新粒子物理模型中的帶電輕子變味反應與雙希格斯粒子生成反應

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(1)Charged Lepton Flavor Violating Processes and Double Higgs Boson Production in New Particle Physics Models. Van Que Tran Supervisor: Professor Chia-Hung Vincent Chang Professor Tzu-Chiang Yuan Department of Physics National Taiwan Normal University. This dissertation is submitted for the degree of Doctor of Philosophy. July 2018.

(2) . Charged Lepton Flavor Violating Processes and Double Higgs Boson Production in New Particle Physics Models. Keywords: Flavor Puzzles, Higgs Physics, Beyond Standard Model , ,.

(3) Dedicated to my loving parents and my wife!.

(4) Declaration I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other University. This work was done wholly or mainly while in candidature for a Ph.D degree at this University. This dissertation is my own work and jointly with others. Van Que Tran July 2018.

(5) Acknowledgements The work presented here would have not been done without the help and support of many people to whom I owe many thanks. Firstly, I would like to express my sincere gratitude to my advisor Prof. T. C. Yuan for his endless support of my PhD study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing this thesis. I could not have imagined having a better advisor and mentor for my PhD study. My deep thanks to my co-advisor, Prof. Chia-Hung Vincent Chang for his care, support for my student life at National Taiwan Normal University (NTNU). He is the kindest person I ever meet in my life. I also would like to thank Prof. Chuan-Ren Chen for his creative discussions. He has taught me a lot of useful knowledge about collider phenomenology. Without him, I could not finish the project on double Higgs boson production. I am strongly indebted to Prof. P. Q. Hung for his guidance and collaborations. Beside my advisors, I am very grateful to the rest of my thesis committee: Prof. ChuanRen Chen, Prof. Hsiang-nan Li and Prof. Kingman Cheung, for giving time out of their schedules to review my thesis, for their insightful comments, suggestions and also for my doctoral defense. I also thank my colleagues and friends Chia-Feng, Trinh Le, Chrisna, Lu Sean, Nhung, Dr. Ray, and Dr. Sming for their collaborations and fruitful discussions regarding physics and beyond. I wish to express deep gratitude to the Department of Physics, NTNU and Institute of Physics, Academia Sinica. I would like to thank professors in Department of Physics, NTNU for their teaching and sharing the knowledge. Furthermore, I am grateful to all members of the High Energy Theory Group at IOP, Academia Sinica. I am also grateful to the administrative staff, especially to Ms. Ming-Fang Lee at NTNU and Ms. Svetlana Sam at Academia Sinica for spending their times to help me with numerous paperworks during my entire study period in Taipei. My deepest gratitude goes to my beloved parents, Noan Tran, Nhan Nguyen and my wife Hue Anh for their moral support throughout my work and studies..

(6) viii Last but not the least, I would like to thank all people whom I could not explicitly mention here but their presences are such valuable memories of my graduate school life..

(7) Abstract In this thesis, I study several important topics in two different frontiers of particle physics. In the high intensity frontier, I study the charged lepton flavor violating radiative decays, muonto-electron conversion in nuclei and electric dipole moments of electron and neutron in the electroweak-scale right-handed neutrino model proposed by Hung. The relevant parameters in the model are constrained by the latest limit from MEG for the radiative decay rate of muon into electron, the current experimental limits (SINDRUM II) and projected sensitivities (Mu2e, COMET and PRISM) for the muon-to-electron conversion rates in various nuclei and the latest limits for electron and neutron electric dipole moments from ACME experiment and ILL collaboration respectively. Overall, depending on the mirror fermion masses and mixing scenarios in the model, it turns out that the most stringent constraint is from the ACME experiment for the electron electric dipole moment which suggests the new Yukawa couplings of leptons should be minuscule of order 10 4 to 10 5 , while the current limit on neutron electric dipole moment from ILL collaboration constrains the new Yukawa couplings of quarks to be of order 10 4 . In the high energy frontier, I investigate the 125 GeV Higgs boson pair production at the Large Hadron Collider (LHC), which is a possible way to measure the trilinear Higgs self-coupling, in the new gauged two Higgs doublet model constructed recently by Huang, Tsai and Yuan. Both theoretical and experimental constraints on the parameter space of the model are taken into account. Theoretical constraints include the tree level vacuum stability and perturbative unitarity, while experimental constraints include the Higgs measurements at the LHC, PLANCK’s relic density and direct search limits at XENON1T and PandaX-II of dark matter. I discuss impacts of these constraints on the total cross section of Higgs boson pair production in the model. I show that with additional contributions from the heavy scalar resonances as well as possible modifications in the 125 GeV Higgs self-coupling and extra self-couplings among different Higgses within the model, the total cross section of this process can be enhanced about one order of magnitude larger ¯ than the Standard Model prediction. Kinematic distributions of the two final states of bbgg ¯ b¯ at the LHC are also discussed. and bbb Keywords: Flavor Puzzles, Higgs Physics, Beyond Standard Model. Thesis Supervisors: Professor Chia-Hung Vincent Chang and Professor Tzu-Chiang Yuan.

(8) Table of contents List of figures. xv. List of tables. xix. 1. Introduction. 1. 2. An Overview of the Standard Model 2.1 Matter Particles and Interactions . . . . . . . . . . . . . . . . . . . . . . . 2.2 The SM Gauge Interactions and Symmetry Breaking . . . . . . . . . . . .. 5 5 7. 3. The Electroweak-Scale nR Model 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Particle Content . . . . . . . . . . . . . . . . . . 3.3 Neutrino Masses and Mixings . . . . . . . . . . . . . 3.4 Charged Fermion Masses and Mixings . . . . . . . . . 3.4.1 Charged Lepton Sector . . . . . . . . . . . . . 3.4.2 Quark Sector . . . . . . . . . . . . . . . . . . 3.5 Phenomenological Constraints . . . . . . . . . . . . . 3.5.1 Electroweak Precision Constraints [24] . . . . 3.5.2 The 125-GeV SM-like Scalar Constraints [25]. 4. Low-energy Constraints in the EW-nR Model 4.1 The µ ! eg Process . . . . . . . . . . . . 4.1.1 Overview . . . . . . . . . . . . . 4.1.2 Analytical Expressions . . . . . . 4.1.3 Numerical Analysis . . . . . . . . 4.1.4 Implications . . . . . . . . . . . 4.1.5 Summary . . . . . . . . . . . . . 4.2 µ e Conversion in Nuclei . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. . . . . . . .. . . . . . . . . .. 13 13 14 17 20 20 22 23 23 24. . . . . . . .. 27 27 27 28 30 41 42 43.

(9) xii. Table of contents. 4.3. 4.4. 4.5 5. 6. 4.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Effective Lagrangian for µ e Conversion . . . . . 4.2.3 The Calculation . . . . . . . . . . . . . . . . . . . . 4.2.4 The Relation Between µ e Conversion and µ ! eg 4.2.5 Numerical Analysis . . . . . . . . . . . . . . . . . . 4.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . Electron Electric Dipole Moment . . . . . . . . . . . . . . . 4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Charged Lepton Electric Dipole Moments . . . . . . 4.3.3 Numerical Analysis . . . . . . . . . . . . . . . . . . 4.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . Neutron EDM . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Neutron EDM Formulas . . . . . . . . . . . . . . . 4.4.3 Numerical Analysis . . . . . . . . . . . . . . . . . . 4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. Double Higgs Boson Production in G2HDM at the LHC 5.1 The G2HDM Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The Matter Content . . . . . . . . . . . . . . . . . . . . . 5.1.3 Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Spontaneous Symmetry Breaking and Scalar Mass Spectrum 5.1.5 Theoretical and Higgs Phenomenological Constraints [125] 5.2 Double Higgs Boson Production in G2HDM at the LHC . . . . . . 5.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Relevant Couplings and Production Cross Section . . . . . 5.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . Summary. Appendix A Formulas for I , J , Ii0 (i = 1, · · · , 5). . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. 43 44 48 59 62 71 72 72 73 74 81 81 81 82 87 90 92. . . . . . . . . . . .. 93 93 93 94 95 97 99 100 100 102 104 117 119 123. Appendix B Useful Formulas used in Sec. 4.3 125 B.1 Decay Length of Mirror Leptons . . . . . . . . . . . . . . . . . . . . . . . 125.

(10) Table of contents. xiii. B.2 Muon Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.3 µ e Conversion and Radiative Decay µ ! eg . . . . . . . . . . . . . . . 126 References. 129.

(11) List of figures 2.1. The Standard Model of elementary particles. (Credit: wikipedia.org) . . . . . . .. 6. 3.1 3.2. Constrained S˜S versus S˜MF . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. 4.1 4.2. ¯ t t, e ! bb, ¯ gg, W +W , ZZ) in the EWPredictions of signal strength µ(H scale nR model for examples 1 and 2 in Dr. Jekyll and example 1, 2 and 3 in Mr. Hyde scenarios as discussed in [25], in comparison with corresponding best fit values by CMS [38–41]. . . . . . . . . . . . . . . . . . . . . . . . One-loop induced Feynman diagram for li ! l j g in EW- nR model. . . . . . . . .. 25 29. Contour plots of Log10 B(µ ! eg) (top panel) and Log10 Daµ (bottom panel) on the. (g0S , Mmirror ) plane for normal (left panel) and inverted (right panel) hierarchy in scenarios 1 (red curves) and 2 (blue curves) with g0S = g00S and g1S = g01S = 0. For. . . . . . .. 34 35 36 37 38 39. conversion in electroweak-scale nR model. . . . . . . . . . . . . . . . . . . . .. 48. details of other input parameters, one can refer to the text in Sec. 4.1.3.. 4.3 4.4 4.5 4.6 4.7 4.8 4.9. Same as Fig. (4.2) with g0S Same as Fig. (4.2) with g0S Same as Fig. (4.2) with g0S Same as Fig. (4.2) with g0S Same as Fig. (4.2) with g0S. = g00S = g00S = g00S = g00S = g00S. . . . . . . .. and g1S = g01S = 10 2 · g0S instead. and g1S = g01S = 10 1 · g0S instead. and g1S = g01S = 0.5 · g0S instead. . = g1S = g01S instead. . . . . . . . = 0 and g1S = g01S instead. . . . .. One-loop induced Feynman diagrams from photon and Z boson exchanges for µ One-loop induced Feynman diagram from CP-even scalar exchanges for µ. . . . . . .. . . . . . . e e. conversion in the electroweak-scale nR model. Diagrams for external leg dressings are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.10 Box diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 A two-loop induced Feynman diagram from scalar and gluonic exchanges for µ e. 54 55. conversion in the electroweak-scale nR model. Diagrams for external leg dressings are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58.

(12) xvi. List of figures 4.12 The dipole coupling coefficients of the photon versus the common mirror lepton mass. All the Yukawa couplings are set to be the same as 10 3 . . . . . . . . . . . 4.13 The vector coupling coefficients for the proton and neutron versus the common. 63. mirror lepton mass. All the Yukawa couplings are set to be the same as 10 3 . Note that, the vector couplings coefficients for the neutron arise only from Z diagrams. .. 64. 4.14 The scalar coupling coefficients and gluonic coefficients for the proton and neutron versus the common mirror lepton mass for the SM-like case. All the Yukawa couplings are set to be the same as 10. 3. and all mirror quark masses are set to be. 500 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.15 Same as Fig. (4.14) for the SM-unlike case. . . . . . . . . . . . . . . . . . . . 4.16 Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10 (g0S ), Mmirror ). 65 66. plane for normal mass hierarchy in Scenario 1 with g0S = g00S and g1S = g01S =. 10 2 g0S . The legend shows current experimental limits and projected sensitivities from COMET, Mu2e, SINDRUM II, PRISM and MEG. For details of other input parameters, one can refer to the text in Sec. 4.1.3. . . . . . . . . . . . . . . . .. 4.17 Contour plots of B(µ. 67. e conversion) and B(µ ! eg) on the (log10 (g0S ), Mmirror ). plane for normal mass hierarchy in Scenario 2 with g0S = g00S and g1S = g01S = 10 2 g0S . 67. 4.18 Contour plots of B(µ. e conversion) and B(µ ! eg) on the (log10 (g0S ), Mmirror ). plane for normal mass hierarchy in Scenario 1 with g0S = g00S = g1S = g01S . . . . .. 4.19 Contour plots of B(µ. e conversion) and B(µ ! eg) on the (log10 (g0S ), Mmirror ). plane for normal mass hierarchy in Scenario 2 with g0S = g00S = g1S = g01S . . . . .. 68 68. 4.20 Contour plots of B(µ e conversion) and B(µ ! eg) on the (log10 (g0S ), log10 (g1S )) plane for normal mass hierarchy in Scenario 1 with g0S = g00S , g1S = g01S and. Mmirror = 500 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.21 Same as Fig. 4.20 for Scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . 4.22 Feynman diagrams contributing to charged lepton EDM in mirror fermion model. (a) one-loop diagram and (b) two-loop Zee-Barr type diagram. . . . . . . . . . . 4.23 Contour plot for decay length of eM ! l + fS on the (log10 |g0S |, mM ) plane with |g0S | = |g00S | = |g1S | = |g01S |. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Electron EDM versus log10 |g0S | in the Scenarios A and normal hierarchy for a =. 70 70 74 78. 4.93, b = 0 and dCP = 3p/2. The current upper limit of the electron EDM from. the ACME Collaboration [84] is indicated by the pink line. The color pattern represents various values of the mirror lepton mass mM in logarithmic scale. We set |g0S | = |g00S | = |g1S | = |g01S |. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.25 Same as Fig. (4.24) for Scenario B with a = p/2. . . . . . . . . . . . . . . . .. 78 79.

(13) xvii. List of figures 4.26 Constraints from the current limits and project sensitivities of µ. e conversion,. µ ! eg and electron EDM on the magnitude of the couplings and their phases in. normal mass hierarchy with Scenario A for b = 0 and dCP = 3p/2. The straight dashed lines are the decay length of various values for the mirror electron. The two orange and blue bands are the allowed regions of the muon anomalous magnetic dipole moment with the Majorana phase a21 = 0 and p/4 respectively. The mirror lepton mass mM is taken to be 800 GeV. . . . . . . . . . . . . . . . . . . . . .. 4.27 The same as Fig. (4.26) but for Scenario B. . . . . . . . . . . . . . . . . . . . 4.28 The diagrams contributing to a) the quark EDM, b) the quark chromo-EDM and c) gluon chromo-EDM in the mirror fermion model. . . . . . . . . . . . . . . . . 4.29 Contour plot for the decay length of qM 1 ! qi + fkS on the (log10 |gS |, mM ) plane. . 4.30 The logarithm of the nEDM versus log10 |gS | in the Scenarios 1 with CP phase. 80 80 84 89. a = p/4 and 3p/2. The pink line is the current upper limit of the nEDM from ILL.. The color pattern represents various values of the mirror quark mass mM as shown at the right palette. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.31 Same as Fig. (4.30) for Scenario 2 with CP phase a = 0, a = p/4 and 3p/2. . . . 4.32 The contour plot for the nEDM set to be the current upper limit at ILL (solid pink line) and decay length of qM 1 ! qi + fkS (red dashed line) on the plane of the phase. a and the logarithm of the magnitude of Yukawa coupling. . . . . . . . . . . . .. 5.1. 5.2 5.3. 89 90. 91. A summary of the parameter space allowed by the theoretical and phenomenological constraints. The red regions show the results from the theoretical constraints (VS+PU). The magenta regions are constrained by Higgs physics as well as the theoretical constraints (HP+VS+PU). . . . . . . . . . . . . . 101 Feynman diagrams for production of a pair of 125 GeV Higgs bosons in G2HDM. 103 The scatter plots of relevant parameters to Higgs boson pair production without the experimental constraints from DM relic density and direct searches. The colour palette indicates the ratio of double Higgs boson production cross sections between G2HDM and SM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. 5.4. The scatter plots for the ratio of production cross sections for a pair of 125 GeV Higgs bosons between G2HDM and SM on the plane of dark matter mass and a) relic density of DM, b) spin-independent cross section of DM and nucleon. The green (yellow) band corresponds to 1s (3s ) range of the PLANCK’s relic density measurement of DM [161]. The green and orange line represent the upper limit on spin-independent cross section of DM and nucleon from XENON1T [162] and PandaX-II Experiment [163], respectively. . . . . . . . . . . . . . . . . . . . . 107.

(14) xviii 5.5. List of figures Same as Fig. 5.3 but after taking into account the experimental constraints from DM relic density from PLANCK [161] and direct searches from XENON1T [162] and. 5.6. PandaX-II [163]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 ¯ channel for a) the invariant mass of bbgg, ¯ The kinematic distributions of the bbgg b) the opening angle DR of two photons and c) of 2 b-jets in SM and in G2HDM with p mh2 = 300, 400, 500 and 600 GeV at s = 13 TeV for ATLAS detectors. . . . . . 112. 5.7. c 2 test for the benchmark point B0 (pseudo data) and SM at 13 TeV LHC with an integrated luminosity of Lint = 3000 fb 1 . . . . . . . . . . . . . . . . . . . . . 113. 5.8. The integrated luminosity versus standard deviation with c 2 test for the benchmark. 5.9. point B0 and SM at 100 TeV LHC. s = sSM means the cross section for the process ¯ of benchmark point B0 same as SM value. . . . . . . . . . . 113 gg ! h1 h1 ! gg bb ¯ b¯ channel for a) invariant mass of bbb ¯ b, ¯ b) The kinematic distributions of the bbb opening angle DR of 2 b-jets associated with leading Higgs boson candidate and c) of 2 b-jets associated with sub-leading Higgs boson candidate with mh2 = 700, 800, p 900 GeV at s = 13 TeV for ATLAS detectors. . . . . . . . . . . . . . . . . . 116.

(15) List of tables 1.1. Table of acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1. Fermion fields and their assignments of Standard Model quantum numbers. The index i stands for the generation of fermions. . . . . . . . . . . . . . .. 8. 3.1. 3.2. 4.1 4.2. 4.3 4.4 5.1 5.2 5.3. Matter field contents and their SM quantum numbers together with the horizontal A4 symmetry assignments. The electric charge Q equals T3 +Y /2 in unit of e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Matrix elements for the four auxiliary M k (k = 0, 1, 2, 3) where w ⌘ exp(i2p/3) and g0S and g1S are complex Yukawa couplings. M 0 k can be obtained from M k with the following substitutions g0S ! g00S and g1S ! g01S . . . . . . . . 21 Mixing parameters from global three-neutrino oscillation data taken from [55, 56]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of the dimensionless overlap integrals for aluminum, titanium and gold, evaluated under the assumption that the proton and neutron distributions within each nuclei are the same [72]. . . . . . . . . . . . . . . . . . . . . . Standard Model values of the capture rates for aluminum, titanium and gold in unit of 106 s 1 taken from Ref. [77]. . . . . . . . . . . . . . . . . . . . The current global fit results (±1s ) of three mixing angles taken from [92]. 33. 47 47 75. Matter content and their quantum number assignments in G2HDM. . . . . 95 Seven benchmark points allowed by the combined (VS+PU+HP+DM) constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Branching ratios of the two body decays of h2 in the seven benchmark points. 110.

(16) Chapter 1 Introduction In particle physics, the Standard Model (SM) [1–3] is the most successful theory which describes the universe in terms of fundamental particles and their interactions of fundamental forces. This theory helps us not only to understand how the particles and forces are related to each other but also predict the existence of many particles and their properties which over time are in excellent agreements with almost all experimental results. For example, the SM has predicted the existence of W , Z bosons, top quark, tau neutrino and their properties with great accuracies. Especially, in 2012, both ATLAS [4] and CMS [5] experiments at the Large Hadron Collider (LHC) has discovered the 125 GeV Higgs boson, which seems compatible with the Higgs mechanism [6–8] that provides masses for all fermions, W and Z bosons and the Higgs boson itself, as predicted in the SM. Ever since the discovery of this SM-like Higgs boson, the spectrum of particle content of the SM of particle physics is completed. In addition, leptonic family numbers are accidentally conserved quantities in SM regarding the fact that neutrinos are strictly massless. However, many questions remain unanswered within the SM. In particular, on the conceptual side, we do not understand: • The decoupling of gravity with the other three fundamental forces. So far we have been omitting its existence in SM. • The instability of the Higgs boson mass under quantum corrections, indicating that SM is very sensitive to new physics beyond the TeV scale. • The origins of mass hierarchies and mixings: “Why are there three generations of quarks and leptons?”, “Why exist very different mass scales of quark and lepton sectors?”. • etc..

(17) 2. Introduction. While on the phenomenological side, we have the following issues: • The observations of neutrino oscillation at several experiments imply that neutrinos have non-zero masses. One can then raise the following questions: “What is the origin of neutrino mass?”, “Why are they so light?”, etc. • The Baryon Asymmetry of the Universe (BAU), or in other words the predominance of matter today and what happened to the antimatter after the Big Bang? • “What is the origin of dark matter?” We know from cosmology that dark matter is account for approximately 80% of the total matter in the universe. Moreover, it is still mysterious about the nature of dark energy which accounts for roughly 72% of the total energy budget in the universe. • etc. Solving these questions will help us to understand the universe to provide a more complete picture of the subatomic world, which will definitely affect future theoretical and experimental developments of the field of particle physics. In order to explain the above questions, one needs to search for new physics Beyond the Standard Model (BSM). Indeed many interesting models have been built for various purposes in mind. For example, supersymmetry model (SUSY) which introduces a supersymmetric partner particle for each particle in the SM. This model addresses to solve the Higgs mass hierarchy problem by the cancelation of quadratic divergencies between new contributions from new SM-partners and SM particles to the Higgs mass quantum corrections. Moreover, this model with a discrete R parity provides a dark matter candidate, the neutralino, which is the lightest, stable, electrically neutral SM-partner and weakly interact with the SM particles. Unfortunately, so far there is no direct experimental observation for the existence of these supersymmetric particles. Another well-known extension of SM is extra dimension model which adding more dimensions to the (3 + 1) spacetime, etc. With the motivation of understanding the mass origins and mixings of neutrinos, Hung [9] has constructed a so-called “electroweak-scale right-handed neutrino” (EW-nR for short) model. The crucial feature of this model is the possibility of producing and detecting righthanded neutrinos at the colliders such as the LHC. On the other hand, motivated by the origin of dark matter, Huang, Tsai and Yuan [10] have proposed a so-called “Gauged Two Higgs Doublet Model” (G2HDM for short) which embeds the two Higgs doublets in the popular two Higgs doublet models into a doublet of a non-abelian gauge group SU(2)H . The neutral component of the second doublet can be a dark matter candidate which is stable under the protection of this new gauge group rather than the ad hoc Z2 symmetry in the usual case..

(18) 3 In this thesis, based on the framework of the EW-nR model, we will study some lowenergy phenomena involving charged lepton flavor violation and electric dipole moments, while for the new G2HDM model we will study the Higgs boson pair production at highenergy collider machine such as the LHC. The thesis has the following structure: • Chapter 2 will give us a brief summary of the SM’s particle content and gauge interactions. • Chapter 3 will discuss in detail the EW-nR model. The possible signatures and phenomenological constraints in the model will be discussed. • Chapter 4 will study the low-energy experimental constraints for the EW-nR model. The charged lepton flavor violation (CLFV), µ ! eg, µ e conversion as well as electric dipole moment (EDM) of electron and neutron are calculated within this model. The latest experimental limits and projected sensitivities are used to constraint the parameter spaces in the model. • Chapter 5 will study the double Higgs boson production in the G2HDM at the LHC. A brief overview of the model will be discussed. The benchmark points which have passed all theoretical constraints, the Higgs phenomenology constraints as well as the relic density and direct search limits of dark matter, are picked up to study behaviors of double Higgs boson production. • We conclude our thesis in Chapter 6. Useful formulas and calculations can be found in the Appendices. The metric and unit used throughout in this thesis are gµn = (+1, 1, 1, 1) and h¯ = c = 1. We follow closely Peskin and Schroeder [11] for other conventions. Some of the acronyms used in this thesis are listed in Table 1.1..

(19) 4. Introduction Acronym 2HDM A4 ACME ATLAS BAU BNL BSM CKM CLFV CMS COMET CP DM EDM EM EW EW-nR G2HDM GM HP IHDM ILL LHC MDM MEG MSSM Mu2e nEDM PandaX PLANCK PMNS PRISM PU QCD SINDRUM SM SSB SUSY VEV VS XENON. Description Two Higgs Doublet Model Tetrahedron Symmetry Group Advanced Cold Molecule Electron EDM A Toroidal LHC ApparatuS Baryon Asymmetry of the Universe Brookhaven National Laboratory Beyond the Standard Model Cabibbo-Kobayashi-Maskawa Charged Lepton Flavor Violation Compact Muon Solenoid COherent Muon to Electron Transition Charge Conjugation and Parity Dark Matter Electric Dipole Moment Electromagnetism Electroweak Electroweak-scale Right-handed Neutrino Model Gauged Two Higgs Doublet Model Georgi-Machacek Model Higgs Physics Inert Higgs Doublet Model The Institut Laue-Langevin Large Hadron Collider Magnetic Dipole Moment Mu to E Gamma Minimal Supersymmetric Standard Model Muon-to-Electron Conversion Neutron Electric Dipole Moment Particle and Astrophysical Xenon Detector https://www.cosmos.esa.int/web/planck/home Pontecorco-Maki-Nakagawa-Sakata Phase Rotated Intense Slow Muon source Perturbative Unitarity Quantum Chromodynamics Muon-to-Electron Conversion Experiment Standard Model Spontaneous Symmetry Breaking Supersymmetry Vacuum Expectation Value Vacuum Stability http://www.xenon1t.org Table 1.1 Table of acronyms..

(20) Chapter 2 An Overview of the Standard Model In this chapter, we will review some basic features of the Standard Model (SM) and their key roles in our understanding of the phenomenology of particle physics.. 2.1. Matter Particles and Interactions. The building blocks of matter are spin 1/2 elementary particles that we call fermions. These fermions are postulated to interact via the exchange of mediators called gauge bosons. The names of these particles, together with their basic properties, are given in Fig. 2.1. There are two basic types of fermions: quarks and leptons which are classified based on their charges under strong and electromagnetic interactions. Indeed, there are totally 12 flavours, six flavours for quarks named as up-quark (u), down-quark (d), charm-quark (c), strange-quark (s), top-quark (t) and bottom-quark (b), and another six flavours for leptons named as electron (e), electron neutrino (ne ), muon (µ), muon neutrino (nµ ), tau (t) and tau neutrino (nt ). These fermions have half-integral intrinsic angular momentum or spin and obey Fermi-Dirac statistics [12]. They formed three families with the identical quantum number but different masses, each family consists of 2 quark and 2 lepton flavours. For example, the first family contains u, d quarks and ne , e leptons. In unit of e > 0, up-type quarks (u, c,t) have fraction electric charges of + 23 , while down-type quarks (d, s, b) of 13 . For the leptons, e, µ, t have electric charges of 1, while neutrinos have charges 0. Masses of these fermions can fall into a range from the sub-eV for neutrino masses to the 173 GeV for top quark mass. Moreover, each family shows such a peculiar mass structure in which the masses are hierarchically separated from one another. For each of the fermions, there is an antiparticle, with identical mass and lifetime but opposite sign of magnetic moment and electric charge. For example, antiparticle of electron is positron. In addition, while the leptons exist as free particles, the quarks do not. They are always bound to other quarks by.

(21) 6. An Overview of the Standard Model. Fig. 2.1 The Standard Model of elementary particles. (Credit: wikipedia.org) the strong force and form as composites called hadrons. The hadrons consist of three quarks are called baryons, while mesons consist of a quark-antiquark pair. For example, proton and neutron are baryons with proton is composited by uud whereas neutron is ddu or pions are mesons with p + = ud¯ and p = ud. ¯ The elementary particles which play the crucial role as mediators between the interacting fermions are called bosons. In contrast to fermions, bosons obey Bose-Einstein statistics [13] and have integral spin quantum numbers (±0, ±1, ±2, ...). There are four known types of interactions, each interaction is characterized by a boson exchange particle. They are as follows: • The electromagnetic interaction is characterized by photon exchange with spin 1 and zero mass. This interaction happens between charged particles and is long range. • The strong interaction is characterized by gluon exchange with also spin 1 and zero mass. This interaction only occurs between the quarks and is responsible for the confining force binding quarks together to form baryons and mesons. • The weak interaction is characterized by massive bosons W (mW = 80 GeV) and Z (mZ = 91 GeV) exchange. These particles also have spin 1. At low energy, the weak interaction is weak because it can be well described by the dimension 6 effective 2 . Weak interac4-fermion operators with coefficients inversely proportional to the mW,Z tion is thus short range..

(22) 2.2 The SM Gauge Interactions and Symmetry Breaking. 7. • The gravitational interaction is characterized by graviton with spin 2 and zero mass. This interaction occurs between all forms of matter and radiation. It is also long range. However, its strength among elementary particles is too weak as compared to other interactions. For example, if one takes the strong interaction strength as 1, then approximately the relative strength for electromagnetic interaction will be 10 2 , for weak interaction it will be 10 7 but for gravitational interaction it will be 10 39 . This is the reason why we can omit gravity interaction among elementary particles in the SM. It is important to note that the electromagnetic interaction and weak interaction are indeed different aspects of a single unified interaction, the so-called electroweak (EW) interaction. The merging of two interactions occurs when the universe is hot enough or the energy is above order of 246 GeV. There is another boson in SM called scalar Higgs boson with zero spin which was discovered in 2012 [4, 5]. Its role is to implement the Higgs mechanism, giving mass to all particles as well as itself. A combined result of the Higgs boson at Run1 LHC showed that it has a mass around 125 GeV [14] and very likely it is the SM Higgs boson but we still do not know for sure. Future experiments carried out at the colliders will hopefully give us the answer soon.. 2.2. The SM Gauge Interactions and Symmetry Breaking. Gauge theory is used to describe all the fundamental interactions in Nature. Gauge symmetries are the invariance of the Lagrangian density under gauge transformations which are realized on the elementary fields as local phase rotations. The SM gauge group is identified as GSM = SU(3)C ⇥ SU(2)L ⇥U(1)Y , (2.1) where the color group SU(3)C describes strong interaction with the gauge coupling gs , while electroweak interaction obeys SU(2)L ⇥U(1)Y with gauge couplings g and g0 , respectively. Because parity is violated in weak interaction as opposed to electromagnetic and strong interactions, one expects to use Weyl spinors in SM. Indeed, within the SM, neutrinos have only left-handed components, in contrast to quarks and charged leptons. The left-handed fermions transform as SU(2)L doublets, while the right-handed components are all singlets. The leptons are not involved with strong interaction, therefore they are singlets under SU(3)C , while the quarks are triplets under SU(3)C . Note that the generator of the U(1)Y hypercharge is given by Y /2 = Q T 3 , where Q is the electric charge in a unit of e and T 3 is the third.

(23) 8. An Overview of the Standard Model Fields ✓ ◆ nL LLi = eL i eRi ◆ ✓ uL QLi = dL i uRi dRi. SU(3)C. SU(2)L. U(1)Y. 1. 2. 1 2. 1. 1. 1. 3. 2. 1 6. 3 3. 1 1. 2 3. 1 3. Table 2.1 Fermion fields and their assignments of Standard Model quantum numbers. The index i stands for the generation of fermions.. component of weak isospin SU(2)L . In Table 2.1, we summarize the fermion fields and their quantum number assignments under the SM gauge group. The Standard Model Lagrangian is defined as LSM = LGauge + LFermion + LHiggs + LYukawa ,. (2.2). where LGauge , LFermion , LHiggs and LYukawa are Lagrangian for the gauge, fermion, Higgs and Yukawa interactions, respectively. The Lagrangian for the gauge interaction could be written down as follows LGauge =. 1 a µn G G 4 µn a. 1 i µn Wµn Wi 4. 1 Bµn Bµn 4. (2.3). i and B where Gaµn , Wµn µn are gauge field strengths defined as. Gaµn ⌘ ∂µ Gan. ∂n Gaµ + gs f abc Gbµ Gcn , ∂n Wµi + ge i jkWµ Wnk ,. (2.5). Bµn ⌘ ∂µ Bn. ∂n B µ ,. (2.6). i Wµn ⌘ ∂µ Wni. j. (2.4). with f abc and e i jk the structure constants of SU(3)C and SU(2)L , respectively. The fermion Lagrangian LFermion is given by LFermion = L¯Li iDL / Li + e¯Ri iDe / Ri + Q¯ Li iDQ / Li + u¯Ri iDu / Ri + d¯Ri iDd / iR ,. (2.7).

(24) 9. 2.2 The SM Gauge Interactions and Symmetry Breaking. where D/ ⌘ Dµ gµ and Dµ is the gauge covariant derivative. For a fermion f under the representation of SM gauge group given in Table 2.1, we have ⇣ Dµ f = ∂µ. igs. la a G 2 µ. igT iWµi. ig0. Yf ⌘ Bµ f , 2. (2.8). with l a (a = 1, · · · , 8) are the eight Gell-Mann matrices, T i (i = 1, 2, 3) are the SU(2) generators (for doublet T i = s i /2 where s i are the Pauli matrices, while for singlet T i = 0) and Y f /2 is given by the last column in Table 2.1. The Higgs Lagrangian LHiggs is defined as ⇣ ⌘† LHiggs = Dµ H Dµ H. V (H). (2.9). with V (H) = µ 2 H † H + lSM (H † H)2 being the Higgs potential for the Higgs doublet H. The shape of the potential V (H) depends on the choice of the parameters µ and lSM . While lSM > 0 will ensure stability of the vacuum, the potential will have a local maximum at the origin and degenerate minima on a circle around it when µ 2 < 0. Because the Higgs field H transforms as SU(3)C singlet, SU(2)L doublet and has U(1)Y hypercharge Y = 1, one can write down its covariant derivative as ⇣ Dµ H = ∂µ. ig. si i W 2 µ. g0 ⌘ i Bµ H . 2. (2.10). The last piece is Yukawa Lagrangian, LYukawa , which is given by ij LYukawa = Q¯ LiYd dR j H. Q¯ LiYui j uR j H˜. L¯ LiYei j eRi H + h.c.. (2.11). where H˜ = is2 H ⇤ . Recall that gauge invariance leads to massless gauge bosons. However it is not the case for the electroweak gauge bosons in reality. The experimentally observed W ± and Z bosons are massive. Thus the EW gauge symmetry SU(2)L ⇥ U(1)Y has to be broken to give masses to these gauge bosons. This can be achieved by giving a non-vanishing p vacuum expectation value (VEV) of the Higgs field, hHi = (0, v/ 2)T . As a result, the gauge symmetry SU(2)L ⇥ U(1)Y breaks down to U(1)EM , the electromagnetism gauge group. This is called spontaneous symmetry breaking (SSB) of the EW gauge group. After.

(25) 10. An Overview of the Standard Model. the SSB, there are three massive vector bosons which are given as follows Wµ±. (Wµ1 ⌥ iWµ2 ) p = 2. Zµ =. with mass. sin qW Bµ + cos qW Wµ3. v mW ± = g , 2 with mass. mZ =. (2.12) p v g2 + g02 , 2. (2.13). where the weak mixing angle qW is related to the coupling constants in the following way:. m. g0 sin qW = p , g2 + g02. (2.14). It is easy to see that cos qW = mWZ± . Furthermore, the photon field defined as. Aµ = cos qW Bµ + sin qW Wµ3 ,. (2.15). remains massless. On the other hand, when the Higgs field develops the VEV, the Yukawa interactions generate mass matrices for the charged leptons and quarks of the form M f = Y f v/2, where f = u, d, e. Because neutrinos have only left-handed components, they are massless in the SM. Note that the fermions mass matrices M f are in general 3 ⇥ 3 complex matrices and can be diagonalized by bi-unitary transformations as VLu Mu (VRu )† = diag(mu , mc , mt ) ,. (2.16). VLd Md (VRd )† = diag(md , ms , mb ) ,. (2.17). VLe Me (VRe )† = diag(me , mµ , mt ) ,. (2.18). with VLu , VRu , VLd , VRd , VLe and VRe are unitary matrices in the flavor space, i.e. uL = VLu u0L , uR = VRu u0R , dL = VLd dL0 , dR = VRd dR0 , eL = VLe e0L , eR = VRe e0R ,. (2.19). where the primed fields are physical fields of definite masses. The electromagnetic and the neutral current interactions are given by Lem =. ⇣ e Qu (u¯L g µ uL + u¯R g µ uR ) + Qd d¯L g µ dL + d¯R g µ dR ⌘ µ µ + Qe (e¯L g eL + e¯R g eR ) Aµ ,. (2.20).

(26) 2.2 The SM Gauge Interactions and Symmetry Breaking. 11. and LNC =. h1 e 1 ¯ µ 1 1 u¯L g µ uL dL g dL + n¯ L g µ nL e¯L g µ eL sin qW cos qW 2 2 2 2 ⇣ sin2 qW Qu (u¯L g µ uL + u¯R g µ uR ) + Qd d¯L g µ dL + d¯R g µ dR ⌘i µ µ + Qe (e¯L g eL + e¯R g eR ) Zµ ,. (2.21). respectively. Here, Qu , Qd and Qe are the electric charges of the up-type quarks, downtype quarks and charged leptons. Note that these interactions are invariant under the field redefinitions in Eq. 2.19. Lastly, under these field redefinitions the charged gauge boson W ± couples to the physical fermions become LCC =. p. ⇣ ⌘ e u¯0 L g µ Wµ+ VCKM dL0 + n¯ L g µ Wµ+ (VLe )† e0L + h.c. 2 sin qW. (2.22). where VCKM = VLu (VLd )† is the famous Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix [15, 16]. In SM where the neutrinos are left-handed and strictly massless, the mixing matrix VLe has no physical consequences since one can define a new neutrino field nL0 = VLe nL to rotate this mixing matrix away. We note that the neutral current interaction of the neutrinos with the Z boson is invariance under this field redefinition. Thus we can drop all the primes in the physical fermion fields in the SM Lagrangian..

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(28) Chapter 3 The Electroweak-Scale nR Model The electroweak-scale right-handed neutrino (EW-nR ) model proposed by Hung [9] will be briefly reviewed in this chapter. The crucial feature of this model is non-sterile right-handed neutrinos having masses at the electroweak scale. In the first section we will review the main motivation of the EW-nR model. Next we will present the gauge structure and particle content of the model. In the third section we will review the SSB of this model which provide masses and mixings of various particles, especially that of Dirac and Majorana neutrinos and their mixings. Then we will mainly focus on low energy phenomenological constraints of the EW-nR model.. 3.1. Motivation. The motivation of introducing mirror fermions in [9] was manifold. First of all, it is aesthetically satisfactory to have parity restoration at a higher energy scale while the maximal parity violating interaction (V A interaction) in the SM emerges from spontaneous symmetry breaking. This is one of the main reasons for various left-right symmetric models in the literature [17–20]. Secondly, it is important to study non-perturbative effects in the SM by discretizing it on the lattice. However it is well known that putting chiral fermion on the lattice is plagued by fermion doubling - an unavoidable consequence of the no-go theorem proved by Nielsen and Ninomiya [21]. Sophisticated techniques like using Wilson fermion, Wilson-Ginsparg fermion, staggered fermion, or domain wall fermion etc., which by violating at least one of the assumptions in the no-go theorem gets rid of the unwanted species, are often employed to handle this problem in practise. For new physics model builders, it is attractive to add mirror fermions to the SM which makes the theory becomes vector-like at a higher scale and hence one can avoid the fermion doubling problem if formulating on the lattice. Chiral gauge anomalies will then be cancelled automatically in this class of.

(29) The Electroweak-Scale nR Model. 14. models. The third motivation is the electroweak scale non-sterile right-handed neutrinos introduced in [9]. For each generation, the right-handed neutrino is introduced together with a right-handed heavy charged fermion partner to form a SM SU(2) doublet. Similarly a left-handed heavy mirror charged lepton will be introduced for each right-handed SM charged lepton. Majorana masses can then be given to these right-handed neutrinos via the vacuum expectation value (VEV) of a hypercharge Y /2 = 1 Higgs triplet with mass at the electroweak scale, rather than the grand unification scale in the usual scheme. Tiny Dirac masses can also be given via small VEVs of Higgs singlets with Y = 0. This is the electroweak scale see-saw mechanism in mirror fermion model which is testable at the LHC [22, 23]. Recently, many phenomenological implications of the mirror model [9] have been explored further. We summarize what we have been done in a series of works involving various collaborations: In [24], the model was challenged by the electroweak precision measurements. It was shown that the dangerously large contributions to the oblique parameters from the mirror fermions (especially the S parameter) can be tamed by the opposite contributions from the Higgs triplets. In [25], the original mirror model was extended by adding a mirror Higgs doublet so as to accommodate the LHC data for the SM Higgs signal strengths of various channels. Searches for mirror fermions at the LHC were studied in [22] for mirror quarks and [23] for mirror leptons. In [28], the neutrino and charged lepton masses and mixings were discussed in the mirror model with a horizontal A4 (the discrete tetrahedron symmetry group) symmetry imposed on the lepton sector. Subsequently, in [37], the charged lepton flavor violating (CLFV) radiative decay µ ! eg was studied in details in this mirror model with the horizontal A4 symmetry extension, updating an earlier calculation [43] done for the original model. Moreover, the µ e conversion in nuclei was also studied [44]. In [45], the CLFV Higgs decay h(125 GeV) ! µt was studied for the extended mirror model with a mirror Higgs doublet [25], while the electron electric dipole moment has been performed in [46] for the model. In [47], a study on top quark rare decays in the EW-nR model has also been investigated. Some of these works will be presented in the following chapter.. 3.2. The Particle Content. • The gauge group in this model is: G = GSM ⇥ A4. (3.1). where GSM = SU(3)c ⇥ SU(2) ⇥ U(1)Y is SM gauge group and A4 is the discrete horizontal symmetry imposed on the lepton and Higgs sectors [28] as an extension.

(30) 15. 3.2 The Particle Content. of the original mirror fermion model [9] to address the issues of neutrino masses and lepton mixings in the lepton sector. We will discuss more about this A4 extension in Sec. 3.3 for the neutrino masses and mixings and in Sec. 3.4 for the charged fermion masses and mixings. • For the fermion sector, new mirror fermions have been introduced. The mirror fermions were constructed in a way that for every SM left-handed doublets there is a corresponding right-handed mirror doublet and for every SM right-handed singlet there is a corresponding left-handed mirror singlet. According to this construction, lepton and quark SU(2) doublets would be: ! ! nL n R M= SM: lLi = ; Mirror: lRi . eL eM R i i ! ! uL uM R SM: qLi = ; Mirror: qM . Ri = dL dRM i. i. And lepton and quark SU(2) singlets would be: M M SM: eRi ; uRi , dRi ; Mirror: eM Li ; uLi , dLi .. where the superscript “M” stands for mirror fermions and “i” stands for the generation of fermions. By this way, the right handed neutrinos are now in the doublet SU(2) and being non-sterile neutrinos, instead of sterile neutrinos as in the usual case. Note that all fermions are assigned to be triplets under A4 symmetry. • On the scalar sector side, in the extended mirror fermion model of [25], it consists of two SU(2) doublets F2 and F2M , and the Georgi-Machacek (GM) SU(2) triplets x ˜ A global symmetry U(1)SM ⇥U(1)MF was also introduced in [25] such that and c. F2 only couples to the SM fermions and F2M only couples to the mirror fermions. Thus there is no flavor changing neutral current interactions at tree level in the Yukawa couplings. The x and c˜ form the cornerstone of the EW-nR model. As shown in [9], the VEV of the c˜ (Y /2 = 1) gives an electroweak-scale Majorana mass to the right-handed neutrinos and the x (Y /2 = 0) is needed to preserve the custodial symmetry so that the r parameter equals unity at tree level. Furthermore, the large contribution to the electroweak oblique parameters from the triplets was shown in [24] to be offset by the opposite contribution from the mirror fermions such that the model is safe against the electroweak precision data..

(31) The Electroweak-Scale nR Model. 16. The form and VEVs of SU(2) doublets and triplets are summarized below. F2 (Y /2 = 1/2) =. F2M (Y /2 = 1/2) =. 1 ce (Y /2 = 1) = p ~t.~c = 2. f20⇤ f2+ f2 f20. !. v2 with hf20 i = p . 2. (3.2). 0⇤ f + f2M 2M 0 f2M f2M. !. v2M 0 with hf2M i= p . 2. (3.3). p1 c + 2 c0. c ++ p1 c + 2. !. with hc 0 i = vM .. x (Y /2 = 0) = (x + , x 0 , x ) with hx 0 i = vM .. (3.4) (3.5). In fact, we will have relation between the VEVs: v22 + v22M + 8v2M = v2 ⇡ (246 GeV)2 .. (3.6). It has a total of 17 real scalar fields. Besides the three Nambu-Goldstone bosons, eaten by the longitudinal components of the W ± and Z bosons after spontaneous symmetry breaking of SU(2) ⇥U(1)Y ! U(1)EM , the remaining fourteen real fields are grouped into 5 + 3 + 3 + 1 + 1 + 1 of a SU(2)D , which is a residual symmetry of the breaking of the global custodial symmetry SU(2)L ⇥ SU(2)R ! SU(2)D . The three p singlets are the CP-even neutral Higgses, Re(F02 ), Re(F02M ) and 13 ( 2Re(c˜ 0 ) + x 0 ). While the states within the 5-plet and 3-plet are degenerate in masses, the three singlets can in general be mixed together. It was shown in [25] that the 125 GeV Higgs is an admixture of these three singlets, and these mixing effects are essential to make the model consistent with the LHC data of the 125 GeV Higgs. All these scalars are A4 singlets. The weak singlet scalars f0S and ~fS introduced in [28] are singlet and triplet under A4 respectively. They are the only fields connecting the SM fermions and their mirror counterparts. The particle content of fermions and bosons of the model and their SM quantum numbers together with the horizontal A4 symmetry assignments are shown in Table 3.1..

(32) 17. 3.3 Neutrino Masses and Mixings Fields ◆ ✓ ◆ nR nL M lLi = , lRi = eL i eM R i M e , e Ri Li ✓ ◆ ✓ M ◆ uL uR M qLi = , qRi = dL i dRM i uRi , uM Li M dRi , dLi f0S fiS F2 , F2M x c˜ ✓. SU(3). SU(2). U(1)Y. 1. 2. 1 2. 3. 1. 1. 1. 3. 3. 2. 1 6. 3 3 1 1 1 1 1. 1 1 1 1 2 3 3. 2 3. 3 1 3. 0 0 1 2. A4. 0 1. 3 3 1 3 1 1 1. Table 3.1 Matter field contents and their SM quantum numbers together with the horizontal A4 symmetry assignments. The electric charge Q equals T3 +Y /2 in unit of e.. 3.3. Neutrino Masses and Mixings. Recall that the tetrahedron symmetry group A4 has four irreducible representations 1, 10 , 100 , and 3. The multiplication rule that is relevant to us is 1 : 3 ⇥ 3 = 31 (23, 31, 12) + 32 (32, 13, 21). + 1(11 + 22 + 33) + 10 (11 + w 2 22 + w33) + 100 (11 + w22 + w 2 33). (3.7). where w = e2pi/3 . In the gauge eigenbasis (fields with superscript 0), one can write down the following A4 invariant Yukawa couplings 2 , LY l. g0S f0S (lL0 lR0M )1 + g1S~fS · (lL0 ⇥ lR0M )31 + g2S~fS · (lL0 ⇥ lR0M )32 + H.c. 0 ~ 0M 0 ~ 0M 0 0 + g00S f0S (e0R e0M L )1 + g1S fS · (eR ⇥ eL )31 + g2S fS · (eR ⇥ eL )32 + H.c.. (3.8). Similar couplings can be written down for the quarks as well and we will describe them later. 1 3 is differ from 3 because A is nonabelian. 4 1 2 2 After spontaneous symmetry breaking the scalar. singlets f0S and ~fS might be mixing among each other as well as with other scalars in the model. We have assumed the quartic couplings responsible to these mixing effects are negligibly small so that f0S and ~fS are the physical states..

(33) The Electroweak-Scale nR Model. 18. As shown in [28], after the scalar singlets develop VEVs with v0S = hf0S i and vkS = hfkS i, one obtains the neutrino mass matrix from the first line of Eq. (3.8) 0. 1 g0S v0S g1S v3S g2S v2S B C MnDirac = @ g2S v3S g0S v0S g1S v1S A . g1S v2S g2S v1S g0S v0S. (3.9). Reality of the mass eigenvalues of MnDirac implies g0S is real and g2S = g⇤1S . Furthermore, if one assumes vkS = vS , MnDirac reduces to 0. 1 g0S v0S g1S vS g⇤1S vS B C MnDirac = @ g⇤1S vS g0S v0S g1S vS A . g1S vS g⇤1S vS g0S v0S. (3.10). The above form of MnDirac can be diagonalized by an unitary matrix Un , i.e. Un† MnDiracUn = Diag Mn with [28] 0 1 1 1 1 1 B C † Un ⌘ UCW = p @ 1 w2 w A , (3.11) 3 2 1 w w. where w is the same as in the multiplication rules of A4 given in Eq. (3.7). The matrix UCW in Eq. (3.11) was first discussed by Cabibbo [29] and also by Wolfenstein [30] in the context of CP violation for three generations of neutrino oscillations. On the other hand, the Majorana mass term for the right-handed neutrinos can be generated by the following A4 invariant Lagrangian [28] ⇣ ⌘ ˜ lR0M + H.c. . LM = gM lR0M,T s2 (it2 c). (3.12). When the neutral component of the A4 singlet c˜ develops a VEV hc0 i = vM , one obtains the Majorana mass matrix MR [28] 0. 1 1 0 0 B C MR = gM vM @ 0 1 0 A . 0 0 1. (3.13). The full neutrino mass matrix is given by Mn =. 0 MnDirac. T. MnDirac MR. !. ,. (3.14).

(34) 19. 3.3 Neutrino Masses and Mixings. with MnDirac and MR given by Eq. (3.10) and Eq. (3.13) respectively. The light neutrino mass matrix is then mn ⇠. MnDirac MR 1. ⇣ ⌘T Dirac Mn =. ⇣ ⌘T 1 Dirac Dirac M Mn , gM vM n. (3.15). where we have used the fact that MR in Eq. (3.13) is proportional to the unit matrix. We note that an unitary matrix Un that diagonalizes the Dirac mass matrix MnDirac in general won’t diagonalize the light neutrino mass matrix mn . However, with MnDirac given by Eq. (3.10) in the model, one can check readily that the light neutrino mass matrix mn in Eq. (3.15) can be diagonalized by Un given by Eq. (3.11) as well. Note also that the true light neutrino masses should be obtained by block-diagonalization [31– 34] of the full neutrino mass matrix Mn in (3.14). The mn given by Eq. (3.15) can only be regarded as an effective light neutrino mass matrix obtained by integrating out the heavy degrees of freedom represented by the heavy Majorana fermions with mass of order MR . Thus it may receive sub-leading corrections upon block-diagonalization of Mn . For the purposes of this work, since MR hf0S i, hfiS i, it is sufficient to consider the effective light neutrino mass matrix. The PMNS (Pontecorco-Maki-Nakagawa-Sakata) neutrino mixing matrix is then given by UPMNS = Un†ULl where ULl is the unitary matrix that diagonalizes the charged lepton mass matrix squared. A phenomenological approach was proposed in [28] to parameterize ULl as deviating from unity in the form of a Wolfenstein-like unitary matrix. Using the experimental input for the matrix elements of UPMNS , the allowed ranges for the Wolfenstein parameters in ULl can be deduced [28]. We note that this discrete A4 symmetry does not forbid the quartic couplings of the Higgs singlets with the doublets and triplets. After symmetry breaking, these would lead to additional scalar mixings not considered before in [25] which can give contributions to the invisible width for the 125 GeV Higgs. As shown in [25], the mixings between the neutral components of the two Higgs doublets and the GM triplets are tightly constrained already by the LHC data for the signal strengths of the 125 GeV Higgs. Including the singlets in the mixings is beyond the scope of this thesis. However they are expected to be tightly constrained as well. We will assume these additional scalar mixings are small enough in order to circumvent the LHC data on the Higgs invisible width and signal strengths. In recent years, advocating A4 symmetry in the lepton sector was mainly due to Ma [35]. For an elementary introduction of the A4 discrete group, see for example [36]..

(35) The Electroweak-Scale nR Model. 20. 3.4 3.4.1. Charged Fermion Masses and Mixings Charged Lepton Sector. From Eq. (3.8), one also can see the mixing between SM charged lepton and mirror lepton l and U l M be the unitary matrices relating through the interaction with singlet Higgs. Let UL,R R,L the gauge eigenstates and the mass eigenstates (fields without superscripts 0) of the SM and mirror fermions defined as M. M. l M lL0 = ULl lL , e0R = URl eR , lRM,0 = URl lRM , eM,0 L = UL eL .. (3.16). Following [37], we express the Yukawa couplings in (3.8) as follows 3. 3. Â Â. LY l. k=0 i,m=1. ⇣ ⌘ M Rk M ¯lLi UimL k lRm + e¯Ri Uim eLm fkS + H.c.. (3.17). The coupling coefficients UimL k and UimR k are given by UimL k. ⇣ ⌘ † k M ⌘ UPMNS · M ·UPMNS. im. 3. =. Â. j,n=1. ,. ⇣ ⌘ † M UPMNS M kjn UPMNS ij. nm. ,. (3.18). nm. ,. (3.19). and UimR k ⌘ =. ⇣ ⌘ 0† 0M UPMNS · M 0 k ·UPMNS. im. 3. Â. j,n=1. ,. ⇣ ⌘ 0† 0M UPMNS M 0jnk UPMNS ij. where the matrix elements for the four auxiliary matrices M k (k = 0, 1, 2, 3) are listed in Table 3.2, and M 0jnk can be obtained from M kjn with the following substitutions for the Yukawa couplings g0S ! g00S and g1S ! g01S ; UPMNS is the usual neutrino mixing matrix defined as UPMNS = Un†ULl ,. (3.20). M 0 0M and its mirror and right-handed counter-parts UPMNS , UPMNS and UPMNS are defined analogously as M M UPMNS = Un†URl , (3.21).

(36) 21. 3.4 Charged Fermion Masses and Mixings. Table 3.2 Matrix elements for the four auxiliary M k (k = 0, 1, 2, 3) where w ⌘ exp(i2p/3) and g0S and g1S are complex Yukawa couplings. M 0 k can be obtained from M k with the following substitutions g0S ! g00S and g1S ! g01S . M kjn. Value. 0 , M0 , M0 , M0 , M0 , M0 M12 13 21 23 31 32 0 , M0 , M0 M11 22 33 1 , M2 , M3 ; M1 , M1 M11 11 23 32 11 1 , M2 , M3 ; M1 , M1 M22 22 13 31 22 1 , M2 , M3 ; M1 , M1 M33 33 12 21 33 2 , M3 M12 21 3 , M2 M12 21 2 , M3 M13 31 3 , M2 M13 31 2 , M3 M23 32 3 , M2 M23 32. 0. 1 3 1 3. 1 3. 1 3. g0S 2 3 Re (g1S ) 2 ⇤ 3 Re (w g1S ) 2 3 Re (wg1S ) g1S + wg⇤1S g⇤1S + w ⇤ g1S g1S + w ⇤ g⇤1S g⇤1S + wg1S 2w ⇤ 3 Re (g1S ) 2w 3 Re (g1S ). 0 UPMNS = Un†URl ,. and. M. 0M UPMNS = Un†ULl .. (3.22) (3.23). Charged fermion masses are obtained by considering the SM Yukawa couplings to the Higgs doublets. As mentioned above, the SM fermions only couples to F2 and the mirror fermions only couples to F2M . Thus, we can write down the Yukawa Lagrangian for lepton sector as follows: LlSM = gl l¯L F2 eR + h.c. , LlM = gM l¯RM F2M eM L + h.c. . l. (3.24) (3.25). where, gl and gM SSB, F2 and F2M acquire their VEV l pare 3 ⇥ 3 real matrices. After p T T as hF2 i = (0, v2 / 2) and hF2M i = (0, v2M / 2) , respectively. If one assumes that the mixing between SM lepton and mirror lepton is neglectable, we can approximately obtain the p p mass of SM charged leptons and mirrors leptons as ml ⇡ gl v2 / 2 and ml M ⇡ gM v / 2, 2M l respectively..

(37) The Electroweak-Scale nR Model. 22. 3.4.2. Quark Sector. For the mixing between SM and mirror quarks through singlets Higgs, in analogy with the lepton sector, we will write down the relevant A4 invariant Yukawa interactions 3. LY q. 3. Â Â Â. q=u,d k=0 i, j=1. where. n o Lqk Rqk qi Vi j PR + Vi j PL qM j fkS + H.c. . q†. qM. q†. qM. V Lqk ⌘ VL M Q,kVR ,. (3.27). V Rqk ⌘ VR M q,kVL . M. (3.26). M. u , V d , V u , V d , are the unitary matrices which transform the fields to the physical Here VL,R L,R L,R L,R basis uM M M dM M uL,0 = VLu uL , dL,0 = VLd dL , uM L,0 = VL uL , dL,0 = VL dL ,. and. M. M. u M M d M uR,0 = VRu uR , dR,0 = VRd dR , uM R,0 = VR uR , dR,0 = VR dR .. The M Q,k in (3.27) are 3 ⇥ 3 matrices which are given by 0. gQ 0S B Q,0 M =@ 0 0 0 0 B Q,2 M =@ 0 gQ 1S. 1 0 0 0 0 0 C B Q Q,1 g0S 0 A , M = @0 0 0 gQ 0 gQ 0S 2S 1 0 Q 0 g2S 0 gQ 1S C B Q Q,3 0 0 A , M = @g2S 0 0 0 0 0. 1 0 C gQ 1S A , 0 1 0 C 0A , 0. (3.28). and similar decompositions for M u,k and M d,k in (3.27) can be obtained by the substitutions Q u u d d of gQ iS ! giS and giS respectively in (3.28). Note that giS , giS and giS are in general complex couplings. Similarly to the lepton sector, the interaction between quarks and Higgs doublets can be written down as ˜ 2 uR + gd q¯L F2 dR + h.c. , LqSM = gu q¯L F M˜ M M M M LqM = gM u q¯R F2M uL + gd q¯R F2M dL + h.c. .. (3.29) (3.30). M M ˜ 2,2M = it2 F⇤ where F 2,2M and gu , gd , gu , gd are 3 ⇥ 3 real matrices. After the SBB, by making an assumption that the mixing between SM quarks and mirror quarks is really tiny,.

(38) 23. 3.5 Phenomenological Constraints. 0.6. + 1 σ constraint × 2 σ constraint. 0.4. ~ SS. 0.2. 0. -0.2. -0.4. -0.6. -0.2. 0. 0.2. ~ SMF. 0.4. 0.6. 0.8. Fig. 3.1 Constrained S˜S versus S˜MF . p we can approximately get the mass of SM quarks and mirror quarks as mu,d ⇡ gu,d v2 / 2 p and muM ,d M ⇡ gM u,d v2M / 2, respectively.. 3.5. Phenomenological Constraints. In this review section, we will discuss two sets of results for the EW-scale nR model obtained in [24] (the electroweak precision constraints) and [25] (constraints from the 125-GeV SM-like scalar).. 3.5.1. Electroweak Precision Constraints [24]. The presence of mirror quark and lepton SU(2)-doublets can, by themselves, seriously affect the constraints coming from electroweak precision data. As noticed in [24], the positive contribution to the S-parameter coming from the extra right-handed mirror quark and lepton doublets could be partially cancelled by the negative contribution coming from the triplet Higgs fields. Ref. [24] has carried out a detailed analysis of the electroweak precision parameters S and T and found that there is a large parameter space in the model which satisfies the present constraints and that there is no fine tuning due to the large size of the allowed parameter space. It is beyond the scope of the thesis to show more details here but a representative plot would be helpful. Fig. 3.1 shows the contribution of the scalar sector versus that of the mirror fermions to the S-parameter within 1s and 2s . In this plot, [24].

(39) The Electroweak-Scale nR Model. 24. took for illustrative purpose 3500 data points that fall inside the 2s blue region with about 100 data points falling inside the 1s red region. More details can be found in [24].. 3.5.2. The 125-GeV SM-like Scalar Constraints [25]. In light of the discovery of the 125-GeV SM-like scalar, it is imperative that any model beyond the SM (BSM) shows a scalar spectrum that contains at least one Higgs field with the desired properties as required by experiment. The present data from CMS and ATLAS only show signal strengths that are compatible with the SM Higgs boson. The definition of a signal strength µ is as follows µ(H-decay) =. s (H-decay) , sSM (H-decay). (3.31). with s (H-decay) = s (H-production) ⇥ B(H-decay) .. (3.32). To really distinguish the SM Higgs field from its impostor, it is necessary to measure the partial decay widths and the various branching ratios. In the present absence of such quantities, the best one can do is to present cases which are consistent with the experimental signal strengths. This is what was carried out in [25]. The minimization of the potential containing the scalars shown above breaks its global symmetry SU(2)L ⇥ SU(2)R down to a custodial symmetry SU(2)D which guarantees at tree 2 /M 2 cos2 q = 1 [25]. The physical scalars can be grouped, based on their level r = MW W Z transformation properties under SU(2)D as follows: five-plet (quintet) ! H5±± , H5± , H50 ; triplet ! H3± , H30 ; ± 0 triplet ! H3M , H3M ; 0 three singlets ! H10 , H1M , H100 .. (3.33). The three custodial singlets are the CP-even states, one combination of which can be the 0 0r 0 0r 00 125-GeV ⇣p scalar.⌘In terms of the original fields, one has H1 = f2 , H1M = f2M and H1 = p1 2c 0r + x 0 . These states mix through a mass matrix obtained from the potential and 3 e H e 0 and H e 00 , with the convention that the lightest of the mass eigenstates are denoted by H, e the next heavier one by H e 0 and the heaviest state by H e 00 . the three is denoted by H,.

(40) 25. 3.5 Phenomenological Constraints. H → bb. CMS: µ = 0.93 ± 0.49. ~ H→ f f. H → ττ. CMS: µ = 0.91 ± 0.27. H → γγ. ~ H → γγ. CMS: µ = 1.13 ± 0.24. +. -. H→ W W. CMS: µ = 0.83 ± 0.21. ~ + H → W W / ZZ. H → ZZ. CMS: µ = 1.00 ± 0.29. 0. 0.5. 1. 1.5. 2. 2.5. Best fit σ / σSM CMS preliminary mH = 125.7 GeV. EWνR "Dr. Jekyll" Ex. 1 mH~ = 125.7 GeV. EWνR "Mr. Hyde" Ex. 1 mH~ = 125.8 GeV. EWνR "Dr. Jekyll" Ex. 2 mH~ = 125.7 GeV. EWνR "Mr. Hyde" Ex. 2 mH~ = 125.2 GeV EWνR "Mr. Hyde" Ex. 3 mH~ = 125.6 GeV. ¯ t t, e ! bb, ¯ gg, W +W , ZZ) in the EW-scale nR Fig. 3.2 Predictions of signal strength µ(H model for examples 1 and 2 in Dr. Jekyll and example 1, 2 and 3 in Mr. Hyde scenarios as discussed in [25], in comparison with corresponding best fit values by CMS [38–41]. e ! ZZ, W +W , gg, bb¯ and t t. ¯ To compute the signal strengths µ, Ref. [25] considers H e related to H e ! gg was also calculated. A scan over In addition, the cross section of gg ! H the parameter space of the model yielded two interesting scenarios for the 125-GeV scalar: e ⇠ H 0 meaning that the SM-like component H 0 = f 0r is 1) Dr. Jekyll’s scenario in which H 1 1 2 00 e dominant; 2) Mr. Hyde’s scenario in which H ⇠ H1 meaning that the SM-like component H10 = f20r is subdominant. Both scenarios give signal strengths compatible with experimental data as shown below in Fig. (2). As we can see from Fig. (2), both SM-like scenario (Dr. Jekyll) and the more interesting scenario which is very unlike the SM (Mr. Hyde) agree with experiment. As stressed in [25], present data cannot tell whether or not the 125-GeV scalar is truly SM-like or even if it has a dominant SM-like component. It has also been stressed in [25] that it is essential to measure the partial decay widths of the 125-GeV scalar to truly reveal its nature. Last but not least, in 0 = f 0r is subdominant but is essential to obtain the agreement with the both scenarios, H1M 2M data as shown in [25]..

(41) 26. The Electroweak-Scale nR Model. As discussed in detail in [25] , for proper vacuum alignment, the potential contains a term proportional to l5 (Eq. (32) of [25]) and it is this term that prevents the appearance of Nambu-Goldstone (NG) bosons in the model. The would-be NG bosons acquire a mass proportional to l5 . 0 and the heavy CP-even states H e0, H e 00 was An analysis of CP-odd scalar states H30 , H3M presented in [25]. The phenomenology of charged scalars including the doubly-charged ones was also discussed in [42]. The phenomenology of mirror quarks and leptons was briefly discussed in [24] and a detailed analysis of mirror leptons and quarks have been presented in [22, 23]. It suffices to mention here that mirror fermions decay into SM fermions through the process qM ! qfS , l M ! lfS with fS “appearing” as missing energy in the detector. Furthermore, the decay of mirror fermions into SM ones can happen outside the beam pipe and inside the silicon vertex detector. Searches for non-SM fermions do not apply in this case. It is beyond the scope of the thesis to discuss these details here..

(42) Chapter 4 Low-energy Constraints in the EW-nR Model In this chapter, we will study the low-energy experimental processes in the EW-nR model. In the first section, we perform an updated analysis for the one-loop induced lepton flavor violating radiative decays li ! l j g. The µ e conversion in nuclei will be presented in the second section. The third section will deal with electron electric dipole moment, while the fourth section will discuss the neutron electric dipole moment. Detailed analytical expressions and numerical results will be presented in sections 3 and 4. We will summarize our work in the last section. Except for the neutron electric dipole moment, our results have been presented in three publications [37, 44, 46].. 4.1 4.1.1. The µ ! eg Process Overview. As is well known, lepton flavor is an accidental conserved quantity in the SM with strictly massless neutrinos. For example, a muon never decays radiatively into an electron plus a photon and neutrinos do not oscillate in the SM. However various experiments have now established firmly that neutrinos do oscillate from one flavor to another. The common wisdom, motivated by the physics of K K oscillation in the kaon system, is to give tiny masses with small mass differences to the various light neutrino species. Radiative decay of the muon into electron is then possible but with an unobservable rate highly suppressed by the minuscule neutrino masses [48, 49]. Searches for lepton flavor violating rare processes in high intensity experiments are thus important for new physics beyond the SM..

(43) 28. Low-energy Constraints in the EW-nR Model The recent limits on the branching ratio B(µ ! eg) is from the MEG experiment [50, 51] B(µ ! eg)  5.7 ⇥ 10 B(µ ! eg)  4.2 ⇥ 10. 13. (90 %C.L.) (MEG 2013) ,. (4.1). 13. (90% C.L.) (MEG 2016) ,. (4.2). and its projected improvement [52] is B(µ ! eg) ⇠ 4 ⇥ 10. 14. (4.3). .. Daµ from E821 experiment [53]: exp. Daµ ⌘ aµ. aSM µ = 288(63)(49) ⇥ 10. 11. .. (4.4). As mentioned above, the EW-nR model entails extra SU(2) chiral doublets (the mirror fermions) which have many consequences. These mirror fermions enter loop corrections to various quantities and processes such as the electroweak precision parameters, rare processes, etc. The first type of effects that needs to be examined is the contributions of these extra chiral doublets to the electroweak precision parameters which have have been performed in [24] and have briefly reviewed in subsection (3.4.1) in this thesis. The next place where mirror fermions enter through loop corrections is rare processes such as µ ! e g and t ! µ g. In [43], such processes have been discussed in a generic fashion, with an emphasis on the possible correlation between the observability of the aforementioned rare processes and the decay lengths of the mirror charged leptons, both of which are of phenomenological interests. In this section, we will present an update of the process µ ! e g taking into account recent developments of the model, including experimental inputs from the recently-discovered 125 GeV SM-like scalar [4, 5]. This section is organized as follows. In Sec. 4.1.2, we present the detailed calculation of process li ! l j g, the anomalous magnetic dipole moment Dali in the model. We then proceed with detailed numerical analysis in Sec. 4.1.3. Implications of our results concerning the possible detection of mirror leptons at the LHC and the ILC are discussed in Sec. 4.1.4. We finally summarize and conclude in Sec. 4.1.5 .. 4.1.2. Analytical Expressions. The one-loop irreducible diagram for li ! l j g is shown in Fig. (4.1). Other two diagrams not shown are reducible associated with the one-loop dressing for the external fermion lines. They are crucial for the cancellation of ultraviolet divergences and gauge invariance in our.

(44) 4.1 The µ ! eg Process. 29 kS. lj. li. M lm. M lm. Fig. 4.1 One-loop induced Feynman diagram for li ! l j g in EW- nR model. calculation. The relevant Yukawa couplings between the leptons, mirror leptons and the A4 singlet and triplet scalars was shown in Eq. (3.17). The process li ! l j g (i 6= j) Lorentz and gauge invariance dictate the form of the amplitude for the process li (p) ! l j (p0 ) + g(q) to be ⇣ ⌘ n h io ij ij M li ! l j g = eµ⇤ (q)u¯ j (p0 ) is µn qn CL PL +CR PR ui (p) ,. (4.5). ij. where PL,R = (1 ⌥ g5 )/2. The coefficients CL,R can be extracted from the one-loop diagram (Fig. (4.1)), ij CL. ij. CR. (. ⇣ ⌘⇤ ⇣ ⌘⇤ i 1 h Rk Rk Lk Lk m U U + m U U I l l jm im jm im i j m2l M m !) 2 ⇣ ⌘⇤ m 1 fkS + U R k UimL k J , mlmM jm m2l M m ( h ⇣ ⌘ ⇣ ⌘⇤ i 3 3 ⇤ e 1 Lk Lk Rk Rk = + m U U + m U U I l j jm Â Â m2M li jm im im 16p 2 k=0 m=1 lm !) ⇣ ⌘⇤ m2fkS 1 Lk Rk + U Uim J . mlmM jm m2l M e 3 3 = + ÂÂ 16p 2 k=0 m=1. m. m2fkS m2l M m. ! (4.6). m2fkS m2l M m. ! (4.7).

(45) 30. Low-energy Constraints in the EW-nR Model. Note that m stands for the generation of mirror leptons, mlmM and mfkS are mirror leptons and scalar Higgs masses respectively, UimL k and UimR k are defined in Eq. (3.18) and Eq. (3.19). Here we have assumed the mirror lepton masses are much larger than the external fermion masses mlmM mli, j and set mli, j ! 0 in the loop functions I (r) and J (r), which are simply given by ⇥ ⇤ 1 6r2 log r + r(2r2 + 3r 6) + 1 , 4 12(1 r) ⇥ ⇤ 1 J (r) = 2r2 log r + r(3r 4) + 1 . 3 2(1 r). (4.8). I (r) =. (4.9). In our numerical work for µ ! eg presented in Sec.4.1.3, we will consider the mirror lepton masses of the order a few hundred GeV and the A4 singlet and triplet scalar masses of the order 10 MeV, thus the ratio r = m2fkS /m2l M ⇠ 10 8 is very tiny. For all practical purposes, one m can replace Eqs. (4.8) and (4.9) by the limits limr!0 I (r) = 1/12 and limr!0 J (r) = 1/2 respectively. Formulas of I and J for the general case of mi, j 6= 0 are given in the Appendix A. The partial width for li ! l j g is given by 1 3 G li ! l j g = m 1 16p li. m2l j m2li. !3. ⇣ ⌘ ij 2 ij 2 |CL | + |CR | .. (4.10). Magnetic Dipole Moment The magnetic dipole moment anomaly for lepton li can be easily extracted from the above calculation with the following result Dali. 2mli = e. ✓. CLii +CRii 2 ( 3. +. 3. ÂÂ. k=0 m=1. 4.1.3. m2fkS. ⇣ ⌘ m2 l Lk 2 Rk 2 Â Â 2 |Uim | + |Uim | m2Mi I k=0 m=1 l 3. 1 = + 16p 2. ◆. 3. Re. m. ⇣. UimL k. ⇣ ⌘⇤ ⌘ m li UimR k J mlmM. m2l M m !). m2fkS m2l M m. ! .. (4.11). Numerical Analysis. The branching ratio B(µ ! eg) is given by B(µ ! eg) = tµ · G (µ ! eg). (4.12).

(46) 4.1 The µ ! eg Process. 31. where tµ is the lifetime of the muon [54] tµ = (2.1969811 ± 0.0000022) ⇥ 10. 6. s .. (4.13). In our numerical analysis, we will adopt the following approach: • For the masses of the singlet scalars fkS , we take mf0S : mf1S : mf2S : mf3S = MS : 2MS : 3MS : 4MS with a fixed common mass MS = 10 MeV. As long as mfkS ⌧ mlmM , our results will not be affected much by the exact mass relations among these singlet scalars. • For the masses of the mirror lepton lmM , we take mlmM = Mmirror + dm with d1 = 0, d2 = 10 GeV, d3 = 20 GeV and vary the common mass Mmirror from 100 GeV to 800 GeV. • We assume all the Yukawa couplings g0S , g1S , g2S , g00S , g01S , and g02S to be all real1 . As mentioned before, g2S = (g1S )⇤ and g02S = (g01S )⇤ due to the reality of the mass eigenvalues of the Dirac neutrino masses. For simplicity, we also take g0S = g00S , g1S = g01S and study the following 6 cases: 1. g0S 6= 0, g1S = 0. The A4 triplet terms are switched off. 2. g1S = 10 ones.. 2 ⇥g. 3. g1S = 10. 1 ⇥g. 0S .. 0S .. The A4 triplet couplings are merely one percent of the singlet The A4 triplet couplings are 10 percent of the singlet ones.. 4. g1S = 0.5 ⇥ g0S . The A4 triplet couplings are one half of the singlet ones. 5. g1S = g0S . Both A4 singlet and triplet terms have the same weight. 6. g0S = 0, g1S 6= 0. The A4 singlet terms are switched off. M 0 0M • For the three unknown mixing matrices UPMNS , UPMNS and UPMNS , we will consider two scenarios:. – Scenario 1 1 We. † M 0 0M UPMNS = UPMNS = UPMNS = UCW. will study the case of complex Yukawa couplings in Sec. 4.3 and 4.4..