Supersymmetry at the Crossroads

SUPERSYMMETRY AT THE CROSSROADS:

HIGGS, DARK MATTER AND COLLIDER SIGNALS

Muhammad Adeel Ajaib

A dissertation submitted to the Faculty of the University of Delaware in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics

Summer 2014

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ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 SUPERSYMMETRY AT THE CROSSROADS:

HIGGS, DARK MATTER AND COLLIDER SIGNALS

Muhammad Adeel Ajaib

Approved: Edmund Nowak, Ph.D. Chair of the Department of Physics

Approved: George Watson, Ph.D. Dean of the College of Arts and Sciences

Approved: James G. Richards, Ph.D. Vice Provost for Graduate and Professional Education I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: Qaisar Shaﬁ, Ph.D. Professor in charge of dissertation

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: Stephen M. Barr, Ph.D. Member of dissertation committee

Signed: John Gizis, Ph.D. Member of dissertation committee

Signed: Nobuchika Okada, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS

My gratitude to Almighty Allah whose blessings always support and strengthen me and whose love always inspire me to think even deeper about the mysteries of the Universe. The verse of the Quran which describes believers as those who “think deeply about the creation of the heavens and the earth” has always inspired me during my graduate studies. His blessings and benevolence be on the Prophet (peace be upon him) who was chosen to show us the way towards the Almighty. Especial thanks to my advisor professor Qaisar Shafi whose help and guidance enabled me to learn so much during my PhD. I am thankful to him for his continued support and encouragement. He has helped me a lot and without his guidance I would not have reached this stage. I am also grateful to Professor Stephen Barr for many useful discussions. He was always welcoming to any questions I had about any topic. I have worked with several postdocs during my PhD who really taught me a lot. I am thankful to Ilia Gogoladze, Tong Li, Hasan Yuksel and Liucheng Wang for teaching me about numerous topics and for many enlightening discussions. In particular, Ilia Gogoladze has inspired me to be more inquisitive by encouraging me to ask questions. He will continue to be an inspiration to me throughout my career. From day one, the physics staff has been really helpful. In particular Maura Perkins and Debra Morris are the best. Whenever I got confused about any official matter they were always there to help. I really appreciate the help provided by Dr Ismat Shah in various matters during my stay here. I am also deeply indebted to many teachers from QAU and UD who have shaped my understanding of this subject. These include Riazuddin, Fayyazuddin, Faheem Hussain, Pervez Hoodbhoy, Michael Shay and many others. There are a lot of friends who I cant thank enough for their

iv help and support. Shahid Hussain was a lot of support for all of us when I ﬁrst came to Delaware. Thanks to my friends Mansoor Rehman, Hassan Chishti, Asif Warsi, Hassnain Khurshid, Evan Kimberly, Amir Javaid for making UD a much more delightful place. I am thankful to my group members Rizwan Khalid, Shabbar Raza Rizvi, Cem Salih Un, Bin He, Mansoor Rehman, Josh Wickman, Matthew Civiletti and Fariha Nasir for their support. In particular Rizwan Khalid, Shabbar Raza Rizvi and Cem Salih Un were really helpful with computational tools and many useful discussions. I am also grateful to Asifa Hasan and Khurram Ashraf for their support and help in taking care of our daughter. There are several other people who supported me a lot. My spiritual mentor, Mufti Saeed Ahmed Khan has been an enormous source of encouragement for me. My uncle Amir Hashmi was also a great support for me. My M.Phil. advisor Riazuddin and my father passed away during my PhD. They will be greatly missed for all their support in my life. My lovely wife, Fariha Nasir, has always been a lot of support. She was always there for me whenever I needed her. I am also thankful to her for all the physics related discussions we had. She taught me a lot. My daughter, Raima, is a continuous source of happiness for me. At times when the going used to get tough spending time with her innocence was a unique source of strength for me. Lots of love for my mother, brothers and sisters who supported me so much during my course of studies.

v This thesis is dedicated to, My Sheikh Mufti Muhammad Saeed Khan Sahab, My Parents and Siblings My Uncles (Amir and Waqar Hashmi) My lovely Wife, adorable Daughter and our newly born son

vi TABLE OF CONTENTS

LIST OF TABLES ...... x LIST OF FIGURES ...... xi ABSTRACT ...... xvi

Chapter

1 INTRODUCTION TO THE STANDARD MODEL OF PARTICLE PHYSICS ...... 1

1.1 Introduction ...... 1 1.2 The Standard Model of Particle Physics ...... 2 1.3 The Higgs Mechanism ...... 5 1.4 The Need to go Beyond the Standard Model ...... 9

2 PHYSICS BEYOND THE STANDARD MODEL ...... 11

2.1 Supersymmetry ...... 11 2.2 The Minimal Supersymmetric Standard Model ...... 12

2.2.1 The MSSM Lagrangian ...... 14 2.2.2 R-parity ...... 16 2.2.3 Soft SUSY Breaking Terms ...... 17 2.2.4 Electroweak Symmetry Breaking in the MSSM ...... 18 2.2.5 Neutralinos and Charginos ...... 21 2.2.6 Squark and slepton masses ...... 22 2.2.7 Renormalization Group Equations ...... 24

2.3 Grand Uniﬁcation ...... 26

2.3.1 SU(5) Grand Uniﬁcation ...... 27 2.3.2 SU(4)c × SU(2)L × SU(2)R ...... 31 2.3.3 SO(10) ...... 31

2.4 Scanning the Parameter Space of SUSY GUTs ...... 32

vii 2.5 Constraining the SUSY GUT Parameter Space ...... 34

2.5.1 Sparticle mass limits ...... 35 2.5.2 b → sγ ...... 35 + − 2.5.3 Bs → µ µ ...... 36 2.5.4 Bu → τντ ...... 36 2.5.5 Ωh2 ...... 36 2.5.6 (g − 2)µ ...... 36

3 SUSY SEARCHES WITH NLSP STOP ...... 38

3.1 Introduction ...... 38 3.2 Brief Review of Collider Physics ...... 39 3.3 Collider Searches for Supersymmetry ...... 42 3.4 Why are NLSP scenarios interesting? ...... 42

3.4.1 Relic abundance ...... 43 3.4.2 NLSP signatures at colliders ...... 44

3.5 NLSP stop in Constrained MSSM with b − τ Yukawa uniﬁcation ... 45

3.5.1 Event Generation and Detector simulation ...... 46 3.5.2 Implementing ATLAS selection requirements ...... 48 3.5.3 Deriving a Limit on the NLSP Stop mass ...... 50 3.5.4 ATLAS search Implications for Direct Detection ...... 52

3.6 Conclusion ...... 52

4 h → γγ CHANNEL AND THE MSSM ...... 54

4.1 Introduction ...... 54 4.2 Higgs mass in the MSSM ...... 54 4.3 Higgs mass in the MSSM with Vector like particles ...... 55 4.4 gg → h → γγ Process ...... 56

4.4.1 gg → h...... 56 4.4.2 h → γγ ...... 58

4.5 Phenomenological Constraints and Scanning Procedure ...... 59 4.6 The gg → h → γγ process and a light stop ...... 60

4.6.1 Decoupling Limit (mA >> mZ ) ...... 60

viii 4.6.2 Low mA Region ...... 64

4.7 Enhancement in the h → γγ channel with a stau ...... 69

4.7.1 Light Stau in the Decoupling limit ...... 69 4.7.2 Light Stau and low mA region ...... 74

4.8 Conclusion ...... 77

5 IMPACT OF THE HIGGS DISCOVERY ON THE mGMSB MODEL ...... 79

5.1 Introduction ...... 79 5.2 The mGMSB model ...... 80 5.3 Parameter Space ...... 81 5.4 Constraining the mGMSB model with a 125 GeV Higgs ...... 82

5.4.1 Sparticle Spectroscopy ...... 82 5.4.2 Gravitino Constraints and the Higgs ...... 88

5.5 Conclusion ...... 89

6 HIGGS MASS PREDICTION IN SO(10) WITH t-b-τ YUKAWA UNIFICATION ...... 92

6.1 Introduction ...... 92 6.2 SO(10) GUT with t − b − τ Yukawa uniﬁcation ...... 93 6.3 Fundamental Parameter Space ...... 94 6.4 RGE running in Isajet and Suspect ...... 97 6.5 Higgs Mass Prediction ...... 98 6.6 Higgs and Sparticle Spectroscopy From Isajet and SuSpect ...... 104 6.7 Fine tuning constraints for little hierarchy ...... 112 6.8 Conclusion ...... 115

BIBLIOGRAPHY ...... 116

ix LIST OF TABLES

1.1 The three generations of particles in the standard model along with their SU(3)c × SU(2)L × U(1)Y quantum number. The W and Z gauge bosons mediate weak interactions between these particles. The electromagnetic interaction is mediated by the photon γ and eight gluons mediate strong interactions between these particles...... 4

2.1 The chiral superﬁelds (χSF ) in the MSSM. Here q = (uL, dL) and ˜ l = (νL, eL) and similarly forq ˜ and l. The two Higgs bosons are + 0 0 − SU(2) doublets, i.e., hu = (hu , hu) and hd = (hd, hd )...... 12

2.2 The vector superﬁelds (VSF) in the MSSM...... 13

3.1 Summary of selection cuts and 95% C.L. upper limits on the eﬀective cross section for non-SM processes for signal region LP, HP and VHP containing ﬁnal states with monojet and missing transverse momentum with 1 fb−1 luminosity, following the ATLAS data analyses [76]...... 51

5.1 Benchmark points for the mGMSB. All masses are in units of GeV. Point 1, 2 and 3 show the lightest neutralino, stau and gravitino (shown in bold) that can be realised in mGMSB for a Higgs mass of 125 GeV. For the three points, mt = 173.3 GeV and cgrav = 1. ... 91

6.1 Benchmark points with good Yukawa uniﬁcation. All the masses are in units of GeV. Point 1, 3 and 4 are generated using Isajet 7.84 whereas point 2 is from SuSpect 2.41. Point 1 and 2 demonstrates how a small value of Rtbτ yields a Higgs mass ∼ 125 GeV. Point 3 exhibits stau coannihilation and has a small Rtbτ that agrees with 2 Ωh < 1. Point 4 has m16 ' m10 with good YU...... 114

x LIST OF FIGURES

1.1 Example of symmetry breaking for the potential V = µ2φ2 + λφ4. For µ2 > 0 the potential has a trivial minimum (red line). For µ2 < 0, the minima of the potential are non-trivial (blue line). Symmetry breaking takes place as the ﬁeld settles down in one of these minima (+v or −v). The Lagrangian when expressed in terms of the physical ﬁeld (φphys = φ − v) will no more have the Z2 symmetry (φ → −φ) of the original Lagrangian...... 6

3.1 σ×acceptance vs. Mt˜1 with horizontal line as the 95% C.L. upper limits on eﬀective non-SM processes cross section for signal region LP (top left), HP (top right), VHP (bottom). Green regions correspond to models with Yukawa uniﬁcation (R ≤ 1.1) and NLSP stop. ... 47

3.2 M 0 vs. M˜ for models with Yukawa uniﬁcation and NLSP stop χ˜1 t1 (green circle), those excluded by ATLAS monojet regions (red triangle) and other excluded ones by ATLAS multi-jets regions (black box) in the framework of b − τ Yukawa uniﬁed mSUGRA/CMSSM. The kinematic limits and coannihilation bounds are also displayed. The blue region refers to the excluded region by Tevatron [54]. ... 48

3.3 m1/2 vs. m0 for models satisfying all low energy experiments (grey box), those with Yukawa uniﬁcation and NLSP stop (green circle) and excluded ones by combined ATLAS monojet and multi-jets searches (red triangle) in the framework of b − τ Yukawa uniﬁed mSUGRA/CMSSM. The most stringent bound on this plane from ATLAS is also displayed [46]...... 49

3.4 σ (left panel) and σ (right panel) vs. M 0 in the framework of SI SD χ˜1 b − τ Yukawa uniﬁed mSUGRA/CMSSM. The excluded region is denoted in red. Current limits from CDMS-II, XENON100, SuperK and IceCube, and future projected sensitivities from XENON1T, SuperCDMS and IceCube DeepCore are also shown...... 50

xi 4.1 mh vs. tan β plane illustrating the contributions of vector-like multilplets to the Higgs mass. The blue curve corresponds to MS = 2TeV and Xt = 6, and the red dashed line corresponds to (MS,MV ,Xk10,Xt) = (200 GeV, 2 TeV, 3, 6) and κ10 = 1. The black dashed line shows mh = 126 GeV...... 55

4.2 Feynman diagram showing the light stop running in the production and decay loops of the gg → h → γγ process. σgg→h can be enhanced due to constructive interference of the top and stop loops. For large At, σgg→h can be suppressed due to destructive interference of the top and stop loops. Γh→γγ diphoton decay can be suppressed by up to 20% due to destructive interference of the W boson and top/stop loops...... 60

4.3 Plots in the Br/BrSM vs. mt˜1 plane for (a) h → γγ and (b) h → ZZ

4.4 Plots in the M3SU vs. M3SQ and At vs. M3SQ planes. Orange points satisfy the constraints described in section 6.4. The brown points form a subset of the orange points and satisfy the current limits on Rgg and RZZ from the CMS experiment given in Eq. (4.2). The black points form a subset of the brown points that satisfy the Higgs mass range given in Eq. (4.18)...... 62

4.5 Plots in the (Br/BrSM)h→γγ vs. mt˜1 ,(Br/BrSM)h→ZZ vs. mt˜1 and

(σ/σSM)gg→h vs. mt˜1 planes. The ratio of the cross sections and branching ratio for the gg → h vs. h → γγ and h → ZZ channels are plotted in panels (d) and (e). Panel (f) shows the plot in the Rγγ vs. RZZ planes. The color coding and deﬁnition of the dashed lines is given in Figure 4.3...... 65

4.6 RXX vs. mA and RXX vs. tan β planes. The red points correspond to Rγγ and the blue points correspond to RZZ ...... 66

xii 4.7 Plots in the M3SU vs. M3SQ, At vs. M3SQ, tan β vs. M3SQ and mA

vs. mt˜1 planes. The green and brown points form a subset of the orange points and satisfy the current limits on Rgg and RZZ from the ATLAS and CMS experiments given in Eq. (4.2). The black points form a subset of the green and brown points and satisfy the Higgs mass bounds given in Eq. (4.18)...... 67

4.8 Feynman diagram showing the light stau running in the decay loop of the gg → h → γγ process. Presence of a light stau will only aﬀect Γh→γγ and not the cross section σgg→h. Enhancement in this corresponds to light stau eﬀects coming from mixing controlled by µ tan β...... 69

4.9 (Br/Br ) vs. m ,(Br/Br ) vs. m and (σ/σ ) SM h→γγ ˜b1 SM h→ZZ ˜b1 SM gg→h

vs. mt˜1 planes. The ratio of the cross section and branching ratio for gg → h vs. h → γγ and h → ZZ channels are plotted in panels (d) and (e). Panel (f) shows the plot in the Rγγ vs. RZZ planes. The deﬁnition of the dashed lines is given in Figure 4.3. All points satisfy the constraints described in section 6.4...... 70

4.10 Plots in MSD3 vs. M3SQ, Ab vs. M3SQ, tan β vs. M3SQ and mA vs. m planes. Color coding is the same as in Figure 4.7...... 71 ˜b1

4.11 (Br/BrSM)h→γγ vs. mt˜1 ,(Br/BrSM)h→ZZ vs. mt˜1 and (σ/σSM)gg→h

vs. mt˜1 planes. Panel (c) shows the plot in the Rγγ vs. RZZ planes. The deﬁnition of the dashed lines is given in Figure 4.3. All points satisfy the constraints described in section 6.4...... 72

4.12 M3SE vs. M3SL and Aτ vs. M3SL planes. Color coding is the same as in Figure 4.7...... 73

4.13 (Br/BrSM)h→γγ vs. mτ˜1 ,(Br/BrSM)h→ZZ vs. mτ˜1 and (σ/σSM)gg→h

4.14 M3SE vs. M3SL, Aτ vs. M3SL, tan β vs. M3SL and mA vs. mτ˜1 planes. Color coding is the same as in Figure 4.7...... 76

xiii 5.1 Plots in Mmess − Λ and tanβ − Λ planes for n5 = 1 and n5 = 5. Gray points are consistent with REWSB. Green points satisfy particle mass bounds and constraints described in section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Red points belong to a subset of green points and satisfy the Higgs mass range 124 GeV ≤ mh ≤ 126 GeV...... 83

5.2 Plots in At − mt˜R and mh − mt˜R planes for n5 = 1 and n5 = 5. Color coding is the same as described in Figure 5.1...... 85

5.3 Plots in m − m 0 , m − m and m − m planes for n = 1 and h χ˜1 h τ˜R h g˜ 5 n5 = 5. Color coding is the same as described in Figure 5.1. .... 86

5.4 Plots in mh − mG˜, planes for n5 = 1 and n5 = 5. Color coding is the same as described in Figure 5.1...... 90

6.1 Plot in the M3 − m16 planes. The panel on the left shows data points collected with Isajet 7.84 whereas the panel on the right shows data from SuSpect 2.41. The light blue points are consistent with REWSB and neutralino LSP. For both the panels, black points are subset of the light blue points and satisfy Rtbτ ≤ 1.03 in panel 6.1(a) and Rtbτ ≤ 1.05 in panel 6.1(b). The unit line is to guide the eye. .... 98 √ √ 6.2 Plot in the mt˜L mt˜R − At/ mt˜L mt˜R planes. The data points shown are collected using Isajet. Color coding is the same as in Figure 6.1. 99

6.3 Plot in the Rtbτ − mh plane. Panel 6.3(a) shows results obtained from Isajet, while 6.3(b) shows results from obtained SuSpect. Gray points in 6.3(a) and 6.3(b) are consistent with REWSB and LSP neutralino. Green points form a subset of the gray and satisfy sparticle mass [45] and B-physics constraints described in Section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Brown points form a subset of the green points and satisfy Ωh2 ≤ 1. Panel 6.3(c) shows data collected with Isajet (blue) and SuSpect (red). The red and blue points satisfy the constraints imposed on the green points in panels 6.3(a) and 6.3(b)...... 100

xiv 6.4 Plots in the M1/2 − m16, A0/m16 − m16, m10 − m16 and µ − m16 planes. All the points shown are consistent with REWSB, LSP neutralino and satisfy sparticle mass [45] and B- physics constraints described in Section 6.4. We also require that the points do no worse than the SM in terms of (g − 2)µ. In addition the satisfy the Higgs mass range 122 GeV < mh < 128 GeV and Rtbτ < 1.2. The purple points show results obtained from SuSpect and yellow points is the data collected using Isajet...... 105

6.5 Plots in the Rtbτ − µ, Rtbτ − m16 planes. Color coding is the same as in Figure 6.3(c)...... 107

6.6 Plot in the R − m , R − m , R − m˜ , R − m , R − m 0 tbτ A tbτ τ˜1 tbτ t1 tbτ tildeg tbτ χ˜1 and Rtbτ − mχ˜± planes. The color coding is the same as in Figure 6.3. 108

6.7 Plot in the Rtbτ − mA planes. Color coding is the same as in Figure 6.3(c)...... 109

6.8 Plots in the σ − m 0 and σ − m 0 planes. The cross sections are SI χ˜1 SD χ˜1 calculated using Isajet. Points shown in gray are consistent with REWSB and LSP neutralino. Green points form a subset of the gray and satisfy sparticle mass [45] and B-physics constraints described in Section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Brown points form a subset of the green points and satisfy Ωh2 ≤ 1. The orange points are a subset of the brown points and satisfy R ≤ 1.1 In the σ - m 0 plane, the tbτ SI χ˜1 current and future bounds from the CDMS experiment are represented as black (solid and dashed) lines and as red (solid and dotted) lines for the Xenon experiment. The right panel shows the σ - m 0 plane with the current bounds from Super K (solid red SD χ˜1 line) and IceCube (solid black line) and future reach of IceCube DeepCore (dotted black line)...... 111

6.9 Plots in the Rtbτ -∆EW and Rtbτ -∆HS planes. Color coding is the same as in Figure 6.3...... 111

xv ABSTRACT

Supersymmetry (SUSY) is one of the most elegant extensions of the standard model (SM) of particle physics. It solves several problems in the SM, supports the idea of grand unification and also provides a candidate for dark matter. The recent discovery of the Higgs boson is consistent with the lightest Higgs boson in Minimal Supersymmetric Standard Model. However, to date, no signals of supersymmetry were found at the LHC. The presence of the Higgs boson and the absence of supersymmetric particles at the LHC can significantly impact various SUSY models. The topic of this dissertation is to demonstrate how supersymmetry is being tested at various fronts in the era of the Large Hadron Collider (LHC). We will employ SUSY searches from the LHC to derive a model dependent limit on the supersymmetric top quark mass when it is next to the lightest to the SUSY dark matter candidate. We will also address the impact of these observations on the SUSY dark matter candidate. Furthermore, if SUSY is present at LHC accessible energies it can effect the rate at which the Higgs boson decays to some final states. We will discuss how the decay rate of the Higgs boson to two photons can be affected in the presence of SUSY and show that the supersymmetric partner of the tau particle, the stau, can account for an enhancement in this channel. One of the popular mechanisms of SUSY breaking is the minimal gauge mediated SUSY breaking model. We will show that a 125 GeV Higgs severely constrains the minimal version of this model. It implies that, with the exception of the stau, all the SUSY particles are heavy enough to evade detection at the LHC. Finally, we demonstrate that a 125 GeV Higgs boson can also hint to a particular grand unified model. We shall discuss the SO(10) grand unified model with t − b − τ Yukawa unification and show that the Higgs boson mass can be predicted in this model.

xvi Chapter 1

INTRODUCTION TO THE STANDARD MODEL OF PARTICLE PHYSICS

In this chapter we introduce the Standard Model (SM) of particle physics. We begin with a brief description of some important developments that laid the foundations of the standard model of particle physics. We will then describe its framework and structure. We will further discuss the unanswered questions in the standard model and the need to go beyond the SM.

1.1 Introduction The Standard Model (SM) of particle physics [5,6,7] is a hallmark of our understanding of the Universe at a fundamental level. Our quest to understand the Universe began from time immemorial and the SM is a colossal step towards the culmination of this quest. We now understand particles called quarks and leptons as being the building blocks of matter. Several nobel prizes have been awarded for the development of its formalism with the ﬁnal one awarded for the discovery of the Higgs boson. The foundations of the SM were laid with the discovery of the electron by Thomson in 1897. Furthermore, the development of special relativity and quantum mechanics are important historical advancements towards the SM. Following are some of the key theoretical ideas that laid the foundations of the SM:

• Symmetry principles in physics: Symmetries play an extremely important role in particle physics. The idea of symmetries became important with the identification of Lorentz invariance in Einstein’s theory of relativity. Symmetries became more important with the identification of internal symmetries of theories describing fundamental particles. A type of internal symmetry first identified was the isospin symmetry which is a symmetry of strong interactions. In the isospin space the proton and neutron can be treated as two states of the same particle. The

1 standard model, in addition to being invariant under space-time symmetries, is also invariant under an internal symmetry group.

• Union of QM and Special Relativity: Relativistic quantum field theory emerged from the union of Quantum mechanics and Einsteins special theory of relativity. Feynman, Schwinger, Tomonaga and Dyson were pioneers in the development of the quantum field theory of electrodynamics. The SM is a quantum field theory which describes the strong, electromagnetic and weak interactions.

• Parity violation: The resolution of the τ-θ puzzle lead to the conclusion that weak interactions violate parity [8]. This paved the way for the introduction of spontaneous symmetry breaking to generate the masses of fermions in the standard model.

• Symmetry breaking: With the identification of symmetries it was also realized that these symmetries can be broken in nature. The concept of broken symmetries was inspired by a similar phenomenon observed in superconductors [9]. The idea is that the Hamiltonian that describes the theory can possess symmetries that the vacuum of the theory might not. If these symmetries are global, breaking them leads to massless spin 0 particles called Goldstone bosons. However, when the broken symmetries are local these massless particles are used up or ‘eaten’ by the gauge bosons. In addition, a field is needed to break these symmetries which is the role of the Higgs field in the SM.

• Renormalizability: With the advent of quantum field theory physicist came across infinities that appeared to sabotage the elegance of the theory. However, it was later shown that these infinities could be ‘tamed’ by, for instance, redefining the physical parameters of the theory. An approach referred to as ‘renormalization’. The standard model was thereby shown to be a renormalizable theory [10].

Therefore, the foundations of the standard model were laid down in the beginning of the 20th century and culminated with the discovery of the Higgs boson in the beginning of the 21st century. This quest however is far from over and there are various motivations to go beyond the standard model as we shall discuss at the end of this chapter.

1.2 The Standard Model of Particle Physics The standard model of particle physics, also called the Glashow-Weinberg-Salam model, is one of the most successful theories that describe the Universe at the fundamental level. The recent discovery of the Higgs boson is yet another success of this

2 model. Excellent introductions to the SM can be found in literature (see for example refs [11, 12, 13]). In the following, we present a brief review:

• The SM is a relativistic quantum field theory invariant under local SU(3)c × SU(2)L × U(1)Y symmetry, where the subscript c stands for color, L stands for weak interactions and Y stands for hypercharge. • It describes strong, weak and electromagnetic interactions. • There are 3 families of fermions and each family contains 15 particles. Each particle has a corresponding anti-particle. • Spin 1 gauge bosons are the force carriers that mediate interactions between these particles. • The fermions acquire mass via spontaneous symmetry breaking when the Higgs field acquires a vacuum expectation value. The symmetry SU(2)L ×U(1)Y breaks to U(1)em when the Higgs field acquires a vacuum expectation value. • It is a chiral theory, which means that the interactions described by the SM differentiate between left and right handed particles. • The gauge couplings in the SM evolve with energy and indicate that there might be a more fundamental grand unified structure at high energies.

We will now brieﬂy describe the mathematical structure and the interactions in the SM. The SU(2)L × U(1)Y invariant Lagrangian of the SM can be written as

LSM = LGM + LG + LY + LH (1.1) where X τ i Y L = if¯ γµ ∂ − ig W i − ig0 B f GM L µ 2 µ 2 µ L generations

Y +if¯ γµ ∂ − ig0 B f (1.2) R µ 2 µ R

1 1 L = − W iµνW i − BµνB (1.3) G 4 µν 4 µν

X ¯ ¯ ¯ ˜ LY = [yeLφeR + ydQφdR + yuQφuR + h.c.] (1.4) generations

2 LH = |Dµφ| − V (φ) (1.5)

3 1st 2nd 3rd Quantum numbers

u c t 1 (3, 2, 3 ) d L s L b L 4 uR cR tR (3, 1, 3 )

2 dR sR bR (3, 1, - 3 )

ν ν ν e µ τ (1, 2, −1) e L µ L τ L

eR µR τR (1, 1, −2)

Table 1.1: The three generations of particles in the standard model along with their SU(3)c × SU(2)L × U(1)Y quantum number. The W and Z gauge bosons mediate weak interactions between these particles. The electromagnetic interaction is mediated by the photon γ and eight gluons mediate strong interactions between these particles.

Here, fL = Q, L, fR = uR, dR, eR and the sum runs over all three generations. The terms in LGM and LG constitute the gauge sector of the theory. LGM contains the kinetic terms of the fermions of each generation and also interaction of fermions and gauge bosons. LG contains the kinetic terms of the gauge bosons. LY contains interaction terms of the fermions with the Higgs field. The only term in the above Lagrangian ˜ ∗ that flips the chirality of a fermion is the Yukawa interaction. The term with φ = iτ2φ is needed to give mass to the up type quark, here τ2 is the second Pauli matrix. LH contain the kinetic term of the Higgs field and the interaction of the Higgs field with electroweak gauge bosons.

Note that we have not displayed the SU(3)c part of the Lagrangian which includes additional terms of interaction for the gluons and fermions. For example, for the left handed quarks the covariant derivative is given by

λa τ i Y D = ∂ − ig Ga − ig W i − ig0 B µ µ s 2 µ 2 µ 2 µ

4 which imply that the quarks participate in strong, electromagnetic and weak interactions. The covariant derivative for leptons would not include the term proportional to

Gµ since leptons do not participate in strong interactions. Similarly, the term proportional to Wµ is absent for right handed leptons and quarks since the W gauge bosons ¯ only couple to left handed particles. The interactions given by the term ∼ ψi Dψ i contain fermion and gauge boson vertices . The ﬁeld strength tensors are given by

a a a abc b c Gµν = ∂µGν − ∂νGµ − gsf GµGν

i i i 0 ijk j k Wµν = ∂µWν − ∂νWµ − g WµWν

Bµν = ∂µBν − ∂νBµ (1.6)

The kinetic term of the gauge fields include self interactions of these fields if the group describing these interactions is non-abelian. Therefore gauge bosons of strong and weak interactions self interact. Since QED is an abelian theory there are no tree level photon-photon interactions. In other words gauge bosons of non-abelian interactions can scatter off each other at tree level whereas this is not the case for abelian interactions. The photon also interacts with the W ± bosons since these are electrically charged.

1.3 The Higgs Mechanism The particles in the SM acquire mass through the Higgs mechanism [14, 15, 16, 17] when the Higgs field φ acquires a vacuum expectation value. Figure 1.1 demonstrates this for a potential with a real scalar field. The Lagrangian for the Higgs field is given by

2 LHiggs = |Dµφ| − V (φ) V (φ) = µ2|φ|2 + λ|φ|4 where µ is the Higgs mass parameter and λ is the Higgs quartic coupling. The parameter λ has to be positive in order for the potential to be bounded from below. Here φ

5 Figure 1.1: Example of symmetry breaking for the potential V = µ2φ2 + λφ4. For µ2 > 0 the potential has a trivial minimum (red line). For µ2 < 0, the minima of the potential are non-trivial (blue line). Symmetry breaking takes place as the ﬁeld settles down in one of these minima (+v or −v). The Lagrangian when expressed in terms of the physical ﬁeld (φphys = φ − v) will no more have the Z2 symmetry (φ → −φ) of the original Lagrangian. is a complex SU(2) doublet given by

+ 1 φ1 + iφ2 φ φ = √ = (1.7) 0 2 φ3 + iφ4 φ

For µ2 > 0 the potential has a trivial minimum at φ = 0. For µ2 < 0, the minima of the theory corresponds to the points satisfying v2 −µ2 hφ†φi = ≡ (1.8) 2 2λ Symmetry breaking in the SM takes place when the Higgs field φ acquires a vacuum expectation value. The minima in this case is infinitely degenerate and corresponds to a ring in the complex plane. We can choose a particular direction (say, θ = 0) signaling the breakdown of the rotational symmetry of the vacuum. This means that the Higgs field chooses a particular direction in the space of internal symmetry group. The breaking of a global symmetry leads to massless Goldstone bosons. The fact that the SM symmetries are local implies that there are no massless Goldstone bosons

6 associated with symmetry breaking in the standard model. These massless bosons provide the longitudinal degrees of freedom to the gauge bosons in local symmetry breaking or as it is commonly described, they are ‘eaten’ by the gauge bosons. We therefore understand the Higgs ﬁeld as permeating all space-time such that its value

0 at each point is the vacuum expectation value hφ i = hφ3i = v. The Higgs boson is an excitation in this Higgs ﬁeld, i.e., H(x) = φ3(x) − v. The VEV is chosen for the neutral component of the ﬁeld so that the vacuum remains neutral. The VEV is given by

1 0 hφi = √ (1.9) 2 v

As the Higgs ﬁeld acquires a VEV the symmetries SU(2)L and U(1)Y are broken, i.e.,

τihφi= 6 0,Y hφi= 6 0 (1.10)

whereas the vacuum remains invariant under U(1)em

1 Qhφi = (τ + Y )hφi = 0 (1.11) 2 3 indicating that the Lagrangian is invariant under U(1)em. Here Yφ = 1. Note also that the Lagrangian in equation (1.2) is not written in terms of the

2 physical fields. Substituting (1.9) in |Dµhφi| we find that the physical fields are given by

W 1 ∓ W 2 W ± = µ√ µ µ 2 ! ! ! 3 Zµ = cos θw − sin θw Wµ (1.12) Aµ sin θw cos θw Bµ where

g g0 cos θw = , sin θw = (1.13) pg2 + g02 pg2 + g02

7 The photon remains massless and the W and Z bosons acquire masses. The masses of the fermions and gauge bosons are related to the VEV of the Higgs ﬁeld as

1 M = g v ' 80.4 GeV W 2 pg2 + g02 MZ = v ' 91.2 GeV √ 2 mh = λ v

mf ∼ yf v (1.14)

The electroweak Lagrangian with fermion and gauge boson interaction terms with physical states of the gauge bosons can be obtained by placing the above equations (1.12) in Lagrangian (1.2).

g µ + µ† − µ 2 02 1/2 µ − Lew = √ (J W + J W ) + eJ Aµ + (g + g ) J Zµ (1.15) 2 cc µ cc µ em nc where e = gg0/(g2 + g02)1/2 and

1 J µ = ψ¯ γµ (τ + iτ )ψ (1.16) cc L 2 1 2 L µ ¯ µ Jem = ψγ Qψ (1.17) 1 J µ = ψγ¯ µ (c − c γ5)ψ (1.18) nc 2 V A

f 2 f f 1 The coeﬃcient cV ≡ T3 − 2Qf sin θw, cA ≡ T3 and T3 = 2 τ3.

It is also important to note that the ﬁelds in the Lagrangian (1.2) are weak eigenstates and not the physical states of the particles. As we rotate the ﬁelds to mass

0 u 0 u eigenstates (e.g. uL = ULuL, uR = URuR) the neutral currents are diagonal whereas the charged currents are not, i.e., ! d µ ¯ µ Jcc = (¯u c¯ t)γ V s (1.19) b

u† u where V = UL UR is a 3 × 3 unitary matrix determined by flavor changing weak processes. This is how the flavor changing effects are understood in charged weak interactions.

8 The W bosons can interact with the Higgs and Z boson at the tree level. How- ever, it only interacts with fermions that are left handed (equation 1.16). The Z interacts with fermions of both chiralities, i.e., left handed and right handed but with diﬀerent coupling strengths (equation 1.18). Weak interactions violate charge conjuga- tion C and parity P . The combined operation CP is violated by a very small amount as is observed in K0 decays.

+ + Strangeness is violated by charged weak interactions such as K → µ νµ whereas there is no evidence of violation of strangeness in neutral weak interaction (mediated by the Z boson). The suppression of Flavor Changing Neutral Currents is an important constraints for physics beyond the SM. One important feature of the SM is that it is anomaly free which means that divergences resulting from anomalous triangular diagrams vanish.

1.4 The Need to go Beyond the Standard Model Despite its success, the SM still has some theoretical problems and questions that motivate us to go beyond the SM.

• The SM explains the electromagnetic, weak and strong force but does not include gravity.

• Why are there three family of particles?

• What is the solution of the gauge hierarchy problem in the SM? The Higgs mass receives large radiative corrections and the SM does not has the solution to this problem.

• Neutrinos are massless in the SM at the renormalizable level. The observed oscillations of neutrinos from the Sun can explained if neutrinos are assumed to have a small mass.

• The standard model does not provide a candidate for cold dark matter. Neutrinos can form fraction of hot dark matter but a cold dark matter candidate is needed to agree with observations.

• Is there a grand unified structure? The three gauge couplings in the SM merge together but do not unify. In theories beyond the SM, the unification can be nearly perfect. This is an indication that there might be a unified structure at higher energies.

9 • The CP violation observed in the quark sector is too small to explain the baryon asymmetry of the Universe.

Therefore there are several factors that motivate us to enter the realm of physics beyond the SM (BSM). There is a vast landscape of ideas that have been developed for BSM physics. These include supersymmetry, extra-dimensions, grand uniﬁed theories, technicolor, etc. In the following chapters we shall mainly focus on Supersymmetry and Grand uniﬁed theories. Supersymmetry, in particular, is one of the most anticipated theories in the BSM physics realm as it can provide answers to several of the questions unanswered by the SM.

10 Chapter 2

PHYSICS BEYOND THE STANDARD MODEL

2.1 Supersymmetry Supersymmetry (SUSY) is another attempt to search for symmetries in nature at the fundamental level (for reviews, see [19, 20, 21, 22, 23, 24, 25]). It is a symmetry which connects fermions and bosons. SUSY combines the notions of space-time and internal symmetries. Prior to its discovery combining these symmetries was believed to be impossible because of the “no-go theorems”. Coleman and Mandula showed in 1967 [18] that it is impossible to combine Lorentz symmetry with internal symmetries in a non-trivial manner. SUSY however escapes the no-go theorem because the generators of SUSY are fermionic. Therefore, if Q is an operator that causes supersymmetric transformations than its operation on a fermionic state would result in a bosonic state or vice versa, i.e.,

Q|F i = |Bi

Q|Bi = |F i (2.1)

It is obvious that the operator Q will carry a spin 1/2 since it changes the spin of the state it acts on. In other words Q is a fermionic operator. If we allow for one set of generators of SUSY, Qα (α = 1,..., 4) the SUSY algebra is given by

¯ µ {Qα, Qβ} = 2(γ )αβPµ (2.2) 1 [Q ,J ] = (σ ) Q (2.3) α µν 2 µν αβ β

[Qα,Pµ] = 0 (2.4)

The ﬁrst anticommutation relation essentially says that two SUSY transformations are equivalent to a space-time translation.

11 χSF s = (1/2, 0) SU(3)c × SU(2)L × U(1)Y Q (q, q˜) (3, 2, 1/6) c U (uR, u˜R) (3,¯ 1, -2/3) c ˜ D (dR, dR) (3,¯ 1, 1/3) L (l, ˜l) (1, 2, -1/2) c E (eR, e˜R) (1, 1, 1) ˜ Hu (hu, hu) (1, 2, 1/2) ˜ Hd (hd, hd) (1, 2, -1/2)

Table 2.1: The chiral superﬁelds (χSF ) in the MSSM. Here q = (uL, dL) and ˜ l = (νL, eL) and similarly forq ˜ and l. The two Higgs bosons are SU(2) + 0 0 − doublets, i.e., hu = (hu , hu) and hd = (hd, hd ).

SUSY is one of the most elegant extensions of the Standard Model (SM) as it solves some of the problems within the SM and also provides a candidate for dark matter. SUSY predicts the presence of new particles that can manifest themselves in nature through a number of ways. These particles can, for example, be produced in high energy collisions, such as those taking place in the Large Hadron Collider (LHC). They can also lead to phenomenon such as neutron-antineutron oscillations. Furthermore, the ﬂux of particles, such as neutrinos, produced at the center of the Sun can also be aﬀected due to the presence of these supersymmetric particles. In addition to being an elegant extension of the idea of symmetry SUSY can also solve a problem that plagues the SM. This problem is referred to as the gauge hierarchy problem. The source of this problem is the presence of a scalar in the theory. The mass of this Higgs in the SM receives large radiative corrections if the scale of

2 2 2 new physics is the Planck scale, i.e., mh = m0 + O(Λ ) where Λ is the scale of new physics. Another way to state the gauge hierarchy problem is as follows: Why are the electroweak scale (∼ 100 GeV) and the Plank scale (∼ 1019 GeV) are so far apart.

2.2 The Minimal Supersymmetric Standard Model If SUSY is a symmetry in nature and all fermions have bosonic partners and vice versa than the particles in the SM should also have their respective superpartners.

12 VSF s = 1/2 s = 1 SU(3)c × SU(2)L × U(1)Y G g˜ Gµ (8,1,0) ˜ µ W Wi Wi (1,3,0) B B˜ Bµ (1,1,0)

Table 2.2: The vector superﬁelds (VSF) in the MSSM.

This guides us to a supersymmetric extension of the SM. A minimal way to do this is the introduce a supersymmetric partner for every particle in the SM, which is called the minimal supersymmetric standard model (MSSM). This is typically done by combining the fermion (ψ) and its scalar partner (φ) in a chiral supermultiplet. A superﬁeld can be written in terms of the Grassmanian variables as √ Φ(y) = φ + 2θψ¯ + iθθF (2.5)

Here F is an auxiliary ﬁeld which does not have a kinetic term and is therefore non- propagating. It can be eliminated using the equations of motion. For example, the electron chiral multiplet contains the electron and its superpartner, the selectron, i.e. √ ¯ E =e ˜L + 2θψeL + iθθFe (2.6)

Similarly, the gauge bosons and their superpartners (gauginos) are contained in vector supermultiplets. The ﬁeld content of the MSSM is given in Table 2.1 and 2.2. Su- persymmetrizing the SM however comes at a price, with the additional ﬁeld content we can now write operators that are singlets under the SM gauge group but violate baryon and lepton numbers. Baryon and lepton number conservation are accidental symmetries in the SM and the MSSM allows for these symmetries to be violated. We will discuss below how a parity can be imposed on the MSSM to forbid such operators.

13 2.2.1 The MSSM Lagrangian For a model with only chiral superﬁelds (Wess-Zumino model) the SUSY La- grangian is sum of F and D terms

LWZ = LF + LD (2.7) where, Z Z 2 † † 2 ¯ h i h † † i LF = d θW (φ) + W (φ )d θ ≡ W (φi) + W (φi ) θθ θ¯θ¯ Z 2 2 ¯ † h † i LD = d θd θφiφi ≡ φi φi (2.8) θθθ¯θ¯

Here LF contains mass and interaction terms of the fermions and scalars. The part

LD contains kinetic terms of the scalars and fermions. For a QED like theory with spin 1 bosons there are additional terms in the SUSY Lagrangian. Using the notation of [19], the SUSY Lagrangian of an abelian theory can be written as

h i h i h i h i h i 1 α 1 ¯ ¯ α˙ † 2gV † † LQED = U Uα + Uα˙ U + φi e φi + W (φi) + W (φi ) (2.9) 4 θθ 4 θθ θθθ¯θ¯ θθ θ¯θ¯

Here Uα is the SUSY field strength for the vector superfield. The first two terms contain the kinetic terms of the gauge bosons and gauginos. The second term, in addition to the kinetic terms of the fermions and scalars, also contains interaction of these particles with the gauge bosons and gauginos. The Lagrangian for a non-abelian theory is similar to (2.9) but more intricate in details. In particular the derivatives in the first two terms of (2.9) have to be replaced by covariant derivatives. These terms would therefore also contain the self interaction of the gauge bosons as in QCD and also gaugino-gaugino-gluon interaction.

The MSSM Lagrangian which is invariant under SU(3)c × SU(2)L × U(1)Y can be constructed based on the above considerations. The following terms constitute the

Lagrangian of the MSSM (LMSSM = L1 + L2 + L3),

14 1 Z L = d2θ(T rGαG + T rW αW + BαB ) + h.c. (2.10) 1 4 α α α

Z X 2 2 ¯ † 2g3V3+2g2V2+2g1V1 L2 = d θd θ Φi e Φi Matter

Z 0 0 2 2 ¯ † 2gsG+2gW +2g B † 2gW +2g B = d θd θ (Qi e Qi + Hi e Hi + ...) (2.11)

Z 2 L3 = (WR + WR/ )d θ + h.c. (2.12)

L1 contains the kinetic terms of the gauge bosons and the gauginos. The terms in

L2 are the kinetic terms of the SM particles and their superpartners. In addition, L2 also contains interaction of these particles with gauge bosons and their superpartners

(the gauginos). In equation (2.12), WR is the part of the MSSM superpotential that conserves baryon and lepton number whereas terms in WR/ violate these accidental symmetries of the SM. L3 also contains Yukawa and other interaction terms of the ﬁelds in the MSSM.

The SU(3)c × SU(2)L × U(1)Y invariant renormalizable superpotential of the MSSM that does not violate the baryon and lepton numbers is given by

c c c WR = yu U QHu + yd D QHd + ye E LHd + µHuHd (2.13)

Each field1 in the superpotential is a chiral superfield with contents given in Table 2.1. The last term in the superpotential is referred to as the µ term and is present due to the additional Higgs superfield in the MSSM. This term gives mass terms of the Higgs bosons in the MSSM. The gauge and family indices have been suppressed in the above

1 In this chapter fields denoted by capital letters represent chiral superfields and the fields with a ∼ on top represent the additional fields introduced in the SM to supersymmetrize it.

15 potential. With these indices shown, the ﬁrst term of the superpotential is written as follows

c c α yu U QHu = yu (U )αQ Hu [SU(3) indices, 3 × 3¯ = 8 + 1]

c i j = yu U Q ij(Hu) [SU(2) indices, 2 × 2 = 3 + 1]

c = (yu)ab (U )aQbHu [Family indices] where the first line shows the SU(3) indices (α=1, 2, 3), the second line shows the SU(2) indices (i, j=1, 2) and the third line displays the family indices (a, b=1, 2, 3). After eliminating the auxiliary fields using their classical equations of motion the interaction terms of the matter fields and their superpartners in the MSSM can be written as

1 L = − (W ψiψj − W ∗ ψi†ψj†) − W iW ∗ (2.14) int 2 ij ij i

If the neutral component of the Higgs acquires a vacuum expectation value, the above Lagrangian results in equal masses for the fermions and their superpartners. This, of course does not agree with observations since we have not observed spins 0 particles with the same mass and charge as the electron. SUSY is therefore broken in nature. We will further discuss this in more detail below. The µ term poses a problem in the MSSM often referred to as the µ problem.

The term µHuHd is not forbidden by any symmetry and therefore µ can even be O(MP ) whereas the more natural value for this parameter is around a TeV. In other words a larger value of µ implies that more ﬁne tuning is needed in the MSSM. So the question of why the µ term is O(T eV ) is referred to as the µ problem.

2.2.2 R-parity The following operators violate baryon and lepton numbers in the MSSM

c 0 c 0 WL/ = λLLE + λ LQD + µ LHu (2.15)

00 c c c WB/ = λ U D D (2.16)

16 These operators can lead to fast proton decay (p → e+π0) which obviously contradicts

34 with observations, since τp & 10 yrs. A way around this problem is to introduce a parity in the MSSM commonly referred to as the R-parity. It is deﬁned as follows

R = (−1)3(B−L)+2s (2.17) where B and L are the baryon and lepton numbers of the particles and s is the spin. Therefore, ordinary particles in the MSSM will carry an even parity and their superpartners will have odd parity. In addition to restricting the baryon and lepton violating operators, R-parity has other important consequences. One signiﬁcant implication is that the lightest supersymmetric particle is stable. In many SUSY models the neutralino, which is a combination of the neutral spin 1/2 states in the MSSM, can be the lightest particle. This therefore implies that imposing R-parity can provide a dark matter candidate. Various experimental searches for this dark matter candidate are under way. We will discuss neutralino dark matter in more detail in later chapters.

2.2.3 Soft SUSY Breaking Terms SUSY breaking takes place when the F or D term in the scalar potential acquire a vev. The potential in SUSY is given by

V = VF + VD 1 = |F |2 + (Da)2 i 2 ∂W 2 1 † a 2 = + (gφi T φi) (2.18) ∂φi 2

So SUSY breaking takes place when Vmin 6= 0. However, D-term and F -term breaking do not work in the MSSM [21]. Therefore, SUSY breaking is typically understood to take place in the hidden sector and the eﬀects are transmitted to the visible sector via mediating interactions. These interactions can be due to gravity or even gauge interactions similar to those in the SM. Soft SUSY Breaking terms are induced as a result of these mediating interactions. In gravity mediated SUSY breaking, soft SUSY

17 breaking terms are induced when the F-term of a hidden sector ﬁeld (X) acquires a VEV. The size of SUSY breaking terms in this case are of the order

2 FX Λ mSUSY ∼ ∼ (2.19) MP MP

11 3 If Λ ∼ 10 GeV then mSUSY ∼ 10 GeV. In chapter 5 we shall study the minimal version of the gauge mediated SUSY breaking scenario in more detail. The soft SUSY breaking (SSB) Lagrangian contains the mass terms of the squarks, Higgs, gauginos and also the interaction terms of the scalars in the MSSM. It is important that these terms are introduced so that the desired features of SUSY, such as solving the hierarchy problem, are not lost. The SSB Lagrangian is therefore given by

LSSB = Lgaugino + Lhiggs + LY uk + Lscalars (2.20) where

1 L = − (M B˜B˜ + M W˜ W˜ + M G˜G˜ + h.c.) gaugino 2 1 2 3 L = −m2 h† h − m2 h† h + (bh h + h.c.) higgs hu u u hd d d u d

˜ ˜ Lcubic = −auu˜Rqh˜ u − addRqh˜ d − aee˜Rlhd + h.c. L = −m2q˜†q˜ − m2 u˜† u˜ − m2 d˜† d˜ scalars q uR R R dR R R −m2˜l†˜l − m2 e˜† e˜ l eR R R where, Lgaugino, Lhiggs and Lscalars contain mass terms of the gauginos, Higgs and scalars in the MSSM. Lcubic contains cubic interactions of the scalars in the MSSM.

2.2.4 Electroweak Symmetry Breaking in the MSSM The mechanism by which electroweak symmetry breaking takes place in the MSSM is similar to the SM but is more intricate since there are two Higgs doublets involved. The Lagrangian of the MSSM alone cannot be employed to implement electroweak symmetry breaking. The mass terms for the Higgs obtained from the µ term

18 in the MSSM are positive and cannot be used to implement radiative electroweak symmetry breaking. The terms in the SSB Lagrangian enable implementing REWSB in the MSSM correctly. In the MSSM the Higgs potential is given by

VHiggs = VF + VSSB + VD (2.21) where

V = m2 |h |2 + m2 |h |2 − b(h h + h.c.) (2.22) SSB hu u hd d u d

2 2 2 VF = |µ| (|hu| + |hd| )

1 a 2 a 2 VD = [(D ) + (D ) ] (2.23) 2 U(1)Y SU(2) !2 !2 1 X 1 X ~τ = g0 φ†Y φ + g φ† φ (2.24) 2 i i 2 i 2 i φi=hu,hd φi=hu,hd g02 1 −1 2 g2 ~τ ~τ 2 = h† h + h† h + h† h + h† h (2.25) 2 u 2 u d 2 d 2 u 2 u d 2 d 1 1 = (g02 + g2)(|h |2 − |h |2)2 + g2|h† h |2 (2.26) 8 u d 2 u d Also in the above equations

2 + 2 0 2 |hu| = |hu | + |hu|

2 0 2 − 2 |hd| = |hd| + |hd |

The freedom to make gauge transformations allows us to choose the following vacuum expectation values of the Higgs bosons

0 hhui = vu

vd hhdi = 0 With these expectation values equation (2.21) gives

02 2 g + g vu V = (v2 − v2)2 + (v v )M2 (2.27) Higgs 8 u d u d H vd

19 where

m2 + |µ|2 −b 2 hu MH = (2.28) −b m2 + |µ|2 hd

In order to obtain EWSB the eigen values of the Higgs mass matrix should be negative

det(MH ) < 0 (2.29)

We can also see from equation (2.26) that the quartic terms from the D-term in the

Higgs potential vanishes if vu = vd. If this is true than the potential can be unbounded from below. Therefore, to avoid this the following condition should be satisﬁed.

m2 + m2 + 2|µ|2 > 2b (2.30) hd hu

The minimization conditions ∂V/∂vu = 0 and ∂V/∂vd = 0 from the potential in (2.27) gives

m2 m2 − m2 tan2 β µ2 = − Z + hd hu (2.31) 2 tan2 β − 1

b = (m2 + m2 + 2|µ|2) sin β cos β (2.32) hd hu where

g02 + g2 (v2 + v2) = m2 = (91.18 GeV)2 (2.33) 2 u d Z v tan β = u (2.34) vd

As in the SM where the Higgs field carries 4 real degrees of freedom to begin with and with symmetry breaking three of these provide longitudinal degrees of freedom to the gauge bosons. In the MSSM we initially have two complex Higgs field with 8 real degrees of freedom. Three of these are eaten by the gauge bosons W +, W − and Z and we are left with five physical Higgs bosons. Two of these are the neutral CP-even

20 Higgs bosons (h0, H0), one is CP-odd (A0) and two are charged Higgs bosons (H±). The masses of these bosons are given by

2 2 2 2 m 0 = 2b/ sin β = 2|µ| + m + m (2.35) A hd hu q 2 1 2 2 2 2 2 2 2 2 m 0 0 = m 0 + m ± (m + m ) − 4m m cos (2β) (2.36) h ,H 2 A Z A0 Z A0 Z 2 2 2 mH± = mA0 + mW (2.37)

Note that for cos2 2β = 1 we get

2 1 2 2 2 2 m 0 = m 0 + m − |m 0 − m | (2.38) h 2 A Z A Z

2 2 • For mA0 > mZ : mh0 = mZ .

2 2 2 • For mA0 < mZ : mh0 = mA < mZ

Therefore, in the MSSM the tree level Higgs mass has to be less than the Z boson mass at the tree level

mh0 ≤ mZ (2.39)

This prediction is modiﬁed with radiative corrections to the Higgs mass with dominant contributions from the top and stop loops (mh0 . 135 GeV). We will present the complete two loop expression of the Higgs mass in section 4.2.

2.2.5 Neutralinos and Charginos There are four gauge eigenstates in the MSSM that are spin 1/2 and neutral. ˜0 ˜0 ˜ ˜ 3 These are the two neutral Higgsino states hu, hd, the bino (B) and the wino (W ). The Lagrangian for these neutral majorana states is given by   B˜   W˜ 3 ˜ ˜ 3 ˜0 ˜0   L ⊃ (B W hd hu)Mχ˜   (2.40)  ˜0   hd    ˜0 hu

21 where Mχ˜ is the neutralino mass matrix given by   M1 0 −mZ sW cβ mZ sW sβ    0 M2 mZ cW cβ −mZ cW sβ Mχ˜ =   (2.41) −m s c m c s 0 −µ   Z W β Z W β  mZ sW sβ −mZ cW sβ −µ 0

Here Mi are the masses of the gauginos from the SSB Lagrangian, µ is from the superpotential and the terms porportinal to mZ arise from the gaugino-Higgs-Higgsino interaction (∼ gψ¯λ¯aT aφ) when the neutral component of the Higgs ﬁeld acquires a vev.

0 The physical mass eigen states of the neutralino (˜χi ) are obtained by diaganolizing the mass matrix and are linear combinations of the bino, wino and higgsino states, i.e.,

0 ˜ ˜ 3 ˜ 0 ˜ 0 χ˜i = Ni1B + Ni2W + Ni3H1 + Ni4H2 (2.42)

where Nij are obtained by diagonalizing the neutralino mass matrix. The physical masses are assumed to obey the following hierarchy: m 0 < m 0 < m 0 < m 0 . χ˜1 χ˜2 χ˜3 χ˜4 ˜ + ˜+ Similarly, the masses of the winos and Higgsinos that are positively (W , hu ) ˜ − ˜− and negatively (W , hd ) charged is described by a 2 × 2 matrix. √ M 2m s W˜ + ˜ − ˜− 2 W β L ⊃ −(W hd ) √ + c.c. (2.43) ˜+ 2mW cβ µ hu

˜− ˜+ Mixing among the Higgsinos hd and hu occurs due to the µ term in the superpotential and the oﬀ diagonal term in this also arise from gaugino-Higgs-Higgsino interaction.

± The physical mass eigenstates for the charginos (˜χi ) are again obtained by diagonalizing the mass matrix in the above equation and by convention m ± < m ± . χ˜1 χ˜2

2.2.6 Squark and slepton masses The Lagrangian describing the stop quark mass matrix in the MSSM is given by

t˜ ˜† ˜† 2 L L ⊃ −(tL tR) m˜t (2.44) t˜R

22 where

m2 + m2 + m2 ( 1 − 2 s2 )c m (A − µ cot β) 2 q3 t Z 2 3 W 2β t t m˜t = (2.45) m (A − µ cot β) m2 + m2 + m2 (− 2 s2 )c t t tR t Z 3 W 2β Following are four important contributions that shape the stop quark mass matrix (see for example [23]):

2 2 • The SSB mass terms contribute mq3 and mu3 .

2 c • The top (mass) arises from the superpotential term ytQ3(U )3Hu

∂W 2 = y2t˜† t˜ |h0 |2 = m2t˜† t˜ (2.46) ˜† t L L u t L L ∂tR

˜ ˜ 0 0 0 • The oﬀ-diagonal mixing term µmt cot β arises from the term yttRtLhu − µhuhd in the MSSM potential as

∂W 2 = |y t˜† t˜ − µh0|2 0 t R L d ∂hu vev ˜† ˜ 2 −−→|yttRtL − µvd|

• The oﬀ-diagonal term proportional to the trilinear coupling arises from ˜† vev ˜† ˜ AttRq˜3hu + h.c. −−→ AtvutRtL + h.c.

2 2 • The D-term contribution is proportional to mZ (I3 − Q sin θW ) cos 2β.

Similar contributions lead to the following mass matrices of the sbottom and stau.

m2 + m2 + m2 (− 1 + 1 s2 )c m (A − µ tan β) 2 q3 b Z 2 3 W 2β b b mb˜ = (2.47) m (A − µ tan β) m2 + m2 + m2 ( 1 s2 )c b b bR b Z 3 W 2β

m2 + m2 + m2 (− 1 + s2 )c m (A − µ tan β) 2 l3 τ Z 2 W 2β τ τ mτ˜ = (2.48) m (A − µ tan β) m2 + m2 + m2 ( 1 s2 )c τ τ τR τ Z 3 W 2β These mass matrices are diagonalized to obtain the physical mass eigenstates of ˜ the third generation sfermions t˜1,2, b1,2 andτ ˜1,2.

23 2.2.7 Renormalization Group Equations Renormalization group equations (RGEs) are essential in understanding the low energy implications of physics at high energies. The gauge couplings in the SM scale with energy and appear to converge to a single value at high energies. The convergence however is not perfect. In the MSSM, due to the richer particle content of the theory, the convergence of gauge couplings works accurately and hint to a grand uniﬁed scale at energy scales around 1016 GeV 2. At one loop the RGE equations are given by

dα b i = − i α2 (2.49) dt 2π i

2 where, αi = gi /4π and i = 1, 2, 3 correspond to the U(1)Y , SU(2)L and SU(3)c groups. Q is the energy scale of the interaction and t = ln Q. In the SU(5) normalization the

0 p U(1)Y gauge coupling is normalized to g = 3/5 g1. The above equation can be integrated to reach the following equation [23] −1 −1 bi Q αi (Q) = αi (Q0) + ln (2.50) 2π Q0

The energy scale Q0 is typically taken be mZ and the known parameters at mZ are −1 2 −1 α1 (mZ ) = 129, sin θW ' 0.232 and αs (mZ ) = 0.119. The coeﬃcients bi depend on the gauge group and the matter multiplets to which the gauge bosons couple. For the SM and MSSM these coeﬃcients are given by

41 19 bSM = , − , −7 (2.51) 19 6 33 bMSSM = , 1, −3 (2.52) 5

Using these beta functions and parameters we can check that the gauge coupling uni- ﬁcation for the SM is not accurate whereas it is nearly perfect for the MSSM and the couplings unify around 1016 GeV. This therefore indicates a grand uniﬁed structure that is supersymmetric in nature.

2 Gauge coupling uniﬁcation can also work in the absence of SUSY, for example in the presence of vector like particles and also in the SO(10) model.

24 Other parameters in the MSSM also scale with energy and lead to interesting eﬀects at low energy. The gaugino masses in the MSSM satisfy the simple equation at one loop order

d M i = 0 (2.53) dt αi

In mSUGRA type models where all the gauginos have the same mass at the GUT scale, the ratio Mi/αi is essentially maintained at any scale. Using the values of the known parameters at mZ it can be shown that

M1 : M2 : M3 ' 1 : 2 : 7 (2.54)

Therefore, the low energy gluino is predicted to be much heavier than the bino (B˜) and wino (W˜ ). This holds for most but not all models of SUSY. There can be scenarios where the gluino is light and is the NLSP. The gaugino contribution to the running of the scalar masses is negative (∼

2 2 −αiMi ) and tends to increase the (mass) of the scalars for low energies. Similarly, 2 2 the contributions from the fermion loops is positive (∼ +yt mscalar). This tends to decrease the low energy (mass)2 of the scalars. These two opposite eﬀects enable the implementation of radiative electroweak symmetry breaking in the MSSM. Due to the

2 absence of gluino contribution in the RGE for mHu and a comparatively bigger eﬀect from the fermion loops its low energy (mass)2 can be negative. As in the SM, this negative (mass)2 is needed to implement REWSB. Lastly, another interesting eﬀect can take place in the running of the RGEs for Yukawa couplings. The RGEs receive a positive contribution from yukawa couplings (∼

2 2 yi ) and a negative contribution from gauge couplings (∼ −gi ). The latter effect tends to decrease the Yukawa couplings at high scales whereas the former would result in an increasing effect. The interplay of the two enable the unification of Yukawa couplings at the GUT scale as well. The unification of Yukawa couplings is also predicted in grand unified theories such as SO(10) (yt = yb = yτ ) and SU(5) (yb = yτ ). We will further discuss Yukawa coupling unification in the next section and also in chapter 6.

25 2.3 Grand Unification In this section we will describe supersymmetric grand unified models (For SUSY GUT review see, for example, [26, 27] and for a general introduction to GUTs see Chap- ter 7 of reference [28]). As we discussed briefly before, there are several motivations to consider grand unified theories. One important indication is that the gauge couplings in the SM, that vary with energy scale of the interaction, appear to unify at a high energy scale. We also described earlier that as symmetry breaking takes place in the SM we are left with strong and electromagnetic interactions, i.e.,

SU(3)c × SU(2)L × U(1)Y −→ SU(3)c × U(1)em (2.55)

It is possible that the SM itself results from symmetry breaking of a grand uniﬁed group G at a higher energy scale and the symmetry breaking may take place in the following manner,

MGUT MW G −−−−→ SU(3)c × SU(2)L × U(1)Y −→ SU(3)c × U(1)em (2.56)

If this is the case than a unified gauge group (and therefore a single gauge coupling) describes the interactions of the particles and a single family of fermions in the SM can be combined in one multiplet of this unified gauge group. This can also help us explain why charge is quantized. The gauge bosons in grand unified theories should be able to interchange particles in a single multiplet. In other words, the gauge bosons would mediate interactions that violate baryon and lepton number conservation. Therefore, proton decay is one of the testable prediction of grand unified theories. If such a grand unified model exists than it should satisfy the following requirements:

• The SM gauge group SU(3)c × SU(2)L × U(1)Y should be a subgroup of this unified group. The particles in the SM should therefore fit in the representations of this model. • The cumulative rank of the SM group is 4. Therefore, the rank of the unified gauge group should atleast be 4. The rank of a group is the number of generators of a group that can be diagonalized simultaneously. For the group SU(N) the rank is N − 1.

26 • The unified group should have a complex representation. This is because the SM is a chiral theory and the left handed fields transform in a different manner from right handed fields. Therefore in order to account for parity violation ψ and ψc should reside in the conjugate representations.

• It should be free of anomalies.

We described in the previous chapters that there are several reasons SUSY can be an interesting extension of the SM. These include resolution of the gauge hierarchy problem and gauge coupling unification. Grand unified theories stand independent of supersymmetry as a possible extension of the SM. However, considering grand unified theories that are supersymmetric can solve some of the problems in grand unified theories as well. These include unification of gauge couplings which is problematic in some GUTs and also the resolution of the gauge hierarchy problem that manifests itself in another way in GUTs.

2.3.1 SU(5) Grand Uniﬁcation One of the earliest attempt to construct such a grand uniﬁed model was made by Georgi and Glashow [30]. In this model the fermions are arranged in a 5¯ and 10 dimensional representation of SU(5). The group SU(5) has n2−1 = 24 generators which are 5 × 5 hermitian and traceless matrices. Out of these, 8 of the SU(3) generators are contained in the following matrices   λi 0   , i = 1,, 8 (2.57) 0 0 5×5 and 3 of the generators of SU(2) are contained in   0 0   , j = 1, 2, 3 (2.58) 0 σ20+j 5×5

27 The normalized hypercharge generator is

 1  − 3    1   − 3    r3 Y   SU(3) =  1  ∼ (2.59)  − 3    5 2   SU(2)  1   2    1 2 p The overall normalization ( 3/5) is chosen so that the SU(5) generators satisfy T r(TiTj) =

1/2δij. This is an important generator and it commutes with the matrices containing SU(3) and SU(2) generators. These matrices will act on objects in a 5×1 matrix which would contain SU(3) and SU(2) multiplets. To see how the SM particle content can be embedded in SU(5) representation we can notice from reading the particle hyper- charges in Table 1.1 that a suitable choice for the 5 dimensional representation is the lepton doublet l and the dc quark, i.e.,   c  d1      c   d2  SU(3)c    5¯ =  c   (2.60)  d3      −   e    o −ν SU(2) L c c T ∗ where ψL ≡ PLψ = Cγ0 ψR. Note that the sum of the charges of the particle in the multiplet is zero, Tr Q = 0. This therefore explains charge quantization. The reason why the quark uc was not chosen is that the charges of the multiplet would not add to zero. Once we have chosen the SM particles in the 5¯ representation we can choose the representation for the remaining particles as the antisymmetric representation of the following product

5 × 5 = 15S + 10A (2.61)

The antisymmetric 10 contains the remaining SM ﬁelds and has the form 1 10αβ = √ (5α5β − 5α5β) (2.62) 2 1 2 2 1

28   c c 0 u3 −u2 u1 d1    c c   −u3 0 u1 u2 d2  1   10αβ = √  c c  (2.63)  u2 −u1 0 u3 d3  2    +   −u1 −u2 −u3 0 −e    + −d1 −d2 −d3 e 0

There are 24 gauge bosons in SU(5) that are in the adjoint representation.

5 × 5¯ = 1 + 24 (2.64) |{z} g,W ±,Z,γ,X,Y,X,¯ Y¯

Therefore, in addition to 12 gauge bosons of the SM there are 12 new boson referred to as the X and Y bosons. The gauge bosons of the SM induce the known transitions such W − g as d −−→ u, u1 −→ u2, etc. The new gauge bosons can mix the leptons and quarks and + X Y can induce transitions such as e −→ d andν ¯ −→ d, where QX = 4/3 and QY = 1/3. Following are some problems in the minimal SU(5) model:

• SU(5) and other grand unified theories typically suffers from the so called doublet triplet splitting problem. The origin of this problem lies in the presence of a colored triplet in the 5-plet Higgs. The colored triplet can mediate fast proton decay and needs to be heavy in order to suppress this process. The Higgs doublet in the multiplet however is light. This seeming unnatural difference in masses is another aspect of the gauge hierarchy problem in the SM.

• Minimal SU(5) appears to be ruled out by the limit on the proton decay rate. Proton decay is induced through the exchange of gauge bosons and higgs boson mediated interactions. The eﬀective dimension 5 operators that describe proton 2 31 decay are of the form qqql/MGUT . The estimated proton decay in SU(5) is . 10 years whereas the current experimental limit is ∼ 1034 years.

• As in the SM, there is no right handed neutrino in SU(5) and the neutrino is massless. One way to solve this problem is to consider larger groups such as SO(10) in which the right handed neutrino is part of the 16 dimensional multiplet containing the SM particles.

In order to supersymmetrize SU(5) we have to introduce chiral and gauge multiplets to ﬁt the particle content of SU(5) and their superpartners. In addition to this we need to introduce another Higgs in the 5¯ representation to cancel anomalies and

29 give masses to the both up and down type quarks. The MSSM Higgs Hu and Hd are contained in the 5 and 5¯ dimensional representions. The fermions acquire mass with the 5 dimensional Higgs through Yukawa couplings given by

(Yu)ij 10i10j5H + (Yd)ij 10i5¯j5¯H + h.c. (2.65)

c c c These terms contain the SM interaction ytu QHu +y(d QHd +LHde ). Therefore at the

GUT scale SU(5) predicts b − τ Yukawa uniﬁcation (YU), yb = yτ ≡ y which implies

me = md, mµ = ms, mτ = mb (2.66) and obviously disagrees with observations. These values however are at the GUT scale and renormalization group running can modify these relations considerably. Moreover a more complicated Higgs sector might be needed to agree with observed mass ratios. In the supersymmetric SU(5) case the uniﬁcation of gauge couplings is nearly perfect. This being possible due to the additional matter ﬁelds that modify the beta functions of the gauge couplings. Although SUSY solves the gauge hierarchy problem it does not solve the doublet-triplet splitting problem in minimal SUSY SU(5). This problem reappears as the µ problem in MSSM, i.e., the unknown reason of why is

µ ∼ MZ and the triplet Higgs mass ∼ MGUT . In SUSY SU(5) dimension 4 proton decay operators are forbidden by R-parity.

However, eﬀective dimension 5 operators (e.g.q ˜qql/M˜ GUT ) are possible and are generated via interactions involving higgsino exchange. The dominant decay channel in SUSY SU(5) is p → K+ν¯. The limit on the proton lifetime through this decay mode

33 is τp→K+ν¯ > 3.33 × 10 yrs and essentially rules out minimal SUSY SU(5). However there are nonminimal versions that survive (see [26] and references therein).

30 2.3.2 SU(4)c × SU(2)L × SU(2)R Pati-Salam model [30] uniﬁes the quark and the leptons of a SM family in one multiplet.   ur ug ub νe Q = (4, 2, 1) ≡   (2.67) dr dg db e L   c c c c c ur ug ub νe Q = (4¯, 1, 2) ≡   (2.68) c c c c dr dg d e b R The particle νc is the right handed neutrino and is absent in the SM. The MSSM Higgs doublet can be contained in the following representation (see for example [31])   + 0 hu hd h = (1, 2, 2) ≡   (2.69) 0 − hu h d L

c In this case the Yukawa coupling (y Q.Q .h) predicts yt = yb = yτ = yντ ≡ y. The PS model is left right symmetric and predicts the presence of a right handed neutrino. SU(5) and the Pati-Salam group are both subgroups of SO(10). However the breaking pattern of the SO(10) is diﬀerent in both cases. Moreover, in the Pati- Salam model the proton is stable whereas SU(5) predicts the proton decay. PS model also predicts neutron-antineutron oscillations ∆B = 2 [33]. Therefore, observation of neutron-antineutron will hint towards the Pati-Salam route to grand uniﬁcation rather than SU(5).

2.3.3 SO(10) In the SO(10) model a single family of the SM ﬁts in a 16 dimensional spinor representation with the addition of a right handed neutrino. With a heavy right handed neutrino in the SO(10) model the light mass of the neutrino is naturally explained through the seesaw mechanism. In addition, exact cancellation of anomalies is obtained in this group. The representation decomposes as follows under SU(5)

16 = 10 + 5¯ + 1 (2.70)

31 where the singlet is interpreted as the right handed neutrino. Under the Pati-Salam model the 16 decomposes as

16 = (4, 2, 1) + (4¯, 1, 2) (2.71)

SO(10) has N(N−1)/2 = 45 generators and there are 45 gauge bosons corresponding to these generators that transform in the adjoint representation. The breaking of SO(10) can proceed in a number of patterns to the SM. For example, SO(10) can break to the SM via the subgroup SU(5) or the Pati-Salam model or SU(5) × U(1)X (Flipped SU(5)). The Yukawa coupling in SO(10) follows from the following product

16 × 16 = 10 + 120 + 126 (2.72) which guides us to assigning the Higgs to the 10, 120 or 126 dimensional representations of SO(10). The Yukawa couplings ( y 10.10.16) imply t − b − τ Yukawa unification in SO(10) in addition to gauge coupling unification. SO(10) can be supersymmetrized in the same manner as SU(5), i.e., by introducing chiral multiplets that contain the SM fields and their superpartners. Gauge coupling unification works in SO(10) even without supersymmetry since there are two or more intermediate symmetry breaking steps to the SM. The two MSSM Higgs doublets Hu and Hd can reside in the 10 and 126 dimensional representation of SO(10). An interesting property of SUSY SO(10) is that it predicts the MSSM at low energies with exact R-parity [32].

2.4 Scanning the Parameter Space of SUSY GUTs In this section we discuss the methods we use to explore the parameter space of SUSY GUTs. In our analysis we shall employ publicly available spectrum calcu- lators ISAJET 7.84 package [35] and Suspect [36] to perform random scans over the fundamental parameter space. The ﬁrst step in analyzing a SUSY GUT model is to identify the appropriate boundary conditions at the GUT scale. The parameters are then randomly scanned using a random number generator. We employ the following steps in order to perform the random scan

32 • A uniform and logarithmic distribution of random points is ﬁrst generated in the parameter space of a particular model. The built-in functions of Fortran are used for this purpose.

• The function RNORMX [37] is then employed to generate a gaussian distribution around each point in the parameter space.

RGE running is performed using these packages to the low scale to test the prediction of a model. Various observables can be calculated at the SUSY scale. These include branching ratios of various processes, relic density of the neutralino, spin dependent and independent cross sections of the neutralino, etc. The next important step is to apply the constraints such as the branching ratios of various processes, limits on sparticle masses, relic density bound, etc. The predictability of the model can then be analyzed by identifying regions interesting for various experiments. This procedure is of course not limited to grand uniﬁed models. Packages such as FeynHiggs and Micromegas can be used to perform a SUSY scale scan as well. The predictions of such an analysis is independent of the GUT scale boundary conditions. In Chapter 4 we shall employ FeynHiggs to perform a SUSY scale scan and identify regions that predict an enhancement in the diphoton channel.

Isajet Spectrum Calculation: It is worthwhile to discuss some of the features of one of the packages, namely, Isajet. Here we describe some important steps incorporated in Isajet to calculate the spectrum. In this package, the weak scale values of gauge and third generation Yukawa couplings are evolved to the GUT scale via the MSSM renormalization group equations (RGEs) in the DR regularization scheme. We do not strictly enforce the unification condition at MGUT, since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [38]. The deviation between g1 = g2 and g3 at MGUT is no worse than 3 − 4%. For simplicity, we do not include the Dirac neutrino Yukawa coupling in the RGEs. For possible values of this coupling, the impact on our calculations is not significant.

33 The various boundary conditions are imposed at the GUT scale and all the SSB parameters, along with the gauge and Yukawa couplings, are evolved back to the weak scale MZ. In the evolution of Yukawa couplings the SUSY threshold corrections [39] √ are taken into account at the common scale MSUSY = mt˜L mt˜R , where mt˜L and mt˜R denote the masses of the third generation left and right-handed stop quarks. The entire parameter set is iteratively run between MZ and MGUT using the full 2-loop RGEs until a stable solution is obtained. To better account for leading-log corrections, one-loop step-beta functions are adopted for the gauge and Yukawa couplings, and the SSB parameters mi are extracted from RGEs at multiple scales mi = mi(mi). The RGE- improved 1-loop eﬀective potential is minimized at MSUSY, which eﬀectively accounts for the leading 2-loop corrections. Full 1-loop radiative corrections are incorporated for all sparticle masses.

2.5 Constraining the SUSY GUT Parameter Space The interactions in the MSSM are strongly constrained by rare processes, especially FCNC such as the b → sγ and µ → eγ decay. These processes are suppressed in the SM and so any physics beyond the SM must not contradict with this. Similarly new sources of CP violation are also severely constrained by observation of the electron and neutron dipole moments. A possible way to avoid conflicts with observations related to flavor changing neutral currents and CP violation is to assume universality of soft parameters at the high energy scale. Another way is to assume that the third generation is light whereas the first two generations are decoupled. Similarly, the mechanism of SUSY breaking can also affect these predictions. A mechanism to break SUSY referred to as the gauge mediated SUSY breaking can naturally account for the suppression in FCNC. Another important constraint arises from the direct and indirect searches for dark matter. We have seen that imposing R-parity in the MSSM can lead to a candidate for dark matter. There are several experiments that are underway to search for dark

34 matter. We shall employ the limits on dark matter relic abundance from the WMAP satellite to constrain the parameter space of the models we study. These constraints also include the limits on the masses of the particles from direct searches for SUSY particles. The observation of the Higgs boson and various direct and indirect searches for SUSY is restricting the MSSM parameter space further. Once the data collection for a particular model is done we impose various constraints on the calculated observables. Following is a list of important constraints on the SUSY parameter space:

2.5.1 Sparticle mass limits There have been various experimental searches for supersymmetric particles and limits have been derived on the masses of these sparticles. The current sparticle limits we employ in our analyses are:

mg˜ & 1.4 TeV (for mg˜ ∼ mq˜)[40, 41]

mg˜ & 0.9 TeV (for mg˜ mq˜)[40, 41]

123 GeV ≤mh ≤ 127 GeV [42, 43]

mt˜1 > 100 GeV [46] m > 100 GeV [44, 60] ˜b1

mτ˜1 > 105 GeV [45].

2.5.2 b → sγ This process, being loop suppressed in the SM, can be modiﬁed with SUSY contributions. The SUSY contributions to this decay arise with loops containing sfermions, charginos and charged Higgs bosons. The branching ratio can be signiﬁcantly enhanced if these sparticles are light. The limit on the branching ratio of this process is

2.99 × 10−4 ≤BR(b → sγ) ≤ 3.87 × 10−4 (2σ)[47]

35 + − 2.5.3 Bs → µ µ This decay is also suppressed in the SM and the SUSY contributions to it scale

6 4 as tan β/mA. Large values of tan β and small mA values can therefore signiﬁcantly enhance the SUSY contributions. The limit on this decay is:

−9 + − −9 0.8 × 10 ≤BR(Bs → µ µ ) ≤ 6.2 × 10 (2σ)[48, 49]

2.5.4 Bu → τντ The SUSY contribution for this process arise from the tree level exchange of the

2 charged Higgs bosons and scale as tan β/mH± . The limit on the branching ratio of this process is BR(B → τν ) 0.15 ≤ u τ MSSM ≤ 2.41 (3σ). [47] BR(Bu → τντ )SM

2.5.5 Ωh2 As discussed earlier, the MSSM provides a promising candidate for dark matter as the LSP. The neutralino relic abundance has to agree with the observed limit of relic density of dark matter. The current limit from WMAP is given by

2 +0.028 ΩCDMh = 0.111−0.037 (5σ)[50]

2.5.6 (g − 2)µ

The SUSY contribution to the muon anomalous magnetic moment aµ scales as 2 mµµMi tan β aµ ∼ 4 (i = 1, 2) (2.73) MSUSY where mµ is the mass of the muon, Mi are the electroweak gaugino masses and MSUSY ± is the characteristic mass scale of the sparticles circulating in the loop (˜µL,R,ν ˜µ,χ ˜ 0 andχ ˜j ). Therefore, light sparticles running in the loops can be employed to explain the g − 2 anomaly. While the anomalous magnetic moment of the electron is in excellent agreement with the observed value, there is a discrepancy in the anomalous magnetic moment of the muon

−10 ∆aµ ≡ aµ(exp) − aµ(SM) = (28.6 ± 8.0) × 10 [51]

36 These constraints can impose severe restrictions on the parameter space of a particular model. After collecting the data, we impose mass bounds on all the particles and various phenomenological constraints as well. We successively apply various experimental constraints on the data that we acquire. In later chapters we shall discuss diﬀerent models and apply these constraints to study the implications on the parameter space of these models.

37 Chapter 3

SUSY SEARCHES WITH NLSP STOP

3.1 Introduction In this chapter we discuss scenarios in SUSY that can lead to distinct signatures at colliders. We will focus on models where one of the sparticles can be next to the lightest SUSY particle (NLSP). We will study the LHC constraints on NLSP stop scenario in b − τ Yukawa uniﬁed mSUGRA/CMSSM using the monojet and multiple jets search results from the LHC. Our analysis [1] is inspired by a study [52] which showed that SU(5) or SO(10) inspired b − τ Yukawa uniﬁcation is compatible within the CMSSM framework with the WMAP dark matter bounds only if there exists NLSP stop - LSP neutralino coannihilation. The search for NLSP stop, especially in the region of nearly degenerate stop and LSP neutralino masses, is challenging and has been implemented by both LEP and Tevatron [45, 53, 54]. In this case, the stop two-body decay channels into a top quark and neutralino, or a bottom quark and chargino, and three-body decay ˜ + 0 ˜ + ˜ ˜+ channels t1 → W bχ˜1, t1 → b` ν˜ and t1 → b` ν are all kinematically forbidden. The loop-induced stop two-body decay into a charm quark and a neutralino is gen- ˜ + 0 0 erally considered to overwhelm the four-body channel t1 → ` ν(qq¯ )bχ˜1 and tends to be the dominant NLSP stop decay mode [55, 56]. The most stringent mass limit on a light stop with decay into charm quark and LSP neutralino comes from the CDF

search for events containing two jets and missing transverse energy, namely Mt˜1 > 180 GeV [54]. However, the Tevatron is not sensitive to stop searches if the stop and LSP neutralino mass diﬀerence is below 40 GeV. Thus the Tevatron bound does not cover

the coannihilation region above the LEP limit of Mt˜1 ≈ 100 GeV [45].

38 3.2 Brief Review of Collider Physics Here we provide a brief review of collider physics. For a detailed description see ref [34]. In the era of the LHC it is crucial to understand how physics beyond the SM can be probed at the LHC. Colliders help us probe the fundamental structure of matter and provide a bridge between theory and experiment. Collisions of particles in colliders can be categorized in the following three ways:

Hadronic Collisions Hadrons are bound states of quarks and its constituents include sea quarks and gluons as well. Therefore the collision of these particles amounts to colliding several particles together. The proton is typically used in hadron colliders such as the LHC (p →← p) and Tevatron (p →← p¯). These colliders are circular since the energy loss due to synchroton radiation is small for massive particles (∼ m−4) and therefore higher accelerator energies can be reached. The partons in the hadrons carry a fraction of its energy and therefore the eﬀective energy reach of a hadron collider is lesser. Since the cross sections of hadronic processes are typically large the event yield of these colliders is also large.

Leptonic collisions Collision of leptons is a more lucid process since these are point-like particles and the entire center of mass energy can be used up in the collision. Examples of linear colliders are LEP I and SLC which are e+e− colliders. These colliders are linear since the energy loss will be large for an electron in a circular collider.

Mixed Collisions Another type of collision involves the scattering of a lepton and a hadron. Ex- periments at SLAC, for example, probed the structure of the proton with collisions of an electron and a proton. These type of collisions were used to probe protons and eventually led to the proof that quarks are the constituents of protons.

39 In collider physics the transverse plane is very important and most of the quantities we list below are deﬁned in the transverse plane. In particular the decision of cuts for the transverse momentum pT and missing energy E T are important in studying the characteristic signal of a particular model and then removing the SM background from these events. We list here some important variables used in collider physics to study various aspects of a collision.

• The Lorentz invariant Mandelston variables s, t and u for two body scattering process 1 + 2 → 3 + 4 are given by

2 s = (p1 + p2) 2 t = (p1 − p3) 2 u = (p1 − p4)

• Luminosity L is a measure of the number of collisions in a collider f n n L = rev 1 2 (3.1) A

where frev is the frequency of beam crossing and (n1, n2) are the number of particles in the colliding beams. The quantity A is a measure of the size of the beam. If the beam size is smaller the number of collisions would be greater. Typical units for luminosity are cm−2 s−1 and barn−1 s−1 (1 barn= 10−24 cm2).

• Transverse momentum, pT = p sin θ, is the momentum perpendicular to the direction of the beam pipe. Since particles in a head on collision have zero initial transverse momentum, to a good approximation the sum of the pT of particles P i produced in a collision is zero, i.e., pT ≈ 0. Note that there are particles that can go along the beam line but these escape detection and have a low pT so the sum of the pT of the visible particles is still approximately zero. Therefore a large missing pT can indicate new phenomenon.

• Missing energy E T , is given by the sum of the pT of visible particles X E T = − pT (3.2) i

• Consider a process in which a particle of mass M is produced in a collision and decays to n particles. The invariant mass of the particle is given by

n X 2 2 ( pi) ≈ M (3.3) i

40 where pi is the four momentum of the ith particle. For example, for the decay Z → e+e− the invariant mass will be

2 2 mee = (pe+ + pe− ) 2 2 2 = (Ee+ + Ee− ) − (~pe+ + ~pe− ) ≈ MZ (3.4)

The invariant mass distribution for this decay will result in a peak at MZ . • Consider the decay of particle to two particles out of which 1 is invisible. For such semi-invisible ﬁnal states the transverse mass is deﬁned as

2 2 2 mT = [ET (1) + ET (2)] − [~pT (1) + ~pT (2)] (3.5)

where, pT (1) = E T . An example of such a decay is W → eν. The Jacobian peak in the invariant mass distribution of the electron and neutrino is used to measure the mass of the W boson. Therefore, new particles typically appear as resonances in the invariant mass distribution or peaks in the transverse mass distribution.

• Eﬀective mass, meff , is the sum of missing energy and the sum of the pT of visible jets X meff = E T + pT (3.6) jets

• The angle θ that a particle produced in a collision makes with the beam line is not Lorentz invariant. A more useful quantity is pseudorapidity η which for relativistic particles is given by θ η = −ln[tan( )] (3.7) 2 For a particle produced perpendicular to the beam line the pseudorapidity is 0. Typical detector coverage for this quantity is |η| . 3. • The separation of jets in the η − φ plane is given by p ∆R = ∆η2 + ∆φ2 (3.8)

where φ is the azimuthal angle. ∆R deﬁnes the cone size of a jet formed by multiple hadrons. • Observed number of events in a collider is given by Z Nobs = ( L dt) × × σ (3.9)

where σ is the cross section of the process and is the detector eﬃciency. The quantity in the bracket is the integrated luminosity and is a measure of the data that has been collected.

41 3.3 Collider Searches for Supersymmetry If SUSY is the solution to the hierarchy problem and sparticle masses lie around the TeV scale than a possible signal of SUSY might be imminent at the LHC. The current searches for SUSY have not yielded any signals at the LHC indicating that SUSY might be decoupled. In addition a 125 GeV Higgs boson also indicates that the stops that contribute to its mass via radiative corrections might be heavy. Several experiments, such as LEP, Tevatron and the LHC have searched for signatures of SUSY. To date, no signals have been found. The signatures of SUSY at colliders depend on several factors such as the type of colliders or the hierarchy of the spectrum. At hadron colliders, SUSY particles are dominantly produced via strong interactions whereas at lepton colliders these particles are produced via electroweak interactions. Furthermore, the manner in which SUSY breaking also effects the signatures at colliders. In gravity mediated and R-partity conserving scenarios, neutralino is the LSP whereas in gauge mediated SUSY breaking scenarios the gravitino is always the LSP. Moreover, the mass pattern can also be different for each scenario. The leads to a variety of possibilities of SUSY particles at colliders. We discussed in chapter 2 that in MSSM with R-parity the lightest sparticle is stable and can be the neutralino. One implication of this is that SUSY particles will be produced in pairs at colliders. Another implication of R-parity is that missing energy is one of the important signals for SUSY at the LHC. Since LHC is a hadron collider we expect colored superparticles to be produced in abundance due to large cross sections. Therefore important channels for SUSY at the LHC are pp → g˜g,˜ g˜q,˜ q˜q˜. These sparticles when produced can decay in cascades and result in various final states as jets, leptons, Z bosons or photons with missing energy accompanying all these states.

3.4 Why are NLSP scenarios interesting? There are several reasons why the NLSP scenario can be interesting. Before describing the motivations for coannihilation lets ﬁrst review how the NLSP scenario

42 can arise in the MSSM. Following are two important factors that can lead to a light sparticle in the MSSM, • The RGE running of the first and second generation sparticle mass parameters is mostly governed by the gauge couplings since the Yukawa couplings are small. For the third generation the Yukawa couplings are large and there is a competing 2 2 2 2 effect of the gauge and Yukawa couplings (schematically: ∼ y m0 − g M ) which leads to a small masses compared to the first and second generation. Within the third generation the right handed squarks are lighter since they have no SU(2)L interactions.

• If the oﬀ diagonal term mt(At − µ cot β) in the stop mass matrix (Eq 2.45) are large the mass matrix mimics the seesaw mechanism and one of the stops can be very light.

3.4.1 Relic abundance When the neutralino is nearly a pure bino (B˜), as is the true for most the CMSSM parameter space, the annihilation cross sections are very small. Remem- ber that the bino has no weak or color charge which implies that its interactions are

0 governed by the U(1)Y gauge coupling g . The relic density for such a dark matter candidate is large (Ωh2 ∝ hσvi−1). However if RGE eﬀects can drive the mass of the third generation sparticles (stau, stop or sbottom) light and close to the mass of the neutralino, the relic density gets additional contributions due to coannihilation [57]. A coannihilating neutralino which is a thermal relic can be pictured in the following way in the early Universe. At high temperatures the neutralino is in equilibrium with the thermal bath of particles. In other words reactions such as χχ → ff¯ take place at the same rate in the forward and backward direction. As the Universe ex- pands and cools down the particles do not have enough energy to create neutralinos and the neutralino density would start decreasing through annihilation. However due to expansion, the rate at which the neutralinos are annihilating will also decrease. This will result in a freeze out of the density of the neutralinos. If the neutralino is a pure bino this density will be too large. However, if there is another super particle such as the stau which has mass very close to the neutralino, the density of the neutralino can ∗ be lowered to the desired range. The neutralinos can annihilate via χτ˜ −→τ τγ. For

43 coannihilation the effective cross section is proportional to exp(−(mi − mχ)/T ) where mi is the mass of the coannihilating particle [58]. Therefore for small mass difference this factor would result in an increased effective cross section and a suppression of neutralino density. This is how coannihilation can play an interesting role in the early Universe.

3.4.2 NLSP signatures at colliders The other interesting implications of the coannihilating scenario include possibly distinct signatures at colliders. Below we discuss three important NLSP scenarios [21]: • NLSP Gluino Gluino pair production is an important channel for SUSY signals at hadron colliders such as the LHC. If produced the gluino can decay into a squark and a quark. The third generation squarks, namely the stop and sbottom can typically be light due to competing effects of the Yukawa and gauge couplings in the RGEs. ˜ ¯ This implies that the dominant decay modes can be t˜1t¯ and b1b. If these squarks 0 are heavy then the possible decay channels are through off-shell squarks to qqχ¯ i and/or qq¯0χ±. If gluino is the NLSP than two decay channels can be important. 0 0 One that proceeds via off-shell squarks (qqχ1) and the other is the decay gχ1 which proceeds through a loop containing squarks and quarks. • NLSP Stop ˜ ˜∗ The squarks if produced via processes such as gg → t1t1 can decay in a number of 0 0 ± ways such asq ˜ → gq˜ , qχi , q χ . If kinematically allowed, the dominant channel will beq ˜ → gq˜ since it is governed by the strong gauge coupling gs. However, for the third generation squarks which can typically be light the channelsq ˜ → qχ0 and q → q0χ± can be more important. For the NLSP stop even these channels can be kinematically forbidden if m˜ −m 0 < m and the important decay mode will t1 χ1 t ˜ 0 be t → cχ1. There are regions of the parameter space however where the three ˜ 0 ˜ ¯0 0 body decay (e.g. t1 → bW χ1) and the four body decay modes (e.g. t1 → bff χ1) can dominate [25]. Stop pair production will appear as jets + missing energy in colliders. We will study the NLSP stop scenario in more detail in the following sections. • NLSP Sbottom ˜ 0 ± The decay modes of the sbottom are similar to the stop, i.e., b1,2 → gb˜ , bχi , tχ . When sbottom is the NLSP the kinematically allowed and dominant channel will 0 be bχi . This decay mode will result in a b-jet + missing energy. Due to the interesting properties of the B-hadron (such as a large cτ) b-jet tagging is a well developed searching strategy to study final states with a b-quark.

44 • NLSP stau The cross sections for stau pair production are small at the LHC. However, staus can be produced at the LHC through cascade decays of the gluinos and squarks 0 0 + 0 0 e.g.q ˜ → qχ2 → qττ˜ andq ˜ → q χ1 → q τν˜ τ . The NLSPτ ˜ can decay to τχ1 leading to signals with multiple taus, jets and missing energy in the ﬁnal state [59]. The signal in the detector depends on whether the staus are long-lived or not. Long lived staus will result in charged tracks similar to slow moving muons in the detector.

In summary, we have seen that in the NLSP scenario, the correct relic density of dark matter can be attained and several interesting channels can lead to interesting signatures at colliders.

3.5 NLSP stop in Constrained MSSM with b − τ Yukawa unification In this section we present results of the analysis performed in ref [60]. Our analysis is the extension of the model dependent stop-neutralino coannihilation scenario explored in [52]. We discussed in chapter 2 that SUSY SU(5) predicts b − τ Yukawa unification (YU) in addition to gauge coupling unification. The RGE running of the Yukawa couplings can also lead to YU. It was shown in ref. [52] that SU(5) or SO(10) inspired b − τ YU is compatible within the Constrained MSSM (CMSSM) framework with the WMAP dark matter bounds only if there exists NLSP stop - LSP neutralino coannihilation. In the previous section we discussed how the NLSP stop can lead to distinct signals at colliders such as the LHC. Possible signals of the NLSP stop include monojet or multiple jets with large missing energy. For this purpose we employ the monojet and multiple jet search performed by ATLAS [61] to constrain the CMSSM with b − τ YU. The NLSP stop in this case is nearly degenerate with the mass difference satisfying the following constraint

M˜ − M 0 t1 χ˜1 . 20%. (3.10) M 0 χ˜1

45 It is known that large SUSY scale threshold corrections to the bottom Yukawa coupling are needed in order to implement b − τ YU

2 2 finite µ g3 Mg˜ yt At δyb ≈ 2 2 + 2 tan β, (3.11) 4π 3 M1 8 M2 We shall discuss threshold corrections in more detail in Chapter 6 when we discuss the role these corrections play in predicting the Higgs mass. The degree of Yukawa unification is quantified by the parameter R [62, 63]

max(y , y ) R ≡ b τ . (3.12) min(yb, yτ ) Following [52] we require R ≤ 1.1 to be the measure of good YU. Following are the features of the spectrum we consider in our analysis: • We focus on the parameter space of the model explored in [52] which satisﬁes

R ≤ 1.1 and Mt˜1 < Mt = 173.3 GeV.

• The mass of the gluino satisﬁes m > 6m 0 . g˜ χ˜1 • The other sparticles from the third generation, namely, M ,M ,M are much t˜2 ˜b1,2 τ˜1,2 heavier than 5 TeV.

• The squarks from the ﬁrst two families are heavier than 10 TeV.

• We assume that the NLSP stop exclusively decays to a charm and a neutralino, ˜ 0 i.e., BR(t1 → cχ1) = 100%. • The decay width of the stops in this model is such that they decay promptly in the detector (Γ ' 10−10 GeV).

3.5.1 Event Generation and Detector simulation Due to the small mass diﬀerence between the neutralino and NLSP stop the ˜ 0 charm jets resulting from the decay t1 → cχ1 are very soft and the missing energy is small. Due to this only a few of the resulting events will be able to pass the stringent cuts incorporated by the ATLAS search for new phenomenon. In order to implement more stringent bounds on the NLSP stop scenario we therefore consider the emission a hard QCD jet. This hard QCD jet can result from the initial state radiation (ISR) emitted in the form of a gluon. From momentum conservation it would also provide

46 Figure 3.1: σ×acceptance vs. Mt˜1 with horizontal line as the 95% C.L. upper limits on eﬀective non-SM processes cross section for signal region LP (top left), HP (top right), VHP (bottom). Green regions correspond to models with Yukawa uniﬁcation (R ≤ 1.1) and NLSP stop. the large missing energy needed to pass the stringent cuts. We implement the following steps for the event generation, hadronization, detector simulation in this scenario

• The SLHA ﬁle and decay table for each model in the parameter space are generated using Isajet 7.80 [35] and are used as input in Madgraph [64, 65, 66].

• Madgraph/Madevent is used to generate events with a pair of gluinos, stops and stops with a hard jet, i.e.,

˜ ˜∗ ˜ ˜∗ pp → g˜g,˜ t1t1, jt1t1 (3.13)

∗ with the gluino decaying to tt˜ 1 and t¯t˜1 each with 50% branching ratio. In Madgraph/Madevent we implement MLM matching with PT -ordered showers and the shower-KT scheme with Qcut = 100 GeV as described in Ref. [67]. • We use Pythia for parton showering and hadronization [68]

47 Figure 3.2: M 0 vs. M˜ for models with Yukawa uniﬁcation and NLSP stop (green χ˜1 t1 circle), those excluded by ATLAS monojet regions (red triangle) and other excluded ones by ATLAS multi-jets regions (black box) in the framework of b − τ Yukawa uniﬁed mSUGRA/CMSSM. The kinematic limits and coannihilation bounds are also displayed. The blue region refers to the excluded region by Tevatron [54].

• PGS-4 is used to simulate the detector eﬀects with ATLAS-like parameters [69]. We take care to correctly match (without double-counting) between matrix ele- ment and showering generation of additional jets.

• The cross sections are normalized to next-to-leading order using Prospino 2.1 [70, 71, 72, 73, 74, 75].

3.5.2 Implementing ATLAS selection requirements The ATLAS experiment has searched for monojet plus missing energy events with 1 fb−1 integrated luminosity [76]. Next, in order to derive a limit on the NLSP stop mass in this scenario we employ the selection cuts from this search. Because of the additional hard jet, our ﬁnal state is sensitive to the search with monojet and missing energy. In the ATLAS monojet search signal events are selected with according to three diﬀerent requirements, namely LP (low pT ), HP (high pT ) and VHP (very high

48 Figure 3.3: m1/2 vs. m0 for models satisfying all low energy experiments (grey box), those with Yukawa uniﬁcation and NLSP stop (green circle) and excluded ones by combined ATLAS monojet and multi-jets searches (red triangle) in the framework of b − τ Yukawa uniﬁed mSUGRA/CMSSM. The most stringent bound on this plane from ATLAS is also displayed [46].

pT ). The selection cuts are shown in Table 3.1. Following is a brief description of the cuts:

• The LP (HP) selection requires a jet with pT > 120 GeV (pT > 250 GeV), jet |η | < 2 in the ﬁnal state, and E T > 120 GeV ( E T > 220 GeV). Events with a second leading jet pT above 30 GeV (60 GeV) in the region |η| < 4.5 are rejected.

• For the HP selection, the pT of the third leading jet must be less than 30 GeV, miss and an additional requirement on the azimuthal separation ∆φ(jet, ~pT ) > 0.5 between the missing transverse momentum and the direction of the second leading jet is required. This cut is used to select events with the ﬁrst and second jets going in roughly the same direction to reduce background from j(W →)τν as stated in Refs. [77, 78].

• The VHP selection is deﬁned with the same requirements as in the HP region, but with thresholds on the leading jet pT and E T increased up to 350 GeV and 300 GeV, respectively.

49 Figure 3.4: σ (left panel) and σ (right panel) vs. M 0 in the framework of b − τ SI SD χ˜1 Yukawa uniﬁed mSUGRA/CMSSM. The excluded region is denoted in red. Current limits from CDMS-II, XENON100, SuperK and IceCube, and future projected sensitivities from XENON1T, SuperCDMS and Ice- Cube DeepCore are also shown.

Events with charged leptons are also rejected. The 95% C.L. upper limits on eﬀective cross section (cross section times acceptance) for non-SM processes for signal region LP, HP, VHP are also shown in the last row of Table 3.1.

Following Ref. [79] we apply σ ×acceptance > σexp as the exclusion requirement for each model, where σ is the relevant total cross section and the acceptance is the ratio of signal events after and before selection cuts which reflects the effects of experimental efficiency.

3.5.3 Deriving a Limit on the NLSP Stop mass

In Fig. 3.1 we show σ×acceptance vs. Mt˜1 for the models with Yukawa uniﬁca- tion and NLSP stop, after applying the requirements in the ATLAS monojet regions LP, HP and VHP. The constrained values of NLSP stop mass increase as the required pT of leading jet gets higher in the three diﬀerent regions as shown in Table 3.1, namely

Mt˜1 ∼ 110 GeV for LP region, and Mt˜1 ∼ 160 GeV for the HP and VHP regions. It is because the emitted jet recoils against the two associated stops in the transverse direction to the beams, and thus its pT is somewhat correlated with the relevant stop

50 LP HP VHP Leading jet pT (GeV) > 120 > 250 > 350 Second jets pT (GeV) < 30 < 60 < 60 Third jets pT (GeV) − < 30 < 30 miss ∆φ(~pT , j2) − > 0.5 > 0.5 E T (GeV) > 120 > 220 > 300 ATLAS σexp (pb) 1.7 0.11 0.035

Table 3.1: Summary of selection cuts and 95% C.L. upper limits on the eﬀective cross section for non-SM processes for signal region LP, HP and VHP containing ﬁnal states with monojet and missing transverse momentum with 1 fb−1 luminosity, following the ATLAS data analyses [76]. mass. Combining the exclusions from the three regions, an NLSP stop mass below 140 GeV is essentially excluded.

Fig. 3.2 shows the exclusion plot in the M 0 −M˜ plane, with Yukawa unification χ˜1 t1 and NLSP stop models (green circle). The top line corresponds to the kinematic bound ˜ 0 of t1 → cχ˜1 channel which is open below this line. The region below the bottom ˜ + 0 most straight line corresponds to the stop decay channel t1 → bW χ˜1. In the region between these two lines, a stop decay into a charm quark and LSP neutralino is the unique channel for our study, since we assume that the 4-body channel is always highly suppressed. The coannihilation bounds from Eq. (3.10) are also displayed in this plot. One can see that Tevatron bound does not cover the coannihilation region. However, the ATLAS monojet search does make additional inroads beyond the Tevatron, denoted by red triangles. For coannihilation region, it is more sensitive as the mass difference between the NLSP stop and LSP neutralino decreases. For the region with Mt˜1 & 140 GeV the monojet search loses its capability when the mass difference is larger than 20 GeV because the charm jets from stop decay in this case become harder and cannot pass the pT selection requirement for the non-leading jets. Besides the monojet channels, we also apply the ATLAS multi-jets search requirements [46] and show the excluded models (but not by monojet search) with black box in Fig. 3.2. The requirement of the additional jet and heavier gluino also provides

51 events with hard jet(s) and large missing energy that pass the multiple energetic jets search cuts. These excluded points gather in the region with M˜ − M 0 20 GeV t1 χ˜1 & because of the induced relatively large pT of jets from stop decay. Based on these features we can clearly identify the region excluded by the LHC in Fig. 3.2.

In Fig. 3.3, the excluded models are displayed in the m1/2 − m0 plane with

µ > 0 and varying A0 and tan β. We display all models that survive the low energy experiments listed in section II (grey color), models with good Yukawa unification and NLSP stop (green circle), and excluded models by the combined monojet and multi- jets searches (red triangle). One can see that the most stringent lower limit on m0 is around 3 TeV for tan β = 10,A0 = 0, µ > 0 case, from the LHC searches corresponding to comparable gluino and the first two family squarks masses [46]. Our models with good Yukawa unification and NLSP stop correspond to m0 > 8 TeV and the above study on the production of NLSP stop approach much larger values of m0, namely

8 TeV < m0 < 16 TeV. A signiﬁcant region of the parameter space is excluded.

3.5.4 ATLAS search Implications for Direct Detection It is both interesting and important to see the implications of LHC data on direct and indirect dark matter detection in this class of Yukawa uniﬁed mSUGRA/CMSSM with NLSP stop. In Fig. 3.4 we display this by plotting the spin-independent and spin-dependent WIMP-nucleon scattering cross section σSI (left panel) and σSD (right panel) vs. M 0 . A signiﬁcant region around M 0 ' 100 GeV is excluded by LHC χ˜1 χ˜1 data, although it is allowed by CDMS-II, XENON100, SuperK and IceCube experiments. This excluded region even lies about one (six) order of magnitude below the expected XENON 1T/SuperCDMS (IceCube DeepCore) bound for spin-independent (spin-dependent) cross section.

3.6 Conclusion We discussed in this chapter how the NLSP scenario can be interesting in terms of yielding the right relic density and also leading to distinct signatures in colliders. We

52 discussed a particular model (CMSSM with b − τ YU) which admits stop NLSP and used the monojet search from the ATLAS experiment to derive a limit on the NLSP stop mass. This coannihilation scenario with NLSP stop decaying into a soft jet evades the previous Tevatron bound. In terms of the emission of a hard QCD jet associated with stop pair production, followed by stop decay into a soft charm quark and LSP neutralino, we observed that the monojet search at ATLAS is sensitive to the region with small mass diﬀerence between NLSP stop and LSP neutralino. The excluded limit can reach 160 GeV for the NLSP stop mass in the coannihilation region, while NLSP stop mass below 140 GeV is essentially excluded. Our results are consistent with the

model independent exclusion limit Mt˜1 < 140 GeV on the stop mass derived in [80]. The analysis for the production of stops based on the above searches excludes a signiﬁcant parameter region in mSUGRA/CMSSM, namely in the region 8 TeV . m0 . 16 TeV. The LHC implications for spin-dependent and spin-independent LSP neutralino- nucleon cross sections are also explored. Regions of the parameter space, some lying well below the much anticipated future bounds from IceCube DeepCore, Xenon 1T and SuperCDMS, are already excluded by utilizing the LHC data.

53 Chapter 4

h → γγ CHANNEL AND THE MSSM

4.1 Introduction In addition to the Higgs discovery the ATLAS and CMS experiments have both observed an excess in Higgs production and decay in the diphoton channel which is a factor 1.4 − 2 times larger than the SM expectations. In this chapter we discuss how SUSY can account for such an enhancement [2]. For the final state consisting of a pair of Z bosons, the ATLAS experiment sees an excess, whereas CMS observes a deficit. However, both are currently consistent with the presence of a SM Higgs boson [81, 82]. The observed signal for these channels is quantified by the ratio of the product of production cross sections times branching ratio to the final state XX compared to the theoretical expectation for the SM. Thus, σ(h) × Br(h → XX) RXX ≡ . (4.1) (σ(h) × Br(h → XX))SM The current values of this ratio for the γγ and ZZ channels are follows:

ATLAS: Rγγ = 1.90 ± 0.5 ,RZZ = 1.3 ± 0.6 ,

CMS: Rγγ = 1.56 ± 0.43 ,RZZ = 0.7 ± 0.5 , (4.2)

ATLAS⊕CMS: Rγγ = 1.71 ± 0.33 ,RZZ = 0.95 ± 0.4 .

4.2 Higgs mass in the MSSM The leading 1- and 2- loop contributions to the lightest CP-even Higgs boson mass in the MSSM is given by [83, 84, 85, 86, 88, 88, 89, 90, 91, 92] 3 m2 m2 = M 2 cos2 2β 1 − t t h MSSM Z 8π2 v2 4 2 3 mt 1 1 3 mt 2 + t + Xt + − 32παs Xtt + t , (4.3) 4π2 v2 2 (4π)2 2 v2

54 Figure 4.1: mh vs. tan β plane illustrating the contributions of vector-like multilplets to the Higgs mass. The blue curve corresponds to MS = 2TeV and Xt = 6, and the red dashed line corresponds to (MS,MV ,Xk10,Xt) = (200 GeV, 2 TeV, 3, 6) and κ10 = 1. The black dashed line shows mh = 126 GeV. where

2 2 2 ! MS 2Aet Aet v = 174.1 GeV, t = log 2 ,Xt = 2 1 − 2 . (4.4) Mt MS 12MS

Also Aet = At − µ cot β, where At denotes the stop left and stop right soft mixing √ parameter and MS = mt˜L mt˜R . Note that one loop radiative corrections to the CP- even Higgs mass depend logarithmically on the stop quark mass and linearly on Xt.

4.3 Higgs mass in the MSSM with Vector like particles It was noticed in refs. [93, 94, 95, 96, 97, 98, 99] that in the presence of a vector like particles around the TeV region and with suitably large couplings to the Higgs ﬁeld, one can have sizable corrections to the light CP-even Higgs mass. As an example, particles which are in the 10 + 10 dimensional representation of the SU(5) symmetry group were introduced. In the superpotential, the coupling κ1010 10 5H contains the interaction κ10Q10 U10 Hu. Here Q10 and U10 stand for vector like particles which have the same MSSM quantum numbers as the left and right handed up type quarks. Hu

55 is the MSSM up type Higgs ﬁeld and κ10 is a dimensionless coupling. In this case the CP-even Higgs boson gets the following additional contribution to its mass [93, 94, 95, 96, 97, 98, 99]:

3 3 1 m2 = −M 2 cos2 2β κ2 t + κ4 v2 sin2 β t + X . (4.5) h 10 Z 8π2 10 V 4π2 10 V 2 κ10

Here Xκ10 and tV are given as follows

2 2 2 4 2 2 4 4Aeκ10 (3MS + 2MV ) − Aeκ10 − 8MSMV − 10MS Xκ = , (4.6) 10 2 2 2 6 (MS + MV ) and 2 2 MS + MV tV = log 2 , (4.7) MV where Aeκ10 = Aκ10 − µ cot β, Aκ10 is the Q10 − U10 trilinear soft mixing parameter and

µ is the MSSM Higgs bilinear mixing term. MS ' pm ˜ m ˜ c , where m ˜ and m ˜ c are Q3 U3 Q3 U3 the stop left and stop right soft SUSY breaking masses at low scale. MV is the mass term for the vector like particles. The total CP-even Higgs mass is therefore given by

2 2 2 mh = mh MSSM + mh 10 . (4.8)

In Figure 4.1 we plot the mass mh vs. tanβ for the MSSM and the MSSM + vector like particle cases. The blue curve corresponds to the upper bound for the CP-even

Higgs mass if MS = 2 TeV and At takes its maximum possible values. It hardly reaches the 126 GeV mass bound. On the other hand, in order to minimize the stop quark contribution to mh we could choose MS = 200 GeV and consider vector like particles with masses around 2 TeV. We choose κ10 = 1 and Xκ10 = 3. The red dashed line shows that in this case the CP-even Higgs mass can be as large as 138 GeV.

4.4 gg → h → γγ Process 4.4.1 gg → h The gluon fusion process is the main production channel of the Higgs at the LHC. In the SM, the leading order (LO) process involves a top quark loop which has the largest Yukawa coupling with the Higgs. The cross-section for this process is known

56 to the next-to-next-to-leading order (NNLO) [100] which can enhance the LO result by 80-100%. Any new particle which strongly couples with the Higgs can signiﬁcantly enhance this cross-section. In the MSSM the stop plays such a role and therefore this process can probe the stop sector with the exception of scenarios when the contribution from sbottom becomes important. The decay width for this process is given by (see [101, 102] and references therein) G α2m3 2 F s h 2 h gg Γ(h → gg) = √ Nc Qt ghtt A 1 (τt) + ASUSY , (4.9) 36 2π3 2 2 2 where ghtt is the coupling of h to the top quark and τi = mh/(4mi ). The form factors are given by

h 2 A 1 (τ) = 2 [τ + (τ − 1)f(τ)] , (4.10) 2 τ 1 Ah(τ) = − [τ − f(τ)] , (4.11) 0 τ 2 1 Ah(τ) = − [2τ 2 + 3τ + 3(2τ − 1)f(τ)] , (4.12) 1 τ 2  2 √  arcsin τ τ ≤ 1 √ 2 f(τ) = 1 1 + 1 − τ −1 (4.13)  − log √ − iπ τ > 1  4 1 − 1 − τ −1 gg The supersymmetric contribution ASUSY is given by X m2 Agg = N Q2 g Z Ah(τ ) . (4.14) SUSY c q˜i hq˜iq˜i m2 0 q˜i i q˜i

The couplings ghq˜ q˜ of the CP-even Higgs boson to the squark mass eigenstates, nor- √ i i 1/2 malized to 2( 2GF ) , are given by [101, 102, 103] 1 2 2 2 2 2 cos α 1 cos α sin α g ˜ ˜ = cθ − swc2θ˜ MZ sin(β + α) − mt − s2θ˜ mt At + µ , ht1t1 2 t˜ 3 t sin β 2 t sin β sin β 1 2 2 2 2 2 cos α 1 cos α sin α g ˜ ˜ = sθ + swc2θ˜ MZ sin(β + α) − mt + s2θ˜ mt At + µ , ht2t2 2 t˜ 3 t sin β 2 t sin β sin β 1 2 1 2 2 2 sin α 1 sin α cos α g ˜ ˜ = cθ − swc2θ˜ MZ sin(β + α) − mb − s2θ˜ mb Ab + µ , hb1b1 2 ˜b 3 b cos β 2 b cos β cos β 1 2 1 2 2 2 sin α 1 sin α cos α g ˜ ˜ = sθ + swc2θ˜ MZ sin(β + α) − mb + s2θ˜ mb Ab + µ , hb2b2 2 ˜b 3 b cos β 2 b cos β cos β

2 where sw ≡ sin θW , cθ ≡ cos θ and θq˜ is the mixing angle between the ﬂavor basis and mass eigenbasis. The couplings for the stau have expressions similar to that of the

57 sbottom with the relevant electric charge for the stau in the ﬁrst parenthesis. The cross section for the gg → h process is directly proportional to the decay width Γ(gg → h).

2 The stop and sbottom loop contribution goes like 1/mq˜ and can signiﬁcantly enhance the cross section for light squarks. Moreover the cross section can also increase from an enhancement in the couplings ghq˜iq˜i . The latter enhancement can arise due to light stops, large values of the mixing parameter At and also large µ tan β. We shall discuss the enhancement and suppression of this cross section in more detail in the following sections where we present our results.

4.4.2 h → γγ The Higgs boson can decay to a pair of gauge bosons, leptons, or quarks. The dominant decay channel for a 126 GeV Higgs is a pair of b quarks (b¯b) at tree level, but is not very useful due to the large QCD background. One of the most promising decay channels is the γγ ﬁnal state which, at leading order, proceeds through a loop containing charged particles, namely the charged Higgs, sfermions and charginos. The dominant contribution to h → γγ decay comes from the W boson loop and the decay width is given by (see [101, 102] and references therein)

2 3 GF α mh 2 h h γγ 2 Γ(h → γγ) = √ Nc Q ghtt A (τt) + ghW W A (τW ) + ASUSY , (4.15) 128 2π t 1/2 1 where ghW W is the coupling of h to the W boson. The supersymmetric contribution γγ ASUSY is given by

2 2 γγ mW h X 2 mZ h A = g + − A (τ ± ) + N Q g A (τ ) + SUSY hH H m2 0 H c f hf˜f˜ m2 0 f˜ H± f f˜ X mW h + − ghχ χ A 1 (τχi ), (4.16) i i m 2 i χi

± ˜ ± where ghXX is the coupling of h to the particle X (= H , f, χi ). The stop and sbottom loop factors have similar contributions as the gluon fusion case. In this case however the stau can also contribute to enhance the decay width without changing the gluon fusion cross section. The chargino contribution to the decay width is known to be

58 less than 10% for m ± 100 GeV. The charged Higgs contribution is even smaller χi & since its coupling to the CP-even Higgs is not proportional to its mass and also due to

2 2 the loop suppression mW /mH± . For a light stop the Higgs production and decay can be signiﬁcantly enhanced. For a light sbottom the enhancement in the gluon fusion production can be large but an overall enhancement in gg → h → γγ is diﬃcult to achieve as we shall see in our analysis.

4.5 Phenomenological Constraints and Scanning Procedure We employ the FeynHiggs 2.9.0 [104, 105, 106, 107] package to perform random scans over the MSSM fundamental parameter space. The range of the parameters we choose in each case are given in subsequent sections. In our analysis the ﬁrst and second generations are decoupled since their masses are assumed to be around 5 TeV.

The gaugino mass parameters M1,M2 and M3 are also taken to be 5 TeV. We set the top quark mass mt = 173.3 GeV [108]. The version of FeyHiggs we employ also tests for color and charge breaking (CCB), and therefore points where color breaking minima is detected are rejected. The scanning procedure we employ has been described in section 2.5. The points with 0.8 < RXX < 3 are scanned more rigorously for this analysis. We successively apply the following experimental constraints on the data that we acquire from FeynHiggs: m > 130 GeV [46,1, 80], m > 100 GeV [44, 60], m > 105 GeV, [45]. The lower t˜1 ˜b1 τ˜1 bound on sfermion masses are consistent with nearly degenerate neutralino-sfermion scenarios, which are very helpful in obtaining the correct relic abundances [109]. We do not apply constraints from B-physics in our analysis since our aim is to highlight the effects of a light third generation on the Higgs production and decay to the γγ and ZZ final states. In each scenario we choose our parameters to make one of the sparticles from the third generation light with all others decoupled so that there effects on B-physics are negligible. In principle other sparticles can also be light and hence give contributions to B-physics. However, such an analysis would involve additional parameters in each case and therefore require a much more extensive analysis.

59 Figure 4.2: Feynman diagram showing the light stop running in the production and decay loops of the gg → h → γγ process. σgg→h can be enhanced due to constructive interference of the top and stop loops. For large At, σgg→h can be suppressed due to destructive interference of the top and stop loops. Γh→γγ diphoton decay can be suppressed by up to 20% due to destructive interference of the W boson and top/stop loops.

4.6 The gg → h → γγ process and a light stop

4.6.1 Decoupling Limit (mA >> mZ ) We ﬁrst consider a scenario with only a pair of light scalar top quarks eﬀectively contributing to new physics via Higgs production and decay processes. We assume the decoupling limit (mA mZ ) in which the lightest Higgs is SM-like and the other Higgs ± bosons are nearly degenerate (mA ' mH ' mH ). For this case we scan the following range of the parameter space,

100 GeV < M3SU ,M3SQ < 5000 GeV,

−4000 GeV < At < 4000 GeV, 3 < tan β < 60, (4.17) where, M3SQ,M3SU are the mass parameters of the third generation left handed squark doublet and right handed top squark, respectively. The parameter At is the coeﬃcient of the trilinear soft term associated with the top quark Yukawa coupling. All other A terms are set equal to zero. tan β is the ratio of the VEVs of the two MSSM Higgs doublets. We assume the neutralino to be the lightest supersymmetric particle (LSP) which is nearly degenerate with the light stop quark. This assumption relaxes the stop mass bound compared with other colored sparticles [80].

60 (a) (b)

(e) (f)

Figure 4.3: Plots in the Br/BrSM vs. mt˜1 plane for (a) h → γγ and (b) h → ZZ

channels. Panel (c) shows the ratio of the cross section σ/σSM vs. mt˜1 for the gluon fusion process. The ratio of the cross section and branching ratio for the h → γγ and h → ZZ vs. gg → h channel are plotted in panels (d) and (e). Panel (f) shows the plot of the product Rγγ vs. RZZ , where R is deﬁned in Eq. (4.1). The purple points satisfy the Higgs mass window given in Eq. (4.18). The vertical dashed line in panel (f) shows the upper bound on RZZ and lower bound on Rγγ from the combined analysis given in Eq. (4.2). All points satisfy the constraints described in section 6.4. 61 Figure 4.4: Plots in the M3SU vs. M3SQ and At vs. M3SQ planes. Orange points satisfy the constraints described in section 6.4. The brown points form a subset of the orange points and satisfy the current limits on Rgg and RZZ from the CMS experiment given in Eq. (4.2). The black points form a subset of the brown points that satisfy the Higgs mass range given in Eq. (4.18).

In Figure 4.3(a) and 4.3(b), we show our results in the Br/BrSM vs. mt˜1 planes for the h → γγ and h → ZZ decay channels. The cross section ratio σ/σSM vs. mt˜1 for the gluon fusion process is shown in Figure 4.3(c). The ratio of the gg → h cross section is plotted versus the branching ratio of the h → γγ and h → ZZ channels in Figures 4.3(d) and 4.3(e). The ratio R is plotted in Figure 4.3(f) for the h → γγ vs. h → ZZ channel and is given by Eq. (4.1). All the points displayed in Figure 4.3 satisfy the experimental constraints described in section 6.4. The points shown in purple in Figure 4.3 satisfy the following Higgs mass window

123 GeV ≤ mh ≤ 127 GeV. (4.18)

The ﬁrst notable feature in these ﬁgures is the large enhancement of the diphoton production and gluon fusion process in Figure 4.3(a) and 4.3(c). Following are some important factors that lead to enhancement in the diphoton channel:

• It has been noted earlier [110] for a light stop and small At, the gluon fusion rate can be enhanced by up to 60% due to constructive interference of the stop and top loops in the gluon fusion cross section. The diphoton decay, however,

62 is suppressed by up to 20% due to destructive interference of the W boson and top/stop loops. Together, this leads to an overall enhancement in the product Rγγ.

• For large values of the parameter At the gluon fusion cross section is suppressed due to destructive interference between the top and stop loops. This cancellation leads to enhancement in the diphoton channel which is now dominated by the W boson loop as seen in Figure 4.3(a). The reduction in the gluon fusion rate however is much stronger, so that the overall enhancement in Rγγ is not large.

The purple points show that the large enhancement in the diphoton production and gluon fusion process through the light stop contribution is drastically reduced once the Higgs mass bound from Eq. (4.18) is applied to the data. Figure 4.3(b) shows that the enhancement in the ZZ production is not large compared to the diphoton case. This is because in the decoupling limit, the coupling of the CP-even Higgs with the gauge boson is proportional to g sin(β − α) ∼ g. From Figure 4.3(d) we can see that the gluon fusion cross section does not vary significantly with change in the branching ratio to a pair of Z bosons. Figure 4.3(e), however, shows that the gluon fusion cross section has an inverse relationship with the diphoton branching ratio. This trend shows that the overall enhancement in Rγγ does not become large over the whole region of the parameter space for this scenario. The reason for this inverse trend is that the enhancement in the gluon fusion rate is from the constructive interference of the top and stop loops, which is accompanied by the cancellation of these with the W boson loop. The enhancement in the diphoton rate, which is due to destructive interference between the top and stop loops, is accompanied by a reduction in the gluon fusion rate. We can notice that the reduction in the gluon fusion rate is much stronger for relatively larger values of the diphoton decay rate. The ATLAS and CMS experiments have seen an enhancement in the γγ final state which is 1.4 − 2 times the SM value. The enhancement seen by ATLAS is accompanied by an enhancement in the ZZ final states, whereas this is not the case for CMS, as can be seen from the current limits given in Eq. (4.2). Clearly, more data is required to settle this. If the enhancement in the γγ channel is accompanied

63 by an enhancement in the ZZ channel the light stop scenario is disfavored. Figure

4.3(f) shows that an enhancement in RZZ is accompanied by a similar but less stronger enhancement in Rγγ for values greater than 1. The dashed lines show the upper bound on RZZ and the lower bound on Rγγ for the combined analysis given in Eq. (4.2). If this bound from the ATLAS and CMS collaborations is conﬁrmed in the near future, it will rule out the light stop scenario.

In Figure 4.4 our results are shown in the M3SU vs. M3SQ and At vs. M3SQ planes. The orange points show the data that is consistent with the bounds discussed in section 6.4. The brown points also form a subset of the orange points satisfying the current limits on the diphoton and ZZ channels from the CMS given in Eq. (4.2). The points shown in black form a subset of the brown points satisfying the limit on the Higgs mass given in Eq. (4.18). As seen from the ﬁgures and also described above, the CMS observations seem to be in favor of the light stop scenario. We can observe a large region of the parameter space consistent with the CMS bound (brown points), whereas there are no points satisfying the ATLAS bound. This is because the central values of ATLAS indicate an enhancement in both the γγ and ZZ channels, which is not favored in this scenario as seen in Figure 4.3(d). We may also note that the Higgs mass constraints is satisﬁed by the very few points shown in black. This shows that requiring the Higgs mass to be ∼ 126 GeV appears to disfavor this scenario. However, as we will discuss in the next section, the contributions of vector-like matter to the Higgs mass can ameliorate this situation.

4.6.2 Low mA Region

For low/moderate values of mA and large/moderate tan β, the bb and ττ channels can be suppressed and this, in turn, can enhance the other decay channels. Simi- larly, an enhancement in the h → bb channels leads to a suppression of the other decay channels.

64 (a) (b)

(e) (f)

Figure 4.5: Plots in the (Br/BrSM)h→γγ vs. mt˜1 ,(Br/BrSM)h→ZZ vs. mt˜1 and

65 Figure 4.6: RXX vs. mA and RXX vs. tan β planes. The red points correspond to Rγγ and the blue points correspond to RZZ .

For this case we scan the following range of the parameter space,

100 GeV < M3SU ,M3SQ < 5000 GeV,

−4000 GeV < At < 4000 GeV,

100 GeV < mA < 2000 GeV, 100 GeV < µ < 1000 GeV,

3 < tan β < 60.

The first and second generation masses are assumed to be 5 TeV. All other A-terms are set to zero. Our results for this case are shown in Figures 4.5 and 4.7. In Fig 4.5 we plot the same variables as in Figure 4.3. Comparing the figures for the two cases we can notice a much broader region showing enhancement in the γγ and ZZ final states. Figures 4.5(a), 4.5(b) show that this enhancement in the γγ and ZZ final states can now accommodate much larger stop masses (mt˜1 . 1 TeV) compared to the decoupling limit. In other words, heavier stops can now accommodate the enhancement and also satisfy the Higgs mass range from Eq.(4.18), as seen by the broader coverage of the purple points in this Figure. The enhancement in the cross section in Figure 4.5(c) shows a similar trend as in the previous case and corresponds to small values of At, resulting from the destructive interference of the stop and top loops. Figures 4.5(d) and

66 (a) (b)

Figure 4.7: Plots in the M3SU vs. M3SQ, At vs. M3SQ, tan β vs. M3SQ and mA vs.

mt˜1 planes. The green and brown points form a subset of the orange points and satisfy the current limits on Rgg and RZZ from the ATLAS and CMS experiments given in Eq. (4.2). The black points form a subset of the green and brown points and satisfy the Higgs mass bounds given in Eq. (4.18).

67 4.5(e) show that for smaller cross section, there are points with larger branching ratio for the two decay channels. In the decoupling case we saw an inverse trend between the Br and cross section, which is not present in this case due to additional enhancement for low values of mA. Figure 4.5(f) again plots the measurable quantities Rγγ vs. RZZ . We can see a large number of data points above the dashed lines and therefore a much broader region is able to satisfy the current bounds on these products.

We also observe from Figure 4.5(f) that an enhancement in Rγγ can be explained by a light stop. The product Rγγ is significantly enhanced for a light stop quark mass, which makes it difficult to get the correct Higgs mass in this scenario. However, as discussed in section 4.3, we can possibly overcome this problem by assuming the presence of vector like particles. In the presence of such particles around the TeV region and with suitably large couplings to the Higgs field, one can have sizable corrections to the light CP-even Higgs mass. This shows that in the presence of vector like particles we can have a stop quark as light as needed, without worrying about the value of the lightest CP-even Higgs mass. Therefore, in the presence of vector like particles the blue points in Figure 4.5(f) can accommodate the bounds from the ATLAS and CMS experiments.

In Figure 4.6 our results are shown in the RXX vs. mA and RXX vs. tanβ planes in order to emphasize the contribution from the MSSM CP-odd Higgs A. The red points show the product RXX for the γγ ﬁnal state, whereas the points in blue show this for the ZZ ﬁnal state. The additional enhancement observed in this case in

Figures 4.5(a) and 4.5(b) corresponds to low values of mA . 600 GeV and tan β & 30. It has been discussed in earlier references [111, 112, 113] that lower/moderate values of mA and tan β can suppress the bb and ττ channels and, as a result, the decays to

γγ and ZZ can be signiﬁcantly enhanced. The sensitivity of Br(h → bb) to mA comes through the coupling ghbb ∝ − sin α/ cos β, where the mixing angle α is a function mA. Moreover, the radiative corrections to the Yukawa couplings of the b quarks and τ leptons (which are employed in FeynHiggs) can suppress these couplings signiﬁcantly for large µ tan β.

68 Figure 4.8: Feynman diagram showing the light stau running in the decay loop of the gg → h → γγ process. Presence of a light stau will only aﬀect Γh→γγ and not the cross section σgg→h. Enhancement in this corresponds to light stau eﬀects coming from mixing controlled by µ tan β.

In Figure 4.7 we show plots of the fundamental parameters in the M3SU vs.

M3SQ, At vs. M3SQ, tan β vs. M3SQ and mA vs. mt˜1 planes. The orange, green and black points satisfy the same conditions as described in section 4.6.1. A much wider expanse of the parameter space now satisﬁes the current bounds from the ATLAS and CMS experiments given in Eq. (4.2). The combination of the two experiments is also satisﬁed as can be seen from the overlap of the two regions. There are also more points shown in black satisfying the Higgs mass for stop mass . 1 TeV. This case therefore provides a much richer parameter space that can accommodate the current bounds from experiments. Hence a spectrum consisting of a light stop with low mA and large tan β with all other particles decoupled can provide a possibility of explaining the current experimental bounds.

4.7 Enhancement in the h → γγ channel with a stau 4.7.1 Light Stau in the Decoupling limit One possible way of explaining the enhancement in the diphoton channel without signiﬁcantly enhancing the gluon fusion rate and the decay widths of the other channels is by assuming the presence of a light stau [114, 115, 116]. In this section, we therefore assume the presence of light tau sleptons and study their eﬀects Higgs production and

69 (a) (b)

(e) (f)

Figure 4.9: (Br/Br ) vs. m ,(Br/Br ) vs. m and (σ/σ ) vs. SM h→γγ ˜b1 SM h→ZZ ˜b1 SM gg→h

mt˜1 planes. The ratio of the cross section and branching ratio for gg → h vs. h → γγ and h → ZZ channels are plotted in panels (d) and (e). Panel (f) shows the plot in the Rγγ vs. RZZ planes. The deﬁnition of the dashed lines is given in Figure 4.3. All points satisfy the constraints described in section 6.4.

70 (a) (b)

Figure 4.10: Plots in M vs. M , A vs. M , tan β vs. M and m vs. m SD3 3SQ b 3SQ 3SQ A ˜b1 planes. Color coding is the same as in Figure 4.7.

71 (a) (b)

(c)

Figure 4.11: (Br/BrSM)h→γγ vs. mt˜1 ,(Br/BrSM)h→ZZ vs. mt˜1 and (σ/σSM)gg→h vs.

mt˜1 planes. Panel (c) shows the plot in the Rγγ vs. RZZ planes. The deﬁnition of the dashed lines is given in Figure 4.3. All points satisfy the constraints described in section 6.4.

72 Figure 4.12: M3SE vs. M3SL and Aτ vs. M3SL planes. Color coding is the same as in Figure 4.7. decay in the decoupling limit. For this case we scan the parameter space as follows,

100 GeV < M3SE,M3SL < 5000 GeV,

−4000 GeV < Aτ < 4000 GeV, 3 < tan β < 60. (4.19)

As before, the first two generation masses are set equal to 5 TeV and all other A-terms are set to zero. The Higgs mass parameter µ and mA are also decoupled to 5 TeV. In Figure 4.11 we plot the same variables as in Figure 4.3 for the light stau scenario. Figure 4.11(a) shows an enhancement in (Br/BrSM )h→γγwhich increases for light stau masses. The branching ratio of the ZZ channel is very close to its SM value (since g sin(β − α) ∼ g) as can be seen from Figures 4.11(b). The stau with no color charge does not affect the gluon fusion cross section. The effect on Br(h → ZZ) can be significant for low values of mA as we shall see in the next section. Earlier references (see for example, [114, 115, 116]) have noted that large values of the mixing parameter

Aτ and moderate values of mA can lead to enhancement or suppression of the h → bb decay which, in turn, can enhance or suppress the h → γγ, W W and ZZ decay modes. Moreover, the eﬀects from a light stau can also be important in enhancing the diphoton branching ratio for suitably large values large µ tan β [114, 115, 116]. Note that we

73 do not apply the Higgs mass bound in this case because we set the mixing parameter

At = 0. Choosing suitably large values of this parameter or the presence of vector like matter can accommodate the Higgs mass in the desired range given in Eq. (4.18).

The plot of RZZ vs. Rγγ in Figure 4.11(c) shows that the product RZZ remains close to the SM value, whereas Rγγ undergoes a strong enhancement. We can also derive an upper limit on the stau mass if the we require an enhancement in the diphoton channel as suggested by current observation. For Rγγ > 1.2 the stau has to be . 300 GeV. This scenario may be favored if in future analyses, the enhancement in the diphoton channel as seen by the CMS and ATLAS experiments persists, with the ZZ channel being closer to its SM values. This can also be seen from the plot of the fundamental parameters in Figure 4.12 where a large region of the parameter space satisﬁes the current limits from the ATLAS and CMS experiments. Moreover, the combination of the two experiments is also satisﬁed by a broad range of the parameter space.

4.7.2 Light Stau and low mA region

Our ﬁnal scenario involves a light stau in the low mA region. For this case we scan the parameter space as follows:

100 GeV < M3SE,M3SL < 5000 GeV,

−4000 GeV < Aτ < 4000 GeV,

100 GeV < mA < 2000 GeV, 100 GeV < µ < 1000 GeV,

3 < tan β < 60. (4.20)

The ﬁrst and second generation masses are decoupled to 5 TeV and all other A-terms are set to zero. The Higgs mass parameter µ and mA are also assumed to be 5 TeV. In Figure 4.13(a) we can again see a large enhancement in the diphoton branching ratio for light stau masses. As described earlier, this enhancement corresponds to large values of µ tan β as shown in previous references [114, 115, 116]. Our results are shown

74 (a) (b)

(c)

Figure 4.13: (Br/BrSM)h→γγ vs. mτ˜1 ,(Br/BrSM)h→ZZ vs. mτ˜1 and (σ/σSM)gg→h vs.

mt˜1 planes. Panel (c) shows the plot in the Rγγ vs. RZZ planes. The deﬁnition of the dashed lines is given in Figure 4.3. All points satisfy the constraints described in section 6.4.

75 (a) (b)

Figure 4.14: M3SE vs. M3SL, Aτ vs. M3SL, tan β vs. M3SL and mA vs. mτ˜1 planes. Color coding is the same as in Figure 4.7.

76 in Figures 4.13 and 4.14. Comparing this case to the previous one we can see that the enhancement is not aﬀected, whereas a large region of the parameter space now corresponds to suppressed values of the branching ratios for the γγ and ZZ channels.

As described in the previous section, lower/moderate values of mA can enhance h → bb decay and therefore lead to a suppression of the diphoton and other decay channels.

This suppression can be seen in Figures 4.13(a) and 4.13(b). The points with Br . 1 for the two channels correspond to mA . 700 GeV. The cross section is also suppressed for lower values of mA.

The plot of RZZ vs. Rγγ in Figure 4.13(c) shows the enhancement seen in the previous case with additional suppression of the two channels corresponding to lower values of mA. We can see that this case contributes more parameter space when

RXX . 1 whereas the enhancement still corresponds to large values of mA. For small values of the ratio R there appears to be a linear relationship between the two products

RZZ and Rγγ which is not present when mA is large. This scenario is therefore more favored compared to the previous one. In Figure 4.14 plots in the fundamental parameter space plots further show the wide range of available parameter space that satisﬁes the current constraints from experiments. The overlap of the green and brown points show that the combination of the two experiments is also broadly satisﬁed. Figure 4.14(d) shows that the current limits from the ATLAS and CMS experiments prefer a stau with mass . 800 GeV.

4.8 Conclusion The ATLAS and CMS experiments have reported some exciting results regard- ing the production and subsequent decays (especially into γγ and ZZ) of a SM-like Higgs boson with mass close to 126 GeV. We have explored their implications in the MSSM framework with relatively light third generation sfermions (stop, sbottom and stau). We also considered scenarios in which TeV scale vector like particles are introduced to make sure that the Higgs boson has the desired mass of around 126 GeV.

In addition, we explored both the decoupling limit (mA >> mZ ) as well as the low

77 mA region, with the first two family sfermions in all cases assumed to be essentially decoupled. For the light stop case we find a wide region of the parameter space that can explain the current observations especially for low values of the pseudo-scalar mass mA. Requiring the Higgs to be 126 GeV constrains the parameter space but the presence of vector-like matter can always ameliorate this situation. The case of light sbottom seems to be disfavored since the sbottom contribution in enhancement of the cross section and branching ratio is not large. For the case of light staus we find a large region of the parameter space which agrees with current observations and also with the combined ATLAS and CMS limits. More data from both experiments will help pin down the eventual scenario but, based on our analysis and also noted before by other authors, a light stau seems to be the most viable scenario in explaining current observed deviations from the SM.

78 Chapter 5

IMPACT OF THE HIGGS DISCOVERY ON THE mGMSB MODEL

5.1 Introduction In this chapter we discuss the implications of a 125 GeV Higgs boson on a model of SUSY breaking called the minimal gauge mediated SUSY breaking (mGMSB) model

[3]. A Higgs boson with mh ∼ 125 GeV places stringent constraints on supersymmetry (SUSY), especially in the context of the minimal supersymmetric standard model (MSSM) [118, 198, 120, 121, 122, 123, 124, 114, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140]. In order to realize a light CP-even Higgs of mass around 125 GeV in the MSSM, we need either a very large, O(10 − 100) TeV, stop quark mass, or a large trilinear soft supersymmetry breaking (SSB) A-term with stop quark mass still around a TeV [95]. Assuming gravity mediated SUSY breaking, it was shown in ref. [118] that a SM-like Higgs boson with mass ∼ 125 GeV is nicely accommodated in SUSY grand unified theory (GUT) models with t-b-τ Yukawa coupling unification at MGUT [141]. Models with gauge mediated SUSY breaking (GMSB) model provide a com- pelling resolution of the SUSY flavor problem, as a consequence of the flavor blind gauge interactions responsible for generating the SSB term [142, 143]. In both the minimal [142, 143] and general [144] GMSB scenarios the trilinear SSB A-terms are relatively small at the messenger scale, even if an additional sector is added to generate the µ/Bµ terms [145, 146]. Although non-zero A-terms are generated at the low scale through renormalization group equation (RGE) running, a significantly high scale for the messenger fields or very heavy gauginos are required, thereby making most of the sparticles very heavy and difficult to access at the LHC.

79 Several studies [147, 148, 149, 150, 151, 152, 140] analyzing the GMSB scenario have been performed. In this chapter we perform a more comprehensive study of the mGMSB model by scanning all the essential parameters characteristic of this scenario. The messenger scale is allowed to be as high as 1016 GeV and we study the resulting sparticle spectrum corresponding to the SM-like light CP-even Higgs mass of mh = 125 ± 1 GeV.

5.2 The mGMSB model Supersymmetry breaking in a typical GMSB scenario takes place in a hidden sector, and this effect is communicated to the visible sector via messenger fields. The messenger fields interact with the visible sector via known SM gauge interactions, and induce the SSB terms in the MSSM through loops. In order to preserve perturbative gauge coupling unification, the minimal GMSB scenario can include n5 5i + 5i (i =

1, ..., n5) or a single 10+10 [143], or 10+10+5+5, or 15+15 [153] multiplets of SU(5).

For simplicity, we only consider the case with n5 5i + 5i vectorlike multiplets. Notice ¯ that 5 + 5 includes SU(2)L doublets (` + `), and SU(3)c triplets (q +q ¯). In order to incorporate SUSY breaking in the messenger sector, the ﬁelds in (5 + 5) multiplets are coupled, say, with the hidden sector gauge singlet chiral ﬁeld S:

¯ W ⊇ y1S ` ` + y2S q q,¯ (5.1) where W denotes the appropriate superpotential. Assuming non-zero vacuum expectation values (VEVs) for the scalar and F components of S, namely S = hSi + θ2hF i, the mass spectrum of the messenger ﬁelds is as follows: r Λ m = M 1 ± , m = M. (5.2) b M f

Here mb and mf denote the masses of the bosonic and fermionic components of the appropriate messenger superﬁeld, M = yhSi and Λ = hF i/hSi. The dimensionless parameter Λ/M determines the mass splitting between the scalars and fermions in the messenger multiplets. This breaking is transmitted to the MSSM particles via loop corrections.

80 At the messenger scale the MSSM gaugino masses are generated at 1-loop level, and assuming hF i hSi2, are given by α M = n Λ i , (5.3) i 5 4π where i = 1, 2, 3 stand for the SU(3)c, SU(2)L and U(1)Y , sectors, respectively. The MSSM scalar masses are induced at two loop level:

3 2 X αi m2(M) = 2 n Λ2 C , (5.4) 5 i 4π i=1 2 where C1 = 4/3, C2 = 3/4 and C3 = (3/5)(Y/2) , and Y denotes the hypercharge. The A-terms in mGMSB models vanish at the messenger scale (except when the MSSM and messenger fields mix [154, 155, 158, 157, 158], which we do not consider in this study). They are generated from the RGE running below the messenger scale. The mGMSB spectrum is therefore completely specified by the following parameters defined at the messenger scale:

Mmess, Λ, tanβ, sign(µ), n5, cgrav. (5.5)

Mmess ≡ M and Λ are the messenger and SSB mass scale deﬁned above , and tanβ is the ratio of the VEVs of the two MSSM Higgs doublets. The magnitude of µ, but not its sign, is determined by the radiative electroweak breaking (REWSB) condition. The parameter cgrav ≥ 1 eﬀects the mass of the gravitino and we set it equal to unity from now on.

5.3 Parameter Space We perform random scans for the following range of the mGMSB parameter space:

0 ≤ Λ ≤ 107 GeV

16 1.01Λ ≤ Mmess ≤ 10 GeV 1.5 ≤ tan β ≤ 60

µ > 0, (5.6)

81 with mt = 173.3 GeV [108]. We have checked that our results are not too sensitive to one or two sigma variation in the value of mt [159]. We use mb(mZ ) = 2.83 GeV which is hard-coded into ISAJET. The scanning procedure we employ is described in section 2.5. The points with CP-even Higgs mass in the range 125 ± 1 GeV are scanned more rigorously using this function. Note that for ∆(g − 2)µ, we only require that the model does no worse than the SM.

5.4 Constraining the mGMSB model with a 125 GeV Higgs 5.4.1 Sparticle Spectroscopy In this section, we present the sparticle spectroscopy which results from the procedure outlined in section 6.4. We focus on the following mass range for the lightest CP-even SUSY Higgs boson:

124 GeV . mh . 126 GeV. (5.7)

In Figure 5.1, we show our results in the Mmess − Λ and Mmess − tanβ planes for n5 = 1 and n5 = 5. The reason we choose these values for the number of messenger fields is that the sparticle spectrum, along with some other salient features, do not change appreciably for the intermediate values n5 = 2, 3 and 4. In the figures the gray points are consistent with REWSB, whereas the green points, a subset of the gray ones, satisfy all the constraints described in section 6.4. The red points correspond to the Higgs mass range given in equation (5.7), and form a subset of the green ones. We see that the desired Higgs mass requires relatively large values of Λ which, in turn, pushes the MSSM sparticle mass spectrum to larger values. The minimum values of the parameters Mmess and Λ, for n5 = 1, and satisfying the bound in Eq. (5.7) is 6 5 ∼ 10 GeV. For n5 = 5, the minimum values of Mmess and Λ are 5 × 10 GeV and 2.5 × 105 GeV, respectively. These can have interesting effects on the mass of the gravitino in mGMSB as we will see below. The tanβ − Λ plot in Figure 5.1 shows a trend in which the red points appear to merge as we approach smaller values of tanβ and large values of Λ, with a minimum

82 Figure 5.1: Plots in Mmess − Λ and tanβ − Λ planes for n5 = 1 and n5 = 5. Gray points are consistent with REWSB. Green points satisfy particle mass bounds and constraints described in section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Red points belong to a subset of green points and satisfy the Higgs mass range 124 GeV ≤ mh ≤ 126 GeV. value of tanβ ∼ 6. From this, we expect the MSSM spectrum to be much heavier for smaller values of tanβ. Note that varying n5 from 1 to 5 does not change the range of tanβ (red points) by much, while the values for Mmess and Λ are more signiﬁcantly altered.

We present in Figure 5.2 the results in mh −mt˜R and At −mt˜R planes for n5 = 1 and n5 = 5. The color coding is the same as in Figure 5.1. Note that since in the

83 mGMSB scenario the non-diagonal elements in the squark and slepton mass matrices are always smaller in comparison to the diagonal elements, we will use left and right handed notations for the third generation squark and slepton masses in our discussion.

The mh − mt˜R panel indicates that the minimal value of mt˜R , which corresponds to mh = 125 ± 1 GeV (red points) is above 6 TeV, for n5 = 1. It is more than 5 TeV for n5 = 5. The lower bound on mt˜R , therefore, is very large. In order to understand our ﬁnding, consider again the CP-even Higgs boson mass given in equation (4.3):

3 m2 3 m4 1 m2 ≈ M 2 cos2 2β 1 − t t + t t + X , (5.8) h MSSM Z 8π2 v2 4π2 v2 2 t

From the At −mt˜R plane in Figure 5.2, we see that the in the mGMSB model, At/mt˜R <

1 is always the case, no matter how large the values of Mmess and Λ. As described in section 5.2, the A-terms in this model vanish at the messenger scale whereas the scalar masses are given by Eq. (5.4). The RGE running, however, can generate large values for the A-terms at low scale. But as we see from Eq. (5.8) and (4.4), for radiative

corrections to the Higgs mass the ratio, At/mt˜R is important and not the actual value of At. As shown explicitly in ref.[95], with At/mt˜R < 1, the radiative corrections to the lightest CP-even Higgs mass are dominantly generated from logarithmic corrections. This explains why the stop quark masses have to lie in the few TeV region in the

mGMSB model. We also observe that mt˜R becomes lighter for the n5 = 5 case. We can understand this from Eq. (5.3) which shows that the MSSM gaugino masses increase by a factor 5 if we increase n5 from 1 to 5. According to the Eq. (5.4), however, the √ √ scalar masses only increase by a factor n5 = 5. This means that the large gaugino, particularly the gluino, mass enhances the low scale value of At through RGE running.

This explains why for n5 = 5, we have more red points around the unit-slope line in the At − mt˜R plane, which, on the other hand, also relaxes the lower bound on the stop quark mass.

In Figure 5.3 we display our results in the m − m 0 , m − m˜ and m − m h χ˜1 h tR h g˜ planes for n = 1 and n = 5. Here m 0 and m denote the lightest neutralino 5 5 χ˜1 g˜

84 Figure 5.2: Plots in At − mt˜R and mh − mt˜R planes for n5 = 1 and n5 = 5. Color coding is the same as described in Figure 5.1. and gluino masses, respectively. The color coding is the same as in Figure 5.1. As √ mentioned earlier, Eqns. (5.3) and (5.4) show that the scalar masses scale as n5, whereas the gaugino masses scale as n5, which is why the scalars are typically lighter than the gauginos for larger n values. This explains why the lower bound on m 0 and 5 χ˜1 mg˜ increases for higher values of n5. Also, the lowering of the bound on mt˜R is strongly related to how the stop mass changes, as discussed above in analyzing Figure 5.2. The lightest gluino for n5 = 1 is ∼ 4.5 TeV, whereas for n5 = 5, the lower bound on mg˜ increases up to ∼ 8 TeV.

85 Figure 5.3: Plots in m − m 0 , m − m and m − m planes for n = 1 and n = 5. h χ˜1 h τ˜R h g˜ 5 5 Color coding is the same as described in Figure 5.1.

86 As all other sparticles, the lightest MSSM neutralino is also heavy with a minimum mass ∼ 1 TeV for n5 = 1, and ∼ 1.8 TeV for n5 = 5. For n5 = 1, the neutralino is typically the NLSP in mGMSB. Since all other sparticles are much heavier, the neutralino, which is essentially a bino, dominantly decays to a gravitino and photon (Gγ˜ ). 0 ˜ 0 ˜ Other decay channelsχ ˜1 → GZ andχ ˜1 → Gh are also open but relatively suppressed. The decay length of the NLSP neutralino for the Gγ˜ channel is given by [160]

1 100 GeV5 Λ 2 M 2 L ∼ 10−6 mess cm, (5.9) κ1γ mNLSP 10 TeV 10 TeV where κ1γ is the photino component of the neutralino. The minimum decay length for the NLSP can vary depending on the choice of the parameters Mmess, Λ and n5. For 6 n5 = 1, (Mmess)min = Λmin ∼ 10 GeV and (mχ)min = 1.5 TeV. Choosing κ1γ ∼ 1 we get Lmin ∼ 10 cm. The NLSP is therefore short lived and would decay promptly. For larger values of Mmess and Λ, the NLSP can be long-lived and results in non-prompt photons or can escape the detector. The collider signals for mGMSB at the Tevatron and LHC were studied in [161, 162]. Neutralino pair production at the LHC can take place via loop suppressed

0 0 gluon fusion or through subprocesses qq → χ˜i χ˜j [163], which yield tiny cross sections for large neutralino and squark masses. Single neutralino production in association with a squark, gluino or a chargino is also suppressed. If produced, the neutralino would lead to ﬁnal states with photons plus missing energy, where the missing energy results from the gravitino. In reference [161], the expected number of events for prompt (non-prompt) di-photon (photon) events were estimated for the NLSP neutralino. It was shown that Nγγ(Nγ) . 1 for neutralino mass ∼ 1 TeV for 14 TeV LHC with 10 fb−1 integrated luminosity. A search for mGMSB model in ﬁnal states with diphoton events and missing transverse energy was performed by the CDF [164] and D0 [165] collaborations and no excess above the SM expectations was observed. The limits on the sparticle masses obtained from the Higgs mass bound that we have found are far more stringent compared to those obtained from these searches.

87 Comparing the m − m 0 and m − m planes in Figure 5.3, we can observe h χ˜1 h τ˜R that for n5 = 5,τ ˜R can be the NLSP, with a minimum value ∼ 600 GeV. The NLSP stau dominantly decays to Gτ˜ . The decay length of the stau is also given by equation

5 (5.9) with κ1γ = 1. For (Mmess)min = Λmin ∼ 10 GeV and (mχ)min = 600 GeV, we −6 get Lmin ∼ 10 cm. Short lived stau would decay in the detector and will result in energetic tau leptons. A long-lived stau would typically show up as charged tracks in the detector.

It was shown in reference [161] that the event yield for mτ˜ > 600 GeV is less than 10 for 14 TeV LHC with 10 fb−1 integrated luminosity, for final states with non-prompt and metastable leptons. Thus, it will be very difficult to see any events characteristic of the mGMSB scenario at the LHC if the current preferred value of the Higgs mass (∼ 124 − 126 GeV) is confirmed. The pseudo-scalar Higgs boson of MSSM turns out to have a mass & 3 TeV. The large limit on mA also implies that the lightest CP-even Higgs h is very much SM-like.

5.4.2 Gravitino Constraints and the Higgs The gravitino, which is the spin 3/2 superpartner of the graviton, acquires mass through spontaneous breaking of local supersymmetry. The gravitino mass in such scenarios can be ∼ 1 eV − 100 TeV. A light gravitino is a plausible dark matter candidate and can also manifest itself through missing energy in colliders [161]. In mGMSB the gravitino mass is given by

√ !2 F F mG˜ = √ = 2.4 eV, (5.10) 3MP 100 TeV

18 where the reduced Planck scale MP = 2.4 × 10 GeV. The lower limit on Λ and Mmess implies a lower limit on the gravitino mass.

We present in Figure 5.4 the results in mh − mG˜ planes for n5 = 1 and n5 = 5.

The color coding is the same as described in Figure 5.1. For n5 = 1, the lower limit (∼ 106 GeV) on these parameters implies that the lightest allowed gravitino mass ∼ 360 eV. The Higgs mass window in Eq. (5.7) therefore excludes very light gravitinos

88 which can be produced in the standard cosmological scenarios. In standard scenarios, the relic density bound (Ωh2 ∼ 0.11 [50]) is satisﬁed with a gravitino mass ∼ 200 eV [161], which makes it a hot dark matter candidate. The hot component of dark matter, however, cannot be more than 15% which in turn implies that the gravitino mass . 30 eV [161]. In standard scenarios, therefore, the gravitino can form only a fraction of dark matter. A gravitino mass & 30 eV requires non-standard scenarios in order to agree with observations. Such non-standard scenarios include gravitino decoupling and freezing out earlier than in the standard scenario, which may be possible in a theory with more degrees of freedom than the MSSM [161]. For n5 = 1, the lightest gravitino can be ∼ 360 eV. For n5 = 5, however, the lower limits on Λ and Mmess are smaller and the gravitino mass can be as light as ∼ 60 eV. A gravitino of mass & keV is still possible for n5 = 1 or 5, and it can be cold enough to constitute all of the dark matter if non-standard scenarios such as early decoupling is assumed. Note that these lower bounds on the gravitino mass apply for cgrav = 1. For cgrav > 1, the lower bound on mG˜ will increase, which will make the gravitino problem more severe. In Table 6.1, we show three benchmark points satisfying the various constraints mentioned in section 6.4. These display the minimal values of the neutralino, stau and gravitino masses in mGMSB that are compatible with a 125 GeV CP-even Higgs boson. Point 1 shows that the lightest NLSP neutralino allowed mass is around 1.4 TeV for n5 = 1. The second point has the lightest stau that can be realized for n5 = 5. The last point shows a 686 eV gravitino, which is the lightest value we found for a Higgs mass ∼ 125 GeV. The rest of the spectrum turns out to be quite heavy, as expected, for all three benchmark points, with the squarks typically heavier than 10 TeV and the sleptons have masses more than 2 TeV.

5.5 Conclusion The ATLAS and CMS experiments at the LHC have presented tantalizing al- beit tentative evidence for the existence of the SM Higgs boson with mass close to 125 GeV. We have explored in this chapter the implications of this observation for the

89 Figure 5.4: Plots in mh − mG˜, planes for n5 = 1 and n5 = 5. Color coding is the same as described in Figure 5.1. sparticle spectroscopy of the minimal gauge mediated supersymmetry breaking scenario. By performing a random scan of the fundamental parameter space, we find that accommodating a 125 GeV Higgs mass in these models typically forces the sparticle spectrum, with few exceptions, to lie in the few to multi-TeV mass range. The colored sparticles, in particular, all have masses in the multi-TeV range. With a single 5+5 pair of SU(5) messenger fields, the lightest MSSM neutralino mass lies close to 1 TeV. As we increase the number of SU(5) messenger multiplets the MSSM gauginos, and hence the neutralino, become heavier. The lightest stau mass is close to 1.4 TeV for the single 5 + 5 models, and it becomes lighter as we increase the number of SU(5) messenger multiplets. Particularly, with five pairs of 5 + 5, stau becomes the NLSP and can be as light as 800 GeV. The detection of a stau at the LHC may shed light on the number of SU(5) messenger multiplets at the messenger scale. A Higgs mass close to 125 GeV also yields lower limits on both the messenger and soft supersymmetry breaking scales which, in turn, constrain the gravitino mass.

A single 5 + 5 pair requires that the gravitino mass & 360 eV. With ﬁve pairs of 5 + 5 messenger ﬁelds, this lower limit on the gravitino mass is reduced to 60 eV. The simplest GMSB models, it appears, require non-standard cosmological scenarios

90 Point 1 Point 2 Point 3 Λ 1 × 106 4.22 × 105 1.5 × 106 14 12 6 Mmess 1.74 × 10 9.02 × 10 1.9 × 10 n5 1 5 1 tan β 42 60 46 µ 7873 5802 3678 mh 125 125.2 125.1 mH 9930 4865 5141 mA 9865 4833 5107 mH± 9930 4866 5142 m 0 1398, 2619 2924, 5307 2405, 3732 χ˜1,2 m 0 7775, 7775 5833, 5836 3735, 4449 χ˜3,4 m ± 2624, 7711 5315, 5840 3811, 4364 χ˜1,2 mg˜ 6689 12312 10613

mu˜L,R 15956, 14113 12014, 11276 14064, 13301

mt˜L,R 13637, 9847 10289, 8994 13027, 11873 m ˜ 15956, 13540 12015, 11151 14064, 13213 dL,R m˜ 12233 9720 12421 bR

mν˜1 9281 4885 5112

mν˜3 8722 4424 5009

me˜L,R 9290, 6774 4900, 2962 5133, 2640

mτ˜L,R 8706, 5109 4407, 783 4991, 2306 −7 mG˜ 42 0.916 6.86 × 10

Table 5.1: Benchmark points for the mGMSB. All masses are in units of GeV. Point 1, 2 and 3 show the lightest neutralino, stau and gravitino (shown in bold) that can be realised in mGMSB for a Higgs mass of 125 GeV. For the three points, mt = 173.3 GeV and cgrav = 1. in order to be in agreement with observations [161].

91 Chapter 6

HIGGS MASS PREDICTION IN SO(10) WITH t-b-τ YUKAWA UNIFICATION

6.1 Introduction In this chapter we discuss the consistency of t-b-τ YU with a 125 GeV Higgs [4]. It was shown in [118] that with non-universal soft supersymmety breaking (SSB) gaugino masses, which can be derived in the framework of SO(10) GUT, and with universal SSB scalar Higgs doublet masses at M (M 2 = M 2 ), the mass of CP- GUT Hu Hd even SM-like Higgs boson can be predicted with t-b-τ YU case. Furthermore, we employ two diﬀerent computing packages, namely Isajet 7.84 and SuSpect 2.41, and show that they agree very well with the 125 GeV Higgs mass prediction for t-b-τ YU. Good agreement is found between these two programs over the entire parameter space, thereby rendering our conclusions more robust. Another motivation to revisit the analysis presented in ref. [118] is the recent

+ − discovery of Bs → µ µ decay by the LHCb collaboration [234]. The branching + − +1.5 −9 fraction BF (Bs → µ µ ) = 3.2−1.2 ×10 is in accord with the SM prediction of (3.2± 0.2) × 10−9 [235]. In SUSY models, this ﬂavor-changing decay receives contributions from the exchange of the pseudoscalar Higgs boson A [236], which is proportional to

6 4 (tan β) /mA. Since tan β ≈ 47 was predicted in ref. [118], it is interesting to see how + − the parameter space is impacted with the Bs → µ µ discovery. We also intend to highlight another interesting feature of this YU model, namely that in addition to the prediction of a CP even Higgs boson mass of around 125 GeV , the model also prefers the CP odd Higgs boson mass of around 600 GeV. This prediction can hopefully be tested at the LHC in the near future [237]. The colored sparticle masses, consistent with good (10% or better) t-b-τ YU, lie well above the current mass

92 limits from the LHC, i.e., mg˜ & 1.4 TeV (for mg˜ ∼ mq˜) and mg˜ & 0.9 TeV (for mg˜ mq˜)[40, 41]. In this chapter we discuss how the grand uniﬁed SO(10) model can predict the Higgs mass. The discovery of a 125 GeV Higgs is one of the most important discovery in modern day particle physics.

6.2 SO(10) GUT with t − b − τ Yukawa unification One of the main motivations of SO(10) GUT, in addition to gauge coupling unification, is matter unification. The spinor representation of SO(10) unifies all matter fermions of a given family in a single multiplet (16i), which also contains the right handed neutrino (νR) that helps to generate light neutrino masses via the see-saw mechanism [166, 167, 168, 169]. The right handed neutrino can also naturally account for the baryon asymmetry of the Universe via leptogenesis [170, 171]. Another virtue of SO(10) is that, in principle, the two MSSM Higgs doublets can be accommodated in a single ten dimensional representation, which then yields the following Yukawa couplings

Yij 16i 16j 10H. (6.1)

Here i, j = 1, 2, 3 stand for family indices and the SO(10) indices have been omitted for simplicity. For the third generation quarks and leptons, the interaction in Eq.(6.1) yields the following Yukawa coupling uniﬁcation condition at the GUT scale [141]

Yt = Yb = Yτ = Yντ . (6.2)

It is interesting to note that in the gravity mediation SUSY breaking scenario [172, 173, 174, 175], t-b-τ YU condition leads to LHC testable sparticle spectrum [176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201] and even predicts a 125 GeV light CP-even Higgs boson mass [118]. On the other hand, it is well known that the interaction in Eq.(6.1) leads to a naive SO(10) relation: N = U ∝ D = L, where U, D, N and L denote the Dirac

93 mass matrices for up and down quarks, neutrinos and charge leptons respectively. U ∝ D would imply vanishing quark ﬂavor (Cabibbo-Kobayashi-Maskawa (CKM))

0 0 0 0 mixing [202], and mc /mt = ms/mb which significantly contradicts with experimental observations. The superscript zero refers to the parameters evaluated at the MGUT . T 0 0 0 0 D = L , which is a naive SU(5) relation, would imply ms = mµ and md = me and this relation also strongly disagrees with the measurements. So, it is obvious the interaction in Eq.(6.1) should be modified in order to accommodate the observed pattern of quarks and mixing. There are two main approaches to avoid these problems and obtain realistic fermion masses and mixings. One way is to extend the Higgs sector and assume that the SM Higgs doublet fields are superposition of fields from the different SO(10) representations [203]. Another way is to introduce additional vector-like matter multiplets at the GUT scale [204, 205, 206, 207, 208, 209, 210, 211] which mix in a nontrivial way with fermions in the 16 dimensional representation. It is equivalent to introducing non-renormalizable couplings which involve a non-singlet SO(10) field that develops a VEV [212]. Both these cases, however, allow t-b-τ YU to be completely destroyed or partially destroyed to yield b-τ YU. Therefore, in order to maintain t-b-τ YU the following two conditions should be satisfied:

• Third generation charged fermions obtain their masses only from Eq.(6.1).

• The MSSM Higgs doublets Hu and Hd reside solely in 10H .

There exist models where t-b-τ YU is maintained to a very good approximation with realistic fermion masses and mixings [213].

6.3 Fundamental Parameter Space It has been pointed out [215, 216, 217, 218, 219] that non-universal MSSM gaugino masses at MGUT can arise from non-singlet F-terms, compatible with the

94 underlying GUT symmetry such as SU(5) and SO(10). The SSB gaugino masses in supergravity [172] can arise, say, from the following dimension five operator: F ab − λaλb + c.c. (6.3) 2MP Here λa is the two-component gaugino field, F ab denotes the F-component of the field which breaks SUSY, the indices a, b run over the adjoint representation of the

ab gauge group. The resulting gaugino mass matrix is hF i/MP, where the supersymmetry breaking parameter hF abi transforms as a singlet under the MSSM gauge group

ab SU(3)c × SU(2)L × U(1)Y . The F ﬁelds belong to an irreducible representation in the symmetric part of the direct product of the adjoint representation of the uniﬁed group. This is a supersymmetric generalization of operators considered a long time ago [220]. In SO(10), for example,

(45 × 45)S = 1 + 54 + 210 + 770. (6.4)

If F transforms as a 54 or 210 dimensional representation of SO(10) [215], one obtains the following relation among the MSSM gaugino masses at MGUT :

M3 : M2 : M1 = 2 : −3 : −1, (6.5) where M1,M2,M3 denote the gaugino masses of U(1), SU(2)L and SU(3)c respectively. The low energy implications of this relation have been investigated in [221] without imposing YU. In this chapter we consider the case with µ > 0 and non-universal gaugino masses deﬁned in Eq.(6.5). In order to obtain the correct sign for the desired contribution to (g − 2)µ, we set M1 > 0, M2 > 0 and M3 < 0. Somewhat to our surprise, we ﬁnd that this class of t-b-τ YU models make a rather sharp prediction for the lightest SM-like Higgs boson mass [118]. In addition, lower mass bounds on the masses of the squarks and gluino are obtained. Notice that in general, if F ab transforms non trivially under SO(10), the SSB terms such as the trilinear couplings and scalar mass terms are not necessarily universal at MGUT . However, we can assume, consistent with SO(10) gauge symmetry,

95 that the coefficients associated with terms that violate the SO(10)-invariant form are suitably small, except for the gaugino term in Eq.(6.5). We also assume that D-term contributions to the SSB terms are much smaller compared with contributions from fields with non-zero auxiliary F-terms. Employing the boundary condition from Eq.(6.5), one can define the MSSM gaugino masses at MGUT in terms of the mass parameter M1/2 :

M1 = M1/2

M2 = 3M1/2

M3 = −2M1/2. (6.6)

Note that M2 and M3 have opposite signs which, as we will show, is important in implementing Yukawa coupling uniﬁcation to a high accuracy. In order to quantify

Yukawa coupling uniﬁcation, we deﬁne the quantity Rtbτ as,

max(yt, yb, yτ ) Rtbτ = . (6.7) min(yt, yb, yτ )

We have performed random scans for the following parameter range:

0 ≤ m16 ≤ 10 TeV

0 ≤ m10 ≤ 10 TeV

0 ≤ M1/2 ≤ 5 TeV 35 ≤ tan β ≤ 55

−3 ≤ A0/m16 ≤ 3 (6.8)

Here m16 is the universal SSB mass for MSSM sfermions, m10 is the universal SSB mass term for up and down MSSM Higgs masses, M1/2 is the gaugino mass parameter, tan β is the ratio of the vacuum expectation values (VEVs) of the two MSSM Higgs doublets, A0 is the universal SSB trilinear scalar interaction (with corresponding Yukawa coupling factored out). We use the central value mt = 173.1 GeV and 1σ deviation

(mt = 174.2 GeV) for top quark in our analysis [108]. We choose a +1σ deviation in mt

96 since it leads to an increase in the Higgs mass and improves the prediction of the Higgs mass in our analysis. Our results however are not too sensitive to one or two sigma variation in the value of mt [159]. We use mb(mZ ) = 2.83 GeV which is hard-coded into Isajet.

6.4 RGE running in Isajet and Suspect We employ Isajet 7.84 [35] and SuSpect 2.41 [36] interfaced with Micromegas 2.4 [222] to perform random scans over the fundamental parameter space. Isajet and

SuSpect employ full two loop RGEs for the SSB parameters between MZ and MGUT . The approach employed by both is similar, but there are some important diﬀerences that have been previously studied [223, 224]. SuSpect assumes that the full set of

MSSM RGEs are valid between MZ and MGUT and uses what is referred to as the ‘common scale approach’ in [224]. In this approach the DR parameters are extracted √ at a common scale MEWSB = mt˜1 mt˜2 . Isajet, on the other hand, uses the ‘step-beta function approach’ which means that it employs one-loop step beta functions for gauge and Yukawa couplings. These two approaches yield very similar results for most of the SUSY parameter space. There are some regions where the discrepancies get magniﬁed [223] and which we observe in our results as well. These diﬀerences will be discussed in section 6.6. An approximate error of around 2 GeV in the estimate of the Higgs mass in Isajet and SuSpect largely arise from theoretical uncertainties in the calculation of the minimum of the scalar potential, and to a lesser extent from experimental uncertainties in the values for mt and αs. An important constraint comes from limits on the cosmological abundance of stable charged particles [45]. This excludes regions in the parameter space where charged SUSY particles become the lightest supersymmetric particle (LSP). We accept only those solutions for which one of the neutralinos is the LSP and saturates the WMAP bound on the relic dark matter abundance. Micromegas is interfaced with

+ − Isajet and SuSpect to calculate the relic density and branching ratios BR(Bs → µ µ )

97 (a) Isajet 7.84 (b) SuSpect 2.41

Figure 6.1: Plot in the M3 − m16 planes. The panel on the left shows data points collected with Isajet 7.84 whereas the panel on the right shows data from SuSpect 2.41. The light blue points are consistent with REWSB and neutralino LSP. For both the panels, black points are subset of the light blue points and satisfy Rtbτ ≤ 1.03 in panel 6.1(a) and Rtbτ ≤ 1.05 in panel 6.1(b). The unit line is to guide the eye. and BR(b → sγ). With these codes we implement the random scanning procedure described in section 2.5.

6.5 Higgs Mass Prediction In this section we discuss the implicit relationship between the SUSY threshold corrections to the Yukawa couplings, REWSB and the light CP-even Higgs mass. We begin with the SUSY threshold corrections to the top, bottom and tau Yukawa couplings which play a crucial role in t-b-τ Yukawa coupling unification. In general, the bottom Yukawa coupling yb can receive large threshold corrections, while the threshold corrections to yt are typically smaller [39]. The scale at which Yukawa coupling unification occurs is identified with MGUT, which is the scale of gauge coupling unification.

Consider first the unification between yt(MGUT) and yτ (MGUT). The SUSY correction to the tau lepton mass is given by δmτ = v cos βδyτ . For the large tan β values of interest here, there is sufficient freedom in the choice of δyτ to achieve yt ≈ yτ at

MGUT. This freedom stems from the fact that cos β ' 1/ tan β for large tan β, and so

98 √ √ Figure 6.2: Plot in the mt˜L mt˜R − At/ mt˜L mt˜R planes. The data points shown are collected using Isajet. Color coding is the same as in Figure 6.1.

we may choose an appropriate δyτ and tan β to give us both the correct τ lepton mass and yt ≈ yτ at MGUT. The SUSY contribution to δyb has to be carefully monitored in order to achieve Yukawa coupling uniﬁcation yt(MGUT) ≈ yb(MGUT) ≈ yτ (MGUT).

We choose the sign of δyi (i = t, b, τ) from the perspective of evolving yi from

MGUT to MZ. With this choice, δyb must receive a negative contribution (−0.27 .

δyb/yb . −0.15) in order to realize Yukawa coupling uniﬁcation. The leading contribution to δyb arises from the ﬁnite gluino and chargino loop corrections, and in our sign convention, it is approximately given by [39]

g2 µm tan β y2 µA tan β δyfinite ≈ 3 g˜ + t t , (6.9) b 12π2 m2 32π2 m2 ˜b t˜ where g3 is the strong gauge coupling, mg˜ is the gluino mass, m˜b (mt˜) is the heaviest sbottom (stop) mass, and At is the top trilinear SSB coupling. The logarithmic corrections to yb are positive, which leaves the finite corrections to provide for the correct overall negative δyb in order to realize YU. For models with gaugino mass unification or same sign gauginos, the gluino contribution (first term in Eq.(6.9)) is positive for µ > 0. Thus, the chargino contribution (second term in Eq.(6.9)) must play an essential role in providing the required negative

99 (a) Isajet 7.84 (b) SuSpect 2.41

Figure 6.3: Plot in the Rtbτ − mh plane. Panel 6.3(a) shows results obtained from Isajet, while 6.3(b) shows results from obtained SuSpect. Gray points in 6.3(a) and 6.3(b) are consistent with REWSB and LSP neutralino. Green points form a subset of the gray and satisfy sparticle mass [45] and B- physics constraints described in Section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Brown points form a subset of the green points and satisfy Ωh2 ≤ 1. Panel 6.3(c) shows data collected with Isajet (blue) and SuSpect (red). The red and blue points satisfy the constraints imposed on the green points in panels 6.3(a) and 6.3(b).

100 contribution to δyb. This can be achieved [62, 227] only for

m16 M1/2, m16 & 6 TeV and A0/m16 ∼ −2.6. (6.10)

One could lower the sparticle mass spectrum by considering opposite sign gaugino SSB terms [201] which is allowed by the 4-2-2 model [226]. In SO(10) GUT, non-universality of SSB gaugino masses with opposite signs can be generated through various SO(10) non singlet representations responsible for SUSY breaking [215].

In particular, for M3 < 0, M2 > 0 and µ > 0, the gluino contribution to δyb has the correct sign to obtain the required b-quark mass, and furthermore, it is not necessary to have very strong relations among SSB fundamental parameters. Yukawa coupling uniﬁcation in this case is achieved for

m16 & 300 GeV and M3(M2) > m16, as well as for M3(M2) 6 m16, (6.11) as opposed to the parameter space given in Eq. (6.10). This enables us to simultaneously satisfy the requirements of t-b-τ YU, neutralino dark matter abundance and constraints from (g − 2)µ, as well as a variety of other bounds. But for the above mentioned cases the relation m2 > m2 was imposed at M . It is well known Hd Hu GUT that REWSB cannot be realized if we require m2 = m2 at M and, in addition, Hd Hu GUT demand exact t-b-τ YU. Let’s brieﬂy review REWSB condition in light of YU. The REWSB minimization condition at tree level requires that m2 − m2 tan2 β M 2 M 2 µ2 = Hd Hu − Z ≈ −m2 − Z . (6.12) tan2 β − 1 2 Hu 2 Here the approximate equality works well for large values for tan β, which is the case

2 for t-b-τ YU. Plugging the tree level expression of the CP-odd Higgs mass mA = m2 + m2 + 2µ2 in Eq. (6.12), we obtain the relation Hd Hu

m2 − m2 M 2 + m2 . (6.13) Hd Hu & Z A

On the other hand, a semi-analytical expression for m2 − m2 at M , in terms of Hd Hu Z GUT scale fundamental parameters, has the following form [228]

m2 − m2 ≈ −0.13m2 + 0.26m2 − 0.04m A − 0.01A2, (6.14) Hd Hu 16 1/2 1/2 0

101 which implies that in order to satisfy the condition in Eq. (6.13), we should require

M1/2 > m16. (6.15)

This requirement clearly contradicts the t-b-τ YU condition obtained in Eq. (6.10) if gaugino mass unification is assumed at MGUT. This is the reason why REWSB cannot occur if precise Yukawa coupling unification and m2 = m2 conditions are imposed Hd Hu at MGUT. On the other hand, the condition from Eq. (6.15) can be consistent with the condition presented in Eq. (6.11), with non-universal gaugino masses at MGUT. The overlap of conditions from Eqs. (6.15) and (6.11) gives very characteristic relations among m16 and gaugino masses. We present in Figure 6.1 the results of the scan over the parameter space listed in Eq.(6.8) in M3 − m16 plane. The light blue points are consistent with REWSB and 0 χ˜1 LSP. Points in black correspond to 3% or better t-b-τ YU (Rtbτ ≤ 1.03) for Isajet data. For SuSpect data, the black points correspond to 5% or better t-b-τ YU. In Section 6.6 we will discuss the factors that can result in a few percent difference in the YU obtained in Isajet and SuSpect. We see that 3% in Isajet’s case (5% in SuSpect’s case ) or better t-b-τ YU can occur for M3 slightly heavier than m16, and M3 > 2

TeV. On the other hand, M3 signiﬁcantly aﬀects the low scale stop quark mass and an approximate semi-analytic expression for m2 for t-b-τ YU case is as follows: t˜R m2 ≈ 0.27m + 5.3M 2 + 0.4M 2 + ... (6.16) t˜R 16 3 2

Using Eq. (6.16) we can obtain a rough lower bound for stop quarks, namely m 4 t˜R & TeV. Next we discuss how the ﬁndings presented in Figure 6.1 can aﬀect the CP-even Higgs boson mass calculation. For the actual calculation, both Isajet and SuSpect employ a more elaborate calculation procedure. From 4.3 we can see that the one loop radiative corrections to the CP-even Higgs mass depend logarithmically on the stop quark mass and linearly on Xt. These two quantities essentially determine the radiative corrections to the CP-even Higgs boson mass. Because of this, it is interesting to present the result from Figure 6.1 in terms of Xt and MS.

102 In Figure 6.2 we show the results in the MS − At/MS plane. The color coding is the same as in Figure 6.1. From the MS − At/MS plane, we see that the black points lie in the interval 5 TeV < MS < 9 TeV. This interval reduces further if we require better YU. This means that for good YU, following Eq.(5.8) the lightest CP even Higgs boson should be relatively heavy owing to the logarithmic dependence on MS. Since the growth or decay of a logarithmic function is slow, the logarithmic dependence of mh on MS nicely explains the shape of colored points in Figure 6.3(a) and 6.3(b). Panel 6.3(a) shows results obtained from Isajet, while 6.3(b) shows results obtained using Suspect. The gray points in Figs. 6.3(a) and 6.3(b) are consistent with REWSB and LSP neutralino. The green points form a subset of the gray points and satisfy the sparticle mass and B-physics constraints described in Section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. The brown points form a subset of the green points and satisfy Ωh2 ≤ 1. An intriguing feature of Figures 6.3(a) and 6.3(b) is that the minima of the distribution occurs at the Higgs mass value very close to the observed mass of the SM-like Higgs at the LHC. In other words, nearly perfect t-b-τ YU prefers the current favored value of mh. We can also understand from Figure 6.2 why the minima in the Figure 6.3(a) and 6.3(b) have relatively small widths. In Figure 6.2, from the

MS − At/MS plane, we see that the ratio At/MS lies in the very small interval 0.3<

At/MS <0.7. On the other hand, it is known [229] that the CP even Higgs boson obtains significant contributions from At if At/MS > 1. We can therefore conclude that there is no significant contribution from the finite corrections to the CP even Higgs boson mass if we have almost perfect YU, and the Higgs mass is mostly generated from the logarithmic corrections. This is why the minima in Figures 6.3(a) and 6.3(b) are not wide. In Figures 6.3(c) we show the overlap of the data from Isajet 7.84 and SuSpect 2.41. All the points shown are consistent with REWSB, LSP neutralino, and satisfy the sparticle mass and B-physics constraints described in Section 6.4. The blue (red) points show results obtained from Isajet (SuSpect). We can see that the minima of the

103 data distribution from Isajet predicts a Higgs mass mh ∼ 124 GeV, whereas SuSpect predicts mh ∼ 126 GeV. For lower values of Rtbτ , the results from the two packages overlap around mh ≈ 125 GeV. This observation makes the predicted value of Higgs mass close to 125 GeV obtained from t-b-τ YU condition quite reliable.

6.6 Higgs and Sparticle Spectroscopy From Isajet and SuSpect In this section we compare the sparticle spectroscopy obtained from two computing packages Isajet and SuSpect. The comparison of the uncertainties in the sparticle spectroscopy of different packages was done in [223] and in particular the threshold effects were compared in [224]. The approach used by the two programs is very similar in that both use two loop RGEs for running gauge and Yukawa coupling. But there are some factors that can lead to numerical differences of a few percent. These factors can include the following [223, 224]:

• The scale at which the sparticles are integrated out of the theory. In Isajet SSB parameters are extracted from RGE running at their respective mass scales m = m (m ) whereas in Suspect these parameters are extracted at the M (≡ √i i i EWSB mt˜1 mt˜2 ).

• Suspect uses αs in the DR scheme whereas Isajet uses the MS value. • Another source of discrepancy can be the use of bottom pole mass by Suspect mb = mb(MZ ) whereas Isajet uses the mass at the SUSY scale mb = mb(MSUSY ). • The default guess of the sparticle masses at the beginning of the RG evolution process is diﬀerent in Isajet and Suspect.

• Suspect assumes that the full set of MSSM RGEs are valid between MZ and MGUT . This is also true for Isajet except that it employs one-loop step-beta functions for gauge and Yukawa couplings.

• Loop corrections in the minimization of the scalar potential, which are eﬀected by the whole SUSY spectrum. To get the SUSY spectrum a loop corrected value of µ has to be guessed. The manner in which this value is guessed by the two codes can lead to a diﬀerence.

In Figure 6.4 we show our results in the M1/2−m16, A0/m16−m16, m10−m16 and

µ−m16 planes. All of the points shown are consistent with REWSB, LSP neutralino and

104 Figure 6.4: Plots in the M1/2 − m16, A0/m16 − m16, m10 − m16 and µ − m16 planes. All the points shown are consistent with REWSB, LSP neutralino and satisfy sparticle mass [45] and B- physics constraints described in Section 6.4. We also require that the points do no worse than the SM in terms of (g − 2)µ. In addition the satisfy the Higgs mass range 122 GeV < mh < 128 GeV and Rtbτ < 1.2. The purple points show results obtained from SuSpect and yellow points is the data collected using Isajet.

105 satisfy sparticle mass and B-physics constraints described in Section 6.4. In addition, the points shown satisfy the condition 122 GeV < mh < 128 GeV and Rtbτ < 1.2. The purple points show results obtained from SuSpect, whereas the yellow points correspond to the data collected using Isajet. We observe that there are some small but notable diﬀerences in the results obtained from the two packages. It is interesting to note that for the regions where the points are more dense there is good overlap between the two packages. In Figure 6.4 under the very dense yellow points there also lie dense purple points. From M1/2 −m16,

A0/m16 − m16, m10 − m16 panels, we see that the best agreement between solutions obtained from Isajet and SuSpect occurs for m16 < 6 TeV, M1/2 < 3 TeV and m10 < 4 TeV.

In the µ − m16 plane of Figure 6.4 we see that the solutions from SuSpect with 20% or better t-b-τ YU have lower values of µ compared to the µ values from Isajet.

To exemplify this further, in Figure 6.5 we show a plot in Rtbτ − µ plane. We see that requiring 20% or better t-b-τ YU allows solutions from SuSpect with µ ≈ 200 GeV, while requiring the same for Isajet yields µ & 1 TeV. On the other hand, requiring YU better then 5% leads to similar limits on the values of µ from Isajet and SuSpect, i.e.,

µ & 2 TeV. It was noted in [225] that the two codes can differ notably in regions with low µ and tan β values. This difference can stem from the factors previously mentioned, and which may have important implications for natural SUSY. We observe from Figure 6.5 that SuSpect does not yield YU better than 5%, with a minimum value of Rtbτ ∼ 1.05. Isajet, however, predicts even better YU with the minimum value of Rtbτ ∼ 1.02. This few percent difference could be due to the way threshold effects are evaluated by the two codes [224]. The Rtbτ −m16 plane shows that in order to have YU better than 5% the results from Isajet require 600 GeV < m16 < 2.5 TeV.

In Figure 6.6 we show results in the Rtbτ −mA, Rtbτ −mτ˜1 , Rtbτ −mt˜1 , Rtbτ −mg˜,

R − m 0 and R − m ± planes. The color coding is the same as in Figure 6.3. tbτ χ˜1 tbτ χ˜ The data collected with Isajet is used to make the plots in this ﬁgure. All panels in

106 Figure 6.5: Plots in the Rtbτ − µ, Rtbτ − m16 planes. Color coding is the same as in Figure 6.3(c).

Figure 6.6 indicate that the model predicts relatively narrow ranges for the sparticle masses corresponding to the best t-b-τ YU. The sparticles are heavy enough to evade observation at current LHC energies, but a signal would hopefully be observed during the 14 TeV LHC run.

From the Rtbτ − mA plane we see that just from the REWSB condition, t-b-τ YU better than 5% predicts that we cannot have a very light CP-odd Higgs boson (A). Similar bounds apply for the heavier CP even H boson and charged H± bosons, since,

2 2 2 2 2 2 in the so-called decoupling limit, mA MZ , we have mH ' mA and mH± ' mA +MW

[21], where MW stands for the W -boson mass. The leading decay modes for the heavy Higgs bosons are H,A → b ¯b and H,A → τ +τ −. The heavy Higgs production cross section and branching ratios depends on tan β and its mass. The MSSM sparticle dependence appears through gluino-squark, Higgsino-squark, bino-sfermion and wino- sfermion loops. The bound on mA in our scenario is more relaxed since we have non universal gauginos with opposite signs for M2 and M3 at the GUT scale. We also ﬁnd that good t-b-τ YU requires M2/M3 > 1.5. This will alter the gluino-squark and wino- sfermion loop contributions to the heavy Higgs production cross section and branching ratios [230].

107 Figure 6.6: Plot in the R − m , R − m , R − m˜ , R − m , R − m 0 tbτ A tbτ τ˜1 tbτ t1 tbτ tildeg tbτ χ˜1 and Rtbτ − mχ˜± planes. The color coding is the same as in Figure 6.3.

108 Figure 6.7: Plot in the Rtbτ −mA planes. Color coding is the same as in Figure 6.3(c).

Applying all the collider and B-physics constraints we obtain a lower bound for mA which is very close to the value corresponding to best YU. Restricting to 5% or better uniﬁcation and including the constraints presented in Section 6.4, we obtain the bound 400 GeV. MA . 1 TeV. The lower bound is very close to the current experimental limit [237] obtained from the GUT scale gaugino uniﬁcation condition and can be further tested in near future.

In the Rtbτ − mτ˜1 plane we can observe that the preferred values for the stau lepton mass from the point of view of good YU (Rtbτ < 1.05) is in the interval 500 GeV

. mτ˜1 . 1.5 TeV. The search for a stau in this mass range is challenging at the LHC. In this model, the stau is the NLSP, and this can yield the correct relic abundance through neutralino-stau coannihilation. The lightest colored sparticle in this scenario is one of the stops. The preferred

mass as seen from the Rtbτ − mt˜1 plane, is around 3-4 TeV. For the gluino, the mass according to the Rtbτ − mg˜ plane is around 5-6 TeV. In principle they can be found at the LHC. We can also see from Figure 6.6 that the lightest neutralino, for Rtbτ < 1.05, is around 500 GeV, and the preferred value from the point of view YU is ∼ 700 GeV. The model also predicts the charginos to be heavier than 2 TeV. Since the pseudoscalar A boson is being searched for at the LHC, we present in Figure 6.7 the combined results from Isajet and SuSpect. The color coding is the same as in Figure 6.6, and we see that the agreement between the two programs is quite

109 satisfying. In Figure 6.8 we show the implication of our analysis for direct detection of dark matter. Plots are shown in the σ − m 0 and σ − m 0 planes. The gray points in SI χ˜1 SD χ˜1 the ﬁgure are consistent with REWSB and LSP neutralino. The green points form a subset of the gray points and satisfy the constraints described in Section 6.4. The brown points form a subset of the green points and satisfy Ωh2 ≤ 1, and the orange points form a subset of the brown points with 10% or better YU (Rtbτ ≤ 1.1). The left panel shows the current and future bounds from CDMS as black (solid and dashed) lines, and as red (solid and dotted) lines for the Xenon experiment. The right panel also shows the current bounds from Super K (solid red line) and IceCube (solid black line), and future reach of IceCube DeepCore (dotted black line). We can see that the parameter space of this model representing neutralino- stau coannihilation can be tested with these experiments. However models with YU better than 10% yield tiny cross sections which are well below the sensitivity of these experiments. We also observe that good YU predicts a heavy neutralino (m 0 χ˜1 & 400 GeV). It is, however, interesting that these orange points also predict a 125 GeV Higgs mass as seen in Figures 6.3(a) and 6.3(b). Therefore, t-b-τ YU not only predicts a 125 GeV Higgs but also a relatively heavy dark matter LSP which coannhilates with the stau to yield the correct relic abundance, and also yields tiny cross sections well below the sensitivity of current experiments. In Table 6.1 we present four benchmark points with good YU and Higgs mass ∼ 125 GeV. The points shown also satisfy the constraints described in Section 6.4. Points 1 and 2 represent solutions that yield the best YU in Isajet and SuSpect. As described earlier, Isajet yields YU as good as ∼ 2%, whereas in SuSpect it is ∼ 5%. Point 3 depicts stau coannihilation in addition to a 124 GeV Higgs and Rtbτ = 1.03. Point 4 shows that good YU can be attained with the sfermions and Higgs nearly degenerate at MGUT , i.e., m16 ' m10.

110 Figure 6.8: Plots in the σ − m 0 and σ − m 0 planes. The cross sections are SI χ˜1 SD χ˜1 calculated using Isajet. Points shown in gray are consistent with REWSB and LSP neutralino. Green points form a subset of the gray and satisfy sparticle mass [45] and B-physics constraints described in Section 6.4. In addition, we require that green points do no worse than the SM in terms of (g − 2)µ. Brown points form a subset of the green points and satisfy Ωh2 ≤ 1. The orange points are a subset of the brown points and satisfy R ≤ 1.1 In the σ - m 0 plane, the current and future bounds from tbτ SI χ˜1 the CDMS experiment are represented as black (solid and dashed) lines and as red (solid and dotted) lines for the Xenon experiment. The right panel shows the σ - m 0 plane with the current bounds from Super K SD χ˜1 (solid red line) and IceCube (solid black line) and future reach of IceCube DeepCore (dotted black line).

Figure 6.9: Plots in the Rtbτ -∆EW and Rtbτ -∆HS planes. Color coding is the same as in Figure 6.3.

111 6.7 Fine tuning constraints for little hierarchy The latest (7.84) version of ISAJET [35] calculates the fine-tuning conditions related to the little hierarchy problem at Electro Weak (EW ) scale and at the GUT scale (HS). We will briefly describe these parameters in this section. After including the one-loop effective potential contributions to the tree level MSSM Higgs potential, the Z boson mass is given by the following relation:

M 2 (m2 + Σd) − (m2 + Σu) tan2 β Z = Hd d Hu u − µ2 . (6.17) 2 tan2 β − 1

The Σ’s stand for the contributions coming from the one-loop effective potential (For more details see ref. [231]). All parameters in Eq. (6.17) are defined at EW scale. In order to measure the EW scale fine-tuning condition associated with the little hierarchy problem, the following definitions are used [231]:

C ≡ |m2 /(tan2 β − 1)|,C ≡ | − m2 tan2 β/(tan2 β − 1)|,C ≡ | − µ2|, (6.18) Hd Hd Hu Hu µ

2 with each C u,d less than some characteristic value of order MZ . Here, i labels the Σu,d(i) SM and supersymmetric particles that contribute to the one-loop Higgs potential. For the ﬁne-tuning condition we have

2 ∆EW ≡ max(Ci)/(MZ /2). (6.19)

Note that Eq. (6.19) defines the fine-tuning condition at EW scale without addressing the question of the origin of the parameters that are involved. In most SUSY breaking scenarios the parameters in Eq. (6.17) are defined at a scale higher than MEW . In order to fully address the fine-tuning condition we need to check the relations among the parameters involved in Eq. (6.17) at high scale. We relate the parameters at low and high scales as follows:

m2 = m2 (M ) + δm2 , µ2 = µ2(M ) + δµ2. (6.20) Hu,d Hu,d HS Hu,d HS

112 Here m2 (M ) and µ2(M ) are the corresponding parameters renormalized at Hu,d HS HS the high scale, and δm2 , δµ2 measure how the given parameter is changed due to Hu,d renormalization group evolution (RGE). Eq. (6.17) can be re-expressed in the form

2 2 2 d 2 2 u 2 m (m (MHS) + δm + Σ ) − (m (MHS) + δm + Σ ) tan β Z = Hd Hd d Hu Hu u 2 tan2 β − 1 2 2 − (µ (MHS) + δµ ) . (6.21)

Following ref. [231], we introduce the parameters:

B ≡ |m2 (M )/(tan2 β − 1)|,B ≡ |δm2 /(tan2 β − 1)|, Hd Hd HS δHd Hd

2 2 2 2 BHu ≡ | − mHu (MHS) tan β/(tan β − 1)|,Bµ ≡ |µ (MHS)|,

2 2 2 2 BδHu ≡ | − δmHu tan β/(tan β − 1)|,Bδµ ≡ |δµ |, (6.22) and the high scale ﬁne-tuning measure ∆HS is deﬁned to be

2 ∆HS ≡ max(Bi)/(MZ /2). (6.23)

The current experimental bound on the chargino mass (m > 103 GeV) [45] Wf indicates that either ∆EW or ∆HS cannot be less than 1. The quantities ∆EW and

∆HS measure the sensitivity of the Z-boson mass to the parameters deﬁned in Eqs.

(6.18) and (6.22), such that (100/∆EW )% ((100/∆HS)%) is the degree of fine-tuning at the corresponding scale. Based on the definition of high and low scale fine tuning described above we show results in the Rtbτ − ∆EW and Rtbτ − ∆HS planes (using Isajet) in Figure 6.9. We see that the low scale little hierarchy problem becomes more severe for the t-b-τ YU case compared to what we have in the constrained MSSM (CMSSM), but high scale fine tuning is at the same level as in the CMSSM [232]. For t-b-τ YU of around 5%, the EW fine tuning parameter ∆EW ∼ 800 and the HS fine tuning parameter is also

∆HS ∼ 800. As mentioned above, the ﬁne tuning condition has to be scale invariant which means that cancellation between parameters at a particular scale cannot be more severe compared to same conditions at another scale. Based on this assumption the little hierarchy problem in this model remains the same as we have when gaugino universality is assumed in the theory [231, 232].

113 Isajet SuSpect Isajet Isajet 2 3 2 3 m10 4.19 × 10 3.82 × 10 4.49 × 10 1.94 × 10 3 3 3 3 m16 2.13 × 10 2.69 × 10 1.91 × 10 2.00 × 10 3 3 3 3 M1 1.89 × 10 2.00 × 10 1.78 × 10 1.51 × 10 3 3 3 3 M2 5.67 × 10 6.00 × 10 5.35 × 10 4.53 × 10 3 3 3 3 M3 −3.78 × 10 −4.00 × 10 −3.57 × 10 −3.02 × 10 A0/m16 2.39 1.37 0.03 1.56 tan β 47.18 48.05 47.93 47.46 mt 174.2 173.1 174.2 173.1 µ 3729 1935 2913 2526 mh 125 126 124 123 mH 747 491 572 558 mA 742 491 568 554 mH± 753 500 580 567 m 0 895, 3739 955, 1935 848, 2932 709, 2540 χ˜1,2 m 0 3742, 4822 1936, 5043 2935, 4562 2543, 3849 χ˜3,4 m ± 3789, 4774 1934, 5043 2978, 4516 2579, 3809 χ˜1,2 mg˜ 7694 7673 7266 6239

mu˜L,R 7667, 6824 8112, 7245 7219, 6415 6295, 5635

mt˜1,2 5331, 6560 5604, 6839 5239, 6367 4390, 5370 m ˜ 7668, 6814 8112, 7236 7220, 6406 6296, 5628 dL,R m 5553, 6526 5870, 6870 5434, 6333 4591, 5341 ˜b1,2

mν˜1,2 4148 4590 3870 3487

mν˜3 3898 4234 3641 3243

me˜L,R 4153, 2234 4590, 2780 3875, 2009 3491, 2068

mτ˜1,2 1094, 3875 1140, 4235 881, 3620 1061, 3225 −11 −11 −11 −11 ∆(g − 2)µ 3.11 × 10 3.36 × 10 3.71 × 10 4.97 × 10 −11 −9 −11 −10 σSI (pb) 1.59 × 10 1.29 × 10 7.08 × 10 1.00 × 10 −10 −9 −9 −9 σSD(pb) 4.69 × 10 1.35 × 10 1.60 × 10 2.89 × 10 2 ΩCDM h 6.5 2.8 0.8 4.0 Rtbτ 1.02 1.05 1.03 1.04

Table 6.1: Benchmark points with good Yukawa uniﬁcation. All the masses are in units of GeV. Point 1, 3 and 4 are generated using Isajet 7.84 whereas point 2 is from SuSpect 2.41. Point 1 and 2 demonstrates how a small value of Rtbτ yields a Higgs mass ∼ 125 GeV. Point 3 exhibits stau coannihilation 2 and has a small Rtbτ that agrees with Ωh < 1. Point 4 has m16 ' m10 with good YU.

114 6.8 Conclusion In this chapter we showed how t-b-τ YU is consistent with a 125 GeV Higgs. Our analysis is an extension of the analysis in ref. [118] in several ways. We have highlighted the effects of threshold corrections on the bottom quark Yukawa coupling in this model, and discussed the implicit relationship between these corrections and the Higgs mass. We showed that for YU better than ∼ 5%, M3 > m16 at MGUT, with M 2 TeV. This, in turn, leads to a heavy stop quark, m 4 TeV. The dominant 3 & t˜R & contribution to the Higgs mass arises from the logarithmic dependence of mh on the stop quark mass. This leads to the prediction mh ≈ 125 GeV, consistent with t-b-τ YU better than 5%. We also compared our results from two different packages, namely Isajet and SuSpect. We found good agreement between the two codes with only a few percent difference between the calculations. One important difference is that Isajet allows YU better than 2%, whereas SuSpect has, at best, 5% YU. Another notable difference is that SuSpect allows for much smaller values of the Higgs mixing parameter µ. This can have implications for natural SUSY since smaller µ values are preferred in resolving the little hierarchy problem. The two codes also agree well in their predictions of sparticle masses. We find that insisting on YU better than 5% implies that the sparticles are heavy enough to evade observation at the current LHC energies, but may be observed during the 14 TeV LHC run. Furthermore, we showed that t-b-τ YU predicts a light CP-odd Higgs boson (A). Restricting to 5% and better YU yields the following bound on the pseudoscalar mass, 400 GeV. MA . 1 TeV. Similarly, the bounds on other sparticle masses are mg˜ & 4 TeV, mτ˜ & 500 GeV and mχ˜± & 2 TeV for YU 5% or better. Finally, we also tested the implications of YU for direct detection of dark matter. We found that stau-neutralino coannihilation can lead to the correct dark matter relic abundance. Moreover, insisting on YU better than 10% implies a heavy dark matter candidate (m 0 400 GeV). The neutralino-nucleon cross sections are found to be χ˜1 & well below the current sensitivity of direct detection experiments.

115 BIBLIOGRAPHY

[1] M. A. Ajaib, T. Li and Q. Shaﬁ, Phys. Rev. D 85, 055021 (2012) [arXiv:1111.4467 [hep-ph]].

[2] M. A. Ajaib, I. Gogoladze and Q. Shaﬁ, Phys. Rev. D 86, 095028 (2012) [arXiv:1207.7068].

[3] M. A. Ajaib, I. Gogoladze, F. Nasir and Q. Shaﬁ, Phys. Lett. B 713, 462 (2012) [arXiv:1204.2856 [hep-ph]].

[4] M. Adeel Ajaib, I. Gogoladze, Q. Shaﬁ and C. S. Un, JHEP 1307, 139 (2013) [arXiv:1303.6964 [hep-ph]].

[5] S.L.Glashow, Nucl.Phys. 22, 579(1961).

[6] A.Salam, in Elementary Particle Physics, edited by N.Svartholm (Almqvist and Wiksell, Stockholm, 1968), p.368

[7] S.Weinberg, Phys.Rev.Lett. 19, 1264(1967).

[8] T. D. Lee and C. N. Yang, Phys. Rev. 104, 254-258 (1956).

[9] Nambu, Y. Phys. Rev. Lett. 4, 380382 (1960).

[10] G. ’t Hooft, Nucl. Phys. B 33, 173 (1971).

[11] A. Pich, hep-ph/0502010.

[12] S. F. Novaes, In *Sao Paulo 1999, Particles and ﬁelds* 5-102 [hep-ph/0001283].

[13] C. Quigg, Ann. Rev. Nucl. Part. Sci. 59, 505 (2009) [arXiv:0905.3187 [hep-ph]].

[14] F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964); P.W. Higgs, Phys. Lett. 12, 132 (1964); Phys. Rev. Lett. 13 508 (1964).

[15] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13 585 (1964).

[16] P. W. Higgs, Phys. Rev. 145 1156 (1966).

[17] T. W. B. Kibble, Phys. Rev. 155 1554 (1967).

[18] S. R. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967).

116 [19] A. Signer, J. Phys. G 36, 073002 (2009) [arXiv:0905.4630 [hep-ph]].

[20] D. I. Kazakov, hep-ph/0012288.

[21] S. P. Martin, In *Kane, G.L. (ed.): Perspectives on supersymmetry II* 1-153 [hep-ph/9709356].

[22] H. Murayama, hep-ph/0002232.

[23] I. Aitchison, Supersymmetry in Particle Physics: An Elementary Introduction (Cambridge. University Press, 2007)

[24] M. Dine, Supersymmetry and String Theory: Beyond the Standard Model, Cam- bridge. University Press, Cambridge (2006).

[25] H. Baer and X. Tata, Weak Scale Supersymmetry: From Superelds to Scattering Events, (Cambridge University Press, 2006).

[26] S. Raby, http://pdg.lbl.gov/2013/reviews/rpp2013-rev-guts.pdf

[27] T. Fukuyama, Int. J. Mod. Phys. A 28, 1330008 (2013) [arXiv:1212.3407 [hep-ph]].

[28] Collins, B. D. B. , Martin, A. D. & Squires, E. J. (1989) Particle Physics and Cosmology. (Wiley, New York)

[29] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974) 438.

[30] J. C. Pati and A. Salam, Phys. Rev. Lett. 31, 661 (1973); Phys. Rev. D 10 (1974) 275.

[31] R. Jeannerot, S. Khalil, G. Lazarides and Q. Shaﬁ, JHEP 0010, 012 (2000) [hep- ph/0002151].

[32] C. S. Aulakh, B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, Phys. Lett. B 588, 196 (2004) [hep-ph/0306242].

[33] R. N. Mohapatra, J. Phys. G 36, 104006 (2009) [arXiv:0902.0834 [hep-ph]].

[34] T. Han, hep-ph/0508097.

[35] F. E. Paige, S. D. Protopopescu, H. Baer and X. Tata, hep-ph/0312045.

[36] A. Djouadi, J. -L. Kneur and G. Moultaka, Comput. Phys. Commun. 176, 426 (2007).

[37] J.L. Leva, A fast normal random number generator, ACM Trans. Math. Softw. 18 (1992) 449-453; J.L. Leva, Algorithm 712. A normal random number generator, ACM Trans. Math. Softw. 18 (1992) 454-455.

117 [38] J. Hisano, H. Murayama , and T. Yanagida, Nucl. Phys. B402 (1993) 46. Y. Ya- mada, Z. Phys. C60 (1993) 83; J. L. Chkareuli and I. G. Gogoladze, Phys. Rev. D 58, 055011 (1998).

[39] D. M. Pierce, J. A. Bagger, K. T. Matchev, and R.-j. Zhang, Nucl. Phys. B491 (1997) 3.

[40] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D 87, 012008 (2013).

[41] S. Chatrchyan et al. [CMS Collaboration], JHEP 1210, 018 (2012).

[42] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012).

[43] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012).

[44] G. Aad et al. [ATLAS Collaboration], arXiv:1103.4344 [hep-ex].

[45] K. Nakamura et al. [ Particle Data Group Collaboration ], J. Phys. G G37, 075021 (2010).

[46] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 710, 67 (2012) [arXiv:1109.6572 [hep-ex]].

[47] E. Barberio et al. [Heavy Flavor Averaging Group], arXiv:0808.1297 [hep-ex].

[48] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 100, 101802 (2008).

[49] Jose Angel Hernando, LHCb-TALK-2012-028.

[50] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009).

[51] G. W. Bennett et al. [Muon G-2 Collaboration], Phys. Rev. D 73, 072003 (2006).

[52] I. Gogoladze, S. Raza and Q. Shaﬁ, Phys. Lett. B 706, 345 (2012) [arXiv:1104.3566 [hep-ph]].

[53] R. Demina, J. D. Lykken, K. T. Matchev and A. Nomerotski, Phys. Rev. D62 (2000) 035011.

[54] CDF Collaboration, CDF Note 9834, see http://www- cdf.fnal.gov/physics/exotic/r2a/20090709.stop-charm/ for details.

[55] K. I. Hikasa and M. Kobayashi, Phys. Rev. D36 (1987) 724.

[56] M. Muhlleitner and E. Popenda, JHEP 1104 (2011) 095.

[57] K. Griest and D. Seckel, Phys. Rev. D 43, 3191-3203 (1991).

[58] C. Boehm, A. Djouadi and M. Drees, Phys. Rev. D 62, 035012 (2000) [hep- ph/9911496].

118 [59] R. L. Arnowitt, B. Dutta, T. Kamon, N. Kolev and D. A. Toback, Phys. Lett. B 639, 46 (2006) [hep-ph/0603128].

[60] M. Adeel Ajaib, T. Li and Q. Shaﬁ, Phys. Lett. B 701, 255 (2011) [arXiv:1104.0251 [hep-ph]].

[61] [ATLAS Collaboration], ATLAS-CONF-2011-096.

[62] H. Baer, S. Kraml, S. Sekmen and H. Summy, JHEP 0803, 056 (2008) [arXiv:0801.1831 [hep-ph]].

[63] H. Baer, M. Haider, S. Kraml, S. Sekmen and H. Summy, JCAP 0902, 002 (2009) [arXiv:0812.2693 [hep-ph]].

[64] T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357.

[65] F. Maltoni and T. Stelzer, JHEP 0302 (2003) 027.

[66] J. Alwall et al., JHEP 0709 (2007) 028.

[67] J. Alwall, S. de Visscher and F. Maltoni, JHEP 0902 (2009) 017; J. Alwall et al., Eur. Phys. J. C 53 (2008) 473.

[68] T. Sjostrand, S. Mrenna and P. Skands, JHEP 0605 (2006) 026.

[69] J. Conway et al., http://www.physics.ucdavis.edu/∼conway/research/software/pgs/pgs4- general.htm.

[70] Prospino 2.1, available at http://www.ph.ed.ac.uk/∼tplehn/prospino/.

[71] W. Beenakker, R. Hopker, M. Spira and P. M. Zerwas, Nucl. Phys. B492 (1997) 51.

[72] W. Beenakker et al, Nucl. Phys. B515 (1998) 3.

[73] W. Beenakker et al, Phys. Rev. Lett. 83 (1990) 3780 [Erratum-ibid. 100 (2008) 029901].

[74] M. Spira, hep-ph/0211145.

[75] T. Plehn, Czech. J. Phys. 55 (2005) B213.

[76] ATLAS Collaboration, ATLAS-CONF-2011-096.

[77] M. Carena, A. Freitas and C. Wagner, JHEP 10 (2008) 109.

[78] L. Vacavant and I. Hinchliﬀe, J. Phys. G27 (2001) 1839; ATL-PHYS-2000-016 [hep-ex/0005033].

119 [79] T. J. LeCompte and S. P. Martin, Phys. Rev. D 84, 015004 (2011) [arXiv:1105.4304 [hep-ph]].

[80] B. He, T. Li and Q. Shaﬁ, JHEP 1205, 148 (2012) [arXiv:1112.4461 [hep-ph]].

[81] CMS Collaboration, CMS-PAS-HIG-12-016.

[82] ATLAS Collaboration, ATLAS-CONF-2012-092.

[83] Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. 85, 1 (1991); Phys. Lett. B 262, 54 (1991).

[84] A. Yamada, Phys. Lett. B 263, 233 (1991).

[85] J.R. Ellis, G. Ridolﬁ and F. Zwirner, Phys. Lett. B 257, 83 (1991); Phys. Lett. B 262, 477 (1991).

[86] H.E. Haber and R. Hempﬂing, Phys. Rev. Lett. 66, 1815 (1991).

[87] M. Carena, J. R. Espinosa, M. Quiros and C. E. M. Wagner, Phys. Lett. B 355, 209 (1995).

[88] M. S. Carena, M. Quiros and C. E. M. Wagner, Nucl. Phys. B 461, 407 (1996) [hep-ph/9508343].

[89] H. E. Haber, R. Hempﬂing and A. H. Hoang, Z. Phys. C 75, 539 (1997) [hep- ph/9609331].

[90] S. Heinemeyer, W. Hollik and G. Weiglein, Phys. Rev. D 58, 091701 (1998) [hep- ph/9803277].

[91] M. S. Carena, H. E. Haber, S. Heinemeyer, W. Hollik, C. E. M. Wagner and G. Weiglein, Nucl. Phys. B 580, 29 (2000) [hep-ph/0001002].

[92] S. P. Martin, Phys. Rev. D 67, 095012 (2003) [hep-ph/0211366].

[93] T. Moroi, Y. Okada, Mod. Phys. Lett. A7 (1992) 187; Phys. Lett. B295 (1992) 73.

[94] K. S. Babu, I. Gogoladze, M. U. Rehman and Q. Shaﬁ, Phys. Rev. D 78, 055017 (2008).

[95] I. Gogoladze, M. U. Rehman and Q. Shaﬁ, Phys. Rev. D 80, 105002 (2009) [arXiv:0907.0728 [hep-ph]].

[96] S. P. Martin, Phys. Rev. D 81, 035004 (2010) [arXiv:0910.2732 [hep-ph]] ;Phys. Rev. D 82, 055019 (2010).

120 [97] P. W. Graham, A. Ismail, S. Rajendran and P. Saraswat, Phys. Rev. D 81, 055016 (2010) [arXiv:0910.3020 [hep-ph]].

[98] T. Li, J. A. Maxin, D. V. Nanopoulos and J. W. Walker, Phys. Lett. B 710, 207 (2012) [arXiv:1112.3024 [hep-ph]].

[99] M. Endo, K. Hamaguchi, S. Iwamoto and N. Yokozaki, Phys. Rev. D 85, 095012 (2012) [arXiv:1112.5653 [hep-ph]].

[100] C. Anastasiou and K. Melnikov, Nucl. Phys. B 646, 220 (2002).

[101] A. Djouadi, Phys. Rept. 459, 1 (2008).

[102] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, “The Higgs Hunter’s Guide”, Addison-Wesley, Reading (USA), 1990.

[103] M. Boonekamp, J. Cammin, S. Lavignac, R. B. Peschanski and C. Royon, Phys. Rev. D 73, 115011 (2006) [hep-ph/0506275].

[104] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, JHEP 0702, 047 (2007).

[105] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Eur. Phys. J. C 28, 133 (2003).

[106] S. Heinemeyer, W. Hollik and G. Weiglein, Eur. Phys. J. C 9, 343 (1999).

[107] S. Heinemeyer, W. Hollik and G. Weiglein, Comput. Phys. Commun. 124, 76 (2000).

[108] [Tevatron Electroweak Working Group and CDF Collaboration and D0 Collab], arXiv:0903.2503 [hep-ex].

[109] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267, 195 (1996).

[110] A. Djouadi, Phys. Lett. B 435, 101 (1998); R. Dermisek and I. Low, Phys. Rev. D 77, 035012 (2008); I. Low and S. Shalgar, JHEP 0904, 091 (2009). I. Low, R. Rattazzi and A. Vichi, JHEP 1004, 126 (2010); I. Low and A. Vichi, Phys. Rev. D 84, 045019 (2011).

[111] M. S. Carena, S. Mrenna and C. E. M. Wagner, Phys. Rev. D 60, 075010 (1999).

[112] M. S. Carena, S. Mrenna and C. E. M. Wagner, Phys. Rev. D 62, 055008 (2000).

[113] J. Cao, Z. Heng, T. Liu and J. M. Yang, Phys. Lett. B 703, 462 (2011).

[114] M. Carena, S. Gori, N. R. Shah and C. E. M. Wagner, JHEP 1203, 014 (2012).

121 [115] M. Carena, S. Gori, N. R. Shah, C. E. M. Wagner and L. -T. Wang, arXiv:1205.5842 [hep-ph].

[116] G. F. Giudice, P. Paradisi and A. Strumia, arXiv:1207.6393 [hep-ph].

[117] For the scientiﬁc programme and slides, see: http://indico.in2p3.fr/conferenceDisplay.py?confId=6001; http://moriond.in2p3.fr/QCD/2012/qcd.html.

[118] I. Gogoladze, Q. Shaﬁ and C. S. Un, arXiv:1112.2206 [hep-ph].

[119] I. Gogoladze, Q. Shaﬁ and C. S. Un, arXiv:1203.6082 [hep-ph].

[120] H. Baer, V. Barger and A. Mustafayev, arXiv:1112.3017 [hep-ph];

[121] L. J. Hall, D. Pinner and J. T. Ruderman, arXiv:1112.2703 [hep-ph];

[122] J. L. Feng, K. T. Matchev and D. Sanford, arXiv:1112.3021 [hep-ph].

[123] S. Heinemeyer, O. Stal and G. Weiglein, arXiv:1112.3026 [hep-ph].

[124] A. Arbey, M. Battaglia and F. Mahmoudi, Eur. Phys. J. C 72 (2012) 1906;

[125] O. Buchmueller, R. Cavanaugh, A. De Roeck, M. J. Dolan, J. R. Ellis, H. Flacher, S. Heinemeyer and G. Isidori et al., arXiv:1112.3564 [hep-ph];

[126] S. Akula, B. Altunkaynak, D. Feldman, P. Nath and G. Peim, arXiv:1112.3645 [hep-ph];

[127] M. Kadastik, K. Kannike, A. Racioppi and M. Raidal, arXiv:1112.3647 [hep-ph];

[128] J. Cao, Z. Heng, D. Li and J. M. Yang, arXiv:1112.4391 [hep-ph];

[129] M. Gozdz, arXiv:1201.0875 [hep-ph];

[130] H. Baer, I. Gogoladze, A. Mustafayev, S. Raza and Q. Shaﬁ, JHEP 1203, 047 (2012);

[131] N. Karagiannakis, G. Lazarides and C. Pallis, arXiv:1201.2111 [hep-ph];

[132] L. Aparicio, D. G. Cerdeno and L. E. Ibanez, arXiv:1202.0822 [hep-ph]

[133] L. Roszkowski, E. M. Sessolo and Y. -L. S. Tsai, arXiv:1202.1503 [hep-ph];

[134] J. Ellis and K. A. Olive, arXiv:1202.3262 [hep-ph];

[135] J. Cao, Z. Heng, J. M. Yang, Y. Zhang and J. Zhu, arXiv:1202.5821 [hep-ph];

[136] L. Maiani, A. D. Polosa and V. Riquer, arXiv:1202.5998 [hep-ph];

122 [137] T. Cheng, J. Li, T. Li, D. V. Nanopoulos and C. Tong, arXiv:1202.6088 [hep-ph];

[138] N. Christensen, T. Han and S. Su, arXiv:1203.3207 [hep-ph];

[139] E. Gabrielli, K. Kannike, B. Mele, A. Racioppi and M. Raidal, arXiv:1204.0080 [hep-ph].

[140] P. Draper, P. Meade, M. Reece and D. Shih, arXiv:1112.3068 [hep-ph].

[141] B. Ananthanarayan, G. Lazarides and Q. Shaﬁ, Phys. Rev. D 44, 1613 (1991) and Phys. Lett. B 300, 24 (1993)5; Q. Shaﬁ and B. Ananthanarayan, Trieste HEP Cosmol.1991:233-244.

[142] S. Dimopoulos, S. D. Thomas and J. D. Wells, Nucl. Phys. B 488, 39 (1997) [hep-ph/9609434].

[143] G. F. Giudice and R. Rattazzi, Phys. Rept. 322, 419 (1999) [hep-ph/9801271].

[144] P. Meade, N. Seiberg and D. Shih, Prog. Theor. Phys. Suppl. 177, 143 (2009).

[145] Z. Komargodski and N. Seiberg, JHEP 0903, 072 (2009).

[146] Z. Kang, T. Li, T. Liu and J. M. Yang, arXiv:1109.4993 [hep-ph].

[147] A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, Phys. Lett. B 708 (2012) 162.

[148] H. Baer, V. Barger and A. Mustafayev, arXiv:1202.4038 [hep-ph].

[149] M. Ibe and T. T. Yanagida, Phys. Lett. B 709, 374 (2012).

[150] M. Ibe, S. Matsumoto and T. T. Yanagida, Phys. Rev. D 85, 095011 (2012);

[151] D. Grellscheid, J. Jaeckel, V. V. Khoze, P. Richardson and C. Wymant, JHEP 1203, 078 (2012).

[152] D. A. Vasquez, G. Belanger, C. Boehm, J. Da Silva, P. Richardson and C. Wymant, Phys. Rev. D 86, 035023 (2012).

[153] F. R. Joaquim and A. Rossi, Phys. Rev. Lett. 97, 181801 (2006).

[154] J. L. Evans, M. Ibe and T. T. Yanagida, Phys. Lett. B 705, 342 (2011).

[155] J. L. Evans, M. Ibe, S. Shirai and T. T. Yanagida, arXiv:1201.2611 [hep-ph].

[156] Z. Kang, T. Li, T. Liu, C. Tong and J. M. Yang, arXiv:1203.2336 [hep-ph];

[157] N. Desai, B. Mukhopadhyaya and S. Niyogi, arXiv:1202.5190 [hep-ph].

[158] P. Byakti and D. Ghosh, arXiv:1204.0415 [hep-ph].

123 [159] I. Gogoladze, R. Khalid, S. Raza and Q. Shaﬁ, JHEP 1106 (2011) 117.

[160] S. Ambrosanio, G. D. Kribs and S. P. Martin, Phys. Rev. D 56, 1761 (1997) [hep-ph/9703211].

[161] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto, Phys. Rev. D 71, 063534 (2005); J. L. Feng, M. Kamionkowski and S. K. Lee, Phys. Rev. D 82, 015012 (2010).

[162] H. Baer, M. Brhlik, C. -h. Chen and X. Tata, Phys. Rev. D 55, 4463 (1997).

[163] G. J. Gounaris, J. Layssac, P. I. Porfyriadis and F. M. Renard, Phys. Rev. D 70, 033011 (2004).

[164] CDF Collaboration, Phys.Rev.Lett. 104 (2010) 011801.

[165] D0 Collaboration, Phys.Rev.Lett. 105 (2010) 221802.

[166] P. Minkowski, Phys. Lett. B 67, 421 (1977); T. Yanagida, in Proceedings of the Workshop on the Uniﬁed Theory and the Baryon Number in the Universe (O. Sawada and A. Sugamoto, eds.), KEK, Tsukuba, Japan, 1979, p. 95.

[167] M. Gell-Mann, P. Ramond, and R. Slansky, Supergravity (P. van Nieuwenhuizen et al. eds.), North Holland, Amsterdam, 1979, p. 315.

[168] S. L. Glashow, The future of elementary particle physics, in Proceedings of the 1979 Carg`eseSummer Institute on Quarks and Leptons (M. L´evyet al. eds.), Plenum Press, New York, 1980, p. 687.

[169] R. N. Mohapatra and G. Senjanovi´c,Phys. Rev. Lett. 44, 912 (1980).

[170] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45.

[171] G. Lazarides, Q. Shaﬁ and , Phys. Lett. B 258, 305 (1991).

[172] A. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49 (1982) 970.

[173] R. Barbieri, S. Ferrara and C. Savoy, Phys. Lett. B119 (1982) 343.

[174] N. Ohta, Prog. Theor. Phys. 70 (1983) 542.

[175] L. J. Hall, J. D. Lykken and S. Weinberg, Phys. Rev. D27 (1983) 2359.

[176] L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50, 7048 (1994)

[177] B. Ananthanarayan, Q. Shaﬁ and X. Wang, Phys. Rev. D 50, 5980 (1994)

[178] R. Rattazzi and U. Sarid, Phys. Rev. D 53, 1553 (1996)

[179] T. Blazek, M. Carena, S. Raby and C. Wagner, Phys. Rev. D 56, 6919 (1997)

124 [180] J. L. Chkareuli and I. G. Gogoladze, Phys. Rev. D 58, 055011 (1998).

[181] T. Blazek, S. Raby and K. Tobe, Phys. Rev. D 62, 055001 (2000).

[182] H. Baer, M. Brhlik, M. Diaz, J. Ferrandis, P. Mercadante, P. Quintana and X. Tata, Phys. Rev. D 63, 015007(2001).

[183] C. Balazs and R. Dermisek, JHEP 0306, 024 (2003)

[184] U. Chattopadhyay, A. Corsetti and P. Nath, Phys. Rev. D 66 035003, (2002).

[185] T. Blazek, R. Dermisek and S. Raby, Phys. Rev. Lett. 88, 111804 (2002)

[186] M. Gomez, T. Ibrahim, P. Nath and S. Skadhauge, Phys. Rev. D 72, 095008 (2005)

[187] K. Tobe and J. D. Wells, Nucl. Phys. B 663, 123 (2003)

[188] I. Gogoladze, Y. Mimura, S. Nandi, Phys. Lett. B562, 307 (2003)

[189] W. Altmannshofer, D. Guadagnoli, S. Raby and D. M. Straub, Phys. Lett. B 668, 385 (2008)

[190] S. Antusch and M. Spinrath, Phys. Rev. D 78, 075020 (2008)

[191] H. Baer, S. Kraml and S. Sekmen, JHEP 0909, 005 (2009)

[192] S. Antusch and M. Spinrath, Phys. Rev. D 79, 095004 (2009)

[193] K. Choi, D. Guadagnoli, S. H. Im and C. B. Park, JHEP 1010, 025 (2010)

[194] M. Badziak, M. Olechowski and S. Pokorski, JHEP 1108, 147 (2011).

[195] S. Antusch, L. Calibbi, V. Maurer, M. Monaco and M. Spinrath, Phys. Rev. D 85, 035025 (2012).

[196] J. S. Gainer, R. Huo and C. E. M. Wagner, JHEP 1203, 097 (2012)

[197] H. Baer, S. Raza and Q. Shaﬁ, Phys. Lett. B 712, 250 (2012)

[198] I. Gogoladze, Q. Shaﬁ, C. S. Un and , JHEP 1207, 055 (2012)

[199] M. Badziak, Mod. Phys. Lett. A 27, 1230020 (2012)

[200] G. Elor, L. J. Hall, D. Pinner and J. T. Ruderman, JHEP 1210, 111 (2012).

[201] I. Gogoladze, R. Khalid, S. Raza and Q. Shaﬁ, JHEP 1012, 055 (2010). I. Gogo- ladze, Q. Shaﬁ and C. S. Un, Phys. Lett. B 704, 201 (2011);

[202] P. Langacker, Phys. Rept. 72, 185 (1981).

125 [203] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 70, 2845 (1993).

[204] E. Witten, Phys. Lett. B91, 81 (1980).

[205] S. M. Barr, Phys. Rev. D 21, 1424 (1980).

[206] Y. Nomura and T. Yanagida, Phys. Rev. D 59, 017303 (1999).

[207] M. Frigerio, P. Hosteins, S. Lavignac and A. Romanino, Nucl. Phys. B 806, 84 (2009).

[208] S. M. Barr, Phys. Rev. D 76, 105024 (2007);

[209] M. Malinsky, Phys. Rev. D 77, 055016 (2008);

[210] M. Heinze and M. Malinsky, Phys. Rev. D 83, 035018 (2011);

[211] K. S. Babu, B. Bajc and Z. Tavartkiladze, Phys. Rev. D 86, 075005 (2012) and references therein.

[212] G. Anderson, S. Raby, S. Dimopoulos, L. J. Hall and G. D. Starkman, Phys. Rev. D 49, 3660 (1994).

[213] For a brief review, see A. S. Joshipura and K. M. Patel, Phys. Rev. D 86, 035019 (2012) and references therein.

[214] S. Dimopoulos and F. Wilczek, Print-81-0600 (Santa Barbara); K. S. Babu and S. M. Barr, Phys. Rev. D 48 (1993) 5354.

[215] B. Ananthanarayan, P. N. Pandita, Int. J. Mod. Phys. A22, 3229-3259 (2007).

[216] S. Bhattacharya, A. Datta and B. Mukhopadhyaya, JHEP 0710, 080 (2007).

[217] S. P. Martin, Phys. Rev. D79, 095019 (2009).

[218] U. Chattopadhyay, D. Das and D. P. Roy, Phys. Rev. D 79, 095013 (2009);

[219] A. Corsetti and P. Nath, Phys. Rev. D 64, 125010 (2001) and references therein.

[220] Q. Shaﬁ and C. Wetterich, Phys. Rev. Lett. 52, 875 (1984); C. T. Hill, Phys. Lett. B 135, 47 (1984).

[221] N. Okada, S. Raza and Q. Shaﬁ, Phys. Rev. D 84, 095018 (2011).

[222] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 180, 747 (2009).

[223] B. Allanach, S. Kraml and W. Porod, JHEP 0303, 016 (2003).

[224] H. Baer, J. Ferrandis, S. Kraml and W. Porod, Phys. Rev. D 73, 015010 (2006).

126 [225] H. Baer, T. Krupovnickas, S. Profumo and P. Ullio, JHEP 0510, 020 (2005).

[226] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974).

[227] I. Gogoladze, R. Khalid and Q. Shaﬁ, Phys. Rev. D 79, 115004 (2009).

[228] R. Hempﬂing, Phys. Rev. D 52, 4106 (1995).

[229] A. Arbey, M. Battaglia, A. Djouadi, F. Mahmoudi and J. Quevillon, Phys. Lett. B 708, 162 (2012) and references therein.

[230] See, for instance, W. Altmannshofer, M. Carena, N. R. Shah and F. Yu, JHEP 1301, 160 (2013) and references therein.

[231] H. Baer, V. Barger, P. Huang, D. Mickelson, A. Mustafayev and X. Tata, arXiv:1210.3019 [hep-ph].

[232] I. Gogoladze, F. Nasir and Q. Shaﬁ, arXiv:1212.2593 [hep-ph].

[233] S. Schael et al. Eur. Phys. J. C 47, 547 (2006).

[234] RAaij et al. [LHCb Collaboration], Phys. Rev. Lett. 110, 021801 (2013).

[235] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).

[236] S. R. Choudhury and N. Gaur, Phys. Lett. B 451, 86 (1999); K. S. Babu and C. F. Kolda, Phys. Rev. Lett. 84, 228 (2000).

[237] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 713, 68 (2012).

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