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ANOMALY MEDIATED SUPERSYMMETRY BREAKING AND NONSTANDARD NEUTRINO OSCILLATIONS By CYRIL OJODUME ANOKA Bachelor of Science Obafemi Awolowo University IleIfe, Nigeria 1995 HEP Diploma International Center for Theoretical Physics Trieste, Italy 1999 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2005 ANOMALY MEDIATED SUPERSYMMETRY BREAKING AND NONSTANDARD NEUTRINO OSCILLATIONS Thesis Approved: Dr. K.S. Babu Thesis Advisor Dr. J. Perk Member Dr. J. Mintmire Member Dr. J. Chandler Outside Member Dr. G. Emslie Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to my advisor Prof. K. S. Babu and my collaborator Dr. I. Gogoladze for their constructive guidance, constant encour agement, kindness and great patience. Without their guidance and collaboration, I would not have been able to ¯nish this work. Their critical reading and precious suggestions greatly enhanced the writing of this thesis. I also would like to express my sincere appreciation to Prof. S. Nandi for his assistance during these years of my study at the Oklahoma State University. My appreciation extends to my other committee members Prof. J. Perk, Prof. J. Mintmire and Prof. J. Chandler whose encouragement has also been invaluable. Moreover, I wish to express my thanks to my colleagues Ts. Enkhbat, A. Bachri, Wang Kai, Dr. G. Seidl and the rest of the members of the High Energy Physics Theory group. I will always miss our trips to conferences and outdoor activities. There are some special friends that I will like to acknowledge for their support during my studies. They are Sylvester Onoyona and family, Kingsley Dike and family, Jude Ulogo, Dr. Solomon Osho and family, Angelica Keng, Dr. Saliki and family and a host of others. I would like to thank my Mother and my late Father and all my Brothers and Sisters to whose support I owe my successes. Finally, I thank the Physics Department for providing the opportunity for my graduate study and the US Department of Energy for providing part of my ¯nancial support. iii iv TABLE OF CONTENTS Chapter Page 1. INTRODUCTION :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.1. The Standard Model ::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.2. Symmetry breaking via the Higgs mechanism ::::::::::::::::: 2 1.3. Gauge hierarchy problem:::::::::::::::::::::::::::::::::::::::::::: 5 1.4. Gauge coupling uni¯cation:::::::::::::::::::::::::::::::::::::::::: 7 2. Supersymmetry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10 2.1. Supersymmetry algebra:::::::::::::::::::::::::::::::::::::::::::::: 11 2.2. Superspace and super¯elds:::::::::::::::::::::::::::::::::::::::::: 12 2.3. Supersymmetric Action :::::::::::::::::::::::::::::::::::::::::::::: 16 2.4. SUSY breaking::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 18 3. The Minimal Supersymmetric Standard Model::::::::::::::::::::: 21 3.1. Electroweak symmetry breaking and the Higgs boson masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23 3.2. The sfermions masses :::::::::::::::::::::::::::::::::::::::::::::::: 25 3.3. Neutralinos:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 26 3.4. Charginos:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 27 4. ANOMALY MEDIATED SUPERSYMMETRY BREAKING::::: 28 4.1. Gravity mediation::::::::::::::::::::::::::::::::::::::::::::::::::::: 29 4.2. Gauge mediation :::::::::::::::::::::::::::::::::::::::::::::::::::::: 30 4.3. Gaugino mediation:::::::::::::::::::::::::::::::::::::::::::::::::::: 31 4.4. Anomaly mediation ::::::::::::::::::::::::::::::::::::::::::::::::::: 32 4.4.1. The negative slepton mass problem of anomaly mediated supersymmetry breaking :::::::::::::::::: 34 4.4.2. Suggested solutions to the AMSB slepton mass problem ::::::::::::::::::::::::::::::::::::::::::::::::::::: 35 iv Chapter Page 5. TeV{Scale Horizontal Symmetry and the Slepton Mass Problem of Anomaly Mediation :::::::::::::::::::::::::::::::::::::::::: 37 5.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 37 5.2. SU(3)H horizontal symmetry :::::::::::::::::::::::::::::::::::::: 38 5.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 41 5.3.1. Constraints on tan ¯ and mh ::::::::::::::::::::::::::: 41 5.3.2. SU(3)H symmetry breaking::::::::::::::::::::::::::::: 44 5.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 45 5.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 46 5.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 46 5.4.3. ´ fermion and ´ scalar masses ::::::::::::::::::::::::: 47 5.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 48 5.6. Experimental signatures ::::::::::::::::::::::::::::::::::::::::::::: 53 5.7. Origin of the ¹ term :::::::::::::::::::::::::::::::::::::::::::::::::: 54 5.8. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 55 6. SU(2)H Horizontal Symmetry as a Solution to the Slep ton Mass Problem of Anomaly Mediation::::::::::::::::::::::::::: 56 6.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 56 6.2. SU(2)H horizontal symmetry :::::::::::::::::::::::::::::::::::::: 57 6.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 59 6.3.1. Lepton masses :::::::::::::::::::::::::::::::::::::::::::::: 61 6.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 62 6.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 62 6.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 63 6.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 64 6.6. Other experimental implications :::::::::::::::::::::::::::::::::: 66 6.7. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 66 7. Constraining Z0 From Supersymmetry Breaking::::::::::::::::::: 68 7.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 68 7.2. U(1)x model :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 70 7.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 72 7.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 76 7.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 76 7.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 77 7.4.3. Heavy sneutrino masses :::::::::::::::::::::::::::::::::: 78 7.5. Numerical results for the spectrum ::::::::::::::::::::::::::::::: 79 7.6. Z0 decay modes and branching ratios :::::::::::::::::::::::::::: 84 7.7. Other experimental signatures ::::::::::::::::::::::::::::::::::::: 94 v Chapter Page 7.7.1. Z decay and precision electroweak data :::::::::::::: 94 7.7.2. Z0 mass limit:::::::::::::::::::::::::::::::::::::::::::::::: 96 7.7.3. h ! h0h0 decay ::::::::::::::::::::::::::::::::::::::::::::: 96 7.7.4. Signatures of SUSY particles ::::::::::::::::::::::::::: 97 7.8. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 97 8. Quark{Lepton Supersymmetry ::::::::::::::::::::::::::::::::::::::::::::: 99 8.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 99 8.2. TeV scale quark{lepton symmetric model::::::::::::::::::::::: 100 8.2.1. Uni¯cation of gauge couplings:::::::::::::::::::::::::: 102 8.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 103 8.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.3. Exotic slepton masses :::::::::::::::::::::::::::::::::::: 109 8.4.4. Exotic lepton masses:::::::::::::::::::::::::::::::::::::: 109 8.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 110 8.5.1. Coupling of light Higgs to SM fermions:::::::::::::: 115 8.5.2. Neutralino schannel annihilation:::::::::::::::::::::: 116 8.6. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 117 9. CP Violation in Neutrino Oscillations from Nonstan dard Physics:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 118 9.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 118 9.2. Neutrino oscillations including new physics :::::::::::::::::::: 119 9.2.1. Neutrino mixing formalism:::::::::::::::::::::::::::::: 119 9.2.2. Two °avor neutrino mixing ::::::::::::::::::::::::::::: 120 9.2.3. Three generation neutrino oscillation::::::::::::::::: 121 9.2.3.1. General formalism in vacuum::::::::::::::: 121 9.2.3.2. Bilarge mixing :::::::::::::::::::::::::::::::::: 124 9.2.3.3. Exact analysis of three generation neutrino oscillation in vacuum::::::::::::::::: 127 9.3. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 131 9.4. Three neutrino oscillations including matter e®ects :::::::::: 148 9.4.1. Formalism ::::::::::::::::::::::::::::::::::::::::::::::::::: 148 9.4.2. Parameter mapping ::::::::::::::::::::::::::::::::::::::: 149 9.5. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 161 BIBLIOGRAPHY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 163 vi Chapter Page APPENDICES::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 171 APPENDIX ATeV scale Horizontal Symmetry :::::::::::::::::::::::::::::: 172 A.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 172 A.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 172 A.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 A.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 A.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 APPENDIX BSU(2)H Symmetry ::::::::::::::::::::::::::::::::::::::::::::::: 175 B.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 175 B.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 177 APPENDIX CU(1)x Model ::::::::::::::::::::::::::::::::::::::::::::::::::::::: 178 C.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 178 C.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 180 APPENDIX DQuarkLepton Supersymmetric Model :::::::::::::::::::::: 181 D.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 181 D.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 182 D.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 182 D.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 183 D.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 183 APPENDIX ETwo Generation Neutrino Oscillation Model :::::::::::::: 184 vii LIST OF TABLES Table Page 1.1. Particle content of the SM and the charge assignment. ::::::::::::::::::::::: 2 3.1. Chiral super¯elds of the MSSM. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21 3.2. Vector Super¯elds of the MSSM.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22 5.1. Particle content and charge assignment of the SU(3)H model. ::::::::::::: 39 5.2. Sparticle masses in the SU(3)H model for one choice of parameters :::::: 50 5.3. Sparticle masses in the SU(3)H model for a second choice of param eters :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 51 5.4. Sparticle masses in the SU(3)H model for a third choice of parameters:: 52 6.1. Particle content and charge assignment of the SU(2)H model. ::::::::::::: 58 6.2. Sparticle masses in the SU(2)H model for one choice of input param eters.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 65 7.1. Particle content and charge assignment of the Z0 model. :::::::::::::::::::: 71 7.2. Sparticle masses in Model 1 with x = 1:3 :::::::::::::::::::::::::::::::::::::::: 81 7.3. Z0 mass and Z ¡ Z0 mixing angle in Model 1 for the same set of input parameters as in Table 7.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 82 7.4. Eigenvectors of the neutralino mass matrix in Model 1. :::::::::::::::::::::: 82 7.5. Eigenvectors of the chargino mass matrix in Model 1. :::::::::::::::::::::::: 82 7.6. Eigenvectors of the CP{even Higgs boson mass matrix in Model 1. ::::::: 83 7.7. Sparticle masses in Model 2 with x = 1:6 :::::::::::::::::::::::::::::::::::::::: 84 7.8. Z0 mass and Z ¡ Z0 mixing angle in Model 2 for the same set of input parameters as in Table 7.7. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 85 viii Table Page 7.9. Eigenvectors of the neutralino mass matrix in Model 2. :::::::::::::::::::::: 85 7.10. Eigenvectors of the chargino mass matrix in Model 2. :::::::::::::::::::::::: 85 7.11. Eigenvectors of the CP{even Higgs boson mass matrix in Model 2. ::::::: 85 7.12. Decay modes for Z0 in Model 1 for the parameters used in Table 7.2. :::: 92 7.13. Decay modes for Z0 in Model 2 for the parameters used in Table 7.7. :::: 93 8.1. Particle content and charge assignment of the quark{lepton symmet ric model.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 100 8.2. Sparticle masses in the quark{lepton symmetric model (Model 1) for one choice of input parameters.:::::::::::::::::::::::::::::::::::::::::::::::::::: 112 8.3. Z0 mass and Z ¡ Z0 mixing angle in Model 1 for the same set of input parameters as in Table 8.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 113 8.4. Eigenvectors of the neutralino mass matrix in Model 1. :::::::::::::::::::::: 113 8.5. Eigenvectors of the chargino mass matrix in Model 1. :::::::::::::::::::::::: 113 8.6. Eigenvectors of SU(2)` chargino mass matrix in Model 1. ::::::::::::::::::: 114 8.7. Sparticle masses in the quark lepton symmetric model (Model 2) for a di®erent choice of parameters.::::::::::::::::::::::::::::::::::::::::::::::::::: 114 8.8. Z0 mass and Z ¡ Z0 mixing angle in Model 2 for the same set of input parameters as in Table 8.7. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 115 ix LIST OF FIGURES Figure Page 1.1. 1loop correction to the mass of a fermion. :::::::::::::::::::::::::::::::::::::: 6 1.2. 1loop corrections to a scalar mass. :::::::::::::::::::::::::::::::::::::::::::::::: 6 1.3. Running of the couplings in the SM (left) and its minimal supersym metric version (right). :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 8 4.1. Leading diagram that contributes to SUSY{breaking scalar masses. The bulk line is a gaugino propagator. :::::::::::::::::::::::::::::::::::::::::: 32 5.1. E®ective operators inducing charged lepton masses.::::::::::::::::::::::::::: 40 5.2. Plot of tan ¯ as a function of M2 ::::::::::::::::::::::::::::::::::::::::::::::::::: 43 8.1. Renomalization group evolution the inverse gauge couplings. ::::::::::::::: 103 8.2. ~W + decay to two leptons and LSP. :::::::::::::::::::::::::::::::::::::::::::::::: 110 8.3. Neutralino annihilation to two charged leptons in the early universe. :::: 111 8.4. Bound state of two x leptons decay to two photons. :::::::::::::::::::::::::: 111 8.5. Bound state of two x leptons decay to two charged leptons via ex change of SU(2)H gauge boson. ::::::::::::::::::::::::::::::::::::::::::::::::::: 111 8.6. Doublet SU(2)H gauge boson decay to two charged leptons via ex change of neutralino LSP. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 112 9.1. CP asymmetry A¹e as a function of energy for two generation neu trino oscillation in vacuum:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 122 9.2. Oscillation probabilities Pe¹ and P¹e as a function of energy for the three generation neutrino oscillations in vacuum ::::::::::::::::::::::::::::: 132 9.3. Oscillation probabilities P¹¹ and P¹¿ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::: 133 x Figure Page 9.4. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 134 9.5. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of energy for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 135 9.6. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of energy for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 136 9.7. Change in oscillation probabilities ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ and ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a function of energy for the same choice of input parameters as in Fig. 9.2, except that ± = 0 and ²± = 0. ::::::: 137 9.8. Oscillation probabilities Pe¹ and P¹e as a function of energy for a ¯xed baseline L = 295 km (a) and L = 730 km (b). All other parameters are as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::: 138 9.9. Oscillation probabilities P¹¹ and P¹¿ as a function of energy for ¯xed baseline L = 295 km (a) and L = 730 km (b). Input parameters are as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 139 9.10. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 140 9.11. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of energy for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 141 9.12. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of energy for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 142 9.13. Oscillation probabilities Pe¹ and P¹e as a function of Length for ¯xed energy E = 5 GeV . All other parameters are the same as in Fig. 9.2. 143 9.14. Oscillation probabilities P¹¹ and P¹¿ as a function of length for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::: 144 xi Figure Page 9.15. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of length for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 145 9.16. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of length for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 146 9.17. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of Length for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 147 9.18. Oscillation probabilities Pe¹ and P¹¹ in matter (assuming constant matter density ½ = 2.8 g/cm3) as a function of energy for ¯xed length L = 2540 km. All other parameters are the same as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 151 9.19. Oscillation probability P¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 152 9.20. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 153 9.21. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡ P¹¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 154 9.22. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡P¹¿ ¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 155 9.23. Oscillation probabilities Pe¹ and P¹¹ in matter as a function of energy for ¯xed length L = 295 km. All other parameters are the same as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 156 9.24. Oscillation probability P¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 157 9.25. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 158 xii Figure Page 9.26. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡ P¹¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 159 9.27. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡P¹¿ ¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 160 xiii xiv CHAPTER 1 INTRODUCTION In this section we give a brief description of the Standard Model of particle physics and reasons for going beyond it. 1.1 The Standard Model The Standard Model (SM) of elementary particle physics has recorded remark able success in describing physics at length scales ranging from atomic scales down to the shortest probed scale of about 10¡18 m. It is a non{abelian gauge theory based on the gauge group [1] SU(3)C £ SU(2)L £ U(1)Y ; where SU(3)C is the color gauge group describing strong interactions and SU(2)L £ U(1)Y is the electroweak gauge group describing weak and electromagnetic interac tions. The SM describes the interactions of quarks, leptons, gauge bosons and the Higgs boson. The ¯eld content and the transformation properties under the gauge symmetries are shown in Table 1. It is important to note that the left{ and the right{handed components of the matter fermions are assigned to di®erent representations (doublets and singlets respectively) of the weak gauge group SU(2)L, thereby allowing a chiral structure for the weak interactions. The Yukawa and Higgs part of the SM Lagrangian is given by LY ukawa = Y ` ®¯`®ec ¯ ~Á + Y d ®¯Q®dc ¯ ~Á + Y u ®¯Q®uc ¯Á + h.c.; (1.1) where ~Á = i¾2Á¤ = Ã ¹ Á0 ¡Á¡ ! . Here generation indices ®; ¯ = 1; 2; 3 are explicitly displayed, while color and SU(2)L indices are suppressed. 1 2 Fields SU(3)C SU(2)W U(1)Y Quarks Qi ® = Ã ui ® di ® ! 3 2 1 6 uci ® ¹3 1 ¡2 3 dci ® ¹3 1 1 3 Leptons `® = Ã º® e® ! 1 2 ¡1 2 ec ® 1 1 2 Gluon Ga ¹ 8 1 0 Intermediate weak bosons Wr ¹ 1 3 0 Hypercharge gauge boson B¹ 1 1 0 Higgs boson Á = Ã Á+ Á0 ! 1 2 1 2 TABLE 1.1. Particle content of the SM and the charge assignment. Here ® = 1; 2; 3 is the generation index, i = 1 ¡ 3 (color), a = 1 ¡ 8 (SU(3)C generators) and r = 1 ¡ 3 (SU(2)L generators). 1.2 Symmetry breaking via the Higgs mechanism If we consider the SU(2)L£U(1)Y part of the Lagrangian, assuming that there is no Higgs ¯eld, all the fermions and the four gauge bosons (Wr ¹; B¹) would be massless. This is unacceptable, for the weak interactions are short range, meaning that the mediators must be massive. We must then break the symmetry spontaneously which will ensure renormalizability. This is achieved through the scalar Higgs doublet Á = Ã Á+ Á0 ! : (1.2) The only observed unbroken local symmetry in Nature is the U(1)em (apart from SU(3)C). Therefore the SU(2)L£U(1)Y symmetry should be broken down to U(1)em. The renomalizable Higgs potential is given by VH ´ ¹2ÁyÁ + ¸(ÁyÁ)2: 3 This has a minimum for ¹2 < 0 at hÁyÁi = ¡ ¹2 2¸ = À2 2 : (1.3) We can choose the vacuum expectation value (VEV) after an SU(2)L transformation in the unitary gauge as hÁ0i = 1 p 2 Ã 0 À ! : (1.4) It is not di±cult to see that the gauge boson associated with the U(1)em subgroup of SU(2)L £ U(1)Y remains massless. The electric charge Qem, the U(1)Y hypercharge and the third component of weak isospin T3L are related by Qem = T3L + Y 2 ; (1.5) and the gauge boson masses are given by MW = gÀ 2 ; MZ = MW cos µW ; MA = 0: (1.6) Here g is the SU(2)L gauge coupling strength and tan µW = g0=g, where g0 is the U(1)Y gauge coupling constant. These masses are obtained from the Lagrangian for the gauge and Higgs ¯eld, given by LgaugeHiggs = ¯¯¯¯ @¹Á ¡ ig 2 ~¿ : ~W ¹Á ¡ ig0 2 B¹Á ¯¯¯¯ 2 ; once the VEV of Á0 is inserted. It is worthwhile to note that the weak mixing angle µW is a parameter of the SM which has been measured to a very high accuracy. Another accurately measured quantity is the ½ parameter (½ ´ M2W M2Z cos2 µW ) which is predicted to be 1 (at tree level) in the SM. New physics can also be severely constrained by the observed value of ½. After symmetry breaking, from the Yukawa interactions in Eq. (1.1), the fermions become massive with masses given by Mu = YuÀ; Md = YdÀ; M` = Y`À: (1.7) Here Yu; d; ` are arbitrary 3 £ 3 complex matrices in generation space. 4 Not all parameters in these matrices are observable in the SM. After fermion ¯eld rede¯nitions, the 3 eigenvalues of each of the matrices, 3 mixing angles and one phase entering in the charged W§ ¹ interactions with quarks become physical quantities. One makes biunitary transformations, Uu LYuUuy R = Y diag u , UdL YdUdy R = Y diag d , U`L Y`U`y R = Y diag ` , in which case the charged W§ current takes the form LW§ ¹ cc = g p 2 ¹uL°¹VCKMdLW¹+ + h.c.; where VCKM = Uuy L UdL is a unitary matrix, the Cabibbo{Kobayashi{Maskawa matrix or the quark mixing matrix. Since there is no right{handed neutrino ¯eld ºR, the neutrinos remain massless. The fermion masses are arbitrary since the Yukawa couplings Y are free parameters. To ¯nd the Higgs boson mass, we write the complex ¯eld Á0 in terms of real ¯elds. The Higgs doublet then takes the form (in unitary gauge) Á = 1 p 2 Ã 0 À + ´ ! ; (1.8) where ´ is the physical Higgs scalar with mass m2 ´ = 2¸À2: (1.9) The Higgs mass is left undetermined since ¸ is a free parameter, with only its sign constrained to be positive. There are several good features of the SM some of which are: 1. All the particles predicted by the SM have been observed except the Higgs boson. 2. Both baryon and lepton number are automatically conserved. This prevents rapid decay of the proton. 3. It has an extremely economical Higgs sector which is responsible for giving masses to all particles. 4. With only two independent parameters MW and sin µW, all the electroweak processes at high energy are correctly described. 5 The SM also has several drawbacks. There are several free parameters in the SM Lagrangian: The Higgs coupling constant ¸, the Higgs mass parameter ¹2, three gauge couplings (g0; g; gs), the number of generations (matter ¯elds) and three Yukawa matrices Y u ®¯, Y d ®¯, Y ` ®¯. Despite the remarkable success of the SM, there are still several questions left unanswered. For example, does the Higgs boson exist? Do the gauge couplings unify? How is gravity incorporated? An attempt to answer these numerous questions will take us to beyond the SM. For example, some earlier attempts tried to unify strong and electroweak forces by embedding the SU(3)C £ SU(2)L £ U(1)Y structure into higher groups such as SU(5) and SO(10). These \Grand Uni¯ed Theories" or GUT's, were only partially successful. Di±culties with the SM and GUT models concerning gauge hierarchy and ¯ne tuning problems led to theoretical remedies such as technicolor, supersymmetry, string theory, etc. The most appealing of these theories is perhaps supersymmetry, which is the main focus of this thesis. 1.3 Gauge hierarchy problem The hierarchy problem is one of the main reasons why we think supersymmetry has something to do with Nature, and that it might be broken at a scale comparable to the scale of weak interactions, rather than at some enormous energy such as the Planck scale MPl » 1019 GeV. The mass hierarchy problem stems from the fact that masses, in particular scalar masses, are not stable to radiative corrections [2]. While fermion masses also receive radiative corrections from diagrams of the form in Figure 1.1, these are only logarithmically divergent (see for example [3]), ±mf ' 3® 4¼ mf ln(¤2=m2f ); (1.10) where ¤ is an ultraviolet cuto®, where we expect new physics to play an important role. As one can see, even for ¤ » MPl, these corrections are small, ±mf <» mf . 6 Figure 1.1. 1loop correction to the mass of a fermion. In contrast, scalar masses are quadratically divergent. 1{loop contributions to scalar masses, such as those shown in Fig. 1.2, are readily computed ±m2 H ' fg2 f ; g2; ¸g Z d4k 1 k2 » O ³ ® 4¼ ´ ¤2; (1.11) due to contributions from fermion loops with coupling gf , from gauge boson loops with coupling g2, and from quartic scalarcouplings ¸. g2 g f g f l Figure 1.2. 1loop corrections to a scalar mass. An alternative and by far simpler solution to this problem exists if one postu lates that there are new particles with similar masses and equal couplings to those responsible for the radiatively induced masses but with a di®erence (by a half unit) in spin. Then, because the contribution to ±m2 H due to a fermion loop comes with a relative minus sign, the total contribution to the 1loop corrected mass2 is ±m2 H ' O ³ ® 4¼ ´ (¤2 + m2 B) ¡ O ³ ® 4¼ ´ (¤2 + m2 F ) = O ³ ® 4¼ ´ (m2 B ¡ m2 F ): (1.12) If in addition, the bosons and fermions all have the same masses, then the radiative corrections vanish identically. The stability of the hierarchy only requires that the weak scale is preserved so that we need only require that jm2 B ¡ m2 F j <» 1 TeV2: (1.13) As we will see latter, supersymmetry o®ers just the framework for including the nec essary new particles and ensures the absence of these dangerous radiative corrections [4]. 7 1.4 Gauge coupling uni¯cation Another motivation for supersymmetry lies in the gauge coupling constant uni ¯cation. In the SM, the gauge couplings do not unify. The solutions to the SM renormalization group equations to one loop accuracy are given by 1 ®i(Q) = 1 ®i(¹) + bi 2¼ log µ ¹ Q ¶ ; where the bi are bi = 0 BB@ b1 b2 b3 1 CCA = 0 BB@ 0 ¡22 3 ¡11 1 CCA + Ng 0 BB@ 4 3 4 3 4 3 1 CCA + Nh 0 BB@ 1 10 1 6 0 1 CCA : Here Ng = 3 is the number of generations and Nh = 1 is the number of Higgs doublets. The numerical values for the bi coe±cients are bi = ( 41 10 ; ¡19 6 ; ¡7). The three gauge coupling constants used as input are ®1 = 5®=(3 cos2 µW); ®2 = ®=sin2 µW; ®3 = g2 s=(4¼); where ®¡1(MZ) = 128:978; sin2 µW = 0:23146 and ®3 = 0:1184: On evolving the inverse of the three coupling constants as a function of logarithm of the uni¯cation scale Q, the result is shown in Fig. 3 (left). These couplings do not meet at a common point, hence uni¯cation does not occur. If we consider supersymmetric grand uni¯ed theory, the beta function coe±cients are modi¯ed due to the quantum corrections involving the superpartners and are given in the Minimal Supersymmetric Standard Model (MSSM) by bi = 0 BB@ b1 b2 b3 1 CCA= 0 BB@ 0 ¡6 ¡9 1 CCA + Ng 0 BB@ 2 2 2 1 CCA + Nh 0 BB@ 3 10 1 2 0 1 CCA : Here Ng = 3 and Nh = 2. The numerical value for bi is bi = ( 33 5 ; 1; ¡3). If we assume that all the SUSY particle masses are around 1 TeV, on evolving the inverse coupling constants, they meet at a point (unify) as shown in Fig. 3 (right). The point at which these particles meet is around 1016 GeV. The SUSY particles are assumed to 8 Figure 1.3. Running of the couplings in the SM (left) and its minimal supersymmetric version (right). contribute only above the e®ective SUSY scale (» 1 TeV) which causes the change of slope in the evolution of the couplings. This is another reason why most high energy physicist believe in supersymmetry. The present thesis contains nine chapters. In the second chapter we review all the basics for Supersymmetry (SUSY), we de¯ne the SUSY algebra and introduce all the tools needed to write down the supersymmetric version of gauge ¯eld theories. In chapter 3, the minimal supersymmetric extension of the Standard Model is intro duced, all the interactions and relevant mass matrices for our analysis are studied. In the fourth chapter we review various symmetry breaking models, here we introduce the Anomaly Mediated Supersymmetry Breaking (AMSB) and review the relevant literature. In chapter 5, we suggest TeV{Scale horizontal symmetry as a solution to the negative slepton mass squared problem of AMSB. In chapter 6, we suggest an SU(2)H model as a solution to the negative slepton mass problem. In chapter 7, we study a speci¯c Z0 model as a solution to the slepton mass problem of AMSB. 9 In Chapter 8, we suggest another model to solve this problem of AMSB with the quarks and the leptons transforming identically under two di®erent SU(3) symmetry group. Finally, we divert from the AMSB to Neutrino Physics, here we suggest a nonstandard neutrino interaction as a solution to the neutrino oscillation problem. CHAPTER 2 Supersymmetry Supersymmetry (SUSY) is often called the last great symmetry of Nature. Rarely has so much e®ort, both theoretical and experimental, been spent to un derstand and discover a symmetry of Nature, which up to the present time lacks concrete evidence. Why SUSY? If for no other reason, it would be nice to understand the origin of the fundamental di®erence between the two classes of particles distinguished by their spin, fermions and bosons. If such a symmetry exists, one might expect that it is represented by an operator which relates the two classes of particles. For example, QjBosoni = jFermioni; QjFermioni = jBosoni: (2.1) However, without a connection to experiment, SUSY would remain a mathematical curiosity and a subject of a very theoretical nature as indeed it stood from its initial description in the early 1970's [5; 6] until its incorporation into a realistic theory of physics at the electroweak scale. One of the ¯rst breakthroughs came with the realization that SUSY could help resolve the di±cult problem of mass hierarchies [2], namely the stability of the electroweak scale with respect to radiative corrections. With precision experiments at the electroweak scale, it has also become apparent that Grand Uni¯cation is not possible in the absence of SUSY [7]. Considering a new class of \fermionic" generators Q, that satisfy anti{commutation relations [Q®; J¹º] = i¾¹º ¯ ® Q¯; 10 11 [Q®; P¹] = 0; £ ¹Q ®_ ; J¹º¤ = i¹¾¹º®_ _¯ ¹Q _¯ ; £ ¹Q ®_ ; P¹¤ = 0; (2.2) where Q® (¹Q ®_ ) is a symmetry operator (SUSY charge), P¹ is the energy{momentum operator and J¹º is the angular momentum operator. The Q's are translationally invariant (no explicit x{dependence) and they satisfy anti{commutation relations fQ®; ¹Q _¯ g = 2¾¹ ® _¯ P¹; (2.3) where the factor 2 is conventional and can be achieved by re{scaling the Q's. There are three main properties of a supermultiplet: (1) All particles belonging to an irreducible representation of SUSY have the same mass, (2) there are equal number of fermionic (NF ) and bosonic (NB) degrees of freedom in a supermultiplet, (3) the energy P0 in a supersymmetric theory is always positive. 2.1 Supersymmetry algebra Combined with the usual Poincar¶e and internal symmetry algebra the Super Poincar¶e Lie algebra contains additional SUSY generators Qi ® and ¹Q i ®_ [8] [P¹; Pº] = 0; [P¹;M½¾] = i(g¹½P¾ ¡ g¹¾P½); [M¹º;M½¾] = i(gº½M¹¾ ¡ gº¾M¹½ ¡ g¹½Mº¾ + g¹¾Mº½); [Br;Bs] = iCt rsBt; [Br; P¹] = [Br;M¹¾] = 0; [Qi ®; P¹] = [¹Q i ®_ ; P¹] = 0; [Qi ®;M¹º] = 1 2 (¾¹º)¯ ®Qi ¯; [¹Q i ®_ ;M¹º] = ¡1 2 ¹Q i _¯ (¹¾¹º) _¯ ®_ ; [Qi ®;Br] = (br)i jQj ®; [¹Q i ®_ ;Br] = ¡ ¹Q j ®_ (br)i j ; fQi ®; ¹Q j _¯ g = 2±ij(¾¹)® _¯ P¹; fQi ®;Qj ¯g = 2²®¯Zij ; Zij = ar ijbr; Zij = Z+ ij ; f¹Q i ®_ ; ¹Q j _¯ g = ¡2² _ ® _¯ Zij ; [Zij ; anything] = 0; ®; ®_ = 1; 2 i; j = 1; 2; : : : ;N: (2.4) 12 Here P¹ and M¹º are fourmomentum and angular momentum operators, re spectively, Br are the internal symmetry generators, Qi and ¹Q i are the spinorial SUSY generators and Zij are the socalled central charges, while ®; _ ®; ¯; _¯ are the spinorial indices. In the simplest case one has one spinor generator Q® (and the conjugated one ¹Q ®_ ) that corresponds to an ordinary or N=1 SUSY. When N > 1 one has an extended SUSY. The constraint on the number of SUSY generators comes from a requirement of consistency of the corresponding quantum ¯eld theory (QFT). The number of supersymmetries and the maximal spin of the particle in the multiplet are related by N · 4S; where S is the maximal spin. Since the theories with spin greater than 1 are non renormalizable and the theories with spin greater than 5/2 have no consistent coupling to gravity, this imposes a constraint on the number of SUSY generators N · 4 for renormalizable theories (YM), N · 8 for (super)gravity: In what follows, we shall consider simple SUSY, or N = 1 SUSY, contrary to extended supersymmetries with N > 1. In this case, one has two types of supermultiplets: the socalled chiral multiplet, which contains two physical states (Á; Ã) with spin 0 and 1/2, respectively, and the vector multiplet with ¸ = 1=2, which also contains two physical states (¸;A¹) with spin 1/2 and 1, respectively. 2.2 Superspace and super¯elds An elegant formulation of SUSY transformations and invariants can be achieved in the framework of superspace [9]. Superspace di®ers from the ordinary Euclidean (Minkowski) space by the addition of two new coordinates, µ® and µ¹®_ , which are Grassmannian, i.e. anticommuting, variables fµ®; µ¯g = 0; f¹µ _ ®; ¹µ _¯ g = 0; µ2® = 0; ¹µ2_ ® = 0; ®; ¯; ®_ ; _¯ = 1; 2: 13 Thus, we go from space to superspace Space ) Superspace x¹ x¹; µ®; µ¹®_ A SUSY group element can be constructed in superspace in the same way as an ordinary translation in the usual space G(x; µ; ¹µ) = ei(¡x¹P¹ + µQ + ¹µ ¹Q ): (2.5) It leads to a supertranslation in superspace x¹ ! x¹ + iµ¾¹¹" ¡ i"¾¹¹µ; µ ! µ + "; ¹µ ! ¹µ + ¹"; (2.6) where " and ¹" are Grassmannian transformation parameters. From Eq. (2.6) one can easily obtain the representation for the supercharges Eq. (2.4) acting on the superspace Q® = @ @µ® ¡ i¾¹ ®®_ ¹µ _ ®@¹; ¹Q ®_ = ¡ @ @µ¹®_ + iµ®¾¹ ®®_ @¹: (2.7) Taking the Grassmannian transformation parameters to be local, or spacetime de pendent, one gets a local translation. As has already been mentioned, this leads to a theory of (super) gravity. To de¯ne the ¯elds on a superspace, consider repre sentations of the SuperPoincar¶e group Eq. (2.4) [10]. The simplest one is a scalar super¯eld F(x; µ; ¹µ) which is SUSY invariant. Its Taylor expansion in µ and ¹µ has only several terms due to the nilpotent character of Grassmannian parameters. However, this super¯eld is a reducible representation of SUSY. To get an irreducible one, we de¯ne a chiral super¯eld which obeys the equation ¹D F = 0; where ¹D = ¡ @ @µ ¡ iµ¾¹@¹ (2.8) is a superspace covariant derivative. In superspace (by Taylor expanding y = x + iµ¾¹µ), a chiral super¯eld is written as ©(y; µ) = A(y) + p 2µÃ(y) + µµF(y) = A(x) + iµ¾¹¹µ@¹A(x) + 1 4 µµ¹µ¹µ2A(x) + p 2µÃ(x) ¡ ip 2 µµ@¹Ã(x)¾¹¹µ + µµF(x): (2.9) 14 Here A is a complex scalar ¯eld (with two bosonic degrees of freedom), Ã is a Weyl spinor ¯eld (with 2 fermionic degrees of freedom)and F is the auxiliary ¯eld (with no physical meaning) which is needed to close the SUSY algebra (2.4). We see from here that a super¯eld contains an equal number of fermionic and bosonic degrees of freedom. Under a SUSY transformation with anticommuting parameter ", the component ¯elds transform as ±"A = p 2"Ã; ±"Ã = i p 2¾¹¹"@¹A + p 2"F; (2.10) ±"F = i p 2¹"¾¹@¹Ã: The antichiral super¯eld ©+ obey the equation D©+ = 0; with D = @ @µ + i¾¹¹µ@¹: The product of chiral (antichiral) super¯elds ©2;©3, etc., is also a chiral (antichiral) super¯eld, while the product of chiral and antichiral ones ©+© is a general super¯eld. For any arbitrary function of chiral super¯elds one has W(©i) = W(Ai + p 2µÃi + µµF) = W(Ai) + @W @Ai p 2µÃi + µµ µ @W @Ai Fi ¡ 1 2 @2W @Ai@Aj ÃiÃj ¶ : (2.11) The W is usually referred to as a superpotential which replaces the usual potential for the scalar ¯elds. The vector super¯eld satis¯es the condition V = V +. They should be understood in terms of their power series expansion in µ and ¹µ as V (x; µ; ¹µ) = C(x) + iµÂ(x) ¡ i¹µ ¹Â(x) + i 2 µµ[M(x) + iN(x)] ¡ i 2 ¹µ¹µ[M(x) ¡ iN(x)] ¡ µ¾¹¹µv¹(x) + iµµ¹µ[¸(x) + i 2 ¹¾¹@¹Â(x)] ¡ i¹µ¹µµ[¸ + i 2 ¾¹@¹ ¹Â(x)] + 1 2 µµ¹µ¹µ[D(x) + 1 2 2C(x)]: (2.12) The component ¯elds C; D; M; N and v¹ must be real for Eq. (2.12) to satisfy V = V +. These vector supermultiplet contains 8 bosonic degrees of freedom (one 15 each for C; D; M; M; N and four from the real vector ¯eld v¹) and 8 fermionic degrees of freedom (from the two component spinors Â and ¸). The physical degrees of freedom corresponding to a real vector super¯eld V are the vector gauge ¯eld v¹ and the Majorana spinor ¯eld ¸. All other components are unphysical and can be eliminated. We now de¯ne the supersymmetric generalization of an Abelian gauge transformation of the super¯eld V as V ! V + © + ©+; where © and ©+ are some chiral super¯elds. Under this transformation, the compo nent transform as C ! C + A + A¤; Â ! Â ¡ i p 2Ã; M + iN ! M + iN ¡ 2iF; v¹ ! v¹ ¡ i@¹(A ¡ A¤); (2.13) ¸ ! ¸; D ! D: We see that there is a special gauge known as the WessZumino gauge [11] in which C; Â; M and N are all zero. Fixing this gauge breaks SUSY but still allows the usual gauge transformation v¹ ! v¹+@¹A. In this gauge, the vector multiplet reduces to 4 bosonic degrees of freedom (1 for D and the three remaining components of v¹) and 4 fermionic degrees of freedom (from the Majorana spinor ¸). In this gauge the vector super¯eld takes the form V = ¡µ¾¹¹µv¹(x) + iµµ¹µ¹¸ (x) ¡ i¹µ¹µµ¸(x) + 1 2 µµ¹µ¹µD(x); V 2 = ¡ 1 2 µµ¹µ¹µv¹(x)v¹(x); V 3 = 0; V n = 0 for n > 3: (2.14) One can de¯ne also a ¯eld strength tensor (as analog of F¹º in gauge theories) W® = ¡ 1 4 ¹D 2eVD®e¡V ; 16 ¹W ®_ = ¡ 1 4 D2eV ¹D ®e¡V ; (2.15) which is a polynomial in the WessZumino gauge. (Here Ds are the supercovariant derivatives.) The strength tensor is a chiral super¯eld ¹D _¯ W® = 0; D¯ ¹W ®_ = 0: In the WessZumino gauge it is a polynomial over component ¯elds: W® = Ta µ ¡i¸a ® + µ®Da ¡ i 2 (¾¹¹¾ºµ)®Fa ¹º + µ2¾¹D¹¹¸ a ¶ ; (2.16) where Fa ¹º = @¹va º ¡ @ºva¹ + gfabcvb¹ vcº ; D¹¹¸ a = @¹¸ a + gfabcvb¹ ¹¸ c: In Abelian case eqs.(2.15) are simpli¯ed and take form W® = ¡ 1 4 ¹D 2D®V; ¹W ®_ = ¡ 1 4 D2 ¹D ®V: 2.3 Supersymmetric Action Using the rules of Grassmannian integration: Z dµ® = 0 Z µ® dµ¯ = ±®¯ we can de¯ne the general form of a SUSY and gauge invariant Lagrangian as [10]: LYM SUSY = 1 4 Z d2µ Tr(W®W®) + 1 4 Z d2¹µ Tr( ¹W ® ¹W ®) (2.17) + Z d2µd2¹µ ©y ia (egV )ab ©bi + Z d2µ W(©i) + Z d2¹µ ¹W (¹© i) ©i are chiral super¯elds which transform as: ©i ! e¡ig¤©i and egV ! eig¤y egV e¡ig¤ where both ¤ and V are matrices: 17 ¤ij = ¿ a ij¤a; Vij = ¿ a ijVa; with ¿ a the gauge generators. The supersymmetric ¯eld strength W® is equal to W® = ¡ 1 4 ¹D ¹D e¡VD®eV and transforms as: W ! e¡i¤Wei¤. W is the superpotential, which should be invariant under the group of symme tries of a particular model. In terms of component ¯elds the above Lagrangian takes the form [12] LYM SUSY = ¡ 1 4 Fa ¹ºFa¹º ¡ i¸a¾¹D¹¹¸ a + 1 2 DaDa + (@¹Ai ¡ igva¹ ¿ aAi)y(@¹Ai ¡ igva¹¿ aAi) ¡ i ¹ Ãi¹¾¹(@¹Ãi ¡ igva¹¿ aÃi) ¡ DaAy i ¿ aAi ¡ i p 2Ay i ¿ a¸aÃi + i p 2 ¹ Ãi¿ aAi¹¸ a + Fy i Fi + @W @Ai Fi + @ ¹W @Ay i Fy i ¡ 1 2 @2W @Ai@Aj ÃiÃj ¡ 1 2 @2 ¹W @Ay i@Ay j ¹ Ãi ¹ Ãj (2.18) Integrating out the auxiliary ¯elds Da and Fi, one reproduces the usual Lagrangian. Contrary to the SM, where the scalar Higgs potential is arbitrary and is de¯ned only by the requirement of the gauge invariance, in supersymmetric theories it is completely de¯ned by the superpotential. It consists of the contributions from the Dterms and Fterms. The kinetic energy of the gauge ¯elds yields the 1 2DaDa term, and the mattergauge interaction yields the gDa¿ a ijA¤i Aj one. Together they give LD = 1 2 DaDa + gDa¿ a ijA¤ iAj : (2.19) The equation of motion reads Da = ¡g¿ a ijA¤ iAj ; (2.20) Substituting it back into Eq. (2.19) yields the Dterm part of the potential LD = ¡ 1 2 DaDa =) VD = 1 2 DaDa; (2.21) 18 where D is given by Eq. (2.20). The Fterm contribution can be derived from the matter ¯eld selfinteraction. For a general type superpotential W one has LF = F¤ i Fi + ( @W @Ai Fi + h:c:) (2.22) Using the equations of motion for the auxiliary ¯eld Fi F¤ i = ¡ @W @Ai (2.23) yields LF = ¡F¤ i Fi; =) VF = F¤ i Fi; (2.24) where F is given by Eq. (2.23). The full potential is the sum of the two contributions V = VD + VF : (2.25) Thus, the form of the Lagrangian is constrained by symmetry requirements. The only freedom is the ¯eld content, the value of the gauge coupling g, Yukawa couplings yijk and the masses. Because of the renormalizability constraint V · A4 the superpoten tial should be limited by W · ©3. All members of a supermultiplet have the same masses, i.e. bosons and fermions are degenerate in mass. This property of SUSY theories contradicts phenomenology and requires SUSY breaking. 2.4 SUSY breaking Since the SUSY algebra leads to mass degeneracy in a supermultiplet, it should be broken to explain the absence of superpartners at accessible energies. There are several ways of SUSY breaking. It can be broken either explicitly or spontaneously. In performing SUSY breaking one has to be careful not to spoil the cancellation of quadratic divergencies which allows one to solve the hierarchy problem. This is achieved by spontaneous breaking of SUSY. It is possible to show that in SUSY models the energy is always nonnegative de¯nite. According to quantum mechanics the energy is equal to E = h0j bH j0i; (2.26) 19 where bH is the Hamiltonian and due to the SUSY algebra fQ®; ¹Q _¯ g = 2(¾¹)® _¯ P¹: (2.27) Taking into account that Tr(¾¹P¹) = 2P0 one gets E = 1 4 X ®=1;2 h0jfQ®; ¹Q ®gj0i = 1 4 X ® kQ®j0ik2 ¸ 0: (2.28) Hence E = h0j bH j0i 6= 0 if and only if Q®j0i 6= 0: Therefore, SUSY is spontaneously broken, i.e. the vacuum is not invariant under Q (Q®j0i 6= 0), if and only if the minimum of the potential is positive (i:e: E ¸ 0) . Spontaneous breaking of SUSY is achieved in the same way as electroweak symmetry breaking. One introduces a ¯eld whose vacuum expectation value is nonzero and breaks the symmetry. However, due to the special character of SUSY, this should be a super¯eld whose auxiliary F or D component acquires nonzero VEVs. Thus, among possible spontaneous SUSY breaking mechanisms one distinguishes the F{ type breaking and the D{type breaking. i) FayetIliopoulos (Dterm) mechanism [12]. In this case the, the linear Dterm is added to the Lagrangian ¢L = »V jµµ¹µ¹µ = » Z d2µ d2¹µ V: (2.29) It is U(1) gauge and SUSY invariant by itself, however, it may lead to spontaneous breaking of both of them depending on the value of ». The drawback of this mecha nism is the necessity of U(1) gauge invariance. It can be used in SUSY generalizations of the SM but not in GUTs. The mass spectrum also causes some troubles since the following sum rule is always valid STrM2 = X J (¡1)2J(2J + 1)m2 J = 0; (2.30) which is bad for phenomenology. ii) O'Raifeartaigh (Fterm) mechanism [12]. 20 In this case, several chiral ¯elds are needed and the superpotential should be chosen in such way that trivial zero VEVs for the auxiliary F¯elds are forbidden. For instance, choosing the superpotential to be W(©) = ¸©3 + m©1©2 + g©3©21 ; (2.31) one gets the equations for the auxiliary ¯elds F¤ 1 = mA2 + 2gA1A3; (2.32) F¤ 2 = mA1; (2.33) F¤ 3 = ¸ + gA21 ; (2.34) which have no solutions with hFii = 0 and SUSY is spontaneously broken. The drawback of this mechanism is that there is a lot of arbitrariness in the choice of the potential. The sum rule (2.30) is also valid here. Unfortunately, none of these mechanisms explicitly works in SUSY generalizations of the SM. None of the ¯elds of the SM can develop nonzero VEVs for their F or D components without breaking SU(3)C or U(1)Y gauge invariance since they are not singlets with respect to these groups. This requires the presence of extra sources for spontaneous SUSY breaking [13{18]. CHAPTER 3 The Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) [19] respects the same gauge symmetry SU(3)C £SU(2)L £ U(1)Y as does the SM. Here SUSY is somehow (softly) broken at the weak scale. The MSSM is the simplest phenomenologically viable supersymmetric theory beyond the SM in that it contains the fewest number of new particles and new interactions. To construct the MSSM [20] we start with the complete set of chiral fermions, and add a scalar superpartner to each Weyl fermion so that each ¯elds represents a chiral multiplet. Similarly we must add a gaugino for each of the gauge bosons in the SM making up the gauge multiplets. The particles necessary to construct the MSSM are shown in Tables 3.1. and 3.2. Super¯eld SU(3)C SU(2)L U(1)Y Particle Content ^Q 3 2 1 6 (uL; dL), (~uL; ~ dL) ^U c 3 1 ¡2 3 uR, ~u¤R ^D c 3 1 1 3 dR, ~ d¤ R ^L 1 2 ¡1 2 (ºL; eL), (~ºL; ~eL) ^E c 1 1 1 eR, ~e¤ R ^H d 1 2 ¡1 2 (Hd; ~H d) ^H u 1 2 1 2 (Hu; ~H u) TABLE 3.1. Chiral super¯elds of the MSSM. The MSSM is de¯ned by its minimal ¯eld content (which accounts for the known SM ¯elds) and minimal superpotential necessary to account for the known Yukawa 21 22 Super¯eld SU(3)C SU(2)L U(1)Y Particle Content ^G a 8 1 0 G¹, ~g¹ ^W r 1 3 0 W¹ r , ~!¹ r ^B 1 1 0 B¹, ~b ¹ TABLE 3.2. Vector Super¯elds of the MSSM. mass terms. Notice that in Table 3.1. and 3.2., we have introduced a partner for every particle of the SM with the same internal quantum number and a spin di®ering by 1 2 . We de¯ne the MSSM by the superpotential W = ²ij [yeHj dLiec + ydHj dQidc + yuHiu Qjuc + ¹Hid Hju ]: (3.1) Here, the indices, fijg, are SU(2)L doublet indices and ¹ is the Higgs mass parameter. The Yukawa couplings, y, are all 3£3 matrices in generation space. Note that there is no generation index for the Higgs multiplets. Color and generation indices have been suppressed in the above expression. There are two Higgs doublets in the MSSM. This is a necessary addition to the SM which can be seen as arising from the holomorphic property of the superpotential. That is, there would be no way to account for all of the Yukawa terms for both uptype and downtype multiplets with a single Higgs doublet. To avoid a massless Higgs state, a mixing term ²ij¹Hid Hju must be added to the superpotential. However, even if we stick to the minimal ¯eld content, there are several other superpotential terms which we can envision adding to Eq. (3.1) since they are con sistent with all of the symmetries of the theory. We could have considered terms like WR = ¹0iLiHu + ¸ijkLiLjec k + ¸0ijkLiQjdc k + ¸00ijkuci dcj dc k; (3.2) where i; j and k are the generation indices and ¸'s are the coupling constants. In Eq. (3.2), the terms proportional to ¸; ¸0, and ¹0, all violate lepton number by one unit. The term proportional to ¸00 violates baryon number by one unit. 23 Each of the terms in Eq. (3.2) predicts new particle interactions and can be to some extent constrained by the lack of observed exotic phenomena. However, the combination of terms which violate both baryon and lepton number can be disastrous. In order to avoid these unwanted terms, we impose a discrete symmetry on the theory called R{parity [21], which can be de¯ned as R = (¡1)3B+L+2s; (3.3) where B; L, and s are the baryon number, lepton number, and spin respectively. With this de¯nition, it turns out that all of the known SM particles have Rparity +1, and all the superpartners of the known SM particles have R = ¡1, since they must have the same value of B and L but di®er by 1/2 unit of spin. 3.1 Electroweak symmetry breaking and the Higgs boson masses We analyze the scalar potential in this section. It is derived from the superpo tential and the terms involving the Higgs in the soft breaking Lagrangian. The part of the scalar potential which involves only the Higgs bosons (Hu and Hd) is given by V = j¹j2(H¤ dHd + H¤ uHu) + 1 8 g02(H¤ uHu ¡ H¤ dHd)2 + 1 8 g2 ¡ 4jH¤ dHuj2 ¡ 2(H¤ dHd)(H¤ uHu) + (H¤ dHd)2 + (H¤ uHu)2¢ +m2 HdH¤ dHd + m2 HuH¤ uHu + (B¹²ijHid Hju + h.c.): (3.4) Here the ¯rst term is the Fterm, derived from j(@W=@Hd)j2 and j(@W=@Hu)j2 setting all sfermion VEV's equal to 0. The next two terms are D{terms, the ¯rst a U(1)Y D{term, recalling that the hypercharges for the Higgses are YHd = ¡1 2 and YHu = 1 2 , and the second is an SU(2)L D{term, taking Ta = ¾a where ¾a are the three Pauli matrices. Finally, the last three terms are the soft SUSY breaking masses mHd and mHu, and the bilinear term B¹. The Higgs doublets can be written as hHdi = Ã H0 d H¡ d ! ; hHui = Ã H+ u H0 u ! ; (3.5) where in Eq. (3.4) by (H¤ dHd), we mean H0 d ¤H0 d + H¡ d ¤ H¡ d etc. 24 The neutral portion of Eq. (3.4) can be expressed more simply as V = g2 + g02 8 ¡ jH0 d j2 ¡ jH0 uj2¢2 + (m2 Hd + j¹j2)jH0 d j2 +(m2 Hu + j¹j2)jH0 uj2 + (B¹H0 dH0 u + h.c.): (3.6) For electroweak symmetry breaking, it will be required that either one (or both) of the soft masses (m2 Hd ;m2 Hu) be negative (as in the SM). From the minimization of the potential Eq. (3.6), we obtain the following two conditions ¡2B¹ = (m2 Hd + m2 Hu + 2¹2) sin 2¯; (3.7) and v2 = 4 ¡ m2 Hd + ¹2 ¡ (m2 Hu + ¹2) tan2 ¯ ¢ (g2 + g02)(tan2 ¯ ¡ 1) ; (3.8) where tan ¯ = Àu Àd . From the potential and these two conditions, the masses of the physical scalars can be obtained. At the tree level, m2 H§ = m2 A + m2 W; (3.9) m2 A = m2 Hd + m2 Hu + 2¹2 = ¡B¹(tan ¯ + cot ¯); (3.10) m2 H;h = 1 2 · m2 A + m2 Z § q (m2 A + m2 Z)2 ¡ 4m2 Am2 Z cos2 2¯ ¸ : (3.11) Notice that these expressions and the above constraints limit the number of free inputs in the MSSM. First, from the mass of the pseudoscalar, we see that B¹ is not independent and can be expressed in terms of mA and tan ¯. Furthermore from the conditions Eqs. (3.7) and (3.8), we see that if we keep tan ¯, we can either choose mA and ¹ as free inputs thereby determining the two soft masses, mHd and mHu, or we can choose the soft masses as inputs, and ¯x mA and ¹ by the conditions for electroweak symmetry breaking. Both choices of parameter ¯xing are widely used in the literature. The tree level expressions for the Higgs masses make some very de¯nite predic tions. The charged Higgs is heavier than MW, and the light Higgs h, is necessarily lighter than MZ. Note if uncorrected, the MSSM would already be excluded (from current accelerator limits). However, radiative corrections to the Higgs masses are 25 not negligible in the MSSM, particularly for a heavy top mass mt » 178 GeV. The leading oneloop corrections to m2 h depend quartically on mt and can be expressed as [22] ¢m2 h = 3m4t 4¼2v2 ln µ m~t1m~t2 m2t ¶ + 3m4t ^ A2t 8¼2v2 h 2h(m2 ~t1 ;m2 ~t2 ) + ^ A2t g(m2 ~t1 ;m2 ~t2 ) i + : : : (3.12) where m~t1;2 are the physical masses of the two stop squarks ~t1;2 to be discussed in more detail shortly, A^t ´ At + ¹ cot ¯, (At is the SUSY breaking trilinear term associated with the top quark Yukawa coupling). The functions h and f are h(a; b) ´ 1 a ¡ b ln ³a b ´ ; g(a; b) = 1 (a ¡ b)2 · 2 ¡ a + b a ¡ b ln ³a b ´¸ : (3.13) Additional corrections to coupling vertices, twoloop corrections and renormalization group resummations have also been computed in the MSSM [23]. With these correc tions one can allow mh <» 130 GeV; (3.14) within the MSSM. While certainly higher than the tree level limit of MZ, the limit still predicts a relatively light Higgs boson, and allows the MSSM to be experimentally excluded (or veri¯ed!) at the LHC. 3.2 The sfermions masses We turn next to the discussion of scalar partners of the quarks and leptons. The mixing matrices for ~m2t ; ~m2b and ~m2¿ are 0 @ ~m2 tL mt(At + ¹ cot ¯) mt(At + ¹ cot ¯) ~m2 tR 1 A; (3.15) 0 @ ~m2 bL mb(Ab + ¹ tan ¯) mb(Ab + ¹ tan ¯) ~m2 bR 1 A; (3.16) 0 @ ~m2¿ L m¿ (A¿ + ¹ tan ¯) m¿ (A¿ + ¹ tan ¯) ~m2¿ R 1 A; (3.17) 26 with ~m2 tL = ~m2 Q + m2t + 1 6 (4M2W ¡M2Z ) cos 2¯; ~m2 tR = ~m2 U + m2t ¡ 2 3 (M2W ¡M2Z ) cos 2¯; ~m2 bL = ~m2 Q + m2b ¡ 1 6 (2M2W +M2Z ) cos 2¯; ~m2 bR = ~m2 D + m2b + 1 3 (M2W ¡M2Z ) cos 2¯; ~m2¿ L = ~m2 L + m2¿ ¡ 1 2 (2M2W ¡M2Z ) cos 2¯; ~m2¿ R = ~m2 E + m2¿ + (M2W ¡M2Z ) cos 2¯: The ¯rst terms here ( ~m2) are the soft ones, which are calculated using the Renor malization Group (RG) equations starting from their values at the GUT (Planck) scale. The second ones are the usual masses of quarks and leptons and the last ones are the D terms of the potential. The o®diagonal mixing term in the mass matrix is negligible for all but the third generation sfermions. The physical sfermion states and their masses are determined by diagonalizing the sfermion mass matrix. 3.3 Neutralinos There are four new neutral fermions in the MSSM which not only receive mass but mix as well. These are the gauge fermion partners of the neutral B and W3 gauge bosons, and the partners of the Higgs. The two gauginos are called the bino, e B, and wino, fW3 respectively. The latter two are the Higgsinos, eH d and eH u. In addition to the SUSY breaking gaugino mass terms, ¡1 2M1 e B e B, and ¡1 2M2fW3fW3, there are supersymmetric mass contributions of the type WijÃiÃj , giving a mixing term between eH d and eH u, 1 2¹eH deH u, as well as terms of the form g(Á¤TaÃ)¸a giv ing the following mass terms after the appropriate Higgs VEVs have been inserted, p1 2 g0vd eH d Be, ¡p1 2 g0vu eH u Be, ¡p1 2 gvd eH dWf3, and p1 2 gvu eH ufW3. These latter terms can be written in a simpler form noting that for example, g0vd= p 2 = MZ sin µW cos ¯. 27 Thus we can write the neutralino mass matrix as (in the ( e B;fW3; eH 0 d ; eH 0 u) basis) [24] 0 BBBBBB@ M1 0 ¡MZsµW cos ¯ MZsµW sin ¯ 0 M2 MZcµW cos ¯ ¡MZcµW sin ¯ ¡MZsµW cos ¯ MZcµW cos ¯ 0 ¡¹ MZsµW sin ¯ ¡MZcµW sin ¯ ¡¹ 0 1 CCCCCCA ; (3.18) where sµW = sin µW and cµW = cos µW. The mass eigenstates (a linear combination of the four neutralino states) and the mass eigenvalues are found by diagonalizing the mass matrix Eq. (3.18). 3.4 Charginos There are two new charged fermionic states which are the partners of the W§ gauge bosons and the charged Higgs scalars, H§, which are the charged gauginos, fW§ and charged Higgsinos, eH §, or collectively charginos. The chargino mass matrix is composed similarly to the neutralino mass matrix. The result for the mass term is ¡ 1 2 (fW¡; eH ¡) Ã M2 p 2mW sin ¯ p 2mW cos ¯ ¹ ! ÃfW+ eH + ! + h.c. (3.19) Note that unlike the case for neutralinos, two unitary matrices must be constructed to diagonalize Eq. (3.19). The result for the mass eigenstates of the two charginos is m2e c1 ;m2e c2 = 1 2 h M2 2 + ¹2 + 2M2W § q (M2 2 + ¹2 + 2M2W )2 ¡ 4(¹M2 ¡M2W sin 2¯)2 i (3.20) Some additional resources on supersymmetry used in this preliminary introduc tion are the classic by Bagger and Wess on supersymmetry [25], the book by Ross on Grand Uni¯cation [26] and some other good reviews by Martin and others [27{33]. CHAPTER 4 ANOMALY MEDIATED SUPERSYMMETRY BREAKING Understanding the origin of Supersymmetry breaking has been one of the main focuses of SUSY phenomenologists. It is highly non{trivial to construct models which break supersymmetry in a generally acceptable way. The most common scenario for producing low{energy Supersymmetry breaking is called the hidden sector. The usual SM matter ¯elds reside in the visible sector and the ¯elds that break supersymmetry reside in the hidden sector. There are no (small) direct couplings between the two sectors. The symmetry breaking which occurs in the hidden sector is communicated to the visible sector via \ messenger " ¯elds. Some of the several competing proposals on what the mediating interaction might be are Gravity mediation (SUGRA), Gauge mediation, Gaugino mediation and Anomaly mediation. Any successful supersymmetry breaking scenario should at least satisfy the fol lowing conditions: ² The theory should give correct masses to the superpartners » 1 TeV, and the scalar mass{squared should be positive, ² The ¹ parameter should be between 100 GeV { 1 TeV and the B¹ parameter should not be too much larger than ¹2, ² There are no large °avor changing neutral currents, ² CP should be approximately conserved (A & B phase should be small, as required by the measurement of the electric dipole moments of neutron and electron), 28 29 ² The model should be simple enough such that it can be tested experimentally. This thesis is based on the Anomaly mediation scenario of SUSY breaking. Before going into any details of the proposed models, I will brie°y review the other three scenarios and what others have done on anomaly mediation. 4.1 Gravity mediation In this scenario, the messenger is gravity. Supersymmetry is broken in the hidden sector by a VEV hFi. The moduli ¯eld T, which appears as a result of compacti¯cation from higher dimensions and the dilaton ¯eld S, which is part of the SUGRA supermultiplet develop a non{zero VEV for their F components which in turn leads to spontaneous SUSY breaking. The soft mass term in the visible sector is roughly msoft » hFi MPl : (4.1) These soft masses should vanish as hFi ! 0 where SUSY remains unbroken. In this scenario, the SUSY sector is completely described by 5 input parameters: Higgs mass parameter (¹), common scalar mass (m0), common gaugino mass (m1=2), common trilinear coupling (A0) and the Higgs mixing parameter (B). When SUSY is broken at a scale p hFi, the graviton will also obtain a mass msoft » m3=2 » hFi MPl : (4.2) Since we argued earlier that for SUSY to solve the hierarchy problem the mass scale should be msoft »1 TeV, therefore SUSY should be broken at a scale p hFi » 1011 GeV. Some of the good features of the models are ² Extremely predictive{ because the entire low energy spectrum is predicted in terms of few input parameters (m0; m1=2; A0; tan ¯ (B) and sign(¹)), where all phenomenological limits can be expressed in terms of these parameters, 30 ² Gauge couplings are uni¯ed and the gaugino masses are predicted to be the ratios of the gauge couplings, ² The ¹ problem is solved through Guidice{Masiero mechanism, where a singlet ¯eld § in the Kahler potential R d4µ§¤HdHu=MPl breaks SUSY, ² It is easy to generate positive scalar mass{squared. ² Hu mass{squared turns negative due to large top Yukawa coupling even if it starts of being positive at the Planck scale. Despite the success of the theory, there are still some problems which are: CP is generally a problem, large freedom of parameters, absence of automatic suppression of °avor violation, lack of consistent theory of quantum supergravity (local symmetry). 4.2 Gauge mediation In this scenario the Supersymmetry breaking is communicated from the hidden sector to the visible sector via gauge interactions. The main idea is to introduce new chiral multiplets (messengers) which couple indirectly to the MSSM ¯elds through the SU(3)C £ SU(2)L £ U(1)Y gauge interactions. The particles ((s)quarks and (s)leptons) gets large mass by coupling to a gauge singlet chiral supermultiplet S. The superpotential for a typical gauge mediation can be written as W = ¸1S`¹` + ¸2Sq¹q: (4.3) The singlet scalar S and the auxiliary component of S (Fs) acquires a VEV by putting the scalar ¯eld into an O'Raifeartaigh{type model or a dynamical mechanism. The gauginos get mass at 1{loop Mi » ®i 4¼ ¤ (i = 1; 2; 3); (4.4) where ¤ = Fs=hSi. The MSSM scalars do not get any radiative corrections to their masses at 1{ loop. Their masses arise at 2{loop level from those diagrams involving the gauge 31 ¯elds and the messengers. The scalar masses are given by ~m » µ ¤ 4¼ ¶2 f®2 3C3 + ®2 2C2 + ®2 1C1g; (4.5) where Ci are the quadratic Casimir operators for the SU(3)C£SU(2)L£U(1)Y gauge group. This implies that the sparticles with the same gauge quantum number will have equal masses (for example: ~me = ~m¹ = ~m¿ ). In order for the gauginos and scalar soft masses to be » 1 TeV (as needed for the hierarchy problem) requires ¤ » 104 ¡ 105 GeV. In most of the gauge mediation models, the slepton and squark masses depend only on their gauge quantum num bers. This leads to the degeneracy of squark and slepton mass which results in the suppression of °avor changing neutral currents (FCNC's). The Lightest Supersym metric Particle (LSP) is usually the gravitino, with mass m3=2 » ¤2=Mpl » 10¡10 GeV, which can be crucial both for cosmology and collider physics. In summary: ² gauge mediated supersymmetry breaking (GMSB) solves the FCNC problem, ² gaugino mass arise at 1{loop while the scalar mass{squared arise at two loop level, ² there is still a problem in the Higgs sector (o®ers no compelling solution to the ¹ problem), ² it does not o®er any solution to the SUSY CP problem. 4.3 Gaugino mediation In this scenario the SM quark and lepton ¯elds are localized on a `3{brane' in extra dimensions, while the gauge and Higgs ¯elds propagate in the bulk. SUSY breaking masses for the gauginos and Higgs ¯elds are generated by higher{dimensional contact terms between the bulk ¯elds and the hidden sector ¯elds, assumed to arise from a more fundamental theory such as string theory [34]. The leading contribution to the SUSY breaking for visible sector ¯elds arises from loops of bulk gauge and Higgs ¯elds as shown in Fig. 4.1 32 Figure 4.1. Leading diagram that contributes to SUSY{breaking scalar masses. The bulk line is a gaugino propagator. The minimal version of gaugino mediation has only three high energy param eters ¹; m1=2 and Mc. Here m1=2 is the universal gaugino mass at the uni¯cation scale and Mc is the compacti¯cation scale where the higher dimensional theory is matched onto the e®ective four{dimensional theory. For sin2 µW prediction to be pre served from gauge coupling uni¯cation requires Mc > MGUT . In some other models of gaugino mediation [35] the ¹ parameter is determined by ¯tting to the Z mass. Such model requires only two free parameters m1=2 and Mc. The gaugino mediation scenario is the least developed in the literature. It does not o®er any real solution to the ¹ problem. 4.4 Anomaly mediation This scenario assumes that supersymmetry breaking takes place in a hidden or sequestered sector. The MSSM super¯elds are con¯ned to a 3{brane in a higher dimensional bulk space{time separated from the sequestered sector by extra dimen sions. A rescaling super{Weyl anomaly generates coupling of the auxiliary ¯eld of the gravity multiplet to the gauginos and the scalars of the MSSM, with the couplings determined by the SUSY renormalization group equations (RGE) [36]. 33 Before going into much details, it is important to give a brief review on how this scenario address the numerous problems associated with the other three scenarios addressed earlier. ² The ¹ parameter can be generated without generating excessively large B¹ due to the constraints from the coupling of the gravitational multiplet. ² The dominant anomaly{mediated contribution to the squark and slepton masses suppresses °avor violation automatically. ² There are no new phases in the A and B terms. This implies a natural solution to the SUSY CP problem. In other words CP can be violated on our 3{brane and nowhere else. ² The model is straightforward in the sense that the basic assumption is that SUSY breaking is derived from higher dimensional theory. ² These SUSY breaking models are very predictive. The ratio of the gaugino masses depends on the beta functions rather than the gauge couplings. The A{ terms are predicted to be proportional to the corresponding Yukawa coupling and there is a nearly degenerate Wino/Zino LSP, of which the Zino is the lighter. ² The gaugino and scalar masses are comparable. ² Since the rescaling anomaly is UV insensitive, the pattern of SUSY breaking masses at any energy scale is governed only by the physics at that scale [36{38]. An arbitrary °avor structure in the SUSY scalar spectrum at high energies gets washed out at low energies. This Ultraviolet (UV) insensitivity provides an elegant solution to the SUSY °avor problem. ² It can naturally solve the cosmological gravitino abundance problem which tends to destroy the success of big bang cosmology in generic supergravity models [39]. 34 ² The decay of the moduli ¯elds present in the model (as well as the gravitino) will produce neutralinos, especially the neutral Winos, with the right abundance to make it a viable cold dark matter candidate [40; 41]. We see from above that this model seems to be a viable (promising) model for understanding the MSSM supersymmetry breaking. It turns out that there is a major problem in this model which is discussed below. 4.4.1 The negative slepton mass problem of anomaly mediated supersymmetry breaking In anomaly mediated supersymmetry breaking models (AMSB), the masses of the scalar components of the chiral supermultiplets are given by [36; 37] (m2)Áj Ái = 1 2 M2 aux · ¯(Y ) @ @Y °Áj Ái + ¯(g) @ @g °Áj Ái ¸ : (4.6) In the above equation summations over the gauge couplings g and the Yukawa cou plings Y are assumed. °Áj Ái are the one{loop anomalous dimensions, ¯(Y ) is the beta function for the Yukawa coupling Y , and ¯(g) is the beta function for the gauge coupling g. Maux is the vacuum expectation value of a \compensator super¯eld" [36] which sets the scale of SUSY breaking. The gaugino mass Mg associated with the gauge group with coupling g is given by [36; 37] Mg = ¯(g) g Maux: (4.7) The trilinear soft supersymmetry breaking term AY corresponding to the Yukawa coupling Y is given by [36; 37] AY = ¡ ¯(Y ) Y Maux: (4.8) In the simplest scenario for generating the ¹ term for a special class of models, the contribution to the Higgs mixing parameter (the Bterm) is given by [36] B = ¡(°Hu + °Hd)Maux: (4.9) Here °Hu and °Hd are the one{loop anomalous dimensions of the Hu and Hd ¯elds. Similar relations hold for other bilinear terms in the SUSY breaking Lagrangian. 35 In the minimal scenario, it turns out that AMSB induces a negative mass{ squared for the sleptons. Such a scenario is excluded since it would break electro magnetism. The reason for the negative mass{squared can be understood as follows. There are two sources for slepton masses in AMSB, the Yukawa part and the gauge part (cf: Eq. (4.6)). For the ¯rst two families the Yukawa couplings are negligible and the dominant contributions arise proportional to the gauge beta function. Since in the MSSM the SU(2)L and the U(1)Y gauge couplings are not asymptotically free, their gauge beta functions are positive. This makes the slepton mass{squared nega tive. In the squark sector, the masses are positive because SU(3)C gauge theory is asymptotically free. 4.4.2 Suggested solutions to the AMSB slepton mass problem Several possible ways of avoiding the slepton mass problem of AMSB have been suggested. A non{decoupling universal bulk contribution to all the scalar masses is a widely studied option [36{42]. While this will make the minimal model phenomeno logically consistent, the UV insensitivity of AMSB is no longer guaranteed. It is therefore interesting to investigate variations of the minimal model which maintain the UV insensitivity but provide positive mass{squared for the sleptons from physics at the TeV scale. One way to avoid the negative slepton mass problem with TeV scale physics is to increase the Yukawa contributions in Eq. (4.6). This can be achieved by introducing new particles at the TeV scale with large Yukawa couplings to the lepton ¯elds. This possibility was studied in Ref. [43] where the MSSM spectrum was extended to include 3 pairs of Higgs doublets, four singlets and a vector{like pair of color{triplets near the weak scale. The Yukawa contributions can also be enhanced by invoking R{parity violating couplings in the MSSM [44]. Unfortunately such a theory would generate unacceptably large neutrino masses. Yet another possibility is to make use of the positive D{term contributions from a U(1) gauge symmetry broken at the weak scale. This was achieved by adding TeV scale Fayet{Iliopoulos terms explicitly to the theory in Ref. [45]. New D{term contributions generated in a controlled fashion 36 by the breaking of U(1)B¡L at an arbitrary high scale may also provide positive contributions to the slepton masses [46; 47]. A low scale ancillary U(1) as a solution to the problem has been studied in Ref. [48]. E®ective supersymmetric theories which are devoid of the negative slepton mass problem of AMSB with new dynamics at the 10 TeV scale have been studied in Ref. [49]. Non{decoupling e®ects of heavy ¯elds at higher orders have been analyzed in AMSB models in Ref. [50] as an attempt to solve the slepton mass problem. CHAPTER 5 TeV{Scale Horizontal Symmetry and the Slepton Mass Problem of Anomaly Mediation 5.1 Introduction As noted in chapter 4, supersymmetry provides an elegant solution to the gauge hierarchy problem of the standard model. To be realistic, it must however be a broken symmetry. There are several ways of achieving supersymmetry (SUSY) breaking. Anomaly mediated SUSY breaking (AMSB) is an attractive and predictive scenario which has the virtue that it can solve the SUSY °avor problem [36; 37]. This scenario assumes that SUSY breaking takes place in a hidden or sequestered sector. The MSSM super¯elds are con¯ned to a 3{brane in a higher dimensional bulk space{ time separated from the sequestered sector by extra dimensions. A rescaling super{ Weyl anomaly generates coupling of the auxiliary ¯eld of the gravity multiplet to the gauginos and the scalars of the MSSM, with the couplings determined by the SUSY renormalization group equations (RGE). Since the rescaling anomaly is UV insensitive, the pattern of SUSY breaking masses at any energy scale is governed only by the physics at that scale [36{38]. Arbitrary °avor structure in the SUSY scalar spectrum at high energies gets washed out at low energies. This ultraviolet insensitivity provides an elegant solution to the SUSY °avor problem. The purpose of this thesis is to suggest and investigate the possibility of solving the negative slepton mass problem by making the gauge contribution in Eq. (4.6) positive. This can only be achieved by introducing a new non{Abelian gauge sym metry for leptons with negative gauge beta function. We point out that an SU(3)H horizontal symmetry acting on the lepton multiplets has all the desired properties 37 38 for achieving this. We show that such an SU(3)H horizontal symmetry broken at the TeV scale is consistent with rare leptonic processes owing to the emergence of approximate global symmetries. The speci¯c AMSB model we study is quite predictive. The lightest Higgs boson mass is predicted to be mh . 120 GeV, and the parameter tan ¯ is found to be tan ¯ ' 4. The model predicts the existence of new particles associated with the SU(3)H symmetry breaking sector. The SU(3)H vector bosons have masses of order 1{4 TeV. These particles should be accessible experimentally at the LHC. The plan of the chapter is as follows. In section 5.2 we introduce our model. In section 5.3 we analyze the Higgs potential of the model. Here we derive the limits on tan ¯ and mh. In section 5.4 we present the SUSY spectrum of the model and show how the sleptons acquire positive masses. Numerical results for the full spectrum of the model are given in section 5.5. In section 5.6 we outline the most signi¯cant experimental consequences of the model. In section 5.7 we comment on the possible origins of the ¹ and the B¹ terms. We summarize in section 5.8. In Appendix A, we give the relevant beta functions, anomalous dimensions as well as the soft masses. 5.2 SU(3)H horizontal symmetry In this section we present our model. Since our aim is to have positive con tributions to the slepton masses from the gauge sector, we are naturally led to a leptonic horizontal symmetry that is asymptotically free. Our model is based on the gauge group SU(3)C £ SU(2)L £ U(1)Y £ SU(3)H, where SU(3)H is a horizontal symmetry acting on the leptons. The left{handed lepton doublets and the antilepton singlets transform as fundamental representations of the SU(3)H gauge symmetry. The theory is made anomaly free by introducing three Higgs multiplets (©1, ©2, ©3) which transform as antifundamental representations of SU(3)H and as singlets of the standard model. These ¯elds are su±cient for breaking the SU(3)H symmetry com pletely near the TeV scale. The particle content of the model and the transformation properties under the gauge group SU(3)C £SU(2)L £U(1)Y £SU(3)H are presented in Table 5.1. It turns out that the Higgs potential involving these ©i ¯elds exhibits 39 a global SU(3)G symmetry. We take advantage of this global symmetry to suppress potentially large °avor changing neutral current processes mediated by the SU(3)H gauge bosons. The last column in Table 5.1 lists the transformation properties under the global SU(3)G symmetry (The Yukawa couplings of the model reduce the global SU(3)G down to U(1).) The ¯elds ´i and ¹´i are introduced to facilitate SU(3)H symmetry breaking within our AMSB framework. Super¯eld SU(3)C SU(2)L U(1)Y SU(3)H SU(3)G Qi 3 2 1 6 1 1 uci ¹3 1 ¡2 3 1 1 dci ¹3 1 1 3 1 1 L® 1 2 ¡1 2 3 1 ec ® 1 1 1 3 1 Hu 1 2 1 2 1 1 Hd 1 2 ¡1 2 1 1 ©®i 1 1 0 ¹3 3 ´i 1 1 0 ¹3 3 ¹´i 1 1 0 3 ¹3 TABLE 5.1. Particle content and charge assignment of the model. SU(3)G in the last column is a softly broken global symmetry present in the model. The indices i and ® take values i; ® = 1 ¡ 3. Note that the quarks are neutral under SU(3)H. This is necessitated by the requirements that SU(3)H be asymptotically free. A separate SU(3)H0 acting on the quarks is a possible quark{lepton symmetric extension of the model. But we do not pursue such an extension here. The superpotential of the model consistent with the gauge symmetries and the global SU(3)G symmetry is given by: W = (Yu)ij QiHuucj + (Yd)ij QiHddcj + ¹HuHd + ·©®1 ©¯ 2©° 3²®¯° + ¸´® a ´¯ b ©°c ²®¯°²abc +M´´a¹´a: (5.1) 40 Here ®, ¯, ° =1, 2, 3 are SU(3)H indices, i; j = 1, 2, 3 are family indices, and a; b; c = 1, 2, 3 are SU(3)G indices. The mass parameters ¹ and M´ are of order TeV, which has a natural origin in AMSB [36]. We will comment on possible origin of these terms in Sec. 5.7. In the SU(3)H symmetric limit the leptons are all massless. They obtain their masses from the e®ective operators Ll eff = L®ec ®©®i ©®i Hd M2 i : (5.2) Such operators can be obtained by integrating ¯elds shown in Fig. 1, for example. The masses of the heavy ¯elds break SU(3)G symmetry softly (the ¹ ÃiÃi and the ¹E iEi mass terms in Fig. 5.1). Note that the mass scale Mi in Eq. (5.2) is of order 5 La Fi a yi yi E Ei i Hd ec a Fi a Figure 5.1. E®ective operators inducing charged lepton masses. TeV for generating realistic ¿{lepton mass, of order 20 TeV for the ¹ mass and of order 300 TeV for the electron mass (assuming that all relevant Yukawa couplings are of order one). Since these masses are all much heavier than the e®ective SUSY breaking scale of order 1 TeV, these heavy ¯elds will have no e®ect in the low energy SUSY phenomenology within AMSB. Note that no generation mixing is induced by these e®ective operators, which will guarantee the approximate conservation of electron number, muon number and tau lepton number. This is what makes the model consistent with FCNC data even when SU(3)H is broken at the TeV scale. Since the Higgs potential respects SU(3)H £ SU(3)G symmetry, after spontaneous 41 symmetry breaking, the diagonal subgroup SU(3)G+H remains as an unbroken global symmetry. This subgroup contains e, ¹ and ¿ lepton numbers. Since right{handed neutrinos are not required to be light for SU(3)H anomaly cancellation, they acquire heavy masses and decouple from the low energy theory. Small neutrino masses are then induced through the seesaw mechanism. Speci¯cally, the following e®ective nonrenormalizable operators emerge after integrating out the heavy right{handed neutrino ¯elds: Lº eff = ¸®¯ ij L®L¯HuHu©®i ©¯ j M3N : (5.3) Here MN represents the masses of the heavy right{handed neutrino ¯elds. For MN » 107 GeV and h©ii » TeV, neutrino masses in the right range for oscillation phe nomenology are obtained. Note that Eq. (5.3) arises from integrating neutral leptons with their masses assumed to break all global symmetries. This enables generation of large neutrino mixing angles, as needed for phenomenology. 5.3 Symmetry breaking The SU(3)H model has two sets of Higgs bosons: the usual MSSM Higgs dou blets Hu and Hd, and the SU(3)H Higgs antitriplets ©i (i = 1; 2; 3). The Higgs potential is derived from the superpotential of Eq. (5.1) and includes the soft terms and the D terms. The tree level potential splits into two pieces: V (Hu;Hd;©i) = V (Hu;Hd) + V (©i); (5.4) enabling us to analyze them independently. The ¯rst part, V (Hu;Hd), is identical to the MSSM potential which is well studied. There are however signi¯cant constraints on the parameters in our AMSB extension, which we now discuss. 5.3.1 Constraints on tan ¯ and mh Minimization of V (Hu;Hd) gives sin 2¯ = 2B¹ 2¹2 + m2 Hu + m2 Hd ; ¹2 = m2 Hd ¡ m2 Hu tan2 ¯ tan2 ¯ ¡ 1 ¡ M2Z 2 : (5.5) 42 Here m2 Hu and m2 Hd are the Higgs soft masses and are given in the Appendix for the AMSB model (see Eqs. (A.19){(A.20).) The constraints on mh and tan ¯ arise since these soft masses and the B parameter are determined in terms of a single parameter Maux in our framework. We eliminate Maux in favor of M2, the Wino mass (M2 = b2g2 2 16¼2Maux). We see from Eqs. (4.9), (5.5) as well as from Eqs. (A.6){(A.7) and Eqs. (A.19){(A.20) of the Appendix that tan ¯ is ¯xed in terms of M2. In Fig. 5.2 we plot tan ¯ as a function of M2. For the experimentally interesting range of M2 & 100 GeV, we ¯nd that tan ¯ ' 3.8 { 4.0. In obtaining the limit on tan ¯, we followed the following procedure. As inputs at MZ we chose [51] ®3(MZ) = 0:119; sin2 µW = 0:2312; ®(MZ) = 1 127:9 : (5.6) Using the central value of Mt = 174:3 GeV, we obtain the running mass mt(Mt) with the 2{loop QCD correction as [52] Mt mt(Mt) = 1 + 4 3 ®3(Mt) ¼ + 10:9 µ ®3(Mt) ¼ ¶2 : (5.7) Using 5{°avor SM QCD beta functions we extrapolated ®3(MZ) and obtained ®3(Mt) = 0.109. The top quark Yukawa coupling is then found to be (for Mt = 174.3 GeV) Y SM t (Mt) = 0.935 corresponding to mt(Mt) = 162:8 GeV. This coupling is then evolved from Mt to 1 TeV where we minimize the MSSM Higgs potential. Using standard model beta function we obtain Y SM t (1 TeV) = 0.851. The corresponding MSSM coupling is Yt(1 TeV) = Y SM t (1 TeV)= sin ¯ , which for tan ¯ ' 4:0 (justi¯ed a{posteriori) is Yt(1 TeV) = 0:824. The gauge couplings at 1 TeV are found to be g1(1 TeV) = 0:466; g2(1 TeV) = 0:642 and g3(1 TeV) = 1:098. With these values of couplings at 1 TeV we obtained Fig. 5.2. Uncertainties in the prediction for tan ¯ are estimated to be §0:5, arising from the error in top quark mass and from the precise scale at which the Higgs potential is minimized. We conclude that tan ¯ = 3.5{4.5 in this model. Since tan ¯ is ¯xed and since the At parameter is not free in AMSB, there is a nontrivial prediction for the lightest Higgs boson mass mh. We use the 2{loop 43 2.5 3 3.5 4 50 100 150 200 M2 (GeV) tanb Figure 5.2. Plot of tan ¯ as a function of M2 radiatively corrected expression for m2 h = (m2 h)o+¢m2 h, where (m2 h)o is the tree{level value of the mass and the radiative correction is given by [53] ¢m2 h = 3m4t 4¼2À2 · t + Xt + 1 16¼2 µ 3 2 m2t À2 ¡ 32¼®3(Mt) ¶ (2Xtt + t2) ¸ : (5.8) Here Xt = A~t 2 m2 ~t Ã 1 ¡ A~t 2 12m2 ~t ! ; A~t = At ¡ ¹ cot ¯; (5.9) and t =log( m2 ~t M2 t ), À = 174 GeV. m2 ~t is the arithmetic average of the diagonal entries of the squared stop mass matrix and At is the soft trilinear coupling associated with the top Yukawa coupling in the superpotential of Eq. (5.1). In these expressions, mt is the one{loop QCD corrected running mass, mt = Mt 1+4 3 ®3(Mt) ¼ , which equals 166.7 GeV for Mt = 174:3 GeV. We ¯nd that mh ' 113 GeV { 120 GeV, depending on the choice of Maux. The larger value mh ' 120 GeV is realized only for larger Mt ' 180 GeV. We list in Tables 5.2{5.4 the value of mh, along with the other sparticle masses. 44 5.3.2 SU(3)H symmetry breaking Let us now analyze the SU(3)H symmetry breaking sector of the potential. The potential V (©i) is given by: V (©i) = m2 Á(©y 1©1 + ©y 2©2 + ©y 3©3) + ·A· ³ ©®1 ©¯ 2©° 3²®¯° + c:c ´ + ·2 £ (©1©2)y(©1©2) + (©1©3)y(©1©3) + (©2©3)y(©2©3) ¤ + g2 4 8 X8 a=1 j©y 1¸a©1 + ©y 2¸a©2 + ©y 3¸a©3j2: (5.10) Here g4 is the gauge coupling of the SU(3)H, A· is the trilinear A{term corresponding to the coupling ·, m2 Á is the soft mass squared for the ©i ¯elds. These soft SUSY breaking parameters are given in the Appendix (Eqs. (A.17), (A.23)). The ·2 term in Eq. (5.10) is the Fterm contribution and the last term in Eq. (5.10) is the SU(3)H D{term with ¸a being the SU(3)H generators. The Higgs potential, Eq. (5.10), has an SU(3)H £ SU(3)G symmetry, with the ©i ¯elds (i = 1¡3) transforming as (¹3 ; 3). This allows for a vacuum which preserves an SU(3)H+G diagonal subgroup. The VEVs of the ©i ¯elds are then given by: h©1i = 0 BB@ u 0 0 1 CCA ; h©2i = 0 BB@ 0 u 0 1 CCA and h©3i = 0 BB@ 0 0 u 1 CCA : (5.11) Using these VEVs the potential becomes hV (©)i = 3m2 Áu2 + 3·2u4 + 2·A·u3: (5.12) Minimization of Eq. (5.12) leads to the condition u = ¡A· § q ¡8m2 Á + A2 · 4· : (5.13) The argument in the square root of Eq. (5.13), which should be positive for a consis tent symmetry breaking, is given by ¡8m2 Á + A2 · = M2 aux (16¼2)2 [15·4 + 56·2¸2 + 304¸4 ¡ 8·2g2 4 ¡ 32¸2g2 4]: (5.14) 45 Positivity of Eq. (5.14) leads to constraints on the parameters f¸; ·g. It can be shown that Eq. (5.14) implies 0 6 j·j 6 0:731g4 and 0 6 j¸j 6 0:324g4. Furthermore, positivity of the slepton masses, along with the experimental limit m2 slepton & (100 GeV)2, require g4 > 0:5. This essentially ¯xes the parameter space of the model. We get the right minimum by choosing the negative sign of the square root in Eq. (5.13) (for positive Maux), with this choice, all the Higgs masses{squared will be positive. Since the symmetry breaking chain is SU(3)H £ SU(3)G ! SU(3)H+G, we can classify the masses of all scalars and fermions as multiplets of SU(3)H+G. The complex ©(¹3; 3) scalar multiplet decomposes into 2 octets and two singlets of SU(3)H+G. One octet gets eaten by the Higgs mechanism. A physical octet remains in the spectrum with a mass given by M2 octet = ¡2·2u2 ¡ 2·uA· + g2 4u2: (5.15) There are two singlets, one scalar (Ás) and one pseudoscalar (Áp) with masses given by m2 Ás = 4·2u2 + ·uA·; (5.16) m2 Áp = ¡3·uA·: (5.17) In the fermionic sector, the octet Higgsino mixes with the octet gaugino with a mixing matrix given by M0 octet = Ã m4 g4u g4u ·u ! : (5.18) In addition, there is a Majorana fermion, a singlet of SU(3)H+G, with a mass of m~Á = 2·u: (5.19) Finally the gauge bosons form an octet with a mass MV = g4u: (5.20) 5.4 The SUSY spectrum We are now ready to discuss the full SUSY spectrum of the model. We will see that the tachyonic slepton problem is cured by virtue of the positive contribution from the SU(3)H gauge sector. 46 5.4.1 Slepton masses The slepton mass{squareds are given by the eigenvalues of the mass matrices (® = e; ¹; ¿ ) M2 ~l = Ã m2 ~L ® mE® ¡ AYE® ¡ ¹ tan ¯ ¢ mE® ¡ AYE® ¡ ¹ tan ¯ ¢ m2 ~ec ® ! : (5.21) Here m2 ~L ® = M2 aux (16¼2) · YE®¯(YE®) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + m2 E® + µ ¡ 1 2 + sin2 µW ¶ cos 2¯M2Z ; (5.22) m2 ~ec ® = M2 aux (16¼2) · 2YE®¯(YE®) ¡ µ 6 5 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + m2 E® ¡ sin2 µW cos 2¯M2Z : (5.23) The o® diagonal terms in Eq. (5.21) are rather small as they are proportional to the lepton masses. The SUSY soft masses are calculated from the RGE give in the Appendix. The last terms of Eqs. (5.22){(5.23) are the D{terms. Note the positive contribution from the SU(3)H gauge sector in Eqs. (5.22){(5.23), given by the term ¡8 3g4¯(g4). In our model g4 is asymptotically free with ¯(g4) = ¡ 3 16¼2 g3 4. This contribution makes the mass{squared of all sleptons to be positive for g4 > 0:5. The left handed sneutrino mass is given by m2 ~ºi = M2 aux (16¼2) · ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + 1 2 cos 2¯M2Z ; (5.24) where i = e; ¹; ¿ . 5.4.2 Squark masses The mixing matrix for the squark sector is similar to the slepton sector. The diagonal entries of the up and the down squark mass matrices are given by [27] m2 ~U i = (m2 soft) ~Q i ~Q i + m2 Ui + 1 6 ¡ 4M2W ¡M2Z ¢ cos 2¯; 47 m2 ~U c i = (m2 soft) ~U c i ~U c i + m2 Ui ¡ 2 3 ¡ M2W ¡M2Z ¢ cos 2¯; m2 ~D i = (m2 soft) ~Q i ~Q i + m2 Di ¡ 1 6 ¡ 2M2W +M2Z ¢ cos 2¯; m2 ~D c i = (m2 soft) ~D c i ~D c i + m2 Di + 1 3 ¡ M2W ¡M2Z ¢ cos 2¯: (5.25) Here mUi and mDi are quark masses of di®erent generations, i = 1, 2, 3. The squark soft masses are obtained from the RGE as (m2 soft) ~Q i ~Q i = M2 aux 16¼2 µ Yui¯(Yui) + Ydi¯(Ydi) ¡ 1 30 g1¯(g1) ¡ 3 2 g2¯(g2) ¡ 8 3 g3¯(g3) ¶ (5; .26) (m2 soft) ~U c i ~U c i = M2 aux 16¼2 µ 2Yui¯(Yui) ¡ 8 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ ; (5.27) (m2 soft) ~D c i ~D c i = M2 aux 16¼2 µ 2Ydi¯(Ydi) ¡ 2 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ : (5.28) 5.4.3 ´ fermion and ´ scalar masses The ¯elds ´ and ¹´ transform as (3; ¹3) and (¹3; 3) under SU(3)H £SU(3)G. After symmetry breaking, ´ and ¹´ both transform as 8 + 1 of the diagonal SU(3)H+G. The octet from ´ mixes with the octet from ¹´, and similarly for the singlets. In the fermionic sector, the octet and the singlet mass matrices are given by M´ octet = Ã ¡2¸u M´ M´ 0 ! ; (5.29) M´ singlet = Ã 4¸u M´ M´ 0 ! : (5.30) In the scalar sector, there are 4 real octets and 4 real singlets from ´ and ¹´ ¯elds. The two scalar octets are mixed, as are the two pseudoscalar octets. The mass squared matrices for the octet are M2 s¡octet = Ã ( ~m2 soft)´´ +M2 ´ + 2¸u(¡A¸ ¡ ·u + 2¸u) M´(B´ ¡ 2¸u) M´(B´ ¡ 2¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (;5.31) M2 p¡octet = Ã ( ~m2 soft)´´ +M2 ´ + 2¸u(A¸ + ·u + 2¸u) ¡M´(B´ + 2¸u) ¡M´(B´ + 2¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (:5.32) 48 The singlet scalar mass matrices are M2 s¡singlet = Ã ( ~m2 soft)´´ +M2 ´ + 4¸u(A¸ + ·u + 4¸u) M´(B´ + 4¸u) M´(B´ + 4¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! ; (5.33) M2 p¡singlet = Ã ( ~m2 soft)´´ +M2 ´ ¡ 4¸u(A¸ ¡ ·u ¡ 4¸u) ¡M´(B´ ¡ 4¸u) ¡M´(B´ ¡ 4¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (5:.34) The soft masses ( ~m2 soft)´´ and ( ~m2 soft)´´ are given in Eqs. (61){(62) of the Appendix. 5.5 Numerical results We are now ready to present our numerical results for the SUSY spectrum. The scale of SUSY breaking, Maux, should be in the range 40{120 TeV for the MSSM particles to have masses in the range 100 GeV { 2 TeV. Note that there is a large hierarchy in the masses of the gluino and the neutral Wino, M3 M2 ' 7:1 (after taking account of radiative corrections), in AMSB models. Furthermore the lightest chargino is nearly mass degenerate with the neutral Wino, so M2 & 100 GeV is required to satisfy the LEP chargino mass bound. The SU(3)H gauge coupling g4 is chosen so that the sleptons have positive mass squared (g4 > 0:5). We allow g4 to take two values, g4 = 0.55 (Tables 5.2 and 5.4) and g4 = 1.0 (Table 5.3). Symmetry breaking considerations constrain the couplings · and ¸ as discussed in Sec. 5.3 after Eq. 5.14. In Tables 5.2 and 5.4 we have taken Maux = 47.112 TeV corresponding to a light spectrum, while in Table we have Maux = 66.695 TeV with an intermediate spectrum. Other input parameters are listed in the respective Table captions. The mass parameter M´ cannot be much larger than 1 TeV, as that would decouple the e®ects of ´, ¹´ ¯elds which are needed for consistent symmetry breaking. We see from Table 5.2 that the lightest Higgs boson mass is mh ' 118 GeV. This is very close to the current experimental limit. If Mt = 180 GeV is used (instead of Mt = 176 GeV), for the same set of input parameters, mh will be 119 GeV. mh being close to the current experimental limit is a generic prediction of our framework. It holds in the spectra of Tables 5.3 and as well. We conclude that mh . 120 GeV in this model. 49 The masses of the sleptons will depend sensitively on the choice of g4. The sleptons are relatively light, mslep . 300 GeV, with g4 = 0:55, while they are heavy, mslep ' 800 GeV, when g4 = 1:0. Note however that there is a correlation in the slepton masses and the SU(3)H gauge boson masses (MV ), with the lighter sleptons corresponding to lighter SU(3)H gauge bosons. It is worth noting that very light sleptons, below the current experimental limits of about 100 GeV, would be incon sistent with the limits on MV arising from e+e¡ ! ¹+¹¡ type processes (see Sec. 6). Note also that the left{handed and the right{handed sleptons are nearly degenerate to within about 10 GeV in this model. This a numerical coincidence having to do with the values of g1 and g2 and the MSSM beta functions (see the last paper of Ref. [39]). The new SU(3)H gauge boson contributions to the slepton masses are the same for the left{handed and the right{handed sleptons. In Tables 5.2{5.4 we have included the leading radiative corrections to the gaugino masses M1, M2 and M3 [54]. Including these radiative corrections we ¯nd (in Table 2) M1 : M2 : M3 = 3:0 : 1 : 7:4. The lightest SUSY particle (LSP) is the neutral Wino, which is nearly mass degenerate with the charged Wino. In Tables 5.2{5.4 the mass splitting is about 60 MeV, but this does not take into account SU(2)L £U(1)Y breaking corrections [55]. These electroweak radiative corrections turn out to be very important, and we ¯nd mÂ§ 1 ¡ mÂ01 ' 235 MeV (with about 175 MeV arising from SU(2)L £ U(1)Y breaking e®ects). The decay Â§ 1 ! Â01 + ¼§ is then kinematically allowed, with the ¼§ being very soft. Once produced, the neutralino Â01 will escape the detector without leaving any tracks. With the decay channel Â§ 1 ! Â01 +¼§ open, the lightest chargino will leave an observable track with a decay length of about a few cm. Search strategies for such a quasi{degenerate pair at colliders have been analyzed in Ref. [54; 56; 57]. In the SU(3)H sector, in Tables 5.2{5.4, the horizontal gauge boson has a mass of 1.5{4.0 TeV. The heavy Higgs bosons, Higgsinos, gauginos, squarks and the ´ ¯elds all have masses . (1 ¡ 2) TeV. 50 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:146; 0:431g Neutralinos fm~Â03 ; m~Â04 g f0:876; 0:878g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:146; 0:880g Gluino M3 1:064 Higgs bosons fmh; mH; mA; mH§g f0:118; 0:878; 0:877; 0:880g R.H sleptons fm~eR; m~¹R; m~¿1g f0:183; 0:183; 0:166g L.H sleptons fm~eL; m~¹L; m~¿2g f0:190; 0:190; 0:203g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:175; 0:175; 0:175g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:017; 1:017; 1:015g L.H down squarks fm~ dL ; m~sL; m~b 2 g f1:008; 1:008; 0:885g R.H up squarks fm~uR; m~cR; m~t1g f1:011; 1:011; 0:669g L.H up squarks fm~uL; m~cL; m~t2g f1:005; 1:005; 0:979g New Particles Symbol Mass (TeV) SU(3)H Gauge boson octet MV 2:213 Singlet Higgsino m~Á 0:402 Octet Higgsino/gaugino m~Á1;2 f1:978; 2:450g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:179; 0:624; 2:253g Fermionic ´ (octet) moctet ´1;2 f0:676; 1:480g Fermionic ´ (singlet) msinglet ´1;2 f0:479; 2:089g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:454; 1:703g Pseudoscalar ´ Higgs (octet) mp¡octet ~´1;2 f0:908; 1:259g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:717; 1:868g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:264; 2:310g TABLE 5.2. Sparticle masses for the choiceMaux = 47:112 TeV, tan ¯ = 3:785, ¹ = ¡0:873 TeV, yb = 0:068, ¸ = 0:1, · = 0:05, g4 = 0:55, u = ¡4:024 TeV, M´ = 1:0 TeV and Mt = 0:176 TeV. 51 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:198; 0:586g Neutralinos fm~Â03 ; m~Â04 g f1:179; 1:181g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:198; 1:182g Gluino M3 1:410 Higgs boson fmh; mH; mA; mH§g f0:119; 1:179; 1:178; 1:181g R.H sleptons fm~eR; m~¹R; m~¿1g f0:245; 0:245; 0:227g L.H sleptons fm~eL; m~¹L; m~¿2g f0:254; 0:254; 0:267g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:242; 0:242; 0:242g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:373; 1:373; 1:193g L.H down squraks fm~ dL ; m~sL; m~b 2 g f1:361; 1:361 1:370g R.H up squarks fm~uR; m~cR; m~t1g f1:365; 1:365; 0:940g L.H up squraks fm~uL; m~cL; m~t2g f1:359 1:359; 1:276g New Particles Symbol Mass (TeV) SU(3)H Gauge boson octet MV 1:871 Singlet Higgsino m~Á 0:544 Octet Higgsino/gaugino m~Á1;2 f1:553; 2:191g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:247; 0:840; 1:955g Fermionic ´ (octet) moctet ´1;2 f0:716; 1:397g Fermionic ´ (singlet) msinglet ´1;2 f0:529; 1:890g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:421; 1:699g Pseudoscalar ´ Higgs (octet) mp¡octet ~´1;2 f1:031; 1:098g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:850; 1:593g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:247; 2:189g TABLE 5.3. Sparticle masses for the choice Maux = 63:695 TeV, tan ¯ = 4:02, ¹ = ¡1:178 TeV, yb = 0:0719, ¸ = 0:1, · = 0:08, g4 = 0:55, u = ¡3:402 TeV, M´ = 1:0 TeV and Mt = 0:1743 TeV. 52 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:148; 0:436g Neutralinos fm~Â03 ; m~Â04 g f0:876; 0:878g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:148; 0:878g Gluino M3 1:064 Higgs boson fmh; mH; mA; mH§g f0:118; 0:878; 0:877; 0:880g R.H sleptons fm~eR; m~¹R; m~¿1g f0:825; 825; 0:821g L.H sleptons fm~eL; m~¹L; m~¿2g f0:827; 0:827; 0:830g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:823; 0:823; 0:823g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:017; 1:017; 1:015g L.H down squraks fm~ dL ; m~sL; m~b 2 g f1:008; 1:008; 0:885g R.H up squarks fm~uR; m~cR; m~t1g f1:011; 1:011; 0:669g L.H up squraks fm~uL; m~cL; m~t2g f1:005; 1:005; 0:979g New Particles Symbol Mass (TeV) SU(3)HGauge boson octet MV 3:779 Singlet Higgsino m~Á 1:058 Octet Higgsino/gaugino m~Á1;2 f3:071; 4:495g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:465; 1:646; 3:940g Fermionic ´ (octet) moctet ´1;2 f0:254; 2:521g Fermionic ´ (singlet) msinglet ´1;2 f0:137; 4:672g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:588; 3:090g Pseudoscalar ´ Higgs (octet) mp¡0ctet ~´1;2 f1:058; 1:952g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:964; 4:116g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:711; 5:224g TABLE 5.4. Sparticle masses for the choiceMaux = 47:112 TeV, tan ¯ = 3:785, ¹ = ¡0:873 TeV, yb = 0:068, ¸ = 0:3, · = 0:14, g4 = 1:0, u = ¡3:779 TeV, M´ = 0:800 TeV and Mt = 0:176 TeV. 53 5.6 Experimental signatures The Lightest SUSY particle in the model is the neutral Wino (Â01 ) which is nearly mass degenerate with the lightest chargino (Â§ 1 ), with a mass splitting of about 235 MeV. At the Tevatron Run 2 as well as at the LHC, the process p¹p (or pp) ! Â01 +Â§ 1 will produce these SUSY particles. Naturalness suggest that mÂ01 , mÂ§ 1 . 300 GeV (corresponding to mgluino . 2 TeV). Strategies for detecting such a quasi{degenerate pair has been carried out in Ref. [54; 56; 57]. In the MSSM sector our model predicts tan ¯ ' 4:0 and mh . 120 GeV, both of which can be tested at the LHC. If the SU(3)H gauge coupling g4 takes small values (g4 ' 0:55), the slepton masses will be near the current experimental limit. For larger values of g4 (' 1:0) the slepton masses are comparable to those of the squarks. The SU(3)H gauge boson masses are in the range MV = 1:5 ¡ 4:0 TeV. Al though relatively light, these particles do not mediate leptonic FCNC, owing to the approximate SU(3)H+G global symmetries present in the model. The most stringent constraint on MV arises from the process e+e¡ ! ¹+¹¡. LEP II has set severe constraints on lepton compositeness [51; 58] from this process. We focus on one such amplitude, involving all left{handed lepton ¯elds. In our model, the e®ective Lagrangian for this process is Leff = ¡ 2g2 4 3M2 V (e¹L°¹eL)(¹¹L°¹¹L): (5.35) Comparing with ¤¡ LL(ee¹¹) > 6:3 TeV [51; 58], we obtain MV g4 ¸ 2:05 TeV. For g4 = 0:55 (1:0) this implies MV > 1.129 (2.052) TeV. From Tables 5.2{5.4 we ¯nd that these constraints are satis¯ed. The model as it stands has an unbroken Z2 symmetry (in addition to the usual R{parity) under which the super¯elds ´; ¹´ are odd and all other super¯elds are even. If this symmetry is exact, the lightest of the ´ and ¹´ ¯elds (a pseudoscalar singlet Higgs in the ¯ts of Tables 5.2{5.3 and a singlet fermion in Table 5.4) will be stable. We envision this Z2 symmetry to be broken by higher dimensional terms of the type L®Hu©®¹´¯©¯=¤2. Such a term will induce the decay ´p¡singlet 1 ! L+Â01 with a lifetime less than 1 second for ¤ · 109 GeV. This would make these ´ particles cosmologically 54 safe. It may be pointed out that the same e®ective operator, along with a TeV scale mass for the ´ ¯elds, can provide small neutrino masses even in the absence of the operators given in Eq. 5.3. 5.7 Origin of the ¹ term Any satisfactory SUSY breaking model should also have a natural explanation for the ¹ term (the coe±cient of HuHd term in Eq. (5.1)). In gravity mediated SUSY breaking models, there are at least three solutions to the ¹ problem. The Giudice{ Masiero mechanism [59] which explains the ¹ term through the Kahler potential R HuHdZ¤d4µ=Mpl is not readily adaptable to the AMSB framework. The NMSSM extension which introduces singlet ¯elds can in principle provide a natural explanation of the ¹ term in the AMSB scenario. We have however found that replacing ¹HuHd by the term SHuHd in the superpotential alone can not lead to realistic SUSY breaking. It is possible to make the NMSSM scenario compatible with symmetry breaking in the AMSB framework by introducing a new set of ¯elds which couple to the singlet S. We do not follow this non{minimal alternative here. There is a natural explanation for the ¹ parameter in the context of AMSB mod els, as suggested in Ref. [36]. It assumes a Lagrangian term L ¾ ® R d4µ (§+§y) MPl HuHd©y©, where § is a hidden sector ¯eld which breaks SUSY and © is the compensator ¯eld. After a rescaling, Hu ! ©Hu, Hd ! ©Hd, this term becomes L ¾ ® R d4µ (§+§y) MPl HuHd ©y © , which generates a ¹ term in a way similar to the Giudice{ Masiero mechanism [59]. The B¹ term is induced only through the super{Weyl anomaly and has the form given in Eq. (4.9). Our predictions for tan ¯ and mh depend sensitively on this assumption. We now point out that the ¹ term may have an alternative explanation in the context of AMSB models. This is obtained by promoting ¹HuHd in the superpotential to the following [60]: W0 = aHuHdS2 MPl + bS2 ¹ S2 MPl : (5.36) 55 Here S and ¹ S are standard model singlet ¯elds. Including AMSB induced soft pa rameters for these singlets (which can arise in a variety of ways), this superpotential will have a minimum where hSi ' ¹ S ® ' p MSUSYMPl. This would induce ¹ term of order MSUSY , as needed. From the e®ective low energy point of view, the superpo tential will appear to have an explicit ¹ term. The B term will have a form as given in Eq. 4.9. 5.8 Summary In this chapter we have suggested a new scenario for solving the tachyonic slep ton mass problem of anomaly mediated SUSY breaking models. An asymptotically free SU(3)H horizontal gauge symmetry acting on the lepton super¯elds provides positive masses to the sleptons. The SU(3)H symmetry must be broken at the TeV scale. Potentially dangerous FCNC processes mediated by the SU(3)H gauge bosons are shown to be suppressed adequately via approximate global symmetries that are present in the model. Our scenario predicts mh . 120 GeV for the lightest Higgs boson mass of MSSM and tan ¯ ' 4.0. The lightest SUSY particle is the neutral Wino which is nearly degenerate with the lightest chargino and is a candidate for cold dark matter. The full spectrum of the model is given in Tables 5.2{5.4 for various choices of input parameters. The very few parameters of our model are highly constrained by the consistency of symmetry breaking. CHAPTER 6 SU(2)H Horizontal Symmetry as a Solution to the Slepton Mass Problem of Anomaly Mediation 6.1 Introduction Family symmetries may give a positive mass{squared contribution to the slep tons in AMSB. The simplest of such symmetry is an SU(2)H non{Abelian symmetry. This symmetry when acting on leptons only can be asymptotically free, hence their beta{function will be negative. This is very important because with this new sym metry, the sleptons enjoys the same freedom as the quarks and hence can solve the negative slepton mass problem of AMSB. The quarks are singlet of SU(2)H but it is possible that they transform under a di®erent SU(2)q H symmetry, so that there is an underlying quark{lepton symmetry. Here we will focus on a model where quarks carry no family symmetry. In this chapter we suggest and investigate the possibility of solving the negative slepton mass problem of AMSB using this SU(2)H symmetry broken at the TeV scale. The leptons of the ¯rst two families transform as a doublet of SU(2)H and those of the third family transform as singlet under this new symmetry. The sleptons of the ¯rst two family gets a large positive contribution to their soft masses from the SU(2)H gauge sector. With e and ¹ forming a doublet of SU(2)H, an important issue is how to split their masses, since in Nature me 6= m¹. We introduce two new vector{like ¯elds that couples to the third family which will help to achieve me 6= m¹. The model is quite predictive. The LSP is the Wino which is nearly mass degenerate with the chargino. The lightest Higgs boson mass is predicted to be mh . 135 GeV, and the parameter tan ¯ is found to be tan ¯ ' 40. This model 56 57 is completely di®erent from the previous model because it also predicts a di®erent mass hierarchy for the ~e; ~¹ and ~¿ . In particular m~e; m~¹ and m~¿ are all quite di®erent, which is a characteristic signature of this model. In addition, this model can easily be tested at the LHC by direct discovery of the gauge bosons associated with SU(2)H. The plan of the chapter is as follows. In section 6.2 we introduce our model. In section 6.3 we analyze the Higgs potential. The SUSY spectrum is presented in section 6.4. We discuss our numerical results in section 6.5. In section 6.6 we discuss the experimental implication of the model. We summarize in section 6.7. 6.2 SU(2)H horizontal symmetry We de¯ne the gauge group symmetry of the model as GH ´ SU(3)C £ SU(2)L £ U(1)Y £ SU(2)H; where SU(2)H is a horizontal symmetry that acts on the ¯rst two families of leptons. The third family is a singlet under this new SU(2)H symmetry. A pair of vector like leptons, E, Ec, which are SU(2)H singlets are needed to ensure me 6= m¹. The spectrum of the model is listed in Table. 6.1. The gauge group SU(2)H de¯ned above is asymptotically free (¯ function is given in Eq. B.20) with this spectrum. The superpotential of the model consistent with the gauge symmetries reads W = (Yu)ij QiHuucj + (Yd)ij QiHddcj + fe¹Ã®Ãc® Hd + f¿Ã¿ ¿ cHd + f¿EÃ¿EcHd + feEEÃcÁu + ¹HuHd + ¹0ÁuÁd +MEEEc (6.1) It turns out that there is a Z4 symmetry present in the Lagrangian, under which Áu ! iÁu; Ád ! ¡iÁd; E ! ¡iE; Ec ! iEc; Ã¿ ! ¡iÃ¿ ; ¿ c ! i¿ c: This Z4 symmetry forbids the term EÃcÁd, which will be important to de¯ne an unbroken muon number. Since SU(2)H is broken at TeV, the gauge bosons of SU(2)H can potentially lead to large FCNC processes. The most dangerous of these are in the muon sector, eg; ¹ ! 3e. Such process are forbidden by an unbroken muon number, making TeV scale horizontal symmetry phenomenologically consistent. 58 Super¯eld SU(3)C SU(2)L U(1)Y SU(2)H Qi 3 2 1 6 1 uci ¹3 1 ¡2 3 1 dci ¹3 1 1 3 1 Ã® 1 2 ¡1 2 2 Ãc® 1 1 +1 2 Ã¿ 1 2 ¡1 2 1 ¿ c 1 1 +1 1 Hu 1 2 1 2 1 Hd 1 2 ¡1 2 1 Áu 1 1 0 2 Ád 1 1 0 2 E 1 1 ¡1 1 Ec 1 1 +1 1 ªN 1 1 0 2 TABLE 6.1. Particle content and charge assignment of the model. The indices i and ® take values i = 1 ¡ 3 and ® = 1 ¡ 2. In the model, the Ã® and Ãc® ¯elds contain the ¯rst two family of leptons (e and ¹) which transforms as a doublet under the SU(2)H gauge group, while the members of the third family (Ã¿ and ¿ c) transform as singlets under the SU(2)H gauge group. The ¯eld ªN, which transforms as a doublet under SU(2)H and as singlet under the SM gauge group, is introduced in order to cancel the Witten anomaly. The neutinos in the model get masses from the following nonrenormalizable operators: Ã¿Ã¿ HuHd M ; Ã®Ã® HuHu M03 Áu;dÁu;d; Ã®Ã¿ HuHu M00 Áu;d: (6.2) These terms will lead to a consistent neutrino oscillation phenomenology. 59 6.3 Symmetry breaking The symmetry breaking is achieved in the form GH ! GSM ! SU(3)C £ U(1)EM; where GSM ´ SU(3)C £ SU(2)L £ U(1)Y . The model has the possibility to be consistent with the known low energy physics. The new Higgs multiplets Áu and Ád are su±cient to break the GH ! GSM completely near the TeV scale. The tree level Higgs potential can be written as V (Hu;Hd; Áu; Ád) = (m2 Hu + ¹2)jHuj2 + (m2 Hd + ¹2)jHdj2 + B¹(HuHd + c.c.) + (g2 2 + g2 1) 8 (jHuj2 ¡ jHdj2)2 + g2 2 2 jHuHdj2 + g2 4 8 (jÁuj2 ¡ jÁdj2)2 + g2 4 2 jÁuÁdj2 + (m2 Áu + ¹02)jÁuj2 + (m2 Ád + ¹02)jÁdj2 + B0¹0(ÁuÁd + c.c.): The soft masses m2 Hu and m2 Hd ;m2 Áu and m2 Ád parameters are determined in terms of the single parameter Maux. The B and B0 parameters are taken to be free in the model but in some special class of models, they are determined also by the same mass parameter Maux. Upon symmetry breaking, the Higgs ¯elds acquire VEV's hHui = Ã 0 Àu ! ; hHdi = Ã Àd 0 ! ; hÁui = Ã 0 uu ! ; hÁdi = Ã ud 0 ! : (6.3) It is desired that the VEVs obey hÁui; hÁdi À hHui; hHdi, in order for the symmetry breaking to be consistent. Minimization of the Higgs potential V (Hu;Hd; Áu; Ád) gives sin 2¯ = ¡2B¹ 2¹2 + m2 Hu + m2 Hd ; ¹2 = m2 Hd ¡ m2 Hu tan2 ¯ tan2 ¯ ¡ 1 ¡ M2Z 2 ; (6.4) sin 2¯0 = ¡2B0¹0 2¹02 + m2 Áu + m2 Ád ; ¹02 = m2 Ád ¡ m2 Áu tan2 ¯0 tan2 ¯0 ¡ 1 ¡ M2Z 0 2 ; (6.5) where we have introduced the notation uu = u sin ¯0, ud = u cos ¯0, u2 = u2 u + u2 d, tan ¯0 = uu ud and M2Z 0 = g2 4 2 (u2 u + u2 d). MZ0 is the mass of the gauge boson associated with the SU(2)H. 60 To ¯nd the physical Higgs boson mass, we parameterize the Higgs ¯elds (in the unitary gauge) as Hu = Ã H+ sin ¯ Àu + p1 2 (Á2 + i cos ¯ Á3) ! ; hHdi = Ã Àd + p1 2 (Á1 + i sin ¯ Á3) H¡ cos ¯ ! ; Áu = Ã Á+ sin ¯0 uu + p1 2 (Á4 + i cos ¯0 Á5) ! ; Ád = Ã ud + p1 2 (Á6 + i sin ¯0 Á5) Á¡ cos ¯0 ! : (6.6) The Higgs masses are obtained by expanding the Higgs potential and keeping only terms quadratic in the ¯elds. The masses of the CP{odd Higgs bosons fÁ3; Á5g are m2 A = ¡2B¹ sin 2¯ ; m2 A0 = ¡ 2B0¹0 sin 2¯0 : (6.7) The mass matrices for the CP{even neutral Higgs bosons fÁ1; Á2g and fÁ4; Á6g are decoupled. They are given by (M2)cp¡even = Ã m2 A cos2 ¯ +M2Z sin2 ¯ ¡fm2 A +M2Z gsin 2¯ 2 ¡fm2 A +M2Z gsin 2¯ 2 m2 A sin2 ¯ +M2Zsin2 ¯ ! ; (6.8) (M02)cp¡even = Ã m2 A0 cos2 ¯0 +M02 Z sin2 ¯0 ¡fm2 A0 +M02 Z gsin 2¯0 2 ¡fm2 A0 +M02 Z gsin 2¯0 2 m2 A0 sin2 ¯0 +M02 Z sin2 ¯0 ! : (6.9) Finally, the charged Higgs boson mass (H§ and Á§) is given by m2 H§ = m2 A +M2W m2 Á§ = m2 A0 +M2Z 0 (6.10) Á§ are electrically neutral, they are \charged" under SU(2)H. The Majorana mass matrix of the neutralinos f~B ; ~W 3; H~0 d ; H~0 u; B~H; Á~0 d; Á~0 ug is M(0) = 0 BBBBBBBBBBBB@ M1 0 ¡pÀd 2 g1 pÀu 2 g1 0 0 0 0 M2 pÀd 2 g2 ¡pÀu 2 g2 0 0 0 ¡pÀd 2 g1 pÀd 2 g2 0 ¡¹ 0 0 0 pÀu 2 g1 ¡pÀu 2 g2 ¡¹ 0 0 0 0 0 0 0 0 M4 pud 2 g4 ¡puu 2 g4 0 0 0 0 pud 2 g4 0 ¡¹0 0 0 0 0 ¡puu 2 g4 ¡¹0 0 1 CCCCCCCCCCCCA ; (6.11) 61 where M1; M2 andM4 are the gaugino masses for U(1)Y ; SU(2)L and SU(2)H which are listed in Appendix B. The physical neutralino masses m~Â0i (i =1{7) are obtained as the eigenvalues of this mass matrix Eq. (6.11). In the basis f ~W +; ~H + u g, f ~W ¡; ~H ¡ d g, the chargino (Dirac) mass matrix is M(c) = Ã M2 g2Àd g2Àu ¹ ! : (6.12) Similarly, for the SU(2)H sector, we have ~M (c) = Ã M4 g4ud g4uu ¹0 ! : (6.13) The three SU(2)H gauge boson masses are given by M2 V = g2 4 2 (u2 u + u2 d): (6.14) 6.3.1 Lepton masses Now we describe brie°y how to obtain the masses of the ordinary leptons. We have introduced E and Ec ¯elds in the superpotential Eq. (6.1) for the purpose of breaking e ¡ ¹ degeneracy. These new ¯elds mix with the usual leptons leading to the mass matrix ( e ¹ ¿ E ) 0 BBBBB@ f¹Àd 0 0 0 0 f¹Àd 0 0 0 0 f¿Àd f¿EÀd feEuu 0 0 ME 1 CCCCCA 0 BBBBB@ ec ¹c ¿ c Ec 1 CCCCCA : (6.15) The muon ¯eld completely decouples with mass m¹ = f¹Àd: (6.16) We are then left with a 3 £ 3 mass matrix for the e, ¿ and E ¯elds. The eigenvalue equation can be easily solved using the hierarchy me ¿ m¿ ¿ mE and the result is m¿ ' f¿Àd s f1 + f2 ¿Ef2 eE f2 ¿ u2 u M2E + f2 eEu2 u g; me ' q m¹ME M2E + f2 eEu2 u + f2 ¿Ef2 eEu2 u f2 ¿ ; mE ' q M2E + f2 eEu2 u: (6.17) Note that me 6= m¹, showing consistency of the model. 62 6.4 The SUSY spectrum We will show in this section that the tachyonic slepton problem is cured by virtue of the positive contribution from the SU(2)H gauge sector to the masses for the ¯rst two family and a large Yukawa coupling for the third family. 6.4.1 Slepton masses The slepton masses are given by a 2 £ 2 mass matrix for the smuon (since it decouples) and a 6 £ 6 mass matrix for the e; ¿;E; ec; ¿ c;Ec ¯elds. The smuon mass{squareds are given by the eigenvalues of the mass matrix M2 ~¹ = Ã m2 ~¹ m¹ ¡ Afe¹ ¡ ¹ tan ¯ ¢ m¹ ¡ Afe¹ ¡ ¹ tan ¯ ¢ m2 ~¹c ! ; (6.18) where the diagonal entries are m2 ~¹ = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + m2 ¹ + g2 4 4 (u2 u ¡ u2 d); m2 ~¹c = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 6 5 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + m2 ¹ + g2 4 4 (u2 u ¡ u2 d): Note that the positive contributions from the SU(2)H gauge sector are provided by the term ¡3 2g4¯(g4), with gauge beta function ¯(g4) = ¡ 3 16¼2 g3 4. This contribution ensures that the mass{squareds of all sleptons are positive when g4 > 0:9. It is important to point out that the SU(2)H D{term contributions to the diagonal entries of the mass matrix Eq. (6.18) can either be positive or negative but it must be such that its overall contribution is rather small compared to the soft mass term. The mass matrix for the other sleptons is in the form 0 BB@ m2 ~e 0 fe¹feEÀduu fe¹(Ae¹Àd + ¹Àu) 0 0 0 m2 ~¿ MEf¿EÀd 0 f¿ (A¿ Àd + ¹Àu) f¿E(A¿EÀd + ¹Àu) fe¹feEÀduu MEf¿EÀd m2 ~E feE(AeEuu + ¹0ud) 0 MEBE fe¹(Ae¹Àd + ¹Àu) 0 feE(AeEuu + ¹0ud) m2 e~c 0 MEfeEuu 0 f¿ (A¿ Àd + ¹Àu) 0 0 m2 ¿~c f¿ f¿EÀ2d 0 f¿E(A¿EÀd + ¹Àu) MEBE MEfeEuu f¿ f¿EÀ2d m2 E~c 1 CCA ; 63 where the diagonal entries of this mass matrix read m2 ~e = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + f2 e¹À2 d + g2 4 4 (u2 d ¡ u2 u); m2 ~ec = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 6 5 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + f2 e¹À2 d + f2 eEu2 u + g2 4 4 (u2 d ¡ u2 u) m2 ~¿ = M2 aux (16¼2) · f¿¯(f¿ ) + f¿E¯(f¿E) ¡ µ 3 10 g1¯(g1) + 3 2 g2¯(g2) ¶¸ + (f2 ¿ + f2 ¿E)À2 d; m2 ~¿c = M2 aux (16¼2) · 2f¿¯(f¿ ) ¡ µ 6 5 g1¯(g1) ¶¸ + f2 ¿ À2 d m2 ~E = M2 aux (16¼2) · feE¯(feE) ¡ µ 6 5 g1¯(g1) ¶¸ + m2 E + f2 eEu2 u; m2 ~E c = M2 aux (16¼2) · f¿e¯(fe¿ ) ¡ µ 6 5 g1¯(g1) ¶¸ + m2 E + f2 ¿EÀ2 d (6.19) The requirement that the slepton masses are positive puts constraints on the couplings f¿ ; feE; f¿e and g4. 6.4.2 Squark masses The mixing matrix for the squark sector is similar to the slepton sector, except that they receive no SU(2)H gauge contributions. The diagonal entries of the up and the down squark mass matrices are given by [61] m2 ~U i = (m2 soft) ~Q i ~Q i + m2 Ui + 1 6 ¡ 4M2W ¡M2Z ¢ cos 2¯; m2 ~U c i = (m2 soft) ~U c i ~U c i + m2 Ui ¡ 2 3 ¡ M2W ¡M2Z ¢ cos 2¯; m2 ~D i = (m2 soft) ~Q i ~Q i + m2 Di ¡ 1 6 ¡ 2M2W +M2Z ¢ cos 2¯; m2 ~Dc i = (m2 soft) ~D c i ~D c i + m2 Di + 1 3 ¡ M2W ¡M2Z ¢ cos 2¯; (6.20) were mUi and mDi are the quark masses of the di®erent generations with i = 1, 2, 3. The squark soft masses are obtained from the RGE as (m2 soft) ~Q i ~Q i = M2 aux 16¼2 µ Yui¯(Yui) + Ydi¯(Ydi) ¡ 1 30 g1¯(g1) ¡ 3 2 g2¯(g2) ¡ 8 3 g3¯(g3) ¶ (6; .21) 64 (m2 soft) ~U c i ~U c i = M2 aux 16¼2 µ 2Yui¯(Yui) ¡ 8 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ ; (6.22) (m2 soft) ~D c i ~D c i = M2 aux 16¼2 µ 2Ydi¯(Ydi) ¡ 2 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ : (6.23) 6.5 Numerical results Here we present our numerical results for the SUSY spectrum. We ¯rst per formed a one{loop accuracy numerical analysis to determine the sparticle and Higgs Spectrum. For experimental inputs for the SM gauge couplings we use the same pro cedure Ref. [61] for the g1; g2; g3 with the central value of the top mass taken to be Mt = 174:3 GeV. In the model presented, the scale of SUSY breaking, Maux should be in the range 40¡100 TeV for the MSSM particles to have masses in the range 0:1¡2 TeV. The gauge coupling g4 ¸ 0:9 in order for the slepton masses for the ¯rst two families to be positive and in the right range. Since the positivity of the mass{squared of the slepton of the third family depends on the Yukawa couplings, we ¯nd that the couplings should obey f¿ ; f¿E ¸ 0:5. For a speci¯c choice of parameters (Table. 6.2), we ¯nd the m¹1 ; m¹2 » 800 GeV for the smuon. There is a signi¯cant mass splitting between the selectron and the stau. The lightest of the sleptons is the lefthanded stau. If SUSY is discovered with a large mass hierarchy between the stau and the selectron (or smauon), this model will be a good candidate. The lightest Higgs mass is found to be around 128 GeV which is consistent with current experimental limit. The lightest supersymmetric particle is Wino which is nearly mass degenerate with the lighter chargino of the SM. The SU(2)H gauge boson mass is found to be » 1:4 TeV. The heavy Higgs bosons, Higgsinos and squarks masses are in the range 0:7 ¡ 2:0 TeV. 65 Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 ; m~Â03 ; m~Â04 g f0:176; 274; 0:726; 1:080g Neutralinos fm~Â05 ; m~Â06 ; m~Â07 g f1:091; 1:096; 2:097g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:176; 1:094g Charginos (SU(2)H) fm~Â§ 1 ; m~Â§ 2 g f1:070; 2:102g Gluino M3 1:556 Neutral Higgs bosons fmh; mH; mAg f0:128; 0:922; 0:922g Neutral Higgs bosons fmh0 ; mH0 ; mA0g f0:143; 2:075; 1:554g Charged Higgs bosons mH§ 0:925 Charged Higgs bosons SU(2)H mH§ 2:080 R.H smuon fm~¹1g f0:867g L.H smuon fm~¹Lg f0:796g R.H sleptons fm~eR; m~¿1 ; mERg f0:947; 0:176; 0:758g L.H sleptons fm~eL; m~¿2 ; m~¹Lg f1:904; 0:533; 0:401g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:464; 1:464; 1:369g L.H down squarks fm~ dL ; m~sL; m~b 2 g f1:451; 1:451; 1:115g R.H up squarks fm~uR; m~cR; m~t1 g f1:454; 1:454; 1:107g L.H up squarks fm~uL; m~cL; m~t2 g f1:449; 1:449; 1:295g SU(2)H gauge boson M0Z 1:382 TABLE 6.2. Sparticle masses in Model 1 for the choice Maux = 67:956 TeV, yb = 0:8, f¿ = 0:53, feE = 1:2, f¿E = 0:51, g4 = 1:0, ME = 10:4 TeV and Mt = 0:174 TeV, u = 1:955 TeV, tan ¯ = 57:4, tan ¯0 = 0:87, ¹ = 1:088 TeV, ¹0 = 0:276 TeV, B = 0:014 TeV, B0 = 4:336 TeV, BE = 0:009 TeV. 66 6.6 Other experimental implications The lightest SUSY particle in the model we considered is the wino ( ~ Â01 ) which is nearly mass degenerate with the chargino. This particle is stable and can be a candidate for cold dark matter. The model predicts the lightest Higgs mass mh · 135 GeV which can be tested at the LHC. Because the SU(2)H gauge bosons do not mix with the SM gauge bosons, elec troweak precision data remains unchanged. Also the second family of leptons do not mix with the ¯rst and third family, this is because of the Z4 symmetry present in the model. The processes ¹ ! 3e and ¹ ! e° are not a problem in the model. The SU(2)H gauge boson masses are degenerate with mass MV = 1:382 TeV for the choice of parameters chosen in model 1. The most stringent constraint on MV arising from the process e+e¡ ! ¹+¹¡. LEP II has set severe constraints on lepton compositeness [51; 58] from this process. The e®ective Lagrangian for the process is given by Leff = g2 4 2 (¹e°¹¹)(¹°¹e) M2 V : Here MV is the gauge boson mass. If we compare the above Lagrangian with the ¤¡ LL (ee¹¹) [51; 58], we obtain the limit MV > 1:2 TeV. This limit is satis¯ed in our model. 6.7 Summary We have suggested in this chapter a new scenario for solving the tachyonic slepton mass problem of AMSB. An asymptotically free SU(2)H horizontal gauge symmetry acting on the lepton super¯elds provides positive masses to the sleptons of the ¯rst two families (~e; ~¹) while the Yukawa couplings associated with the third family (¿ ) ¯eld gives a large positive contribution to the ~¿ mass. We have a large mass splitting between the ~e; ~¹; and; ~¿ , due to the transformation properties under the new SU(2)H symmetry. This is how our model di®ers from the other models. The SU(2)H symmetry must be broken at the TeV scale for consistency and our model predicts mh . 135 GeV for the lightest MSSM Higgs boson mass and tan ¯ ' 40. The 67 LSP is the neutral wino which is nearly mass degenerate with the lightest chargino and is a candidate for cold dark matter. CHAPTER 7 Constraining Z0 From Supersymmetry Breaking 7.1 Introduction One of the simplest extensions of the Standard Model (SM) is obtained by adding a U(1) factor to the SU(3)C £SU(2)L £U(1)Y gauge structure [62; 63]. Such U(1) factors arise quite naturally when the SM is embedded in a grand uni¯ed group such as SO(10), SU(6), E6, etc. While it is possible that such U(1) symmetries are broken spontaneously near the grand uni¯cation scale, it is also possible that some of the U(1) factors survive down to the TeV scale. In fact, if there is low energy supersymmetry, it is quite plausible that the U(1) symmetry is broken along with supersymmetry at the TeV scale. The Z0Â and Z0Ã models arising from SO(10) ! SU(5) £ U(1)Â and E6 ! SO(10) £ U(1)Ã are two popular extensions which have attracted much phenomenological attention [62{69]. Z0 associated with the left{ right symmetric extension of the Standard Model does not require a grand uni¯ed symmetry. Other types of U(1) symmetries, which do not resemble the ones with a GUT origin, are known to arise in string theory, free{fermionic construction as well as in orbifold and D{brane models [70{72]. Gauge kinetic mixing terms of the type B¹ºZ0 ¹º [73] which will be generated through renormalization group °ow below the uni¯cation scale can further disguise the couplings of the Z0. The properties of the Z0 gauge boson { its mass, mixing and couplings to fermions { associated with the U(1) gauge symmetry are in general quite arbitrary [74]. This is especially so when the low energy theory contains new fermions for anomaly cancellation. In this chapter we propose and analyze a special class of U(1) models wherein the Z0 properties get essentially ¯xed from constraints of SUSY 68 69 breaking. We have in mind the anomaly mediated supersymmetric (AMSB) frame work [36; 37]. In its minimal version, with the Standard Model gauge symmetry, it turns out that the sleptons of AMSB become tachyonic. We suggest the U(1) symme try, identi¯ed as U(1)x = xY ¡(B¡L), where Y is the Standard Model hypercharge, as a solution to the negative slepton mass problem of AMSB. This symmetry is auto matically free of anomalies with the inclusion of right{handed neutrinos. It is shown that the D{term of this U(1)x provides positive contribut
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Title  Anomaly Mediated Supersymmetry Breaking and Nonstandard Neutrino Oscillations 
Date  20050701 
Author  Anoka, Cyril Ojodume 
Department  Physics 
Document Type  
Full Text Type  Open Access 
Abstract  In this thesis, we propose four different scenarios that solves the tachyonic slepton mass problem of Anomaly Mediated Supersymmetry Breaking (AMSB). We also address the question of neutrino oscillation using non standard interactions. In the first two chapters we introduce the Standard Model (SM) of particle physics and Supersymmetry (SUSY). We review models of SUSY breaking in the third chapter. Chapters four, five, six and seven have our various models that address the negative slepton mass problem of AMSB. In chapter 8, we propose a simple solution to the neutrino oscillation problem based on nonstandard interactions. AMSB is an attractive scenario which can neatly solve the flavor changing neutral current problem of SUSY models. However, the simplest such model has tachyonic sleptons, which is unacceptable. The first model we propose is based on a nonAbelian horizontal gauge symmetry broken at the TeV scale. In this model the sleptons receive positive masssquared from the asymptotically free SU/(3)sub# H/sub#/ gauge sector. The second model is a class of supersymmetric Z/sup#′/sup# models based on the gauge symmetry U/(1)sub#x/sub# = xY/  (B  L/), where Y/ is the Standard Model hypercharge. For 1 < x/ < 2, the U/(1)sub#x /sub# D/term generates positive contribution to the slepton masses. The third model is the quarklepton symmetric model based on leptonic SU/(3)sub#?/sub# gauge symmetry. The negative slepton mass problem is cured by virtue of the positive contribution to the slepton masses from the SU/(3)sub#?/sub# gauge sector. This model also leads to unification of Standard Model gauge couplings in a non trivial way. The fourth model is based on an asymptotically free SU/(2)sub# H/sub#/ gauge symmetry broken at the TeV scale. This model is viable and also solves the tachyonic slepton mass problem of AMSB. Finally in chapter 8, we show how the Liquid Scintillator Neutrino Detector (LSND) experiment puzzle may be solved by adding new physics terms to the standard interactions. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  ANOMALY MEDIATED SUPERSYMMETRY BREAKING AND NONSTANDARD NEUTRINO OSCILLATIONS By CYRIL OJODUME ANOKA Bachelor of Science Obafemi Awolowo University IleIfe, Nigeria 1995 HEP Diploma International Center for Theoretical Physics Trieste, Italy 1999 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2005 ANOMALY MEDIATED SUPERSYMMETRY BREAKING AND NONSTANDARD NEUTRINO OSCILLATIONS Thesis Approved: Dr. K.S. Babu Thesis Advisor Dr. J. Perk Member Dr. J. Mintmire Member Dr. J. Chandler Outside Member Dr. G. Emslie Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to my advisor Prof. K. S. Babu and my collaborator Dr. I. Gogoladze for their constructive guidance, constant encour agement, kindness and great patience. Without their guidance and collaboration, I would not have been able to ¯nish this work. Their critical reading and precious suggestions greatly enhanced the writing of this thesis. I also would like to express my sincere appreciation to Prof. S. Nandi for his assistance during these years of my study at the Oklahoma State University. My appreciation extends to my other committee members Prof. J. Perk, Prof. J. Mintmire and Prof. J. Chandler whose encouragement has also been invaluable. Moreover, I wish to express my thanks to my colleagues Ts. Enkhbat, A. Bachri, Wang Kai, Dr. G. Seidl and the rest of the members of the High Energy Physics Theory group. I will always miss our trips to conferences and outdoor activities. There are some special friends that I will like to acknowledge for their support during my studies. They are Sylvester Onoyona and family, Kingsley Dike and family, Jude Ulogo, Dr. Solomon Osho and family, Angelica Keng, Dr. Saliki and family and a host of others. I would like to thank my Mother and my late Father and all my Brothers and Sisters to whose support I owe my successes. Finally, I thank the Physics Department for providing the opportunity for my graduate study and the US Department of Energy for providing part of my ¯nancial support. iii iv TABLE OF CONTENTS Chapter Page 1. INTRODUCTION :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.1. The Standard Model ::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.2. Symmetry breaking via the Higgs mechanism ::::::::::::::::: 2 1.3. Gauge hierarchy problem:::::::::::::::::::::::::::::::::::::::::::: 5 1.4. Gauge coupling uni¯cation:::::::::::::::::::::::::::::::::::::::::: 7 2. Supersymmetry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10 2.1. Supersymmetry algebra:::::::::::::::::::::::::::::::::::::::::::::: 11 2.2. Superspace and super¯elds:::::::::::::::::::::::::::::::::::::::::: 12 2.3. Supersymmetric Action :::::::::::::::::::::::::::::::::::::::::::::: 16 2.4. SUSY breaking::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 18 3. The Minimal Supersymmetric Standard Model::::::::::::::::::::: 21 3.1. Electroweak symmetry breaking and the Higgs boson masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23 3.2. The sfermions masses :::::::::::::::::::::::::::::::::::::::::::::::: 25 3.3. Neutralinos:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 26 3.4. Charginos:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 27 4. ANOMALY MEDIATED SUPERSYMMETRY BREAKING::::: 28 4.1. Gravity mediation::::::::::::::::::::::::::::::::::::::::::::::::::::: 29 4.2. Gauge mediation :::::::::::::::::::::::::::::::::::::::::::::::::::::: 30 4.3. Gaugino mediation:::::::::::::::::::::::::::::::::::::::::::::::::::: 31 4.4. Anomaly mediation ::::::::::::::::::::::::::::::::::::::::::::::::::: 32 4.4.1. The negative slepton mass problem of anomaly mediated supersymmetry breaking :::::::::::::::::: 34 4.4.2. Suggested solutions to the AMSB slepton mass problem ::::::::::::::::::::::::::::::::::::::::::::::::::::: 35 iv Chapter Page 5. TeV{Scale Horizontal Symmetry and the Slepton Mass Problem of Anomaly Mediation :::::::::::::::::::::::::::::::::::::::::: 37 5.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 37 5.2. SU(3)H horizontal symmetry :::::::::::::::::::::::::::::::::::::: 38 5.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 41 5.3.1. Constraints on tan ¯ and mh ::::::::::::::::::::::::::: 41 5.3.2. SU(3)H symmetry breaking::::::::::::::::::::::::::::: 44 5.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 45 5.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 46 5.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 46 5.4.3. ´ fermion and ´ scalar masses ::::::::::::::::::::::::: 47 5.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 48 5.6. Experimental signatures ::::::::::::::::::::::::::::::::::::::::::::: 53 5.7. Origin of the ¹ term :::::::::::::::::::::::::::::::::::::::::::::::::: 54 5.8. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 55 6. SU(2)H Horizontal Symmetry as a Solution to the Slep ton Mass Problem of Anomaly Mediation::::::::::::::::::::::::::: 56 6.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 56 6.2. SU(2)H horizontal symmetry :::::::::::::::::::::::::::::::::::::: 57 6.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 59 6.3.1. Lepton masses :::::::::::::::::::::::::::::::::::::::::::::: 61 6.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 62 6.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 62 6.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 63 6.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 64 6.6. Other experimental implications :::::::::::::::::::::::::::::::::: 66 6.7. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 66 7. Constraining Z0 From Supersymmetry Breaking::::::::::::::::::: 68 7.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 68 7.2. U(1)x model :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 70 7.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 72 7.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 76 7.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 76 7.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 77 7.4.3. Heavy sneutrino masses :::::::::::::::::::::::::::::::::: 78 7.5. Numerical results for the spectrum ::::::::::::::::::::::::::::::: 79 7.6. Z0 decay modes and branching ratios :::::::::::::::::::::::::::: 84 7.7. Other experimental signatures ::::::::::::::::::::::::::::::::::::: 94 v Chapter Page 7.7.1. Z decay and precision electroweak data :::::::::::::: 94 7.7.2. Z0 mass limit:::::::::::::::::::::::::::::::::::::::::::::::: 96 7.7.3. h ! h0h0 decay ::::::::::::::::::::::::::::::::::::::::::::: 96 7.7.4. Signatures of SUSY particles ::::::::::::::::::::::::::: 97 7.8. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 97 8. Quark{Lepton Supersymmetry ::::::::::::::::::::::::::::::::::::::::::::: 99 8.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 99 8.2. TeV scale quark{lepton symmetric model::::::::::::::::::::::: 100 8.2.1. Uni¯cation of gauge couplings:::::::::::::::::::::::::: 102 8.3. Symmetry breaking ::::::::::::::::::::::::::::::::::::::::::::::::::: 103 8.4. The SUSY spectrum:::::::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.1. Slepton masses ::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.2. Squark masses :::::::::::::::::::::::::::::::::::::::::::::: 108 8.4.3. Exotic slepton masses :::::::::::::::::::::::::::::::::::: 109 8.4.4. Exotic lepton masses:::::::::::::::::::::::::::::::::::::: 109 8.5. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 110 8.5.1. Coupling of light Higgs to SM fermions:::::::::::::: 115 8.5.2. Neutralino schannel annihilation:::::::::::::::::::::: 116 8.6. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 117 9. CP Violation in Neutrino Oscillations from Nonstan dard Physics:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 118 9.1. Introduction :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 118 9.2. Neutrino oscillations including new physics :::::::::::::::::::: 119 9.2.1. Neutrino mixing formalism:::::::::::::::::::::::::::::: 119 9.2.2. Two °avor neutrino mixing ::::::::::::::::::::::::::::: 120 9.2.3. Three generation neutrino oscillation::::::::::::::::: 121 9.2.3.1. General formalism in vacuum::::::::::::::: 121 9.2.3.2. Bilarge mixing :::::::::::::::::::::::::::::::::: 124 9.2.3.3. Exact analysis of three generation neutrino oscillation in vacuum::::::::::::::::: 127 9.3. Numerical results :::::::::::::::::::::::::::::::::::::::::::::::::::::: 131 9.4. Three neutrino oscillations including matter e®ects :::::::::: 148 9.4.1. Formalism ::::::::::::::::::::::::::::::::::::::::::::::::::: 148 9.4.2. Parameter mapping ::::::::::::::::::::::::::::::::::::::: 149 9.5. Summary :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 161 BIBLIOGRAPHY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 163 vi Chapter Page APPENDICES::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 171 APPENDIX ATeV scale Horizontal Symmetry :::::::::::::::::::::::::::::: 172 A.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 172 A.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 172 A.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 A.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 A.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 173 APPENDIX BSU(2)H Symmetry ::::::::::::::::::::::::::::::::::::::::::::::: 175 B.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 175 B.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 B.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 177 APPENDIX CU(1)x Model ::::::::::::::::::::::::::::::::::::::::::::::::::::::: 178 C.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 178 C.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 179 C.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 180 APPENDIX DQuarkLepton Supersymmetric Model :::::::::::::::::::::: 181 D.1. Anomalous dimensions ::::::::::::::::::::::::::::::::::::::::::::::: 181 D.2. Beta functions:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 182 D.3. A terms :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 182 D.4. Gaugino masses :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 183 D.5. Soft SUSY masses ::::::::::::::::::::::::::::::::::::::::::::::::::::: 183 APPENDIX ETwo Generation Neutrino Oscillation Model :::::::::::::: 184 vii LIST OF TABLES Table Page 1.1. Particle content of the SM and the charge assignment. ::::::::::::::::::::::: 2 3.1. Chiral super¯elds of the MSSM. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21 3.2. Vector Super¯elds of the MSSM.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22 5.1. Particle content and charge assignment of the SU(3)H model. ::::::::::::: 39 5.2. Sparticle masses in the SU(3)H model for one choice of parameters :::::: 50 5.3. Sparticle masses in the SU(3)H model for a second choice of param eters :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 51 5.4. Sparticle masses in the SU(3)H model for a third choice of parameters:: 52 6.1. Particle content and charge assignment of the SU(2)H model. ::::::::::::: 58 6.2. Sparticle masses in the SU(2)H model for one choice of input param eters.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 65 7.1. Particle content and charge assignment of the Z0 model. :::::::::::::::::::: 71 7.2. Sparticle masses in Model 1 with x = 1:3 :::::::::::::::::::::::::::::::::::::::: 81 7.3. Z0 mass and Z ¡ Z0 mixing angle in Model 1 for the same set of input parameters as in Table 7.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 82 7.4. Eigenvectors of the neutralino mass matrix in Model 1. :::::::::::::::::::::: 82 7.5. Eigenvectors of the chargino mass matrix in Model 1. :::::::::::::::::::::::: 82 7.6. Eigenvectors of the CP{even Higgs boson mass matrix in Model 1. ::::::: 83 7.7. Sparticle masses in Model 2 with x = 1:6 :::::::::::::::::::::::::::::::::::::::: 84 7.8. Z0 mass and Z ¡ Z0 mixing angle in Model 2 for the same set of input parameters as in Table 7.7. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 85 viii Table Page 7.9. Eigenvectors of the neutralino mass matrix in Model 2. :::::::::::::::::::::: 85 7.10. Eigenvectors of the chargino mass matrix in Model 2. :::::::::::::::::::::::: 85 7.11. Eigenvectors of the CP{even Higgs boson mass matrix in Model 2. ::::::: 85 7.12. Decay modes for Z0 in Model 1 for the parameters used in Table 7.2. :::: 92 7.13. Decay modes for Z0 in Model 2 for the parameters used in Table 7.7. :::: 93 8.1. Particle content and charge assignment of the quark{lepton symmet ric model.:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 100 8.2. Sparticle masses in the quark{lepton symmetric model (Model 1) for one choice of input parameters.:::::::::::::::::::::::::::::::::::::::::::::::::::: 112 8.3. Z0 mass and Z ¡ Z0 mixing angle in Model 1 for the same set of input parameters as in Table 8.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 113 8.4. Eigenvectors of the neutralino mass matrix in Model 1. :::::::::::::::::::::: 113 8.5. Eigenvectors of the chargino mass matrix in Model 1. :::::::::::::::::::::::: 113 8.6. Eigenvectors of SU(2)` chargino mass matrix in Model 1. ::::::::::::::::::: 114 8.7. Sparticle masses in the quark lepton symmetric model (Model 2) for a di®erent choice of parameters.::::::::::::::::::::::::::::::::::::::::::::::::::: 114 8.8. Z0 mass and Z ¡ Z0 mixing angle in Model 2 for the same set of input parameters as in Table 8.7. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 115 ix LIST OF FIGURES Figure Page 1.1. 1loop correction to the mass of a fermion. :::::::::::::::::::::::::::::::::::::: 6 1.2. 1loop corrections to a scalar mass. :::::::::::::::::::::::::::::::::::::::::::::::: 6 1.3. Running of the couplings in the SM (left) and its minimal supersym metric version (right). :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 8 4.1. Leading diagram that contributes to SUSY{breaking scalar masses. The bulk line is a gaugino propagator. :::::::::::::::::::::::::::::::::::::::::: 32 5.1. E®ective operators inducing charged lepton masses.::::::::::::::::::::::::::: 40 5.2. Plot of tan ¯ as a function of M2 ::::::::::::::::::::::::::::::::::::::::::::::::::: 43 8.1. Renomalization group evolution the inverse gauge couplings. ::::::::::::::: 103 8.2. ~W + decay to two leptons and LSP. :::::::::::::::::::::::::::::::::::::::::::::::: 110 8.3. Neutralino annihilation to two charged leptons in the early universe. :::: 111 8.4. Bound state of two x leptons decay to two photons. :::::::::::::::::::::::::: 111 8.5. Bound state of two x leptons decay to two charged leptons via ex change of SU(2)H gauge boson. ::::::::::::::::::::::::::::::::::::::::::::::::::: 111 8.6. Doublet SU(2)H gauge boson decay to two charged leptons via ex change of neutralino LSP. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 112 9.1. CP asymmetry A¹e as a function of energy for two generation neu trino oscillation in vacuum:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 122 9.2. Oscillation probabilities Pe¹ and P¹e as a function of energy for the three generation neutrino oscillations in vacuum ::::::::::::::::::::::::::::: 132 9.3. Oscillation probabilities P¹¹ and P¹¿ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::: 133 x Figure Page 9.4. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 134 9.5. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of energy for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 135 9.6. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of energy for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 136 9.7. Change in oscillation probabilities ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ and ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a function of energy for the same choice of input parameters as in Fig. 9.2, except that ± = 0 and ²± = 0. ::::::: 137 9.8. Oscillation probabilities Pe¹ and P¹e as a function of energy for a ¯xed baseline L = 295 km (a) and L = 730 km (b). All other parameters are as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::: 138 9.9. Oscillation probabilities P¹¹ and P¹¿ as a function of energy for ¯xed baseline L = 295 km (a) and L = 730 km (b). Input parameters are as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 139 9.10. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of energy for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 140 9.11. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of energy for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 141 9.12. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of energy for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 142 9.13. Oscillation probabilities Pe¹ and P¹e as a function of Length for ¯xed energy E = 5 GeV . All other parameters are the same as in Fig. 9.2. 143 9.14. Oscillation probabilities P¹¹ and P¹¿ as a function of length for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::: 144 xi Figure Page 9.15. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹¡P¹¹ as a function of length for the same choice of input parameters as in Fig. 9.2. ::::::::::::::::::::::::::::::::::::::::::::::: 145 9.16. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡P¹¹¿ as a func tion of length for the same choice of input parameters as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 146 9.17. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡ P¹¿ ¹ as a function of Length for the same choice of input parameters as in Fig. 9.2.::::::::::::::::::::::::::::::::::::::: 147 9.18. Oscillation probabilities Pe¹ and P¹¹ in matter (assuming constant matter density ½ = 2.8 g/cm3) as a function of energy for ¯xed length L = 2540 km. All other parameters are the same as in Fig. 9.2. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 151 9.19. Oscillation probability P¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 152 9.20. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 153 9.21. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡ P¹¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 154 9.22. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡P¹¿ ¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 155 9.23. Oscillation probabilities Pe¹ and P¹¹ in matter as a function of energy for ¯xed length L = 295 km. All other parameters are the same as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 156 9.24. Oscillation probability P¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 157 9.25. Change in oscillation probabilities ¢Pe¹ (CP) = Pe¹ ¡ P¹e¹ and ¢P¹¹ (CP) = P¹¹ ¡ P¹¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 158 xii Figure Page 9.26. Change in oscillation probability ¢P¹¿ (CP) = P¹¿ ¡ P¹¹¿ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 159 9.27. Apparent CPT violation parameters ¢Pe¹ (CPT) = Pe¹ ¡ P¹¹e and ¢P¹¿ (CPT) = P¹¿ ¡P¹¿ ¹ in matter as a function of energy for the same choice of input parameters as in Fig. 9.18.:::::::::::::::::::::::::::::: 160 xiii xiv CHAPTER 1 INTRODUCTION In this section we give a brief description of the Standard Model of particle physics and reasons for going beyond it. 1.1 The Standard Model The Standard Model (SM) of elementary particle physics has recorded remark able success in describing physics at length scales ranging from atomic scales down to the shortest probed scale of about 10¡18 m. It is a non{abelian gauge theory based on the gauge group [1] SU(3)C £ SU(2)L £ U(1)Y ; where SU(3)C is the color gauge group describing strong interactions and SU(2)L £ U(1)Y is the electroweak gauge group describing weak and electromagnetic interac tions. The SM describes the interactions of quarks, leptons, gauge bosons and the Higgs boson. The ¯eld content and the transformation properties under the gauge symmetries are shown in Table 1. It is important to note that the left{ and the right{handed components of the matter fermions are assigned to di®erent representations (doublets and singlets respectively) of the weak gauge group SU(2)L, thereby allowing a chiral structure for the weak interactions. The Yukawa and Higgs part of the SM Lagrangian is given by LY ukawa = Y ` ®¯`®ec ¯ ~Á + Y d ®¯Q®dc ¯ ~Á + Y u ®¯Q®uc ¯Á + h.c.; (1.1) where ~Á = i¾2Á¤ = Ã ¹ Á0 ¡Á¡ ! . Here generation indices ®; ¯ = 1; 2; 3 are explicitly displayed, while color and SU(2)L indices are suppressed. 1 2 Fields SU(3)C SU(2)W U(1)Y Quarks Qi ® = Ã ui ® di ® ! 3 2 1 6 uci ® ¹3 1 ¡2 3 dci ® ¹3 1 1 3 Leptons `® = Ã º® e® ! 1 2 ¡1 2 ec ® 1 1 2 Gluon Ga ¹ 8 1 0 Intermediate weak bosons Wr ¹ 1 3 0 Hypercharge gauge boson B¹ 1 1 0 Higgs boson Á = Ã Á+ Á0 ! 1 2 1 2 TABLE 1.1. Particle content of the SM and the charge assignment. Here ® = 1; 2; 3 is the generation index, i = 1 ¡ 3 (color), a = 1 ¡ 8 (SU(3)C generators) and r = 1 ¡ 3 (SU(2)L generators). 1.2 Symmetry breaking via the Higgs mechanism If we consider the SU(2)L£U(1)Y part of the Lagrangian, assuming that there is no Higgs ¯eld, all the fermions and the four gauge bosons (Wr ¹; B¹) would be massless. This is unacceptable, for the weak interactions are short range, meaning that the mediators must be massive. We must then break the symmetry spontaneously which will ensure renormalizability. This is achieved through the scalar Higgs doublet Á = Ã Á+ Á0 ! : (1.2) The only observed unbroken local symmetry in Nature is the U(1)em (apart from SU(3)C). Therefore the SU(2)L£U(1)Y symmetry should be broken down to U(1)em. The renomalizable Higgs potential is given by VH ´ ¹2ÁyÁ + ¸(ÁyÁ)2: 3 This has a minimum for ¹2 < 0 at hÁyÁi = ¡ ¹2 2¸ = À2 2 : (1.3) We can choose the vacuum expectation value (VEV) after an SU(2)L transformation in the unitary gauge as hÁ0i = 1 p 2 Ã 0 À ! : (1.4) It is not di±cult to see that the gauge boson associated with the U(1)em subgroup of SU(2)L £ U(1)Y remains massless. The electric charge Qem, the U(1)Y hypercharge and the third component of weak isospin T3L are related by Qem = T3L + Y 2 ; (1.5) and the gauge boson masses are given by MW = gÀ 2 ; MZ = MW cos µW ; MA = 0: (1.6) Here g is the SU(2)L gauge coupling strength and tan µW = g0=g, where g0 is the U(1)Y gauge coupling constant. These masses are obtained from the Lagrangian for the gauge and Higgs ¯eld, given by LgaugeHiggs = ¯¯¯¯ @¹Á ¡ ig 2 ~¿ : ~W ¹Á ¡ ig0 2 B¹Á ¯¯¯¯ 2 ; once the VEV of Á0 is inserted. It is worthwhile to note that the weak mixing angle µW is a parameter of the SM which has been measured to a very high accuracy. Another accurately measured quantity is the ½ parameter (½ ´ M2W M2Z cos2 µW ) which is predicted to be 1 (at tree level) in the SM. New physics can also be severely constrained by the observed value of ½. After symmetry breaking, from the Yukawa interactions in Eq. (1.1), the fermions become massive with masses given by Mu = YuÀ; Md = YdÀ; M` = Y`À: (1.7) Here Yu; d; ` are arbitrary 3 £ 3 complex matrices in generation space. 4 Not all parameters in these matrices are observable in the SM. After fermion ¯eld rede¯nitions, the 3 eigenvalues of each of the matrices, 3 mixing angles and one phase entering in the charged W§ ¹ interactions with quarks become physical quantities. One makes biunitary transformations, Uu LYuUuy R = Y diag u , UdL YdUdy R = Y diag d , U`L Y`U`y R = Y diag ` , in which case the charged W§ current takes the form LW§ ¹ cc = g p 2 ¹uL°¹VCKMdLW¹+ + h.c.; where VCKM = Uuy L UdL is a unitary matrix, the Cabibbo{Kobayashi{Maskawa matrix or the quark mixing matrix. Since there is no right{handed neutrino ¯eld ºR, the neutrinos remain massless. The fermion masses are arbitrary since the Yukawa couplings Y are free parameters. To ¯nd the Higgs boson mass, we write the complex ¯eld Á0 in terms of real ¯elds. The Higgs doublet then takes the form (in unitary gauge) Á = 1 p 2 Ã 0 À + ´ ! ; (1.8) where ´ is the physical Higgs scalar with mass m2 ´ = 2¸À2: (1.9) The Higgs mass is left undetermined since ¸ is a free parameter, with only its sign constrained to be positive. There are several good features of the SM some of which are: 1. All the particles predicted by the SM have been observed except the Higgs boson. 2. Both baryon and lepton number are automatically conserved. This prevents rapid decay of the proton. 3. It has an extremely economical Higgs sector which is responsible for giving masses to all particles. 4. With only two independent parameters MW and sin µW, all the electroweak processes at high energy are correctly described. 5 The SM also has several drawbacks. There are several free parameters in the SM Lagrangian: The Higgs coupling constant ¸, the Higgs mass parameter ¹2, three gauge couplings (g0; g; gs), the number of generations (matter ¯elds) and three Yukawa matrices Y u ®¯, Y d ®¯, Y ` ®¯. Despite the remarkable success of the SM, there are still several questions left unanswered. For example, does the Higgs boson exist? Do the gauge couplings unify? How is gravity incorporated? An attempt to answer these numerous questions will take us to beyond the SM. For example, some earlier attempts tried to unify strong and electroweak forces by embedding the SU(3)C £ SU(2)L £ U(1)Y structure into higher groups such as SU(5) and SO(10). These \Grand Uni¯ed Theories" or GUT's, were only partially successful. Di±culties with the SM and GUT models concerning gauge hierarchy and ¯ne tuning problems led to theoretical remedies such as technicolor, supersymmetry, string theory, etc. The most appealing of these theories is perhaps supersymmetry, which is the main focus of this thesis. 1.3 Gauge hierarchy problem The hierarchy problem is one of the main reasons why we think supersymmetry has something to do with Nature, and that it might be broken at a scale comparable to the scale of weak interactions, rather than at some enormous energy such as the Planck scale MPl » 1019 GeV. The mass hierarchy problem stems from the fact that masses, in particular scalar masses, are not stable to radiative corrections [2]. While fermion masses also receive radiative corrections from diagrams of the form in Figure 1.1, these are only logarithmically divergent (see for example [3]), ±mf ' 3® 4¼ mf ln(¤2=m2f ); (1.10) where ¤ is an ultraviolet cuto®, where we expect new physics to play an important role. As one can see, even for ¤ » MPl, these corrections are small, ±mf <» mf . 6 Figure 1.1. 1loop correction to the mass of a fermion. In contrast, scalar masses are quadratically divergent. 1{loop contributions to scalar masses, such as those shown in Fig. 1.2, are readily computed ±m2 H ' fg2 f ; g2; ¸g Z d4k 1 k2 » O ³ ® 4¼ ´ ¤2; (1.11) due to contributions from fermion loops with coupling gf , from gauge boson loops with coupling g2, and from quartic scalarcouplings ¸. g2 g f g f l Figure 1.2. 1loop corrections to a scalar mass. An alternative and by far simpler solution to this problem exists if one postu lates that there are new particles with similar masses and equal couplings to those responsible for the radiatively induced masses but with a di®erence (by a half unit) in spin. Then, because the contribution to ±m2 H due to a fermion loop comes with a relative minus sign, the total contribution to the 1loop corrected mass2 is ±m2 H ' O ³ ® 4¼ ´ (¤2 + m2 B) ¡ O ³ ® 4¼ ´ (¤2 + m2 F ) = O ³ ® 4¼ ´ (m2 B ¡ m2 F ): (1.12) If in addition, the bosons and fermions all have the same masses, then the radiative corrections vanish identically. The stability of the hierarchy only requires that the weak scale is preserved so that we need only require that jm2 B ¡ m2 F j <» 1 TeV2: (1.13) As we will see latter, supersymmetry o®ers just the framework for including the nec essary new particles and ensures the absence of these dangerous radiative corrections [4]. 7 1.4 Gauge coupling uni¯cation Another motivation for supersymmetry lies in the gauge coupling constant uni ¯cation. In the SM, the gauge couplings do not unify. The solutions to the SM renormalization group equations to one loop accuracy are given by 1 ®i(Q) = 1 ®i(¹) + bi 2¼ log µ ¹ Q ¶ ; where the bi are bi = 0 BB@ b1 b2 b3 1 CCA = 0 BB@ 0 ¡22 3 ¡11 1 CCA + Ng 0 BB@ 4 3 4 3 4 3 1 CCA + Nh 0 BB@ 1 10 1 6 0 1 CCA : Here Ng = 3 is the number of generations and Nh = 1 is the number of Higgs doublets. The numerical values for the bi coe±cients are bi = ( 41 10 ; ¡19 6 ; ¡7). The three gauge coupling constants used as input are ®1 = 5®=(3 cos2 µW); ®2 = ®=sin2 µW; ®3 = g2 s=(4¼); where ®¡1(MZ) = 128:978; sin2 µW = 0:23146 and ®3 = 0:1184: On evolving the inverse of the three coupling constants as a function of logarithm of the uni¯cation scale Q, the result is shown in Fig. 3 (left). These couplings do not meet at a common point, hence uni¯cation does not occur. If we consider supersymmetric grand uni¯ed theory, the beta function coe±cients are modi¯ed due to the quantum corrections involving the superpartners and are given in the Minimal Supersymmetric Standard Model (MSSM) by bi = 0 BB@ b1 b2 b3 1 CCA= 0 BB@ 0 ¡6 ¡9 1 CCA + Ng 0 BB@ 2 2 2 1 CCA + Nh 0 BB@ 3 10 1 2 0 1 CCA : Here Ng = 3 and Nh = 2. The numerical value for bi is bi = ( 33 5 ; 1; ¡3). If we assume that all the SUSY particle masses are around 1 TeV, on evolving the inverse coupling constants, they meet at a point (unify) as shown in Fig. 3 (right). The point at which these particles meet is around 1016 GeV. The SUSY particles are assumed to 8 Figure 1.3. Running of the couplings in the SM (left) and its minimal supersymmetric version (right). contribute only above the e®ective SUSY scale (» 1 TeV) which causes the change of slope in the evolution of the couplings. This is another reason why most high energy physicist believe in supersymmetry. The present thesis contains nine chapters. In the second chapter we review all the basics for Supersymmetry (SUSY), we de¯ne the SUSY algebra and introduce all the tools needed to write down the supersymmetric version of gauge ¯eld theories. In chapter 3, the minimal supersymmetric extension of the Standard Model is intro duced, all the interactions and relevant mass matrices for our analysis are studied. In the fourth chapter we review various symmetry breaking models, here we introduce the Anomaly Mediated Supersymmetry Breaking (AMSB) and review the relevant literature. In chapter 5, we suggest TeV{Scale horizontal symmetry as a solution to the negative slepton mass squared problem of AMSB. In chapter 6, we suggest an SU(2)H model as a solution to the negative slepton mass problem. In chapter 7, we study a speci¯c Z0 model as a solution to the slepton mass problem of AMSB. 9 In Chapter 8, we suggest another model to solve this problem of AMSB with the quarks and the leptons transforming identically under two di®erent SU(3) symmetry group. Finally, we divert from the AMSB to Neutrino Physics, here we suggest a nonstandard neutrino interaction as a solution to the neutrino oscillation problem. CHAPTER 2 Supersymmetry Supersymmetry (SUSY) is often called the last great symmetry of Nature. Rarely has so much e®ort, both theoretical and experimental, been spent to un derstand and discover a symmetry of Nature, which up to the present time lacks concrete evidence. Why SUSY? If for no other reason, it would be nice to understand the origin of the fundamental di®erence between the two classes of particles distinguished by their spin, fermions and bosons. If such a symmetry exists, one might expect that it is represented by an operator which relates the two classes of particles. For example, QjBosoni = jFermioni; QjFermioni = jBosoni: (2.1) However, without a connection to experiment, SUSY would remain a mathematical curiosity and a subject of a very theoretical nature as indeed it stood from its initial description in the early 1970's [5; 6] until its incorporation into a realistic theory of physics at the electroweak scale. One of the ¯rst breakthroughs came with the realization that SUSY could help resolve the di±cult problem of mass hierarchies [2], namely the stability of the electroweak scale with respect to radiative corrections. With precision experiments at the electroweak scale, it has also become apparent that Grand Uni¯cation is not possible in the absence of SUSY [7]. Considering a new class of \fermionic" generators Q, that satisfy anti{commutation relations [Q®; J¹º] = i¾¹º ¯ ® Q¯; 10 11 [Q®; P¹] = 0; £ ¹Q ®_ ; J¹º¤ = i¹¾¹º®_ _¯ ¹Q _¯ ; £ ¹Q ®_ ; P¹¤ = 0; (2.2) where Q® (¹Q ®_ ) is a symmetry operator (SUSY charge), P¹ is the energy{momentum operator and J¹º is the angular momentum operator. The Q's are translationally invariant (no explicit x{dependence) and they satisfy anti{commutation relations fQ®; ¹Q _¯ g = 2¾¹ ® _¯ P¹; (2.3) where the factor 2 is conventional and can be achieved by re{scaling the Q's. There are three main properties of a supermultiplet: (1) All particles belonging to an irreducible representation of SUSY have the same mass, (2) there are equal number of fermionic (NF ) and bosonic (NB) degrees of freedom in a supermultiplet, (3) the energy P0 in a supersymmetric theory is always positive. 2.1 Supersymmetry algebra Combined with the usual Poincar¶e and internal symmetry algebra the Super Poincar¶e Lie algebra contains additional SUSY generators Qi ® and ¹Q i ®_ [8] [P¹; Pº] = 0; [P¹;M½¾] = i(g¹½P¾ ¡ g¹¾P½); [M¹º;M½¾] = i(gº½M¹¾ ¡ gº¾M¹½ ¡ g¹½Mº¾ + g¹¾Mº½); [Br;Bs] = iCt rsBt; [Br; P¹] = [Br;M¹¾] = 0; [Qi ®; P¹] = [¹Q i ®_ ; P¹] = 0; [Qi ®;M¹º] = 1 2 (¾¹º)¯ ®Qi ¯; [¹Q i ®_ ;M¹º] = ¡1 2 ¹Q i _¯ (¹¾¹º) _¯ ®_ ; [Qi ®;Br] = (br)i jQj ®; [¹Q i ®_ ;Br] = ¡ ¹Q j ®_ (br)i j ; fQi ®; ¹Q j _¯ g = 2±ij(¾¹)® _¯ P¹; fQi ®;Qj ¯g = 2²®¯Zij ; Zij = ar ijbr; Zij = Z+ ij ; f¹Q i ®_ ; ¹Q j _¯ g = ¡2² _ ® _¯ Zij ; [Zij ; anything] = 0; ®; ®_ = 1; 2 i; j = 1; 2; : : : ;N: (2.4) 12 Here P¹ and M¹º are fourmomentum and angular momentum operators, re spectively, Br are the internal symmetry generators, Qi and ¹Q i are the spinorial SUSY generators and Zij are the socalled central charges, while ®; _ ®; ¯; _¯ are the spinorial indices. In the simplest case one has one spinor generator Q® (and the conjugated one ¹Q ®_ ) that corresponds to an ordinary or N=1 SUSY. When N > 1 one has an extended SUSY. The constraint on the number of SUSY generators comes from a requirement of consistency of the corresponding quantum ¯eld theory (QFT). The number of supersymmetries and the maximal spin of the particle in the multiplet are related by N · 4S; where S is the maximal spin. Since the theories with spin greater than 1 are non renormalizable and the theories with spin greater than 5/2 have no consistent coupling to gravity, this imposes a constraint on the number of SUSY generators N · 4 for renormalizable theories (YM), N · 8 for (super)gravity: In what follows, we shall consider simple SUSY, or N = 1 SUSY, contrary to extended supersymmetries with N > 1. In this case, one has two types of supermultiplets: the socalled chiral multiplet, which contains two physical states (Á; Ã) with spin 0 and 1/2, respectively, and the vector multiplet with ¸ = 1=2, which also contains two physical states (¸;A¹) with spin 1/2 and 1, respectively. 2.2 Superspace and super¯elds An elegant formulation of SUSY transformations and invariants can be achieved in the framework of superspace [9]. Superspace di®ers from the ordinary Euclidean (Minkowski) space by the addition of two new coordinates, µ® and µ¹®_ , which are Grassmannian, i.e. anticommuting, variables fµ®; µ¯g = 0; f¹µ _ ®; ¹µ _¯ g = 0; µ2® = 0; ¹µ2_ ® = 0; ®; ¯; ®_ ; _¯ = 1; 2: 13 Thus, we go from space to superspace Space ) Superspace x¹ x¹; µ®; µ¹®_ A SUSY group element can be constructed in superspace in the same way as an ordinary translation in the usual space G(x; µ; ¹µ) = ei(¡x¹P¹ + µQ + ¹µ ¹Q ): (2.5) It leads to a supertranslation in superspace x¹ ! x¹ + iµ¾¹¹" ¡ i"¾¹¹µ; µ ! µ + "; ¹µ ! ¹µ + ¹"; (2.6) where " and ¹" are Grassmannian transformation parameters. From Eq. (2.6) one can easily obtain the representation for the supercharges Eq. (2.4) acting on the superspace Q® = @ @µ® ¡ i¾¹ ®®_ ¹µ _ ®@¹; ¹Q ®_ = ¡ @ @µ¹®_ + iµ®¾¹ ®®_ @¹: (2.7) Taking the Grassmannian transformation parameters to be local, or spacetime de pendent, one gets a local translation. As has already been mentioned, this leads to a theory of (super) gravity. To de¯ne the ¯elds on a superspace, consider repre sentations of the SuperPoincar¶e group Eq. (2.4) [10]. The simplest one is a scalar super¯eld F(x; µ; ¹µ) which is SUSY invariant. Its Taylor expansion in µ and ¹µ has only several terms due to the nilpotent character of Grassmannian parameters. However, this super¯eld is a reducible representation of SUSY. To get an irreducible one, we de¯ne a chiral super¯eld which obeys the equation ¹D F = 0; where ¹D = ¡ @ @µ ¡ iµ¾¹@¹ (2.8) is a superspace covariant derivative. In superspace (by Taylor expanding y = x + iµ¾¹µ), a chiral super¯eld is written as ©(y; µ) = A(y) + p 2µÃ(y) + µµF(y) = A(x) + iµ¾¹¹µ@¹A(x) + 1 4 µµ¹µ¹µ2A(x) + p 2µÃ(x) ¡ ip 2 µµ@¹Ã(x)¾¹¹µ + µµF(x): (2.9) 14 Here A is a complex scalar ¯eld (with two bosonic degrees of freedom), Ã is a Weyl spinor ¯eld (with 2 fermionic degrees of freedom)and F is the auxiliary ¯eld (with no physical meaning) which is needed to close the SUSY algebra (2.4). We see from here that a super¯eld contains an equal number of fermionic and bosonic degrees of freedom. Under a SUSY transformation with anticommuting parameter ", the component ¯elds transform as ±"A = p 2"Ã; ±"Ã = i p 2¾¹¹"@¹A + p 2"F; (2.10) ±"F = i p 2¹"¾¹@¹Ã: The antichiral super¯eld ©+ obey the equation D©+ = 0; with D = @ @µ + i¾¹¹µ@¹: The product of chiral (antichiral) super¯elds ©2;©3, etc., is also a chiral (antichiral) super¯eld, while the product of chiral and antichiral ones ©+© is a general super¯eld. For any arbitrary function of chiral super¯elds one has W(©i) = W(Ai + p 2µÃi + µµF) = W(Ai) + @W @Ai p 2µÃi + µµ µ @W @Ai Fi ¡ 1 2 @2W @Ai@Aj ÃiÃj ¶ : (2.11) The W is usually referred to as a superpotential which replaces the usual potential for the scalar ¯elds. The vector super¯eld satis¯es the condition V = V +. They should be understood in terms of their power series expansion in µ and ¹µ as V (x; µ; ¹µ) = C(x) + iµÂ(x) ¡ i¹µ ¹Â(x) + i 2 µµ[M(x) + iN(x)] ¡ i 2 ¹µ¹µ[M(x) ¡ iN(x)] ¡ µ¾¹¹µv¹(x) + iµµ¹µ[¸(x) + i 2 ¹¾¹@¹Â(x)] ¡ i¹µ¹µµ[¸ + i 2 ¾¹@¹ ¹Â(x)] + 1 2 µµ¹µ¹µ[D(x) + 1 2 2C(x)]: (2.12) The component ¯elds C; D; M; N and v¹ must be real for Eq. (2.12) to satisfy V = V +. These vector supermultiplet contains 8 bosonic degrees of freedom (one 15 each for C; D; M; M; N and four from the real vector ¯eld v¹) and 8 fermionic degrees of freedom (from the two component spinors Â and ¸). The physical degrees of freedom corresponding to a real vector super¯eld V are the vector gauge ¯eld v¹ and the Majorana spinor ¯eld ¸. All other components are unphysical and can be eliminated. We now de¯ne the supersymmetric generalization of an Abelian gauge transformation of the super¯eld V as V ! V + © + ©+; where © and ©+ are some chiral super¯elds. Under this transformation, the compo nent transform as C ! C + A + A¤; Â ! Â ¡ i p 2Ã; M + iN ! M + iN ¡ 2iF; v¹ ! v¹ ¡ i@¹(A ¡ A¤); (2.13) ¸ ! ¸; D ! D: We see that there is a special gauge known as the WessZumino gauge [11] in which C; Â; M and N are all zero. Fixing this gauge breaks SUSY but still allows the usual gauge transformation v¹ ! v¹+@¹A. In this gauge, the vector multiplet reduces to 4 bosonic degrees of freedom (1 for D and the three remaining components of v¹) and 4 fermionic degrees of freedom (from the Majorana spinor ¸). In this gauge the vector super¯eld takes the form V = ¡µ¾¹¹µv¹(x) + iµµ¹µ¹¸ (x) ¡ i¹µ¹µµ¸(x) + 1 2 µµ¹µ¹µD(x); V 2 = ¡ 1 2 µµ¹µ¹µv¹(x)v¹(x); V 3 = 0; V n = 0 for n > 3: (2.14) One can de¯ne also a ¯eld strength tensor (as analog of F¹º in gauge theories) W® = ¡ 1 4 ¹D 2eVD®e¡V ; 16 ¹W ®_ = ¡ 1 4 D2eV ¹D ®e¡V ; (2.15) which is a polynomial in the WessZumino gauge. (Here Ds are the supercovariant derivatives.) The strength tensor is a chiral super¯eld ¹D _¯ W® = 0; D¯ ¹W ®_ = 0: In the WessZumino gauge it is a polynomial over component ¯elds: W® = Ta µ ¡i¸a ® + µ®Da ¡ i 2 (¾¹¹¾ºµ)®Fa ¹º + µ2¾¹D¹¹¸ a ¶ ; (2.16) where Fa ¹º = @¹va º ¡ @ºva¹ + gfabcvb¹ vcº ; D¹¹¸ a = @¹¸ a + gfabcvb¹ ¹¸ c: In Abelian case eqs.(2.15) are simpli¯ed and take form W® = ¡ 1 4 ¹D 2D®V; ¹W ®_ = ¡ 1 4 D2 ¹D ®V: 2.3 Supersymmetric Action Using the rules of Grassmannian integration: Z dµ® = 0 Z µ® dµ¯ = ±®¯ we can de¯ne the general form of a SUSY and gauge invariant Lagrangian as [10]: LYM SUSY = 1 4 Z d2µ Tr(W®W®) + 1 4 Z d2¹µ Tr( ¹W ® ¹W ®) (2.17) + Z d2µd2¹µ ©y ia (egV )ab ©bi + Z d2µ W(©i) + Z d2¹µ ¹W (¹© i) ©i are chiral super¯elds which transform as: ©i ! e¡ig¤©i and egV ! eig¤y egV e¡ig¤ where both ¤ and V are matrices: 17 ¤ij = ¿ a ij¤a; Vij = ¿ a ijVa; with ¿ a the gauge generators. The supersymmetric ¯eld strength W® is equal to W® = ¡ 1 4 ¹D ¹D e¡VD®eV and transforms as: W ! e¡i¤Wei¤. W is the superpotential, which should be invariant under the group of symme tries of a particular model. In terms of component ¯elds the above Lagrangian takes the form [12] LYM SUSY = ¡ 1 4 Fa ¹ºFa¹º ¡ i¸a¾¹D¹¹¸ a + 1 2 DaDa + (@¹Ai ¡ igva¹ ¿ aAi)y(@¹Ai ¡ igva¹¿ aAi) ¡ i ¹ Ãi¹¾¹(@¹Ãi ¡ igva¹¿ aÃi) ¡ DaAy i ¿ aAi ¡ i p 2Ay i ¿ a¸aÃi + i p 2 ¹ Ãi¿ aAi¹¸ a + Fy i Fi + @W @Ai Fi + @ ¹W @Ay i Fy i ¡ 1 2 @2W @Ai@Aj ÃiÃj ¡ 1 2 @2 ¹W @Ay i@Ay j ¹ Ãi ¹ Ãj (2.18) Integrating out the auxiliary ¯elds Da and Fi, one reproduces the usual Lagrangian. Contrary to the SM, where the scalar Higgs potential is arbitrary and is de¯ned only by the requirement of the gauge invariance, in supersymmetric theories it is completely de¯ned by the superpotential. It consists of the contributions from the Dterms and Fterms. The kinetic energy of the gauge ¯elds yields the 1 2DaDa term, and the mattergauge interaction yields the gDa¿ a ijA¤i Aj one. Together they give LD = 1 2 DaDa + gDa¿ a ijA¤ iAj : (2.19) The equation of motion reads Da = ¡g¿ a ijA¤ iAj ; (2.20) Substituting it back into Eq. (2.19) yields the Dterm part of the potential LD = ¡ 1 2 DaDa =) VD = 1 2 DaDa; (2.21) 18 where D is given by Eq. (2.20). The Fterm contribution can be derived from the matter ¯eld selfinteraction. For a general type superpotential W one has LF = F¤ i Fi + ( @W @Ai Fi + h:c:) (2.22) Using the equations of motion for the auxiliary ¯eld Fi F¤ i = ¡ @W @Ai (2.23) yields LF = ¡F¤ i Fi; =) VF = F¤ i Fi; (2.24) where F is given by Eq. (2.23). The full potential is the sum of the two contributions V = VD + VF : (2.25) Thus, the form of the Lagrangian is constrained by symmetry requirements. The only freedom is the ¯eld content, the value of the gauge coupling g, Yukawa couplings yijk and the masses. Because of the renormalizability constraint V · A4 the superpoten tial should be limited by W · ©3. All members of a supermultiplet have the same masses, i.e. bosons and fermions are degenerate in mass. This property of SUSY theories contradicts phenomenology and requires SUSY breaking. 2.4 SUSY breaking Since the SUSY algebra leads to mass degeneracy in a supermultiplet, it should be broken to explain the absence of superpartners at accessible energies. There are several ways of SUSY breaking. It can be broken either explicitly or spontaneously. In performing SUSY breaking one has to be careful not to spoil the cancellation of quadratic divergencies which allows one to solve the hierarchy problem. This is achieved by spontaneous breaking of SUSY. It is possible to show that in SUSY models the energy is always nonnegative de¯nite. According to quantum mechanics the energy is equal to E = h0j bH j0i; (2.26) 19 where bH is the Hamiltonian and due to the SUSY algebra fQ®; ¹Q _¯ g = 2(¾¹)® _¯ P¹: (2.27) Taking into account that Tr(¾¹P¹) = 2P0 one gets E = 1 4 X ®=1;2 h0jfQ®; ¹Q ®gj0i = 1 4 X ® kQ®j0ik2 ¸ 0: (2.28) Hence E = h0j bH j0i 6= 0 if and only if Q®j0i 6= 0: Therefore, SUSY is spontaneously broken, i.e. the vacuum is not invariant under Q (Q®j0i 6= 0), if and only if the minimum of the potential is positive (i:e: E ¸ 0) . Spontaneous breaking of SUSY is achieved in the same way as electroweak symmetry breaking. One introduces a ¯eld whose vacuum expectation value is nonzero and breaks the symmetry. However, due to the special character of SUSY, this should be a super¯eld whose auxiliary F or D component acquires nonzero VEVs. Thus, among possible spontaneous SUSY breaking mechanisms one distinguishes the F{ type breaking and the D{type breaking. i) FayetIliopoulos (Dterm) mechanism [12]. In this case the, the linear Dterm is added to the Lagrangian ¢L = »V jµµ¹µ¹µ = » Z d2µ d2¹µ V: (2.29) It is U(1) gauge and SUSY invariant by itself, however, it may lead to spontaneous breaking of both of them depending on the value of ». The drawback of this mecha nism is the necessity of U(1) gauge invariance. It can be used in SUSY generalizations of the SM but not in GUTs. The mass spectrum also causes some troubles since the following sum rule is always valid STrM2 = X J (¡1)2J(2J + 1)m2 J = 0; (2.30) which is bad for phenomenology. ii) O'Raifeartaigh (Fterm) mechanism [12]. 20 In this case, several chiral ¯elds are needed and the superpotential should be chosen in such way that trivial zero VEVs for the auxiliary F¯elds are forbidden. For instance, choosing the superpotential to be W(©) = ¸©3 + m©1©2 + g©3©21 ; (2.31) one gets the equations for the auxiliary ¯elds F¤ 1 = mA2 + 2gA1A3; (2.32) F¤ 2 = mA1; (2.33) F¤ 3 = ¸ + gA21 ; (2.34) which have no solutions with hFii = 0 and SUSY is spontaneously broken. The drawback of this mechanism is that there is a lot of arbitrariness in the choice of the potential. The sum rule (2.30) is also valid here. Unfortunately, none of these mechanisms explicitly works in SUSY generalizations of the SM. None of the ¯elds of the SM can develop nonzero VEVs for their F or D components without breaking SU(3)C or U(1)Y gauge invariance since they are not singlets with respect to these groups. This requires the presence of extra sources for spontaneous SUSY breaking [13{18]. CHAPTER 3 The Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) [19] respects the same gauge symmetry SU(3)C £SU(2)L £ U(1)Y as does the SM. Here SUSY is somehow (softly) broken at the weak scale. The MSSM is the simplest phenomenologically viable supersymmetric theory beyond the SM in that it contains the fewest number of new particles and new interactions. To construct the MSSM [20] we start with the complete set of chiral fermions, and add a scalar superpartner to each Weyl fermion so that each ¯elds represents a chiral multiplet. Similarly we must add a gaugino for each of the gauge bosons in the SM making up the gauge multiplets. The particles necessary to construct the MSSM are shown in Tables 3.1. and 3.2. Super¯eld SU(3)C SU(2)L U(1)Y Particle Content ^Q 3 2 1 6 (uL; dL), (~uL; ~ dL) ^U c 3 1 ¡2 3 uR, ~u¤R ^D c 3 1 1 3 dR, ~ d¤ R ^L 1 2 ¡1 2 (ºL; eL), (~ºL; ~eL) ^E c 1 1 1 eR, ~e¤ R ^H d 1 2 ¡1 2 (Hd; ~H d) ^H u 1 2 1 2 (Hu; ~H u) TABLE 3.1. Chiral super¯elds of the MSSM. The MSSM is de¯ned by its minimal ¯eld content (which accounts for the known SM ¯elds) and minimal superpotential necessary to account for the known Yukawa 21 22 Super¯eld SU(3)C SU(2)L U(1)Y Particle Content ^G a 8 1 0 G¹, ~g¹ ^W r 1 3 0 W¹ r , ~!¹ r ^B 1 1 0 B¹, ~b ¹ TABLE 3.2. Vector Super¯elds of the MSSM. mass terms. Notice that in Table 3.1. and 3.2., we have introduced a partner for every particle of the SM with the same internal quantum number and a spin di®ering by 1 2 . We de¯ne the MSSM by the superpotential W = ²ij [yeHj dLiec + ydHj dQidc + yuHiu Qjuc + ¹Hid Hju ]: (3.1) Here, the indices, fijg, are SU(2)L doublet indices and ¹ is the Higgs mass parameter. The Yukawa couplings, y, are all 3£3 matrices in generation space. Note that there is no generation index for the Higgs multiplets. Color and generation indices have been suppressed in the above expression. There are two Higgs doublets in the MSSM. This is a necessary addition to the SM which can be seen as arising from the holomorphic property of the superpotential. That is, there would be no way to account for all of the Yukawa terms for both uptype and downtype multiplets with a single Higgs doublet. To avoid a massless Higgs state, a mixing term ²ij¹Hid Hju must be added to the superpotential. However, even if we stick to the minimal ¯eld content, there are several other superpotential terms which we can envision adding to Eq. (3.1) since they are con sistent with all of the symmetries of the theory. We could have considered terms like WR = ¹0iLiHu + ¸ijkLiLjec k + ¸0ijkLiQjdc k + ¸00ijkuci dcj dc k; (3.2) where i; j and k are the generation indices and ¸'s are the coupling constants. In Eq. (3.2), the terms proportional to ¸; ¸0, and ¹0, all violate lepton number by one unit. The term proportional to ¸00 violates baryon number by one unit. 23 Each of the terms in Eq. (3.2) predicts new particle interactions and can be to some extent constrained by the lack of observed exotic phenomena. However, the combination of terms which violate both baryon and lepton number can be disastrous. In order to avoid these unwanted terms, we impose a discrete symmetry on the theory called R{parity [21], which can be de¯ned as R = (¡1)3B+L+2s; (3.3) where B; L, and s are the baryon number, lepton number, and spin respectively. With this de¯nition, it turns out that all of the known SM particles have Rparity +1, and all the superpartners of the known SM particles have R = ¡1, since they must have the same value of B and L but di®er by 1/2 unit of spin. 3.1 Electroweak symmetry breaking and the Higgs boson masses We analyze the scalar potential in this section. It is derived from the superpo tential and the terms involving the Higgs in the soft breaking Lagrangian. The part of the scalar potential which involves only the Higgs bosons (Hu and Hd) is given by V = j¹j2(H¤ dHd + H¤ uHu) + 1 8 g02(H¤ uHu ¡ H¤ dHd)2 + 1 8 g2 ¡ 4jH¤ dHuj2 ¡ 2(H¤ dHd)(H¤ uHu) + (H¤ dHd)2 + (H¤ uHu)2¢ +m2 HdH¤ dHd + m2 HuH¤ uHu + (B¹²ijHid Hju + h.c.): (3.4) Here the ¯rst term is the Fterm, derived from j(@W=@Hd)j2 and j(@W=@Hu)j2 setting all sfermion VEV's equal to 0. The next two terms are D{terms, the ¯rst a U(1)Y D{term, recalling that the hypercharges for the Higgses are YHd = ¡1 2 and YHu = 1 2 , and the second is an SU(2)L D{term, taking Ta = ¾a where ¾a are the three Pauli matrices. Finally, the last three terms are the soft SUSY breaking masses mHd and mHu, and the bilinear term B¹. The Higgs doublets can be written as hHdi = Ã H0 d H¡ d ! ; hHui = Ã H+ u H0 u ! ; (3.5) where in Eq. (3.4) by (H¤ dHd), we mean H0 d ¤H0 d + H¡ d ¤ H¡ d etc. 24 The neutral portion of Eq. (3.4) can be expressed more simply as V = g2 + g02 8 ¡ jH0 d j2 ¡ jH0 uj2¢2 + (m2 Hd + j¹j2)jH0 d j2 +(m2 Hu + j¹j2)jH0 uj2 + (B¹H0 dH0 u + h.c.): (3.6) For electroweak symmetry breaking, it will be required that either one (or both) of the soft masses (m2 Hd ;m2 Hu) be negative (as in the SM). From the minimization of the potential Eq. (3.6), we obtain the following two conditions ¡2B¹ = (m2 Hd + m2 Hu + 2¹2) sin 2¯; (3.7) and v2 = 4 ¡ m2 Hd + ¹2 ¡ (m2 Hu + ¹2) tan2 ¯ ¢ (g2 + g02)(tan2 ¯ ¡ 1) ; (3.8) where tan ¯ = Àu Àd . From the potential and these two conditions, the masses of the physical scalars can be obtained. At the tree level, m2 H§ = m2 A + m2 W; (3.9) m2 A = m2 Hd + m2 Hu + 2¹2 = ¡B¹(tan ¯ + cot ¯); (3.10) m2 H;h = 1 2 · m2 A + m2 Z § q (m2 A + m2 Z)2 ¡ 4m2 Am2 Z cos2 2¯ ¸ : (3.11) Notice that these expressions and the above constraints limit the number of free inputs in the MSSM. First, from the mass of the pseudoscalar, we see that B¹ is not independent and can be expressed in terms of mA and tan ¯. Furthermore from the conditions Eqs. (3.7) and (3.8), we see that if we keep tan ¯, we can either choose mA and ¹ as free inputs thereby determining the two soft masses, mHd and mHu, or we can choose the soft masses as inputs, and ¯x mA and ¹ by the conditions for electroweak symmetry breaking. Both choices of parameter ¯xing are widely used in the literature. The tree level expressions for the Higgs masses make some very de¯nite predic tions. The charged Higgs is heavier than MW, and the light Higgs h, is necessarily lighter than MZ. Note if uncorrected, the MSSM would already be excluded (from current accelerator limits). However, radiative corrections to the Higgs masses are 25 not negligible in the MSSM, particularly for a heavy top mass mt » 178 GeV. The leading oneloop corrections to m2 h depend quartically on mt and can be expressed as [22] ¢m2 h = 3m4t 4¼2v2 ln µ m~t1m~t2 m2t ¶ + 3m4t ^ A2t 8¼2v2 h 2h(m2 ~t1 ;m2 ~t2 ) + ^ A2t g(m2 ~t1 ;m2 ~t2 ) i + : : : (3.12) where m~t1;2 are the physical masses of the two stop squarks ~t1;2 to be discussed in more detail shortly, A^t ´ At + ¹ cot ¯, (At is the SUSY breaking trilinear term associated with the top quark Yukawa coupling). The functions h and f are h(a; b) ´ 1 a ¡ b ln ³a b ´ ; g(a; b) = 1 (a ¡ b)2 · 2 ¡ a + b a ¡ b ln ³a b ´¸ : (3.13) Additional corrections to coupling vertices, twoloop corrections and renormalization group resummations have also been computed in the MSSM [23]. With these correc tions one can allow mh <» 130 GeV; (3.14) within the MSSM. While certainly higher than the tree level limit of MZ, the limit still predicts a relatively light Higgs boson, and allows the MSSM to be experimentally excluded (or veri¯ed!) at the LHC. 3.2 The sfermions masses We turn next to the discussion of scalar partners of the quarks and leptons. The mixing matrices for ~m2t ; ~m2b and ~m2¿ are 0 @ ~m2 tL mt(At + ¹ cot ¯) mt(At + ¹ cot ¯) ~m2 tR 1 A; (3.15) 0 @ ~m2 bL mb(Ab + ¹ tan ¯) mb(Ab + ¹ tan ¯) ~m2 bR 1 A; (3.16) 0 @ ~m2¿ L m¿ (A¿ + ¹ tan ¯) m¿ (A¿ + ¹ tan ¯) ~m2¿ R 1 A; (3.17) 26 with ~m2 tL = ~m2 Q + m2t + 1 6 (4M2W ¡M2Z ) cos 2¯; ~m2 tR = ~m2 U + m2t ¡ 2 3 (M2W ¡M2Z ) cos 2¯; ~m2 bL = ~m2 Q + m2b ¡ 1 6 (2M2W +M2Z ) cos 2¯; ~m2 bR = ~m2 D + m2b + 1 3 (M2W ¡M2Z ) cos 2¯; ~m2¿ L = ~m2 L + m2¿ ¡ 1 2 (2M2W ¡M2Z ) cos 2¯; ~m2¿ R = ~m2 E + m2¿ + (M2W ¡M2Z ) cos 2¯: The ¯rst terms here ( ~m2) are the soft ones, which are calculated using the Renor malization Group (RG) equations starting from their values at the GUT (Planck) scale. The second ones are the usual masses of quarks and leptons and the last ones are the D terms of the potential. The o®diagonal mixing term in the mass matrix is negligible for all but the third generation sfermions. The physical sfermion states and their masses are determined by diagonalizing the sfermion mass matrix. 3.3 Neutralinos There are four new neutral fermions in the MSSM which not only receive mass but mix as well. These are the gauge fermion partners of the neutral B and W3 gauge bosons, and the partners of the Higgs. The two gauginos are called the bino, e B, and wino, fW3 respectively. The latter two are the Higgsinos, eH d and eH u. In addition to the SUSY breaking gaugino mass terms, ¡1 2M1 e B e B, and ¡1 2M2fW3fW3, there are supersymmetric mass contributions of the type WijÃiÃj , giving a mixing term between eH d and eH u, 1 2¹eH deH u, as well as terms of the form g(Á¤TaÃ)¸a giv ing the following mass terms after the appropriate Higgs VEVs have been inserted, p1 2 g0vd eH d Be, ¡p1 2 g0vu eH u Be, ¡p1 2 gvd eH dWf3, and p1 2 gvu eH ufW3. These latter terms can be written in a simpler form noting that for example, g0vd= p 2 = MZ sin µW cos ¯. 27 Thus we can write the neutralino mass matrix as (in the ( e B;fW3; eH 0 d ; eH 0 u) basis) [24] 0 BBBBBB@ M1 0 ¡MZsµW cos ¯ MZsµW sin ¯ 0 M2 MZcµW cos ¯ ¡MZcµW sin ¯ ¡MZsµW cos ¯ MZcµW cos ¯ 0 ¡¹ MZsµW sin ¯ ¡MZcµW sin ¯ ¡¹ 0 1 CCCCCCA ; (3.18) where sµW = sin µW and cµW = cos µW. The mass eigenstates (a linear combination of the four neutralino states) and the mass eigenvalues are found by diagonalizing the mass matrix Eq. (3.18). 3.4 Charginos There are two new charged fermionic states which are the partners of the W§ gauge bosons and the charged Higgs scalars, H§, which are the charged gauginos, fW§ and charged Higgsinos, eH §, or collectively charginos. The chargino mass matrix is composed similarly to the neutralino mass matrix. The result for the mass term is ¡ 1 2 (fW¡; eH ¡) Ã M2 p 2mW sin ¯ p 2mW cos ¯ ¹ ! ÃfW+ eH + ! + h.c. (3.19) Note that unlike the case for neutralinos, two unitary matrices must be constructed to diagonalize Eq. (3.19). The result for the mass eigenstates of the two charginos is m2e c1 ;m2e c2 = 1 2 h M2 2 + ¹2 + 2M2W § q (M2 2 + ¹2 + 2M2W )2 ¡ 4(¹M2 ¡M2W sin 2¯)2 i (3.20) Some additional resources on supersymmetry used in this preliminary introduc tion are the classic by Bagger and Wess on supersymmetry [25], the book by Ross on Grand Uni¯cation [26] and some other good reviews by Martin and others [27{33]. CHAPTER 4 ANOMALY MEDIATED SUPERSYMMETRY BREAKING Understanding the origin of Supersymmetry breaking has been one of the main focuses of SUSY phenomenologists. It is highly non{trivial to construct models which break supersymmetry in a generally acceptable way. The most common scenario for producing low{energy Supersymmetry breaking is called the hidden sector. The usual SM matter ¯elds reside in the visible sector and the ¯elds that break supersymmetry reside in the hidden sector. There are no (small) direct couplings between the two sectors. The symmetry breaking which occurs in the hidden sector is communicated to the visible sector via \ messenger " ¯elds. Some of the several competing proposals on what the mediating interaction might be are Gravity mediation (SUGRA), Gauge mediation, Gaugino mediation and Anomaly mediation. Any successful supersymmetry breaking scenario should at least satisfy the fol lowing conditions: ² The theory should give correct masses to the superpartners » 1 TeV, and the scalar mass{squared should be positive, ² The ¹ parameter should be between 100 GeV { 1 TeV and the B¹ parameter should not be too much larger than ¹2, ² There are no large °avor changing neutral currents, ² CP should be approximately conserved (A & B phase should be small, as required by the measurement of the electric dipole moments of neutron and electron), 28 29 ² The model should be simple enough such that it can be tested experimentally. This thesis is based on the Anomaly mediation scenario of SUSY breaking. Before going into any details of the proposed models, I will brie°y review the other three scenarios and what others have done on anomaly mediation. 4.1 Gravity mediation In this scenario, the messenger is gravity. Supersymmetry is broken in the hidden sector by a VEV hFi. The moduli ¯eld T, which appears as a result of compacti¯cation from higher dimensions and the dilaton ¯eld S, which is part of the SUGRA supermultiplet develop a non{zero VEV for their F components which in turn leads to spontaneous SUSY breaking. The soft mass term in the visible sector is roughly msoft » hFi MPl : (4.1) These soft masses should vanish as hFi ! 0 where SUSY remains unbroken. In this scenario, the SUSY sector is completely described by 5 input parameters: Higgs mass parameter (¹), common scalar mass (m0), common gaugino mass (m1=2), common trilinear coupling (A0) and the Higgs mixing parameter (B). When SUSY is broken at a scale p hFi, the graviton will also obtain a mass msoft » m3=2 » hFi MPl : (4.2) Since we argued earlier that for SUSY to solve the hierarchy problem the mass scale should be msoft »1 TeV, therefore SUSY should be broken at a scale p hFi » 1011 GeV. Some of the good features of the models are ² Extremely predictive{ because the entire low energy spectrum is predicted in terms of few input parameters (m0; m1=2; A0; tan ¯ (B) and sign(¹)), where all phenomenological limits can be expressed in terms of these parameters, 30 ² Gauge couplings are uni¯ed and the gaugino masses are predicted to be the ratios of the gauge couplings, ² The ¹ problem is solved through Guidice{Masiero mechanism, where a singlet ¯eld § in the Kahler potential R d4µ§¤HdHu=MPl breaks SUSY, ² It is easy to generate positive scalar mass{squared. ² Hu mass{squared turns negative due to large top Yukawa coupling even if it starts of being positive at the Planck scale. Despite the success of the theory, there are still some problems which are: CP is generally a problem, large freedom of parameters, absence of automatic suppression of °avor violation, lack of consistent theory of quantum supergravity (local symmetry). 4.2 Gauge mediation In this scenario the Supersymmetry breaking is communicated from the hidden sector to the visible sector via gauge interactions. The main idea is to introduce new chiral multiplets (messengers) which couple indirectly to the MSSM ¯elds through the SU(3)C £ SU(2)L £ U(1)Y gauge interactions. The particles ((s)quarks and (s)leptons) gets large mass by coupling to a gauge singlet chiral supermultiplet S. The superpotential for a typical gauge mediation can be written as W = ¸1S`¹` + ¸2Sq¹q: (4.3) The singlet scalar S and the auxiliary component of S (Fs) acquires a VEV by putting the scalar ¯eld into an O'Raifeartaigh{type model or a dynamical mechanism. The gauginos get mass at 1{loop Mi » ®i 4¼ ¤ (i = 1; 2; 3); (4.4) where ¤ = Fs=hSi. The MSSM scalars do not get any radiative corrections to their masses at 1{ loop. Their masses arise at 2{loop level from those diagrams involving the gauge 31 ¯elds and the messengers. The scalar masses are given by ~m » µ ¤ 4¼ ¶2 f®2 3C3 + ®2 2C2 + ®2 1C1g; (4.5) where Ci are the quadratic Casimir operators for the SU(3)C£SU(2)L£U(1)Y gauge group. This implies that the sparticles with the same gauge quantum number will have equal masses (for example: ~me = ~m¹ = ~m¿ ). In order for the gauginos and scalar soft masses to be » 1 TeV (as needed for the hierarchy problem) requires ¤ » 104 ¡ 105 GeV. In most of the gauge mediation models, the slepton and squark masses depend only on their gauge quantum num bers. This leads to the degeneracy of squark and slepton mass which results in the suppression of °avor changing neutral currents (FCNC's). The Lightest Supersym metric Particle (LSP) is usually the gravitino, with mass m3=2 » ¤2=Mpl » 10¡10 GeV, which can be crucial both for cosmology and collider physics. In summary: ² gauge mediated supersymmetry breaking (GMSB) solves the FCNC problem, ² gaugino mass arise at 1{loop while the scalar mass{squared arise at two loop level, ² there is still a problem in the Higgs sector (o®ers no compelling solution to the ¹ problem), ² it does not o®er any solution to the SUSY CP problem. 4.3 Gaugino mediation In this scenario the SM quark and lepton ¯elds are localized on a `3{brane' in extra dimensions, while the gauge and Higgs ¯elds propagate in the bulk. SUSY breaking masses for the gauginos and Higgs ¯elds are generated by higher{dimensional contact terms between the bulk ¯elds and the hidden sector ¯elds, assumed to arise from a more fundamental theory such as string theory [34]. The leading contribution to the SUSY breaking for visible sector ¯elds arises from loops of bulk gauge and Higgs ¯elds as shown in Fig. 4.1 32 Figure 4.1. Leading diagram that contributes to SUSY{breaking scalar masses. The bulk line is a gaugino propagator. The minimal version of gaugino mediation has only three high energy param eters ¹; m1=2 and Mc. Here m1=2 is the universal gaugino mass at the uni¯cation scale and Mc is the compacti¯cation scale where the higher dimensional theory is matched onto the e®ective four{dimensional theory. For sin2 µW prediction to be pre served from gauge coupling uni¯cation requires Mc > MGUT . In some other models of gaugino mediation [35] the ¹ parameter is determined by ¯tting to the Z mass. Such model requires only two free parameters m1=2 and Mc. The gaugino mediation scenario is the least developed in the literature. It does not o®er any real solution to the ¹ problem. 4.4 Anomaly mediation This scenario assumes that supersymmetry breaking takes place in a hidden or sequestered sector. The MSSM super¯elds are con¯ned to a 3{brane in a higher dimensional bulk space{time separated from the sequestered sector by extra dimen sions. A rescaling super{Weyl anomaly generates coupling of the auxiliary ¯eld of the gravity multiplet to the gauginos and the scalars of the MSSM, with the couplings determined by the SUSY renormalization group equations (RGE) [36]. 33 Before going into much details, it is important to give a brief review on how this scenario address the numerous problems associated with the other three scenarios addressed earlier. ² The ¹ parameter can be generated without generating excessively large B¹ due to the constraints from the coupling of the gravitational multiplet. ² The dominant anomaly{mediated contribution to the squark and slepton masses suppresses °avor violation automatically. ² There are no new phases in the A and B terms. This implies a natural solution to the SUSY CP problem. In other words CP can be violated on our 3{brane and nowhere else. ² The model is straightforward in the sense that the basic assumption is that SUSY breaking is derived from higher dimensional theory. ² These SUSY breaking models are very predictive. The ratio of the gaugino masses depends on the beta functions rather than the gauge couplings. The A{ terms are predicted to be proportional to the corresponding Yukawa coupling and there is a nearly degenerate Wino/Zino LSP, of which the Zino is the lighter. ² The gaugino and scalar masses are comparable. ² Since the rescaling anomaly is UV insensitive, the pattern of SUSY breaking masses at any energy scale is governed only by the physics at that scale [36{38]. An arbitrary °avor structure in the SUSY scalar spectrum at high energies gets washed out at low energies. This Ultraviolet (UV) insensitivity provides an elegant solution to the SUSY °avor problem. ² It can naturally solve the cosmological gravitino abundance problem which tends to destroy the success of big bang cosmology in generic supergravity models [39]. 34 ² The decay of the moduli ¯elds present in the model (as well as the gravitino) will produce neutralinos, especially the neutral Winos, with the right abundance to make it a viable cold dark matter candidate [40; 41]. We see from above that this model seems to be a viable (promising) model for understanding the MSSM supersymmetry breaking. It turns out that there is a major problem in this model which is discussed below. 4.4.1 The negative slepton mass problem of anomaly mediated supersymmetry breaking In anomaly mediated supersymmetry breaking models (AMSB), the masses of the scalar components of the chiral supermultiplets are given by [36; 37] (m2)Áj Ái = 1 2 M2 aux · ¯(Y ) @ @Y °Áj Ái + ¯(g) @ @g °Áj Ái ¸ : (4.6) In the above equation summations over the gauge couplings g and the Yukawa cou plings Y are assumed. °Áj Ái are the one{loop anomalous dimensions, ¯(Y ) is the beta function for the Yukawa coupling Y , and ¯(g) is the beta function for the gauge coupling g. Maux is the vacuum expectation value of a \compensator super¯eld" [36] which sets the scale of SUSY breaking. The gaugino mass Mg associated with the gauge group with coupling g is given by [36; 37] Mg = ¯(g) g Maux: (4.7) The trilinear soft supersymmetry breaking term AY corresponding to the Yukawa coupling Y is given by [36; 37] AY = ¡ ¯(Y ) Y Maux: (4.8) In the simplest scenario for generating the ¹ term for a special class of models, the contribution to the Higgs mixing parameter (the Bterm) is given by [36] B = ¡(°Hu + °Hd)Maux: (4.9) Here °Hu and °Hd are the one{loop anomalous dimensions of the Hu and Hd ¯elds. Similar relations hold for other bilinear terms in the SUSY breaking Lagrangian. 35 In the minimal scenario, it turns out that AMSB induces a negative mass{ squared for the sleptons. Such a scenario is excluded since it would break electro magnetism. The reason for the negative mass{squared can be understood as follows. There are two sources for slepton masses in AMSB, the Yukawa part and the gauge part (cf: Eq. (4.6)). For the ¯rst two families the Yukawa couplings are negligible and the dominant contributions arise proportional to the gauge beta function. Since in the MSSM the SU(2)L and the U(1)Y gauge couplings are not asymptotically free, their gauge beta functions are positive. This makes the slepton mass{squared nega tive. In the squark sector, the masses are positive because SU(3)C gauge theory is asymptotically free. 4.4.2 Suggested solutions to the AMSB slepton mass problem Several possible ways of avoiding the slepton mass problem of AMSB have been suggested. A non{decoupling universal bulk contribution to all the scalar masses is a widely studied option [36{42]. While this will make the minimal model phenomeno logically consistent, the UV insensitivity of AMSB is no longer guaranteed. It is therefore interesting to investigate variations of the minimal model which maintain the UV insensitivity but provide positive mass{squared for the sleptons from physics at the TeV scale. One way to avoid the negative slepton mass problem with TeV scale physics is to increase the Yukawa contributions in Eq. (4.6). This can be achieved by introducing new particles at the TeV scale with large Yukawa couplings to the lepton ¯elds. This possibility was studied in Ref. [43] where the MSSM spectrum was extended to include 3 pairs of Higgs doublets, four singlets and a vector{like pair of color{triplets near the weak scale. The Yukawa contributions can also be enhanced by invoking R{parity violating couplings in the MSSM [44]. Unfortunately such a theory would generate unacceptably large neutrino masses. Yet another possibility is to make use of the positive D{term contributions from a U(1) gauge symmetry broken at the weak scale. This was achieved by adding TeV scale Fayet{Iliopoulos terms explicitly to the theory in Ref. [45]. New D{term contributions generated in a controlled fashion 36 by the breaking of U(1)B¡L at an arbitrary high scale may also provide positive contributions to the slepton masses [46; 47]. A low scale ancillary U(1) as a solution to the problem has been studied in Ref. [48]. E®ective supersymmetric theories which are devoid of the negative slepton mass problem of AMSB with new dynamics at the 10 TeV scale have been studied in Ref. [49]. Non{decoupling e®ects of heavy ¯elds at higher orders have been analyzed in AMSB models in Ref. [50] as an attempt to solve the slepton mass problem. CHAPTER 5 TeV{Scale Horizontal Symmetry and the Slepton Mass Problem of Anomaly Mediation 5.1 Introduction As noted in chapter 4, supersymmetry provides an elegant solution to the gauge hierarchy problem of the standard model. To be realistic, it must however be a broken symmetry. There are several ways of achieving supersymmetry (SUSY) breaking. Anomaly mediated SUSY breaking (AMSB) is an attractive and predictive scenario which has the virtue that it can solve the SUSY °avor problem [36; 37]. This scenario assumes that SUSY breaking takes place in a hidden or sequestered sector. The MSSM super¯elds are con¯ned to a 3{brane in a higher dimensional bulk space{ time separated from the sequestered sector by extra dimensions. A rescaling super{ Weyl anomaly generates coupling of the auxiliary ¯eld of the gravity multiplet to the gauginos and the scalars of the MSSM, with the couplings determined by the SUSY renormalization group equations (RGE). Since the rescaling anomaly is UV insensitive, the pattern of SUSY breaking masses at any energy scale is governed only by the physics at that scale [36{38]. Arbitrary °avor structure in the SUSY scalar spectrum at high energies gets washed out at low energies. This ultraviolet insensitivity provides an elegant solution to the SUSY °avor problem. The purpose of this thesis is to suggest and investigate the possibility of solving the negative slepton mass problem by making the gauge contribution in Eq. (4.6) positive. This can only be achieved by introducing a new non{Abelian gauge sym metry for leptons with negative gauge beta function. We point out that an SU(3)H horizontal symmetry acting on the lepton multiplets has all the desired properties 37 38 for achieving this. We show that such an SU(3)H horizontal symmetry broken at the TeV scale is consistent with rare leptonic processes owing to the emergence of approximate global symmetries. The speci¯c AMSB model we study is quite predictive. The lightest Higgs boson mass is predicted to be mh . 120 GeV, and the parameter tan ¯ is found to be tan ¯ ' 4. The model predicts the existence of new particles associated with the SU(3)H symmetry breaking sector. The SU(3)H vector bosons have masses of order 1{4 TeV. These particles should be accessible experimentally at the LHC. The plan of the chapter is as follows. In section 5.2 we introduce our model. In section 5.3 we analyze the Higgs potential of the model. Here we derive the limits on tan ¯ and mh. In section 5.4 we present the SUSY spectrum of the model and show how the sleptons acquire positive masses. Numerical results for the full spectrum of the model are given in section 5.5. In section 5.6 we outline the most signi¯cant experimental consequences of the model. In section 5.7 we comment on the possible origins of the ¹ and the B¹ terms. We summarize in section 5.8. In Appendix A, we give the relevant beta functions, anomalous dimensions as well as the soft masses. 5.2 SU(3)H horizontal symmetry In this section we present our model. Since our aim is to have positive con tributions to the slepton masses from the gauge sector, we are naturally led to a leptonic horizontal symmetry that is asymptotically free. Our model is based on the gauge group SU(3)C £ SU(2)L £ U(1)Y £ SU(3)H, where SU(3)H is a horizontal symmetry acting on the leptons. The left{handed lepton doublets and the antilepton singlets transform as fundamental representations of the SU(3)H gauge symmetry. The theory is made anomaly free by introducing three Higgs multiplets (©1, ©2, ©3) which transform as antifundamental representations of SU(3)H and as singlets of the standard model. These ¯elds are su±cient for breaking the SU(3)H symmetry com pletely near the TeV scale. The particle content of the model and the transformation properties under the gauge group SU(3)C £SU(2)L £U(1)Y £SU(3)H are presented in Table 5.1. It turns out that the Higgs potential involving these ©i ¯elds exhibits 39 a global SU(3)G symmetry. We take advantage of this global symmetry to suppress potentially large °avor changing neutral current processes mediated by the SU(3)H gauge bosons. The last column in Table 5.1 lists the transformation properties under the global SU(3)G symmetry (The Yukawa couplings of the model reduce the global SU(3)G down to U(1).) The ¯elds ´i and ¹´i are introduced to facilitate SU(3)H symmetry breaking within our AMSB framework. Super¯eld SU(3)C SU(2)L U(1)Y SU(3)H SU(3)G Qi 3 2 1 6 1 1 uci ¹3 1 ¡2 3 1 1 dci ¹3 1 1 3 1 1 L® 1 2 ¡1 2 3 1 ec ® 1 1 1 3 1 Hu 1 2 1 2 1 1 Hd 1 2 ¡1 2 1 1 ©®i 1 1 0 ¹3 3 ´i 1 1 0 ¹3 3 ¹´i 1 1 0 3 ¹3 TABLE 5.1. Particle content and charge assignment of the model. SU(3)G in the last column is a softly broken global symmetry present in the model. The indices i and ® take values i; ® = 1 ¡ 3. Note that the quarks are neutral under SU(3)H. This is necessitated by the requirements that SU(3)H be asymptotically free. A separate SU(3)H0 acting on the quarks is a possible quark{lepton symmetric extension of the model. But we do not pursue such an extension here. The superpotential of the model consistent with the gauge symmetries and the global SU(3)G symmetry is given by: W = (Yu)ij QiHuucj + (Yd)ij QiHddcj + ¹HuHd + ·©®1 ©¯ 2©° 3²®¯° + ¸´® a ´¯ b ©°c ²®¯°²abc +M´´a¹´a: (5.1) 40 Here ®, ¯, ° =1, 2, 3 are SU(3)H indices, i; j = 1, 2, 3 are family indices, and a; b; c = 1, 2, 3 are SU(3)G indices. The mass parameters ¹ and M´ are of order TeV, which has a natural origin in AMSB [36]. We will comment on possible origin of these terms in Sec. 5.7. In the SU(3)H symmetric limit the leptons are all massless. They obtain their masses from the e®ective operators Ll eff = L®ec ®©®i ©®i Hd M2 i : (5.2) Such operators can be obtained by integrating ¯elds shown in Fig. 1, for example. The masses of the heavy ¯elds break SU(3)G symmetry softly (the ¹ ÃiÃi and the ¹E iEi mass terms in Fig. 5.1). Note that the mass scale Mi in Eq. (5.2) is of order 5 La Fi a yi yi E Ei i Hd ec a Fi a Figure 5.1. E®ective operators inducing charged lepton masses. TeV for generating realistic ¿{lepton mass, of order 20 TeV for the ¹ mass and of order 300 TeV for the electron mass (assuming that all relevant Yukawa couplings are of order one). Since these masses are all much heavier than the e®ective SUSY breaking scale of order 1 TeV, these heavy ¯elds will have no e®ect in the low energy SUSY phenomenology within AMSB. Note that no generation mixing is induced by these e®ective operators, which will guarantee the approximate conservation of electron number, muon number and tau lepton number. This is what makes the model consistent with FCNC data even when SU(3)H is broken at the TeV scale. Since the Higgs potential respects SU(3)H £ SU(3)G symmetry, after spontaneous 41 symmetry breaking, the diagonal subgroup SU(3)G+H remains as an unbroken global symmetry. This subgroup contains e, ¹ and ¿ lepton numbers. Since right{handed neutrinos are not required to be light for SU(3)H anomaly cancellation, they acquire heavy masses and decouple from the low energy theory. Small neutrino masses are then induced through the seesaw mechanism. Speci¯cally, the following e®ective nonrenormalizable operators emerge after integrating out the heavy right{handed neutrino ¯elds: Lº eff = ¸®¯ ij L®L¯HuHu©®i ©¯ j M3N : (5.3) Here MN represents the masses of the heavy right{handed neutrino ¯elds. For MN » 107 GeV and h©ii » TeV, neutrino masses in the right range for oscillation phe nomenology are obtained. Note that Eq. (5.3) arises from integrating neutral leptons with their masses assumed to break all global symmetries. This enables generation of large neutrino mixing angles, as needed for phenomenology. 5.3 Symmetry breaking The SU(3)H model has two sets of Higgs bosons: the usual MSSM Higgs dou blets Hu and Hd, and the SU(3)H Higgs antitriplets ©i (i = 1; 2; 3). The Higgs potential is derived from the superpotential of Eq. (5.1) and includes the soft terms and the D terms. The tree level potential splits into two pieces: V (Hu;Hd;©i) = V (Hu;Hd) + V (©i); (5.4) enabling us to analyze them independently. The ¯rst part, V (Hu;Hd), is identical to the MSSM potential which is well studied. There are however signi¯cant constraints on the parameters in our AMSB extension, which we now discuss. 5.3.1 Constraints on tan ¯ and mh Minimization of V (Hu;Hd) gives sin 2¯ = 2B¹ 2¹2 + m2 Hu + m2 Hd ; ¹2 = m2 Hd ¡ m2 Hu tan2 ¯ tan2 ¯ ¡ 1 ¡ M2Z 2 : (5.5) 42 Here m2 Hu and m2 Hd are the Higgs soft masses and are given in the Appendix for the AMSB model (see Eqs. (A.19){(A.20).) The constraints on mh and tan ¯ arise since these soft masses and the B parameter are determined in terms of a single parameter Maux in our framework. We eliminate Maux in favor of M2, the Wino mass (M2 = b2g2 2 16¼2Maux). We see from Eqs. (4.9), (5.5) as well as from Eqs. (A.6){(A.7) and Eqs. (A.19){(A.20) of the Appendix that tan ¯ is ¯xed in terms of M2. In Fig. 5.2 we plot tan ¯ as a function of M2. For the experimentally interesting range of M2 & 100 GeV, we ¯nd that tan ¯ ' 3.8 { 4.0. In obtaining the limit on tan ¯, we followed the following procedure. As inputs at MZ we chose [51] ®3(MZ) = 0:119; sin2 µW = 0:2312; ®(MZ) = 1 127:9 : (5.6) Using the central value of Mt = 174:3 GeV, we obtain the running mass mt(Mt) with the 2{loop QCD correction as [52] Mt mt(Mt) = 1 + 4 3 ®3(Mt) ¼ + 10:9 µ ®3(Mt) ¼ ¶2 : (5.7) Using 5{°avor SM QCD beta functions we extrapolated ®3(MZ) and obtained ®3(Mt) = 0.109. The top quark Yukawa coupling is then found to be (for Mt = 174.3 GeV) Y SM t (Mt) = 0.935 corresponding to mt(Mt) = 162:8 GeV. This coupling is then evolved from Mt to 1 TeV where we minimize the MSSM Higgs potential. Using standard model beta function we obtain Y SM t (1 TeV) = 0.851. The corresponding MSSM coupling is Yt(1 TeV) = Y SM t (1 TeV)= sin ¯ , which for tan ¯ ' 4:0 (justi¯ed a{posteriori) is Yt(1 TeV) = 0:824. The gauge couplings at 1 TeV are found to be g1(1 TeV) = 0:466; g2(1 TeV) = 0:642 and g3(1 TeV) = 1:098. With these values of couplings at 1 TeV we obtained Fig. 5.2. Uncertainties in the prediction for tan ¯ are estimated to be §0:5, arising from the error in top quark mass and from the precise scale at which the Higgs potential is minimized. We conclude that tan ¯ = 3.5{4.5 in this model. Since tan ¯ is ¯xed and since the At parameter is not free in AMSB, there is a nontrivial prediction for the lightest Higgs boson mass mh. We use the 2{loop 43 2.5 3 3.5 4 50 100 150 200 M2 (GeV) tanb Figure 5.2. Plot of tan ¯ as a function of M2 radiatively corrected expression for m2 h = (m2 h)o+¢m2 h, where (m2 h)o is the tree{level value of the mass and the radiative correction is given by [53] ¢m2 h = 3m4t 4¼2À2 · t + Xt + 1 16¼2 µ 3 2 m2t À2 ¡ 32¼®3(Mt) ¶ (2Xtt + t2) ¸ : (5.8) Here Xt = A~t 2 m2 ~t Ã 1 ¡ A~t 2 12m2 ~t ! ; A~t = At ¡ ¹ cot ¯; (5.9) and t =log( m2 ~t M2 t ), À = 174 GeV. m2 ~t is the arithmetic average of the diagonal entries of the squared stop mass matrix and At is the soft trilinear coupling associated with the top Yukawa coupling in the superpotential of Eq. (5.1). In these expressions, mt is the one{loop QCD corrected running mass, mt = Mt 1+4 3 ®3(Mt) ¼ , which equals 166.7 GeV for Mt = 174:3 GeV. We ¯nd that mh ' 113 GeV { 120 GeV, depending on the choice of Maux. The larger value mh ' 120 GeV is realized only for larger Mt ' 180 GeV. We list in Tables 5.2{5.4 the value of mh, along with the other sparticle masses. 44 5.3.2 SU(3)H symmetry breaking Let us now analyze the SU(3)H symmetry breaking sector of the potential. The potential V (©i) is given by: V (©i) = m2 Á(©y 1©1 + ©y 2©2 + ©y 3©3) + ·A· ³ ©®1 ©¯ 2©° 3²®¯° + c:c ´ + ·2 £ (©1©2)y(©1©2) + (©1©3)y(©1©3) + (©2©3)y(©2©3) ¤ + g2 4 8 X8 a=1 j©y 1¸a©1 + ©y 2¸a©2 + ©y 3¸a©3j2: (5.10) Here g4 is the gauge coupling of the SU(3)H, A· is the trilinear A{term corresponding to the coupling ·, m2 Á is the soft mass squared for the ©i ¯elds. These soft SUSY breaking parameters are given in the Appendix (Eqs. (A.17), (A.23)). The ·2 term in Eq. (5.10) is the Fterm contribution and the last term in Eq. (5.10) is the SU(3)H D{term with ¸a being the SU(3)H generators. The Higgs potential, Eq. (5.10), has an SU(3)H £ SU(3)G symmetry, with the ©i ¯elds (i = 1¡3) transforming as (¹3 ; 3). This allows for a vacuum which preserves an SU(3)H+G diagonal subgroup. The VEVs of the ©i ¯elds are then given by: h©1i = 0 BB@ u 0 0 1 CCA ; h©2i = 0 BB@ 0 u 0 1 CCA and h©3i = 0 BB@ 0 0 u 1 CCA : (5.11) Using these VEVs the potential becomes hV (©)i = 3m2 Áu2 + 3·2u4 + 2·A·u3: (5.12) Minimization of Eq. (5.12) leads to the condition u = ¡A· § q ¡8m2 Á + A2 · 4· : (5.13) The argument in the square root of Eq. (5.13), which should be positive for a consis tent symmetry breaking, is given by ¡8m2 Á + A2 · = M2 aux (16¼2)2 [15·4 + 56·2¸2 + 304¸4 ¡ 8·2g2 4 ¡ 32¸2g2 4]: (5.14) 45 Positivity of Eq. (5.14) leads to constraints on the parameters f¸; ·g. It can be shown that Eq. (5.14) implies 0 6 j·j 6 0:731g4 and 0 6 j¸j 6 0:324g4. Furthermore, positivity of the slepton masses, along with the experimental limit m2 slepton & (100 GeV)2, require g4 > 0:5. This essentially ¯xes the parameter space of the model. We get the right minimum by choosing the negative sign of the square root in Eq. (5.13) (for positive Maux), with this choice, all the Higgs masses{squared will be positive. Since the symmetry breaking chain is SU(3)H £ SU(3)G ! SU(3)H+G, we can classify the masses of all scalars and fermions as multiplets of SU(3)H+G. The complex ©(¹3; 3) scalar multiplet decomposes into 2 octets and two singlets of SU(3)H+G. One octet gets eaten by the Higgs mechanism. A physical octet remains in the spectrum with a mass given by M2 octet = ¡2·2u2 ¡ 2·uA· + g2 4u2: (5.15) There are two singlets, one scalar (Ás) and one pseudoscalar (Áp) with masses given by m2 Ás = 4·2u2 + ·uA·; (5.16) m2 Áp = ¡3·uA·: (5.17) In the fermionic sector, the octet Higgsino mixes with the octet gaugino with a mixing matrix given by M0 octet = Ã m4 g4u g4u ·u ! : (5.18) In addition, there is a Majorana fermion, a singlet of SU(3)H+G, with a mass of m~Á = 2·u: (5.19) Finally the gauge bosons form an octet with a mass MV = g4u: (5.20) 5.4 The SUSY spectrum We are now ready to discuss the full SUSY spectrum of the model. We will see that the tachyonic slepton problem is cured by virtue of the positive contribution from the SU(3)H gauge sector. 46 5.4.1 Slepton masses The slepton mass{squareds are given by the eigenvalues of the mass matrices (® = e; ¹; ¿ ) M2 ~l = Ã m2 ~L ® mE® ¡ AYE® ¡ ¹ tan ¯ ¢ mE® ¡ AYE® ¡ ¹ tan ¯ ¢ m2 ~ec ® ! : (5.21) Here m2 ~L ® = M2 aux (16¼2) · YE®¯(YE®) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + m2 E® + µ ¡ 1 2 + sin2 µW ¶ cos 2¯M2Z ; (5.22) m2 ~ec ® = M2 aux (16¼2) · 2YE®¯(YE®) ¡ µ 6 5 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + m2 E® ¡ sin2 µW cos 2¯M2Z : (5.23) The o® diagonal terms in Eq. (5.21) are rather small as they are proportional to the lepton masses. The SUSY soft masses are calculated from the RGE give in the Appendix. The last terms of Eqs. (5.22){(5.23) are the D{terms. Note the positive contribution from the SU(3)H gauge sector in Eqs. (5.22){(5.23), given by the term ¡8 3g4¯(g4). In our model g4 is asymptotically free with ¯(g4) = ¡ 3 16¼2 g3 4. This contribution makes the mass{squared of all sleptons to be positive for g4 > 0:5. The left handed sneutrino mass is given by m2 ~ºi = M2 aux (16¼2) · ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 8 3 g4¯(g4) ¶¸ + 1 2 cos 2¯M2Z ; (5.24) where i = e; ¹; ¿ . 5.4.2 Squark masses The mixing matrix for the squark sector is similar to the slepton sector. The diagonal entries of the up and the down squark mass matrices are given by [27] m2 ~U i = (m2 soft) ~Q i ~Q i + m2 Ui + 1 6 ¡ 4M2W ¡M2Z ¢ cos 2¯; 47 m2 ~U c i = (m2 soft) ~U c i ~U c i + m2 Ui ¡ 2 3 ¡ M2W ¡M2Z ¢ cos 2¯; m2 ~D i = (m2 soft) ~Q i ~Q i + m2 Di ¡ 1 6 ¡ 2M2W +M2Z ¢ cos 2¯; m2 ~D c i = (m2 soft) ~D c i ~D c i + m2 Di + 1 3 ¡ M2W ¡M2Z ¢ cos 2¯: (5.25) Here mUi and mDi are quark masses of di®erent generations, i = 1, 2, 3. The squark soft masses are obtained from the RGE as (m2 soft) ~Q i ~Q i = M2 aux 16¼2 µ Yui¯(Yui) + Ydi¯(Ydi) ¡ 1 30 g1¯(g1) ¡ 3 2 g2¯(g2) ¡ 8 3 g3¯(g3) ¶ (5; .26) (m2 soft) ~U c i ~U c i = M2 aux 16¼2 µ 2Yui¯(Yui) ¡ 8 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ ; (5.27) (m2 soft) ~D c i ~D c i = M2 aux 16¼2 µ 2Ydi¯(Ydi) ¡ 2 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ : (5.28) 5.4.3 ´ fermion and ´ scalar masses The ¯elds ´ and ¹´ transform as (3; ¹3) and (¹3; 3) under SU(3)H £SU(3)G. After symmetry breaking, ´ and ¹´ both transform as 8 + 1 of the diagonal SU(3)H+G. The octet from ´ mixes with the octet from ¹´, and similarly for the singlets. In the fermionic sector, the octet and the singlet mass matrices are given by M´ octet = Ã ¡2¸u M´ M´ 0 ! ; (5.29) M´ singlet = Ã 4¸u M´ M´ 0 ! : (5.30) In the scalar sector, there are 4 real octets and 4 real singlets from ´ and ¹´ ¯elds. The two scalar octets are mixed, as are the two pseudoscalar octets. The mass squared matrices for the octet are M2 s¡octet = Ã ( ~m2 soft)´´ +M2 ´ + 2¸u(¡A¸ ¡ ·u + 2¸u) M´(B´ ¡ 2¸u) M´(B´ ¡ 2¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (;5.31) M2 p¡octet = Ã ( ~m2 soft)´´ +M2 ´ + 2¸u(A¸ + ·u + 2¸u) ¡M´(B´ + 2¸u) ¡M´(B´ + 2¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (:5.32) 48 The singlet scalar mass matrices are M2 s¡singlet = Ã ( ~m2 soft)´´ +M2 ´ + 4¸u(A¸ + ·u + 4¸u) M´(B´ + 4¸u) M´(B´ + 4¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! ; (5.33) M2 p¡singlet = Ã ( ~m2 soft)´´ +M2 ´ ¡ 4¸u(A¸ ¡ ·u ¡ 4¸u) ¡M´(B´ ¡ 4¸u) ¡M´(B´ ¡ 4¸u) ( ~m2 soft)¹´ ¹´ +M2 ´ ! (5:.34) The soft masses ( ~m2 soft)´´ and ( ~m2 soft)´´ are given in Eqs. (61){(62) of the Appendix. 5.5 Numerical results We are now ready to present our numerical results for the SUSY spectrum. The scale of SUSY breaking, Maux, should be in the range 40{120 TeV for the MSSM particles to have masses in the range 100 GeV { 2 TeV. Note that there is a large hierarchy in the masses of the gluino and the neutral Wino, M3 M2 ' 7:1 (after taking account of radiative corrections), in AMSB models. Furthermore the lightest chargino is nearly mass degenerate with the neutral Wino, so M2 & 100 GeV is required to satisfy the LEP chargino mass bound. The SU(3)H gauge coupling g4 is chosen so that the sleptons have positive mass squared (g4 > 0:5). We allow g4 to take two values, g4 = 0.55 (Tables 5.2 and 5.4) and g4 = 1.0 (Table 5.3). Symmetry breaking considerations constrain the couplings · and ¸ as discussed in Sec. 5.3 after Eq. 5.14. In Tables 5.2 and 5.4 we have taken Maux = 47.112 TeV corresponding to a light spectrum, while in Table we have Maux = 66.695 TeV with an intermediate spectrum. Other input parameters are listed in the respective Table captions. The mass parameter M´ cannot be much larger than 1 TeV, as that would decouple the e®ects of ´, ¹´ ¯elds which are needed for consistent symmetry breaking. We see from Table 5.2 that the lightest Higgs boson mass is mh ' 118 GeV. This is very close to the current experimental limit. If Mt = 180 GeV is used (instead of Mt = 176 GeV), for the same set of input parameters, mh will be 119 GeV. mh being close to the current experimental limit is a generic prediction of our framework. It holds in the spectra of Tables 5.3 and as well. We conclude that mh . 120 GeV in this model. 49 The masses of the sleptons will depend sensitively on the choice of g4. The sleptons are relatively light, mslep . 300 GeV, with g4 = 0:55, while they are heavy, mslep ' 800 GeV, when g4 = 1:0. Note however that there is a correlation in the slepton masses and the SU(3)H gauge boson masses (MV ), with the lighter sleptons corresponding to lighter SU(3)H gauge bosons. It is worth noting that very light sleptons, below the current experimental limits of about 100 GeV, would be incon sistent with the limits on MV arising from e+e¡ ! ¹+¹¡ type processes (see Sec. 6). Note also that the left{handed and the right{handed sleptons are nearly degenerate to within about 10 GeV in this model. This a numerical coincidence having to do with the values of g1 and g2 and the MSSM beta functions (see the last paper of Ref. [39]). The new SU(3)H gauge boson contributions to the slepton masses are the same for the left{handed and the right{handed sleptons. In Tables 5.2{5.4 we have included the leading radiative corrections to the gaugino masses M1, M2 and M3 [54]. Including these radiative corrections we ¯nd (in Table 2) M1 : M2 : M3 = 3:0 : 1 : 7:4. The lightest SUSY particle (LSP) is the neutral Wino, which is nearly mass degenerate with the charged Wino. In Tables 5.2{5.4 the mass splitting is about 60 MeV, but this does not take into account SU(2)L £U(1)Y breaking corrections [55]. These electroweak radiative corrections turn out to be very important, and we ¯nd mÂ§ 1 ¡ mÂ01 ' 235 MeV (with about 175 MeV arising from SU(2)L £ U(1)Y breaking e®ects). The decay Â§ 1 ! Â01 + ¼§ is then kinematically allowed, with the ¼§ being very soft. Once produced, the neutralino Â01 will escape the detector without leaving any tracks. With the decay channel Â§ 1 ! Â01 +¼§ open, the lightest chargino will leave an observable track with a decay length of about a few cm. Search strategies for such a quasi{degenerate pair at colliders have been analyzed in Ref. [54; 56; 57]. In the SU(3)H sector, in Tables 5.2{5.4, the horizontal gauge boson has a mass of 1.5{4.0 TeV. The heavy Higgs bosons, Higgsinos, gauginos, squarks and the ´ ¯elds all have masses . (1 ¡ 2) TeV. 50 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:146; 0:431g Neutralinos fm~Â03 ; m~Â04 g f0:876; 0:878g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:146; 0:880g Gluino M3 1:064 Higgs bosons fmh; mH; mA; mH§g f0:118; 0:878; 0:877; 0:880g R.H sleptons fm~eR; m~¹R; m~¿1g f0:183; 0:183; 0:166g L.H sleptons fm~eL; m~¹L; m~¿2g f0:190; 0:190; 0:203g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:175; 0:175; 0:175g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:017; 1:017; 1:015g L.H down squarks fm~ dL ; m~sL; m~b 2 g f1:008; 1:008; 0:885g R.H up squarks fm~uR; m~cR; m~t1g f1:011; 1:011; 0:669g L.H up squarks fm~uL; m~cL; m~t2g f1:005; 1:005; 0:979g New Particles Symbol Mass (TeV) SU(3)H Gauge boson octet MV 2:213 Singlet Higgsino m~Á 0:402 Octet Higgsino/gaugino m~Á1;2 f1:978; 2:450g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:179; 0:624; 2:253g Fermionic ´ (octet) moctet ´1;2 f0:676; 1:480g Fermionic ´ (singlet) msinglet ´1;2 f0:479; 2:089g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:454; 1:703g Pseudoscalar ´ Higgs (octet) mp¡octet ~´1;2 f0:908; 1:259g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:717; 1:868g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:264; 2:310g TABLE 5.2. Sparticle masses for the choiceMaux = 47:112 TeV, tan ¯ = 3:785, ¹ = ¡0:873 TeV, yb = 0:068, ¸ = 0:1, · = 0:05, g4 = 0:55, u = ¡4:024 TeV, M´ = 1:0 TeV and Mt = 0:176 TeV. 51 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:198; 0:586g Neutralinos fm~Â03 ; m~Â04 g f1:179; 1:181g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:198; 1:182g Gluino M3 1:410 Higgs boson fmh; mH; mA; mH§g f0:119; 1:179; 1:178; 1:181g R.H sleptons fm~eR; m~¹R; m~¿1g f0:245; 0:245; 0:227g L.H sleptons fm~eL; m~¹L; m~¿2g f0:254; 0:254; 0:267g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:242; 0:242; 0:242g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:373; 1:373; 1:193g L.H down squraks fm~ dL ; m~sL; m~b 2 g f1:361; 1:361 1:370g R.H up squarks fm~uR; m~cR; m~t1g f1:365; 1:365; 0:940g L.H up squraks fm~uL; m~cL; m~t2g f1:359 1:359; 1:276g New Particles Symbol Mass (TeV) SU(3)H Gauge boson octet MV 1:871 Singlet Higgsino m~Á 0:544 Octet Higgsino/gaugino m~Á1;2 f1:553; 2:191g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:247; 0:840; 1:955g Fermionic ´ (octet) moctet ´1;2 f0:716; 1:397g Fermionic ´ (singlet) msinglet ´1;2 f0:529; 1:890g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:421; 1:699g Pseudoscalar ´ Higgs (octet) mp¡octet ~´1;2 f1:031; 1:098g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:850; 1:593g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:247; 2:189g TABLE 5.3. Sparticle masses for the choice Maux = 63:695 TeV, tan ¯ = 4:02, ¹ = ¡1:178 TeV, yb = 0:0719, ¸ = 0:1, · = 0:08, g4 = 0:55, u = ¡3:402 TeV, M´ = 1:0 TeV and Mt = 0:1743 TeV. 52 MSSM Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 g f0:148; 0:436g Neutralinos fm~Â03 ; m~Â04 g f0:876; 0:878g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:148; 0:878g Gluino M3 1:064 Higgs boson fmh; mH; mA; mH§g f0:118; 0:878; 0:877; 0:880g R.H sleptons fm~eR; m~¹R; m~¿1g f0:825; 825; 0:821g L.H sleptons fm~eL; m~¹L; m~¿2g f0:827; 0:827; 0:830g Sneutrinos fm~ºe ; m~º¹; m~º¿ g f0:823; 0:823; 0:823g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:017; 1:017; 1:015g L.H down squraks fm~ dL ; m~sL; m~b 2 g f1:008; 1:008; 0:885g R.H up squarks fm~uR; m~cR; m~t1g f1:011; 1:011; 0:669g L.H up squraks fm~uL; m~cL; m~t2g f1:005; 1:005; 0:979g New Particles Symbol Mass (TeV) SU(3)HGauge boson octet MV 3:779 Singlet Higgsino m~Á 1:058 Octet Higgsino/gaugino m~Á1;2 f3:071; 4:495g Á Higgs bosons fmÁs ;mÁp ;mÁ¡octetg f0:465; 1:646; 3:940g Fermionic ´ (octet) moctet ´1;2 f0:254; 2:521g Fermionic ´ (singlet) msinglet ´1;2 f0:137; 4:672g Scalar ´ Higgs (octet) ms¡octet ~´1;2 f0:588; 3:090g Pseudoscalar ´ Higgs (octet) mp¡0ctet ~´1;2 f1:058; 1:952g Scalar ´ Higgs (singlet) ms¡singlet ~´1;2 f0:964; 4:116g Pseudoscalar ´ Higgs (singlet) mp¡singlet ~´1;2 f0:711; 5:224g TABLE 5.4. Sparticle masses for the choiceMaux = 47:112 TeV, tan ¯ = 3:785, ¹ = ¡0:873 TeV, yb = 0:068, ¸ = 0:3, · = 0:14, g4 = 1:0, u = ¡3:779 TeV, M´ = 0:800 TeV and Mt = 0:176 TeV. 53 5.6 Experimental signatures The Lightest SUSY particle in the model is the neutral Wino (Â01 ) which is nearly mass degenerate with the lightest chargino (Â§ 1 ), with a mass splitting of about 235 MeV. At the Tevatron Run 2 as well as at the LHC, the process p¹p (or pp) ! Â01 +Â§ 1 will produce these SUSY particles. Naturalness suggest that mÂ01 , mÂ§ 1 . 300 GeV (corresponding to mgluino . 2 TeV). Strategies for detecting such a quasi{degenerate pair has been carried out in Ref. [54; 56; 57]. In the MSSM sector our model predicts tan ¯ ' 4:0 and mh . 120 GeV, both of which can be tested at the LHC. If the SU(3)H gauge coupling g4 takes small values (g4 ' 0:55), the slepton masses will be near the current experimental limit. For larger values of g4 (' 1:0) the slepton masses are comparable to those of the squarks. The SU(3)H gauge boson masses are in the range MV = 1:5 ¡ 4:0 TeV. Al though relatively light, these particles do not mediate leptonic FCNC, owing to the approximate SU(3)H+G global symmetries present in the model. The most stringent constraint on MV arises from the process e+e¡ ! ¹+¹¡. LEP II has set severe constraints on lepton compositeness [51; 58] from this process. We focus on one such amplitude, involving all left{handed lepton ¯elds. In our model, the e®ective Lagrangian for this process is Leff = ¡ 2g2 4 3M2 V (e¹L°¹eL)(¹¹L°¹¹L): (5.35) Comparing with ¤¡ LL(ee¹¹) > 6:3 TeV [51; 58], we obtain MV g4 ¸ 2:05 TeV. For g4 = 0:55 (1:0) this implies MV > 1.129 (2.052) TeV. From Tables 5.2{5.4 we ¯nd that these constraints are satis¯ed. The model as it stands has an unbroken Z2 symmetry (in addition to the usual R{parity) under which the super¯elds ´; ¹´ are odd and all other super¯elds are even. If this symmetry is exact, the lightest of the ´ and ¹´ ¯elds (a pseudoscalar singlet Higgs in the ¯ts of Tables 5.2{5.3 and a singlet fermion in Table 5.4) will be stable. We envision this Z2 symmetry to be broken by higher dimensional terms of the type L®Hu©®¹´¯©¯=¤2. Such a term will induce the decay ´p¡singlet 1 ! L+Â01 with a lifetime less than 1 second for ¤ · 109 GeV. This would make these ´ particles cosmologically 54 safe. It may be pointed out that the same e®ective operator, along with a TeV scale mass for the ´ ¯elds, can provide small neutrino masses even in the absence of the operators given in Eq. 5.3. 5.7 Origin of the ¹ term Any satisfactory SUSY breaking model should also have a natural explanation for the ¹ term (the coe±cient of HuHd term in Eq. (5.1)). In gravity mediated SUSY breaking models, there are at least three solutions to the ¹ problem. The Giudice{ Masiero mechanism [59] which explains the ¹ term through the Kahler potential R HuHdZ¤d4µ=Mpl is not readily adaptable to the AMSB framework. The NMSSM extension which introduces singlet ¯elds can in principle provide a natural explanation of the ¹ term in the AMSB scenario. We have however found that replacing ¹HuHd by the term SHuHd in the superpotential alone can not lead to realistic SUSY breaking. It is possible to make the NMSSM scenario compatible with symmetry breaking in the AMSB framework by introducing a new set of ¯elds which couple to the singlet S. We do not follow this non{minimal alternative here. There is a natural explanation for the ¹ parameter in the context of AMSB mod els, as suggested in Ref. [36]. It assumes a Lagrangian term L ¾ ® R d4µ (§+§y) MPl HuHd©y©, where § is a hidden sector ¯eld which breaks SUSY and © is the compensator ¯eld. After a rescaling, Hu ! ©Hu, Hd ! ©Hd, this term becomes L ¾ ® R d4µ (§+§y) MPl HuHd ©y © , which generates a ¹ term in a way similar to the Giudice{ Masiero mechanism [59]. The B¹ term is induced only through the super{Weyl anomaly and has the form given in Eq. (4.9). Our predictions for tan ¯ and mh depend sensitively on this assumption. We now point out that the ¹ term may have an alternative explanation in the context of AMSB models. This is obtained by promoting ¹HuHd in the superpotential to the following [60]: W0 = aHuHdS2 MPl + bS2 ¹ S2 MPl : (5.36) 55 Here S and ¹ S are standard model singlet ¯elds. Including AMSB induced soft pa rameters for these singlets (which can arise in a variety of ways), this superpotential will have a minimum where hSi ' ¹ S ® ' p MSUSYMPl. This would induce ¹ term of order MSUSY , as needed. From the e®ective low energy point of view, the superpo tential will appear to have an explicit ¹ term. The B term will have a form as given in Eq. 4.9. 5.8 Summary In this chapter we have suggested a new scenario for solving the tachyonic slep ton mass problem of anomaly mediated SUSY breaking models. An asymptotically free SU(3)H horizontal gauge symmetry acting on the lepton super¯elds provides positive masses to the sleptons. The SU(3)H symmetry must be broken at the TeV scale. Potentially dangerous FCNC processes mediated by the SU(3)H gauge bosons are shown to be suppressed adequately via approximate global symmetries that are present in the model. Our scenario predicts mh . 120 GeV for the lightest Higgs boson mass of MSSM and tan ¯ ' 4.0. The lightest SUSY particle is the neutral Wino which is nearly degenerate with the lightest chargino and is a candidate for cold dark matter. The full spectrum of the model is given in Tables 5.2{5.4 for various choices of input parameters. The very few parameters of our model are highly constrained by the consistency of symmetry breaking. CHAPTER 6 SU(2)H Horizontal Symmetry as a Solution to the Slepton Mass Problem of Anomaly Mediation 6.1 Introduction Family symmetries may give a positive mass{squared contribution to the slep tons in AMSB. The simplest of such symmetry is an SU(2)H non{Abelian symmetry. This symmetry when acting on leptons only can be asymptotically free, hence their beta{function will be negative. This is very important because with this new sym metry, the sleptons enjoys the same freedom as the quarks and hence can solve the negative slepton mass problem of AMSB. The quarks are singlet of SU(2)H but it is possible that they transform under a di®erent SU(2)q H symmetry, so that there is an underlying quark{lepton symmetry. Here we will focus on a model where quarks carry no family symmetry. In this chapter we suggest and investigate the possibility of solving the negative slepton mass problem of AMSB using this SU(2)H symmetry broken at the TeV scale. The leptons of the ¯rst two families transform as a doublet of SU(2)H and those of the third family transform as singlet under this new symmetry. The sleptons of the ¯rst two family gets a large positive contribution to their soft masses from the SU(2)H gauge sector. With e and ¹ forming a doublet of SU(2)H, an important issue is how to split their masses, since in Nature me 6= m¹. We introduce two new vector{like ¯elds that couples to the third family which will help to achieve me 6= m¹. The model is quite predictive. The LSP is the Wino which is nearly mass degenerate with the chargino. The lightest Higgs boson mass is predicted to be mh . 135 GeV, and the parameter tan ¯ is found to be tan ¯ ' 40. This model 56 57 is completely di®erent from the previous model because it also predicts a di®erent mass hierarchy for the ~e; ~¹ and ~¿ . In particular m~e; m~¹ and m~¿ are all quite di®erent, which is a characteristic signature of this model. In addition, this model can easily be tested at the LHC by direct discovery of the gauge bosons associated with SU(2)H. The plan of the chapter is as follows. In section 6.2 we introduce our model. In section 6.3 we analyze the Higgs potential. The SUSY spectrum is presented in section 6.4. We discuss our numerical results in section 6.5. In section 6.6 we discuss the experimental implication of the model. We summarize in section 6.7. 6.2 SU(2)H horizontal symmetry We de¯ne the gauge group symmetry of the model as GH ´ SU(3)C £ SU(2)L £ U(1)Y £ SU(2)H; where SU(2)H is a horizontal symmetry that acts on the ¯rst two families of leptons. The third family is a singlet under this new SU(2)H symmetry. A pair of vector like leptons, E, Ec, which are SU(2)H singlets are needed to ensure me 6= m¹. The spectrum of the model is listed in Table. 6.1. The gauge group SU(2)H de¯ned above is asymptotically free (¯ function is given in Eq. B.20) with this spectrum. The superpotential of the model consistent with the gauge symmetries reads W = (Yu)ij QiHuucj + (Yd)ij QiHddcj + fe¹Ã®Ãc® Hd + f¿Ã¿ ¿ cHd + f¿EÃ¿EcHd + feEEÃcÁu + ¹HuHd + ¹0ÁuÁd +MEEEc (6.1) It turns out that there is a Z4 symmetry present in the Lagrangian, under which Áu ! iÁu; Ád ! ¡iÁd; E ! ¡iE; Ec ! iEc; Ã¿ ! ¡iÃ¿ ; ¿ c ! i¿ c: This Z4 symmetry forbids the term EÃcÁd, which will be important to de¯ne an unbroken muon number. Since SU(2)H is broken at TeV, the gauge bosons of SU(2)H can potentially lead to large FCNC processes. The most dangerous of these are in the muon sector, eg; ¹ ! 3e. Such process are forbidden by an unbroken muon number, making TeV scale horizontal symmetry phenomenologically consistent. 58 Super¯eld SU(3)C SU(2)L U(1)Y SU(2)H Qi 3 2 1 6 1 uci ¹3 1 ¡2 3 1 dci ¹3 1 1 3 1 Ã® 1 2 ¡1 2 2 Ãc® 1 1 +1 2 Ã¿ 1 2 ¡1 2 1 ¿ c 1 1 +1 1 Hu 1 2 1 2 1 Hd 1 2 ¡1 2 1 Áu 1 1 0 2 Ád 1 1 0 2 E 1 1 ¡1 1 Ec 1 1 +1 1 ªN 1 1 0 2 TABLE 6.1. Particle content and charge assignment of the model. The indices i and ® take values i = 1 ¡ 3 and ® = 1 ¡ 2. In the model, the Ã® and Ãc® ¯elds contain the ¯rst two family of leptons (e and ¹) which transforms as a doublet under the SU(2)H gauge group, while the members of the third family (Ã¿ and ¿ c) transform as singlets under the SU(2)H gauge group. The ¯eld ªN, which transforms as a doublet under SU(2)H and as singlet under the SM gauge group, is introduced in order to cancel the Witten anomaly. The neutinos in the model get masses from the following nonrenormalizable operators: Ã¿Ã¿ HuHd M ; Ã®Ã® HuHu M03 Áu;dÁu;d; Ã®Ã¿ HuHu M00 Áu;d: (6.2) These terms will lead to a consistent neutrino oscillation phenomenology. 59 6.3 Symmetry breaking The symmetry breaking is achieved in the form GH ! GSM ! SU(3)C £ U(1)EM; where GSM ´ SU(3)C £ SU(2)L £ U(1)Y . The model has the possibility to be consistent with the known low energy physics. The new Higgs multiplets Áu and Ád are su±cient to break the GH ! GSM completely near the TeV scale. The tree level Higgs potential can be written as V (Hu;Hd; Áu; Ád) = (m2 Hu + ¹2)jHuj2 + (m2 Hd + ¹2)jHdj2 + B¹(HuHd + c.c.) + (g2 2 + g2 1) 8 (jHuj2 ¡ jHdj2)2 + g2 2 2 jHuHdj2 + g2 4 8 (jÁuj2 ¡ jÁdj2)2 + g2 4 2 jÁuÁdj2 + (m2 Áu + ¹02)jÁuj2 + (m2 Ád + ¹02)jÁdj2 + B0¹0(ÁuÁd + c.c.): The soft masses m2 Hu and m2 Hd ;m2 Áu and m2 Ád parameters are determined in terms of the single parameter Maux. The B and B0 parameters are taken to be free in the model but in some special class of models, they are determined also by the same mass parameter Maux. Upon symmetry breaking, the Higgs ¯elds acquire VEV's hHui = Ã 0 Àu ! ; hHdi = Ã Àd 0 ! ; hÁui = Ã 0 uu ! ; hÁdi = Ã ud 0 ! : (6.3) It is desired that the VEVs obey hÁui; hÁdi À hHui; hHdi, in order for the symmetry breaking to be consistent. Minimization of the Higgs potential V (Hu;Hd; Áu; Ád) gives sin 2¯ = ¡2B¹ 2¹2 + m2 Hu + m2 Hd ; ¹2 = m2 Hd ¡ m2 Hu tan2 ¯ tan2 ¯ ¡ 1 ¡ M2Z 2 ; (6.4) sin 2¯0 = ¡2B0¹0 2¹02 + m2 Áu + m2 Ád ; ¹02 = m2 Ád ¡ m2 Áu tan2 ¯0 tan2 ¯0 ¡ 1 ¡ M2Z 0 2 ; (6.5) where we have introduced the notation uu = u sin ¯0, ud = u cos ¯0, u2 = u2 u + u2 d, tan ¯0 = uu ud and M2Z 0 = g2 4 2 (u2 u + u2 d). MZ0 is the mass of the gauge boson associated with the SU(2)H. 60 To ¯nd the physical Higgs boson mass, we parameterize the Higgs ¯elds (in the unitary gauge) as Hu = Ã H+ sin ¯ Àu + p1 2 (Á2 + i cos ¯ Á3) ! ; hHdi = Ã Àd + p1 2 (Á1 + i sin ¯ Á3) H¡ cos ¯ ! ; Áu = Ã Á+ sin ¯0 uu + p1 2 (Á4 + i cos ¯0 Á5) ! ; Ád = Ã ud + p1 2 (Á6 + i sin ¯0 Á5) Á¡ cos ¯0 ! : (6.6) The Higgs masses are obtained by expanding the Higgs potential and keeping only terms quadratic in the ¯elds. The masses of the CP{odd Higgs bosons fÁ3; Á5g are m2 A = ¡2B¹ sin 2¯ ; m2 A0 = ¡ 2B0¹0 sin 2¯0 : (6.7) The mass matrices for the CP{even neutral Higgs bosons fÁ1; Á2g and fÁ4; Á6g are decoupled. They are given by (M2)cp¡even = Ã m2 A cos2 ¯ +M2Z sin2 ¯ ¡fm2 A +M2Z gsin 2¯ 2 ¡fm2 A +M2Z gsin 2¯ 2 m2 A sin2 ¯ +M2Zsin2 ¯ ! ; (6.8) (M02)cp¡even = Ã m2 A0 cos2 ¯0 +M02 Z sin2 ¯0 ¡fm2 A0 +M02 Z gsin 2¯0 2 ¡fm2 A0 +M02 Z gsin 2¯0 2 m2 A0 sin2 ¯0 +M02 Z sin2 ¯0 ! : (6.9) Finally, the charged Higgs boson mass (H§ and Á§) is given by m2 H§ = m2 A +M2W m2 Á§ = m2 A0 +M2Z 0 (6.10) Á§ are electrically neutral, they are \charged" under SU(2)H. The Majorana mass matrix of the neutralinos f~B ; ~W 3; H~0 d ; H~0 u; B~H; Á~0 d; Á~0 ug is M(0) = 0 BBBBBBBBBBBB@ M1 0 ¡pÀd 2 g1 pÀu 2 g1 0 0 0 0 M2 pÀd 2 g2 ¡pÀu 2 g2 0 0 0 ¡pÀd 2 g1 pÀd 2 g2 0 ¡¹ 0 0 0 pÀu 2 g1 ¡pÀu 2 g2 ¡¹ 0 0 0 0 0 0 0 0 M4 pud 2 g4 ¡puu 2 g4 0 0 0 0 pud 2 g4 0 ¡¹0 0 0 0 0 ¡puu 2 g4 ¡¹0 0 1 CCCCCCCCCCCCA ; (6.11) 61 where M1; M2 andM4 are the gaugino masses for U(1)Y ; SU(2)L and SU(2)H which are listed in Appendix B. The physical neutralino masses m~Â0i (i =1{7) are obtained as the eigenvalues of this mass matrix Eq. (6.11). In the basis f ~W +; ~H + u g, f ~W ¡; ~H ¡ d g, the chargino (Dirac) mass matrix is M(c) = Ã M2 g2Àd g2Àu ¹ ! : (6.12) Similarly, for the SU(2)H sector, we have ~M (c) = Ã M4 g4ud g4uu ¹0 ! : (6.13) The three SU(2)H gauge boson masses are given by M2 V = g2 4 2 (u2 u + u2 d): (6.14) 6.3.1 Lepton masses Now we describe brie°y how to obtain the masses of the ordinary leptons. We have introduced E and Ec ¯elds in the superpotential Eq. (6.1) for the purpose of breaking e ¡ ¹ degeneracy. These new ¯elds mix with the usual leptons leading to the mass matrix ( e ¹ ¿ E ) 0 BBBBB@ f¹Àd 0 0 0 0 f¹Àd 0 0 0 0 f¿Àd f¿EÀd feEuu 0 0 ME 1 CCCCCA 0 BBBBB@ ec ¹c ¿ c Ec 1 CCCCCA : (6.15) The muon ¯eld completely decouples with mass m¹ = f¹Àd: (6.16) We are then left with a 3 £ 3 mass matrix for the e, ¿ and E ¯elds. The eigenvalue equation can be easily solved using the hierarchy me ¿ m¿ ¿ mE and the result is m¿ ' f¿Àd s f1 + f2 ¿Ef2 eE f2 ¿ u2 u M2E + f2 eEu2 u g; me ' q m¹ME M2E + f2 eEu2 u + f2 ¿Ef2 eEu2 u f2 ¿ ; mE ' q M2E + f2 eEu2 u: (6.17) Note that me 6= m¹, showing consistency of the model. 62 6.4 The SUSY spectrum We will show in this section that the tachyonic slepton problem is cured by virtue of the positive contribution from the SU(2)H gauge sector to the masses for the ¯rst two family and a large Yukawa coupling for the third family. 6.4.1 Slepton masses The slepton masses are given by a 2 £ 2 mass matrix for the smuon (since it decouples) and a 6 £ 6 mass matrix for the e; ¿;E; ec; ¿ c;Ec ¯elds. The smuon mass{squareds are given by the eigenvalues of the mass matrix M2 ~¹ = Ã m2 ~¹ m¹ ¡ Afe¹ ¡ ¹ tan ¯ ¢ m¹ ¡ Afe¹ ¡ ¹ tan ¯ ¢ m2 ~¹c ! ; (6.18) where the diagonal entries are m2 ~¹ = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + m2 ¹ + g2 4 4 (u2 u ¡ u2 d); m2 ~¹c = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 6 5 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + m2 ¹ + g2 4 4 (u2 u ¡ u2 d): Note that the positive contributions from the SU(2)H gauge sector are provided by the term ¡3 2g4¯(g4), with gauge beta function ¯(g4) = ¡ 3 16¼2 g3 4. This contribution ensures that the mass{squareds of all sleptons are positive when g4 > 0:9. It is important to point out that the SU(2)H D{term contributions to the diagonal entries of the mass matrix Eq. (6.18) can either be positive or negative but it must be such that its overall contribution is rather small compared to the soft mass term. The mass matrix for the other sleptons is in the form 0 BB@ m2 ~e 0 fe¹feEÀduu fe¹(Ae¹Àd + ¹Àu) 0 0 0 m2 ~¿ MEf¿EÀd 0 f¿ (A¿ Àd + ¹Àu) f¿E(A¿EÀd + ¹Àu) fe¹feEÀduu MEf¿EÀd m2 ~E feE(AeEuu + ¹0ud) 0 MEBE fe¹(Ae¹Àd + ¹Àu) 0 feE(AeEuu + ¹0ud) m2 e~c 0 MEfeEuu 0 f¿ (A¿ Àd + ¹Àu) 0 0 m2 ¿~c f¿ f¿EÀ2d 0 f¿E(A¿EÀd + ¹Àu) MEBE MEfeEuu f¿ f¿EÀ2d m2 E~c 1 CCA ; 63 where the diagonal entries of this mass matrix read m2 ~e = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 3 2 g2¯(g2) + 3 10 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + f2 e¹À2 d + g2 4 4 (u2 d ¡ u2 u); m2 ~ec = M2 aux (16¼2) · 2fe¹¯(fe¹) ¡ µ 6 5 g1¯(g1) + 3 2 g4¯(g4) ¶¸ + f2 e¹À2 d + f2 eEu2 u + g2 4 4 (u2 d ¡ u2 u) m2 ~¿ = M2 aux (16¼2) · f¿¯(f¿ ) + f¿E¯(f¿E) ¡ µ 3 10 g1¯(g1) + 3 2 g2¯(g2) ¶¸ + (f2 ¿ + f2 ¿E)À2 d; m2 ~¿c = M2 aux (16¼2) · 2f¿¯(f¿ ) ¡ µ 6 5 g1¯(g1) ¶¸ + f2 ¿ À2 d m2 ~E = M2 aux (16¼2) · feE¯(feE) ¡ µ 6 5 g1¯(g1) ¶¸ + m2 E + f2 eEu2 u; m2 ~E c = M2 aux (16¼2) · f¿e¯(fe¿ ) ¡ µ 6 5 g1¯(g1) ¶¸ + m2 E + f2 ¿EÀ2 d (6.19) The requirement that the slepton masses are positive puts constraints on the couplings f¿ ; feE; f¿e and g4. 6.4.2 Squark masses The mixing matrix for the squark sector is similar to the slepton sector, except that they receive no SU(2)H gauge contributions. The diagonal entries of the up and the down squark mass matrices are given by [61] m2 ~U i = (m2 soft) ~Q i ~Q i + m2 Ui + 1 6 ¡ 4M2W ¡M2Z ¢ cos 2¯; m2 ~U c i = (m2 soft) ~U c i ~U c i + m2 Ui ¡ 2 3 ¡ M2W ¡M2Z ¢ cos 2¯; m2 ~D i = (m2 soft) ~Q i ~Q i + m2 Di ¡ 1 6 ¡ 2M2W +M2Z ¢ cos 2¯; m2 ~Dc i = (m2 soft) ~D c i ~D c i + m2 Di + 1 3 ¡ M2W ¡M2Z ¢ cos 2¯; (6.20) were mUi and mDi are the quark masses of the di®erent generations with i = 1, 2, 3. The squark soft masses are obtained from the RGE as (m2 soft) ~Q i ~Q i = M2 aux 16¼2 µ Yui¯(Yui) + Ydi¯(Ydi) ¡ 1 30 g1¯(g1) ¡ 3 2 g2¯(g2) ¡ 8 3 g3¯(g3) ¶ (6; .21) 64 (m2 soft) ~U c i ~U c i = M2 aux 16¼2 µ 2Yui¯(Yui) ¡ 8 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ ; (6.22) (m2 soft) ~D c i ~D c i = M2 aux 16¼2 µ 2Ydi¯(Ydi) ¡ 2 15 g1¯(g1) ¡ 8 3 g3¯(g3) ¶ : (6.23) 6.5 Numerical results Here we present our numerical results for the SUSY spectrum. We ¯rst per formed a one{loop accuracy numerical analysis to determine the sparticle and Higgs Spectrum. For experimental inputs for the SM gauge couplings we use the same pro cedure Ref. [61] for the g1; g2; g3 with the central value of the top mass taken to be Mt = 174:3 GeV. In the model presented, the scale of SUSY breaking, Maux should be in the range 40¡100 TeV for the MSSM particles to have masses in the range 0:1¡2 TeV. The gauge coupling g4 ¸ 0:9 in order for the slepton masses for the ¯rst two families to be positive and in the right range. Since the positivity of the mass{squared of the slepton of the third family depends on the Yukawa couplings, we ¯nd that the couplings should obey f¿ ; f¿E ¸ 0:5. For a speci¯c choice of parameters (Table. 6.2), we ¯nd the m¹1 ; m¹2 » 800 GeV for the smuon. There is a signi¯cant mass splitting between the selectron and the stau. The lightest of the sleptons is the lefthanded stau. If SUSY is discovered with a large mass hierarchy between the stau and the selectron (or smauon), this model will be a good candidate. The lightest Higgs mass is found to be around 128 GeV which is consistent with current experimental limit. The lightest supersymmetric particle is Wino which is nearly mass degenerate with the lighter chargino of the SM. The SU(2)H gauge boson mass is found to be » 1:4 TeV. The heavy Higgs bosons, Higgsinos and squarks masses are in the range 0:7 ¡ 2:0 TeV. 65 Particles Symbol Mass (TeV) Neutralinos fm~Â01 ; m~Â02 ; m~Â03 ; m~Â04 g f0:176; 274; 0:726; 1:080g Neutralinos fm~Â05 ; m~Â06 ; m~Â07 g f1:091; 1:096; 2:097g Charginos fm~Â§ 1 ; m~Â§ 2 g f0:176; 1:094g Charginos (SU(2)H) fm~Â§ 1 ; m~Â§ 2 g f1:070; 2:102g Gluino M3 1:556 Neutral Higgs bosons fmh; mH; mAg f0:128; 0:922; 0:922g Neutral Higgs bosons fmh0 ; mH0 ; mA0g f0:143; 2:075; 1:554g Charged Higgs bosons mH§ 0:925 Charged Higgs bosons SU(2)H mH§ 2:080 R.H smuon fm~¹1g f0:867g L.H smuon fm~¹Lg f0:796g R.H sleptons fm~eR; m~¿1 ; mERg f0:947; 0:176; 0:758g L.H sleptons fm~eL; m~¿2 ; m~¹Lg f1:904; 0:533; 0:401g R.H down squarks fm~ dR ; m~sR; m~b 1 g f1:464; 1:464; 1:369g L.H down squarks fm~ dL ; m~sL; m~b 2 g f1:451; 1:451; 1:115g R.H up squarks fm~uR; m~cR; m~t1 g f1:454; 1:454; 1:107g L.H up squarks fm~uL; m~cL; m~t2 g f1:449; 1:449; 1:295g SU(2)H gauge boson M0Z 1:382 TABLE 6.2. Sparticle masses in Model 1 for the choice Maux = 67:956 TeV, yb = 0:8, f¿ = 0:53, feE = 1:2, f¿E = 0:51, g4 = 1:0, ME = 10:4 TeV and Mt = 0:174 TeV, u = 1:955 TeV, tan ¯ = 57:4, tan ¯0 = 0:87, ¹ = 1:088 TeV, ¹0 = 0:276 TeV, B = 0:014 TeV, B0 = 4:336 TeV, BE = 0:009 TeV. 66 6.6 Other experimental implications The lightest SUSY particle in the model we considered is the wino ( ~ Â01 ) which is nearly mass degenerate with the chargino. This particle is stable and can be a candidate for cold dark matter. The model predicts the lightest Higgs mass mh · 135 GeV which can be tested at the LHC. Because the SU(2)H gauge bosons do not mix with the SM gauge bosons, elec troweak precision data remains unchanged. Also the second family of leptons do not mix with the ¯rst and third family, this is because of the Z4 symmetry present in the model. The processes ¹ ! 3e and ¹ ! e° are not a problem in the model. The SU(2)H gauge boson masses are degenerate with mass MV = 1:382 TeV for the choice of parameters chosen in model 1. The most stringent constraint on MV arising from the process e+e¡ ! ¹+¹¡. LEP II has set severe constraints on lepton compositeness [51; 58] from this process. The e®ective Lagrangian for the process is given by Leff = g2 4 2 (¹e°¹¹)(¹°¹e) M2 V : Here MV is the gauge boson mass. If we compare the above Lagrangian with the ¤¡ LL (ee¹¹) [51; 58], we obtain the limit MV > 1:2 TeV. This limit is satis¯ed in our model. 6.7 Summary We have suggested in this chapter a new scenario for solving the tachyonic slepton mass problem of AMSB. An asymptotically free SU(2)H horizontal gauge symmetry acting on the lepton super¯elds provides positive masses to the sleptons of the ¯rst two families (~e; ~¹) while the Yukawa couplings associated with the third family (¿ ) ¯eld gives a large positive contribution to the ~¿ mass. We have a large mass splitting between the ~e; ~¹; and; ~¿ , due to the transformation properties under the new SU(2)H symmetry. This is how our model di®ers from the other models. The SU(2)H symmetry must be broken at the TeV scale for consistency and our model predicts mh . 135 GeV for the lightest MSSM Higgs boson mass and tan ¯ ' 40. The 67 LSP is the neutral wino which is nearly mass degenerate with the lightest chargino and is a candidate for cold dark matter. CHAPTER 7 Constraining Z0 From Supersymmetry Breaking 7.1 Introduction One of the simplest extensions of the Standard Model (SM) is obtained by adding a U(1) factor to the SU(3)C £SU(2)L £U(1)Y gauge structure [62; 63]. Such U(1) factors arise quite naturally when the SM is embedded in a grand uni¯ed group such as SO(10), SU(6), E6, etc. While it is possible that such U(1) symmetries are broken spontaneously near the grand uni¯cation scale, it is also possible that some of the U(1) factors survive down to the TeV scale. In fact, if there is low energy supersymmetry, it is quite plausible that the U(1) symmetry is broken along with supersymmetry at the TeV scale. The Z0Â and Z0Ã models arising from SO(10) ! SU(5) £ U(1)Â and E6 ! SO(10) £ U(1)Ã are two popular extensions which have attracted much phenomenological attention [62{69]. Z0 associated with the left{ right symmetric extension of the Standard Model does not require a grand uni¯ed symmetry. Other types of U(1) symmetries, which do not resemble the ones with a GUT origin, are known to arise in string theory, free{fermionic construction as well as in orbifold and D{brane models [70{72]. Gauge kinetic mixing terms of the type B¹ºZ0 ¹º [73] which will be generated through renormalization group °ow below the uni¯cation scale can further disguise the couplings of the Z0. The properties of the Z0 gauge boson { its mass, mixing and couplings to fermions { associated with the U(1) gauge symmetry are in general quite arbitrary [74]. This is especially so when the low energy theory contains new fermions for anomaly cancellation. In this chapter we propose and analyze a special class of U(1) models wherein the Z0 properties get essentially ¯xed from constraints of SUSY 68 69 breaking. We have in mind the anomaly mediated supersymmetric (AMSB) frame work [36; 37]. In its minimal version, with the Standard Model gauge symmetry, it turns out that the sleptons of AMSB become tachyonic. We suggest the U(1) symme try, identi¯ed as U(1)x = xY ¡(B¡L), where Y is the Standard Model hypercharge, as a solution to the negative slepton mass problem of AMSB. This symmetry is auto matically free of anomalies with the inclusion of right{handed neutrinos. It is shown that the D{term of this U(1)x provides positive contribut 



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