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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 6 7 7 9 9 12 13 18 19 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 24 24 26 26 27 27 27 28 . . . . . . 30 30 31 36 36 38 40 . . . . . . . . . . . . . . . . . . . . . . . . . . 40 41 42 43 44 44 44 45 . . . . . . . . . . . 47 47 49 51 51 51 54 55 56 56 57 58 6 Software techniques 6.1 Monte Carlo event generation . . . . . . . . . . . . . . . . . . . . . . 6.2 CMS detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . 60 60 62 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 64 64 64 66 70 70 70 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 80 83 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 76 84 86 86 90 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 92 92 93 95 95 95 96 97 99 100 106 108 110 113 117 117 120 122 122 124 124 126 127 134 137 137 138 9 Conclusions and outlook 140 10 Nederlandstalige samenvatting 142 Appendices 144 A Simulation samples 145 B Data samples and luminosity sections 148 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 4 9 12 13 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] . . . . . . . 25 26 28 29 4.1 5.1 Triangle diagram, leading to a chiral gauge anomaly [50]. . . . . . . v 32 33 34 36 37 39 39 40 41 42 52 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] . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 61 65 67 69 73 74 75 77 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 √s = 13TeV CMS Simulation 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 √s = 13TeV CMS Simulation 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 √s = 13TeV 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 √s = 13TeV CMS Simulation normalized events /1.8GeV normalized events /1.8GeV 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 √s = 13TeV CMS Simulation normalized events /1.9GeV normalized events /1.9GeV CMS Simulation WZ 0.15 Wγ 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 √s = 13TeV CMS Simulation 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 √ s = 13TeV normalized events /20GeV CMS Simulation 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 √s = 13TeV normalized events /12GeV 0.15 √s = 13TeV CMS Simulation 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 √ s = 13TeV 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 10−4 10−4 −5 10 0 5 10 (reco MET - gen MET)/gen MET (a) 0 5 10 (reco MET - gen MET)/gen MET (b) 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. 11 119 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 800 600 400 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /12.5GeV events /12.5GeV CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 400 200 50 1.5 100 150 200 250 300 1 obs/pred obs/pred 200 0.5 0 50 50 1.5 100 150 200 100 150 200 250 300 250 300 1 0.5 100 150 200 250 300 0 50 PT(γ ) (GeV) (a) PT(γ ) (GeV) (b) 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 120 √s = 13TeV CMS Simulation WZ 10−1 Wγ + Wjets 10−2 normalized events /20GeV normalized events /20GeV CMS Simulation WZ 10−1 Wγ + Wjets 10−2 10−3 30 100 200χ2 / ndf 6.573 / 14300 p0 2 0.9559 ± 0.0527 1 0 0 100 200 300 WZ/Wγ + Wjets 10−3 WZ/Wγ + Wjets √s = 13TeV 0 1.5 100 200χ2 / ndf p0 12.99 / 14300 0.8965 ± 0.0514 1 0.5 0 0 100 200 300 MT(lepton + MET) (GeV) MT(lepton + MET) (GeV) (a) (b) 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. √s = 13TeV WZ 10−1 Wγ + Wjets 10−2 √s = 13TeV WZ 10−1 Wγ + Wjets 10−2 10−3 10−3 WZ/Wγ + Wjets normalized events /20GeV CMS Simulation 0 3 200χ2 / ndf p0 100 3.964 / 14300 1.12 ± 0.07 2 1 0 0 100 200 300 WZ/Wγ + Wjets normalized events /20GeV CMS Simulation 0 3 100 200χ2 / ndf p0 5.277 / 14300 1.058 ± 0.069 2 1 0 0 100 200 300 MT(lepton + MET) (GeV) MT(lepton + MET) (GeV) (a) (b) 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 √s = 13TeV CMS Simulation trigger efficiency trigger efficiency CMS Simulation 1 data MC 20 40 60 80 100 1.5 1 0.5 0 data MC 0.5 data/MC data/MC 0.5 √s = 13TeV 1 20 40 60 80 100 20 40 60 80 100 1.5 1 0.5 20 40 60 80 100 0 PT(µ) (a) PT(e) (b) 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, 124 √s = 13TeV, ∫ Ldt = 2.26fb −1 events /3.4GeV CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 2000 1500 1000 obs/pred 500 50 100 150 200 50 100 150 200 1.5 1 0.5 0 Mlγ (GeV) 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. √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 600 400 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /3.4GeV events /3.4GeV CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 200 100 50 100 150 200 1.5 1 obs/pred obs/pred 200 0.5 0 50 100 150 50 100 150 200 1.5 1 0.5 50 100 150 200 Mlγ (GeV) 0 200 Meγ (GeV) 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. 125 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 1000 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /1 events /1 CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 800 600 400 500 0 1.5 2 4 6 8 10 1 obs/pred obs/pred 200 0.5 0 0 0 1.5 2 4 6 2 4 6 8 10 8 10 1 0.5 2 4 6 8 10 0 0 number of b jets (a) number of b jets (b) 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 126 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 400 100 150 200 250 300 1 0.5 0 50 data Wγ Wjets TT Gjets Zγ DYjets rare SM 200 100 obs/pred obs/pred 200 50 1.5 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /12.5GeV events /12.5GeV CMS Preliminary 50 1.5 100 150 200 100 150 200 250 300 250 300 1 0.5 100 150 200 250 300 0 50 PT(γ ) (GeV) (a) PT(γ ) (GeV) (b) 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 127 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 400 300 200 200 data Wγ Wjets TT Gjets Zγ DYjets rare SM 150 100 50 100 200 300 1.5 1 obs/pred 100 obs/pred √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /13.85GeV events /14GeV CMS Preliminary 0.5 100 200 100 200 300 1.5 1 0.5 0 100 200 0 300 PT(µ) (GeV) 300 PT(e) (GeV) (a) (b) Figure 8.35: Lepton PT distribution compared in data and MC, in the muon channel (a), and the electron channel (b). √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 400 300 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /12.5GeV events /12.5GeV CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 300 200 200 100 50 1.5 100 150 200 250 300 1 obs/pred obs/pred 100 0.5 0 50 50 1.5 100 150 200 100 150 200 250 300 250 300 1 0.5 100 150 200 250 300 0 50 MET (GeV) (a) MET (GeV) (b) Figure 8.36: MET distribution compared in data and MC, in the muon channel (a), and the electron channel (b). 128 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 60 40 data Wγ Wjets TT Gjets Zγ DYjets rare SM 40 30 20 20 10 0 1.5 1 2 obs/pred obs/pred √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /0.213333 events /0.213333 CMS Preliminary 3 1 0.5 0 1.5 1 2 1 2 3 1 0.5 0 0 1 2 0 3 0 3 ∆Φ(µ, MET) ∆Φ(e, MET) (a) (b) 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). √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 150 100 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /0.146667 events /0.146667 CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 100 50 1 1.5 2 2.5 obs/pred obs/pred 50 3 1.5 1 1 0.5 0 0 1.5 2 2.5 3 2 2.5 1.5 2 2.5 1 ∆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 data Wγ Wjets TT Gjets Zγ DYjets rare SM 300 200 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /0.466667 events /0.466667 CMS Preliminary 400 data Wγ Wjets TT Gjets Zγ DYjets rare SM 300 200 100 0 1.5 2 4 obs/pred obs/pred 100 6 1 0.5 0 1.5 2 4 2 4 6 1 0.5 0 0 2 4 0 6 0 6 ∆R(µ, γ ) ∆R(e, γ ) (a) (b) 300 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 200 100 1 2 3 4 5 1 0 1.5 0.5 0 0 1 2 3 4 5 1 2 3 4 1 2 3 4 0 ∆η(µ, γ ) (a) 5 1 0.5 0 data Wγ Wjets TT Gjets Zγ DYjets rare SM 150 50 obs/pred obs/pred 100 0 1.5 √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 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 400 2 4 6 8 10 obs/pred obs/pred 100 1 0.5 0 0 data Wγ Wjets TT Gjets Zγ DYjets rare SM 300 200 200 0 1.5 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /1 events /1 CMS Preliminary 600 0 1.5 2 4 6 2 4 6 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 Zγ DYjets rare SM 200 data Wγ Wjets TT Gjets Zγ DYjets rare SM 150 100 50 200 400 600 1.5 1 obs/pred obs/pred 100 0.5 0 √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /38GeV events /38GeV CMS Preliminary 300 200 400 200 400 600 1.5 1 0.5 200 400 600 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 √s = 13TeV, ∫ Ldt = 2.26fb −1 data Wγ Wjets TT Gjets Zγ DYjets rare SM 300 200 data Wγ Wjets TT Gjets Zγ DYjets rare SM 150 100 50 0 1.5 100 200 300 1 obs/pred 100 obs/pred √s = 13TeV, ∫ Ldt = 2.26fb −1 CMS Preliminary events /20GeV events /20GeV CMS Preliminary 0.5 0 1.5 100 200 300 200 300 1 0.5 0 0 100 200 300 0 0 100 MT(µ+ MET) (GeV) 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 TT Gjets Zγ DYjets rare SM 10 2 10 obs/pred 1 0 1.5 100 200 300 100 200 300 1 0.5 0 0 MT(µ+ MET) (GeV) Figure 8.44: Comparison of the MT distribution in data and MC in the muon channel on a logarithmic scale. 132 √s = 13TeV, ∫ Ldt = 2.26fb −1 events /20GeV CMS Preliminary data Wγ Wjets TT Gjets Zγ DYjets rare SM 10 2 10 obs/pred 1 0 1.5 100 200 300 100 200 300 1 0.5 0 0 MT(e+ MET) (GeV) 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 rare SM 300 200 obs/pred 100 0 1.5 100 200 300 200 300 1 0.5 0 0 100 MT(µ+ MET) (GeV) 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 PT(e)>23GeV 10−2 10−3 10−3 100 200 300 1.5 T 1 0.5 0 0 100 200 300 MT(µ + MET) (GeV) (a) PT(e)>10GeV 10−1 10−2 T PT (µ )>10GeV/P (µ )>20GeV PT(µ)>20GeV 0 √s = 13TeV CMS Simulation PT (e)>10GeV/P (e)>23GeV events /20GeV CMS Simulation 0 100 200 300 200 300 1.5 1 0.5 0 0 100 MT(e + MET) (GeV) (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. 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