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On the Renormalization of the Two-Higgs-Doublet Model On the Renormalization of the Two-Higgs-Doublet Model Masterarbeit von Marcel Krause An der Fakultät für Physik Institut für Theoretische Physik Referent: Prof. Dr. M. M. Mühlleitner (KIT, Karlsruhe) Korreferent: Prof. Dr. R. Santos (ISEL & ULISBOA, Lissabon) Bearbeitungszeit: 11. Mai 2015 – 11. Mai 2016 KIT — Die Forschungsuniversität in der Helmholtz-Gemeinschaft www.kit.edu Hiermit versichere ich, dass ich die vorliegende Arbeit selbst¨andig verfasst und keine anderen als die angegebenen Hilfsmittel benutzt, die w¨ortlich oder inhaltlich ubernommenen¨ Stellen als solche kenntlich gemacht und die Satzung des Karlsruher Instituts fur¨ Technologie (KIT) zur Sicherung guter wissenschaftlicher Praxis in der aktuell gultigen¨ Fassung beachtet habe. Karlsruhe, den 11. Mai 2016 ............................................ (Marcel Krause) Als Masterarbeit anerkannt. Karlsruhe, den 11. Mai 2016 ............................................ (Prof. Dr. M. M. Muhlleitner)¨ \Es irrt der Mensch so lang er strebt." Johann Wolfgang von Goethe (Faust. Der Trag¨odie erster Teil) Contents 1. Introduction 1 2. Introduction to the Two-Higgs-Doublet Model 5 2.1. Motivation . .5 2.2. Constraints on Theories Beyond the Standard Model . .6 2.2.1. The ρ Parameter . .6 2.2.2. Flavor-Changing Neutral Currents . .7 2.2.3. Unitarity Constraints . .7 2.3. The Electroweak 2HDM Lagrangian . .7 2.4. The Scalar 2HDM Potential . .8 2.5. The Scalar Lagrangian . 12 2.6. The Yukawa Lagrangian and FCNCs . 14 2.7. Gauge-Fixing Procedure and Faddeev-Popov Ghosts . 15 2.8. Set of Independent Parameters . 16 3. Partial Decay Widths of One-to-Two Processes at Next-to-Leading Order 17 3.1. Kinematics of Decay Processes at Leading-Order . 17 3.2. Partial Decay Width at Next-to-Leading Order . 19 4. Renormalization of the Two-Higgs-Doublet Model 21 4.1. Divergences in Classical and Quantum Field Theories . 21 4.2. Regularization and Renormalization . 23 4.3. On-Shell Renormalization . 24 4.4. Tadpole Renormalization . 29 4.4.1. Standard Tadpole Scheme . 30 4.4.2. Alternative Tadpole Scheme . 32 4.5. Renormalization of the Gauge Sector . 39 4.6. Renormalization of the Fermion Sector . 41 4.7. Renormalization of the Scalar Fields and Masses . 43 4.8. Renormalization of the Scalar Mixing Angles α and β ............. 45 4.8.1. Minimal Subtraction Scheme . 46 4.8.2. Kanemura's Scheme . 46 4.8.3. Pinched Scheme . 50 4.8.4. Process-Dependent Scheme . 53 4.9. Renormalization of Λ5 ............................... 58 4.9.1. MSScheme ................................. 59 4.9.2. Improved MSScheme ........................... 59 4.9.3. Process-Dependent Scheme . 60 4.10. Renormalization of the Gauge-Fixing Lagrangian . 64 viii Contents 5. The Decays H+ W + h0=H0 at Next-to-Leading Order 65 −! 5.1. The Partial Decay Width at LO . 66 5.2. NLO Virtual Corrections . 66 5.3. IR Divergences and Real Corrections . 69 5.4. Gauge-Dependence of the NLO Amplitude . 71 6. The Decay H0 Z0 Z0 at Next-to-Leading Order 75 −! 6.1. The Partial Decay Width at LO . 76 6.2. The Partial Decay Width at NLO . 76 6.3. Gauge-Dependence of the NLO Amplitude . 79 7. The Decay H0 h0 h0 at Next-to-Leading Order 81 −! 7.1. The Partial Decay Width at LO . 82 7.2. The Partial Decay Width at NLO . 82 7.3. Gauge-Dependence of the NLO Amplitude . 84 8. Numerical Results 87 8.1. Used Software Packages . 87 8.2. Input Parameter Sets . 88 8.3. Numerical Results for the Decays H+ W + h0=H0 ............ 92 −! 8.4. Numerical Results for the Decay H0 Z0 Z0 ................ 96 −! 8.5. Numerical Results for the Decay H0 h0 h0 ................ 97 −! 9. Conclusion and Outlook 103 A. Alternative Parametrization of the 2HDM Potential 107 B. Introduction to the Pinch Technique 109 B.1. Motivation . 109 B.2. Basic Principles of the Pinch Technique . 110 B.2.1. Cancellation of Gauge-Dependences within One-Loop Amplitudes . 110 B.2.2. The Pinch Technique in Practice . 112 B.3. Gauge-Independent Quark-Quark Scattering Amplitude . 113 B.3.1. Preliminary Remarks . 113 B.3.2. The Scattering Amplitude at LO . 113 B.3.3. Pinch Contributions from the Box Diagrams . 114 B.3.4. Pinch Contributions from the Triangle Diagrams . 117 B.3.5. Pinch Contributions from the Quark Self-Energy Diagrams . 120 B.3.6. Pinch Contributions from the Gluon Self-Energy Diagrams . 120 B.3.7. The Full One-Loop Scattering Amplitude . 121 B.3.8. Additional Pinch Contributions from the Three-Gluon Vertex . 123 B.3.9. Creation of a Pinched Gluon Self-Energy . 126 C. The Pinch Technique in the 2HDM 129 C.1. The Pinch Technique in the 2HDM in Practice . 129 C.2. The Pinch Technique in the CP-Even Sector . 131 C.2.1. Gauge-Dependence of the CP-Even Self-Energies . 131 C.2.2. Derivation of the CP-Even Pinched Self-Energies . 134 C.3. The Pinch Technique in the CP-Odd and Charged Sectors . 139 Acknowledgement / Danksagungen 141 References 143 CHAPTER 1 Introduction Even centuries before the advent of high-energy physics, the concept that ordinary matter might consist of smaller building blocks has driven scientific and philosophic curiosity alike. The ancient Greek word ´atomos, literally translated to \indivisible" [1], is a remnant of this belief which was applied in the last two centuries in chemistry by structuring the atoms of the periodic table of elements as the fundamental components of all matter. It was not until the end of the 19th century that scientists realized that atoms are not truly indivisible, but consist of smaller building blocks themselves. The electron was the first of these sub-atomic particles to be discovered and the detection of protons and neutrons soon followed, enabling a complete explanation of the inner structure of atoms in the picture of the Rutherford model [2,3]. Due to the development of quantum mechanics in the early 20th century, the duality between particles and waves became clear, and the division between particles according to one of their intrinsic quantum mechanical properties, called the spin, was presented in a mathematically rigorous way. Subsequently, the unification of quantum mechanics with the theory of physical fields was achieved in the form of quantum field theories. The first of these theories to be mathematically complete, called quantum electrodynamics, explains among other phenomena the electromagnetic interaction of charged particles and photons [4,5]. The enormous success of quantum electrodynamics led to the search for more general field theories, trying to explain not only the electromagnetic interaction, but additionally, the weak interaction, which explains radioactive decays, and the inner structure of the nuclei, described by the strong interaction. In the middle of the 20th century, the development of particle accelerators allowed for the observation of particle collisions at higher energies. This in turn led to the observation that all baryons, e.g. protons and neutrons, are composed of quarks as their fundamental elementary particles, and the interaction between those particles is mediated by photons γ, electroweak gauge bosons Z0, W ± and gluons g as the carriers of the fundamental forces [3]. In order to arrive at a more fundamental theory of particle physics, the electromagnetic and weak interactions were unified to the so-called electroweak interactions. This theory, together with quantum electrodynamics, was used in the late 1970s to build up the Standard Model (SM) of particle physics [6{8]. Within the SM, all interactions and particles of nature known to date (apart from gravity) arise in a mathematically rigorous way [3, 5]. 2 1. Introduction Through the course of the second half of the 20th century, every component of the SM was observed at a variety of collider experiments, and with the discovery of the top quark in 1995 [9, 10] and of the τ neutrino in 2000 [11], the observation of the lepton, hadron and gauge boson sectors of the SM was completed. The last missing piece of the SM was a scalar particle, called the Higgs boson, being predicted as a fundamental building block of the SM as early as 1968 [12{16]. Over four centuries later, such a scalar particle was discovered at the Large Hadron Collider (LHC) in 2012 [17, 18], finally completing the Standard Model of particle physics. Although the theory of the SM turned out to be successful in the last decades and electroweak precision measurements suggest a so-far very good agreement with SM predictions [19{26], several phenomena exist that are not explainable with our current understanding of particle physics. Among these is the dominance of matter over antimatter in our universe [27] and the absence of a valid candidate for dark matter within the SM framework, while experimental data establish an abundance of dark matter with respect to ordinary matter in our universe [28]. Therefore, it is necessary to explore physics beyond the Standard Model (BSM) in order to propose solutions to the so-far open problems within the SM. The Two-Higgs-Doublet Model (2HDM), which is investigated in this thesis, is a BSM theory which is capable of solving some of the problems mentioned above while still preserving the good agreement between the SM and experiments. The 2HDM provides a simple extension of the SM through the introduction of an additional scalar doublet in the Higgs sector, giving rise to a possible candidate for dark matter [29,30] and mechanisms to explain the dominance of matter over antimatter [31]. Due to the second Higgs doublet, the phenomenology and scalar particle content is extended in comparison to the SM, containing one light and one heavy neutral CP-even Higgs boson (h0 and H0, respectively), one neutral CP-odd Higgs A0 and two charged scalar Higgs bosons H± [32,33].
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