Durham E-Theses Simplied Models and Eective Theories in Searches for New Physics WANG, QI How to cite: WANG, QI (2018) Simplied Models and Eective Theories in Searches for New Physics, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/12723/ Use policy This work is licensed under a Creative Commons Public Domain Dedication 1.0 (CC0) Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk Simplified Models and Effective Theories in Searches for New Physics Qi Wang A Thesis presented for the degree of Doctor of Philosophy Institute for Particle Physics Phenomenology Department of Physics University of Durham England February 2018 ii Dedicated to my parents iv Simplified Models and Effective Theories in Searches for New Physics Qi Wang Submitted for the degree of Doctor of Philosophy February 2018 Abstract In this thesis two different types of effective field theories (EFTs) have been consid- ered to give an interpretation to effects of physics beyond the Standard Model. The first type of EFT is a so-called Simplified Models, in which the Standard Model is extended by new scalar degrees of freedom A (CP-odd) and S (CP-eve), which give rise to interactions of the type ttS¯ and ttA¯ with subsequent decay of S=A b¯b. We ! study the phenomenology of these processes at the LHC and find that the produc- tion of ttA¯ is suppressed compared to that of ttS¯ . Using the Weyl-van-der-Waerden spinor formalism we analyse the helicity amplitudes in order to explain this phe- nomenon. In the second part of this thesis, we focus on the dimension-six Standard Model Effective Field Theory (SMEFT) to calculate the next-to-leading order (NLO) elec- troweak corrections to the forward-backward asymmetry in the process e+e− b¯b. ! We find that the NLO EFT corrections can not be neglected compared to the LO EFT contributions. We show that relevant numerical results have been obtained, which can be used to constrain the Wilson Coefficients involved in this process. We calculate the amplitude of the Z b¯b process and renormalise it together with the ! e+e− b¯b process. We have also analysed the NLO EFT corrections to the cross- ! section of the e+e− b¯b process, which can be used to set limits on the Wilson ! coefficients with future collider experiments. vi Declaration The work in this thesis is based on research carried out in the Institute for Particle Physics Phenomenology, Department of Physics at Durham University. No part of this thesis has been submitted elsewhere for any degree or qualification. The work presented in Chapter 3 is part of the research undertaken by the author in collaboration with Matthew J. Dolan, Michael Spannowsky and Zhao-Huan Yu. The original work can be found in: Determining the Quantum Numbers of Simplified Models in ttX¯ production at the LHC Matthew J. Dolan, Michael Spannowsky, Qi Wang, Zhao-Huan Yu Phys. Rev. D 94, 015025 The work of Chapter 4 which involves NLO calculations of the forward-backward asymmetry in the dimension-six SMEFT framework. It was undertaken by the author in collaboration with Rhorry Gauld, Michael Spannowsky and Ben Pecjak. The results have not been published on the date of submission of this thesis. Copyright c 2018 by Qi Wang. \The copyright of this thesis rests with the author. No quotations from it should be published without the author's prior written consent and information derived from it should be acknowledged". vii viii Acknowledgements I have had a very memorisable time during the past few years in which I carried on my research work included in this thesis. The words are always too limited to express the human emotions and feelings. There are too many people I want to thank in a short acknowledgement. First and foremost I offer my sincerest gratitude to my supervisor, Michael Span- nowsky, who has guided me and encouraged me during my PhD period with his rich knowledge and patience. What's more, the freedom he gave me that allow me to work in my own way is very precious. I would also like to thank Ben Pecjak and Rhorry Gauld, who provided me a lot of help in my research. All the people in IPPP are very nice and they deserve my regards. Special thanks to my friends Yang Lei, Xin Tang, Yan Xu and Ye-Ling Zhou for friendship and accompany, and all the good memories. Finally, I want to thank my parents who allow me to pursue my degree in a foreign country especially in the hard times. ix x Contents Abstract v Declaration vii Acknowledgements ix 1 An introduction to the Standard Model of particle physics 1 1.1 Gauge Symmetries and Lagrangian . .1 1.2 Spontaneous Symmetry Breaking and Electroweak Theory . .7 1.3 Two Processes in Electroweak Physics . 13 1.3.1 e+e− b¯b ............................. 13 ! 1.3.2 The Muon Decay . 16 1.4 Divergences and Renormalisation . 18 1.4.1 Introduction . 18 1.4.2 Renormalisation in Electroweak Standard Model . 19 1.5 Outline of the Thesis . 24 2 Effective Field Theories 27 2.1 Standard Model Effective Fields Theory . 28 2.1.1 Higgs and Electroweak Phenomenology in the Tree-Level Dimension- six SMEFT . 29 2.2 Simplified Models . 32 3 Simplified Models in searches for dark sectors 37 3.1 Motivation and Introduction . 37 3.2 The Amplitudes in the WvdW Formalism . 40 xi Contents xii 3.2.1 A Brief Introduction to the Weyl-van-der-Waerden Spinor Tech- nique . 41 3.2.2 The T-channels and U-channels Amplitudes . 44 3.2.3 The S-channel Amplitudes . 52 3.3 Spin and Parity Discriminating Variables . 56 3.4 Detector-Level Analysis . 59 3.4.1 Expected Sensitivity for Discrimination among Simplified Mod- els ................................. 61 4 Forward-Backward Asymmetry in NLO SMEFT 67 4.1 Tree-Level Matrix Element . 69 4.2 Renormalisation Procedure . 73 4.3 The One-loop SM Counterterms . 79 4.4 The Class 8 Matrix Element . 80 4.4.1 Bare matrix element . 80 4.4.2 Class 8 Counterterms and Renormalised Amplitudes . 90 4.5 The Class 7 Matrix Element . 93 4.6 The Class 4 Matrix Element . 97 4.7 GF as an Input Parameter . 99 4.8 Phenomenology and Numerical Results . 100 4.8.1 The Forward-Backward Asymmetry at Z peak . 104 4.8.2 The EFT Effects on The e+e− b¯b Cross-Section . 114 ! 5 Conclusions 119 Appendix 123 A Basic and Auxiliary Results 123 A.1 Independent Dimension-six Operators in the Warsaw Basis . 123 A.2 Large-mt Limit and Vanishing-mb Limit . 126 A.3 Cancellation of UV Divergence for Class 8 Operators in the Z b¯b ! Process . 129 List of Figures 1.1 The tree-level diagram for e+e− Z/γ ff reactions . 14 ! ! 1.2 The tree level diagram for µ decay process in the Standard Model . 17 ¯ 0 0 3.1 This figure shows the ttX + jets (X = S; A; ZV ; orZA) production cross sections as functions of mX at the 14 T eV LHC for the four simplified models. 40 3.2 LO ggtHt diagrams . 41 3.3 This figure shows the normalised distributions of mtt¯ (a), pT;X (b), CM CM ¯ θt (c) and Θ (d) for parton-level ttX production with ps = 0 0 14 TeV and mX = 50 GeV. Here X = S, A, ZV, or ZA. The CM CM angular variables θt and Θ are defined in the CM frame of the ¯ CM ttX system. As shown in Fig. 3.4, θt is the angle between t and the beamline, while ΘCM is the angle between the normal vector to the ttX¯ system and the beamline. 57 CM CM 3.4 This figure shows the definitions of θt and Θ in the CM frame ¯ CM of the ttX system. θt is the angle between the t-quark and the beamline and ΘCM is the angle between the normal vector to the ttX¯ system and the beamline. 58 3.5 This figure shows the normalised mbb distributions after applying all the selection cuts except the cut on mbb. For all the signals we take mX = 50 GeV. 59 xiii List of Figures xiv 3.6 This figure shows the expected 95% CL exclusion limits on the signal ¯ strength σ(pp ttX¯ ) BR(X bb) with mX = 50 GeV as functions ! · ! of the integrated luminosity at the 14 TeV LHC for ttS¯ (a), ttA¯ (b), ¯ 0 ¯ 0 ttZV (c), and ttZA (d) production. The dot-dashed lines denote the ¯ signal strengths for the gt values labelled and BR(X bb) = 100%. 62 ! 3.7 This figure shows the normalised distributions of mtt (a), pT;X (b), CM CM θt;had (c), and Θ (d) at detector level for the 14 TeV LHC and mX = 50 GeV. 63 3.8 This figure shows the expected 95% CL exclusion limits on the visi- ble cross section σvis as functions of the integrated luminosity at the CM 14 TeV LHC based on the variables mtt (a), pT;X (b), θt;had (c), and ΘCM (d). The lines denote the median value of the limit, while the coloured bands denote the 1σ range. \s1 vs s2" corresponds to the ± exclusion of the signal hypothesis s2 in favour of the signal hypothesis s1, assuming both hypotheses yield the same visible cross section.