Studies of events with a Z or a photon vector boson in association with a large invariant mass dijet system, produced with electroweak and strong interactions, in proton-proton collisions at 13 TeV center of mass energy.

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Studies of events with a Z or a 𝛾 vector boson in

association with a large invariant mass dijet system,

produced with electroweak and strong interactions,

in proton-proton collisions at

√

𝑠 = 13 TeV.

by

Negin Shafiei

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

Doctor of Science in experimental of particle physics

at the

University of Pisa, INFN sezione di Pisa

March 2021

© University of Pisa 2021. All rights reserved.

Author . . . .

Department of Physics

Certified by. . . .

Paolo Azzurri Supervisor

Associate Professor

Thesis Supervisor

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Studies of events with a Z or a 𝛾 vector boson in association

with a large invariant mass dijet system, produced with

electroweak and strong interactions, in proton-proton

collisions at

√

𝑠 = 13 TeV.

by

Negin Shafiei

Submitted to the Department of Physics on , in partial fulfillment of the

requirements for the degree of

Doctor of Science in experimental of particle physics

Abstract

This thesis present studies of 𝛾 or Z plus two jet events produced with strong and

electroweak interactions in proton-proton collisions at √𝑠=13 TeV. Events with a 𝛾

or Z and a pair of jets with a large dijet invariant mass, are selected. This thesis makes use of data collected by CMS detector, in 2016, 2017 and 2018 with integrated

luminosities of 35.9 f𝑏−1_{, 41.4 f𝑏}−1 _{and 59.73 f𝑏}−1_{, respectively. In this work the}

distribution of data/MC comparison with different generators will be shown. The signal can be established based on the fit to a multivariate discriminator, in two independent data samples according to their trigger paths, with the medium photon

transverse momentum 75 < 𝑝𝛾

𝑇 < 200 𝐺𝑒𝑉, and large photon transverse momentum

𝑝𝛾_𝑇 > 200 𝐺𝑒𝑉.

Thesis Supervisor: Paolo Azzurri Supervisor Title: Associate Professor

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Acknowledgments

This thesis brings an end to three interesting years in which I had the opportunity to meet and collaborate with many brilliant and nice people. Hereby I would like to express my gratitude to everybody who helped me to complete my PhD research.

Special thanks to my supervisor Paolo Azzurri without his guidance and support, this thesis and analysis would not have been done. Also thanks to him to give me the chance to start a PhD and for welcoming me into the friendly atmosphere of the University of Pisa particle physics group and INFN CMS group. Also I would like to thanks Pedro Silvia and Hamed Bakhshian who helped me to develop this thesis with their great ideas and their involvement in every aspect of my research and was always available to discuss results.

Many thanks to the people involved with the measurement of the EW 𝛾 jj study. A special mention to Abideh Jafari, the force behind the data-driven analysis. Also special thanks to Marco Zaro for great physics discussion and techniques of generating events by MadGraph5_aMCNLO. Also thanks to Giulio Mandorli and Leonardo Giannini who were not only an amazing office mate, but also helped me in research along the journey.

In addition to people already mentioned above, I would like to thank the other members of CMS group of INFN and Physics group of university of Pisa, Andrea Rizzi, Andrea Venturi and Dario Pisignona. Also thanks to the many people in the wider CMS collaboration which helped me one way or another.

Finally, thanks to my family and friends for the support they gave me all these years. A special mention goes to Notash, Iman, Sara, Giulia, Suvankar.

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List of Figures

1 Diagrams that produce 𝛾𝑗𝑗 final states in proton-proton collisions with

purely electroweak interactions (𝛼3

𝑊). (Left) Vector Boson Fusion.

(Right) bremsstrahlung. . . 21

2 Typical photon plus two jets diagrams that produce 𝛾𝑗𝑗 final states in

proton-proton collisions with mixed electroweak and QCD interactions

(𝛼𝑊𝛼2𝑆). (Left) A process that may interfere with the pure electroweak

production. (Right) A process that does not interfere with the pure

electroweak production. . . 21

3 Representative Feynman diagrams for dilepton production in

associa-tion with two jets from purely electroweak contribuassocia-tions: (left) vector boson fusion, (right) bremsstrahlung- like, and (bottom)

multiperiph-eral production. . . 23

4 Representative diagrams for order 𝛼𝑆2 corrections to DY production

that comprise the main background. . . 23

5 Feynman diagrams for self-interaction of SU(N) gauge bosons. . . 38

6 Candidates for a Higgs produced with a Z. ATLAS (l): both decay

ultimately to leptons, leaving two electrons (green) and four muons (red). CMS (r): the Higgs decays to two charm quarks forming jets (cones); the Z decays to electrons (green) (Image: ATLAS/CMS/CERN) 40

7 The lowest-order Feynman diagram for the Drell-Yan process. . . 41

8 Feynman diagrams for consecutive Loops contributes to Log coefficient. 43

9 The LHC ring with four detectors. . . 46

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11 CMS detector . . . 48

12 𝜂 range of CMS detector . . . 49

13 A longitudinal section of CMS ECAL layout . . . 50

14 𝜂 range of inner Tracker . . . 50

15 ECAL energy resolution as a function of electron energy, measured from a beam test. The stochastic (S), noise (N) and constant (C) terms of the fit are given. . . 52

16 Longitudinal view of the CMS detector showing the locations of the HB, HE, HF and HO detectors. . . 54

17 Jet transverse energy resolution as a function of the transverse energy for the barrel, endcap and forward region [1] . . . 55

18 Overview of the CMS trigger system . . . 58

19 Specific particle interactions in a transverse slice of the CMS detector, from the beam interaction region to the muon detector. . . 59

20 Event display of an illustrative jet made of five particles only in the (x, y) view. . . 62

21 An illustration of two overlapping Multi 5 × 5 clusters. Crystals indi-cated in yellow are eligible to seed further Multi 5×5 clusters, provided they are local maxima in energy . . . 64

22 Shower shape of 𝑍 → 𝑒+_𝑒− _{sample as signal with quit good agreement} in 2017 data simulation with MC. . . 68

23 Relative electron (ECAL) energy resolution plotted for data and Monte Carlo events, unfolded in bins of pseudorapidity (𝜂) for low bremsstrahlung 𝑍 → 𝑒+_𝑒− _{electrons with 𝑅} 9 > 0.94 (left) and for bremsstrahlung electrons with 𝑅9 < 0.94 (right) . The data resolution obtained after corrections for certain time-dependent effects is shown in gray. The res-olution obtained from a full recalibration using the entire 2017 dataset is shown in blue. . . 71

24 Example of fits to the 𝑍 → 𝑒+_𝑒− _{invariant mass distribution,electrons} satisfy pT > 25(20) GeV and medium cut based ID. . . 72

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25 A simple example of an event showing the point of collision, the frag-mentation and hadronization of the quarks and gluons and the result-ing jet found through the detection of the stable particles. Calojets are those jets created using the calorimeter output whereas Genjets are jets created using stable simulated particles. The dashed line represents the

direction of the missing energy. . . 73

26 matching reco and gen particles . . . 75

27 The plot of QCD sample for 𝑝𝑇 80-120 GeV with hadronic final state.

It shows, the both cuts has good response. . . 76

28 The four main jet reconstruction algorithms areas, performed on the same data with the same input radius. Noted features are the high irregularity in the the Kt algorithms area, the conical shape of the Anti-Kt’s jets illustrating this algorithms preference for hard radiation and the smaller effective radius of the SIScone, due to the split merge procedure, which can be observed via smaller jet areas and two jets being resolved in the place of just the one grey jet. The different

colours are used to represent the different jets and their areas. . . 80

29 Fraction of jets which are a light quark jet rather than gluon jet. . . . 83

30 Schematic view of a decision tree. Starting from the root node, a sequence of binary splits using the discriminating variables xi is applied to the data. Each split uses the variable that at this node gives the best separation between signal and background when being cut on. The same variable may thus be used at several nodes, while others might not be used at all. The leaf nodes at the bottom end of the tree are labeled "S" for signal and "B" for background depending on the majority of events that end up in the respective nodes. For regression trees, the node splitting is performed on the variable that gives the maximum decrease in the average squared error when attributing a constant value of the target variable as output of the node, given by

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31 Feynman diagrams illustrating the main prompt photons production

processes at LO and NLO. . . 95

32 The 𝑝𝑇 of the vector boson in the HighVPt(left) and LowVPt (right)

categories. categories definition are in section 11.3 . . . 96

33 The 𝑝𝑇 of the Leading jet in the HighVPt(left) and LowVPt (right)

categories. . . 97

34 The Δ𝜑 between vector boson and Leading jet in the HighVPt(left)

and LowVPt (right) categories. . . 97

35 The 𝑝𝑇 of vector boson in the HighVPt(left) and LowVPt (right)

cat-egories. . . 102

36 The 𝑝𝑇 of Leading jet in the HighVPt(left) and LowVPt (right) categories.103

37 The Δ𝜑 between vector boson and Leading jet in the HighVPt(left) and LowVPt (right) categories. . . 103 38 The Δ𝜑 between vector boson and sub-Leading jet in the HighVPt(left)

and LowVPt (right) categories. . . 104 39 The Δ𝜑 between vector boson and two hardest jets in the HighVPt(left)

and LowVPt (right) categories. . . 104 40 The Δ𝜂 between two hardest jets in the HighVPt(left) and LowVPt

(right) categories. . . 105 41 The scalar sum of pt of vector boson and two hardest jets in the

High-VPt(left) and LowVPt (right) categories. . . 105

42 The 𝑃𝑇 of the vector boson in the LowVPt (Left) and HighVPt(Right)

categories where the vector boson is a photon (first, third and fifth rows) and a Z boson (second, fourth and sixth rows). The first (second and last) two rows are for the 2016 (2017 , 2018). Also The shaded area belongs to systematics. . . 113

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43 The vector boson rapidity in dijet rest frame in the LowVPt (Left) and HighVPt (Right) categories where the vector boson is a photon (first, third and fifth rows) and a Z boson (second, fourth and sixth rows). The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 114 44 The vector boson 𝜂 in the LowVPt (Left) and HighVPt (Right)

cate-gories where the vector boson is a photon (first, third and fifth rows) and a Z boson (second, fourth and sixth rows). The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 115

45 The 𝑃𝑇 of the leading jet in the HighVPt(right) and LowVPt (Left)

categories in the signal (first, third and fifth rows) and the Z + jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 116

46 The 𝑃𝑇 of the sub-leading jet in the HighVPt(right) and LowVPt(Left)

categories in the signal (first, third and fifth rows) and the Z + jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 117 47 The quark-gluon jet discriminator in the HighVPt (right) and LowVPt

(Left) categories in the signal region. The first (second,last) row is for the 2016 (2017, 2018) analysis. Also The shaded area belongs to systematics. . . 118

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48 The 𝑚jj distribution in the HighVPt (Right) and LowVPt(Left) cate-gories in the signal (first, third and fifth rows) and the Z+jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics.The cut on dijet invariant mass leads to separation be-tween the signal and the background in high category. The signal will appear as an excess over the background, in the tail of very high value in High category. . . 119

49 The distribution of Δ𝑅jj _{between two hard jets in the HighVPt(right)}

and LowVPt (Left) categories in the signal (first, third and fifth rows) and the Z + jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systemics. The peak of events around 3, shows two partons are well separated. . . 120

50 The Δ𝜂jj _{distribution in the HighVPt(right) and LowVPt (Left)}

cate-gories in the signal (first, third and fifth rows) and the Z+jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 121 51 The distribution of Δ𝜑 between leading jet and vector boson in the

HighVPt(right) and LowVPt (Left) categories in the signal (first, third and fifth rows) and the Z+jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systemic’s. . . 122

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52 The distribution of Δ𝜑 between sub-leading jet and vector boson in the HighVPt(right) and LowVPt (Left) categories in the signal (first, third and fifth rows) and the Z + jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systemics. The azimuthal distribution shows about 2/3 of signal events have photon collinear. . . 123 53 The distribution of Δ𝜑 between the vector boson and the di-jet system

in the HighVPt(right) and LowVPt (Left) categories in the signal (first, third and fifth rows) and the Z + jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systemics. . . 124

54 The 𝑝𝑇 of the system of two leading jet and the vector boson, in the

HighVPt (right) and LowVPt(Left) categories where the vector boson is a photon (first, third and fifth rows) or a Z boson (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics.125 55 The event isotropy in the HighVPt (right) and LowVPt (Left)

cate-gories in the signal (first, third and fifth rows) and the Z+jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics. . . 126 56 The jet multiplicity (event circularity) in the HighVPt (right) and LowVPt

(Left) categories in the signal (first, third and fifth rows) and the Z+jets (second, fourth and sixth rows) regions. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area be-longs to systematics. . . 127 57 The BDT output in the HighVPt (right) and LowVPt (Left) categories

in the signal region. The first (second and last) two rows are for the 2016 (2017 , 2018) analysis. Also The shaded area belongs to systematics.128

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58 The response plots for the BDT output (left) and the corresponding ROC curves(right) for lowpt category 2016 dataset. . . 130 59 The response plots for the BDT output (Right) and the corresponding

ROC curves(right) for Highpt category 2016 dataset. . . 131 60 The response plots for the BDT output (left) and the corresponding

ROC curves(right) for lowpt category 2017 dataset. . . 131 61 The response plots for the BDT output (Right) and the corresponding

ROC curves(right) for Highpt category 2017 dataset. . . 132 62 The response plots for the BDT output (left) and the corresponding

ROC curves(right) for lowpt category 2018 dataset. . . 132 63 The response plots for the BDT output (Right) and the corresponding

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List of Tables

1 The twelve fundamental fermions divided into quarks and leptons. The

masses of the quarks are the current masses. . . 26

2 the cross-sections of 𝛾 + 2jet producing by MG 5_aMC@NLO-v2.6.3

in two different channel with and without exclusion of s-channel for further optimization. . . 100

3 cross-sections of 𝑍+jets and 𝑊 +jets producing by MG

5_aMC@NLO-v2.6.3 and for 𝑊± _{decay to 𝜇}−/+ _{the precision is → −2.2% and the}

correction for off-shell contributions is → 5% . . . 101

4 Signal and interference MC samples, all generated with Madgraph

and interfaced, as indicated, with 𝑃 𝑦𝑡ℎ𝑖𝑎 − 8.2 or HERWIG++ for hadronization and showering. . . 106

5 Summary of HLT paths and the corresponding integrated luminosity

for the Single-Photon primary datasets. . . 108

6 Input variables for N+1 test. . . 130

7 Variables as input to the MVA discriminator. In event shape variables,

𝑞𝑖’s are eigenvalues of the momentum tensor

∑︀ 𝑝𝑗[𝑎]*𝑝𝑗[𝑏]

∑︀ 𝑝2

𝑗 normalized to

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1 Summary

This thesis is an investigation of events with Z or 𝛾 boson with two jets in the LHC

proton-proton collision at√𝑠 = 13TeV, with CMS experiment. Final goal is to study

the Electroweak (EWK) production of 𝛾 with 2 jets and compare it with the Standard Model. The multivariate analysis technique is developed to extract pure EWK signal from main backgrounds.

This thesis starts with introduction which contains brief overview of analysis, then in the chapter 2, a brief introduction on Standard Model is given, with consideration of an electroweak and strong interactions with involving Z/𝛾+ 2jets processes. In the chapter 3, a brief description of the LHC and CMS detector is presented. The algorithms of reconstruction and identification of photon and jets are described in chapter 4. The theory of MonteCarlo simulation and multivariate techniques are mentioned in chapters 5 and 6. Chapter 7 starts with the characteristics of Vector Boson Fusion, then in addition to different verities of signal, background modeling is mentioned. In chapter 8, the method and scheme of MC simulations and the generator with which this thesis work has been made, is explained. In the last chapters, 9 and 10, I am going through the analysis which contains data samples used in the analysis and the required trigger features, a series of preselections on reconstructed photon and Z and jets, In the end, the steps for simulation samples is described. The distributions of Data/MC comparison is shown in chapter 11. Finally, the prospect of signal measurement with training via Multivariate methods are mentioned in chapter 12.

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2 Introduction

The Standard Model (SM) of particle physics is the most successful theory to repre-sent known elementary particles that are found in nature and the interactions between them. Remarkably, the SM provides a successful description of all current experimen-tal data. It contains fermions (quarks and leptons), which form observable matter and

bosons (photons, gluons, 𝑊± _{and 𝑍}

0), which are responsible for their interactions.

After the discovery of the Higgs boson, all particles in the SM have been observed. However, there are many open questions which are compelling to search beyond the SM, namely the origin of the asymmetry between matter and anti-matter, neutrino masses,etc.

The Large Hadron Collider provides us with proton-proton collisions at energies never seen before, and collects huge amounts of new data. This allows us to study the Standard Model from a new angle, as some processes are becoming experimentally accessible for the first time. The Vector Boson Fusion (VBF), is one of the such classes of process for LHC at CERN. In a p-p collision, this process is identified by two high energy jets in a forward region of detector and reduced jet activity in the central region of the detector. In this process, a quark or anti-quark scatter away from beam axis with other quarks (anti-quark) by exchanging weak bosons (W and Z boson), inside the detector acceptance, and reveal as hadronic jets. The distinctive feature of VBF processes is, therefore, the presence of two energetic jets, often referred to as the

"tagging jets"1, which are found at small angles concerning to the proton beam axis

on opposite sides of the detector. As a result, VBF processes are characterized by a

large pseudorapidity separation, Δ𝜂𝑗𝑗, between the two tagging jets, as well as a large

dijet invariant mass, 𝑚𝑗𝑗. A VBF process does not involve QCD interactions hence,

no colour flow is exchanged between the two quarks. This originate the rapidity

gap (Δ𝜂𝑗𝑗) which is expected to be depleted in the hadronic final state. Different

interactions are possible between the colliding electroweak bosons, and a wide range of VBF processes can be defined, all sharing the same typical VBF dijet signature

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described above.

Following the discovery of Higgs boson at LHC in 2012, researchers want to mea-sure Higgs boson properties as precisely as possible, therefore one of the production modes which can be exploited, is the Vector Boson Fusion production mode. The mea-surement of properties of Higgs boson helps to derive its couplings to weak bosons and fermions. So VBF is a way to probe the dynamics of electroweak symmetry breaking. Higgs production in the VBF channel is a pure EWK process at leading order (LO) involving only quark and antiquark parton distributions. So the process 𝑞𝑞 → 𝑞𝑞𝐻 [2] can be viewed as quark scattering via t-channel with the Higgs boson radiated off the W or Z propagator. Alternatively, one may view this process as two weak bosons fusing to form the Higgs boson.

In this work, the measurement of the VBF with 𝛾 and Z final states with two jets is considered. The measurement of the VBF process with a photon final state (VBF 𝛾 or EW 𝛾jj) has also been proposed [3] but not measured yet. With respect to W and Z boson, the VBF process with photon may have a largely expected total cross-section and this process provides access to the coupling of a photon to weak bosons.

The most relevant diagram for signal, EW 𝛾𝑗𝑗 definition are shown in Figure 1. In the case of single gauge vector boson productions " VBF-V process", other non-VBF electroweak diagrams with identical final states contribute, while they have negative interference so they can not be neglected as additional diagrams. Furthermore, As a kinematic cut for the VBF selection, the signal process needs to introduce a cut on the invariant mass of the two tagging jets. To avoid the on-shell hadronic decay of Z and W bosons we have to put cut on the W and Z bosons poles (to avoid 𝛾 Final State Radiation (FSR) from Z resonance) and the background originated from EW and QCD processes Figure 2. We have to be aware of possible interference effects between signal and background processes in the case of identical initial and final states. Further motivation to study VBF processes is to consider the event hadronization properties via VBF production colour structure. Colour coherence between initial and final-state gluon bremsstrahlung leads to a suppression of hadron production in the central

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region (central rapidity), between the two tagging-jet candidates, this also known as "rapidity gap". In contrast to the main background, there is a colour exchange and particle production in the central region. Measurements of the additional hadronic activity in the rapidity gap can provide important validation of the Monte Carlo models simulations and benchmark results for the use of jet vetoes in independent VBF event productions (e.g. for Higgs selections).

Figure 1: Diagrams that produce 𝛾𝑗𝑗 final states in proton-proton collisions with purely electroweak interactions (𝛼3_𝑊). (Left) Vector Boson Fusion. (Right) bremsstrahlung.

Figure 2: Typical photon plus two jets diagrams that produce 𝛾𝑗𝑗 final states in proton-proton collisions with mixed electroweak and QCD interactions (𝛼_𝑊𝛼2𝑆). (Left) A process

that may interfere with the pure electroweak production. (Right) A process that does not interfere with the pure electroweak production.

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are helpful to improve the modelling on VBF-H. Analogous to Higgs boson production via VBF, the electroweak production of a W or Z plus two jets, with the requirement that the weak boson is centrally produced and that the two jets are well separated in rapidity, will proceed with a sizable cross-section at the LHC. Single vector boson VBF processes have been also measured at the LHC with final states with Z bosons in p-p collisions at 7 TeV [4], at 8 TeV [5, 6], and at 13 TeV [7, 8]. The electroweak production of a W boson in association with two jets (VBF W, or EW Wjj) has also been measured in p-p collisions at 7 and 8 TeV [9,10] and at 13 TeV [11].

The control region is the dominant production of a Z boson in association with 2 jets in a p-p collision. This process is predicted by two jets with large invariant mass and large rapidity separation which also can be occupied with two charged leptons. The pure EW process with the leptonic final state is also expected with a small cross-section. So this process originated as "EWK-Zjj" [3]. Figure 3 shows representative Feynman diagrams for the EW Zjj processes, namely (left) vector boson fusion (VBF), (right) Z-bremsstrahlung, and (bottom) multiperipheral production. Detailed calculations reveal the presence of a large negative interference between the pure VBF process [12]and the two other categories.

The most important background to VBF-Z process is Drell-Yan (DY)+jets since it has the same final state as VBF-Z. The DY process has a larger cross-section compared to the VBF-Z since it receives higher order radiative QCD corrections.

The different orders of QCD (𝛼𝑆2) corrections to DY is shown in Fig 4.

Even though only a small fraction of the Z-bosons decays through electrons and muons, those objects provide a clear signature in the detector, which allows for an easy reconstruction of the dilepton invariant mass. By selecting a mass window around

the Z pole (𝑚𝑍 ≃ 91 GeV), one can select a region in the dilepton invariant mass

spectrum where the Z-resonance dominates over the contribution of the photon, and other dilepton production mechanisms.

The study of the EW W+2 jets process is part of a more general investigation of the SM VBF process. The W production association with two jets via the t-channel in VBF process has the same pattern as VBF Z. These EW processes have been

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Figure 3: Representative Feynman diagrams for dilepton production in association with two jets from purely electroweak contributions: (left) vector boson fusion, (right) bremsstrahlung- like, and (bottom) multiperipheral production.

Figure 4: Representative diagrams for order 𝛼𝑆2 corrections to DY production that

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used to investigate the rapidity gaps at hadron colliders, as a background to Higgs boson measurements in the VBF channel. The VBF W is challenging because of large backgrounds, so it requires to study additional quark and gluon emission in quantum chromodynamics (QCD). At tree level the EW Wjj production can appear as two fermion lines with identical quark flavours. One contains vector-boson pair production with subsequent decay of one of the weak bosons to a pair of jets. The other one is interchange of identical initial- or final-state anti quarks, such as in the

𝑢𝑢 → 𝑑𝑢𝑙+_{+ 𝜈}

𝑙 or 𝑑𝑢 → 𝑑𝑑𝑙++ 𝜈𝑙 subprocesses [13].

The details of the analysis strategy with two trigger path HighPT and LowPT and control region Z+2 jets will be presented in chapter7.

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3 Standard Model

3.1 Introduction

The simple physics model with just four Fundamental particles and four fundamental forces (if we accept gravity) named Standard Model (SM). The Quantum electrody-namics (QED) is the low-energy manifestation of the fundamental theory of electro-magnetism. The strong nuclear force which bounds proton and neutron together, constructs the properties of strong interaction between constituents of proton and neutron namely quarks. The fundamental interactions of particle physics are com-pleted by the weak force, which is responsible for the nuclear 𝛽-decays of certain radioactive isotopes and the nuclear fusion processes that fuel the Sun. There are three generations of elementary particles include leptons and quarks named fermions, The properties of the twelve fundamental fermions are categorised by the types of interaction that they experience, as summarized in Table 1. With the exception of the electrically neutral neutrinos, the other nine particles are electrically charged and participate in the electromagnetic interaction of QED. Only the quarks carry the QCD equivalent of electric charge, called colour charge. Consequently, only the quarks feel a strong force. Because of the nature of the QCD interaction, quarks are never ob-served as free particles but are always confined to bound states called hadrons, such as the proton and neutron. Because the quarks feel a strong force, their properties are very different from those of the electron, muon, tau-lepton and the neutrinos, which

are collectively referred to as the leptons, up-type quarks (𝑢, 𝑐, 𝑡) has charge −2

3 and

the charge 1

3 refer to down-type quarks (d, s, b) [14].

In modern particle physics, each force is described by a Quantum Field Theory (QFT). In the case of electromagnetism, this is the theory of Quantum Electrody-namics (QED) where the interactions between charged particles are mediated by the exchange of a spin-1 force-carrying particle, known as a gauge boson, The familiar spin-1 photon is the gauge boson of QED. In the case of the strong interaction, the force-carrying particle is called the gluon which, like the photon, is massless. The

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Leptons Quarks

Particle Q mass/GeV Particle Q mass/GeV

First generation electron (𝑒 −₎ _-1 _0.0005 _{down (d)} ₋1 3 0.003 neutrino (𝜈𝑒) 0 < 10−9 up (u) +2₃ 0.005 Second generation muon (𝜇 −₎ _-1 _0.106 _{strange (s) −}1 3 0.1 neutrino (𝜈𝜇) 0 < 10−9 charm (c) +2₃ 1.3 Third generation tau (𝜏 −₎ _-1 _1.78 _{bottom (b) −}1 3 4.5 neutrino (𝜈𝜏) 0 < 10−9 top (t) +2₃ 174

Table 1: The twelve fundamental fermions divided into quarks and leptons. The masses of the quarks are the current masses.

weak charged-current interaction, is mediated by the charged 𝑊+ _{and 𝑊}− _bosons,

which are approximately eighty times more massive than the proton. There is also a weak neutral-current interaction, closely related to the charged current, which is mediated by the electrically neutral Z boson. The relative strengths of the forces associated with the different gauge bosons.

The final element of the Standard Model is the Higgs boson, which was discovered by the ATLAS and CMS experiments at the Large Hadron Collider (LHC) in 2012. The Higgs boson, which has a mass:

𝑚𝐻 = 125𝐺𝑒𝑉 (1)

differs from all other Standard Model particles. Unlike, the fundamental fermions and the gauge bosons, which are respectively spin-half and spin-1 particles, the Higgs

boson is spin-0 scalar particle. The masses of the 𝑊±_{, Z and H bosons are all of the}

order of 100 GeV, which is known as the electroweak scale.

3.2 Symmetries in Standard Model

In this chapter, we discuss the symmetry and symmetry breaking of particle physics. the most general symmetries for particle physics in the context of quantum field the-ory is Lorentz symmetry. Quantum electrodynamics, is the quantum version of the gauge invariant electromagnetic theory with Developments in both theory and exper-iment. Many short lived particles were discovered and classified by their quantum

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numbers manifested in the strong interactions. Gell-Mann [15] was led to the quark structure of the particles and successfully predicted the existence of some new par-ticle states. The three quarks form the fundamental representation of SU(3), while many mesons, which are quark-antiquark bound states, were seen to be in the adjoint octet representation and some triple quark states in a decouplet representation of the group. The gauge theory of electromagnetism involves U(1) gauge transforma-tions, where a complex wave function or field may be multiplied by a phase factor, which is a U(1) or 1 × 1 unitary matrix. It was generalized to theories involving more complicated gauge transformations, like SU(2) or SU(3). This would require fields with internal SU(2) or SU(3) symmetry groups. Weinberg and Salam [16, 17], following suggestions by Glashow [18], developed an SU(2)⊗ U(1) gauge theory to describe electroweak interactions. Doublet representations of SU(2) are provided by the quark pairs u and d, c and s, t and b, as well as by the lepton pairs. The stan-dard model including both electroweak theory and quantum chromodynamics with its SU(3) colour symmetry became established, but some puzzles remained. The major question was how the weak gauge bosons become massive and thereby lead to short range interactions. Apart from the weak gauge boson masses, the masses of other particles too were difficult to explain in the chiral theory of SU(2) involving only left-handed doublets of quarks and leptons. These can be explained by a breaking of the SU(2)⊗ U(1) to U(1) in a special way. Through the indirect breaking of the symmetry of the ground state rather than the direct breaking of the symmetry of the interaction. This is called spontaneous symmetry breaking. Spontaneous symme-try breaking was implemented in the standard model of electroweak interactions by breaking of the electroweak gauge symmetry within Higgs boson discovery [19]. So the Standard Model is, therefore, a SU(3)⊗ SU(2) ⊗ U(1) gauge theory.

3.3 Electroweak interaction

At the fundamental level, QED and QCD share a number of common features. Both interactions are mediated by massless neutral spin-1 bosons and the spinor part of

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weak interaction does not obey the four vector 𝑢(𝑝′_)𝛾𝜇_𝑢(𝑝) _{rule, as a result of, parity} violating nature of the interaction. The charged-current weak interaction is mediated

by massive charged 𝑊± _{bosons and consequently couples together fermions differing}

by one unit of electric charge. It is also the only place in the Standard Model where parity is not conserved.

3.4 V- A structure of the weak interaction

From experiments of 𝛽-decay of polarised cobalt-60, Wu and collaborators detected

Parity is not conserved in the weak interaction. The 𝐶𝑜60 _{nuclei, were aligned in a}

strong magnetic field B and the 𝛽-decay electrons were detected at different polar angles with respect to this axis. So if parity were conserved in the weak interaction, the rate at which electrons were emitted at a certain direction relative to the axial-vector B-field would be identical to the rate in the opposite direction. Experimentally, it was observed that more electrons were emitted in the hemisphere opposite to the direction of the applied magnetic field than in the hemisphere in the direction of the applied field. Thus this experiment provides a clear demonstration of parity violation in weak interaction. Chirality is another symmetry in this theory , So the usual kind of chiral structure with fermions which have no mass terms is given by

Ψ → (1 − 𝑖𝜖𝛾5)Ψ (2) 𝛾5 = ⎛ ⎝ 0 𝐼 𝐼 0 ⎞ ⎠ (3)

This is essentially a phase transformation, but the left-handed and right-handed components have different phases; hence it is called a chiral transformation. Mass terms, if present, break this symmetry, which is satisfied by kinetic terms as well as gauge interactions. Right-handed particles and left-handed antiparticles, is a clear violation of parity invariance. Then, by taking addition and subtraction of two

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equa-tions in equation 4: ⃗ 𝜎′ _{· ⃗}_𝑃 𝑃 Ψ = 𝛾 5_Ψ, ⃗ 𝜎′ _{· ⃗}_𝑃 𝑃 𝛾 5 Ψ = Ψ, (4)

and introducing the helicity operator defined by ℎ = _𝜎⃗′_{· ⃗}_𝑃

2𝑃 we obtain ℎΨ𝑅= 1 2Ψ𝑅 ℎΨ𝐿= − 1 2Ψ𝐿 (5) Chiral structure of the weak interaction is introduced with two Left and right handed chiral projection operators:

𝑃𝑅= 1 2(1 + 𝛾 5₎ 𝑃𝐿= 1 2(1 − 𝛾 5₎ (6)

The fermion field here has been assumed to have both left-handed and right-handed components. It is also possible to have a theory with just a left-right-handed fermion without mass.

¯

Ψ𝐿𝛾𝜇Ψ𝐿 (7)

By replacing the two-component spinors, Ψ1(𝑝)and Ψ2(𝑝)in above equation (with

definition of the first one describing left-handed particles and right- handed antipar-ticles, the second one describing instead right-handed particles and left-handed

an-tiparticles), the 𝑃𝐿projector acts for example on the neutrino spinor, in the following

way: 1 2(1 − 𝛾 5_)𝑢 𝜈 = ⎛ ⎝ 𝐼 0 0 0 ⎞ ⎠ ⎛ ⎝ Ψ1 Ψ2 ⎞ ⎠= ⎛ ⎝ Ψ1 0 ⎞ ⎠= (𝑢𝜈)𝐿 (8)

The most general Lorentz-invariant form for the interaction between a fermion and a boson is a linear combination of the bilinear covariants. If this is restricted to

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the exchange of a spin-1 (vector) boson, the most general form for the interaction is a linear combination of vector and axial-vector currents, equation 10, the current has been composed in two vector and axial-vector

𝑗_𝑣𝜇= ¯𝑢(𝑝′)𝛾𝜇𝑢(𝑝)

𝑗_𝐴𝜇 = ¯𝑢(𝑝′)𝛾𝜇𝛾5𝑢(𝑝),

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𝑗𝜇∝ ¯𝑢(𝑝′)(𝑔𝑣𝛾𝜇+ 𝑔𝐴𝛾𝜇𝛾5)𝑢(𝑝) = 𝑔𝑣𝑗𝑣𝜇+ 𝑔𝐴𝑗_𝐴𝜇 (10)

Simply from equation 7 we can draw four-vector current with 𝑔𝑤as a weak coupling

constant for left handed particles:

𝑗𝜇 = √𝑔𝑤 2𝑢(𝑝¯ ′ )1 2𝛾 𝑚𝑢_{(1 − 𝛾}5_)𝑢(𝑝) ₍₁₁₎

Now it is time to expand this ideas to fermion in the GWS Model, so we can start with one of family of lepton. We assign these particles to appropriate representation

of 𝑆𝑈(2)𝐿× 𝑈 (1)𝑌gauge symmetry.

𝜇−→ 𝑒−+ ¯𝜈𝑒+ 𝜈𝜇 (12)

only left-handed leptons and right-handed anti-leptons take part in, so we can write down charge current :

𝐽𝜇(𝑥) ≡ 𝐽𝜇(𝑥)+ = ¯𝜈𝑒𝐿(𝑥)𝛾𝜇𝑒𝐿(𝑥) = 1 2𝜈¯𝑒(𝑥)𝛾𝜇(1 − 𝛾5)𝑒(𝑥) (13) 𝐽_𝜇†(𝑥) ≡ 𝐽𝜇(𝑥)− = ¯𝑒𝐿(𝑥)𝛾𝜇𝜈𝑒𝐿(𝑥) = 1 2¯𝑒(𝑥)𝛾𝜇(1 − 𝛾5)𝜈𝑒(𝑥) (14)

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Via using Lepton doublets composed of the left-handed components of fermions and weak isospin space we can re-write the charge currents:

𝐽𝜇+= ¯𝐿𝛾𝜇𝑇+𝐿 (15)

𝐽_𝜇− = ¯𝐿𝛾𝜇𝑇−𝐿 (16)

These forms suggest that the weak currents make an SU(2) group by introducing an additional neutral current,

𝐽_𝜇3 = ¯𝐿𝛾𝜇 𝑇3 2 𝐿 = 1 2𝜈¯𝑒𝛾𝜇𝜈𝑒− 1 2𝑒¯𝐿𝛾𝜇𝑒𝐿 (17)

Then, we have 2 charged and 1 neutral currents, 𝐽𝜇±and 𝐽𝜇3 , which couple to the

weak bosons 𝑊𝜇± and 𝐴𝜇3, respectively, just as the electromagnetic current 𝐽𝜇𝑒𝑚(𝑥)

couples to photon 𝐴𝜇(𝑥).

equation 17 satisfies the 𝑆𝑈(2)𝐿 algebra

[𝑇𝑖, 𝑇𝑗] = 𝑖𝜀𝑖𝑗𝑘𝑇𝑘 (18)

Besides, we also need another gauge field 𝐵𝜇 associated with the new 𝑈(1)𝑌

symmetry. This new 𝑈(1)𝑌 group should be independent of 𝑆𝑈(2)𝐿 group and thus

its generator should commute with the generators of 𝑆𝑈(2)𝐿, 𝑇𝑖 (i = 1,2,3). The

gauge group is thus extended to the direct product of 𝑆𝑈(2)𝐿 and 𝑈(1)𝑌. So the

form of electromagnetic current for fermion goes to

𝐽𝜇𝑒𝑚(𝑥) = ¯Ψ𝛾𝜇𝑄Ψ (19)

Furthermore, 𝑄 − 𝑇3 commutes with 𝑇𝑖 (i = 1,2,3), that is, [𝑄 − 𝑇3_{, 𝑇}𝑗_{] = 0},

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the 𝑈(1)𝑌 group named weak hypercharge,

𝑄 = 𝑇3+𝑌

2 (20)

In summary, the GWS model is the 𝑆𝑈(2)𝐿⊗ 𝑈 (1)𝑌 gauge theory a left-handed

doublet L and a right-handed singlet R of SU(2) with weak hypercharge operators.

3.5 Spontaneous breaking of 𝑆𝑈 (2)

𝐿

⊗ 𝑈 (1)

𝑌

symmetry with

Higgs mechanism

A field theory is usually described by a Lagrangian density, which is a local function of fields and their partial derivatives. Just as a Lagrangian L in ordinary mechanics is

a function of generalized coordinates and time derivatives, so that one writes 𝐿(𝑞, 𝑞′₎_,

ignoring explicit time dependences, similarly, in field theory, a Lagrangian is a function

of fields and time derivatives, 𝐿[𝜑, 𝜑′_]_{. The functional is usually local, but may include}

first-order spatial derivatives [20]: ℒ =

∫︁

(𝑑3⃗𝑥ℒ(𝜑(𝑥), 𝜕𝜇𝑥)) (21)

So regarding the general introduction of Lagrangian, the gauge-invariant La-grangian with SU(2) symmetry with generalizing the U(1) model, is given by 22.

ℒ = (𝐷†_𝜇)(𝐷𝜇) − 1 4 − 𝑉 (Φ † Φ) (22) with : 𝑉 (Φ†Φ) = −𝜇2†ΦΦ + 𝜆(Φ†Φ)2, (23)

The potential 23 with positive 𝜆 and negative 𝑚2 _{= −𝜇}2_(𝜇2 _{> 0)} _{has minimum}

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Φ†Φ = | Φ |2 = 𝑣 2 2, 𝑤𝑖𝑡ℎ 𝑉 = √︂ 𝜇2 𝜆 , (24)

spontaneous symmetry breaking occurs when the scalar doublet of Φ = ⎛ ⎝ 𝜑† 𝜑0 ⎞ ⎠

develops a vacuum expectation value Likes:

Φ0 = < 0 | Φ | 0 > = ⎛ ⎝ 0 𝑣 √ 2 ⎞ ⎠ (25)

Therefore, even after the symmetry breaking, there remains a symmetry associated

with the charge operator Q of 𝑈(𝐼)𝑒𝑚 being compatible with our real world. It is

convenient to parametrize the scalar doublet with 4 degrees of freedom in terms of the fields denoting the shifts from the vacuum state

Φ = ⎛ ⎝ 𝜑+ 𝜑0 ⎞ ⎠= 𝑒 𝑖𝑇 ·⃗𝜀 2𝑣 ⎛ ⎝ 0 (𝑣+𝐻)_√ 2 ⎞ ⎠ (26)

H as Higgs boson and 𝜀 are so-called Goldstone bosons being absorbed into the longitudinal components W and Z boson. Equation 25 leads to zero vacuum expec-tation values for all of the fields above.

3.6 Electroweak lagrangian

It is a non-Abelian gauge theory with 𝑆𝑈(2)𝐿⊗ 𝑈 (1)𝑌 gauge symmetry accompanied

by the Higgs mechanism. It is the first successful model toward the unified theory of

elementary particle interactions. The discovery of 𝑊± _{and 𝑍}0 _{bosons with expected}

masses and a weak neutral current mediated by a massive neutral vector boson 𝑍0 is

a great triumph of the model. So here we start with it’s Lagrangian.

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gauge-invariant Lagrangian with 𝑆𝑈(2)𝐿⊗ 𝑈 (1)𝑌 of GWS model, given by the sum of:

ℒ = ℒ𝐹 + ℒ𝐺+ ℒ𝑆+ ℒ𝑌 (27)

This Lagrangian contains Lagrangian for left-handed fermions, Yukawa interaction term which is coupling terms between fermions and scalars, Scalar field terms and

also kinetic terms. The gauge-invariant Lagrangian with 𝑆𝑈(2)𝐿⊗ 𝑈 (1)𝑌 symmetry

for fermions is constructed as:

ℒ𝐹 = ¯𝐿𝑖𝛾𝜇(𝜕𝜇− 𝑖𝑔 ⃗ 𝜏 2 · ⃗𝐴𝜇+ 𝑖 2𝑔 ′ 𝐵𝜇)𝐿 + ¯𝑅𝑖𝛾𝜇(𝜕𝜇+ 𝑖𝑔′𝐵𝜇)𝑅, (28) where 𝐴𝑖

𝜇(i=1,2,3) and 𝐵𝜇 are gauge boson fields associated with 𝑆𝑈(2)𝐿 and

𝑈 (1)𝑌, respectively. 𝑔 and 𝑔′ are the gauge coupling constants corresponding to

𝑆𝑈 (2)𝐿⊗ 𝑈 (1)𝑌 invariance. R is a singlet of 𝑆𝑈(2)𝑅, the fermion mass term which

connects the left and right-handed fields violate the invariance of 𝑆𝑈(2)𝐿×𝑈 (1)𝑌 [21].

𝒟𝜇= 𝜕𝜇− 𝑖𝑔 ⃗ 𝜏 2 · ⃗𝐴𝜇+ 𝑖 2𝑔 ′ 𝐵𝜇, (29)

The kinematic terms are includes

𝐿𝐺= − 1 4𝐹𝜇𝜈 𝑖 𝐹𝜇𝜈 𝑖−1 4𝐵𝜇𝜈𝐵 𝜇𝜈 , (30)

And the scalar fields generate masses of gauge bosons and those of quarks and leptons via the Higgs mechanism :

𝐿𝑠= (𝐷𝜇Φ)(𝐷𝜇Φ†) − 𝑉 (Φ+Φ), (31) With (𝐷𝜇Φ) = (𝜕𝜇− 𝑖𝑔 ⃗ 𝑇 2 · ⃗𝐴𝜇− 𝑖 2𝑔 ′ 𝐵𝜇)Φ = (𝜕𝜇− 𝑖𝑔 ⃗ 𝑇 2 · ⃗𝐴𝜇− 𝑖 2𝑔 ′ 𝐵𝜇) 1 √ 2(𝑣 + 𝐻)𝜒, (32)

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𝐿𝑌 = −𝐺𝑒( ¯𝐿Φ𝑅 + 𝑅Φ†𝐿) + ℎ.𝑐, (33)

With define charged boson fields 𝑊±

𝑊±𝜇=

(𝐴𝜇1∓ 𝑖𝐴𝜇2)2

√

2 , (34)

By expanding the equation 31 with replacing Φ we can conclude that charged

vector boson 𝑊± _{are massive with}

𝑀𝑊 =

1

2𝑔𝑣, (35)

And the diagonalized neutral filed can be written as

𝑣2 8 ⎛ ⎝ 𝑔2_{+ 𝑔}′2 ₀ 0 0 ⎞ ⎠= ⎛ ⎝ 𝑍𝜇 𝐴𝜇 ⎞ ⎠= 𝑣2 8(𝑔 2 + 𝑔′2)𝑍𝜇𝑍𝜇+ 0 · 𝐴𝜇𝐴𝜇 (36) As orthogonal transformation ⎛ ⎝ 𝑍𝜇 𝐴𝜇 ⎞ ⎠= ⎛ ⎝ cos 𝜃𝑤 − sin 𝜃𝑤 sin 𝜃𝑤 cos 𝜃𝑤 ⎞ ⎠= ⎛ ⎝ 𝐴3 𝜇 𝐵𝜇 ⎞ ⎠ (37) Thus we define 𝑍𝜇≡ cos 𝜃𝑤𝑊𝜇3− sin 𝜃𝑤𝐵𝜇 𝐴𝜇 ≡ sin 𝜃𝑤𝑊𝜇3+ cos 𝜃𝑤𝐵𝜇 ⇔ 𝐵𝜇 = cos 𝜃𝑤𝐴𝜇− sin 𝜃𝑤𝑍𝜇 𝑊_𝜇3 = sin 𝜃𝑤𝐴𝜇+ cos 𝜃𝑤𝑍𝜇 (38)

Where 𝜃𝑤 is called the weak mixing angle or Weinberg angle, the diagonalization

leads to:

tan 𝜃𝑊 =

𝑔′

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Then we can see the neutral Z boson become massive 𝑀𝑧 = 1 2𝑣 √︀ (𝑔′₎2_{+ 𝑔}2 (40)

The gauge coupling constants could be defined as Weinberg angles so Z boson

becomes massive as terms of gauge coupling constants, and another neutral boson 𝐴𝜇

is massless and hence can be identified with the real photon. Note that in the GWS

model the mass of Z boson is related to the one of 𝑊± _boson.

𝑀𝑧 =

𝑀𝑊

cos 𝜃𝑊

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In particular since the photon is part of 𝑊3

𝜇, its couplings to the 𝑊𝜇𝑎 are

deter-mined by 𝑔[𝐴𝜇, 𝑊𝑎𝑇𝑎] = 𝑔 sin 𝜃𝑤𝑊3𝑊𝑎[𝑇3, 𝑇𝑎], according to equation 18. Thus the

electromagnetic coupling sets by

𝑒 = 𝑔 sin 𝜃𝑤 = 𝑔′cos 𝜃𝑤 (42)

To define the W boson charges, 𝑇± ₌ _√1

2(𝑇

1_{± 𝑖𝑇}2₎. Since [𝑇3_{, 𝑇}±_{] = ±𝑇}± _the

w boson couples to the 𝑇 ± has charge ±1. So the equation 34 have charges ±1 with

𝐴𝜇 → 𝑊𝜇.

In conclusion by using all equation above and obtain sin 𝜃𝑤 in experiment which

is around 0.23, the two Z and W mass goes to 𝑀𝑊 ≃ 80𝐺𝑒𝑉 and 𝑀𝑍 ≃ 90GeV.

In this study, we focus on the 𝑊+_𝑊−_𝛾 _{production via Pure electroweak}

interac-tion so If we want to write down the Lagrangian for this process we have,

𝐿 = −1 4𝐹 2 𝜇𝜈− 1 2(𝜕𝜇𝑊 + 𝜈 − 𝜕𝜈𝑊𝜇+)(𝜕𝜇𝑊𝜈−− 𝜕𝜈𝑊𝜇−) + 𝑚 2 𝑤𝑊 + 𝜇𝑊 − 𝜇− 𝑖𝑒[𝜕𝜇𝐴𝜈(𝑊𝜇+𝑊 − 𝜈 − 𝑊 + 𝜈 𝑊 − 𝜇) + 𝐴𝜈(−𝑊𝜇+𝜕𝜈𝑊𝜇−+ 𝑊 − 𝜇𝜕𝜈𝑊𝜇++ 𝑊 + 𝜇 𝜕𝜇𝑊𝜈−− 𝑊 − 𝜇 𝜕𝜇𝑊𝜈+] (43)

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3.7 Strong interaction

The theory of strong interaction is Yang-Milles Theory [22], which is a generalization of QED with multi massless Spin-1 particle can interact among themselves. The

Yang-Milles Theory follows 𝑆𝑈(3)𝐶 symmetry and Lagrangians for Yang-Mills theories are

strongly constrained by a generalization called non-Abelian gauge invariance.

3.7.1 Gauge symmetry for non-abelian groups

The free theory of N complex fields which is invariant under 𝑆𝑈(𝑁) × 𝑆𝑈(1) are the groups play special roles in Quantum Filed theory which we talked about it in chapter

3.4. So the SU(N) groups represent the 𝑁 × 𝑁 Hermitian matrices with N fields 𝜑𝑖

which transform under infinitesimal group transformation.

𝜑𝑖 → 𝜑𝑖+ 𝑖𝛼𝑎(𝑇𝑎)𝑖𝑗𝜑𝑗 (44)

Which satisfy the algebra [𝑇𝑎_{, 𝑇}𝑏_{] = 𝑖ℱ}𝑎𝑏𝑐_𝑇𝑐_{. On the other hand SU(3) the}

generators are often written as basis 𝑇𝑖 ₌ 1

2𝜆

𝑖 _{they are 𝑁}2 _{− 1} _{generators of the}

symmetry groupas named Gell-Man matrices. Then we can re-write the 44 as :

𝜑𝑖 → 𝑒𝑖𝛼

𝑎_(𝑥)𝑇𝑎

𝜑𝑖 (45)

The transformation of the covariant derivative associated with this transformation is :

𝐷𝜇= 𝜕𝜇− 𝑖𝑔𝐴𝑎𝜇𝑇𝑎, (46)

𝐴𝑎

𝜇 is a set of three gauge bosons, which transform as :

𝐴𝑎_𝜇(𝑥) → 𝐴𝑎_𝜇+1

𝑔𝜕𝜇𝛼

𝑎_{(𝑥) − ℱ}𝑎𝑏𝑐_𝛼𝑏_(𝑥)𝐴𝑐

𝜇, (47)

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deriva-tive: −𝑖𝑔𝐹𝜇𝜈 ≡ [𝐷𝜇, 𝐷𝜈], (48) 𝐹_𝜇𝜈𝑎 = 𝜕𝜇𝐴𝑎𝜈 − 𝜕𝜈𝐴𝑎𝜇+ 𝑔ℱ 𝑎𝑏𝑐_𝐴𝑏 𝜇𝐴 𝑐 𝜈, (49)

The field tensor in the non-abelian case contains an additional term which

indi-cates the vector fields are self-interacting. Because of this term, the 𝐹𝑎

𝜇𝜈𝐹𝑎𝜇𝜈 term in

the Lagrangian will expand in triplet and quartic terms of gauge bosons, resulting in vertices as shown in Figure 5.

Figure 5: Feynman diagrams for self-interaction of SU(N) gauge bosons.

We can now write down a locally SU(N) invariant Lagrangian:

ℒ = −1 4(𝐹 𝑎 𝜇𝜈) 2₊ 𝑁 ∑︁ 𝑖,𝑗=1 ¯ 𝜓𝑖(𝛿𝑖𝑗𝑖 /𝜕 + 𝑔 /𝐴 𝑎 𝑇_𝑖𝑗𝑎 − 𝑚𝛿𝑖𝑗)𝜓𝑗, (50)

These calculations bring us to describe Quantum Chromodynamics (QCD) as the

strong interaction. The theory follows a 𝑆𝑈(3)𝑐 symmetry and is therefore described

by: ℒ𝑄𝐶𝐷 = − 1 4(𝐺 𝑎 𝜇𝜈) 2 + ¯𝜓(𝑖𝛾𝜇𝐷𝜇− 𝑚)𝜓, (51)

Here 𝐺𝜇𝜈 takes a form of equation 49 also 𝜓 are triplets of fermion fields, the

covariant derivative formed in Gelman matrices as equation 46:

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Furthermore, Theory 𝑆𝑈(3)𝑐 leads to eight gauge bosons, the gluons, which inter-act with the quark fields. Each of the three quark fields in the multiplet is represented by a color charge: green (𝑔), red (𝑟), blue (𝑏), and corresponding anti-colors ¯𝑔, ¯𝑟 and

¯_𝑏_{. The eight gluons carry the combinations of colour and anti-colour charge as a}

prop-agator of this theory. We can classify QCD processes, refers to its coupling constants

𝛼𝑠, in two regimes:

• Hard scattering processes, characterized by a large momentum transfer between the colliding partons. Those could be accurately described by perturbation theory in which the involved quarks and gluons are treated as free particles. • The soft QCD regime, with interactions at low momentum exchange and long

distances for which the strong coupling rapidly increases. This leads to the confinement of quarks into hadrons, colourless bound states of quarks. These bound states could be mesons, which are combinations of a quark and an anti-quark (𝑔¯𝑔, 𝑟¯𝑟 or 𝑏¯𝑏), or they could be baryons, in which three anti-quarks or three anti-quarks are grouped together, each of different color (𝑔𝑟𝑏 or ¯𝑔¯𝑟¯𝑏).

3.8 Hadron collision

In hadron-hadron collisions, the momentum fractions 𝑥1 and 𝑥2 of the two interacting

partons are unknown, and the event kinematics have to be described by three

vari-ables, for example, 𝑄2_{, 𝑥}

1 and 𝑥2. These three independent kinematic variables can

be related to three experimentally well-measured quantities. In hadron collider ex-periments, the scattered partons are observed as jets. In a process such as 𝑝𝑝 → two jets + X, the angles of the two-jets with respect to the beam axis are relatively well measured. Consequently, differential cross sections are usually described in terms of these two jet angles and the component of momentum of one of the jets in the plane transverse to the beam axis, referred to as the transverse momentum:

𝑃𝑇 =

√︁

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Where the Z-axis defines the beam direction. At a hadron-hadron collider, such as the LHC, the collisions take place in the centre-of-mass frame of the pp system, which is not the centre-of-mass frame of the colliding partons. The net longitudinal

momentum of the colliding parton-parton system is given by (𝑥1, 𝑥2)𝐸𝑝, where 𝐸𝑝 is

the energy of the proton. Consequently, in a process such 𝑝𝑝 →2 jets + X, the two final-state jets are boosted along the beam direction. At chapter 4 we will discuss it. The invariant mass of the system of particles forming a jet is referred to as the jet mass. The jet mass is not the same as the mass of the primary parton; it is mainly generated in the hadronisation process. For high-energy jets, the jet mass is usually small compared to the jet energy. Broadly speaking, the differential cross sections for jet production in hadron-hadron collisions are approximately constant in pseudorapidity, implying that roughly equal numbers of jets are observed in each interval of pseudorapidity region, reflecting the forward nature of jet production in pp and pp collisions.

Figure 6: Candidates for a Higgs produced with a Z. ATLAS (l): both decay ultimately to leptons, leaving two electrons (green) and four muons (red). CMS (r): the Higgs de-cays to two charm quarks forming jets (cones); the Z dede-cays to electrons (green) (Image: ATLAS/CMS/CERN)

The most common, although not the most interesting, high-energy process at the LHC is the QCD production of two-jets. Figure 6 shows an example of a two-jet event recorded at the CMS experiment. Since the colliding partons have no momentum

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transverse to the beam axis, the jets are produced back to back in the transverse plane and have equal and opposite transverse momenta, pT. In the other view, the jets are not back to back due to the boost of the final-state system from the net

momentum of the colliding partons along the beam axis, (𝑥1, 𝑥2)

√ 𝑠

2 . Initial-state and

final-state radiations originate from respectively the ingoing and outgoing partons of the hard scattering. This radiation gives rise to parton showers, as described in the next section, before reaching the non-perturbative regime where hadronization sets in. The coloured remnants of the protons involve additional radiation and hadronization in the event, forming the Underlying Event (UE). Sometimes, one or more additional hard interactions could occur in the same pp-collision, resulting in a multi-parton interaction.

In the end, the QED production of a pair of leptons in hadron-hadron collisions from the annihilation of an antiquark and a quark, shown in Figure 7, is known as the Drell-Yan process. It provides a useful example of a cross-section calculation for

hadron-hadron collisions, in this case, 𝑝𝑝 → 𝜇+_𝜇−_𝑋_{, where X represents the remnants}

of the colliding hadrons. [23].

Figure 7: The lowest-order Feynman diagram for the Drell-Yan process.

3.8.1 Renormalization and the running coupling constant

At low-energy scales, the coupling constant of QCD is large, 𝛼𝑠∼O(1). Consequently,

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QCD processes are not calculable using traditional perturbation theory due to large

value of 𝛼𝑠 Fortunately, it turns out that 𝛼𝑠 is not constant; its value depends on the

energy scale of the interaction being considered. At high energies, 𝛼𝑠 becomes

suffi-ciently small that perturbation theory can again be used. In this way, QCD divides into a non-perturbative low-energy regime, where first-principles calculations are not currently possible, such as the hadronisation process, and a high-energy regime where perturbation theory can be used. Therefore, the higher-order contribution depends on the total transversed momentum of the system, to prevent the divergence which comes out from the momentum in loops, renormalization introduces a

renormaliza-tion scale of 𝜇2. Thus the renormalization scale depends on 𝜇2. This dependency

described by 𝛽 function: 𝛽(𝛼) = 𝜇2 𝜕𝛼(𝜇) 𝜕𝜇2 = 𝜕𝛼(𝜇) 𝜕𝑙𝑛𝜇2 (54)

Through calculation a perturbation series, the lowest order for a 𝑈(1) theory given by:

𝛽(𝛼) = 2𝛼

2

3𝜋 𝑛𝑓 + 𝑂(𝛼3) (55)

and for non-abelain SU(N) is going :

𝛽(𝛼) = 𝛼 2 2𝜋(− 11 3 𝑁 + 2 3𝑛𝑓)𝑂(𝛼3) (56)

In the non-abelian case, a negative contribution to the b-function is added as a

result of the self-interaction of the gauge boson, which will dominate if 11𝑁 > 2𝑛𝑓.

Running Coupling Constant 𝛼𝑠

The behavior of the running of coupling constant of QCD turns out different from QED in the Logarithmic part of equation.

𝛼(𝑞2) = 𝛼(𝜇 2₎ 1 − 𝛼(𝜇2₎ 1 3𝜋𝑙𝑛( 𝑞2 𝜇2) (57) Then, with respect to the Gluon-Gluon self-interaction (the color self-coupling of

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gluon) with the additional loop diagrams Figure 8:

Figure 8: Feynman diagrams for consecutive Loops contributes to Log coefficient.

The equation 59 turns to QCD form, and for values of 𝑞2 _{and 𝜇}2 _{larger than the}

confinement scale, the difference between the gluon self-energy grows logarithmically, where the 𝛽 depends to the numbers of fermionic (quark) and bosonic (gluon) loops.

For 𝑛𝑓 quark flavours and N colours,

𝛽 = 11𝑁 − 2𝑛𝑓

12𝜋 (58)

The effect of the bosonic loops enters the expression for the 𝑞2 _{evolution of 𝛼}

𝑠 with the opposite sign to the pure fermion loops, with the fermion loops leading to a negative contributions (which was also the case for QED) and the gluon loops leading

to positive contributions. The corresponding evolution of 𝛼𝑠(𝑞2) is

𝛼𝑠(𝑞2) = 𝛼𝑠(𝜇2) 1 + 𝛽𝛼𝑠(𝜇2)𝑙𝑛(𝑞 2 𝜇2) (59)

For N = 3 colours and 𝑛𝑓 ≥ 6quarks, 𝛽 is greater than zero and hence 𝛼𝑠decreases

with increasing 𝑞2, for more details we can follow [21]

3.8.2 Hadronic jet

When a high-energy gluon or quark is produced in high-energy collisions, it will reduce its energy by emitting additional partons: gluons and quarks can emit a gluon, or a gluon can split into two quarks. The radiated partons are mostly soft, i.e. they carry a small fraction of the initial parton’s momentum and are therefore emitted

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at small angles. These partons can, in turn, emit other partons, and this avalanche effect creates a parton shower in which the resulting partons are found in a rather narrow cone. This parton shower development stops when the partons reach the non-perturbative regime, at an energy of about 1 GeV, for which the strong coupling constant becomes too large to use perturbation theory. At this point, the coloured partons are clustered into colour-singlet hadrons, a process called hadronization. The initial parton coming from the hard scattering will therefore be represented by a collimated spray of energetic hadrons, called a jet.

4 The CMS experiment at the Large Hadron

Col-lider

4.1 Large Hadron Collider

The Large Hadron Collider is the proton-proton collider which is located at CERN, which is the world’s largest and most powerful particle accelerator. The Maximum achievable energy in the LHC tunnel with a circumference of 26.659 km depends on the maximum magnetic field which cause to stay protons on a perpendicular (circular) path. It accelerates and circulates bunch of protons in two counter rotating beams, they travel at close to the speed of light. The beams travel in opposite directions in separate beam pipes. The two tubes kept at ultrahigh vacuum. The beams are brought into collisions in four interactions points by superconducting electromagnets. For efficient conducting electricity without resistance or loss of energy the magnets requires to achieve -271.3 C, For this reason accelerator connected to a distribution system of liquid helium, which cools the magnets.

Thousands of magnets of different varieties and sizes are used to direct the beams around the accelerator. These include 1232 dipole magnets 15 metres in length which bend the beams, and 392 quadrupole magnets, each 57 metres long, which focus the beams. Just prior to collision, another type of magnet is used to "squeeze" the particles closer together to increase the chances of collisions. These dipoles spread

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around the 27 km circumference which produce a magnetic field up to 8.33 Tesla. Therefore, the energy of each beam is 7 TeV and the proton-proton center-of-mass energy is 14 TeV. Since the LHC is a proton-proton collider, two beam pipes and two opposite direction magnetic field configurations are necessary. The four collision points where the beams accelerated by the LHC will collide and four experiments have been built at the collision points. ATLAS (A Toroidal LHC Apparatus) and CMS are two large multi-purpose detectors. LHCb has been dedicated to study B-physics and CP violation and ALICE (A Large Ion Collider Experiment) is supposed to provide the possibility of study the heavy ion physics when the LHC is running in the mode of Pb-Pb collisions with a center-of-mass energy of 5.5 TeV per nucleon-nucleon pair. After the energy, the other important feature of an accelerator is the luminos-ity.The Luminosity gives a measure of how many collisions are happening in a parti-cle accelerator, it measures how many partiparti-cles were able to squeeze through a given space in a given time. The luminosity of a collider which collides bunches containing

𝑁1 and 𝑁2 particles at a frequency of f is given by:

𝐿 = 𝑓 𝑁1𝑁2

4𝜋𝜎𝑥𝜎𝑦

(60)

where 𝜎𝑥 and 𝜎𝑦 are the Gaussian transverse beam profiles, which allows us to

calculate the predict number of events of process in given dataset. The total inelastic cross section of proton-proton scattering at the LHC is expected to be 60 mb. Hence, the event rate R (the number events produced per second by the proton-proton in-teractions) at high luminosity is:

𝑅 = 𝐿 × 𝜎 = 1034𝑐𝑚−2𝑠−1× 60𝑚𝑏 6 × 108𝑒𝑣𝑒𝑛𝑡𝑠

𝑠 (61)

4.2 Pile-up interactions

The LHC is designed to deliver a very high peak luminosity, this one has the ad-vantage to improve the collisions. When the machine is running at high luminosity

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Figure 9: The LHC ring with four detectors.

(1034_𝑐𝑚−2_𝑠−1_{), on average 20 minimum bias}2 _{events occur simultaneously at each}

bunch crossing. Therefore, when a high-𝑝𝑇 event is produced during a bunch

cross-ing, this event is overlapped with on average additional 20 soft events. This is called event pile-up and the hard interaction events include the soft additional interactions Which cause extra tracks and energy deposits in the detector. Pile-up events origi-nate from their own interaction vertex, and a good vertex reconstruction is therefore important to distinguish particles originating from the hard interaction from those originating in pile-up interaction. Pile-up is one of the difficulties of the experimen-tal operations at the LHC and has had a major impact on the detector design. For MC simulations, the simulation of the sample of interest is overlaid with events from minimum-bias simulation. This happens after the detector simulation (as described in 4.8) where the detector hits of the minimum-bias sample is simply superimposed with those of the main interaction. The number of additional generated interactions follows an assumed distribution, expected to be close to the true pile-up distribution as measured in data. The MC events are then reweighted in order to fully match the true pile-up distribution of any given data sample.

2_{The events which are due to ”𝑙𝑎𝑟𝑔𝑒 − 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒” collisions between two incoming protons. The}

momentum transfer is small and the produced particles have large longitudinal momenta and small transverse momenta (𝑝𝑇 = 𝑝2𝑥 + 𝑝2𝑦) relative to the beam axis. These events are called

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Figure 10: Illustrative hard scattering event.

4.3 Compact Muon Solenoid

The CMS detector is designed to measure and detect particles based on their inter-actions with detector materials. CMS consists of many layers of silicon as tracking system, an electromagnetic calorimeter, a hadronic calorimeter and a muon system, within an onion-like design. Every layer takes a cylindrical shape in which the compo-nents parallel to the beam line are called the barrel regions, and compocompo-nents closing the detector on both sides are usually referred as the endcaps. The tracker and calorimeter barrel layers are installed inside the solenoid, hence they have to be very compact. The outermost layer is the muon system, which is able to measure the direction and momenta of the muons. Between the hadronic calorimeter and the muon system, a superconducting magnet is located, which is capable of reaching a magnetic field of 4.0 T in its contained volume. The large bending power provided by the solenoidal magnetic field, operating at 3.8 T, is needed to bend the tracks of charged particles in the transverse plane, which allows for a precise measurement of the charge and momentum of these particles. The tracker and calorimeter barrel layers are installed inside the solenoid, hence they have to be very compact. The muon detectors are installed between the different layers of the iron return yoke.

The origin of the CMS coordinate system is taken at the nominal collision point, with the x-axis pointing towards the centre of the LHC ring, the y-axis pointing

Studies of events with a Z or a photon vector boson in association with a large invariant mass dijet system, produced with electroweak and strong interactions, in proton-proton collisions at 13 TeV center of mass energy.

Studies of events with a Z or a 𝛾 vector boson in

association with a large invariant mass dijet system,

produced with electroweak and strong interactions,

in proton-proton collisions at

√

𝑠 = 13 TeV.

by

Negin Shafiei

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

Doctor of Science in experimental of particle physics

at the

University of Pisa, INFN sezione di Pisa

March 2021

© University of Pisa 2021. All rights reserved.

Author . . . .

Department of Physics

Certified by. . . .

Paolo Azzurri Supervisor

Associate Professor

Thesis Supervisor

Studies of events with a Z or a 𝛾 vector boson in association

with a large invariant mass dijet system, produced with

electroweak and strong interactions, in proton-proton

collisions at

√

𝑠 = 13 TeV.

by

Negin Shafiei

Abstract

Acknowledgments

Contents

List of Figures

List of Tables

1

Summary

2

Introduction

3

Standard Model

3.1

Introduction

3.2

Symmetries in Standard Model

3.3

Electroweak interaction

3.4

V- A structure of the weak interaction

3.5

Spontaneous breaking of 𝑆𝑈 (2)

⊗ 𝑈 (1)

symmetry with

Higgs mechanism

3.6

Electroweak lagrangian

3.7

Strong interaction

3.8

Hadron collision

4

The CMS experiment at the Large Hadron

Col-lider

4.1

Large Hadron Collider

4.2

Pile-up interactions

4.3

Compact Muon Solenoid