20 GeV on this electron in the event selection we employed in earlier
sections. This should not be a problem when just using a trigger, and one can just
expect very low trigger efficiencies for events with an electron PT below 23 GeV.
When applying efficiency scale factors however, these events become problematic.
At these low values, the efficiency uncertainties are close to 100%, and some very
large scale factors with dreadfully large uncertainties might result. To get rid of
these pathological efficiency corrections, the PT threshold on electrons had to be
increased to 23 GeV when comparing data and MC.
123
8.12.2
Drell-Yan background in the electron channel
Any photon we select must pass a veto on a pixel hit matched to its ECAL deposit
and a conversion safe electron veto, as disclosed in the section on object selection.
Both of these cuts are intended keep electrons from faking the photons we intend to
select. When selecting photons, fakes entering the event selection are far more likely
to come from electrons, than any other object because both photons and electrons
rely on the ECAL for their reconstruction. But even applying these two separate
vetoes to reduce electron fakes, it will prove insufficient to reduce the background
from Drell-Yan events in the electron channel, because of the very large Drell-Yan
cross section. We select one photon and one electron in this channel, whereas electronically decaying Drell-Yan events have two electrons. One of these electrons can
pass the photon selection, while the other fakes the photon. These events have a
cross section of thousands of pb, so even a small efficiency of passing the electron
+ photon selection might give a significant background. This effect can be clearly
illustrated in a beautiful, but unfortunate, plot of the invariant mass constructed
from the electron and photon passing our event selection, shown in figure 8.31. One
can see a clear peak at the mass of the Z boson in this invariant mass distribution
in both data and MC, indicative of the fact that the photon is in fact faked by an
electron from a Z boson decay. The MET and PT (γ) thresholds were lowered from
50 GeV to 30 GeV to clearly illustrate the Z mass peak. The same plot is shown
in the muon channel on the left of figure 8.32, and even with the lowered MET and
PT (γ) threshold, the Drell-Yan contribution is extremely small here. This is exactly
what we expect, because muons interact relatively little with the ECAL, making
it hard for them to fake photons. An invariant mass distribution of the electronphoton system, with the actual event selection we use in the analysis is shown on the
right of figure 8.32. Because Drell-Yan events have no neutrinos, the MET threshold
reduces their dominance, but their yields remain significant.
In the end we have two options if we want to measure the MT shape of Wγ in the
electron channel. Either we subtract the Drell-Yan MC prediction, and introduce
a serious MC reliance into our WZ background prediction that is supposed to be
data-driven. Or we find a way to reduce this Drell-Yan background. The only
way available at the moment is to remove any electron channel event in which the
electron-photon system has an invariant mass inside the Z boson mass window. To
be precise, we will remove events for which: 75 GeV < Meγ < 105 GeV. Such a cut
will influence the MT shape in a way that is very hard to anticipate, and we will have
to take its influence on Wγ’s MT shape into account as a systematic uncertainty.
8.12.3
tt background
One of the major backgrounds from which the Wγ signal suffers in both the muon
and the electron channel is top quark pair production. While many backgrounds
fall off near the tails of the MT distribution, tt is still prominent at the highest MT
values because it usually goes hand in hand with a large MET. As mentioned before,
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Figure 8.31: Invariant mass distribution of the electron-photon system, compared
in data and MC. A clear Z boson mass peak can be seen in data and MC, while
both the pixel hit veto, and the conversion safe electron veto were applied in the
photon object selection. From this figure it becomes extremely clear that electrons
and photons are hard to distinguish. For dramatic effect, the MET and PT (γ)
thresholds have both been lowered to 30 GeV, coming from 50 GeV in our actual
event selection.
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Figure 8.32: Invariant mass distributions of the lepton-photon system, in the muon
channel with MET and PT (γ) cuts of 30 GeV, and in the electron channel with both
cuts at 50 GeV.
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Figure 8.33: Number of b-tagged jets in data and MC, in the muon channel (a), and
the electron channel (b).
top quarks almost invariably decay to beauty quarks, so we should be able to heavily
reduce this background by vetoing the presence of jets that have been identified as
coming from beauty quarks. The distribution of the number of b-tagged jets is
shown in figure 8.33, and one can see that the largest fraction of the tt background
populates the region with more than one b-tagged jet, while only a small part of the
signal resides here. So in order to reduce the MC reliance of our WZ background
estimation further, we chose to veto events with more than one b-tagged jet.
8.12.4
Data versus MC
Now that a few problems have been dealt with, it is finally time to look at the lepton + photon control sample in data. The simulation predictions are compared to
the data collected by CMS in several kinematic distributions in figures 8.34 to 8.43.
Data and MC predictions can be seen to match well, both in terms of total yields
and in most kinematic distributions in both the muon and electron channel. The
most important distribution for our purposes is obviously the MT , and it is shown,
on a linear scale in figure 8.43, and on a logarithmic scale for respectively the muon
and electron channel in figures 8.44 and 8.45.
In both channels the data and MC prediction of the MT distribution match relatively well, though there is one bin in the muon channel that catches the eye,
namely the first one. It is not immediately clear why there is a seemingly large
data excess in this bin. There is no such excess in the electron channel, making
it unlikely that some relevant MC sample has been forgotten. The fact that the
excess is residing at very low MT values indicates it originates from fake objects.
This hypothesis can be tested by tightening the identification and isolation requirements on the muon. If the excess is truly caused by fakes, it should become smaller
when tightening the muon object selection, whereas it should remain the same if it
is caused by true prompt muons. The MT plot after tightening the muon selection
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Figure 8.34: Photon PT distribution compared in data and MC, in the muon channel
(a), and the electron channel (b).
is shown in figure 8.46 and we see that the excess does indeed become smaller. A
muon enriched QCD simulation sample was then added to the list of MC samples in
an attempt to fill up this excess, but no single event out of this sample passed the
event and object selection we employ. The reason for this is possibly linked to the
fact that QCD simulation samples have far less events than there are QCD events
in data, and compensate for this by assigning large statistical weights to each event.
Even one event from the muon enriched QCD sample would give a contribution
much larger than the discrepancy we observe. So it is quite likely that fakes from
QCD events explain the excess, but the available MC samples do not have enough
events to be used in this context. In order to get rid of this discrepancy, data-driven
fake-rate estimations would have to be implemented, which is beyond the scope of
this thesis.
In the MT distribution of the electron channel, the MC seems to underestimate
the data in the high MT tails. It is hard to tell whether this is statistical, or
whether there is a true discrepancy between data and MC. This is a feature not
present at all in the muon channel, and no explanation has been found so far. It
seems unlikely to just be a statistical fluctuation though, because almost every high
MT bin experiences this issue. The fact that data overshoots MC in the MT tails
will directly result in a data-driven WZ background estimate in the electron channel
that overshoots the WZ MC, as shown below. Whether there is a mismodeling in
the MT tails of the MC, or if we are missing some process in our list of MC samples,
or if the discrepancy is caused by another effect has not been figured out so far and
requires further study.
8.12.5
Extracting the data driven WZ prediction
So now that we have compared data to MC in the lepton + photon + MET control
sample, we need to extract the MT distribution of WZ from this measurement. To
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Figure 8.35: Lepton PT distribution compared in data and MC, in the muon channel
(a), and the electron channel (b).
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Figure 8.36: MET distribution compared in data and MC, in the muon channel (a),
and the electron channel (b).
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Figure 8.37: Comparison of the distribution of the Azimuthal angular separation
∆Φ between the lepton and the MET in data and MC, in the muon channel (a),
and the electron channel (b).
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∆Φ(µ, γ )
(a)
3
1
0.5
1
1.5
1.5
3
∆Φ(e, γ )
(b)
Figure 8.38: Comparison of the distribution of the Azimuthal angular separation
∆Φ between the lepton and the photon in data and MC, in the muon channel (a),
and the electron channel (b).
129
√s = 13TeV, ∫ Ldt = 2.26fb −1
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∆R(e, γ )
(a)
(b)
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√s = 13TeV, ∫ Ldt = 2.26fb −1
CMS Preliminary
events /0.333333
events /0.333333
Figure 8.39: Comparison of the angular separation ∆R distribution between the
lepton and the photon in data and MC, in the muon channel (a), and the electron
channel (b).
5
∆η(e, γ )
(b)
Figure 8.40: Comparison of the distribution of the pseudorapidity separation ∆η
between the lepton and the photon in data and MC, in the muon channel (a), and
the electron channel (b).
130
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events /1
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8
10
8
10
1
0.5
2
4
6
8
0
10
0
number of jets
number of jets
(a)
(b)
Figure 8.41: Distribution of the number of jets, compared in data and MC, in the
electron channel (a), and the muon channel (b).
√s = 13TeV, ∫ Ldt = 2.26fb −1
data
Wγ
Wjets
TT
Gjets
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CMS Preliminary
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600
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1
0.5
200
400
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0
HT (GeV)
(a)
600
HT (GeV)
(b)
Figure 8.42: HT distribution compared in data and MC, in the muon channel (a),
and the electron channel (b).
131
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MT(e+ MET) (GeV)
(a)
(b)
Figure 8.43: MT distribution compared in data and MC, in the muon channel (a),
and the electron channel (b).
√s = 13TeV, ∫ Ldt = 2.26fb −1
events /20GeV
CMS Preliminary
data
Wγ
Wjets
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Gjets
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200
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Figure 8.44: Comparison of the MT distribution in data and MC in the muon channel
on a logarithmic scale.
132
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events /20GeV
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Figure 8.45: Comparison of the MT distribution in data and MC in the electron
channel on a logarithmic scale.
√s = 13TeV, ∫ Ldt = 2.26fb −1
events /20GeV
CMS Preliminary
data
Wγ
Wjets
TT
Gjets
Zγ
DYjets
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Figure 8.46: Comparison of the MT distribution in data and MC in the muon
channel, after requiring the muon to pass a very tight multiisolation working point,
and the tight muon identification criteria as listed in [70]. The excess in the first bin
has been significantly reduced compared to the plot using looser criteria, indicating
that we are missing a contribution from fake objects in our simulation prediction.
133
this end, we reweigh all MC and data events according to the reweighing scale factors
determined by reweighing the lepton PT distributions of WZ and Wγ after which all
backgrounds to Wγ, except Wjets, will be subtracted. It was shown earlier in this
chapter that Wjets can be included in the background prediction. The distribution
that remains is the reweighed Wγ + Wjets prediction from data, which should have
the same shape as the WZ MT distribution in data. The only thing that remains is
scaling this distribution to the yields we expect from WZ. One way to do this is by
relying on the theoretical cross section of WZ production. Following this procedure,
one could just use the ratio of the expected amount of WZ events, as estimated from
MC, to the amount of Wγ + Wjets events measured in data to scale the reweighed
Wγ + Wjets MT prediction. Another option is to prove that the WZ and Wγ MT
shapes can be made to match in a control region that is orthogonal to our signal
region. Once the principle is proven in such a control region, the WZ and Wγ can
be measured in data, and the ratio of the yields can be used to scale the Wγ MT
measurement in the signal regions. One needs to be careful however that there is
no SUSY signal expected in the WZ control region one chooses. If on expects signal
contamination, the difference between the Wγ to WZ scale factor as estimated by
using this control region, and as estimated by using MC might have to be used as
a nuisance parameter instead. Preliminary studies suggest this scaling method can
indeed be done by using a control region defined by 30 GeV < MET < 50 GeV, but
it warrants further research.
For now we relied on the theoretical WZ cross section, and the resulting prediction of the WZ MT distribution is shown and compared to the MC prediction in
figure 8.47 for the muon channel, and figure 8.48 for the electron channel. The
data-driven prediction matches the MC relatively well in the muon channel, except
for in the first bin, which is a direct result from the discrepancy we saw in the first
MT bin when comparing data and MC in the muon + photon control sample. Note
that even though we compare the data-driven estimate to MC, there is no reason
to assume that the MC prediction is the correct one, or is better than our new
prediction. The new prediction comes directly from data after all! In the electron
channel the comparison does not look as nice as in the muon channel. In particular,
the data-driven estimate seems to predict far stronger MT tails than the MC. This
is a direct result of the data excess we saw in the high MT of the electron + photon
control sample. As mentioned above, it is as of this moment unclear where this data
excess comes from.
8.13
Systematic uncertainties
Below we will shortly discuss the systematic uncertainties on the data-driven WZ
background estimation that have so far been identified . We will for now limit
ourselves to a qualitative discussion of these uncertainties, and determining their
exact effects requires further study.
134
events /20GeV
CMSPreliminary
−1
√ s = 13TeV, ∫ Ldt = 2.26fb
µ + γ data
10
WZ MC
1
µ + γ data/WZ MC
10−1
0
1.5
100
200
300
100
200
300
1
0.5
0
0
MT(µ+ MET) (GeV)
Figure 8.47: Comparison of the muon channel WZ MT distribution. as determined
from the muon + photon control sample, to the MC prediction.
135
events /20GeV
CMSPreliminary
−1
√ s = 13TeV, ∫ Ldt = 2.26fb
e + γ data
10
WZ MC
1
e + γ data/WZ MC
10−1
40
3
2
1
0
0
100
200
300
100
200
300
MT(e+ MET) (GeV)
Figure 8.48: Comparison of the electrons channel WZ MT distribution. as determined from the electron + photon control sample, to the MC prediction.
136
√s = 13TeV
events /20GeV
PT(µ)>10GeV
10−1
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T
1
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(a)
PT(e)>10GeV
10−1
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PT (µ )>10GeV/P (µ )>20GeV
PT(µ)>20GeV
0
√s = 13TeV
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PT (e)>10GeV/P (e)>23GeV
events /20GeV
CMS Simulation
0
100
200
300
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300
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(b)
Figure 8.49: Comparison of the MT distribution, with and without explicitly requiring the lepton from the W’s decay to have a PT greater than 20 GeV in the muon
channel (a), and 23 GeV in the electron channel (b).
8.13.1
Lepton PT thresholds in the 3 lepton signal sample
The event selection used in three lepton electroweakino searches uses a PT threshold
of 10 GeV on the lowest PT lepton, one of 10 GeV on the middle lepton if it is a
muon, and 15 GeV if it is an electron, and a PT threshold of 20 GeV on the lepton
with the highest PT . The reason for allowing leptons with PT values this low is
that the available dilepton triggers allow it, and the more signal events there are,
the better the reach of the search. When we compared the mass shapes of WZ and
Wγ in earlier sections, the lepton coming from the W’s decay in WZ was allowed to
pass any of the three PT thresholds, while the one lepton in Wγ events had a PT
threshold of 20 GeV, needed to pass single lepton triggers. So the difference in the
MT shape of WZ requiring the lepton coming from W to have a PT in excess to 20
GeV, and the MT shape when it can pass any of the thresholds, should be taken
into account as a systematic uncertainty. In the electron channel have to go even
further, and require the lepton from the W decay to have a PT in excess of 23 GeV
when studying the systematics, since this is the PT threshold of the single electron
trigger used in Wγ. The shape differences associated with these PT thresholds on
the leptons are shown in figure 8.51. It is clear that the MT distribution is indeed
influenced by increasing the PT threshold on the lepton from the W decay in WZ to
the same threshold value the lepton has in the case of Wγ.
8.13.2
Meγ requirement
To reduce the Drell-Yan background in the electron + photon channel, events in
which the electron-photon system had an invariant mass between 75 GeV and 105
GeV, close to the mass of the Z boson, were rejected. This might have some unforeseen consequences on the MT shape, so the difference in Wγ’s lepton PT reweighed
137
√s = 13TeV
events /20GeV
CMS Simulation
inclusive
10−1
Meγ veto
10−2
10−3
inclusive/Meγ veto
10−4
0
1.5
100
200
300
200
300
1
0.5
0
0
100
MT(e + MET) (GeV)
Figure 8.50: Influence on the MT shape of vetoing events in which Meγ resides
within the Z-mass window.
MT shape before and after this cut should be taken into account as a systematic
uncertainty. The reweighed MT shape of Wγ events, before and after applying this
cut is shown in figure 8.50 and the shape difference should be taken into account as
a systematic uncertainty on the WZ background prediction in the electron channel.
8.13.3
Simulation uncertainties
Even though we do not rely on the MC MT shapes of Wγ and WZ directly in the
final WZ background estimation, the entire proof of principle, as well as the lepton
PT dependent scale factors used for reweighing, are based on these simulations.
Both the Wγ and WZ simulations were produced by the MC@NLO matrix element
generator with a precision up to NLO in perturbation theory. To get a sense of
any potential mismodeling of the MT shape, we can compare the simulation sample
used to one from another NLO generator. For WZ, such a sample, generated with
the Powheg matrix element generator, was available, and the MT shapes predicted
by the different matrix element generators are compared in figure 8.35. The same
test would ideally have to be done in the case of Wγ as well, but no other NLO
Wγ sample was available at the moment. In the muon channel there seems to be no
statistically significant difference between the two MT shapes, but in the electron
channel one can see a significant difference in the high MT tails. It is however
hard to draw any definite conclusions on this systematic uncertainty, because it was
impossible to use events with Mll values below 30 GeV in the MC@NLO sample due
to a bug in the simulation, and this might influence the MT shape. This bug will
probably be solved in a version of the sample, made for a newer release of CMSSW12 ,
and it is possible that this systematic uncertainty will then disappear.
12
CMS Software (CMSSW) is a multipurpose software framework used in CMS for particle reconstruction, analyses and more.
138
√s = 13TeV
MC@NLO
Powheg
events /20GeV
CMS Simulation
10−1
10−3
200
300
1
0.5
0
0
100
200
300
MT(µ + MET) (GeV)
(a)
MC@NLO/Powheg
10−3
100
MC@NLO
Powheg
10−2
0
1.5
√s = 13TeV
10−1
10−2
MC@NLO/Powheg
events /20GeV
CMS Simulation
0
1.5
100
200
300
200
300
1
0.5
0
0
100
MT(e + MET) (GeV)
(b)
Figure 8.51: MT shape in WZ events, as simulated by the Powheg and MC@NLO matrix
element generators, in the muon channel (a), and the electron channel (b).
139
Chapter 9
Conclusions and outlook
The dominant background in searches for electroweakino production in the three
lepton final state, is SM production of a W and a Z boson. A novel method for
estimating the yields of this background as a function of MT , by measuring its MT
shape in Wγ events has been discussed and developed. It was shown in NLO MC
simulations that Wγ’s MT shape does in fact match to that of WZ after removing
the FSR contribution with kinematic cuts, and applying reweighing scale factors
derived from the ratio of the lepton PT distribution shapes. Once it was established
that the WZ MT shape could be measured in a lepton + photon final state, this
measurement was performed. A good match between MC predictions and data was
found in the muon + photon control sample, except for one bin which had a discrepancy that could be attributed to missing fake objects in the simulation. The
resulting MT shape prediction for WZ was in good agreement with the MC. In the
electron + photon control sample, the bulk of the MT distribution was described
well by simulations, but the MC prediction seemed to underestimate the high MT
tails, directly translating into much stronger MT tails in the data-driven WZ prediction than in its MC prediction. The exact origin of this discrepancy has not been
established as of now. Several sources of systematic uncertainties were identified
and discussed. While the systematic uncertainties seem to be very limited in the
muon channel, the electron channel suffers from some additional uncertainties due
to higher trigger thresholds, additional backgrounds from electrons faking photons
and potential mismodeling in MC. The exact size of the systematic uncertainties
has yet to be established, but it is expected that the new method will improve upon
the old one, and especially in the muon channel.
At the end of the road, there is still much more that could have been done. First of
all, data driven methods to estimate the contribution from fake objects in the lepton
+ photon control sample can be used to further improve the result. This might for
instance solve the discrepancy in the first MT bin of the muon channel. Secondly,
the mismatch of data and MC in the electron MT tails definitely warrants a further
investigation, before drawing conclusions. The systematic uncertainties have been
discussed, but their study is still in its infant stages. Their effects still have to be
quantified and studied in more depth. A last thing which might have been done,
140
but for which time was alas too short, is an inclusive search for new physics in the
three lepton MT distribution in data. We have made an inclusive estimate of the
dominant WZ background to such a search after all!
The results attained in this thesis have been presented at the Electroweak and
Compressed SUSY Event at the LHC Physics Center in the Fermi National Accelerator Laboratory (Fermilab), and will be used in the upcoming Run II search for
electroweak SUSY production in the three lepton final state at CMS. It has yet to
be decided whether the method will be used to validate the MC prediction, or to
directly provide the background yields in several search regions.
141
Chapter 10
Nederlandstalige samenvatting
Onze diepste inzichten in de natuur zijn samengevat in het Standaard Model van de
deeltjesfysica. Hoewel deze theorie voorlopig bijna elke meting in de hoge energie
deeltjesfysica kan verklaren, zijn er toch enkele observaties die ons doen vermoeden
dat er meer moet zijn. Één van de voornaamste uitbreidingen op het Standaard
Model is een theorie genaamd Supersymmetrie, die de aanwezigheid van fermionische partners voorspeld voor elk boson in het Standaard Model, en vice versa voor
de fermionen in het Standaard Model. De Large Hadron Collider is momenteel de
krachtigste en grootste deeltjesversneller in de wereld, en de ideale plaats voor de ontdekking van nieuwe deeltjes, zoals die voorspeld door Supersymmetrie. Één van de
best gemotiveerde plaatsen om te zoeken naar Supersymmetrie aan de Large Hadron
Collider, is in evenementen waarin drie leptonen, en ontbrekende transversale energie voorkomen. Het meest bepalende aspect voor het bereik van een zoektocht naar
nieuwe deeltjes, is de precisie waarmee het aantal Standaard Model evenementen dat
hetzelfde signaal geeft, de achtergrond, kan worden bepaald. De voornaamste achtergrond in een zoektocht in een finale toestand met drie leptonen is productie van een
W en Z boson. In vroegere analysen aan het CMS experiment werd deze achtergrond geschat, vertrekkende vanuit Monte Carlo simulaties, waarop enkele correcties
afgeleid vanuit data werden toegepast. Doorheen deze thesis werd echter een nieuwe
methode ontwikkeld, die het mogelijk maakt om de achtergrond als een functie van
de transversale massa rechtstreeks vanuit data te bepalen. Om dit te verwezelijken,
werd er naar een proces gezocht waarvan er verwacht werd dat het een gelijkaardige
transversale massa verdeling heeft. Het proces dat we besloten te gebruiken was Wγ,
of de productie van een W boson en een foton. Wγ en WZ hebben zeer gelijkaardige
productie kanalen, met het verschil dat een foton kan worden geproduceerd door
middel van foton straling van de finale toestand. Dit verschil werd weggewerkt door
verschillende kinematische voorwaarden op te leggen aan de Wγ evenementen. In
Monte Carlo simulaties, berekend tot op tweede orde in perturbatietheorie, werd
dan aangetoond dat de transversale massa distributies van Wγ and WZ inderdaad
vrij goed overeenkomen na het verweideren van de finale toestands straling. Door
dan schaalfactoren, afgeleid vanuit de verhouding van de genormalizeerde distributies in WZ en Wγ van de transversale impuls van het lepton afkomstig van het W
boson verval, toe te passen, kwamen de transversale massa distributies quasi perfect
142
overeen in zowel het elektron, als het muon kanaal. Wanneer het principe van de
analyse, namelijk dat we de transversale massa verdeling van WZ kunnen meten in
in een finale toestand met een lepton en een foton, dusdanig was aangetoond, werd
de meting effectief uitgevoerd. Hiervoor werd de transversale massa distributie in
een lepton + photon + ontbrekende transversale energie finale toestand gemeten,
en vergeleken met de som van de Monte Carlo simulaties van alle processen die aanleiding kunnen geven to zo een evenement. De data en Monte Carlo voorspelling
kwamen grotendeels goed overeen, in een groot aantal kinematische disributies. In
het muon kanaal kwam de transversale massa voorspelling ook goed overeen met de
data, maar in het elektron kanaal leek de Monte Carlo voorspelling de data te onderschatten bij hoge transversale massa waarden. Uit deze transversale massa verdeling
werd dan de meting voor de WZ achtergrond als functie van de transversale massa
afgeleid, en deze kwam goed overeen met de WZ Monte Carlo voor muonen, maar
gaf een hogere schatting voor hoge transversale massa waarden in het geval van elektronen. Dit laatste was een direct gevolg van het feit dat de Monte Carlo de data
onderschatte in de staarten van de transversale massa distributie in het electron
kanaal. Er werden verschillende bronnen van systematische onzekerheden geı̈dentificeerd, die vooral een effect hadden in het elektron kanaal, maar een kwantitatieve
voorspelling van hun effecten vergt nog meer studie. Al bij al werd aangetoond dat
de nieuwe methode werkt, en verwachten we vooral in het geval van muonen een verbetering te kunnen geven ten opzichte van de oude methode om de WZ achtergrond
te bepalen in functie van de transversale massa. Dit resultaat zal dan ook gebruikt
worden in de toekomstige zoektocht van de CMS collaboratie naar de elektrozwakke
productie van Supersymmetrie in de finale toestand met drie leptonen.
143
Appendices
144
Appendix A
Simulation samples
A list of all the samples used to prove the principle of the analysis is laid out below:
• WZ:
/WZJets TuneCUETP8M1 13TeV-amcatnloFXFXpythia8/RunIISpring15DR74-Asympt25ns MCRUN2 74 V9v2/MINIAODSIM
• Wγ:
/WGToLNuG TuneCUETP8M1 13TeV-amcatnloFXFXpythia8/RunIISpring15DR74-Asympt25ns MCRUN2 74 V9v1/MINIAODSIM
• Wjets:
/WJetsToLNu TuneCUETP8M1 13TeV-amcatnloFXFXpythia8/RunIISpring15DR74-Asympt25ns MCRUN2 74 V9v1/MINIAODSIM
As these names suggest, these samples are simulations of proton proton collisions at
13 TeV resulting in respectively leptonically decaying WZ, Wγ and Wjets events.
The WZ sample does not have ”To3LNu” in its name, which would typically indicate
that not only leptonically decaying W and Z’s were simulated, but also other decays
of no interest to this analysis, but the simulated decays are purely leptonic nonetheless. All of the samples listed above were generated with the MC@NLO matrix element
generator, meaning they are accurate up to NLO in perturbation theory. This matrix element generator was interfaced with Pythia8 in all three cases to handle the
hadronization and parton showering steps of the simulation. The TuneCUETP8M1
stands for the particular tune that was used with Pythia, but we shall not wade
into more details about it here. All simulations were used in the MINIAOD format,
which is a format for storing the variables necessary for most analyses, widely used
within the CMS collaboration. RunIISpring15 indicates the campaign in which the
samples were produced, namely in the spring of 2015 for the LHC’s Run II, and
DR74 indicates the CMSSW version the samples were made for, being in this case
7.4.X.
145
All the samples used to compare data and MC in the lepton + photon control
sample are:
• Wγ:
/WGToLNuG TuneCUETP8M1 13TeV-amcatnloFXFXpythia8/RunIISpring15DR74-Asympt25ns MCRUN2 74 V9v1/MINIAODSIM
• Wjets:
/RunIISpring15DR74/WJetsToLNu TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v1/
• Drell-Yan + jets:
– 10 GeV < Mll < 50 GeV:
RunIISpring15DR74/DYJetsToLL M-10to50 TuneCUETP8M1 13TeVamcatnloFXFX-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
– Mll > 50 GeV:
RunIISpring15DR74/DYJetsToLL M-10to50 TuneCUETP8M1 13TeVamcatnloFXFX-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
• γ + jets :
– 40 GeV < HT < 100 GeV:
/RunIISpring15DR74/GJets HT-40To100 TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v2/
– 100 GeV < HT < 200 GeV:
/RunIISpring15DR74/GJets HT-100To200 TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v2/
– 200 GeV < HT < 400 GeV:
/RunIISpring15DR74/GJets HT-200To400 TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v2/
– 400 GeV < HT < 600 GeV:
/RunIISpring15DR74/GJets HT-400To600 TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
– HT > 600 GeV:
/RunIISpring15DR74/GJets HT-600ToInf TuneCUETP8M1 13TeVmadgraphMLM-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
146
• Zγ:
/RunIISpring15DR74/ZGTo2LG TuneCUETP8M1 13TeV-amcatnloFXFXpythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v1/
• tt:
RunIISpring15DR74/TT TuneCUETP8M1 13TeV-powhegpythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v2/
• rare SM processes:
– ttγ:
RunIISpring15DR74/TTGJets TuneCUETP8M1 13TeVamcatnloFXFX-madspin-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
– ttZ:
/RunIISpring15DR74/TTZToLLNuNu M-10 TuneCUETP8M1 13TeVamcatnlo-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v1/
– ttW:
/RunIISpring15DR74/TTWJetsToLNu TuneCUETP8M1 13TeVamcatnloFXFX-madspin-pythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9v1/
– ZZ:
/RunIISpring15DR74/ZZ TuneCUETP8M1 13TeVpythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v3/
– ZZZ:
/RunIISpring15DR74/ZZZ TuneCUETP8M1 13TeV-amcatnlopythia8/MINIAODSIM/Asympt25ns MCRUN2 74 V9-v2/
When available, MC@NLO or Powheg NLO samples were used, but for some samples
only LO simulations were readily available. The generator, and parton showering
program with which the different samples were produced can be deduced from their
names. The same Wγ sample was used as during the proof of principle when looking
at data, but an LO Wjets sample was used instead of the NLO sample used earlier.
This was done in order to gain a bit of statistics, because only very few of the Wjets
events pass the event selection and sample cleaning that was used, and the available
MadGraph sample was about three times as large as the MC@NLO sample. When
showing that the MT shapes in WZ and Wγ + Wjets can be made to match, both
Wjets samples were tested, and no significant difference in the result was found, and
both had extremely low statistics.
147
Appendix B
Data samples and luminosity
sections
The data that was used for this measurement were the SingleMuon and SingleElectron datasets, collected during 2015’s LHC operation at a center of mass energy of
13 TeV with 25ns of time between bunch crossings. These datasets are separated
into different subsections of data taken during a run in which the instantaneous
luminosity remained unchanged, called luminosity sections. The luminosity sections
for which all the CMS detector subsystems were performing adequately are listed
in Java Script Object Notation (JSON) files, and the one used for this thesis was:
Cert_246908-260627_13TeV_PromptReco_Collisions15_25ns_JSON_v2.JSON, often called the ”golden JSON file”. [80] The luminosity sections contained in this
JSON file correspond to an integrated luminosity of 2.26 fb−1 . Other JSON files are
available, which allow luminosity sections in which different detector systems such as
the magnet, or the forward hadronic calorimeter did not work or their information
was not available. These can can be found at [81], but they were of no interest to
this analysis however.
148
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