Soumitra SenGupta, Prof. Shudhanshu Sekhar Mandal, Prof. Koushik Ray ... and Dr. Arnab Sen for their encouragement. I am also thankful to the staff members.
Some aspects of the phenomenology of supersymmetric theories and the Large Hadron Collider
Thesis Submitted to The University of Calcutta for The Degree of Doctor of Philosophy (Science)
By Sabyasachi Chakraborty Department of Theoretical Physics Indian Association for the Cultivation of Science 2A & 2B, Raja S.C.Mullick Road, Jadavpur Kolkata - 700 032, India 2015
Abstract In spite of its enormous success, the Standard Model (SM) of particle physics suffers from a few drawbacks. For example, the SM cannot account for the observations of neutrino mass and mixing and the existence of dark matter (DM). Supersymmetry (SUSY) is capable of addressing these issues in an elegant manner. Nevertheless, the non-observations of the superpartners to date has put stringent constraints on their masses. In the light of this, in this thesis, we consider a U(1)R symmetric model where the R-charges are identified with lepton numbers in such a way that one of the leftchiral sneutrinos can acquire a large vacuum expectation value (vev) and can play the role of a down type Higgs field. Models with U(1)R symmetry and Dirac gauginos are well motivated since they can address the issues relating to the Higgs boson mass, DM and neutrino mass generation. Most importantly, the presence of Dirac gluinos can relax the bound on superpartner masses. We augment the model with a right handed neutrino superfield to have a small Dirac neutrino mass at the tree level with an order one neutrino Yukawa coupling f . We observe, that the Higgs boson mass receives an additional contribution at the tree level, proportional to f , which can ameliorate the Higgs naturalness problem when f ∼ O(1). In the context of a locally supersymmetric theory, R-symmetry is mildly broken by a non-zero gravitino mass. In such a scenario, neutrino Majorana masses can be generated at the tree level as well as at the one loop level. In addition, for an order one f , we also obtain a very light bino-like neutralino with mass around a few hundred MeV. Such a light neutralino has important implications at the LHC as well as in the cosmological sector. On the other hand, the small f (∼ 10−4 ) case is also interesting as we obtain a sterile neutrino with mass around a few keV which can be an excellent warm dark matter candidate. The model also fares very well when studied in conjunction with the latest LHC results pertaining to different decay modes of the Higgs boson. Finally, an added advantage of this model is that it can conceive light top squarks. We also study some interesting and distinct signatures of these top squarks at the LHC.
ii
To Baba & Ma
Acknowledgment I am grateful to the Council of Scientific and Industrial Research, Government of India for providing me the financial assistance for my thesis work (Award No. 09/080(0721)/2010EMR-I (Date: 10.09.2010)). I offer my gratitude to my home institute, Indian Association for the Cultivation of Science and particularly to the Department of Theoretical Physics, for providing me with an invaluable academic environment and all the support during the last five years. Completing a PhD is truly a marathon event, and I would not have been able to complete this journey without the aid and support of countless people over the past five years. I must first express my sincerest gratitude towards Professor Sourov Roy. His leadership, support, attention to detail, hard work, inspiration and scholarship have set an example which I hope to match some day. I am also grateful to Prof. Utpal Chattopadhyay, Prof. Soumitra SenGupta, Prof. Shudhanshu Sekhar Mandal, Prof. Koushik Ray, Prof. Krishnendu Sengupta, Dr. Pushan Majumdar, Dr. Arnab Das and Dr. Arnab Sen for their encouragement. I am also thankful to the staff members of my department, Mr. Tapan Moulik, Mr. Bhudeb Ghosh, Mr. Suresh Mandal, Mr. Bikash Darjee and in particular Mr. Subrata Balti. I would also like to thank my collaborators Prof. Katri Huitu, Prof. AseshKrishna Datta, Prof. Rathin Adhikari, Prof. Dilip Kumar Ghosh, Dr. Biplob Bhattacherjee, Arnab Dasgupta, Swagata Mukherjee and Harri Waltari for their scientific guidance and support. It is a pleasure to thank my senior Dr. Subhadeep Mondal for his assistance throughout my PhD career. I am also fortunate to have departmental seniors like Dr. Pradipta Ghosh, Dr. Joydip Mitra, Dr. Sudipta Palchowdhury, Dr. Dwipesh Majumdar, Dr. Kush Saha, Dr. Amit Chakraborty, Dr. Ashmita Das, Dr. Soumya Rao, Dr. Swapan Majhi for their encouragements. Thanks to my colleagues Sourav Mondal, Anirban Dutta, Sutirtha Mukherjee, Manimala Chakraborty, Ipsita Saha, Abhishek Dey, Nivedita Ghosh, Tanmoy Pal, Asmi Haldar, Hirak Banerjee, Hiya Mukherjee, Bhaskar Mukherjee (Bhesho) and Sourav Nandy for their cheerful presence and support. Life without them would surely have been dull. Most importantly, none of this would have been possible without the love and iv
patience of my family and friends. The continuous care and backing of my parents and my elder brother (dadabhai) is the only reason of what I am today. Finally this journey would not have been possible without the love and support of my best friend and partner Swati.
Sabyasachi Chakraborty
Date & Place
v
List of Publications In refereed journals.
• h → γγ in U (1)R− lepton number model with a right handed neu-
trino
Sabyasachi Chakraborty, AseshKrishna Datta, Sourov Roy JHEP 02 (2015) 124 [Erratum-ibid. 09 (2015) 077] [arXiv:1411.1525]
• 7 keV Sterile neutrino dark matter U (1)R− lepton number model Sabyasachi Chakraborty, Dilip Kumar Ghosh, Sourov Roy JHEP 10 (2014) 146 [arXiv:1405.6967]
• Higgs boson mass, neutrino masses and mixing and keV dark matter in an U (1)R − lepton number model Sabyasachi Chakraborty, Sourov Roy JHEP 01 (2014) 101 [arXiv:1309.6538]
vi
Preprints. • Novel top squark signatures in U (1)R − lepton number model with
a right handed neutrino
Sabyasachi Chakraborty, AseshKrishna Datta, Katri Huitu, Sourov Roy, Harri Waltari arXiv:1508.01875
In proceedings. • Some phenomenological studies of a U (1)R -lepton number model
with a right handed neutrino
Sabyasachi Chakraborty J. Phys. Conf. Ser. 627 (2015)1, 012008
vii
Contents 1 The Standard Model: success and drawbacks 1.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Framework of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Kinetic energies and self interactions of the gauge bosons. . . . . 1.2.2 Lepton and quark kinetic energies and their interactions with gauge bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Masses of the gauge and Higgs boson and their couplings . . . . 1.2.4 Lepton and quark masses and coupling to Higgs . . . . . . . . . 1.3 Success of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Pull values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Oblique Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Higgs signal strengths . . . . . . . . . . . . . . . . . . . . . . . 1.4 Drawbacks of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Naturalness problem . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Neutrino mass and mixing . . . . . . . . . . . . . . . . . . . . . 1.4.3 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 3 6 7 7 10 11 13 13 14 15
2 Supersymmetry and MSSM 2.1 Motivation . . . . . . . . . . . . . . . . 2.2 Naturalness solution . . . . . . . . . . 2.3 Supersymmetry Algebra . . . . . . . . 2.4 Superspace and Superfields . . . . . . . 2.4.1 Grassmann algebra . . . . . . . 2.4.2 Supersymmetry transformation 2.4.3 Chiral superfields . . . . . . . . 2.4.4 Vector superfields . . . . . . . . 2.5 Supersymmetric Lagrangian . . . . . . 2.6 R-Symmetry . . . . . . . . . . . . . . . 2.7 Supersymmetry breaking . . . . . . . .
17 17 17 19 21 22 23 24 25 26 29 30
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2.7.1 F -type SUSY breaking . . . . . . . . . . . . 2.7.2 D-type SUSY breaking . . . . . . . . . . . . Minimal Supersymmetric Standard Model (MSSM) 2.8.1 Broken MSSM: soft terms . . . . . . . . . . 2.8.2 Higgs sector: MSSM . . . . . . . . . . . . . 2.8.3 Neutralino mass matrix . . . . . . . . . . . . 2.8.4 Chargino mass matrix . . . . . . . . . . . . 2.8.5 Sfermion mass matrix . . . . . . . . . . . . 2.8.6 Superpartner searches . . . . . . . . . . . .
3 Physics of neutrinos and dark matter 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Neutrino oscillation . . . . . . . . . . . . . 3.2.1 Solar neutrino problem . . . . . . . 3.2.2 Atmospheric neutrino problem . . . 3.3 Neutrino mass . . . . . . . . . . . . . . . . 3.3.1 Dirac neutrino masses in SUSY . . 3.3.2 Majorana neutrino masses in SUSY 3.4 Dark Matter . . . . . . . . . . . . . . . . . 3.5 DM candidates in SUSY . . . . . . . . . . 3.5.1 Missing satellite problem . . . . . . 3.5.2 Core-cusp problem . . . . . . . . . 3.6 Sterile neutrino DM . . . . . . . . . . . . . 3.6.1 The Dodelson-Widrow mechanism . 3.6.2 The Shi-Fuller mechanism . . . . . 3.7 Detection of WDM . . . . . . . . . . . . .
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4 U(1)R -lepton number model with a right handed neutrino superfield 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 U(1)R -lepton number model with a right handed neutrino . . . . . . 4.3 The scalar sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Symmetry breaking and minimisation conditions . . . . . . . 4.3.2 CP-even neutral scalar sector . . . . . . . . . . . . . . . . . 4.3.3 CP-odd neutral scalar sector . . . . . . . . . . . . . . . . . . 4.3.4 Charged scalar sector . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The fermionic sector . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The neutralino sector: R-conserving case . . . . . . . . . . . ix
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46 46 47 47 47 51 53 54 58 59 60 61 61 61 63 63
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66 66 68 72 75 76 77 78 79 81 81
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4.4.2 The neutralino sector: R-breaking case . . . . . . . . . . . . . . 4.4.3 The chargino sector . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83 88 89
5 Higgs boson mass, neutrino masses and mixing and keV dark matter in U(1)R -lepton number model 91 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Large neutrino Yukawa coupling: f ∼ O(1) . . . . . . . . . . . . . . . . 92 5.2.1 Tree level mass bound on mh . . . . . . . . . . . . . . . . . . . 93 5.3 Small neutrino Yukawa coupling : f ∼ 10−4 . . . . . . . . . . . . . . . 96 5.3.1 eV scale sterile neutrino . . . . . . . . . . . . . . . . . . . . . . 96 5.3.2 Right handed neutrino as a keV warm dark matter . . . . . . . 99 5.4 One loop effects to generate neutrino mass . . . . . . . . . . . . . . . . 101 5.4.1 Charged lepton-slepton loop . . . . . . . . . . . . . . . . . . . . 101 5.4.2 Squark-quark loop . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.3 Neutralino-Higgs boson loop . . . . . . . . . . . . . . . . . . . . 103 5.5 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5.1 Neutrino masses and mixing: Inverted Hierarchy . . . . . . . . . 112 5.5.2 Neutrino masses and mixing: Normal hierarchy . . . . . . . . . 115 5.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6 7 keV Sterile neutrino dark matter in U (1)R− lepton number 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bounds on the parameter space . . . . . . . . . . . . . . . 6.3 Right handed neutrino as a keV warm dark matter . . . . 6.3.1 Sterile neutrino decay . . . . . . . . . . . . . . . . . 6.4 Gravitino cosmology . . . . . . . . . . . . . . . . . . . . . 6.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . .
model . . . . . . . . . . . . . . . . . . . . . . . .
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7 h → γγ in U(1)R − lepton number model with a right-handed neutrino 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Contributions to µγγ . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The decay h → gg . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 The decay h → γγ . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Higgs boson decaying to charginos and neutralinos . . . . . . . 7.2.4 The total decay width of the Higgs boson . . . . . . . . . . . . 7.3 Impact of the LHC results . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 The case of large neutrino Yukawa coupling, f ∼ O(1) . . . . x
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127 127 128 129 130 133 135 136 136
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7.3.2 The case of small Yukawa coupling, f ∼ O(10−4 ) . . . . . . . . 141 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 Light top squarks in U(1)R -lepton number model with a right handed neutrino and the LHC 148 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.2 The top squark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2.1 Decay rates of e t1 (≈ e tR ) . . . . . . . . . . . . . . . . . . . . . . 152 e e 8.2.2 Decay rates of t2 (≈ tL ) . . . . . . . . . . . . . . . . . . . . . . 153 8.2.3 Bounds on top squarks . . . . . . . . . . . . . . . . . . . . . . . 155 8.3 The benchmarks and the final states . . . . . . . . . . . . . . . . . . . 156 8.3.1 Case 1: µu < met1,2 . . . . . . . . . . . . . . . . . . . . . . . . 157 8.3.2 Case 2: µu > met1,2 . . . . . . . . . . . . . . . . . . . . . . . . 161 8.4 Collider (LHC) analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.4.1 The simulation setup and reconstructing the physics objects . . 163 8.4.2 Top squark pair-production cross section . . . . . . . . . . . . . 164 8.4.3 The Standard Model backgrounds . . . . . . . . . . . . . . . . . 165 8.4.4 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.4.5 Signal significance and the reach . . . . . . . . . . . . . . . . . . 172 8.5 The ‘stealth’ top squark scenario . . . . . . . . . . . . . . . . . . . . . 173 8.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9 Summary and Conclusion
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183 A.1 Quartic terms in the potential . . . . . . . . . . . . . . . . . . . . . . . 183
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185 B.1 The Higgs-chargino-chargino coupling . . . . . . . . . . . . . . . . . . . 185
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189 C.1 The Higgs-neutralino-neutralino coupling . . . . . . . . . . . . . . . . . 189
xi
Chapter 1 The Standard Model: success and drawbacks 1.1 The Standard Model The Standard Model (SM) [1–3] (for reviews see [4–9]) of particle physics remains one of the most successful theories ever. It describes the nature of the fundamental particles and their interactions with enormous precision. Banking on Lorentz invariance, the SM Lagrangian is based on the following ideas. The first one is the concept of local gauge symmetry which determines the form of the interaction between elementary particles carrying a charge associated with that symmetry. However, such a Lagrangian fails to generate masses of the fundamental particles. Therefore, one has to resort to some back door process, namely, spontaneous symmetry breaking (SSB) [10–14]. In such a scenario, the ground state does not remain invariant under the action of the symmetry. This chapter is organised as follows. After discussing the framework of the SM in a brief but comprehensive manner, we will look at the success of the SM, followed by some serious drawbacks.
1.2 Framework of the SM The SM comprises of matter particles or fermions which are of two types: quarks and leptons. Leptons interact through electromagnetic interaction (if they are charged) and weak interaction. The quarks interact through strong interactions in addition to the electromagnetic and weak interactions. The electromagnetic and strong interactions are mediated by photons and gluons respectively and the weak interactions are mediated by gauge bosons W ± and Z. Being short ranged, the weak interaction mediators are
1
massive. We note down the particle content of the SM in table 1.1 Fields uiL diL
QiL = uiR diR
νiL liL
LiL = liR Wµi Bµ Gaµ
! !
SU(3)C
SU(2)L
U(1)Y
3
2
1/3
3 3
1 1
4/3 -2/3
1
2
-1
1 1 1 8
1 3 1 1
-2 0 0 0
Table 1.1: Gauge quantum numbers of the fermions and gauge bosons with the definition Q = T3 + Y , where Q is the electromagnetic charge, T3 is the isospin and Y is the weak hypercharge. The subscript i in the fermionic fields goes from 1 to 3 and signifies three generations. The subscript L and R represents left and right chiral fermions. Wµi and Bµ are gauge fields before SSB. The subscript µ in the gauge fields are Lorentz indices and take values from 1 to 4. For SU(2)L , the superscript i in Wµi runs from 1 to 3 and for SU(3)c , a represents the colour index in the gluon fields Gaµ which take values from 1 to 8.
1.2.1
Kinetic energies and self interactions of the gauge bosons.
The gauge structure of the SM is SU(3)c × SU(2)L × U(1)Y , comprising the strong interactions (subscript c for color), weak isospin (L denotes the left chiral fermions it couples to) and hypercharge (Y for the hypercharge operator) respectively. The dynamics of the gauge bosons are encoded in the Lagrangian 1 i 1 1 µν µν Lgauge = − Gaµν Gµν a − Wµν Wi − Bµν B . 4 4 4
(1.1)
For SU(3)C , the field strength tensor takes the form Gaµν = ∂µ Gaν − ∂ν Gaµ + gS f abc Gbµ Gcν ,
(1.2)
where a, b, c=1...8, f abc is the anti-symmetric structure constants of SU(3)C , defined in terms of the group generators ta according to [ta , tb ] = if abc tc . In case of SU(2), 2
f abc = ǫabc , the anti-symmetric levi-cevita symbol defined as ǫ123 = 1. The field strength tensor for SU(2) can be written as i Wµν = ∂µ Wνi − ∂ν Wµi + gǫijk Wµj Wνk .
(1.3)
Finally, for U(1)Y interaction the field strength tensor takes the form similar to electromagnetism as Bµν = ∂µ Bν − ∂ν Bµ .
1.2.2
(1.4)
Lepton and quark kinetic energies and their interactions with gauge bosons.
The Lagrangian containing the information for lepton and quark kinetic energies and their interactions with gauge bosons are embedded in the term Lf = iψ L D L ψL + iψ R DR ψR .
(1.5)
The gauge interactions of fermions and scalars are included in covariant derivatives in the form Dµ = ∂µ − ig ′ Bµ Y − igWµi T i − igS Gaµ ta ,
(1.6)
and DL , DR have different forms corresponding to distinct gauge charges. g ′, g, gS are the coupling strengths of U(1)Y , SU(2)L and SU(3)C interactions respectively. The generators of these groups are Y, T i and ta . Under the gauge group, the fermionic fields and the gauge fields transform as U(1)Y :
ψ → exp iλY (x)Y ψ;
SU(2)L : ψ → exp iλiL (x)T i ψ; SU(3)C : ψ → exp iλac (x)ta ψ;
1.2.3
1 ∂µ λY (x), g′ 1 Wµi → Wµi + ∂µ λiL (x) + ǫijk Wµj λkL (x), g 1 Gaµ → Gaµ + ∂µ λaC (x) + f abc Gbµ λcC (x). gS (1.7) Bµ → Bµ +
Masses of the gauge and Higgs boson and their couplings
The Lagrangian responsible for generating gauge and Higgs boson masses and their couplings after electroweak symmetry breaking is noted as LHiggs = (Dµ Φ)† (D µ Φ) − V (Φ), 3
(1.8)
where the Y = 1/2, SU(2)L − doublet of complex scalar fields, causing spontaneous breaking of the SU(2)L × U(1)Y gauge symmetry is of the form, ! ! φ+ φ1 + iφ2 1 =√ Φ= , (1.9) φ0 2 φ3 + iφ4 and V (Φ) = −µ2 Φ† Φ + λ(Φ† Φ)2 .
(1.10)
The potential respects a φ ↔ −φ discrete symmetry and therefore forbids any odd powers of φ. The Lagrangian in eq. (1.8) has a global SO(4)(≡ SU(2) × SU(2)) symmetry. When µ2 > 0 and λ > 0, the vacuum state is not unique and there exists a full circle of P4 2 2 degenerate minima for the potential V (Φ), governed by the eq. i=1 |h0|φi |0i| = v , p where v = µ2 /λ is the radius of the four dimensional circle. Without any loss of generality and to preserve U(1)em we choose the entire vacuum expectation value (vev) in the φ3 direction, i.e. h0|φj |0i = 0, for j = 1, 2, 4 and h0|φ3 |0i = v (φ3 has non-zero U(1)Y and SU(2)L charges, therefore SU(2)L × U(1)Y breaks to U(1)em ). Once this particular direction for the vev is chosen, the SO(4) symmetry reduces to SO(3). The group SO(3) is isomorphic to SU(2). This remaining SU(2) symmetry is a global symmetry and known as the custodial SU(2) symmetry [15], which remains unbroken even after the vev is generated1 . Therefore, after the electroweak symmetry breaking (EWSM), the scalar field becomes ! φ1 + iφ2 1 , (1.11) Φ= √ 2 v + h(x) + iφ4 where h(x) is the Higgs field around the stable minima. The potential, when expanded using eq. (1.11), becomes V =−
λ 2 µ2 2 φ1 + φ22 + (h + v)2 + φ24 + φ21 + φ22 + (h + v)2 + φ24 . 2 4
(1.12)
Expanding eq. (1.12) and substituting µ2 = λv 2 , obtained from minimising the potential, one obtains V = λv 2 h2 + 0.(φ21 + φ22 + φ24 ) + .... Thus, the Higgs field receives a mass mh = massless. 1
√
For a review on custodial symmetry see [16, 17].
4
(1.13)
2λv 2 and all the other scalar fields remain
We can also write Φ in the form as iξ a σ a 1 Φ = √ exp v 2
0 v+h
!
.
(1.14)
h and ξ a are fields and σ a (a = 1, 2, 3) are the Pauli matrices written in the basis where σ3 is diagonal. Under gauge transformation, Φ transforms as 1 U(1)Y : Φ → exp iλY (x). Φ, 2 σa a Φ. (1.15) SU(2)L : Φ → exp iλL (x) 2 Choosing λaL (x) = −2ξ a /v at each point in spacetime, we find ! 0 1 Φ= √ . 2 v+h
(1.16)
Eq. (1.16) implies we are able to gauge away the fields ξ a or φ1 , φ2 , φ4 equivalently. Hence, these fields are not physical degrees of freedom. The corresponding gauge choice is known as the unitary gauge. Let us now concentrate on the gauge kinetic term given in eq. (1.8). Expanding that one would obtain 1 1 ∂µ h ∂ µ h + g 2(v + h)2 (Wµ1 − iWµ2 )(W 1µ + iW 2µ ) 2 8 1 + (v + h)2 (−g ′ Bµ + gWµ3 )2 . (1.17) 8 √ = (Wµ1 ∓ iWµ2 )/ 2, from the second term of eq. (1.17), we obtain
(Dµ φ)† (D µ φ) =
Using Wµ±
L⊃
g 2 v 2 + µ− g 2 v g 2 h2 + µ− Wµ W + hWµ+ W µ− + Wµ W . 4 2 4
(1.18)
From eq. (1.18), it is straightforward to extract the mass of the W boson, mW = gv/2. Moreover, hW W and hhW W couplings are uniquely predicted in SM. This is one of the most extraordinary prediction of the SM. Before expanding the third term of eq. (1.17) we observe # " ′ p g g Wµ3 − p Bµ (gWµ3 − g ′ Bµ ) = g 2 + g ′2 p 2 ′2 g +g g 2 + g ′2 " # p g 2 + g ′2 cos θW Wµ3 − sin θW Bµ = =
p
g 2 + g ′2 Zµ . 5
(1.19)
where θW is the weak mixing angle. The field combination Zµ , which will receive mass from the Higgs vev, is identified as the Z boson. The corresponding orthogonal state (sin θW Wµ3 + cos θW Bµ ) ≡ Aµ is identified with the photon. This state is devoid of any coupling with the Higgs field and hence remains massless. The third term in eq. (1.17) further simplifies to 1 L ⊃ (v + h)2 (−g ′ Bµ + gWµ3)2 8 (g 2 + g ′2)v (g 2 + g ′2 ) (g 2 + g ′2 )v 2 Zµ Z µ + hZµ Z µ + hhZµ Z µ . (1.20) = 8 4 8 The first term in eq. (1.20) is identified with the Z boson mass as m2Z = (g 2 + g ′2 )v 2 /4. The rest of the terms give interactions relating to the Higgs and Z boson. Similar to the previous scenario concerning the interaction between the W and h boson, all the interactions and the strengths between the Z and h boson are precisely known.
1.2.4
Lepton and quark masses and coupling to Higgs
The fermions i.e. the leptons and the quarks get masses through the Yukawa terms and after the EWSB. The most general gauge-invariant renormalisable Lagrangian terms involving the Higgs doublet and fermions are noted down as " # 3 3 X X e † QLj + y d dRi Φ† QLj + y l lRi Φ† LLj + h.c. (1.21) y u uRi Φ LYukawa = − ij
ij
ij
i=1 j=1
e ≡ iσ 2 Φ∗ , where yiju , yijd and yijl represents the elements of a 3 × 3 complex matrix and Φ ! 0 −i σ2= . After the EWSB, up type, down type and charged leptons generate i 0 √ u/d u/d a mass through eq. (1.21). The mass matrices are denoted as mij = yij v/ 2 and √ mlij = yijl v/ 2. However, the quark mass matrices are in the flavor basis and not in the mass basis. Therefore, we need to diagonalise the two complex matrices mu and md . Such matrices can be transformed into a real diagonal matrix by multiplying it on the left and right by appropriate unitary transformation matrices. For example, we define four unitary matrices UL , UR , DL and DR according to d d1 u u1 (1.22) = DL,R s , = UL,R c , d2 u2 b L,R d3 L,R t L,R u3 L,R where u, c, t, d, s, b are the mass eigenstates of the quarks, such that md 0 0 mu 0 0 −1 d UR−1 mu UL = 0 mc 0 , DR m DL = 0 ms 0 . 0 0 mb 0 0 mt 6
(1.23)
The left handed up-type and down-type quarks are rotated by UL and DL respectively. These two matrices show up in the charged-current weak interactions which change quark flavors d1 d d µ +µ † µ µ JL = (u1 , u2 , u3 )L γ d2 = u, c, t L UL γ DL s = u, c, t L γ V s , d3 L b L b L
(1.24)
where the combination UL† DL ≡ V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [18, 19]. We note in passing that it is often convenient to work in the basis in which the up-type quarks are the mass eigenstates and all the rotation can be acted upon the down type quark states as d′ d ′ (1.25) s = V s . ′ b b Finally, the complete SM Lagrangian can be written as
LSM = Lgauge + Lf + LHiggs + LY ukawa .
(1.26)
1.3 Success of SM The principal predictions of the SM were the weak neutral current (WNC) [20] and the presence of massive bosons namely W ± and Z. The precision programme started with the discovery of WNC in 1973 and is going on to date after the discovery of the Higgs boson [21, 22]. The incredible consistency of the SM predictions with the electroweak precision data firmly established the non-abelian structure of the gauge group and the one loop corrections of SM. Therefore, validating the basic principles of renormalisation, which in turn allowed for a prediction of the top quark mass.
1.3.1
Pull values
The Z boson mass and its total decay width can be reconstructed very precisely from its various decay channels. An important outcome of this analysis is the determination of the total number of neutrino flavours coupling to the Z boson. The result yields Nν = 2.9841 ± 0.0083. This depicts there are only three flavors of neutrinos and hence
7
three known sequential generations of fermions. Furthermore, at the Z pole, the crosssection for e+ e− → f f is dominated by the Z diagram. For f 6= e, one obtains [23] " # sΓee Γf f /m2Z dσZf 9 = (1 + cos2 θ)(1 − Pe Ae ) + 2 cos θAf (−Pe + Ae ) , dΩ 4 (s − m2Z )2 + s2 Γ2Z /m2Z
(1.27)
where Pe is the polarisation of the electron beam, s is the square of the centre of mass (CM) energy, ΓZ is the total decay width of the Z boson and Γee , Γf f are the partial widths for Z → e+ e− , Z → f f , respectively. θ is the angle between the incident electron and the outgoing fermion and Af is the left-right coupling constant asymmetry, defined in terms of the left (gL ≡ (gV + gA )/2) and right (gR ≡ (gV − gA )/2) coupling constants as Af =
(gLf )2 − (gRf )2
≃
(gLf )2 + (gRf )2
2gVf gAf (gVf )2 + (gAf )2
,
(1.28)
where gVf , gAf are the vector and axial vector coupling constants. Assuming f = q, the total cross-section σhad can be obtained for s = m2Z by integrating eq. (1.27) over the full solid angle and keeping in mind LEP beams are not polarised, therefore, Pe = 0, which yields σh0 = 12π
Γee Γhad , m2Z Γ2Z
Rl0 =
Γhad , Γl
Rb0 =
Γbb , Γhad
Rc0 =
Γcc , Γhad
(1.29)
with l = {e, µ, τ }. Besides the cross-sections and ratios of different decay widths, various asymmetries have been measured. The results of all the asymmetry measurements are expressed in terms of the asymmetry parameter Af defined in eq. (1.28). For example, the unpolarised forward-backward asymmetry is defined as A0,f FB
≡
σFf − σBf σFf
+
σBf
3 ≃ Ae Af . 4
(1.30)
Hence, the set of asymmetries defined in eq. (1.28) and eq. (1.30) are functions of the vector and axial vector coupling constants. Their precise measurements allow to extract the effective coupling constants. These coupling constants are then used to determine the effective weak angle θeff , from the relation 1 − 4|qf | sin2 θeff = gV /gA , where |qf | is the electric charge of the fermion. Subsequently, in LEP-II, a pair of W ’s were produced. The three diagrams which contribute to W -pair production are: t-channel ν exchange, and s-channel γ and Z exchange. The latter two diagrams contain triple gauge boson couplings, hence LEP also tested the non-abelian nature of the SM gauge interactions. The W mass measurement 8
SM
Mar ’15
G fitter
MH
0.0
MW
-1.4
ΓW
0.2
MZ
0.2
ΓZ
0.0
σ0had 0 Rlep
-1.5 -0.9
AFB
0,l
-0.8
Al(LEP)
0.2
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lept
sin2Θeff (Q )
-0.7
FB 0,c AFB 0,b AFB
0.9 2.5
Ac
0.0
Ab
0.6
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0.0
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0.0
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∆αhad(M )
-0.2
(5)
2
Z
-3
-2
-1
0
1
2
3
(O - Omeas) / σmeas fit
Figure 1.1: Deviations between the SM predictions and the measured parameters in units of the uncertainty described in terms of pull variables. requires the reconstruction of the invariant mass from the W decay products. The final state topology would change depending on the hadronic, semi-leptonic or leptonic decays of the two W bosons. For example, the invariant mass distribution in the channel
9
qqµνµ helped to extract the mass and the total decay width of the W boson, i.e. mW and ΓW . On the other hand, being heavy, the top quark could only be produced at Tevatron. The dominant mechanism for pair production is qq → tt, with a virtual s-channel gluon. The top quark almost in all cases decays to t → W + b. Therefore, the final state comprises of 2-b jets and W + W − . There are three discovery channels depending on the W decays (leptonically, semi-leptonically, hadronically). Both experiments (CDF and D0) measured [24, 25] the mass of the top quark and it is in very well agreement with the value of mt extracted from electroweak precision fits. The deviations of the electroweak precision observables (EWPOs) from their SM best-fit predictions are shown in fig. 1.1 [26]. The deviations are parametrised in terms of the pull values, defined as: pull =
Ofit − Omeas , ∆σmeas
(1.31)
where Ofit are computed in the paradigm of SM, Omeas are the experimental measured values and ∆σmeas represents the error in experimental measurements. Using these “pulls” the SM theoretical expectations can be compared with the experimental results. From fig. 1.1 it is evident that no major discrepancies are present with so many measurements. Therefore, one is tempted to conclude that SM is enormously successful.
1.3.2
Oblique Parameters
If the scale of new physics (NP) is much higher than the W and Z boson masses then NP appears dominantly through vacuum polarisation corrections. These corrections are also known as the oblique corrections. Their effects in the EWPO can be parametrised in terms of three parameters, S, T and U, first introduced by Peskin and Takeuchi [27, 28] (also see [29–31]). S describes NP contributions to the neutral current process, T is sensitive to isospin violation and U(S + U) is sensitive to the NP contributions to charged currents. U is only sensitive to the W mass and width and therefore is usually very small in NP models (often approximated to 0). Constraints on these parameters are derived from the difference between the oblique vacuum corrections determined from the experimental data and the corrections in a reference SM (defined for a fixed mh and mt ). For mh = 126 GeV and mt = 173 GeV, one finds [32] S = 0.03 ± 0.10,
T = 0.05 ± 0.12,
U = 0.03 ± 0.10.
(1.32)
By fixing U = 0, one further obtains S|U =0 = 0.05 ± 0.09 and T |U =0 = 0.08 ± 0.07. Figure 1.2 shows the constraints on S and T parameters with the well motivated choice 10
0.4
T
T
0.5 68%, 95%, 99% CL fit contours, U free (SM : MH=126 GeV, m t =173 GeV)
0.5 0.4
ref
ref
0.3
0.3
0.2
0.2
0.1
0.1
0
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
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-0.4 -0.5 -0.5
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-0.3
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0
0.1
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B SM
Sep 12
-0.4
B SM
SM Prediction with MH ∈ [100,1000] GeV
-0.3 Sep 12
G fitter
MH
-0.2
SM Prediction with MH ∈ [100,1000] GeV
-0.4
SM Prediction MH = 125.7 ± 0.4 GeV mt = 173.18 ± 0.94 GeV
-0.1
MH
-0.3
-0.5 -0.5
0
SM Prediction MH = 125.7 ± 0.4 GeV mt = 173.18 ± 0.94 GeV
-0.1 -0.2
68%, 95%, 99% CL fit contours, U=0 (SM : MH=126 GeV, m t =173 GeV)
0.5
S
Figure 1.2: Experimental constraints on the S and T parameters with respect to the SM with the reference mh = 126 GeV and mt = 173 GeV. The different contours represent 68%, 95% and 99% CL allowed regions, where U is chosen to be unconstrained in the left figure, and U = 0 for the right figure. The SM prediction is shown in thin black stroke. of U = 0. The SM prediction within uncertainties is indicated by the thin black stroke. We again observe a brilliant fit between the experimental data and the SM predictions.
1.3.3
Higgs signal strengths
Very recently the two CERN based experiments have confirmed the existence of a neutral boson, widely expected to be the SM Higgs boson with mass around 125 GeV. All the decay modes have been measured with moderate accuracy and with more data one would expect a significant increase in the precision. The signal strengths of the Higgs boson are defined as µh→channel = σ(pp → h) × BR(h → channel).
(1.33)
As depicted in fig. 1.3 that all the decay modes have been probed by both ATLAS [33] √ and CMS [34], with combined data of centre of mass energy (com) ( s) 7 and 8 TeV. An analysis has also been performed by combining the data samples of ATLAS √ and CMS experiments in the channel h → γγ and h → ZZ → 4l with s = 7, 8 TeV [35] as shown in fig. 1.4. The signal strengths and the measured masses from these two separate channels and from the two experiments are consistent with each other and the best fit value of the Higgs boson mass, when extracted becomes mh = 125.09 ± 0.21(stat.) + ±0.11(syst.) GeV. Figure 1.4 depicts the signals strengths as a function of the Higgs boson mass combined from both ATLAS and CMS. Although, the best fit values seem to deviate from the SM expectation, but they are well in agreement within 1 σ deviations. 11
Signal strength (µ )
Figure 1.3: Signal strength concerning different decay modes of the Higgs boson from CMS (left) and ATLAS (right). 3 2.5
ATLAS H → γ γ ATLAS H → ZZ → 4l CMS H → γ γ CMS H → ZZ → 4l All combined
ATLAS and CMS LHC Run 1
Best fit 68% CL
2 1.5
1 0.5
124
124.5
125
125.5
126
126.5
127
mH [GeV]
Figure 1.4: Summary of signal strengths µ versus Higgs boson mass mh is shown for both ATLAS and CMS experiments. The dashed curves show the 68% CL regions from individual measurements. The overall combination is represented by the solid curve. The cross markers indicate the respective best-fit values.
12
t h
h − √yt2
− √yt2
Figure 1.5: Top loop contributing to the Higgs mass
1.4 Drawbacks of SM Although, so much successful, still the SM suffers from a few drawbacks. In this section we note down those issues categorically.
1.4.1
Naturalness problem
From a theoretical perspective, the naturalness problem remains a grave issue. To elaborate, the Higgs boson mass in the SM is controlled by a free parameter λ. Being a scalar, Higgs boson mass is not protected by any symmetry (unlike gauge bosons and chiral fermions), and is subject to large radiative corrections mainly coming from the top quark loop. This correction is both logarithmic and quadratically divergent. Therefore, if SM is assumed to be the theory up to a very high scale, then these radiative corrections are massive. One has to resort to a unnatural amount of fine tuning between the one loop correction and the bare mass term in order to fit the Higgs boson mass at 125 GeV. Looking back at eq. (1.21), we find the Higgs coupling to the top quark can be noted down as yt yt Ltth = − √ htt − √ vtt, 2 2
(1.34)
√ where mt = yt v/ 2. The dominant diagram contributing to the Higgs mass via radiative correction is shown in fig. 1.5 The self energy correction to the Higgs boson mass is computed as # " !2 Z 4 i i −y d k √ t (−1) Tr Πfhh (0) = k − mt ✓ k − p✁ − mt (2π)4 ✓ 2 # " Z 4 2 d k 2m 1 t = −2λ2f , (1.35) + (2π)4 k 2 − m2t (k 2 − m2t )2 | {z } | {z } quadratically divergent
13
logarithmically divergent
√ where λf = yt / 2. Assuming the integral in eq. (1.35) is valid up to a cut-off scale Λ ≡ MPl , where MPl ∼ 1019 GeV, the Planck scale, the quantum corrections to the Higgs boson mass is more than 30 order of magnitude larger than m2h itself. Hence, an unnatural amount of fine tuning is required to fit the Higgs mass.
1.4.2
Neutrino mass and mixing
SM written in its purest form did not have any neutrino masses. Later on, neutrino oscillation experiments provide concrete evidence in favor of neutrino masses and mixing [36]. Therefore, to explain neutrino masses and mixing in the paradigm of SM, the simplest extension would be to include a right handed neutrino field. The modified Lagrangian concerning only leptons would now look like " # 3 X 3 X e † LLj + y l l Ri Φ† LLj + h.c., Ll =− y ν ν Ri Φ (1.36) Yukawa
ij
ij
i=1 j=1
νR being the right handed neutrino field. The term comprising this field would generate a Dirac mass for the neutrinos. The mass eigenstates of the neutrinos are different from their flavor eigenstates and they are related to each other by the lepton analogue of the CKM matrix, called the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS) [37, 38]. We note that the Yukawa coupling required to generate an appropriately small neutrino mass of 0.1 eV, would be yν mν √ = ≃ 4 × 10−13 . v 2
(1.37)
The addition of new right handed neutrino fields pose a serious problem. One can always write down a Majorana mass term for the right handed neutrinos mN νR νR . This mass term introduces a new scale in the theory and compels us to go beyond the SM. On the other hand, the Majorana mass term for the neutrinos violates lepton number by two units. The SM Lagrangian has a remarkable feature which enforces B − L (B: baryon number, L: lepton number) symmetry accidentally. These symmetries are accidental because they are not imposed on the SM Lagrangian by hand, but are just the consequences of the particle content of the SM, its gauge invariance, renormalisability and Lorentz invariance. The Majorana mass term breaks that symmetry and takes us beyond the realms of SM. One can however forbid this Majorana mass term by imposing a U(1)B−L symmetry by hand, which again takes us beyond SM. Hence, neutrino masses and mixing remains a natural testing ground for physics beyond the SM (BSM).
14
1.4.3
Dark matter
The deviation from the galactic rotation curve [39] and bullet clusters [40, 41] provide concrete evidence for dark matter. The only possible candidate are the neutrinos in the SM. However, we shall now discuss that even neutrinos are not viable DM candidates for the following reasons. • The upper limit on the neutrino mass comes from the laboratory based experiments (tritium beta decay), where m(νe ) < 2 eV at 95% C.L [42]. Cosmic Microwave Background (CMB) and large scale structures put a more stringent P limit on i mνi ≤ 0.22 eV [43]. On the other hand, neutrinoless double beta decay experiments which are sensitive to effective neutrino mass (hmν i) provide < a range on the neutrino masses as hmν i ∼ 0.13 − 0.31eV [44]. If neutrinos are a component of dark matter, then its contribution to the relic density can be computed using2 P mν 2 (1.38) Ων h = i i . 94 eV Even using the most relaxed estimate on the neutrino mass we find Ων h2 < 0.06, which is much smaller than the recent relic density measurement by Planck [45] (Ων h2 ≃ 0.1198). • Neutrinos propagate at a speed which is very close to the speed of light. Such relativistic neutrinos have a large free-streaming length and they erase fluctuations on small scales. As a result, relativistic neutrinos could form large structures like superclusters but would have trouble forming smaller structures like galaxies. Without going into details, let us list a few other drawbacks of the SM. • Gravity is not included in SM as the impact of gravity on particle physics experiments at the accelerators is negligible. However, if one is interested to have a theory of all matter and forces up to the Planck scale then one must include gravity. This posses a serious problem since quantum gravity is non-renormalisable. Thereby, one has to look for BSM physics. • There are no theoretical reasoning for the presence of only three generation of fermions in the framework of SM. In addition, the fermion masses in SM are controlled by the free Yukawa parameter. The huge hierarchy between the masses of the fermions remains a serious question, unanswered by SM. 2
A more detailed discussion regarding this would be presented later in the context of sterile neutrino warm dark matter.
15
• The non-vanishing phase in the CKM matrix accounts for the charge-parity (CP) violating processes in SM. However, this is not enough to explain the huge matterantimatter asymmetry associated with our universe. One requires additional sources of CP violation which is absent in SM. • From an aesthetic point of view, the non-unification of the gauge couplings at a higher scale forces one to think of SM as an effective theory only. In the next chapter we will discuss supersymmetry (SUSY) and the solution of the naturalness problem in the framework of SUSY. We will also deliberate on the minimal supersymmetric standard model (MSSM) and the present bounds on the masses of the superpartners as obtained from different collider experiments.
16
Chapter 2 Supersymmetry and MSSM 2.1 Motivation SUSY [46–52]1 is one of the most celebrated BSM theories mainly because it is able to solve almost every drawback of the SM. In particular, supersymmetric field theories provide an elegant solution to the naturalness problem and therefore remains a cynosure to date. In a nutshell, SUSY is a symmetry relating bosonic and fermionic degrees of freedom. Although, the idea is remarkable and exciting, but its implementation is technically rather complicated. In this chapter we will first address the solution of the naturalness problem and then provide a brief description of the supersymmetric algebra. We will then discuss the concept of superfields and the supersymmetric Lagrangian in a concise manner. The later part of this chapter is dedicated to the MSSM, which by definition is a realistic SUSY model constructed with the minimal number of superfields. We also discuss the bounds on the superpartner masses, obtained from the two CERN based LHC experiments, namely ATLAS and CMS, pertaining to a simplified MSSM scenario.
2.2 Naturalness solution In supersymmetric field theories, the naturalness problem is resolved because additional contributions from the bosonic degrees of freedom. For example, introduce two complex scalar fields feL , feR , which couples to the Higgs field manner h e e∗ 1 e 2 e 2 2 2 2 e e e e √ Af fL fR + h.c. . Lhfefe = λf h |fL | + |fR | + vhλf |fL | + |fR | + 2 2 1
For books and reviews see [53–66].
17
of the let us in the
(2.1)
feL(R)
feL(R) h
Af (v λe f )
Af (v λe f ) h
λe f
h
feR(L)
h
Figure 2.1: New diagrams contributing to the Higgs mass correction in SUSY theories. The scalars feL , feR , also contribute to the Higgs boson mass via radiative corrections, in addition to the top quark loop. Figure 2.1, when evaluated gives Z i 1 1 d4 k h fe ef + Πhh (0) = −λ (2π)4 (k 2 − m2fe ) (k 2 − m2fe ) L R Z i h 4 dk 1 + |Af |2 2 4 2 (2π) (k − mfe )(k 2 − m2fe ) L R Z i h 4 1 1 dk ef v)2 + . (2.2) + (λ (2π)4 (k 2 − m2fe )2 (k 2 − m2fe )2 L
R
The first term in eq. (2.2) can resolve the naturalness problem by completely canceling the quadratic divergence emerging from eq. (1.35) provided ef = −λ2 . λ f
(2.3)
To simplify the other terms in eq. (2.2), we shall require the following relations Z 4 h m2 i dk 1 2 , = m 1 − log iπ 2 k 2 − m2 µ2 Z 4 m2 1 dk = − log , (2.4) iπ 2 (k 2 − m2 )2 µ2 where µ is the normalisation scale and in addition we shall also assume mfeL = mfeR ≡ mfe to simplify matters. Hence, by adding eq. (1.35), (2.2) and using eq. (2.4) we find m e m2 m e λ2f m2f λ2f m2f |Af |2 f f f log +i log −i log . = −i 2 2 2 2 2 2 4π µ 4π µ 16π µ 2
e Πfhh+f (0)
2
(2.5)
We observe, the scalar loop can cancel the logarithmic divergence coming from the top quark loop absolutely when mfe = mf and Af = 0. However, this is certainly not the case since we have not observed any scalars (superpartners) with masses exactly equal 18
to the SM fermions experimentally. Therefore, one is compelled to consider the idea that SUSY must be broken. In such a scenario, mf 6= mfe and Af 6= 0. Let us assume the SUSY breaking to be characterised in terms of two small parameters δm2 and Af , where δm2 = m2fe − m2f is the mass splitting between SM fermion and its superpartner, with the assumption that |δm2 |, |A2f | 1 is considered as extended SUSY. In our discussions we will confine ourselves only in N = 1 case. The
19
SUSY generators follow the algebra i h Qα , M µν h α˙ i µν Q ,M i h Qα , P µ n o Qα , Qβ n o Qα , Qβ˙
= (σ µν )βα Qβ , β˙
= (σ µν )αβ˙˙ Q , h i = Qα˙ , P µ = 0, o n = Qα˙ , Qβ˙ = 0,
= 2(σ µ )αβ˙ Pµ .
(2.9)
The last relation in eq. (2.9) shows that two symmetry transformations Qα Qβ˙ have the same effect as translation. The action of these operators on a bosonic and fermionic state |Bi and |F i is Qα |F i = |Bi ,
Qβ˙ |Bi = |F i ,
(2.10)
which implies QQ : |Bi → |B(translated)i. The mass-squared operator P 2 ≡ Pµ P µ is a Casimir operator of the Poincare also an operator of i h ialgebra h and subsequently 2 2 the SUSY algebra as because P , Qα = P , Qα˙ = 0. This means equal mass for SM particle and its superpartners. In addition, the other Casimir operator for the Poincare group is W 2 = −m2 J 2 , where J 2 = j(j + 1) is the angular momentum eigenvalue, m2 is the mass-squared eigenvalue and W is the Pauli-Lubanski spin vector W µ = 12 ǫµνρσ Pν Mρσ . Evidently h i W 2 , Qα 6= 0. (2.11)
Thus the irreducible representation of the SUSY algebra will contain different spins compared to its SM partners. There are two important artifacts of the anticommutation relation between Qα and Qβ˙ . Firstly, it is straightforward to show that in a supersymmetric theory the energy of any non-vacuum state is positive definite and the vanishing of the vacuum energy is a necessary and sufficient condition for the existence of a unique vacuum. Secondly, we will use the previous identities to show that the number of bosonic (nB ) and fermionic degrees of freedom (nF ) are equal in a SUSY theory. We consider a fermionic operator of the form (−1)F , which acts as (−1)F |Bi = |Bi,
(−1)F |F i = −|F i.
(2.12)
It is straightforward to show that this fermionic operator anticommutes with Qα , i.e., n o (−1)F , Qα |F i = 0. (2.13) 20
With the help of eq. (2.13), we evaluate o n Tr (−1)F {Qα , Qβ˙ } = 0.
Again, using the last expression of eq. (2.9) we find n o o n F µ F Tr (−1) {Qα , Qβ˙ } = 2(σ )αβ˙ pµ Tr (−1) ,
(2.14)
(2.15)
where Pµ is replaced by pµ for a specific state. Finally, eq. (2.14) and eq. (2.15) yields o n X X = hB|(−1)F |Bi + hF |(−1)F |F i = 0, (2.16) Tr (−1)F bosons
fermions
this implies nB = nF .
(2.17)
As a result, the number of bosonic and fermionic degrees of freedom are equal to each other in a SUSY theory.
2.4 Superspace and Superfields To write down a manifestly invariant supersymmetric theory, one needs to go beyond the Minkowski space and introduce the concept of superspace [68]. The elements of the superspace are so-called supercoordinates, consisting of the usual four Minkowski spacetime coordinates and four constant Grassmann parameters, {θα }α=1,2 and {θβ˙ }β= ˙ 1, ˙ 2˙ . These parameters follow the anticommutation relations {θα , θβ } = 0,
{θα˙ , θβ˙ } = 0,
{θα , θβ˙ } = 0.
(2.18)
The elements of the superspace, thus contains (xµ , θα , θα˙ ). With the help of Grassmanian coordinates [69], one can transform the graded Lie algebra (involving both commutators and anticommutators) into a regular Lie algebra (involving only commuα˙ tators), by modifying the spinor generators as θα Qα and θα˙ Q . The anticommutation relations of the SUSY algebra defined in eq. (2.9) gets modified to β˙
α˙
[θα Qα , θβ Qβ ] = [θα˙ Q , θβ˙ Q ] = 0, β˙
β˙
[θα Qα , θβ˙ Q ] = 2 θα σαµβ˙ θ Pµ .
(2.19)
Before going into the details of the SUSY generators, let us look at some Grassmann algebras which we will use extensively later. 21
2.4.1
Grassmann algebra
Let us consider one single Grassmann variable θ. Using eq. (2.18) (i.e., θ2 = 0), the generic analytic function in θ can be expanded as [69] f (θ) =
∞ X
fk θk = f0 + f1 θ.
(2.20)
k=0
So the most general function f (θ) is linear. Further, its derivative is given by df /dθ = f1 . For integrals, one defines Z df dθ := 0, (2.21) dθ R R this implies dθ = 0 and we again define dθθ := 1. The integral over the function f (θ) thus becomes equal to its derivative Z df (2.22) dθf (θ) = f1 = . dθ Next, let θα , θα˙ be spinors of Grassmann numbers. Their squares are defined as α˙
θθ := θα θα ,
θθ := θα˙ θ ,
(2.23)
1 ˙ α˙ β˙ θ θ := ǫα˙ β θ θ, 2
(2.24)
and 1 θα θβ := − ǫαβ θθ, 2
where ǫ represents the antisymmetric tensor. Derivatives work in analogy to the Minkowski coordinates as ∂θβ = δαβ , ∂θα ∂θ ∂θ As to multi integrals Z
dθ
1
Z
β˙ α˙
1 dθ θ θ = 2 2 2 1
This justifies the definition Z Z Z 1 2 1 dθ := d2 θ, dθ 2
Z
˙
= δαβ˙ . Z
dθ
1
Z
2
d θθθ = 1,
(2.25)
dθ2 θθ = 1. Z
2
dθ
Z
(2.26)
d2 θθθθθ = 1.
(2.27)
In terms of ǫ, these measures can be expanded as 1 d2 θ = − dθα dθβ ǫαβ , 4
1 α˙ β˙ d2 θ = dθ dθ ǫα˙ β˙ . 4 22
(2.28)
2.4.2
Supersymmetry transformation
Using the properties of the Grassmann coordinates, the most general superfield can be Taylor expanded in terms of the θ, θ coordinates, which looks like S(xµ , θα , θα˙ ) = φ(x) + θψ(x) + θχ(x) + (θθ)m(x) + (θθ)n(x) + (θσ µ θ)Vµ (x) + (θθ)θλ + (θθ)θρ(x) + (θθ)(θθ)d(x),
(2.29)
where all the spinor indices are contracted. Amongst the components fields φ(x), m(x), n(x), d(x) are complex scalar fields. ψ(x), χ(x), λ(x) and ρ(x) are spinors and Vµ (x) defines a complex vector field. A finite SUSY transformation on a function of superspace, i.e., the general superfield takes the remarkable simple form h i S(xµ , θα , θα˙ ) → exp i(ǫQ + ǫQ) S(xµ , θα , θα˙ ) = S(xµ + iǫσ µ θ + iθσ µ , θ + ǫ, θ + ǫ),
(2.30)
where ǫ denotes a parameter and Q, Q represents differential operator representation of SUSY generators. This also implies δS = i(ǫQ + ǫQ)S.
(2.31)
Upon Taylor expanding the first and second line of eq. (2.30) and equating the lowest order in ǫ and ǫ from both sides we find the SUSY generators in a compact form as ∂ β˙ µ + (σ ) θ ∂µ , ˙ α β ∂θα ∂ = −i α˙ + θβ (σ µ )β α˙ ∂µ . ∂θ
Qα = −i Qα˙
(2.32)
Finally, the superspace anticommute with the SUSY generators i.e., n o n o derivatives n o o which n Dα , Qβ = Dα , Qβ˙ = Dα˙ , Qβ = Dα˙ , Qβ˙ =0, looks like ∂ β˙ µ − i(σ ) θ ∂µ , ˙ α β ∂θα ∂ β µ = α˙ − iθ (σ )β α˙ ∂µ . ∂θ
Dα = D α˙
Below we note down some basic remarks on superfields. • If S1 and S2 are superfields, then so is the product of S1 S2 . • Linear combinations of superfields are also superfields. 23
(2.33)
• ∂S/∂xµ is a superfield but ∂S/∂θα is not a superfield. It is again trivial to show that Dα S is a superfield. However, S is not an irreducible representation of supersymmetry. In the next part we shall see that one can impose constraints on S leading to superfields, which can be irreducible representation of the supersymmetry algebra. For example • Chiral/Left-chiral superfield Φ such that Dα˙ Φ = 0. • Anti-chiral/Right-chiral superfield Φ such that Dα Φ = 0. • Vector superfield V = V † .
2.4.3
Chiral superfields
We reiterate, a chiral superfield is an irreducible representation of a general superfield, which satisfies the constraint DΦ = 0.
(2.34)
Therefore, to identify a chiral superfield in terms of components [64], we first note that Dα˙ (xµ + iθσ µ θ) = 0,
and D α˙ θ = 0.
(2.35)
Thus a chiral superfield is a function of y µ = xµ + iθσ µ θ and θ. Then expanding as before in powers of θ we find √
2θψ(y) + θ2 F (y) √ 1 2 i = φ(x) − iθσ µ θ∂µ φ − θ2 θ ∂ 2 φ + 2θψ + √ θ2 ∂µ ψσ µ θ + θ2 F. (2.36) 4 2
Φ(y, θ) = φ(y) +
We see that chiral superfield contains a complex scalar φ, a Weyl fermion ψ and another complex scalar F , that we shall refer to as an auxiliary field. It is also important to check the mass dimension of the respective fields. The mass dimension of the scalar field [φ] = 1 and this implies the chiral superfield also has the same mass dimension. [ψ] = 3/2 and thus [θ] = −1/2. The mass dimension of the auxiliary field turns out to be [F ] = 2. Under SUSY transformation, defined in eq. (2.31), the component fields transform as √
2ǫψ, √ µ √ δψ = i 2σ ǫ∂µ φ + 2ǫF, √ δF = i 2ǫσ µ ∂µ ψ. δφ =
24
(2.37)
An important aspect of these transformation properties given in eq. (2.37) is that F transforms as a total derivative under SUSY transformation. In addition the auxiliary field has no dynamics of its own and can be replaced with the dynamical fields by using the equation of motion. We also note down some interesting features of the chiral superfields. • The product of chiral superfields is also chiral superfields. • If Φ is a chiral field, then Φ = Φ† is an anti-chiral field. • Φ† Φ and Φ† + Φ are real superfields but not chiral or anti-chiral.
2.4.4
Vector superfields
After implementing the constraint of reality on a generic superfield leads to a vector superfield [64] V
= V †.
(2.38)
The full vector superfield has many components but one can use the advantage of gauge symmetry to gauge some components away. In such a gauge, more commonly known as the Wess-Zumino gauge [47–49], the vector superfield, consisting of a vector, a fermion and a real scalar field (another auxiliary field), looks like h i WZ µ (2.39) V = (θσ θ) Vµ (x) + (θθ)θλ(x) + (θθ)θλ(x) + (θθ)(θθ)D(x). The Vµ is the gauge field, λ is its supersymmetric partner known as the gaugino. Once again the auxiliary field D transforms as a total derivative under SUSY transformation in the following way o in ∂µ λ(x)σ µ α − ∂µ λ(x)σ µ α . (2.40) δD(x) = 2
We note that the mass dimension of the gauge field is [Vµ ] = 1. The vector superfield has mass dimension [V ] = 0. Since [θ] = [θ] = −1/2, implies the mass dimension of the gauginos [λ] = 3/2 and [D] = 2. Some salient features of the vector superfields are noted below.
• Using the anticommutation relation of the Grassmann variables, it is straightfor2 n ward to show that VWZ (x, θ, θ) = 21 (θθ)(θθ)Vµ (x)V µ (x) and VWZ (x, θ, θ) = 0 for all n > 2.
25
• The property which makes the Wess-Zumino gauge a particularly convenient gauge to work in is 1 exp(V ) = 1 + V + V 2 2 = 1 + (θσ µ θ)Vµ (x) + (θθ)θλ(x) + (θθ)θλ(x) o n 1 + (θθ)(θθ) D(x) + Vµ (x)V µ (x) . 4
(2.41)
2.5 Supersymmetric Lagrangian We are now in a position to construct the Lagrangian of a supersymmetric field theory. We know that the action has to be invariant under SUSY transformation which means Z δS d4 xL(x) = 0. (2.42) This expression is satisfied if L transforms as a total derivative. We have seen the highest components (those with the largest number of θ, θ factors) of chiral and vector superfields satisfy this requirement. As a result, these terms can be used to construct a SUSY Lagrangian. We can thus write the action A as Z Z Z 4 2 A = dx d θLF + d2 θd2 θLD . (2.43) LF and LD are Lagrangian densities constructed from chiral and vector superfields which gives rise to ‘F-terms’ and ‘D-terms’ respectively. In order to be more explicit and write down the action in a more detailed manner, let us compute the product of two chiral superfields from eq. (2.36) Φ1 Φ2 = φ1 φ2 +
√
2θ(ψ1 φ2 + φ1 ψ2 ) + θθ(φ1 F2 + F1 φ2 − ψ1 ψ2 ).
(2.44)
Note that we have used the identity (θψ1 )(θψ2 ) = − 21 (θθ)(ψ1 ψ2 ) to deduce eq. (2.44). Of course, if the product of two chiral superfields is a chiral superfield, by induction the same must be true for the product of any number of chiral superfields. Similarly, the highest component of three chiral superfields can be extracted as h i Φ1 Φ2 Φ3 |θθ = (φ1 φ3 F2 + φ3 F1 φ2 + F3 φ1 φ2 ) − (φ3 ψ1 ψ2 + ψ1 φ2 ψ3 + ψ2 φ1 ψ3 ) . (2.45)
The last three terms describe Yukawa interactions between one scalar and two fermions. In SM, such interactions produce quark and lepton masses. For example, if φ1 is a Higgs field and ψ2 , ψ3 are the left and right handed components of the top quark, 26
then this term produce the desired Higgs-top-top interaction. In addition, this term also generates the interaction between the top squark, higgsino and the top quark with equal strength. So far we have only identified terms which gives rise to explicit fermion masses and their interactions. However, kinetic energy terms can not be obtained from this prescription. In order to evaluate those, let us consider the term Φ† Φ. Clearly, this term is self conjugate and hence a vector superfield. It is therefore a candidate to contribute to the ‘D-terms’ in the action as Z d2 θd2 θ Φ† Φ = F F ∗ − φ∂µ ∂ µ φ∗ − iψσµ ∂ µ ψ. (2.46)
Clearly, eq. (2.46) contains kinetic energy terms for scalars and fermions but not for F . Hence, F is merely an auxiliary field, which can be integrated out exactly using purely algebraic equation of motion. In order to understand how the F fields can be removed from the Lagrangian, let us introduce the superpotential 1 1 (2.47) W (Φi ) = ki Φi + mij Φi Φj + λijk Φi Φj Φk . 2 3 The superpotential W (Φi ) not only determines the scalar masses of the theory, but also determines the fermion masses and their interactions as we will see. The contributions to the Lagrangian that we have identified so far can be written compactly as hZ i XZ † 2 2 d2 θW (Φi ) + h.c. L = d θd θΦi Φi + i
=
X i
+
(Fi Fi∗ + |∂µ φ|2 − iψ i σµ ∂ µ ψi )
h X ∂W (φ ) i
j
∂φj
Fj −
i 1 X ∂ 2 W (φi ) ψj ψk + h.c. . 2 j,k ∂φj ∂φk
(2.48)
In the last line of eq. (2.48), W (φi ) is understood to be a function of the scalar fields φi rather than the superfields Φi . It is easy to convince that the last line indeed reproduces the highest order, i.e., θθ term of eq. (2.47). Integrating out the auxiliary fields Fj with their corresponding equations of motion ∂L/∂Fj = 0, yields X ∂W (φi ) X h ∂W (φi ) i∗ ∗ Fj = − and Fj = − . (2.49) ∂φj ∂φj j j Plugging this back to eq. (2.48), one obtains h X ∂ 2 W (φ) i − ψj ψk + h.c. − L= Lkin |{z} ∂φj ∂φk j,k kinetic energy {z } | fermion masses and interactions
27
X ∂W (φi ) 2 ∂φj j {z } |
scalar masses and interactions
.
(2.50)
We can identify the scalar masses and interactions as VF = Fi∗ Fi . Let us now introduce gauge interactions. In order to obtain the gauge invariant terms with matter chiral superfields, one has to consider their respective gauge transformations which are Φ → e−igΛ Φ,
†
Φ† → Φ† eigΛ ,
V → V + i(Λ − Λ† ).
Λ designates a chiral superfield. The gauge kinetic term is modified as † gV † → Φi e Φi . Φi Φi ¯¯ ¯¯ θθ θθ
θθ θ θ
(2.51)
(2.52)
Further, the Lagrangian transforms as Z Z 2 2 † d θd θ Φ Φ → d2 θd2 θ Φ† egV Φ
√ = |Dµ φ|2 − iψσµ D µ ψ + gφ∗Dφ + i 2g(φ∗λψ − λψφ) + |F |2.
(2.53)
We have used the W Z gauge to expand the vector superfield in the exponential factor. Also Dµ = ∂µ +igAaµ T a , where T a are the group generators. This part of the Lagrangian not only describes the interactions of the matter fields with gauge fields, but also contains gauge-strength interactions between fermions- sfermions and gauginos. Finally, the kinetic energy terms of the gauge fields can be described with the help of the superfield ˙
Wα = (Dα˙ Dβ˙ ǫα˙ β )e−gV Dα egV ,
(2.54)
Dα ’s are the SUSY covariant derivatives defined earlier. For abelian symmetries, Wα simplifies to ˙
Wα = (D α˙ D β˙ ǫα˙ β )Dα V.
(2.55)
Since Dα˙ Dα˙ = 0 and Dα˙ Wα = 0, therefore, Wα is considered to be a chiral superfield. One can further show that Wα W α is gauge invariant and also a chiral superfield. Hence, the θθ component may appear in the Lagrangian i 1 a µν 1 1 abc 1 µ α a a µ W W = − F F + D D + − λ σ ∂ gf λ σ A λ + λ + h.c. α a µ a µ b c a 32g 2 4 µν a 2 2 2 θθ (2.56) In addition to the kinetic energy terms for the gauge fields and gauginos eq. (2.56) also gives the canonical coupling of the gauginos to the gauge fields, determined by the 28
group structure functions f abc . Again, eq. (2.56) does not contain any kinetic energy term for Da fields. They are therefore auxiliary fields and can easily be integrated out. Similar to the F -case, the equation of motion for the D-fields are noted down as X Da = −g φ∗i Taij φj . (2.57) ij
Replacing eq. (2.57) in the Lagrangian, the D-term contribution can be extracted in the form 1 X X ∗ a 2 VD = (2.58) gφi Tij φj . 2 a ij
Hence, VD also contributes to the scalar interactions and are completely determined by the gauge couplings. The full scalar potential, which has contributions from both F and D terms are written as V
= VF + VD .
Similarly, the complete SUSY Lagrangian looks like Z Z Z 1 1 α˙ 2 α 2 d θW Wα + d θ W α˙ W + d2 θd2 θΦ†i egV Φi L = 4 4 Z 1 1 + d2 θ ki Φi + mij Φi Φj + λijk Φi Φj Φk + h.c. 2 3
(2.59)
(2.60)
2.6 R-Symmetry Some SUSY Lagrangians are also invariant under a global U(1)R symmetry [70, 71]. Under this symmetry the Grassmann coordinates θ and θ transform with charges +1 and -1 respectively as θ → eiα θ,
θ → e−iα θ,
(2.61)
where α is a real parameter. Under R−symmetry, the chiral superfields transforms as RΦ(x, θ, θ) = e2inΦ α Φ(x, e−iα θ, eiα θ), RΦ† (x, θ, θ) = e−2inΦ α Φ† (x, e−iα θ, eiαθ ).
(2.62)
nΦ is the R−charge of the chiral superfields. It is quite straightforward to compute the transformation rules of the component fields which are as follows A → e2inΦ α A, 1
ψ → e2i(nΦ − 2 )α ψ,
F → e2i(nΦ −1)α F. 29
(2.63)
On the other hand, the vector superfield being real, transforms as RV (x, θ, θ) = V (x, e−iα θ, eiα θ),
(2.64)
In a similar manner the transformation rules of the component fields are Vµ → Vµ ,
λ → eiα λ,
D → D.
(2.65)
From eq. (2.63) and (2.65), it is quite clear that the component fields transform in a different manner than the superfield. The scalar has the same and the fermion has one less R−charge than the chiral superfield. Similarly, the R−charge of gauge bosons is zero, while gauginos have R−charge of one unit.
2.7 Supersymmetry breaking Supersymmetry should be broken in realistic models since we do not see superpartners with degenerate mass with their SM counterparts, as elaborated later. In this section, we will discuss how SUSY breaking can occur. Let us consider the possibility of SUSY breaking spontaneously. The criterion for spontaneous SUSY breaking implies that the vacuum state |0i should not remain invariant under a general supersymmetry transformation, i.e. Qα |0i = 6 0 or Qα˙ |0i = 6 0.
(2.66)
This has serious implications for the energy of the ground state as the Hamiltonian operator H is related to the SUSY generators through 1 (2.67) H = P 0 = (Q1 Q1˙ + Q1˙ Q1 + Q2 Q2˙ + Q2˙ Q2 ). 4 If SUSY is unbroken in the vacuum state then H |0i = 0 and the vacuum has zero energy. However, if SUSY is broken spontaneously then 1 † |Q1 |0i |2 + |Q1 |0i |2 + |Q†2 |0i |2 + |Q2 |0i |2 > 0, (2.68) h0| H |0i = 4 as eq. (2.68) reflects, that the vacuum must have positive energy. Another important perspective on spontaneous SUSY breaking is that it must respect Lorentz invariance. This can only occur if the auxiliary fields F and D acquire a vev. In a SU(2)L × U(1)Y theory, the scalar potential is given as 1 (2.69) = F ∗ F + (D a D a + D ′2 ). 2 Hence, if SUSY is broken then h0|V |0i = 6 0, which implies hF i = 6 0 and/or hDi = 6 0. V
30
2.7.1
F -type SUSY breaking
The models where spontaneous SUSY breaking occurs through a non-zero F term vev is known as the O’Raifeartaigh model [72]. Let us consider the superpotential as λ (2.70) W = Φ1 Φ22 − µ2 + mΦ2 Φ3 , 2
where Φ1 , Φ2 , Φ3 are chiral superfields. When SUSY is conserved the vev of the F -terms are zero, i.e., D ∂W E λ 2 2 hFΦ1 i = = Φ − µ = 0, ∂Φ1 2 2 D ∂W E = λΦ2 Φ1 + mΦ3 = 0, hFΦ2 i = ∂Φ2 D ∂W E hFΦ3 i = = mΦ2 = 0. (2.71) ∂Φ3 We observe that hFΦ1 i = 0 and hFΦ3 i = 0 are not compatible to each other. Hence, SUSY must be broken. In such a scenario, the scalar potential becomes ∂W 2 ∂W 2 ∂W 2 (2.72) V = + + > 0. ∂Φ1 ∂Φ2 ∂Φ3
Therefore, the scalar potential of the theory has no supersymmetric ground state. Collecting the quadratic terms in the fields from the classical potential we find λ 2 ∗ 2 ∗ Vquad = − µ2 (Φ22 + Φ∗2 2 ) + m Φ2 Φ2 + m Φ3 Φ3 . 2
(2.73)
Further expanding Φ2 = a+ ib, we obtain the following scalar masses as m2Φ1 = 0, m2a = 2 2 m2 − λµ2 , m2b = m2 +Dλµ2 and E mΦ3 = m . In a similar manner, the fermion masses
can be obtained from looks like
∂2W ∂Φi ∂Φj
ψi ψj . The fermion mass matrix in the basis (ψ1 , ψ2 , ψ3 )
0 0 0 mij = 0 0 m 0 m 0
(2.74)
The eigenvalues of the corresponding fermions become mψ1 = 0, mψ2 = mψ3 = m. We conclude that SUSY is spontaneously broken and as a result fermion and scalar masses are non-degenerate.
2.7.2
D-type SUSY breaking
SUSY breaking with a non-zero vev of the D-term can occur through the FayetIliopoulos mechanism [73,74]. This is an example of a U(1) gauge theory where one can 31
introduce a term linear in the auxiliary field of the corresponding gauge supermultiplet LF I = −κD.
(2.75)
κ is a constant and has the mass dimension of 2. This term is gauge invariant and supersymmetric. If one includes this term in the Lagrangian, then D may be forced to get a non-zero vev. The relevant part of the scalar potential takes the form X 1 qi |φi |2 . (2.76) V = κD − D 2 − gD 2 i qi are the charges of the scalar fields φi . In the presence of this Fayet-Iliopoulos term, the D-term equation of motion turns out to be X D =κ−g qi |φi |2 . (2.77) i
Thus, the potential becomes X 1 (κ − g qi |φi |2 )2 . 2 i
VD =
(2.78)
Although the condition VD = 0 implies SUSY remains unbroken but U(1) symmetry suffers a spontaneous breakdown. Otherwise, when VD 6= 0, the minimisation condition implies hφi i = 0 and the minimum of the potential occurs at VD = 12 κ2 . The scalar component φi acquires a mass −κgqi , whereas the fermionic superpartner remains massless and is identified as the goldstino. Hence, in this scenario we also observe that the D-type SUSY breaking results in a non-degeneracy between the masses of the scalars and the gauginos. For both F type and D type SUSY breaking mechanisms, the SUSY particle spectrum follows certain sum rules, known as the supertrace sum rules which states that the supertrace of the tree level squared mass eigenvalues, defined with a weighted sum over all particles with spin j must vanish, i.e., X (−1)j (2j + 1)Tr(mj )2 = 0. (2.79) STr(m2 ) = j
A vanishing supertrace requires some of the SUSY particles to be lighter compared to their SM counterparts if the SUSY breaking order parameter belongs to any of the MSSM supermultiplets. This is clearly not the case as such SUSY particles are not observed experimentally. However, this sum rule holds only at the tree level and only for non-renormalisable theories. Thus, there exists a possibility that spontaneous SUSY breaking occurs in a hidden sector which couples to the MSSM sector only via loops or via non-renormalisable interactions. 32
2.8 Minimal Supersymmetric Standard Model (MSSM) The supersymmetric model in which SM can be embedded with the minimal number of fields is known as the MSSM. The gauge structure of MSSM is SU(3)C × SU(2)L × U(1)Y , which is exactly same to the SM. The simplest way to supersymmetrise a theory is to place all the SM fields into superfields and introduce necessary superpartners. For example, in MSSM, every SM fermion comes with a scalar superpartner dubbed as sfermions. Similarly the superpartners of a gauge boson is a fermion and known as the gauginos. However, one faces a serious problem by introducing a single chiral fermion as the superpartner of the Higgs. This chiral fermion is charged under SU(2)L × U(1)Y and makes the gauge symmetries anomalous. Another problem is that the superpotential has to be a holomorphic function of the chiral superfields and hence some of the necessary Yukawa couplings are forbidden. Both of these problems can be avoided if one introduces a second chiral superfield [70, 75, 76]. Thus there are now b u and H b d . The superfields along with their gauge charges are two Higgs superfields H shown below. Superfields biL Q bc U i b Dic
u e∗iR de∗iR νeiL e liL
biL L
bd = H
bu = H
bic E ci W µ bµ B ba G µ
Bosons ! u eiL deiL
b h0d b h−
d b h+ u b h0u
!
!
ee∗Ri Wµi Bµ Gaµ h0d h− d h+ u h0u
!
!
!
Fermions ! uiL diL
SU(3)C , SU(2)L , U(1)Y
u†iR d†iR
3, 1, -4/3 3, 1, 2/3
νiL liL
!
e†R fi W e B ea G ! e h0d e h− d ! e h+ u e h0u
3, 2, 1/3
1, 2, -1 1, 1, 1, 8,
1, 3, 1, 1,
2 0 0 0
1, 2, -1 1, 2, 1
Table 2.1: MSSM particle content with the corresponding gauge quantum numbers. i(1, 2, 3) indicates the three generations of SM fermions and W -triplets, a(1, ..8) indicates eight gluons. After knowing the particle content, the gauge groups and the gauge transformation 33
properties, all one needs to do to propose a model is to write a superpotential W . The MSSM superpotential is specified as bi H buU bjc + y ij H bdQ bi D b jc + yeij H bdL bi E bjc + µH buH bd. WMSSM = yuij Q d
(2.80)
In eq. (2.80) the flavor and gauge indices are suppressed. The second Higgs field is essential as it gives mass to some of the SM fermions. The µ-term is a mass term for the Higgsinos and will contribute, through F -terms to the scalar potential. Expanding the superfields in WMSSM in their component fields gives the Feynman rules concerning the SM particles and the superpartners. In addition to the terms given in eq. (2.80), there are some other renormalisable operators allowed by the gauge symmetries which can be added to the superpotential. bi L bj E b c + λ′ L b b b c 1 ′′ b c b c b c bi H b u + 1 λijk L (2.81) W∆B,L = ǫi L k ijk i Qj Dk + λijk Ui Dj Dk . 2 2 Note that λijk and λ′′ijk are antisymmetric under (i ↔ j) and (j ↔ k) respectively. However, all the operators in eq. (2.81) violate either lepton number (L) or baryon number (B). In the paradigm of SM, lepton and baryon numbers are accidental symmetries. Hence the operators which violate either B or L quantum numbers are actually forbidden. In MSSM this no longer happens because superfields allow us to construct these operators by respecting renormalisability and gauge invariance. The effects of these operators are phenomenologically catastrophic, leading to unacceptable rate of proton decay. One can always set the coefficients of these operators to be zero, but the idea is not appealing. These operators could be forbidden by introducing a new symmetry, for example a U(1)R -symmetry. However, it turns out that U(1)R -symmetry is too restrictive and would forbid mass terms for gauginos also. Instead the Z2 subgroup of the U(1)R symmetry, called R-parity [77–79] (see [80] for a review) was introduced. Under the R-parity the fields have charge RP = (−1)3(B−L)+2s ,
(2.82)
where s is the spin of the particle. It can be readily shown that all of the SM particles have even R-parity (RP = +1), while all its SUSY partners have odd R-parity (RP = −1). A very important outcome of this extension is when R-parity is conserved, the lightest supersymmetric particle (LSP) is stable and sparticles are produced in pairs in each interaction vertex. It turns out that the LSP is often a neutral state and has the right properties to be the candidate for DM. In a realistic SUSY model one should also include SUSY breaking terms since we know that superpartners are not degenerate in mass with their SM partners. Hence, in the next section we shall consider SUSY breaking terms. 34
2.8.1
Broken MSSM: soft terms
We have already seen that both the F type and D type SUSY breaking involving the MSSM superfields does not lead to an acceptable spectrum. It is now clear that spontaneous SUSY breaking requires an extension of the MSSM. A way to solve this problem is to consider a hidden sector, singlet under the SM gauge group, where SUSY is broken. The SUSY breaking information has to be transferred from the hidden sector to the visible sector, consisting of MSSM superfields, through some messenger fields. In this visible sector, the explicit SUSY breaking terms are known as soft SUSY breaking parameters as they do not cause ultraviolet divergences to reappear in the scalar masses. In a way, all our ignorance about the SUSY breaking is concealed in the soft SUSY breaking terms. In MSSM, the most general set of soft SUSY breaking terms are noted down as 1 SSM f f e e M g e g e + M W W + M B B + c.c. LM = − 3 2 1 soft 2 e u −e e d −e e d + c.c. e u QH − ua dad QH eae LH † e† m2 Q e−L e† m2 L e e 2 e† e 2 e e 2e† − Q Q L − umu u − dmd d − eme e
− m2Hu Hu∗ Hu − m2Hd Hd∗ Hd − (BHu Hd + c.c.),
(2.83)
where M3 , M2 and M1 are the gluino, wino and bino mass terms. au , ad and ae are the trilinear scalar couplings or the ‘A-terms’. m2Q , m2L , m2u , m2d , me2 , m2Hu , m2Hd are the scalar soft masses. Let us look at some of the possibilities by which these soft SUSY breaking terms can be generated at the Hidden sector through a spurion 2 superfield via F or D type breaking [81–85]. These terms are • Scalar mass: The scalar mass terms are generated through a spurion superfield b = θ2 F and by the operator X c
ij
Z
b †X b † X Qi Qj = −(m2 )ij qei∗ qej , dθ 2 M 4
(2.84)
after the spurion superfield generates a F term vev. cij also have new flavor structure and can lead to visible flavour violating effects. M is the mass scale of the messenger field which depend on the type of SUSY breaking. 2
A spurion superfield is an auxiliary non-dynamical superfield with non-vanishing F or D-term components.
35
• Gaugino mass: The Majorana gaugino mass terms are generated via the following operator Z b X 1 eα e 1 ci d2 θ W α Wα = − mi λ λα . (2.85) 2 M 2 b is a MSSM singlet and i runs over the three gauge groups of the MSSM Here X and mi = −ci F/M.
• A term: The trilinear scalar couplings are not invariant under the SUSY transformation. In a similar manner the A-terms are generated through the following operator Z b X bcQ bj H b u − Aij D b cQ bj H b d − Aij E bcL bj H bd d2 θ Aij U u i i e i d M ec ej hd − aij eece = aij eci qej hu − aij (2.86) uu e i lj hd , d di q where aij = Aij F/M. It is important to note that A-terms are another new source of flavor violation.
• B term: Finally the B term is generated via Z b †X b X buH b d = −Bhu hd , H b d4 θ M2
(2.87)
with B = −b(F/M)2 . Another interesting possibility is the SUSY breaking by D-term spurion. Such a term generates a Dirac gaugino mass which requires additional chiral superfields living in the adjoint representation of the SM gauge group. We will discuss about the origin of such a mass term in due course.
We observe that soft SUSY breaking MSSM Lagrangian induces many new parameters (a careful count reveals that there are 105 soft parameters including masses, phases and mixing angles). This, in principle, introduces a tremendous arbitrariness into the theory. As a result, it would be interesting to see if these parameters can be correlated in any way, which also depends on the mediation mechanism which carries the information of SUSY breaking from the hidden sector to the visible sector. This leads to characteristic soft terms and consequently sparticles masses and collider signals in different models. Let us now discuss a few of the most popular SUSY mediation mechanisms. • Gravity mediation: The most popular example of mediating SUSY breaking is the mechanism in which gravitational interactions [86–91] play the role of the messenger. Here, new 36
physics appears near the Planck scale where gravitational interactions become important. If SUSY is broken in the hidden sector by a vev hF i, where F is a hidden sector field, then the soft terms in the visible sector should be msoft ∼ hF i/MP , where MP ∼ 1019 GeV, is the Planck mass. Hence, if we demand msoft ∼ 1 TeV, then the scale associated with the origin of SUSY breaking in the p hidden sectors should be hF i ∼ 1011 GeV or so. Another important point here is gravitational interactions do not distinguish between different types of scalars or fermions. Therefore, all the scalars (fermions) can be considered to have the same mass m0 (m1/2 ) at high scale. • Gauge mediation The basic idea in this case is to introduce new supermultiplets [92–94], not included in the MSSM as the messenger fields. These messenger fields couple to the hidden sector and carry SM quantum numbers. Consequently, these fields couple directly to the SM gauge fields and the corresponding gauginos. This way the gauginos of the MSSM can receive masses at one loop. Although, the messenger fields do not couple to the sfermions at the lowest order, these couplings arise in higher orders of perturbation theory. Thus, the scalars of the MSSM get leading contribution to their masses at the two loop level. The SUSY breaking in the hidden sector is thus communicated to the observable sector and soft masses and other SUSY breaking terms are generated. • Anomaly mediation Anomaly mediated SUSY breaking (AMSB) [95,96] involves a higher dimensional supergravity theory where the hidden sector and visible sector superfields are localised on two distinct parallel branes separated by a distance rc , where rc is the compactification radius in the extra dimension. This can be also thought of as a special case of gravity mediation where there is no direct tree level coupling that transmits the SUSY breaking from the hidden sector to the visible one. In such a scenario, the masses of the gauginos are generated at one loop while those of the scalars are generated at two loop level because of the superconformal anomaly that breaks the scale invariance.
2.8.2
Higgs sector: MSSM
There are three contributions to the Higgs scalar potential arising from F -terms, Dterms and soft SUSY breaking terms. VHiggs = VF + VD + Vsoft . 37
(2.88)
From eq. (2.80) we find − 2 2 0 2 . | + |h | + |h | VF = |µ|2 |h0u |2 + |h+ u d d
The D-term contributions are obtained as a ∗ a ∗ a D2 = −g hu τ hu + hd τ hd , g′ + 2 2 |hu | + |h0u |2 − |h0d |2 − |h− | . D1 = − d 2
(2.89)
(2.90)
where τ a = σ a /2 are the generators of SU(2). These D-terms contribute to the Higgs potential in the form 2 g 2 g 2 + g ′2 + 2 0 2 0 2 − 2 0∗ 0 −∗ 2 VD = |hu | + |hu | − |hd | − |hd | + |h+ u hd + hu hd | . 8 2
Finally the soft terms contributing to the Higgs potential is − 2 2 2 0 2 + m + (Bhu .hd + h.c.). | |h | + |h | Vsoft = m2hu |h0u |2 + |h+ hd u d d
(2.91)
(2.92)
The neutral components of the Higgs field can now acquire nonzero vevs, which in turn breaks the gauge symmetry of MSSM. The potential is required to be bounded from below. This gives the following condition (when VD → 0) 2B < 2|µ|2 + m2Hu + m2Hd .
(2.93)
The existence of a minima for the Higgs potential also requires at least one of the Higgs mass squared to be negative giving the condition 2 2 2 2 2 B > |µ| + mHu |µ | + mHd . (2.94)
After solving the minimisation equations ∂V /∂Hu0 and ∂V /∂Hd0 , the CP-even Higgs mass matrix in the basis (h0u , h0d ) turns out to be ! 2 2 −B |µ| + m hu . (2.95) Mh2 = −B |µ|2 + m2hd It is important to emphasise that there are two Higgs doublets in MSSM3 , which corresponds to 8 real degrees of freedom. Out of those 8, 3 degrees of freedom are eaten up to give masses to W ± and Z boson, leaving 5 physical degrees of freedom. These are two real scalar fields h0 (the lighter), H 0 (the heavier), a pair of charged 3
For a review see [97].
38
Higgses h± and CP odd scalar A0 . The respective masses of these Higgs bosons can be found out from the Higgs mass matrix given in eq. (2.95): 2B = 2|µ|2 + m2hu + m2hd , sin β = m2A + m2W , q 1 2 mA + m2Z ∓ (m2A − m2Z )2 + 4m2Z m2A sin2 2β , = 2
m2A = m2H ± m2h0 ,H 0
(2.96)
where m2Z = (g 2 + g ′2 )v 2 /2 and m2W = g 2 v 2 /2. In the decoupling limit when mA → ∞, the lightest Higgs boson mass at the tree level is bounded from above by [98–101] m0h ≤ mZ | cos 2β|.
(2.97)
Hence, at the tree level the lightest CP-even scalar mass, i.e., the Higgs boson mass can only be as large as the MZ mass, i.e., roughly around 90 GeV. However, there are radiative corrections to the Higgs boson mass. The one-loop diagrams involving top quark and top squark would contribute significantly to the Higgs quartic term in the potential as these particles have the largest coupling to the Higgs. If SUSY was unbroken, these corrections would have cancelled each other, which is clearly not the case. After taking into account the contributions coming from these particles the lightest CP-even Higgs mass can be written as [102–106] ! 2 m 3 e t . (2.98) m2h0 → m2h0 + 2 yt2m2t log 4π m2t tree
For a top squark mass of ∼ 1 TeV, the upper bound on the Higgs mass is roughly around m0h
2.8.3
j I(Uαi Jarlskog invariant [145]. So far we have only concentrated on neutrino oscillations in vacuum. However, for a realistic solar and atmospheric neutrino experiment, the neutrino has to traverse long distances through matter. Hence, one should include matter interactions which would modify the neutrino mass squared difference and the mixing angles resulting in a reformed neutrino oscillation probability. The modified mass squared difference and the mixing angles are q ∆m2M = ∆m2 sin2 2θ + (cos 2θ − x)2 , sin2 2θM =
sin2 2θ , sin2 2θ + (cos 2θ − x)2
(3.10)
√ where x = 2 2GF Ne E/∆m2 parametrises the matter interaction. Another important artifact of matter interactions in neutrino oscillations is when the resonant condition is reached, i.e., x ≡ cos 2θ, the modified mixing angle gets enhanced and in turn enhances the neutrino oscillation probability significantly. This phenomenon in known as the Mikheyev-Smirnov-Wolfenstein (MSW) effect [146, 147]. As mentioned in the beginning, recent experiments are enormously successful in deducing the oscillation parameters with a very high degree of precision. The global 49
fit [148, 149] of these parameters for both normal and inverted hierarchy are noted in table 3.1 Parameter
Best fit ±1σ
2σ range
3σ range
∆m221 [10−5 eV2 ] |∆m231 |[10−3 eV2 ] (NH) |∆m231 |[10−3 eV2 ] (IH) sin2 θ12 /10−1 θ12 /◦ 2 sin θ23 /10−1 (NH) θ23 /◦ 2 sin θ23 /10−1 (IH) θ23 /◦ 2 sin θ13 /10−2 (NH) θ13 /◦ 2 sin θ13 /10−2 (IH) θ13 /◦ δ/π (NH) δ/◦ δ/π (IH) δ/◦
7.60+0.19 −0.18 2.48+0.05 −0.07 2.38+0.05 −0.06 3.23±0.16 34.6±1.0 5.67+0.32 −1.24 48.9+1.8 −7.2 +0.25 5.73−0.39 49.2+1.5 −2.3 2.26±0.12 8.6+0.3 −0.2 2.29±0.12 8.7±0.2 1.41+0.55 −0.40 254+99 −72 1.48±0.31 266±56
7.26-7.99 2.35-2.59 2.26-2.48 2.92-3.57 32.7-36.7 4.14-6.23 40.0-52.1 4.35-6.21 41.3-52.0 2.02-2.50 8.2-9.1 2.05-2.52 8.2-9.1 0.0-2.0 0-360 0.0-0.09 & 0.86-2.0 0-16 & 155-360
7.11-8.18 2.30-2.65 2.20-2.54 2.78-3.75 31.8-37.8 3.93-6.43 38.8-53.3 4.03-6.40 39.4-53.1 1.90-2.62 7.9-9.3 1.93-2.65 8.0-9.4 0.0-2.0 0-360 0.0-2.0 0-360
Table 3.1: Neutrino oscillation parameters summary from the global analysis [148,149]. However, we observe that the neutrino (anti-neutrino) oscillation probability shown in eq. (3.8) and (3.9) is only sensitive to the mass squared difference between different neutrino mass eigenstates. Hence one has to resort to some other experiments to obtain the absolute mass of the neutrinos. For example, the tritium beta decay experiment [150–152] puts an upper bound on the neutrino mass close to mν ≡ q P 2 2 i=1,3 |Uei | mi < 2.2 eV at 95% CL. A more stringent limit is obtained from the neuP trinoless beta decay experiment [153–159] which gives mν ≡ i=1,3 Uei2 mi < 0.2 − 0.6 eV. Similar bound on the neutrino masses can be obtained from cosmological observations. To elaborate, assume that the neutrinos are massive and their masses are of eV order, then these neutrinos would constitute a hot DM component. These type of DM washes out small scale structures. Hence by studying density fluctuations in the CMB [160] and the large scale structure distributions of galaxies, a bound [161] on the P sum of the neutrino masses have been drawn, which is roughly around 3i=1 mν ≤ 0.22 eV. 50
The neutrino mixing angles, θ12 , θ13 and θ23 are most sensitive to three different types of neutrino experiments. θ23 can be best measured with the atmospheric neutrino experiments [162–165] and hence known as the atmospheric neutrino angle. The best fit value is close to 49.2◦ which signifies maximal mixing. θ12 is identified as the solar neutrino angle and is close to 34◦ [166, 167]. Most recently, θ13 was discovered which is close to 9◦ . This angle is referred to as the reactor neutrino angle as it is sensitive only to reactor neutrino oscillation setups [168–170]. It is important to note that the non-zero value of θ13 also gives us an opportunity to probe the extent of CP-violation which is hinting towards a large value [149]. Although, we have some knowledge about the neutrino mass scale, but we have no idea about how this mass is being generated. In the next section we will consider a few processes which can generate neutrino masses in the framework of a SUSY model.
3.3 Neutrino mass As discussed earlier, neutrinos are massless in the paradigm of SM due to the absence of right handed neutrinos. However, adding a right handed neutrino to the field content of SM and thus extending the SM would generate masses to the neutrinos. Neutrinos can have both Dirac as well as Majorana masses. Let us discuss them one by one. • Dirac masses: We have already addressed this issue in section 1.4.2. However, for completeness, let us reiterate that in order to have Dirac neutrino mass in SM, one needs to add right handed neutrinos which are singlet under the SM gauge group. The corresponding Yukawa term is yijν ν Ri φe† LLj +h.c. After EWSB, when the Higgs √ field (φ) acquires a vev, results in a Dirac neutrino mass (mD )ij = (yijν )v/ 2. To have an appropriate neutrino mass of 0.1 eV, (yijν ) should be roughly around 10−11 . • Majorana masses: The mass term for Majorana particles is noted as i 1h T ψL CMψL + h.c. , −Lm = 2
(3.11)
where ψ = (ψ1 , ...ψn )T is a vector in flavor space and M is a n × n matrix. C is T the charge conjugation operator which is defined as: ψ c = Cψ . An important point to note is that unlike the Dirac mass term, the Majorana mass [171–174] term does not remain invariant under U(1) transformations. In fact, such a mass 51
term breaks the U(1) symmetry by two units. Hence, to conserve U(1)em , only neutral particles, such as neutrinos can have Majorana masses, although it breaks lepton number by two units. Once we accept non-conservation of the lepton numbers, we find an elegant solution to generate neutrino Majorana masses through the Weinberg operator [175], defined as Lm = λij
Li Lj φφ , Λ
(3.12)
clearly this represents a dimensional 5 operator, where Li,j are the SM lepton doublets and φ is the standard Higgs doublet. Λ is the mass scale where this operator is generated. Clearly, this term breaks the lepton number by 2 units. After EWSB, i.e., when the Higgs field acquires a vev, the relevant Majorana neutrino mass term gets generated, which is λij v 2 /Λ. Let us now look at one mechanism in which this operator can be realised. Seesaw mechanism: The most popular way to realise the effect of the Weinberg operator is by adding a right handed neutrino field νR to the SM particle content for each generation. This mechanism is known as the Type-I seesaw [176–180]. The Lagrangian relevant to the neutrino mass terms looks like −Lmass ν
3 3 X 1X T e Rj + h.c. = MNij νRi CνRj + yijν Li Φν 2 i,j=1 i,j=1
(3.13)
Here MN is the 3 × 3 complex symmetric Majorana mass matrix for the νR fields. √ e ≡ iσ 2 Φ∗ . (mD )ij = yijν v/ 2, are the elements of the Dirac mass matrix and Φ
The Lagrangian in eq. (3.13) can be further written as
1 T η CMηL + h.c., 2 L
−Lmass = ν
(3.14)
where ηL =
νL (νR )c
!
The matrix M is given as 0 mD mTD MN
M =
52
.
!
(3.15)
.
(3.16)
Assuming MR ≫ mD , the matrix given in eq. (3.16) can be block diagonalised to good approximation, which yields ! m 0 light c = M , (3.17) 0 Mheavy with
mlight = −mTD .MN−1 .mD
Mheavy = MN .
(3.18)
The masses of the light neutrinos are obtained by diagonalising the matrix mlight . Neutrino mass generation via this process is known as the seesaw mechanism or more precisely Type-I seesaw. For one generation of left and right handed neutrinos, the mass of the light active neutrino turns out to be mν ∼ m2D /MN . Hence, we observe that due to a mixing, the massless neutrino acquires a non-zero Majorana mass. The appropriate active neutrino mass of 0.1 eV can be obtained by making mD small or by making MN large. It is again trivial to show that by integrating out the heavy right handed neutrino field, one gets back the Weinberg operator. We also note in passing that there are other realisations of the dimension 5 operator, defined in eq. (3.12). For example one could add Higgs/fermionic triplets under SU(2)L to generate Majorana neutrino masses. These mechanisms are known as Type-II [181–184] /Type-III seesaw [185]. However, SUSY theories are attractive as they can address the naturalness and the DM issues. Hence, we will now pay attention to some of the possible SUSY alternatives which can deal with the neutrino mass generation mechanism. We first consider the generation of Dirac neutrino masses in SUSY.
3.3.1
Dirac neutrino masses in SUSY
Dirac masses of the neutrinos can be generated in MSSM by adding a right handed b to the field contained already declared in table 2.1. The generic neutrino superfield N, term in the superpotential responsible for generating such Dirac masses looks like bi L bj H bu. WN = yijν N
(3.19)
yν being the neutrino Yukawa coupling. To fit the neutrino mass around 0.1 eV, one needs to have the neutrino Yukawa coupling roughly around 10−12 , which seems rather 53
unnatural and hard to justify from an aesthetic perspective. On the other hand, one can forbid such a term with the help of some discrete symmetry ZN and assume instead a higher dimensional term [186–190] WN = yijν
K bb b b Z Ni Lj Hu . MP
(3.20)
b = 1, Here Zb is the superfield which is invariant under the SM gauge group and ZN (Z) b = 1, ZN (Fb) = 0, where Fb being any of the chiral superfields in the MSSM. ZN (N) The higher dimensional operator suppressed by a Planck scale given in eq. (3.20) can generate a Dirac neutrino mass of the form (mν )ij =
yijν K AZ vu . MP
(3.21)
√ b If AZ ∼ FZ ∼ Where AZ is the vev of the scalar component of the superfield Z. p m3/2 MP ∼ 1011 GeV, then an appropriately small Dirac mass can be obtained without assuming very small neutrino Yukawa coupling. In fact yijν ∼ ye ∼ 10−6 is enough to fit the neutrino mass.
3.3.2
Majorana neutrino masses in SUSY
Majorana neutrino masses in a SUSY theory can be generate either by seesaw mechanism or by invoking RPV. We will now discuss these concepts categorically. • Seesaw mechanism : The idea of generating neutrino Majorana masses in the framework of SUSY follows the same procedure as discussed previously for SM. Only the right handed neutrino fields νRi -s are to be replaced with right handed neutrino superfields bi -s. The neutrino masses turns out to be exactly equal to the masses shown in N eq. (??). • Bilinear RPV at tree level: RPV is an intrinsic way to generate neutrino masses both at the tree level as well as at the one loop level [77, 80, 191–198]. Let us discuss the consequences of bilinear RPV violation first. In addition to the MSSM superpotential described in eq. (2.80) the bilinear RPV term would also include the following term B b b W✚ Rp = µi Hu Li .
(3.22)
Once R-parity is violated, there exists no conserved quantum number that distinguishes between the lepton supermultiplets and the down-type Higgs supermultiplet. In such a scenario, it is convenient to denote the four supermultiplets 54
bα (α = 0, 1, 2, 3) with L b0 ≡ H b d . The inclusion of this RPV vioby one symbol L lating term in the superpotential shown in eq. (3.22) would result in an extended soft SUSY breaking sector compared to MSSM. Thus, one must also add new ei L ej e e i ej A (trilinear term: Aαβm L α β Em ) and B (bilinear term: Bα Hu Lα ) terms in the scalar sector.
The bilinear RPV violating term also has some interesting consequences. the left handed sneutrinos can acquire vev, resulting in a mixing between the charged leptons and the charginos and similarly between neutrinos and the neutralinos. This also results in an extended neutralino and chargino mass matrix as opposed to MSSM. The lightest eigenvalue of the extended neutralino mass matrix would then correspond to the Majorana mass of the neutrinos. The neutralino mass e W f, H e u , νi ) takes the shape [199–202] matrix in the basis (B, M1 0 mZ sW sβ −mZ cβ sW 0 0 M2 −mZ cW sβ mZ cβ cW 0 MN = 0 −µ −µi , (3.23) mZ sβ sW −mZ cW sβ −µ 0 0 −mZ cβ sW mZ cβ cW 0 0 −µi 0 0
p where tan β = vu /vα and vα = vd2 + vi2 , vi designates sneutrino vevs. The lightest eigenvalue of the neutralino mass matrix given in eq. (3.23) are the Majorana mass of the neutrinos. Assuming the mixing between the neutrinos and the neutral gauginos to be small, one finds m2Z µi µj cos2 βmγ , (mν )ij = µ(m2Z mγ sin 2β − M1 M2 µ)
(3.24)
where mγ = M1 cos2 θ + M2 sin2 θ. Assuming all the masses are of the same order, i.e., m, e the neutrino Majorana masses can be further simplified to (mν )ij ∼ µi µj cos2 β/m. e Although, only one neutrino acquires a Majorana mass at the tree level by this mechanism. • Bilinear RPV at one loop level: In addition to the tree level mass, bilinear RPV can also generate neutrino Majorana masses at one loop [203–208]. At first, we look into the BB diagram shown in the left panel of fig. 3.1, where the blobs denote the mixing between the sneutrinos and the Higgs bosons and are proportional to the soft SUSY breaking B terms. The cross in the internal line refers to the Majorana masses of the neutralinos. Another important artifact of the mass generated via this loop is, the 55
h, H, A
h, H, A Bj
Bi νei νi
χe α
νej
νej
µi νi
νj
Bj
χe α
χe β
νj
Figure 3.1: Neutrino Majorana mass generation at the one loop through bilinear RPV couplings. two B insertions in the scalar line generates a splitting between the two sneutrino mass eigenstates [209]. Therefore, the contribution of the BB loop vanishes once the sneutrino mass splitting becomes zero. Assuming all the mass scales involved in this diagram are equal (m), e then one can approximately write [203] (mν )BB ij ≃
g2 Bi Bj . 2 2 64π cos β m e3
(3.25)
We observe that the BB loop is suppressed not only by the RPV term Bi but also by the generic loop factor and the heavy scalar masses running in the loop. Another important diagram which could generate neutrino Majorana masses at one loop is shown in the right panel of fig. 3.1. The blob on the external fermion line signifies mixing between the neutrinos and the neutralinos. Similar to the BB loop, the blob on the scalar line demonstrates mixing between the sneutrinos and the Higgs boson. Again assuming all the masses are at the weak scale and equal to m, e the approximate expression for the neutrino Majorana mass turns out to be [203] (mν )µB ij ≃
µi Bj + µj Bi g2 . 2 64π cos β m e2
(3.26)
• Trilinear RPV at one loop level:
The neutrino Majorana mass also receives contribution through trilinear RPV couplings (λ′ , λ) at the one loop level. These terms originate from the generic RPV superpotential shown in eq. (2.81). The bounds on λ′ couplings are much less stringent [210–220] and in addition diagrams involving these coupling come with an addition colour factor. Hence, their contribution to the neutrino masses are more profound compared to the same, coming from λ couplings. The relevant diagram is shown in fig. 3.2. The contribution to neutrino masses can be approximated as [203] X 3 md md ′ ′ (mν )λij λ ≃ λ′ilk λ′jkl l k . (3.27) 2 8π m e l,k 56
dekR
dekL dlR
dlL νi
λ′jkl
λ′ilk
νj
Figure 3.2: Trilinear RPV couplings generating neutrino masses at the one loop level. dekL
µi νi
dkL
dkR λ′jkk
χe α
νj
Figure 3.3: Bilinear+trilinear RPV couplings generating neutrino masses at the one loop level. m e signifies the squark masses propagating in the loop. The blob in the scalar line indicates mixing between the left-handed and the right-squarks which are essentially controlled by the A terms. The mass insertion on the internal quark propagator is denoted by a cross. • Bilinear+trilinear RPV at one loop level: Finally we note that both bi and trilinear RPV coupling might also contribute to neutrino mass generation. The relevant diagram is shown in fig. 3.3. The approximate expression for the neutrino mass can be written as [203] ′ (mν )µλ ij
=
X k
µi λ′jkk + µj λ′ikk 3 gmdk . 16π 2 m e
(3.28)
In addition to generating neutrino masses, RPV violation has other serious implications. In ref. [221], it was shown for the first time that in a SUSY scenario where R-parity is explicitly broken through a bilinear term in the lepton and Higgs superfields would lead to interactions similar to those which are proportional to the trilinear RPV couplings. From the perspective of collider physics, characteristic decay signatures of the lightest neutralino to a final state comprising of τ W and ντ Z were discussed in this context. In addition, atmospheric neutrino mixing angle dictates maximal mixing in the neutrino sector. Hence, a comparable number of muons or taus are expected from the decays of the lightest neutralino. This is the first example [222] where the physics of the neutrino sector can be probed at the colliders. In the presence of only trilinear 57
RPV terms in the superpotential, the lightest neutralino can have various three body decay modes which can be generically described by χ e0 → νf f , lf1 f 2 . Detailed collider signatures in models with RPV have been studied extensively in the literature and can be found in [223–225]. Bilinear R-parity violation introduces a mixing in the scalar sector between the Higgs bosons and the sleptons, which leads to Rp decay modes of these scalar as h, H → χ e0 ν, χ e+ ℓ− for neutral CP-even Higgs bosons [226–228] or τe1 → ℓ− ν, qq′ for the lightest stau [229]. Finally the top squark, which is now being searched at the present experiments extensively, would also have interesting and rather clean decay channels [230–232], for example e t → bℓ+ .
3.4 Dark Matter
Over the years, a wide variety of evidences have been accumulated in support of the existence of DM. For example, galactic rotation curves [233], the gravitational lensing of distant galaxies by foreground structure [234] and bullet clusters [235] are considered as concrete testimonies in favour of DM. An important observable in this context is the relic density of the DM, defined in terms of the ratio of its density to the critical matter density as ΩDM =
ρDM , ρc
(3.29)
where ρc = 3H 2/8πGN , H being the Hubble expansion rate and GN is the Newton’s gravitational constant. Very recently the PLANCK experiment [236] measured the relic abundance with very high degree of precision, i.e., ΩDM h2 = 0.1199 ± 0.0027,
(3.30)
h = H/100(km/s/Mpc)−1 , where 1 pc = 3.26 light year and h takes the value 0.705 ± 0.013. However, the particle identity and the nature of the DM remains unknown to date2 . For example, the DM can be ‘hot’, ‘warm’ or ‘cold’. ‘Hot’ DM (HDM) is composed of particles which are massless or has very small mass (e.g, neutrinos). Hence, the move at nearly the speed of light. ‘Cold’ DM (CDM) particles on the other hand are massive and move slowly compared to the speed of light. They also interact very weakly with the visible matter. ‘Warm’ DM (WDM) particles have properties which are intermediate between CDM and HDM particles. Out of these candidates CDM is the most popular and widely accepted. 2
For reviews on DM see [237–239].
58
Among several candidates for a CDM, the most popular is the weakly interacting massive particles (WIMPs). The WIMP class of particles are those which are considered to be created thermally in the early universe. If thermal equilibrium was to maintained, then the forward and the backward processes of WIMP annihilation and creation would have been same. However, as the universe cools down with decreasing temperature, the annihilation rate dominates over the creation rate of WIMPs. As a result, the number density of the WIMPs would have fallen exponentially and resulted in a much smaller relic density as opposed to the observed value. However, as the density decreased, the annihilation rate became small compared to the cosmological expansion, and the WIMP experienced a ‘thermal freeze out’. Therefore, one would be left with a substantial number of WIMPs today. Finally, the annihilation cross section for any thermally created particles turns out to be just what predicted for particles with electroweak interactions. SUSY models with R-parity conservation provide just the perfect nominee, in the form of the lightest neutralino. Let us now discuss the DM candidates in a SUSY framework.
3.5 DM candidates in SUSY In the framework of R-parity conserving SUSY, the superpartners need to be created or destroyed in pairs, leading to the lightest supersymmetric particle (LSP) to be absolutely stable. Hence, the LSP turns out to be a natural choice for a DM candidate. The identity of the LSP depends on the spectrum of the particular SUSY model, which in turn is determined by the details of how SUSY is broken. The list of potential LSPs which could become a plausible DM candidate is somewhat limited in SUSY and particularly in MSSM. The only electrically neutral and colourless superpartners in MSSM are the four neutralinos (e χ01 ) [240–246], three sneutrinos3 (e νi ) [247–250] 4 e [251–256]. Amongst them, the lightest neutralino is a very and the gravitino (G) attractive and thoroughly studied candidate for DM. In MSSM, the lightest neutralino mass eigenstate is actually an admixture of the four gauge eigenstates, i.e., the bino, the wino and the two neutral higgsinos. The lightest neutralino, in principle, can annihilate through a wide variety of Feynman diagrams. The resulting relic density depends on a 3
Sneutrino DM is ruled out in the MSSM as because the sneutrinos interact via Z boson exchange and would have been detected by now if it makes up the dark matter. Although, extended models with right handed sneutrinos can reopen the possibility of sneutrino DM. 4 For example, gravitino DM is a possibility in SUSY models in which the scale of SUSY breaking is very low, i.e., around 100 TeV. In such models, gravitinos are very light with mass around tens of eV. As DM, gravitino is sometimes called the super-WIMP because its interaction strength is much weaker than that of other SUSY dark matter candidates.
59
large number of model parameters, for example the relevant couplings and the elements of the mixing matrices. Over most of the MSSM parameter space, the relic abundance of neutralinos are predicted to be in excess of the observed dark matter density. To avoid such a disaster, one is forced to consider the regions which leads to efficient neutralino annihilation in the early universe. We note down the following scenarios which are efficient to produce a phenomenologically viable density of neutralino dark matter. • If the lightest neutralino has a significant higgsino or wino fraction then it can annihilate very efficiently and provide the appropriate relic abundance. • If the mass of the lightest neutralino is near a resonance, for example the CP-odd Higgs pole, it can annihilate efficiently even in the presence of relatively small couplings. • If the lightest neutralino happens to be slightly lighter than another superpartner, i.e., NLSP, such as the lightest stau, co-annihilation between these two states can very efficiently deplete the dark matter abundance. Up until now we have only mentioned CDM candidate in the form of lightest neutralinos. However, at present, we do not seem to have any solid direct evidence for such a DM candidate. Even worse, recent experiments [257–268] are ruling out more and more portions of the allowed parameter space. As a result, a vast number of CDM candidates have started being in trouble. In addition, CDM is also in tension with some of the recent astrophysical and observational findings. In this light a SUSY model with a warm dark matter candidate is very much welcome. We will now discuss a few advantages of a WDM candidate as opposed to a CDM5 .
3.5.1
Missing satellite problem
Structure formation at relatively small scales mainly differentiates between a CDM and a WDM candidate. Smaller galaxies, and in particular, the so called dwarf galaxies are considered to be the key to distinguish between these two types of DM. These differences arise only at scales of about 0.1 Mpc, where WDM being relativistic, typically washes out structures. In comparison, CDM only washes out structures of roughly a tenth of that size or below. As a result, WDM simulations tend to yield a smaller number of dwarf satellites, which seems to coincide with the observations as opposed to CDM simulations. This is often quoted in the literature as the missing satellite 5
For a review on WDM see [269, 270].
60
problem [271–275]. It is important to note that structure formation of galaxies at the large scale cannot distinguish between these two types of DM.
3.5.2
Core-cusp problem
The core-cusp problem remains one of the unsolved discrepancies between the observations and the predictions of the CDM simulations. For example, if there exists only CDM then simulations predict that the center of the Milky way would have a higher DM density [276–281]. However, as far as the observational evidences are concerned, it seems that the centre of the galaxies have no cusp like structure at all, rather the DM distribution is distributed equally all throughout the galaxy. This problem can be somewhat ameliorated by assuming a tepid DM candidate instead of a CDM candidate. Possible candidates for WDMs are sterile neutrinos, axinos and light gravitinos (with mass in the keV range). In this thesis we will only discuss about sterile neutrino as it has direct relevance to the analysis and the discussion presented later.
3.6 Sterile neutrino DM Sterile neutrinos are one of the most attractive examples of a WDM candidate. Typically the mass of this particular DM particle can vary from a few keV to a few hundred MeV. The most model independent lower limit on sterile neutrino DM candidate stems from the Tremaine-Gunn bound [282, 283], which can be obtained by demanding that the maximal (Fermi) velocity of the degenerate fermionic gas in the dwarf spheroidal galaxies is less than the escape velocity leads to a bound on the sterile neutrino mass, > which is roughly around MN ∼ 0.4 keV. It is important to note that a keV mass of the DM particle does not ensure it to be a WDM. The decisive point is the velocity profile of the particular DM particle. On the other hand, the exact velocity profile depends heavily on the production mechanism. The most generic DM production mechanism is the so-called thermal freeze-out. However, for this production mechanism to work, the interaction strength is required to be comparable to the weak interactions, which is rather difficult for sterile neutrinos. We shall now discuss two alternative production mechanisms for sterile neutrinos to be a valid WDM candidate.
3.6.1
The Dodelson-Widrow mechanism
In the presence of negligible lepton asymmetry, sterile neutrinos can be produced through mixing with the active neutrinos, as proposed by Dodelson and Widrow [284].
61
In such a scenario, the production of sterile neutrinos depend only on the mass of the sterile neutrinos and its mixing with the active neutrinos [285–290]. We start with the simplified Boltzmann equation for the sterile neutrinos which essentially gives the evolution of the phase space density distribution function fS (E, T ) as ∂f S = ΓS fSeq (E, T ), (3.31) −HT ∂T E/T where
1 2 sin 2θm dα G2F ET 4 , (3.32) 4 fSeq (E, T ) is the Fermi-Dirac distribution that heavy neutrinos would have if they were in thermal equilibrium. ΓS (T ) designates the interaction of sterile neutrino interactions with active neutrinos. dα = 1.13 for sterile neutrino mixing with να = νe and dα = 0.79 for να = νµ,τ . θm parametrises the active-sterile mixing which is related to the vacuum angle by ΓS (T ) ≃
sin2 2θm =
∆2 (p) sin2 2θ h i2 , 2 2 ∆ (p) sin 2θ + ∆(p) cos 2θ − V (p, T )
(3.33)
∆p ≡ δm2 /2p and V (p, T ) reflects the total weak potential experienced by an active neutrino να and written approximately as T 4 L T 3 − Bp . (3.34) V (p, T ) ≃ (40.2 eV) 10−2 GeV GeV In the absence of any primordial lepton asymmetry, i.e., L ∼ 0, the total weak potential given in eq. (3.34) reduces to a simpler form, which is the case for DW mechanism. For neutrino of flavor e, the coefficient B ≡ 10.79 eV. Solving eq. (3.31) yields a rather simpler form of " # π Mpl dα G2F mS sin2 2θ(GeV)2 fS √ = . (3.35) fSeq 48 1.66 2Bg∗ Progressing further and using the fact that fSeq and fνeq have the same functional form, where fνeq is the Fermi-Dirac distribution for active neutrinos in thermal equilibrium, one obtains mS fSeq ΩS = . (3.36) Ων mν fνeq Again, using the relation mν /Ων ≃ 92h2 eV, g∗ = 10.75 and the reduced Planck Mass as Mpl = 2.435 × 1018 GeV one finds [291] 2 sin2 2θ m S . (3.37) ΩS h2 ≃ 0.3 100 keV 10−10 62
Despite its simplicity, the DW mechanism is very tightly constrained from the strong X-ray bounds, discussed later. Hence, one must look for other production processes of the sterile neutrino DM, for example the Shi-Fuller mechanism [292]. However, one must note that DW contribution to the DM relic density is unavoidable as long as active-sterile mixing is not completely switched off. As a result, this would always contribute to the DM relic abundance, albeit small.
3.6.2
The Shi-Fuller mechanism
The production mechanism proposed by Dodelson and Widrow is altered in the presence of a primordial lepton asymmetry of the universe. In such a scenario, the production of sterile neutrinos can be enhanced by MSW effect. Shi-Fuller deduced [292] that the MSW resonance makes the production more efficient for small active-sterile mixing angles. As a result, regions with smaller mixing angles opens up which are less constrained by the X-ray data. In addition, the momentum distribution of non-thermal sterile neutrinos produced through such a mechanism are cooler compared to sterile neutrinos produced via DW mechanism. The MSW effect reaches it optimum when the resonant condition is reached (see eq. (3.33)), i.e., ∆(p) cos 2θ ≃ V (p, T ).
(3.38)
This translates into the fact that for Shi-Fuller mechanism to work, the lepton asymmetry should be fairly large, i.e., L ∼ O(10−3 ) or so. It was shown [293–300] that active-sterile neutrino oscillations can themselves create a cosmological lepton number of this magnitude assuming that the number of sterile neutrinos is negligible to start with. This ‘cool’ dark matter is also consistent with the structure formation evidences as a too warm spectrum may washout structures larger than dwarf satellites, which is certainly not observed experimentally.
3.7 Detection of WDM The primary decay mode of the sterile neutrinos in the keV mass range is νS → 3ν, which is a tree level process and essentially contributes to unobservable missing energy in the final state. In addition to this, there exists a one-loop diagram that results in a νa + γ final state. The partial decay width for photon+neutrino final state is m 5 1 S 2 sin θ . (3.39) ΓνS →γνa = 21 1.8 × 10 sec keV 63
The lifetime of the sterile neutrino DM turns out to be much larger than the age of the universe. If sterile neutrinos constitute the whole DM of the universe, then a small decay width can make them observable via the photons produced in the radiative decay. This offers the best opportunity to detect these particles. Furthermore, as it is a two-body decay, the resulting photon would have the energy Eγ =
mS . 2
(3.40)
This decay results in a photon line and can be observed with photon energy of a few keV using an X-ray telescope. Although the same signal could have other astrophysical origins, hence, it is of utmost importance to distinguish the background from the signal, which is a vast area of research by itself. In fig. 3.4 we show the present X-ray bounds in the mass-mixing plane of sterile neutrinos. For example, the region in red stripes is ruled out from the data coming from cosmic X-ray background (CXB) [301]. Constraints from M31 as observed by the Chandra telescope rules out the region in grey [302]. In addition, XMM-Newton observations from Coma and Virgo clusters rule out the narrow patch in green [303]. The horizontal yellow band is ruled out by the Tremaine Gunn bound [282, 283] Finally, the light blue line refers to a 100% relic density of the sterile neutrino dark matter, produced via DW mechanism. The light blue region above this line leads to over abundance of the sterile neutrino WDM. It is clear from the same figure that a vast region in the small active-sterile mixing domain is still allowed. We have seen thus far that SUSY is an attractive choice which can not only address the naturalness problem but can also generate neutrino masses and mixing and provide a suitable DM candidate. Moreover, we have also seen that the present astrophysical results indicate that the potential DM candidate should be rather tepid compared to the usual CDM. However, the present LHC run has found no direct evidence for SUSY particles. This has put stringent limits on the superpartner masses. In this light, models with U(1)R symmetry and Dirac gluinos/gauginos are well motivated as they can lower the bound on the first two generation squark masses considerably. We cultivate such a model giving special emphasis to the Higgs sector, generation of neutrino masses and mixing and keV dark matter in the form of a sterile neutrino. Finally, we also perform a detailed collider study to show that such models can have novel and interesting collider signatures which can distinguish it from other models.
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Diffuse X-ray background Cluster X-ray
101
CXB Milky Way
MR N (keV)
M31
DW
100
Tremaine Gunn 10-1 10-12
10-11
10-10
10-9
2
10-8
10-7
sin 2θ
Figure 3.4: Constraint on the mass-mixing plane for sterile neutrinos as depicted. The yellow region is ruled out from the Tremaine Gunn bound [282, 283]. Cosmic X-ray background (CXB) rules out the region in red stripes [301]. Constraints from M31, observed by Chandra telescope rules out the region in grey [302]. The blue region is ruled out from the diffuse X-ray background observations. XMM-Newton observations from Coma and Virgo clusters rule out the small patch in green [303]. Finally, the light blue line represents the 100% relic density of the sterile neutrino dark matter, produced via DW mechanism. The light blue region above this line leads to over abundance of the sterile neutrino WDM.
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Chapter 4 U (1)R-lepton number model with a right handed neutrino superfield 4.1 Introduction The observation of a new neutral boson, widely believed to be the first elementary scalar boson of nature, by the ATLAS and CMS experimental collaborations at the CERN LHC is perhaps the most important discovery in high energy physics in recent times [21,22]. The mass of this particle is measured to be ∼ 125 GeV. Obviously, more data and analysis can confirm whether this is the Higgs boson of the SM or not. On the other hand, supersymmetric particle searches by ATLAS and CMS for pp collision √ at center-of-mass energy, s = 7 and 8 TeV, has observed no significant excess over the expected SM background. This has set stringent limits on the superparticle masses (particularly on the masses of squarks and gluinos) for a number of supersymmetric models/scenarios [107, 108]. At the same time, we have very strong experimental evidences in favor of neutrino oscillation [148, 149]. These results have firmly established the existence of massive neutrinos and non-trivial mixing pattern in the neutrino sector (including the recent discovery [169, 170] of a small but non-zero mixing angle θ13 ). Non-vanishing neutrino masses and mixing are very important indications of new physics. However, the mechanism which generates neutrino masses remains a mystery to date. Naturally, the neutrino sector is a testing ground for various models going beyond the SM. There is also compelling evidence for the existence of DM and cosmological observations have measured the relic density of DM with a high degree of precision [236]. Nevertheless, the identity of the DM remains unknown to date and the potential candidates are, for example, the lightest neutralino in an R-parity conserving supersym-
66
metric theory, the gravitino, the axino, the axion and the keV sterile neutrino [246]. On the theoretical frontier, SUSY is a very popular choice for new physics. The MSSM with RPV is an intrinsically supersymmetric way of generating observed neutrino masses and mixing pattern. There are extensive studies involving MSSM with R-parity violation on neutrino masses and mixing, under various assumptions, both at the tree and the loop level [80]. It is, therefore, tempting to see whether there exist supersymmetric models which can naturally explain the observed mass of the new scalar boson at ∼ 125 GeV, relax the strong constraints on SUSY particle masses coming from the LHC, provide a suitable dark matter candidate and at the same time produce neutrino masses and mixing consistent with current data. In this direction a class of very interesting models are those with a global continuous U(1)R symmetry [304–312]. Models with R-symmetry have Dirac gauginos instead of Majorana gauginos and the bounds on the first two generation squarks are somewhat relaxed compared to MSSM because of the presence of a Dirac gluino [70,313–366]. Flavor and CP violating constraints are also suppressed in this class of models [323]. Let us mention at this stage that extensive studies have been performed with Dirac gaugino masses in the R-symmetric limit. In order to have a Dirac gaugino mass, one needs to incorporate a singlet superfield Sb in the adjoint representation of U(1)Y , an b SU(2)L triplet superfield Tb (with zero hypercharge), and an SU(3)C octet superfield O. The Dirac gaugino masses have also been motivated from “supersoft” supersymmetry breaking [317]. Another notable feature of these models are, the absence of trilinear scalar interactions (A terms) and also the µ term, when the R-symmetry is preserved. However one can reintroduce these terms by considering the breaking of R-symmetry [337]. This chapter is organised as follows. In section 4.2 we introduce the U(1)R -symmetric model with a right handed neutrino superfield. The R-charges are identified with the lepton number in such a manner that left chiral sneutrinos do not carry any lepton number. Hence, the sneutrino vevs can be large and are not constrained by the Majorana mass of the neutrinos. In such a scenario, the left chiral sneutrino can play the role of a down type Higgs field. The main motivation of augmenting a right handed neutrino superfield is to have a tree level mass for the neutrinos. In the later chapters we will see that such an extension has enormous importance and novelty as we will discuss later. In section 4.3 the scalar sector concerning the Higgs boson has been discussed in great detail. Apart from the mass matrices, many interesting sum rules are also obtained. Later on the fermionic sector has been deliberated in section 4.4, with special importance given to neutrino masses in the R-symmetry conserving as well as
67
violating schemes.
4.2 U (1)R -lepton number model with a right handed neutrino We consider a minimal extension of an R-symmetric model, first discussed in [304,306], by extending the field content with a single right handed neutrino superfield [308–312]. bi , H bu, H bd, U b c, D b c, L bi , E bc , two inert doublet Along with the MSSM superfields, Q i i i b b superfields Ru and Rd with opposite hypercharge are considered in addition to the b c . These two doublets R bu and R bd carry non zero right handed neutrino superfield N R-charges and therefore, in order to avoid spontaneous R-breaking and the emergence bu and R bd do not receive any nonzero vevs and of R-axions, the scalar components of R as a result they are coined as inert doublets. The SM gauge quantum numbers and U(1)R charges of the chiral superfields are shown in table 4.1. Superfields b Q bic U bc D i bi L bc E i b Hu bd H bu R bd R Sb Tb b O bc N
SU(3)C , SU(2)L , U(1)Y (3, 2, 31 ) (3, 1, − 34 ) (3, 1, 23 ) (1, 2, −1) (1, 1, 2) (1, 2, 1) (1, 2, −1) (1, 2, 1) (1, 2, −1) (1, 1, 0) (1, 3, 0) (8, 1, 0) (1, 1, 0)
U(1)R 1 1 1 0 2 0 0 2 2 0 0 0 2
Table 4.1: SM gauge quantum numbers and U(1)R charge assignments of the chiral superfields [308].
R-symmetry prohibits the gauginos to have Majorana mass term and trilinear scalar interactions (A-terms) are absent in a U(1)R invariant scenario. However, the gauginos can acquire Dirac masses. We reiterate that in order to have Dirac gaugino masses 68
one needs to include chiral superfields in the adjoint representations of the standard b an SU(2)L triplet Tb and an octet O b under model gauge group. Namely a singlet S, SU(3)c . These chiral superfields are essential to provide Dirac masses to the bino, wino and gluinos respectively. In this model the lepton numbers have been identified with the (negative) of R-charges such that the lepton number of the SM fermions are the usual ones whereas the superpartners of the SM fermions carry non-standard lepton numbers. With such lepton number assignments this R-symmetric model is also lepton number conserving [306–312]. Thus, one cannot generate neutrino Majorana masses which violate lepton number by two units. The R-charge assignments given in table 4.1 have an interesting consequence for the left chiral sneutrinos which now do not carry any lepton number. Hence, although in this model left chiral sneutrinos get non-zero vev in general, the latter do not get constrained from neutrino Majorana masses. The generic superpotential carrying an R-charge of two units can be written as buQ bi U b c + µu H buR bd + fi L bi H buN b c + λS SbH buR bd + 2λT H b u TbR bd − MR N b c Sb W = yiju H j bu H b d + λ′ SbR bu H b d + λijk L bi L bj E b c + λ′ L bi Q bj D b c + 2λ′ R bu TbH bd + yd H bdQ bi D bc + µd R S
k
ijk
bdL bi E b c + λN N b cH buH bd. + yije H j
k
T
ij
j
(4.1)
b c SbS, b ηN b c, ξN b c T r(TbTb) and For simplicity, in this work we have omitted the terms κN b c T r(O bO) b from the superpotential. As long as η ∼ M 2 ζN SU SY , κ, ξ, ζ ∼ 1 and the < vevs of the scalar components of Sb and Tb are very small ( ∼ O(10−2) GeV, as we shall consider later), we do not expect any significant changes in the analysis and the results presented in this work. In order to have a realistic model one should also include supersymmetry breaking terms, which are the scalar and the gaugino mass terms. The Lagrangian containing the Dirac gaugino masses can be written as [337, 339] Z √ √ W′ √ Dirac b + h.c., Lgaugino = d2 θ α [ 2κ1 W1α Sb + 2 2κ2 tr(W2α Tb) + 2 2κ3 tr(W3α O)] Λ (4.2) where Wα′ = λα + θα D ′ is a spurion superfield parametrising D-type supersymmetry breaking and Wiα contains the gauginos of the MSSM vector superfields. This results in Dirac gaugino masses as D ′ acquires vev and are given by MiD = κi
hD ′ i , Λ
(4.3)
where Λ denotes the scale of SUSY breaking mediation and κi are order one coefficients. It is worthwhile to note that these Dirac gaugino mass terms have been dubbed as ‘supersoft’ terms because the Majorana gaugino mass terms generate logarithmic 69
divergence to the scalar masses whereas in ref. [317], it was shown that the purely scalar loop, obtained from the adjoint superfields cancels this logarithmic divergence in the case of Dirac gauginos. Hence it is not unnatural to consider the Dirac gaugino masses to be rather large. In models where soft SUSY breaking terms are generated only by D-type SUSY breaking, one gets a vanishing D-term contribution to the Higgs quartic coupling at tree level. This is a disaster since without such a contribution Higgs boson mass cannot come even near to the observed value of 125 GeV. However, in our situation the problem can be circumvented by assuming F -type SUSY breaking. Detailed calculation in this regard is relegated to Appendix A. The R-conserving but soft supersymmetry b where breaking terms in the scalar sector are generated from a spurion superfield X, b = x + θ2 FX such that R[X] b = 2 and hxi = 0, hFX i = X 6 0. The non-zero vev of FX generates the scalar soft terms and the corresponding potential is given by 2 e† e e† L e Vsoft = m2Hu Hu† Hu + m2Ru Ru† Ru + m2Hd Hd† Hd + m2Rd Rd† Rd + m2Lei L ei lRi lRi i i + mR
e c† N e c + m2 S † S + 2m2 tr(T † T ) + 2m2 tr(O †O) + (BµHu Hd + h.c.) + MN2 N S T O 1 ei + h.c.) + (tS S + h.c.) + bS (S 2 + h.c.) + bT (tr(T T ) + h.c.) − (BµiL Hu L 2 + BO (tr(OO) + h.c.). (4.4)
It has been argued in ref. [346] that the dangerous tS parameter in scenarios with Dirac gaugino masses originating from the tadpole term tS S are suppressed and that is what we shall consider henceforth. Thus, this term does not introduce quadratic divergence e c +h.c.) is leading to phenomenological disaster. Note that the tadpole term (tNe c N absent from the scalar potential because of R-symmetry. We also neglect the U(1)R symmetric scalar trilinear terms from eq. (4.4) as because they are assumed to be suppressed by the factor msusy /Λ, where msusy ∼ 1 TeV and Λ ≫ msusy . ei implies that all the three left handed The presence of the bilinear terms BµiL Hu L sneutrinos can acquire non-zero vev’s. As emphasised earlier in the introduction that the sneutrino vevs (vi ) can be large since they are not constrained by neutrino Majorana bu and H b d by choosing masses. In such a situation, one can integrate out the superfields R bu H bd a large value of µd (µ2 ≫ m2 ), where µd is the coefficient of the bilinear term µd R d
e L
in the superpotential and m2Le is the soft mass squared of the left handed sleptons, which simplifies the superpotential and the soft supersymmetry breaking terms given in eqns. (4.1) and (4.4), respectively. This again allows us to rotate the sneutrino vevs in such a way that only one of the left handed sneutrinos get a non-zero vev and one must keep in mind that the physics is independent of this basis choice. This also ensures that the particular sneutrino plays the role of a down type Higgs field. As a result, the masses of the charged lepton and down type quarks arise because of the 70
non-zero vev of the left-handed sneutrino. Such a rotation can be defined as X ba + bb , bi = vi L eib L L va b
(4.5)
ba is the combination of the L bi superfields whose neutral scalar component gets where L bb , a non-zero vev va , a = 1(e) whereas the other sneutrino fields corresponding to L b = 2, 3(µ, τ ) do not acquire any vev, that is to say vb = 0 for b = 2, 3(µ, τ ). Here pP 2 b va ≡ i vi and the superfield La is defined as X bi . ba = 1 vi L (4.6) L va i
Here eib are the elements of the rotation matrix which connect the two different bases such that the vectors {ei2 } and {ei3 } are orthogonal to each other and normalised to unity. In addition, they are also orthogonal to the vector {vi }. ba H buN bc + bi H buN b c in the superpotential transforms into fi vi L In this basis the term fi L va bb H buN b c . Using the freedom to choose fi such that fi eib = 0, the modified neutrino fi eib L ba H buN b c , where f ≡ fi vi /va . Yukawa coupling term in the superpotential looks like f L b c couples only Therefore, in this rotated basis the right handed neutrino superfield N ba , a = 1(e), with a coupling strength f . Note that in this single sneutrino vev with L basis the soft supersymmetry breaking bilinear term in the scalar potential involving b u field appears as ǫij Bµa H i L ej the doublet slepton field and the H L u a + h.c. [a = 1(e)], where {i, j} are SU(2) indices with ǫ12 = −ǫ21 = 1. Furthermore, the trilinear RPV interactions in the superpotential looks like X vi ejb 1 ba Q bj D bc bi L bj E b c + λ′ L bi Q bj D bc = ba L bb E bc + λ′ vi L λijk L λijk L k k ijk k k ijk 2 v v a a b=2,3 X 1 bb L bc E bkc + bb Q bj D b kc . (4.7) + (eib ejc λijk )L eib λ′ijk L 2 b=2,3 From eq. (4.7) we can identify the Yukawa couplings and the trilinear R-parity violating couplings in the single sneutrino vev basis as X vi ejb X vi l d fbk = λijk , fjk = λ′ijk , (4.8) v v a a ij i λbck =
X
eib ejc λijk ,
λ′bjk =
ij
X
eib λ′ijk .
(4.9)
i
In view of the above discussion it is easy to see that the superpotential include the following terms buR bd + f L ba H buN b c + λS SbH buR bd + 2λT H b u TbR bd buQ bi U b c + µu H W = yiju H j b c Sb + W diag . − MR N 71
(4.10)
bb , b = 2, 3) and down type Yukawa couplings In the basis where the charged lepton (L are diagonal, W diag is given as X X X 1 d b b ′ b ′c e ba L b′ E b ′c + b′ L b′ b′c W diag = fbl L f L Q D + λ23k L a b b k k k 2 3 Ek 2 b=2,3 k=1,2,3 k=1,2,3 X e b′ Q b′ b ′c + λ′bjk L (4.11) b j Dk . j,k=1,2,3;b=2,3
b′ , b = 2, 3) and quark superfields denotes that they are Here the prime on the lepton (L b 1 e λ e′ are the trilinear R-parity violating couplings in in the mass-eigenstate basis and λ, that basis. In our subsequent analysis we shall work in this mass eigenstate basis and e λ e′ → λ, λ′ . Remember that we are also remove the prime from the fields along with λ, working in a basis where only one left-handed sneutrino (corresponding to flavor a) gets a vev. To reiterate, we observe that these trilinear RPV operators are consistent with the R-symmetric superpotential. Nevertheless, this superpotential conserves lepton number because of the identification of lepton number with R-charges and hence the lepton number violating processes do not constrain these trilinear couplings. The flavor structures of these trilinear R-parity violating couplings in this model will have important implications in the context of neutrino masses and other phenomenology as we shall discuss later. In this rotated basis the soft supersymmetry breaking terms look like X 2 e c† e c 2 e† e e† L ea + e† L e m2Le L Vsof t = m2Hu Hu† Hu + m2Rd Rd† Rd + m2Lea L e lRi lRi a b b + MN N N + mR b
i
b=2,3
ea + h.c.) + (tS S + h.c.) + +m2S S † S + 2m2T tr(T † T ) + 2m2O tr(O † O) − (BµL Hu L 1 bS (S 2 + h.c.) + bT (tr(T T ) + h.c.) + BO (tr(OO) + h.c.). (4.12) + 2 With this description of the theoretical framework let us now explore the scalar and the fermionic sectors in some detail.
4.3 The scalar sector The scalar potential comprises of four different terms. V
= VF + VD + Vsoft + Vone−loop ,
(4.13)
where VF is the F -term contribution to the scalar potential, VD is the D-term contribution, Vsoft is the soft supersymmetry breaking part and Vone−loop is the one-loop 1
Note, however, that the mass of the lepton of flavor a cannot be generated from the trilinear R-parity violating operators and one must invoke R-symmetry preserving supersymmetry breaking operators to generate a small mass [306].
72
contribution to the scalar potential. The relevant part of the F -term contribution is [308] X ∂W 2 VF = ∂φi i √ e c + 2λT T+ R− |2 + |(µu + λS S + λT T0 )H 0 = |(µu + λS S + λT T0 )Rd0 − f νeL N u d √ + 2 0 0 c + − 2 0 0 + − 2 e − λS Hu R | + |λT (Hu Rd + Hu R )| − 2λT T− Hu | + |λS Hu Rd + MR N d
+
e c |2 |f Hu0N
+
e c |2 |f Hu+ N
+
|f (e νL Hu0
−e lL− Hu+ )
d
2
− MR S| + |(µu + λS S − λT T0 )Hu+
√ √ 2λT Hu0 T+ |2 + | 2λT Hu+ Rd0 |2 + | 2λT Hu0Rd− |2 + |(µu + λS S − λT T0 )Rd− √ e c + 2λT T− R0 |2 , − fe lL− N (4.14) d −
√
and the D-term contribution is given by VD =
1X a a 1 D D + D Y DY , 2 a 2
(4.15)
where i h i √ h e† τ a L ei + qe† τ a qeiL + T † λa T + 2 M D T a + M D T a† , D a = g Hu† τ a Hu + L 2 2 i iL h 1 † 4 † 2 e† e i 1 ∗ e† L ei + 2e d diR q e u e e e + q e − u + D Y = − g ′ Hu† Hu − L iL iR iR iR i 2 3 iL 3 iR 3 iR h i √ − 2M1D S + S † . (4.16)
The τ a ’s are the SU(2) generators in the fundamental representation, whereas λa ’s are the three generators of the SU(2) group in adjoint representation. Here g and g ′ are SU(2)L and U(1)Y gauge couplings respectively. Therefore, using eq. (4.16), we expand eq. (4.15) and obtain the contribution to the
73
scalar potential from D-terms as VD = + + + − + − +
g ′2 (|Hu+ |2 + |Hu0|2 − |e νi0 |2 − |e li− |2 )2 + (M1D )2 (S + S † )2 + (M2D )2 (T0 + T0† )2 8√ ′ g 2 D M1 (S + S † )(|Hu+ |2 + |Hu0 |2 − |e νi0 |2 − |e li− |2 ) 2 g2 (|Hu+ |2 − |Hu0 |2 + |e νi0 |2 − |e li− |2 + 2|T+ |2 − 2|T− |2 )2 8 √ g2 (M2D )2 ((Hu+ )∗ Hu0 + (e li− )∗ νei0 + 2(T− − T+ )T0∗ + h.c.)2 − ((T+ − T− ) − h.c.)2 8 2 √ (M2D )2 g2 ((Hu0 )∗ Hu+ + (e li− )∗ νei0 + 2T0 (T+∗ + T−∗ ) − h.c.)2 + ((T+ + T− ) + h.c.)2 8 2 √ gM2D ((T+ + T− ) + h.c.)((Hu+ )∗ Hu0 + (e li− )∗ νei0 + 2(T− − T+ )T0∗ + h.c.) 2 √ gM2D ((T+ − T− ) − h.c.)((Hu0 )∗ Hu+ + (e li− )∗ νei0 + 2T0 (T+∗ + T−∗ ) − h.c.) √2 2gM2D (T0 + h.c.)(|Hu+ |2 − |Hu0 |2 + |e νi0 |2 − |e li− |2 + 2|T+ |2 − 2|T− |2 ). (4.17) 2
The soft supersymmetry breaking part of the scalar potential is explicitly shown in eq. (4.12) and the dominant radiative corrections to the quartic potential are of the νa |2 )2 , and 21 δλ3 |Hu0 |2 |e νa |2 . The coefficients of these quartic form 12 δλu (|Hu |2 )2 , 21 δλν (|e terms are [328] 2 2 met1 met2 3yt4 λ4S 5λ4T mT mS δλu = + ln ln ln + 2 2 2 2 2 16π mt 16π v 16π v2 m2S λ2S λ2T m2T 1 2 2 − 1} − mS {ln − 1}), (4.18) (m {ln − 16π 2 m2T − m2S T v2 v2 δλν
2 2 meb1 meb2 3yb4 mT λ4S mS 5λ4T ln ln + ln = + 2 2 2 2 2 16π mb 16π v 16π v2 1 m2T m2S λ2S λ2T 2 2 − (m {ln − 1} − mS {ln − 1}), 16π 2 m2T − m2S T v2 v2
(4.19)
and finally, δλ3
2 2 5λ4T mT 1 4 mS = ln + λS ln 2 2 2 32π v 32π v2 2 mS m2T λ2S λ2T 1 2 2 − 1} − mS {ln − 1}). (mT {ln + 2 2 2 2 32π mT − mS v v2
(4.20)
These contributions to the Higgs quartic couplings can be very important for the lightest CP-even Higgs boson to have a mass ∼ 125 GeV for large stop masses and/or large values of the couplings λT and λS . 74
4.3.1
Symmetry breaking and minimisation conditions
In minimizing the scalar potential we assume that the neutral scalar fields Hu0 , νea (a = 1(e)), S and T acquire real vacuum expectation values vu , va , vs and vT , respectively. e c carry R-charge 2 and they decouple from the scalar fields The scalar fields Rd and N mentioned above carrying R-charge 0. In order to write down the minimisation conditions, first we split the fields in terms of their real and imaginary parts: Hu0 = hR + ihI , νe = νeRa + ie νIa , S = SR + iSI and T = TR + iTI . The resulting minimisation equations with respect to hR , νeR , TR , and SR fields, are
(m2Hu + µ2u ) + (BµaL − f MR vS )(tan β)−1 + λ2S vS2 + λ2T vT2 + 2µu λS vS + 2µu λT vT √ 2δλu + δλ3 2 v cos2 β + 2λS λT vS vT + f 2 v 2 cos2 β + 2(g ′M1D vS − gM2D vT ) + 2 (g 2 + g ′2 + 4δλu ) 2 − v cos 2β = 0, (4.21) 4 m2Lea + (BµaL − f MR vS ) tan β + f 2 v 2 sin2 β + + (
g 2 + g ′2 − δλ3 + 2δλν 2 v cos 2β 4
δλ3 + 2δλν 2 √ )v + 2(gM2D vT − g ′M1D vS ) = 0, 4
m2TR + µu λT
(4.22)
v2 g vS v2 sin2 β + λS λT v 2 sin2 β + λ2T v 2 sin2 β + √ M2D cos 2β = 0, vT vT vT 2 (4.23)
g′ vS (m2SR + λ2S v 2 sin2 β) + (µu λS v 2 sin2 β + λS λT vT v 2 sin2 β + tS − √ M1D v 2 cos 2β 2 2 f MR v sin 2β ) = 0, (4.24) − 2 where we identify m2TR = m2T + bT + 4(M2D )2 , m2SR = m2S + bS + 4(M1D )2 + MR2 , tan β = vu /va and v 2 = v12 + v22 . The W- and the Z-boson masses can be written as 1 m2W = g 2(v 2 + 4vT2 ), 2 1 2 2 2 mZ = g v / cos2 θW . 2
(4.25)
The tree level ρ-parameter comes out to be ρ≡
m2W 4vT2 = 1 + . m2Z cos2 θW v2
(4.26)
Electroweak precision measurements of the ρ-parameter constrain the triplet vev vT to < be ∼ 3 GeV [367] and can be taken to be zero in the first approximation. 75
4.3.2
CP-even neutral scalar sector
With the help of these minimisation equations, it is straightforward to write down the neutral CP even scalar squared-mass matrix in the basis (hR , νeR , SR , TR ). The CP even scalar squared-mass matrix, thus, would be a symmetric 4 × 4 matrix. Note that we are working in the R-symmetry conserving case. The elements of the 4 × 4 CP-even scalar squared-mass matrix MS2 are given by (MS2 )11 = (MS2 )12 = (MS2 )13 = (MS2 )14 = (MS2 )22 = (MS2 )23 = (MS2 )24 = (MS2 )33 = (MS2 )34 = (MS2 )44 =
(g 2 + g ′2 ) 2 2 v sin β + (f MR vS − BµaL )(tan β)−1 + 2δλu v 2 sin2 β, 2 (g 2 + g ′2 − 2δλ3 ) 2 f 2 v 2 sin 2β + BµaL − v sin 2β − f MR vS , 4 √ 2λ2S vS v sin β + 2µu λS v sin β + 2λS λT vvT sin β + 2g ′ M1D v sin β − f MR v cos β, √ 2λ2T vT v sin β + 2µu λT v sin β + 2λS λT vS v sin β − 2gM2D v sin β, (g 2 + g ′2 ) 2 v cos2 β + (f MR vS − BµaL ) tan β + 2δλν v 2 cos2 β, 2 √ − 2g ′ M1D v cos β − f MR v sin β, √ 2gM2D v cos β, v 2 sin2 β λS λT vT v 2 sin2 β tS g ′ M1D v 2 cos 2β f MR v 2 sin 2β √ + −µu λS − − + , vS vS vS 2vS 2vS λS λT v 2 sin2 β, v2 gM D v 2 v2 sin2 β − λS λT vS sin2 β − √ 2 cos 2β. (4.27) −µu λT vT vT 2 vT
Since we want to have the lightest CP-even Higgs boson to be doublet-like and with a mass around 125 GeV, we would require small vevs, vS and vT of the singlet S and triplet T respectively, as well as large radiative corrections to the Higgs boson mass. Because of the choices of R-charges of various fields in this model, one cannot get tree level contributions to the lightest Higgs boson mass proportional to λ2S and λ2T as obtained in [328, 337]. However, there can be an additional contribution to the lightest Higgs boson mass at the tree level proportional to the square of the neutrino Yukawa coupling f and that can be significant when f is O(1). We shall discuss more on this scenario at a later stage. Note also that the smallness of vS and vT can be easily obtained by keeping the corresponding soft supersymmetry breaking mass terms > mS and mT somewhat larger ( ∼ a TeV). In such a scenario, when the singlet and the triplet fields are effective decoupled the CP-even mass matrix becomes a 2 × 2 one. In this case the angle α represents the mixing angle between hR and νeR and can be
76
expressed in terms of other parameter as follows (g 2 +g ′2 −2δλ3 ) 2 v sin 2β 4 − 2v 2 {δλu sin2 β − δλν
f 2 v 2 sin 2β + BµaL −
tan 2α = −2 (g2 +g′2 )v2 cos 2β 2
4.3.3
+ 2BµaL cot 2β
cos2 β} (4.28)
CP-odd neutral scalar sector
In the basis (hI , νeI , SI , TI ) the elements of the tree-level neutral CP-odd symmetric scalar squared-mass matrix MP2 are (MP2 )11 = (f MR vS − BµaL )(tan β)−1 , (MP2 )12 = −BµaL + f MR vS , (MP2 )13 = −f MR v cos β,
(MP2 )14 = 0,
(MP2 )22 = (f MR vS − BµaL ) tan β,
(MP2 )23 = −f MR v sin β,
(MP2 )24 = 0,
(MP2 )33 = λ2S v 2 sin2 β + m2SR − 2bS − 4(M1D )2 , (MP2 )34 = λS λT v 2 sin2 β,
(MP2 )44 = λ2T v 2 sin2 β + m2TR − 2bT − 4(M2D )2 .
(4.29)
The eigenvalues of the CP-odd scalar squared-mass matrix consists of a massless Goldstone boson and three physical CP-odd Higgs bosons. Out of these three physical Higgs bosons, one is essentially the linear combination of hI and νeI whereas the other two eigenstates are composed mainly of SI and TI , the imaginary parts of the singlet S and the triplet T . One can perform the following rotation to separate out the Goldstone mode
G − sin β cos β 0 A cos β sin β 0 ′ = SI 0 0 1 TI′ 0 0 0
0 hI 0 νeI . 0 SI 1
(4.30)
TI
The 4 × 4 squared-mass matrix then reduces to a 3 × 3 matrix structure from which one can find out the physical CP-odd Higgs bosons.
77
4.3.4
Charged scalar sector
In this case the elements of the tree-level charged scalar squared-mass matrix in the + − ∗ e−∗ basis (Hu+ , L a , T , (T ) ) are given by (a = 1(e))
√ 1 ±2 M11 = 2 2gM2D vT − 4vS vT λS λT − 4vT λT µu − f 2 v 2 cos2 β + g 2 v 2 cos2 β 2 + (−BµaL + f MR vS ) cot β, 1 1 ±2 M12 = −BµaL + f MR vS − f 2 v 2 sin 2β + g 2 v 2 sin 2β, 2 4 2 √ g vv sin β T ±2 √ − 2vλT (µu + vS λS − vT λT ) sin β, M13 = gM2D v sin β − 2 g 2 vvT sin β √ ±2 √ − 2vλT (µu + vS λS + vT λT ) sin β, M14 = gM2D v sin β + 2 √ 1 ±2 M22 = −2 2gM2D vT − f 2 v 2 sin2 β + g 2 v 2 sin2 β + (−BµaL + f MR vS ) tan β, 2 2 g vvT cos β ±2 √ , M23 = gM2D v cos β − 2 g 2 vvT cos β ±2 √ M24 = gM2D v cos β + , 2 gM2D v 2 cos 2β v 2 vS λS λT sin2 β 1 ±2 √ − M33 = −bT − 2(M2D )2 + g 2vT2 + g 2 v 2 cos 2β − 2 vT 2vT 2 2 v λT µu sin β + v 2 λ2T sin2 β − , vT ±2 M34 = bT + 2(M2D )2 − g 2 vT2 , 1 2 2 gM2D v 2 cos 2β v 2 vS λS λT sin2 β ±2 D 2 2 2 √ M44 = −bT − 2(M2 ) + g vT − g v cos 2β − − 2 vT 2vT 2 2 v λT µu sin β . (4.31) − v 2 λ2T sin2 β − vT
In the limit where the vev of the neutral component of the triplet is very small, the triplet essentially decouples from the doublet fields. Considering that, the Goldstone mode can be written as [368, 369] + + −∗ + − ∗ e (4.32) G = − sin βHu + cos β La + aT + b(T ) , where a and b represents small admixtures of the triplet fields with the doublet Higgssneutrino block. In order to evaluate the coefficients a and b, we note that the charged scalar squared-mass matrix follows the eigenvalue equation ±2 ±2 ±2 ±2 −M11 sin β + M12 cos β + M13 a + M14 b = 0,
±2 ±2 ±2 ±2 −M12 sin β + M22 cos β + M23 a + M24 b = 0.
78
(4.33)
Solving for a and b in terms of the charged scalar squared-mass matrix elements, we √ find a = b = 2vT /v and finally the expression for the Goldstone mode becomes √ √ 2v 2vT − ∗ 1 T + −∗ + + e (4.34) T + (T ) , G = √ − sin βHu + cos β La + ρ v v
where ρ is the appropriate normalisation factor defined in eq. (4.26). The Goldstone boson G+ gives a mass to W + and G− ≡ (G+ )∗ gives a mass to W − . The other states orthogonal to G+ are √ √ 2vT + 2vT − ∗ 1 + + −∗ e T − (T ) , H = √ cos βHu + sin β La + ρ v v √ √2v 1 2vT e−∗ T TP+ = √ La + sin βT + + cos β(T − )∗ , Hu+ − ρ v v √ √ 2vT + 2vT e−∗ 1 + − ∗ − ∗ (4.35) La − cos βT + sin β(T ) . Hu + (TP ) = √ ρ v v Once again we can separate out the Goldstone mode and write down the resulting 3 ×3 symmetric charged scalar squared-mass matrix in the basis of these orthogonal states (and their charge conjugates) to find out the physical charged scalar states.
4.3.5
Sum rules
We will conclude the discussion on scalar sector by presenting various sum rules for this model. Let us look at the CP-even neutral scalar squared-mass matrix once again and assume that the singlet and triplet vevs are very small. In such a situation these two fields are effectively decoupled from the theory and as a result the scalar squared-mass matrix becomes, a 2 × 2 matrix. Under these assumptions, we can write down the elements of the neutral CP-even squared-mass matrix (for the MSSM case see [60]) in a compact form as (see eq. (4.27)), 2 M11 = m2Z sin2 β + ξ cot β, 1 2 2 M12 = −ξ + m2Z (α − 1) sin 2β = M21 , 2 2 M22 = ξ tan β + m2Z cos2 β,
(4.36)
where we have defined α = 2f 2 v 2 /m2Z and ξ = f MR vS − BµaL . Note that we have kept small terms proportional to vS in this (2 × 2) light CP-even squared-mass matrix. The eigenvalues of this matrix represent the square of the masses of the two physical doublet-like Higgs bosons (remember that in this model the sneutrino of flavor a plays the role of the down type Higgs) and they are given by 1 2 (MZ + ζ) ± ∆ , (4.37) λ± = 2 79
where ζ = 2ξ/ sin 2β and h 2 i 21 2 ∆ = m2Z − ζ cos2 2β + m2Z (1 − α) + ζ sin2 2β .
(4.38)
Similarly, in the decoupling limit of the singlet and triplet fields, the CP odd scalar mass matrix has two eigenvalues. One of which corresponds to the massless Goldstone boson, whereas the other eigenvalue being ζ =
2(−bµaL + f MR vS ) ≡ m2A . sin 2β
(4.39)
The upper bound on the squared-mass of the lightest CP-even Higgs boson (λ− ≡ m2h ) will depend on the value of ∆. With the help of the inequality [368] 2 1 (4.40) a cos2 2β + b2 sin2 2β 2 > a cos2 2β + b sin2 2β ,
we can write down the tree level upper bound on the lightest CP-even Higgs boson mass depending on whether α < 1 or α > 1. However, as long as the quantity ζ ≡ m2A > m2Z , we find that the tree level upper bound on the lightest CP-even Higgs boson mass is m2h 6 m2Z cos2 2β + f 2 v 2 sin2 2β , (4.41)
irrespective of whether α < 1 or α > 1. It is very interesting to note that the neutrino Yukawa coupling f provides a tree level correction to the lightest Higgs boson mass. We shall discuss later that in our model f can be as large as O(1) and in that case this large f would certainly provide a significant correction to the tree level mass of the lightest Higgs boson, requiring very small radiative corrections via the triplet and the singlet as well as from the top squark loop. In a similar way we can obtain a lower bound on the heavy Higgs boson mass irrespective of α for ζ ≡ m2A > m2Z and is given by m2H > m2Z sin2 2β + m2A − f 2 v 2 sin2 2β . (4.42)
Finally, we also obtain a relation between the trace of the CP-even scalar squared-mass matrix and the trace of the CP-odd scalar squared-mass matrix, which differs from that of the MSSM Tr(MS2 ) = Tr(MP2 ) + m2Z + 2(bS + bT ) + 4 (M1D )2 + (M2D )2 . (4.43) Looking at the charged Higgs boson squared-mass matrix in the limit of very heavy triplet, we can see that the charged Higgs boson mass (mH ± ) can be written in terms of the CP-odd scalar mass (mA ) and the W boson mass as m2H ± = m2A + m2W − f 2 v 2 − 4vT λT (µu + λS vS ). 80
(4.44)
Let us also emphasise that we have checked that all the eigenvalues of the CP-even, CP-odd and charged scalar squared-mass matrices (leaving aside the Goldstone bosons) come as positive for a minimum.
4.4 The fermionic sector The fermionic sector of the scenario, involving the neutralinos and the charginos, has rich new features. In the context of the present study, when analysed in conjunction with the scalar sector of the scenario, this sector plays a pivotal role by presenting the defining issues for the phenomenology of this scenario. Its influence ranges over physics of the Higgs boson at current experiments and the physics of the neutrinos before finally reaching out to the domain of astrophysics and cosmology by offering a possible warm dark matter candidate whose actual presence may find support in the recent observations of a satellite-borne X-ray experiment. Thus, it is of crucial importance to study the structure and the content of this sector in appropriate detail. A natural consequence of such a U(1)R -lepton number model with a right-handed neutrino is that one of the left-handed neutrinos (the electron-type active one) and the right-handed (sterile) neutrino become parts of the extended neutralino mass matrix. The electron-type neutrino of the SM can be identified with the lightest neutralino eigenstate. Subsequently, we show that in certain region of the parameter space, the next-to lightest neutralino is actually bino-dominated and can be very light (with a mass of order 100 MeV).
4.4.1
The neutralino sector: R-conserving case
In the neutral fermion sector we have mixing among the Dirac gauginos, the higgsinos, the active neutrino of flavor ‘a’ (i.e., νe ) and the single right-handed neutrino N c once the electroweak symmetry is broken. The part of the Lagrangian that corresponds to the neutral fermion mass matrix is given by L = (ψ 0+ )T MχD (ψ 0− ) where ψ 0+ = e0 , N c ), with R-charges +1 and ψ 0− = (S, e Te0 , H e u0, νe ) with R-charges -1. In (eb0 , w e0 , R d principle, νµ and ντ would also appear in the basis of ψ 0− . In the absence of any mixing, these two neutrinos would remain massless. The neutral fermion mass matrix MχD is given by
MχD
M1D
′v g√ u 2 √u − gv 2
0
′
− g√v2a
gv 0 √a M2D 2 = λ v λ v µ + λ v + λ v 0 S S T T S u T u u MR 0 −f va −f vu 81
.
(4.45)
The above matrix can be diagonalised by a biunitary transformation involving two uni! e0+ ψ i , tary matrices V N and U N and results in four Dirac mass eigenstates χ e0+ i ≡ ψe0− i
with i = 1, 2, 3, 4 and ψei0+ = VijN ψj0+ , ψei0− = UijN ψj0− . The lightest mass eigenstate χ e0+ 4 is identified with the light Dirac neutrino. In order to obtain an analytical expression of this small mass, we make a series expansion of
which implies
b = Det(M D )Tr[(M D )−1 ] = 0, Det(MχD − λI) χ χ λ =
1 . Tr[(MχD )−1 ]
(4.46)
(4.47)
From eq. (4.47) we obtain the light Dirac neutrino mass as D 3 M γτ + v f ω sin β 2 i, mD √ νe = h γ(τ + 2M2D (M1D − f v sin β)) + M2D τ + (v 3 f sin β)(g ′λS − gλT ) − v 2 ω sin2 β
(4.48)
where τ = v cos β(g tan θW MR −
√
2f M1D tan β),
ω = g(M2D λS tan θW − M1D λT ), γ = (µu + λS vS + λT vT ).
(4.49)
From eq. (4.48) one can verify that the generic spectrum of the model would include a Dirac neutrino of mass in the range of a few eV to tens of MeV. However, by suitable choices of other parameters one can also accommodate a mass of 0.1 eV or smaller for the Dirac neutrino. This can be achieved, for example, by assuming the following relationships, which are λT = λS tan θW , and MR =
√
2f M1D tan β . g tan θW
(4.50)
(4.51)
With these choices and assuming (M2D − M1D ) ≪ M1D , M2D and M1D ≫ f v sin β, the expression in eq. (4.48) can be further simplified and the Dirac mass of the neutrino can be written as v 3 f g sin β √ λT (M2D − M1D ). = mD νe 2γM1D M2D 82
(4.52)
It is straightforward to check from eq. (4.52) that by suitable choices of the parameters f , λT and ǫ ≡ (M2D − M1D ) one can have a Dirac neutrino mass in the right ballpark < of ∼ 0.1 eV. Note that a choice of large f ∼ O(1) is possible for a small λT (∼ 10−6 ) and < nearly degenerate Dirac gauginos (ǫ ∼ 10−1 GeV) assuming µu , M2D , M1D in the few hundred GeV range. The near degeneracy between the Dirac gaugino masses can be lifted by assuming f, λT ∼ O(10−4 ). However, the order one Yukawa coupling plays an important role to enhance the lightest Higgs boson mass at the tree level, as discussed in section 4.3.5.
4.4.2
The neutralino sector: R-breaking case
Recent cosmological observations imply a positive but very small vacuum energy or cosmological constant associated with our universe [45]. In the context of a spontaneously broken supergravity theory in a hidden sector, having a very small vacuum energy would require a non-zero value of the superpotential in vacuum (hW i) and that will break R-symmetry because the superpotential carry non-zero R-charges. Since a non-zero gravitino mass also requires a non-zero hW i, one can consider the gravitino as the order parameter of the R-symmetry breaking. The breaking of R-symmetry has to be communicated to the visible sector and in this work we consider anomaly mediation of supersymmetry breaking playing the role of the messenger of R-breaking. This is known as anomaly mediated R-breaking (AMRB) [305]. A non-zero gravitino mass generates Majorana gaugino masses and trilinear scalar couplings. We shall consider the R-breaking effects to be small thus limiting the gravitino mass (m3/2 ) around 10 GeV. The R-breaking Lagrangian contains the following terms X X ea L eb E ec + ea Q ek D e c + Hu QA e uU ec L = M1eb0eb0 + M2 w e0 w e0 + M3 e ge g+ Alb L Adk L b k b=2,3
k=1,2,3
X X 1 ′ eb Q ej D e c + Aν Hu L ea N e c, e2 L e3 E ec + Aλbjk L Aλ23k L + k k 2 j,k=1,2,3;b=2,3 k=1,2,3
(4.53)
where M1 , M2 and M3 are the Majorana mass parameters corresponding to U(1), SU(2) and SU(3) gauginos, respectively and A’s are the scalar trilinear couplings, given by [370, 371] Mi = bi
gi2 m3/2 , 16π 2
(4.54)
where i = 1, 2, 3 for bino, wino and gluinos respectively. The coefficients are b1 = p 33 5/3g ′. The third generation trilinear , b = 1, b = −3 and one has g = g, g = 2 3 2 1 5 83
scalar couplings are At =
βˆht m3/2 βˆhb m3/2 βˆhτ m3/2 v , A = v , A = va , u b a τ mt 16π 2 mb 16π 2 mτ 16π 2
(4.55)
ˆ are written in terms of the usual beta functions as, βˆ = β/16π 2 and are where the β’s given by [370, 371] 16 13 2 2 2 2 2 βˆht = ht − g1 − 3g2 − g3 + 6ht + hb , 15 3 7 2 16 2 2 2 2 2 ˆ βhb = hb − g1 − 3g2 − g3 + ht + 6hb + hτ , 15 3 9 2 2 2 2 ˆ (4.56) βhτ = hτ − g1 − 3g2 + 3hb + 4hτ . 5 The trilinear scalar couplings for the first two generations can be obtained in a straightforward way by replacing the Yukawa couplings appropriately. The (Majorana) neutralino mass matrix containing R-breaking effects can be write w ed , H e u0, Nc , νe )T as ten in the basis ψ 0 = (eb0 , S, e0 , Te, R 1 = (ψ 0 )T MχM ψ 0 + h.c. Lmass χ e0 2
(4.57)
where the symmetric (8 × 8) neutralino mass matrix MχM is given by ′v ′ g√ u M1 M1D 0 0 0 0 − g√v2a 2 M1D 0 0 0 λS vu 0 MR 0 0 gv gv √a 0 M2 M2D 0 − √2u 0 2 D 0 0 M 0 λ v 0 0 0 T u 2 MχM = 0 λS vu 0 λT vu 0 µu + λS vS + λT vT 0 0 g′ vu gv u √2 0 − √2 0 µu + λS vS + λT vT 0 −f va 0 0 MR 0 0 0 −f va 0 −f vu ′v gv g√ √a 0 0 0 0 −f vu 0 − 2a 2 In the absence of Majorana gaugino masses (M1 = M2 = 0), the pure Dirac neutrino case discussed in section 4.4.1 is recovered from eq. (4.58) and we have one light Dirac neutrino of mass mD νe . This is equivalent to saying that we have two Majorana neutrinos D of mass −mνe and mD νe with opposite CP parities [136]. If the gaugino Majorana mass parameters M1 and M2 are non-zero but small compared to the corresponding Dirac gaugino mass parameters M1D and M2D then the pair of light Majorana neutrinos will be quasi-degenerate and sometimes called a pseudoDirac neutrino. By increasing the gravitino mass (which means larger M1 and M2 ) 84
.
(4.58)
one can generate a larger splitting between these two light Majorana neutrino states. We will discuss these two cases in the context of our model, in detail. Note that in the absence of N c , the neutralino-neutrino mass matrix cannot produce a non-zero mass of the light neutrino even if the gaugino Majorana mass parameters M1 and M2 are non-zero. The above mass matrix can be diagonalised by a unitary transformation given by N ⋆ MχM N † = (Mχ )diag .
(4.59)
The two-component mass eigenstates are defined as χ0i = Nij ψj0 ,
i, j = 1, ..., 8
(4.60)
and one can arrange them in Majorana spinors defined by ! χ0i 0 χ ei = , i = 1, ...8. χ0i
(4.61)
Similar to the Dirac case, the lightest eigenvalue (mχe08 ) of this neutralino mass matrix corresponds to the Majorana neutrino mass. Let us discuss the characteristics of lightest neutralinos in two different cases corresponding to (i) small R-breaking effects and (ii) R-breaking effects are comparatively larger. Case - 1 In this subsection, we consider a case, where R-breaking effects are very small. The two light mass eigenstates of the neutralino-neutrino matrix in eq. (4.58) are almost degenerate, maximally mixed and they combine to form a (pseudo)Dirac neutrino. We can evaluate the product of these two mass eigenvalues by calculating the ratio of the determinants of the full 8 × 8 matrix and that of the upper 6 × 6 block of MχM , without the (N c , νe ) sector. Assuming small mixing between this neutrino sector with other neutral fermions we end up with −λ
2
v 3 sin βf g = − √ 2γ(M2D M1D ) 2 = −(mD νe ) ,
2
λ2T M2D − M1D
2
, (4.62)
where γ is defined previously in eq. (4.49) and we have used the relations in eq. (4.50) and eq. (4.51).
85
Case - 2 Here we are going to consider a relatively lager value of m3/2 . We observe that, with this choice, there is a splitting in masses of the two light Majorana neutrinos with a relatively smaller mixing between the two states. The light neutrinos are predominantly right handed or left handed and the mass eigenstate N c′ which is mostly a right handed neutrino is heavier than the mass eigenstate νe′ with a large left handed component2 . We will explicitly show this in the section on numerical analysis, but first let us evaluate the lightest Majorana neutrino mass, which corresponds to the mass of νe′ . This can be done by calculating the ratio of the determinant of the 8 × 8 neutralino-neutrino mass matrix MχM , to that of the 7 × 7 upper block of MχM . For a very small neutrino mass we can assume that the eigenvalues of the 7 × 7 matrix remain unchanged from the seven heavier eigenvalues of the 8 × 8 matrix. This approximation can be safely g 2 va2 g ′2 va2 implemented as long as M1 ≫ 2M D and M2 ≫ 2M D . We shall choose the mass of the 1 2 gravitino in such a way that these conditions are satisfied. Therefore the light active Majorana neutrino mass at the tree level, in the R-symmetry breaking scenario is 2 2 D D 2 gλT v (M2 − M1 ) sin β (mν )Tree = −v , (4.63) [M1 α2 + M2 δ 2 ] where 2M1D M2D γ tan β √ 2 + 2v λS tan β(M1D sin2 β + M2D cos2 β), g tan θw √ δ = 2M1D v 2 λT tan β.
α =
(4.64)
In order to obtain eq. (4.63) we have used once again the relations in eq. (4.50) and eq. (4.51) and γ has been defined previously. We can see from eq. (4.63) that mν = 0 (at the tree level) when M2D = M1D and a small splitting of these Dirac gaugino mass parameters will result in a value of (mν )Tree in the right ballpark provided M1D , M2D are of the order of a few hundred GeV or 1 TeV with the couplings λT , λS ∼ 10−4 or so. It is very interesting to note that (mν )Tree is independent of the neutrino Yukawa coupling f . This is an artifact of the relation we have used in eq. (4.51) and thus even for a large f ∼ O(1), the tree level Majorana mass of the active neutrino can be kept very small with the above choices of parameters. Our approximate analytical result matches very well with the full numerical analysis as described later in this work. To derive an expression for the mass of the sterile neutrino, we work in the region of parameter space, where the active neutrino becomes a pure left handed neutrino state and f ∼ O(10−4 ). Thus by excluding this left handed neutrino state, we are left with a 2
The prime indicates a superposition of the states N c and νe due to mixing.
86
7 × 7 neutralino mass matrix and the lightest eigenvalue then would correspond to the mass of the sterile neutrino. In the limit of large M1D , together with small couplings λS , λT , f and considering only the dominant contributions, we eventually obtain the sterile neutrino mass as MR M1 R MR . (4.65) MN ≃ M1D M1D Substituting the expression of MR , given in eq. (4.51), we reduce the sterile neutrino mass in the form MNR
2f 2 tan2 β = M1 , g′2
(4.66)
which is independent of M1D . For a wide range of parameters, the active-sterile mixing angle, denoted as θ14 , can be estimated as 2 θ14 =
(mν )Tree . MNR
(4.67)
We note that the neutrino Yukawa coupling ‘f ’ has some interesting effects on the next-to-lightest eigenstates of the mass matrix. The following situations are phenomenologically important: • A large value of f ∼ O(1) generates a very light bino-like neutralino (e χ07 ) with mass around a few hundred MeV. In this case, this is the lightest supersymmetric particle (LSP) and its mass is mainly controlled by the R-breaking Majorana gaugino mass parameter M1 . This is because of the presence of MR N c S term in the Lagrangian, where the coefficient MR becomes very large (∼ 105 GeV) for an order one ‘f ’. This results in forming a heavy pseudo-Dirac pair with mass ∼ MR and makes the lightest eigenvalue very small and predominantly bino-like. A very light neutralino has profound consequences in both cosmology as well as in collider physics [372–381]. In the context of the present model one can easily satisfy the stringent constraint coming from the invisible decay width of the Z boson because the light neutralino is predominantly a bino. One should also take into account the invisible decay branching ratio of the lightest Higgs boson as noted in detail later. • For small f ∼ O(10−4 ), χ e07 is a sterile neutrino state, which is a plausible warm dark matter candidate with appropriate relic density. The mass of the sterile neutrino and active-sterile mixing is given in eq. (4.66) and (4.67). Furthermore, the sterile neutrino can be identified with a warm dark matter candidate only if the following requirements are fulfilled. These are: (i) it should be heavier than 87
0.4 keV, which is the bound obtained from a model independent analysis [283] and (ii) the active-sterile mixing needs to be small enough to satisfy the stringent constraint coming from different X-ray experiments [382]. Under the circumstances, the lightest neutralino-like state is the next-to-next-tolightest eigenstate (e χ06 ) of the neutralino mass matrix. Its composition is mainly controlled by the parameter µu , chosen to be rather close to the electroweak scale (M1D , M2D > µu ). The masses of the heavier neutralino states for this case (small f ) will be presented later.
4.4.3
The chargino sector
We shall now discuss the chargino sector in some detail. The relevant Lagrangian after R-breaking in the AMRB scenario obtains the following form: √ − e− + gvu H e +w e + e− Lch = M2 w e+ w e− + M2D − gvT Ted− w e+ + 2λT vu Teu+ R u e − µu Hu Rd d e u+ R e− − λS vS H e u+ R e− + gva w + λT vT H e+ e− + M2D + gvT Teu+ w e− + me ecR e− + h.c. d
L
d
L
(4.68)
e + , ec ) and (w e− , e− ), is writThe chargino mass matrix, in the basis (w e+ , Teu+ , H e− , Ted− , R u R L d 3 ten as M2 M2D − gvT 0 gva √ M D + gv 0 2vu λT 0 2 T (4.69) Mc = . gvu 0 −µu − λS vS + λT vT 0 0 0 0 me
This matrix can be diagonalised by a biunitary transformation, UMc V T = MD± . The chargino mass eigenstates are related to the gauge eigenstates by these two matrices U and V . The chargino mass eigenstates (two-component) are written in a compact form as χ− = Uij ψj− , i χ+ = Vij ψj+ , i where
− w e+ w e Te+ Te− ψi+ = u+ , ψi− = d− . e e Hu R d e− ecR L
(4.70)
(4.71)
− c c ψi− and ψi+ would also include µ− L , τL and µR , τR respectively. However, since only the electron type sneutrino acquires a vev, µ and τ do not mix with other chargino states. 3
88
The four-component Dirac spinors can be written in terms of these two-component spinors as ! + χ i , (i = 1, ..., 4). (4.72) χ e+ i = χ− i c It is to be noted that χ eci ≡ (e χ+ e− i ) = χ i is a negatively charged chargino. Hence, the − lightest chargino (e χ4 ) corresponds to the electron and the structure of the chargino mass matrix ensures (see eq. (4.69)) that the lightest mass eigenvalue remains unaltered from the input mass parameter for the electron , i.e., me = 0.5 MeV. Let us now analyse the composition of different chargino states:
• In the limit when M2D >> µu , the next-to-lightest chargino, χ− 3 (which is actually e− with the lightest chargino-like state in the MSSM sense), comprises mainly of R d e + with a small admixture a very little admixture of Ted− while χ+ is dominated by H u 3 of w e+ .
e− while χ+ • For M2D ≪ µu , χ− 3 is composed predominantly of 3 is predominantly w + Teu . • Finally, for M2D ≈ µu , χ− e− and χ+ 3 is also predominantly w 3 is also predominantly made up of Teu+ .
Apart from the electron, the mass of the chargino states are controlled mainly by the parameters M2D and µ. The chargino mass spectra corresponding to different benchmark points will be presented later.
4.5 Concluding remarks This chapter mainly deals with the theoretical framework of the proposed U(1)R -lepton number model with a right handed neutrino superfield. We present detailed calculations in the scalar sector as well as in the fermionic sector. We observe that the right handed neutrino superfield not only provides a tree level mass to the active neutrinos, but also produces a correction to the tree level mass of the Higgs boson (see eq. (4.41)). Such a correction can be important in the presence of an order one neutrino Yukawa coupling, i.e., when f ∼ O(1). In addition, such an enhancement in tree level Higgs mass might also help to ameliorate the Higgs ‘naturalness’ problem, as less radiative corrections are required to fit the Higgs boson mass to the observed value of 125 GeV. In this regime, the lightest neutralino turns out to be bino-like with mass around a few hundred MeV. On the other hand, we have shown that for f ∼ O(10−4 ), the right handed neutrino 89
can be identified with a keV dark matter candidate. In the next chapter, we have cultivated these issues extensively and in a quantitative manner. We also note that by adding a single right handed neutrino superfield, only one neutrino acquires a mass at the tree level. Therefore, to fit the neutrino mass and mixing results, we have taken into account radiative corrections which only appear when R-symmetry is broken.
90
Chapter 5 Higgs boson mass, neutrino masses and mixing and keV dark matter in U (1)R-lepton number model 5.1 Introduction In this chapter we look into some of the phenomenological aspects of the model discussed in the previous chapter. We first look into the Higgs boson mass both at the tree level as well as at the one loop level. The presence of large neutrino Yukawa coupling f provides an additional tree level contribution to the Higgs boson mass and ameliorates the naturalness argument. This extra contribution usually saturates for smaller values of tan β, i.e., tan β ≤ 5. For larger tan β, one has to resort to radiative corrections, mainly the quartic corrections. These are significant for λS , λT ∼ O(1). The large f scenario is also interesting because the spectrum includes a light bino-like neutralino with mass around a few hundred MeV. We shall also discuss in detail that this model can produce one very light Dirac neutrino at the tree level with an Yukawa coupling as large as ∼ 10−4 and in some cases even with an Yukawa coupling of O(1) (see eq. (4.52)). In the presence of only a single right-handed neutrino the low energy spectrum includes two massless neutrinos and one must think of some other mechanism to generate non-zero mass to at least one of these massless neutrinos. This can be achieved by introducing a small breaking of U(1)R symmetry. We know that a non-zero gravitino mass m3/2 implies breaking < of U(1)R symmetry. In this chapter we shall consider a small gravitino mass m3/2 ∼ 10 GeV in the context of anomaly mediated supersymmetry breaking. This ensures that the effects of U(1)R symmetry breaking are also not very large. In fact, the small
91
breaking of R-symmetry generates small Majorana masses for the gauginos as well as trilinear scalar interactions or the A-terms [305], as shown in eqs. (4.54) and (4.55). We shall quantify in our subsequent analysis the non-zero Majorana mass terms for the neutrinos at the tree level as well as at the one-loop level. One important thing to note is that in this model we can have a sterile neutrino with mass of the order of a few keV for f ∼ 10−4 . This can be identified as a WDM candidate [383] with appropriate relic density. We have checked that the active sterile mixing is small and consistent with the experimental observations of satellite based Xray telescopes. Thus the spectrum include appropriate values of light neutrino masses along with a warm dark matter candidate in the form of sterile neutrino.
5.2 Large neutrino Yukawa coupling: f ∼ O(1) In the previous chapter, while discussing the sum rules in the scalar sector (see eq. (4.41)), we observed that the lightest Higgs boson mass receives an additional tree level conbuL ba N b c in the superpotribution due to the presence of the neutrino Yukawa term f H tential. In the minimal supersymmetric standard model (MSSM) one requires a very large loop correction in order to fit the Higgs boson mass in the range of (123 – 127) GeV [384]. In the next-to-minimal supersymmetric model (NMSSM), the µ term is dynamically generated through a λS SHu Hd term in the superpotential and the tree level Higgs boson mass receives a correction proportional to λ2S [384]. Similarly in the singlet-triplet extension of the MSSM, a tree level correction to the Higgs boson mass proportional to λ2S and λ2T is obtained [369]. However, in this model these tree level contributions to the lightest Higgs boson mass are absent but because of the presence of the neutrino Yukawa coupling f an additional contribution (∆m2h )Tree = f 2 v 2 sin2 2β is obtained (see eq. (4.41)). In fig. 5.1 we show the variation of the lightest Higgs boson mass in this model as a function of tan β. We observe from this figure that for a low value of tan β the Higgs boson mass of ∼ 125 GeV can be achieved with f = 0.9, even at the tree level. Moreover, we find that met1 = met2 = 500 GeV is sufficient enough to provide the correct Higgs boson mass through radiative corrections for a slightly larger value of tan β. In chapter 4, we have discussed that the spectrum of the model for f ∼ O(1) includes a light bino-like neutralino with mass around a few hundred MeV. The lightest neutralino corresponds to the active neutrino with an appropriate mass of around 0.1 eV. The benchmark point corresponding to the large f scenario is shown in table 5.1. However, in our case the MeV neutralino LSP can decay through R-parity violating channels. Note, that in this case the gravitino with a mass of ∼ 10 GeV decays 92
140
mt1 = mt2 = 500 GeV
mhHGeVL
130 120 110 100 90 80
f=0.9, Tree f=0.8, Tree f=0.9, 1-loop f=0.8, 1-loop 0 5
10
15
20
tan Β Figure 5.1: The variation of the lightest Higgs boson mass with tan β. The dashed lines represent the Higgs boson mass at the tree level and the continuous lines represent the Higgs boson mass after radiative correction is added for a top squark mass of 500 GeV. Here red corresponds to f = 0.9 whereas blue corresponds to f = 0.8. mainly to the lightest neutralino + photon final state and has a lifetime of ∼ 1012 sec. Such a gravitino will decay after the big-bang nucleosynthesis (BBN) producing an unacceptable amount of entropy. This conflicts with the predictions of BBN if one assumes the standard big-bang cosmology and results in a constraint on the gravitino mass to be m3/2 > 10 TeV [385]. However, this constraint on the gravitino mass can be avoided if one assumes that the universe had gone through an inflationary phase and in order to avoid the strong constraints obtained from the photo-dissociation of the light elements because of the radiative decay of the gravitino, one arrives at the < upper bound on the reheating temperature of the universe TR ∼ 106 GeV [376,386]. In this case the gravitino is a stable particle in the collider time scale. However, it cannot be a candidate for dark matter because of its small lifetime in the cosmological time scale. Implications of such a scenario at the LHC and in the context of dark matter requires further investigations.
5.2.1
Tree level mass bound on mh
In this section, we look into the tree level mass of the Higgs boson in detail. The < bound in eq. (4.41) is saturated for vs ∼ 10−3 GeV, i.e., when the singlet has a large soft supersymmetry breaking mass and is effectively integrated out. The f 2 v 2 term
93
Parameters M1D M2D tan β λS λT µ tS bµL mS mT vS vT f MR met1 = met2 bS bT m3/2 mh (mν )Tree m0χe
BP-1 800 GeV 580 GeV 2.6 10−5 λS tan θW ∼ 5.5 × 10−6 200 GeV (200)3 −(200)2 (GeV)2 7.39 TeV 7.7 TeV 0.5 GeV 0.1 GeV 0.9 7.4 TeV 500 GeV 1 TeV 1 TeV 20 GeV 125.5 GeV 0.049 eV 167 MeV
Table 5.1: A benchmark point with large f and small λS and λT grows at small tan β and thus the largest Higgs boson mass is obtained with low tan β and large values of f . Therefore, for f ∼ O(1), the tree level Higgs boson mass can be as large as ∼ 125 GeV where the peak in the diphoton invariant mass has been observed and no radiative corrections are required. This means that in this scenario one can still afford a top squark mass as small as 350 GeV or so and couplings λT and λS can be small (∼ 10−4 ) as well. This is illustrated in fig. 5.2 where, the lightest Higgs boson mass is shown as a function of vS for f = 1.5, tan β = 4. One can see < that for a very small vS ( ∼ 10−3 GeV) the tree level Higgs boson mass is 150 GeV and is reduced to 125 GeV for a vS ∼ 0.2 GeV. As vS increases further, (mh )Tree starts decreasing rapidly and the Higgs boson mass becomes lighter than 100 GeV. In such a case one requires larger radiative corrections to the Higgs boson mass. For example, with a choice of λS = 0.91 and λT = 0.5, the one-loop quartic corrections
94
200
180
mh HGeVL
160
140
120
100
Hmh LTree-bound 80
Hmh LTree mh =125 GeV
60 10-4
0.001
0.01
0.1
1
vS HGeVL Figure 5.2: The tree level mass of the lightest Higgs boson as a function of the singlet (S) vacuum expectation value vS with f = 1.5, tan β =4 and other parameter choices are as described in the text. The upper bound on the tree level mass of the Higgs boson from eq. (4.41) is also shown.
to the Higgs boson mass arising from these two couplings are sizable1 . In order to have a 125 GeV Higgs boson in the presence of order one λS and λT , one requires the > tree level contribution to be small for very small vS (∼ 10−4 GeV) and large f ( ∼ 1). The one loop corrections from the top squark loop must also be small and this is realised for small met and large tan β. This is illustrated in fig. 5.3 where we plot contours for the lightest Higgs boson with a mass of 125 GeV in the met – tan β plane for different choices of f and vS . One can see from this figure the effect of a larger vS , which requires a larger top squark loop contribution to have a Higgs boson mass of 125 GeV. However, this tree level contribution gets diluted at large values of tan β. There, the one loop quartic corrections [306, 328] can come into play and can substantially enhance the Higgs boson mass in the presence of order one couplings, λS and λT , as shown in eqs. (4.18)-(4.20). Thus, even for larger values of tan β, one can easily find a Higgs boson as heavy as observed at the LHC experiments, when the top squarks are relatively light. 1
These choices of λT and λS are not completely independent. Rather they follow a relationship derived from the requirement of small tree level mass of the active neutrino shown in eq. (4.50).
95
40
f =2, vs =10-4 GeV f =1, vs =10-4 GeV
35
f =2, vs =10-2 GeV f =1, vs =10-2 GeV
30
tanΒ
25
20
15
10
5
200
400
600
mt
800
1000
HGeVL
Figure 5.3: Mass-contours for the lightest Higgs boson with mh = 125 GeV in the met –tan β plane for large values of f and λT = 0.5.
5.3 Small neutrino Yukawa coupling : f ∼ 10−4 In the small f (∼ O(10−4 )) case, the Higgs boson mass is devoid of any additional contribution at the tree level. Hence, one is compelled to consider large radiative corrections either from the top squark loop or from the one-loop quartic terms defined in eqs. (4.18)- (4.20). However, the physics in this case is also interesting from the point of view of neutrinos and DM as we will discuss.
5.3.1
eV scale sterile neutrino
The right handed sterile neutrino, introduced in our model can be at the eV scale or at the keV scale depending on the relevant model parameters. We have discussed in the previous chapter that there are two different cases, one where the active and sterile neutrinos mix maximally to form a (pseudo)Dirac neutrino, and in the other case there is a relatively large mass splitting between the sterile neutrino and the active neutrino with a very small mixing. In the latter situation there are two distinct Majorana neutrinos in the spectrum. Let us now discuss these two cases separately in the light of the LSND anomaly [387–389].
96
Pseudo-Dirac case: When the R-breaking effects are small, the light neutrinos are almost degenerate in mass at the tree level and with near maximal mixing between the two states. In this case, taking into account the possible loop contributions for the active neutrinos as well as the sterile neutrino, the neutrino mass matrix has a two texture zero structure in the basis (NR′ , νe′ , νµ , ντ ), where the prime signifies that these two states combine to form a (pseudo)Dirac neutrino
× ⋆ 0 0
⋆ × × ×
0 × × ×
0 × × ×
.
(5.1)
The asterisks in the (12) and (21) elements symbolise the Dirac neutrino mass obtained at tree level from the neutralino-neutrino mass matrix MχM . The crosses in the mass matrix signify the contributions to neutrino masses via loop corrections which we shall discuss elaborately in the next section. Finally we have a texture two zero structure of the neutrino mass matrix, in the 3+1 scenario2 . In order to check whether such a texture of neutrino mass matrix is ruled out or ′ ′ not, we consider a general neutrino mass matrix in the basis (NR , νe , νµ , ντ )
Mν =
Mss Mes Mµs Mτ s
Mse Mee Mµe Mτ e
Msµ Meµ Mµµ Mτ µ
Msτ Meτ Mµτ Mτ τ
.
(5.2)
This mass matrix can be diagonalised by a 4 × 4 PMNS matrix U which can be constructed with 6 orthogonal rotation matrices. For simplicity, let us consider the scenario with no CP violating phases. The neutrino mass matrix can be obtained from
Mν =
Us1 Ue1 Uµ1 Uτ 1
Us2 Ue2 Uµ2 Uτ 2
Us3 Ue3 Uµ3 Uτ 3
Us4 Ue4 Uµ4 Uτ 4
m1 0 0 0 0 m2 0 0 0 0 m3 0 0 0 0 m4
Us1 Us2 Us3 Us4
Ue1 Ue2 Ue3 Ue4
Uµ1 Uµ2 Uµ3 Uµ4
Uτ 1 Uτ 2 Uτ 3 Uτ 4
,
(5.3)
where m1 , m2 , m3 , m4 are the physical neutrino masses and m1 ≫ m2 , m3 , m4 . We compare this with the two texture zero structure given in eq. (5.1) and obtain two 2
A detailed study of two texture zero neutrino mass matrix structure has been performed in [390].
97
equations corresponding to the zeros in the mass matrix. They are as follows Msµ = m1 Us1 Uµ1 + m2 Us2 Uµ2 + m3 Us3 Uµ3 + m4 Us4 Uµ4 = 0, Msτ = m1 Us1 Uτ 1 + m2 Us2 Uτ 2 + m3 Us3 Uτ 3 + m4 Us4 Uτ 4 = 0.
(5.4)
These equations can be further simplified with the assumption that the lightest neutrino mass m4 could be zero. This choice is justified as the oscillation experiments are sensitive to the mass squared differences. With this simplification, eq. (5.4) reduces to m1 Us1 Uµ1 + m2 Us2 Uµ2 + m3 Us3 Uµ3 = 0, m1 Us1 Uτ 1 + m2 Us2 Uτ 2 + m3 Us3 Uτ 3 = 0.
(5.5)
We notice that eq. (5.5) contains m1 Us1 , which is much larger than all the other terms. Thus, no cancellation between the first term and the rest can satisfy eq. (5.5). Hence, we conclude that this texture is not viable to explain LSND anomaly. Majorana case: In this section we shall consider a different texture of the light neutrino mass matrix, where we have one Majorana neutrino with a tree level mass ∼ 1.2 eV, but is composed mainly of the right handed sterile neutrino. The other Majorana neutrino has a very small mass at the tree level and it is essentially an active neutrino. Again taking into account possible loop contributions to the active neutrinos, the three texture zero structure3 of the neutrino mass matrix in the basis (NR′ , νe′ , νµ , ντ ) (the prime indicates a mixed state of N c and νe ), is given by ⋆ 0 0 0 0 ⋆ × × (5.6) . 0 × × × 0 × × ×
The asterisks in the (11) and (22) elements represent the tree level Majorana masses of NR′ and νe′ (with additional loop contribution in the (22) element) whereas all the other masses are generated at the one-loop level. The state NR′ is mostly a right handed sterile neutrino and the active sterile mixing in this case is negligible. Comparing the neutrino mass matrix obtained in eq. (5.3), with the three texture zero structure of eq. (5.6), we find m1 Us1 Ui1 + m2 Us2 Ui2 + m3 Us3 Ui3 + m4 Us4 Ui4 = 0 3
(i = e, µ, τ ). (5.7)
A detailed study of three texture zero neutrino mass matrix structure has been performed in [391, 392].
98
Again with the assumption of the lightest neutrino mass, m4 = 0, this expression can be simplified further. However, as argued in the context of Pseudo-Dirac case, eq. (5.7) cannot be solved by taking into consideration the neutrino oscillation parameters which satisfy the LSND anomaly. Thus we see that this model as it is, cannot solve the LSND anomaly. Nevertheless, in the next section we shall see that by appropriate choice of parameters we can fit the three flavor global neutrino data in this model and at the same time the sterile neutrino can be accommodated as a keV warm dark matter candidate.
5.3.2
Right handed neutrino as a keV warm dark matter
We are considering a situation where the Majorana sterile neutrino acquires a tree level mass of the order of a few keV and the gravitino mass is around a few GeV (m3/2 ∼ 10 GeV). There has been a lot of work on model building aspects of keV sterile neutrino dark matter. For example, keV sterile neutrino dark matter has been discussed in gauge extensions of the SM [393, 394], models of composite Dirac neutrinos [395, 396], 331 models [397,398], models involving Froggatt-Nielsen mechanism [399] and in several other contexts [400–405]. A review of different models/mass generation mechanisms can be found in [270]. Various other issues related to keV sterile neutrinos can be found in [269, 406]. In this case, the neutrino mass matrix in the basis (NR′ , νe′ , νµ , ντ ) looks like
⋆ 0 0 0
0 ⋆ × ×
0 × × ×
0 × × ×
,
(5.8)
where the stars and crosses have the same meaning as given in eq. (5.6). However, here we have considered a set up in which the sterile neutrino has a mass around a few keV. We also have to make sure that the active-sterile mixing is very small, and within the valid range given by different X-ray experiments [285–290]. A very rough bound on the active-sterile mixing angle can be written as [382] 5 1 keV 2 −5 , (5.9) θ14 ≤ 1.8 × 10 MNR where MNR represents the Majorana mass of the right handed sterile neutrino. Therefore, we can treat the right handed neutrino as a decoupled state and work with the effective 3 × 3 matrix of the active Majorana neutrinos. Note that the (11) element of this 3 × 3 neutrino mass matrix in the basis (νe′ , νµ , ντ ) receives tree level as well as 99
one-loop level contributions whereas the other entries in this mass matrix comes only through various loop corrections. The size of this tree level contribution to (mν )11 is controlled by the model parameters and for suitable choices of the parameters one can < obtain a tree level value (mν )Tree ∼ 0.1 eV. Combining with the loop contributions one can then perform a fit to the three flavor global neutrino data. However, if we wish the keV sterile neutrino to be a candidate for dark matter then it should have the correct relic density (ΩN h2 ∼ 0.1) and must satisfy the constraints coming from X-ray experiments. An approximate formula for the relic density of sterile neutrinos via the Dodelson-Widrow (DW) [284] mechanism is [284, 285] 2 2 MNR sin 2θ 2 , (5.10) ΩN h ≈ 0.3 10−10 100 keV where ΩN is the ratio of density of sterile neutrinos to the total density of the universe and the present value of h is 0.673 [236]. There have been different experimental observations which put lower limits on the mass of the keV warm dark matter. For fermionic dark matter particles, a very robust lower bound on their mass comes from Pauli exclusion principle. By demanding that the maximal (Fermi) velocity of the degenerate fermionic gas in the dwarf spheroidal galaxies is less than the escape velocity leads to a lower bound on the mass of the sterile neutrino dark matter MNR > 0.41 keV [283]. This is the only model independent mass bound which holds for any fermionic dark matter. Model dependent bounds such as the ones coming from phase space density considerations have put strong lower bounds on the mass of the sterile neutrino acting as a warm dark matter candidate [283, 404, 405]. The authors of [407, 408] put a more stringent lower bound on the warm dark matter mass (MNR > 8–14 keV) by analyzing Lyman-α experimental data. In the context of left-right symmetric model a lower bound of 1.6 keV on the mass of the sterile neutrino warm dark matter has been discussed in [393]. In ref. [394], a lower limit of 0.5 keV on the sterile neutrino dark matter mass has been advocated in low scale left-right theory. In the present work we shall stick to the model independent lower bound of 0.4 keV as discussed above. Moreover, our parameter choices are such that the active sterile neutrino mixing is within the valid range of experimental observations. In order to get some idea about the numbers involved let us take two examples. With a choice of M1D = 805 GeV, M2D = 800 GeV, the R-symmetry breaking order parameter m3/2 = 5 GeV, tan β = 5.5, λS = 10−4 , f = 1.5 × 10−4 one produces a tree level mass of the active Majorana neutrino (mν )Tree ≃ 2.06 × 10−4 eV, and a sterile neutrino of mass around 0.47 keV. The active-sterile mixing is close to 4.35 × 10−7 , which is within the acceptable limit as observed by different X-ray experiments and 100
the relic density of the sterile neutrinos comes out to be ΩN h2 = 0.117. Again with another set of parameters such as M1D = 1200.001 GeV, M2D = 1200 GeV, m3/2 = 5 GeV, tan β = 5, λS = 1.2 and f = 1.55 × 10−4 , we obtain a sterile neutrino of 0.42 keV mass and the tree level active Majorana neutrino mass (mν )Tree ≃ 2.2 × 10−4 eV with an active-sterile mixing 3.64 × 10−7 and ΩN h2 = 0.114. In our scenario, the sterile neutrino fulfills the requirements of a good warm dark matter with appropriate mass of a few keV. In addition, the active-sterile mixing is also in the right ballpark to produce the correct relic abundance via DW mechanism. We shall now discuss different loop contributions to the neutrino mass matrix, which provide Majorana masses for the light active neutrinos with appropriate mixing between them.
5.4 One loop effects to generate neutrino mass In our model only the electron neutrino acquires a mass at the tree level. The other two neutrinos obtain their masses via one loop diagrams. At one loop level, the neutrino masses are generated from diagrams involving charged lepton-slepton loop, quarksquark loop and neutralino-Higgs loop respectively.
5.4.1
Charged lepton-slepton loop
We first consider the charged lepton-slepton loop which will generate Majorana mass terms for the neutrinos of all flavors [201]. We consider only the tau-stau loop as other charged lepton-slepton loops have very mild effect as far as neutrino mass is concerned. The contribution of the stau-tau loop (see, fig. 5.4) to the one loop neutrino mass
l˜
ν
λ
λ
ν
l Figure 5.4: Charged lepton-slepton loop
101
matrix is ml−s = ν
mτ m3/2 va 1 2 2 (16π ) m2τe
λ2133 λ133 λ233 0 2 mτe1 2 b βτ λ233 λ133 , (5.11) λ233 0 ln m2τe2 0 0 0
where we have used the expression of Aτ from eq. (4.55), which provide the necessary lepton number violation of two units in the scalar propagator. Here m2τe2 > m2τe1 represent the physical squared-masses of the staus and m2τe ≃ m2τe2 . In the above mass matrix we considered e = 1, and µ, τ = 2, 3 respectively, keeping in mind that λ is antisymmetric in the first two indices. Because of this antisymmetry property of the coupling λ, some of the elements in ml−s are zero. ν
5.4.2
Squark-quark loop
The squark-quark loop will also contribute to the light neutrino Majorana mass matrix [201]. Here we have taken into account bottom and strange squark-quark loop as shown in fig. 5.5. The contribution of quark-squark loop to the one loop neutrino mass matrix
d˜
ν
λ
′
λ
′
ν
d Figure 5.5: Quark-squark loop for d = b, s quarks and de = eb, se squarks.
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is " # m v 3 3/2 a βbb mνq−s = (16π 2 )2 me2b ! 0 λ′2 λ′133 λ′233 λ′133 λ′333 2 133 meb ′ ′ 1 ′2 ′ ′ + ms 0 λ233 λ233 λ333 ln mb λ133 λ233 me2b ′ ′ ′ ′ ′2 2 0 λ333 λ133 λ333 λ233 λ333 m3/2 va b 3 + βs 2 2 (16π ) m2se λ′2 2 0 0 0 122 m ′ ′ se1 ′ ′ ′ ′ ln + m m λ 0 λ λ λ λ b s 122 λ222 232 223 232 323 2 m s e 2 λ′322 λ′122 0 λ′332 λ′223 λ′332 λ′323
0 0 λ′223 λ′232 λ′223 λ′332 ln λ′323 λ′232 λ′323 λ′332
me2b 1 me2b 2
e b1
se2
se1
e b
e b2
se
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Neutralino-Higgs boson loop
We now consider the loop consisting of neutralino and Higgs propagators to generate Majorana mass of neutrinos [204,205]. This is shown in fig. 5.6. The loop contribution
h,H,A
ν
ν χ˜0
Figure 5.6: Neutralino-Higgs boson loop is proportional to the Majorana gaugino mass parameters M1 and M2 , which are much smaller than the mass of the corresponding physical neutralino states. In order to compute this loop we consider a simplified scenario where the singlet and the triplet states are integrated out and we are therefore left only with the Hu and νea fields as 103
(5.12)
mb ≫ ms , the dominant contribution to the neutrino mass matrix arises from the first two terms in eq. (5.12) as long as we assume m2se ≫ me2b . The other terms have a sub dominant contribution to the neutrino masses and therefore, it is safe to consider only the first two terms for computing the neutrino mass eigenvalues.
5.4.3
2 λ′122 λ′222 λ′122 λ′322 mse1 λ′2 λ′222 λ′322 ln . 222 2 m s e 2 λ′322 λ′222 λ′2 322
me2b ,eb m2se1 ,es2 are the physical squared-masses of the bottom squarks and strange squarks 1 2 respectively with m2 > m2 , m2 > m2 and m2 ≃ m2 , m2 ≃ m2 . Since βbb ≫ βbs and e b2
!
considered earlier in the discussion of the scalar sector, generating the CP-even physical states h, H and the CP-odd state A. Majorana mass term of a neutrino implies lepton number violation by two units. This is provided by the Majorana mass insertion in the neutralino propagator. The contribution to the neutrino mass matrix from this loop is given by (mν )11 =
M1 g2 X [Zγ2 − tan θW Zγ1 ]2 2 64π γ=1,2 2
2 cos αB0 (0, m2H , m2χe0 ) + sin2 αB0 (0, m2h , m2χe0 ) − sin2 βB0 (0, m2A , m2χe0 ) g2 X 2 M2 + [Z − tan θ Z ] γ2 W γ1 64π 2 γ=3,4 2 2 cos αB0 (0, m2H , m2χe0 ) + sin2 αB0 (0, m2h , m2χe0 ) − sin2 βB0 (0, m2A , m2χe0 ) ,(5.13)
where we have used
1 νeRa ≃ v1 + √ (H cos α − h sin α), 2 1 Hu0 ≃ v2 + √ (H sin α + h cos α), 2 1 νeIa ≃ √ (G cos β + A sin β). 2
(a = 1(e))
(5.14)
The summation in eq. (5.13) is taken over two pairs of nearly degenerate pseudoDirac heavier neutralino states mχe1,2 and mχe3,4 , which are predominantly bino (eb0 ) M1 and wino (w e0 ) respectively. Here we have assumed that |mχe1,2 | ≃ M1D ± and 2 M2 |mχe3,4 | ≃ M2D ± and for a given pair the neutralino mixing matrix elements Zγ2 2 and Zγ1 does not change for γ = (1, 2) and (3, 4). B0 is a Passarino-Veltman function and follow its definition as mentioned in [201, 204, 205]. It is important to note that this one loop contribution adds only to the (11) element of the effective 3×3 neutrino mass matrix. The other neutrino flavors do not get any contribution to their masses from this loop because the corresponding sneutrinos do not mix with Hu .
5.5 Numerical analysis We now present the results of our detailed numerical investigations to fit the lightest Higgs boson mass, neutrino masses and mixing angles as well as the keV sterile neutrino mass and its mixing with the active neutrino. As mentioned earlier in the text, we analyse two situations, one with small singlet and triplet couplings (λS and λT respectively), which would imply heavy top squarks to fit the lightest Higgs boson 104
mass whereas the other case with light top squark mass requires large λS and λT , which would provide significant quartic corrections generated at one loop to the lightest Higgs boson mass. A set of benchmark points for the latter case is provided below in table 5.2. Parameters M1D M2D tan β λS λT µ tS BµL mS mT vS vT f MR met1 = met2 bS bT m3/2 mh MNR (mν )Tree 2 θ14 ΩN h2
BP-2 1200.001 GeV 1200 GeV 5 1.25 λS tan θW ∼ 0.69 590 GeV (200)3 (GeV)3 −(200)2 (GeV)2 7.6 TeV 5.46 TeV -0.6 GeV 0.1 GeV 1.55 × 10−4 3.67 GeV 600 GeV 1 TeV 1 TeV 5 GeV 125.15 GeV 0.42 keV 2.17 × 10−4 eV 5.05×10−7 0.1121
BP-3 1000.001 GeV 1000 GeV 7 1.1 0.6 530 GeV (200)3 (GeV)3 −(200)2 (GeV)2 10 TeV 5.8 TeV -0.3 GeV 0.1 GeV 1.1 × 10−4 3 GeV 900 GeV 1 TeV 1 TeV 6 GeV 124.9 GeV 0.51 keV 1.86 × 10−4 eV 3.64×10−7 0.114
BP-4 800.001 GeV 800 GeV 10 0.98 0.54 650 GeV (200)3 (GeV)3 −(200)2 (GeV)2 18 TeV 1.9 TeV -0.1 GeV 0.05 GeV 1.0 × 10−4 3.16 GeV 1.2 TeV 1 TeV 1 TeV 3 GeV 123.7 GeV 0.43 keV 2.4 × 10−4 eV 5.53 × 10−7 0.122
Table 5.2: Benchmark points (with large λS and λT ) to calculate the lightest Higgs boson mass, light active neutrino mass, mass of the sterile neutrino as well as its mixing with active neutrino and the relic density of sterile neutrino dark matter via DW mechanism. An important fact to note down is the mixing between the charged leptons and charginos gives rise to a deviation in the couplings of the Z to charged leptons [304,306], which is constrained by electroweak precision measurements. Here, only one charged 105
lepton (e− ) mixes with the charginos, giving rise to a deviation in the vector and axial-vector coupling to the Z from those in SM. One obtains [304, 306] δgVi
= δgAi =
sin2 φ , 2
(5.15)
where sin φ = −
(m2e + g 2 va2 − (M2D )2 ) +
p [m2e + g 2 va2 + (M2D )2 ]2 − 4m2e (M2D )2 .(5.16) 2gva M2D
From the measured value of gAe = −0.5011 ± 0.00035, one obtains va ≤ 0.07. M2D
(5.17)
For a heavy Dirac wino mass ∼ 800 GeV, one finds an upper bound on tan β ≥ 2.7. On the other hand, the lower bound on tan β arises from leptonic Yukawa coupling, in particular fτ . In our scenario fτ ≡ λ133 , contributes to the ration Rτ ≡ Γ(τ → eν e ντ )/Γ(τ → µν µ ντ ). The resulting constraint is [304, 306] m τeR fτ < 0.07 , (5.18) 100GeV
considering stau mass close to 280 GeV, translates into an lower limit on tan β ∼ 19. For higher values of stau mass, this bound gets relaxed. For example, choosing stau mass to be around 1 TeV corresponds to tan β ≤ 70. In addition, in order to fit the neutrino data, we choose the Dirac bino mass very close to the Dirac wino mass. In table 5.3 we show benchmark points corresponding to small λT ∼ 10−4 . In this case, in order to fit the neutrino data, one does not require a strong degeneracy between M1D and M2D . It is worth mentioning once again that we have reduced the number of independent parameter of the model by assuming certain relations between some of them as shown in eqs. (4.51) and (4.50). One can observe from these two tables that the benchmark points provide a lightest Higgs boson mass around 125 GeV, a sterile neutrino mass in the keV range along with a very small active-sterile mixing and a very small tree level active neutrino Majorana mass. The mass and mixing of the sterile neutrino are in the allowed range of values coming from X-ray observations and it can be accommodated as a warm dark matter candidate in our model. In fig. 5.7(a) the contours of the tree level mass (mν )Tree of the light active neutrino in the (M1D -M2D ) plane exhibits the degeneracy required for these two parameters in order to have a small neutrino mass. Fig. 5.7(b) shows that the active-sterile mixing is also dependent on the degeneracy of M1D and M2D . Since the X-ray experiments provide very stringent constraints on the mixing, one is compelled to choose the Dirac gaugino masses close to each other. For these two plots, all the other parameters are 106
Parameters M1D M2D tan β λS λT µ tS BµL mS mT vS vT f MR met1 = met2 bS bT m3/2 mh MNR (mν )Tree 2 θ14 ΩN h2
BP-5 1018 GeV 1000 GeV 10 10−4 λS tan θW ∼ 5.5 × 10−5 700 GeV (200)3 (GeV)3 −(200)2 (GeV)2 12 TeV 11 TeV -0.1 GeV 0.1 GeV 0.92 × 10−4 3.69 GeV 6.5 TeV 1 TeV 1 TeV 3.5 GeV 126 GeV 0.41 keV 2.41 × 10−4 eV 5.85×10−7 0.119
BP-6 805 GeV 800 GeV 5.5 10−4 5.5 × 10−5 500 GeV (200)3 (GeV)3 −(200)2 (GeV)2 11.6 TeV 10.14 TeV -0.1 GeV 0.1 GeV 1.5 × 10−4 2.62 GeV 6.5 TeV 1 TeV 1 TeV 5 GeV 123.1 GeV 0.47 keV 2.06 × 10−4eV 4.35×10−7 0.117
BP-7 604 GeV 600 GeV 7 10−4 5.5 × 10−5 580 GeV (200)3 (GeV)3 −(200)2 (GeV)2 11 TeV 9 TeV -0.1 GeV 0.1 GeV 1.2 × 10−4 2 GeV 6.5 TeV 1 TeV 1 TeV 6 GeV 124.9 GeV 0.59 keV 1.6 × 10−4 eV 2.7 × 10−7 0.114
Table 5.3: Benchmark points with small λS and λT . fixed at the values of BP-5. In fig. 5.7(c) we show the variation of the sterile neutrino mass in the (f -m3/2 ) plane. The figure shows that for a fixed f , a larger gravitino mass produces a larger mass of the sterile neutrino. Again we expect this to happen because the gravitino is the order parameter of R-breaking and therefore, a larger gravitino mass creates a larger mass splitting between the sterile and the active neutrino, which would be zero in the absence of gravitino mass. This way the sterile neutrino mass gets more enhanced whereas the active neutrino mass becomes smaller. On the contrary, the active-sterile mixing decreases with m3/2 for a fixed f as shown in fig. 5.7(d). This is also expected, as a larger gravitino mass increases the mass of the sterile neutrino and thus reduces its mixing with the active neutrino. In figs. 5.7 (c) and 5.7(d), we
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Figure 5.7: Figure (a) represents contours of tree level light neutrino mass (mν )Tree in the (M1D –M2D ) plane. The black thick line represents (mν )Tree = 0.1 eV. The blue dotted line and the red dashed line represent (mν )Tree = 5 × 10−3 eV and 5 × 10−5 eV, 2 respectively. Figure (b) represents active-sterile mixing (θ14 ) in the same (M1D –M2D ) plane. The black thick line represents the contour of 5 × 10−6 and the blue dotted line and the red dashed line represent contours of 5 × 10−7 and 10−8 , respectively. 2 Figures (c) and (d) show the contours of sterile neutrino mass (MNR ) and θ14 in the (f –m3/2 ) plane. In fig. (c) the black thick line corresponds to MNR = 10 keV whereas the blue dotted line and the red dashed line show contours of MNR = 5 keV and 1 keV, respectively. In fig. (d) the black thick line shows a mixing of 6 × 10−8 and the blue 2 and the red line show θ14 = 3 × 10−8 and 10−8 , respectively. have fixed M2D at 1 TeV and M1D at 1.018 TeV, corresponding to BP-5 in table 5.3. In 2 fig. 5.8 the contours of (a) MNR , (b) θ14 and (c) (mν )Tree are shown in the (M1D –m3/2 ) 2 plane and in (d) contours of θ14 are presented in the (f –M1D ) plane for other parameter choices shown in BP-5. One can see from fig. 5.8(a) that for large values of M1D , the 108
sterile neutrino mass MNR is almost insensitive to M1D as expected from eq. (4.66). 2 However, the mixing θ14 increases with M1D for a fixed m3/2 and this is because of the fact that the light neutrino mass mν also increases with M1D for a fixed M2D and 2 2 m3/2 (see fig. 5.8(c)) and thus leads to an increase in θ14 . The variation of θ14 in the D (f –M1 ) plane can also be explained in a similar way by looking at eq. (4.63). 40
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Figure 5.8: Figure (a) represents contours of MNR in the (M1D –m3/2 ) plane. The red dashed line shows a sterile neutrino of mass 1.6 keV whereas the blue dotted and the thick black line shows a sterile neutrino mass of 2.5 keV and 4 keV, respectively. In fig. 2 (b) θ14 contours are shown in the same plane. The contours are 5 × 10−7 (black-thick), 8 × 10−8 (blue-dotted) and 3 × 10−8 (red-dashed), respectively. In fig. (c) we show the variation of tree level active neutrino mass (mν )Tree in the (M1D –m3/2 ). The outermost 2 contours represent (mν )Tree = 10−3 eV. Finally in fig. (d) we plot the contours of θ14 in the (M1D –f ) plane. 2 We have also presented two scatter plots in the (MNR –θ14 ) plane in figs. 5.9 and
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5.10 showing the allowed region after taking into account the constraints from the Xray experiments as well as the lower bound of 0.4 keV on the sterile neutrino mass, discussed earlier . On top of that we have also shown the points satisfying the correct dark matter relic density at 3σ (ΩDM h2 = 0.1199 ± 0.0027 at 1σ) as obtained from the recent observations of the PLANCK experiment [236]. In order to generate figs. 5.9 and 5.10 we have varied all the parameters, which play an important role in sterile neutrino mass and active sterile mixing. This plot 0
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MN (keV) Figure 5.9: Scatter plot in sterile neutrino mass and active-sterile mixing plane showing the allowed regions, in the heavy top squark scenario. The grey region shows the part of the parameter space excluded by the X-ray experiments. Lower bound on sterile neutrino mass excludes the blue region to the left of the vertical line and the thick red band represents the parameter points which satisfy correct dark matter relic density, at 3σ. has been generated by varying the model parameters in the following range: 800 GeV ≤ M1D , M2D ≤ 850 GeV, 1 GeV ≤ m3/2 ≤ 40 GeV, 10−5 ≤ f ≤ 8 × 10−4 and 2.7 ≤ tan β ≤ 17. We have kept λS ∼ 10−4 and so obviously these points represent the heavy top squark scenario. The grey region is disallowed by the constraints from X-ray observations whereas the red line at 0.4 keV and the blue region to its left is ruled out by the lower bound on sterile neutrino mass. Finally, note that by varying the top squark mass we ensured that all the scattered points in this plot produced the lightest 110
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MN (keV) 2 Figure 5.10: Scatter plot in the (MNR –θ14 ) plane showing the allowed regions, in the light top squark scenario. The colored(shaded) regions are the same as described in fig. 5.9.
Higgs boson mass in the range (123 − 127) GeV. In fig. 5.10, we show the results of our parameter space scan in the light top squark scenario. In this plot we have used λS ∼ 1.1 and 1 GeV ≤ m3/2 ≤ 40 GeV whereas f and tan β are varied in the same range as before. We discussed earlier that for large λS , the Dirac gaugino masses M1D and M2D need to be almost degenerate in order to fit a small tree level mass of the active neutrino. Therefore in this plot we fixed M1D = 1000.001 GeV and M2D = 1000 GeV. The grey and the blue regions again represent the parameter points ruled out by X-ray experiments and lower limit on the sterile neutrino mass respectively. We have also ensured that each and every point in this scattered plot produce a Higgs boson mass in the range (123 − 127) GeV. In fig. 5.11 we showed the variation of the relic density of the sterile neutrino with its mass. The blue scattered points respect the X-ray constraints and the Higgs boson mass within the range (123 − 127) GeV. The grey region shows the parameter space disfavored by the Pauli exclusion principle discussed earlier. The red-circle, green-triangle and orange-square points represent tree level neutrino mass greater than 10−5 , 10−4 and 10−3 eV respectively. We observe that in order to have a sterile neutrino as a warm dark matter candidate in our model, the neutrino mass at 111
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MN (keV) Figure 5.11: Scatter plot in the (MNR –ΩN h2 ) plane showing the allowed regions, in the heavy top squark scenario. The grey region describes the lower bound of the sterile neutrino mass, for it to become a warm dark matter candidate. All the scattered points satisfy the X-ray constraints. The red-circle scattered points show (mν )Tree > 10−5 eV. The green(triangle) and the orange(square) points represent (mν )Tree > 10−4 , 10−3 eV respectively. The horizontal band is the 3σ allowed region for the dark matter relic abundance. the tree level has to be very small.
5.5.1
Neutrino masses and mixing: Inverted Hierarchy
For inverted hierarchy the best-fit values of solar and the atmospheric neutrino mass squared differences and the three mixing angles are as follows [148,149] ∆m221 = 7.62 × 10−5 eV2 , |∆m231 | = 2.43 × 10−3 eV2 , θ12 = 34.4◦ , θ23 = 50.8◦ and θ13 = 9.1◦ , where ∆m2ij ≡ m2i − m2j . The neutrino mass matrix can be obtained using m1 0 0 mν = UP M N S 0 m2 0 UPT M N S , 0 0 m3
112
(5.19)
where the standard PMNS matrix UPMNS , with vanishing CP violating phases is of the form c12 c13 s12 c13 s13 (5.20) UPMNS = −s12 c23 − c12 s23 s13 c12 c23 − s12 s23 s13 s23 c13 , s12 s23 − c12 c23 s13 −c12 s23 − s12 c23 s13 c23 c13
and m1 , m2 and m3 are the neutrino mass eigenvalues. Since the oscillation experiments are sensitive only to the mass squared differences, therefore for simplicity, we can assume the lightest neutrino mass m3 to be zero in this case. Thus we have m21 = |∆m231 | and m22 = ∆m221 + m21 . For example, using the central values of the oscillation parameters mentioned above, the three flavor neutrino mass matrix in the inverted hierarchy case comes out to be 0.049 −0.0059 −0.0052 (5.21) mIH 0.0211 −0.024 . ν = −0.0059 −0.0052 −0.024 0.0311
The three flavor active neutrino mass matrix in our model is composed mainly of the one-loop radiative corrections as discussed above because the tree level contribution to (mν )11 is very small in order to have the correct relic density of the keV sterile neutrino dark matter. We shall now present the results of our numerical analysis in order to fit the three-flavor global neutrino data in our model in the inverted hierarchy scenario. We shall confine ourselves in the parameter region which will produce the correct value for the lightest Higgs boson mass and where the sterile right handed neutrino can be a good candidate for keV warm dark matter. Note that there are contributions from the tau-stau, quark-sqaurk and neutralinoHiggs loop to the (11) element of the neutrino mass matrix (neglecting the tree level contribution). The trilinear R-parity violating couplings involved in these loop contributions are λ133 and λ′133 , which are identified with the tau and the bottom Yukawa couplings. The other parameters which play a crucial role in order to fit the (11) element of the neutrino mass matrix are tan β, m3/2 and me2b (assuming that the stau-tau loop contribution is smaller than the other loop contributions). However, for a fixed value of tan β the trilinear couplings λ133 and λ′133 are fixed and thus this leaves us with only two parameters (m3/2 and me2b ) in terms of which (mν )11 can be fitted. Figure 5.12 presents the contour plots of (mν )11 in the (m3/2 –meb ) plane. Here the blue-dotted line corresponds to the maximum value of (mν )11 whereas the red-dashed line corresponds to the minimum value of (mν )11 . These maximum and minimum values are obtained by varying the oscillation parameters within the 3σ range. Moreover, we also draw a third contour (the black-bold line) which represents the upper bound on (mν )11 113
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8
10
m32 HGeVL Figure 5.12: Contours of (mν )11 in the (m3/2 –meb ) plane for inverted hierarchy and tan β = 10. See text for details. as obtained by the neutrinoless double beta decay experiments kamLAND-Zen and EXO-200 [409, 410]. In order to produce this figure we fixed all the other parameters at values corresponding to BP-5. The grey line represents a gravitino mass of 3.5 GeV. On the right hand side of this grey vertical line the mass of the sterile neutrino is ≥ 0.41 keV. Moreover, this plot shows the allowed range of bottom squark mass, required to fit (mν )11 for fixed values of tan β and m3/2 . For example, we see that for tan β = 10, m3/2 = 3.5 GeV, the bottom squark mass is allowed in the range (1228-1242) GeV. For larger values of tan β the allowed range of meb increases. Bounds on the trilinear RPV couplings for inverted hierarchy By varying the neutrino oscillation parameters within their 3σ allowed ranges we can get maximum and minimum values for different neutrino mass matrix elements. These allowed ranges of neutrino mass matrix elements can be translated into a lower and an upper bound on different trilinear R-parity violating couplings involved. With the choice of other parameters as presented in BP-5, we present in table 5.4 the bounds on λ and λ′ type couplings as functions meb and m3/2 for a particular value of tan β = 10. Note that these bounds are independent of the choices of other parameters shown in different benchmark points because they are calculated from the neutrino mass matrix 114
′
Table 5.4: Bounds on λijk and λijk couplings for tan β = 10 and for inverted hierarchy Couplings |λ′233 | |λ′333 | |λ′232 λ′223 | |λ′223 λ′332 | |λ′323 λ′332 |
Bounds for BP-5
2 10 GeV meb (2.37 × 10−7 − 1.03 × 10−6 ) 100 GeV m3/2 2 10 GeV meb −7 −6 (2.84 × 10 − 1.04 × 10 ) 100 GeV m3/2 2 10 GeV meb −5 (2.11 − 4.3) × 10 100 GeV m3/2 2 10 GeV meb −5 (2.82 − 3.34) × 10 100 GeV m3/2 2 m e 10 GeV b (2.38 − 4.64) × 10−5 100 GeV m3/2
Existing bounds 6.8 × 10−3 cos β 1.305 cos β (2 × 10−3 ) cos2 β (e νL2 u eL3 )2 -
elements which get contributions only from the one loop corrections.
5.5.2
Neutrino masses and mixing: Normal hierarchy
In the case of normal hierarchy best-fit values of the neutrino oscillation parameters are given as [148, 149] ∆m221 = 7.62 × 10−5 eV2 , |∆m231 | = 2.55 × 10−3 eV2 and the three mixing angles are θ12 = 34.4◦ , θ23 = 51.5◦ and θ13 = 9.1◦ . With these values and assuming that m1 = 0, m22 = ∆m221 and m23 = |∆m231 | the neutrino mass matrix in the case of normal hierarchy turns out to be 0.0039 0.0082 0.0014 H (5.22) mN = 0.0082 0.0318 0.021 . ν 0.0014 0.021 0.023 Figure 5.13 presents the contour of (mν )11 in the (m3/2 –meb ) plane in the case of normal hierarchy. Here the blue-dotted line corresponds to the maximum value of (mν )11 = 0.005 eV whereas the red-dashed line corresponds to the minimum value of (mν )11 = 0.003 eV. Once again these maximum and minimum values are obtained by varying the oscillation parameters within their 3σ range. The right side of the 3.5 GeV gravitino mass line can produce a keV sterile neutrino warm dark matter with a mass greater than 0.41 keV. The values of other parameters correspond to BP-5. Here we have chosen a small λS = 10−4, which requires heavy top squarks to produce a ∼ 125 GeV Higgs boson. Looking at this figure one can also see the range of bottom squark mass required to fit the value of (mν )11 for a fixed value of tan β and m3/2 . If we take a large value of λs , then the meb mass range changes slightly but the essential feature remains the same.
115
HmΝ L11 =.003 eV HmΝ L11 =.005 eV m32 =3.5 GeV
1235.5
mb HGeVL
1235.0
1234.5
1234.0
1233.5
1233.0 0
2
4
6
8
10
m32 HGeVL Figure 5.13: Contours of (mν )11 in the (m3/2 –meb ) plane for normal hierarchy case and tan β = 10. Bounds on the trilinear RPV couplings for normal hierarchy One can also constrain the trilinear R-parity violating couplings in the case of normal hierarchy after analyzing the other elements of the neutrino mass matrix in the light of neutrino data. The resulting bounds are shown in table 5.5. ′
Table 5.5: Bounds on λijk and λijk couplings for tan β = 10 and for normal hierarchy Couplings
Bounds for BP-5
|λ′233 |
meb GeV 100 GeV m3/2 2 10 GeV meb −8 −7 (3.74 × 10 − 6.11 × 10 ) 100 GeV m3/2 2 10 GeV meb −5 (2.5 − 4.7) × 10 100 GeV m3/2 2 10 GeV meb −5 (2.4 − 3.0) × 10 100 GeV m3/2 2 m e 10 GeV b (2.5 − 4.69) × 10−5 100 GeV m3/2
|λ′333 | |λ′232 λ′223 | |λ′223 λ′332 | |λ′323 λ′332 |
(8.07 × 10−7 − 1.2 × 10−6 )
2 10
Existing Constraints 6.8 × 10−3 cos β 1.305 cos β (2 × 10−3 ) cos2 β (e νL2 u eL3 )2 -
5.6 Concluding remarks In this chapter we have seen that the Higgs boson mass at 125 GeV can be accommodated even at the tree level for order one neutrino Yukawa coupling, i.e., f ∼ O(1) and 116
low tan β (tan β ≤ 5). Larger values of tan β or smaller f dilutes the tree level contribution rapidly. In such a scenario, the quartic corrections generated at one-loop level becomes important. This radiative corrections are substantial when λS , λT ∼ O(1), which alleviates the Higgs boson mass. An order one f also gives a very light bino-like neutralino with mass around a few hundred MeV. Some cosmological implications of this light neutralino are also mentioned in this chapter. As explained in great detail that this model provides a tree level mass to one of the active neutrinos. To explain the recent neutrino data, one has to give mass to at least two neutrinos. Hence, we took into account all the possible radiative corrections. To fit the neutrino data, we were able to put constraint on the relevant parameters, i.e., trilinear RPV couplings. Albeit model dependent, these bounds appear to be much more stringent than the available ones. Another important aspect of this model lies in the DM sector. It is shown that for f ∼ O(10−4 ), we obtain a sterile neutrino with mass around a few keV. Such sterile neutrinos are excellent candidates for WDM and can produce the appropriate relic density. Detail scan of the parameter space reveals that for a large portion of the parameter space, this keV DM candidate can easily be accommodated in the present context. Recently, two independent studies [411, 412] have analysed the data from XMMNewton X-ray observatory and identified a weak line at energy close to 3.5 keV. Such an X-ray line might emerge from a decaying DM of mass around 7 keV. In the next chapter, we will try to fit this result with our sterile neutrino DM candidate. We will also discuss that such a precise measurement of the DM mass and active-sterile mixing helps to constrain the parameter space in a significant way.
117
Chapter 6 7 keV Sterile neutrino dark matter in U (1)R− lepton number model 6.1 Introduction We are fortunate to live in an era enriched with many experimental breakthroughs and results especially in the area of astro-particle physics and cosmology. The most recent one is the identification of a weak line at E ∼ 3.5 keV in the X-ray spectra of the Andromeda galaxy and many other galaxy clusters including the Perseus galaxy cluster, observed by XMM-Newton X-ray Space observatory [411, 412]. Subsequently, it was found that this result is also fully consistent with the data coming from the centre of Milky way galaxy [413]. The observed flux and the best fit energy peak are at Φγ = 4 ± 0.8 × 10−6 photons cm−2 sec−1 ,
Eγ = 3.57 ± 0.02 keV.
(6.1)
Atomic transitions in thermal plasma may or may not account for this energy. Nevertheless, the concept of a dark matter, providing the possible explanation regarding the appearance of this photon line becomes extremely important. This result can be explained by a sterile neutrino [414–419], axion or axion like warm dark matter [420–423], axino [424–426], excited dark matter [427, 428], gravitino [429, 430] and keV scale LSP [431] as decaying dark matter. Other interesting scenarios with an annihilating scalar dark matter [432], decaying Majoron [433] and a keV scale dark gaugino [434] have also been considered in this context. In this chapter we consider sterile neutrino in a U(1)R− lepton number model, which could provide a possible explanation for the emergence of the photon line. The observed flux and the peak of the energy readily 118
translates to an active-sterile mixing in the range 2.2 × 10−11 < sin2 2θ14 < 2 × 10−10 and the mass of the sterile neutrino dark matter MNR = 7.06 ± 0.05 keV [412]. The plan of this chapter is as follows. In section 6.2 we present bounds on the parameter space pertaining to the measurements of the sterile neutrino mass and estimated active sterile mixing. In section 6.3 the essential features of the sterile neutrino as a keV warm dark matter candidate are discussed and its production mechanism and the dominant decay modes relevant to our model are highlighted. In section 6.4 we briefly present a discussion related to the cosmology of the gravitino in this model with a few GeV mass and finally in section 6.5 we summarise our results with some concluding remarks.
6.2 Bounds on the parameter space Such a precise measurements of sterile neutrino mass and active sterile mixing also helps to constrain the parameter space as we will discuss now. In fig. (6.1), we show in the (f − tan β) plane the contour of sterile neutrino mass fixed at 7.06 keV and also two different contours of sin2 2θ14 , fixed at the lower and upper limit of 2.2 × 10−11 and 2 × 10−10 respectively. We have chosen the gravitino mass, m3/2 to be 10 GeV and M1D = 900 GeV, keeping a degeneracy between the Dirac gaugino masses, ǫ ≡ (M2D − M1D ) = 10−4 GeV. We have also fixed µu = 750 GeV, λS = 1.1, vS = −0.1 GeV and vT = 0.1 GeV. The sterile neutrino mass contour can be easily explained by looking at eq. (4.66). Similarly from eq. (4.63), eq. (4.66) and eq. (4.67), it is straightforward to show that sin2 2θ14 goes as 1/(1 + tan2 β). This means that for smaller tan β one would expect larger mixing angle for fixed values of other parameters. This is also evident from fig. 6.1. Furthermore, for larger Dirac gaugino masses, the active neutrino mass gets reduced (see eq. (4.63)), which also implies a reduction in the active-sterile mixing. Looking at fig. 6.1, we observe that the largest value of the active-sterile mixing, required to explain the observed photon line flux at an energy E ≈ 3.5 keV, corresponds to the minimum value of tan β. In fact, for this particular case shown in fig. 6.1, (tan β)min ≈ 11.3. Similarly the smallest active-sterile mixing (sin2 2θ14 = 2.2 × 10−11 ) provides the maximum allowed value of tan β, which in this case turns out to be (tan β)max ≈ 33. In order to obtain an analytical relationship between the lower limit of tan β and M2D , we can solve for tan β using eq. (4.67), with sin2 2θ14 = 2 × 10−10
119
40
sin 2 2Θ 14 =210-10 35
sin 2 2Θ 14 =2.210-11 M NR =7.06 ke V
30
tan Β
25
20
15
10
5
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
f Figure 6.1: The contour in the black thick line represents a sterile neutrino mass of 7 keV. Contours in red (dotted) and blue (dashed) colours show active-sterile mixing 2.2 × 10−11 and 2 × 10−10 respectively. and MNR = 7.06 keV. This gives rise to (tan2 β)min =
4v 2 {gλT v 2 (M2D − M1D )}2 − 1, (1.4 × 10−15 GeV)[M1 α′ 2 + M2 δ ′ 2 ]
(6.2)
where 2(M2D )2 µu √ 2 + 2v λs M2D , g′ √ ≃ 2M2D v 2 λT .
α′ ≃ δ′
(6.3)
In a similar way an analytical expression for the upper limit of tan β can also be derived. Figure 6.2 shows the lower and upper limits of tan β as a function of M2D , for µu = 700 GeV, m3/2 = 10 GeV and ǫ = 10−4 GeV. We have fixed λS at the previously mentioned value. The horizontal grey line shows the upper limit on tan β arising from the contribution of the leptonic Yukawa coupling, fτ ≡ λ133 to the ratio Rτ ≡ Γ(τ → eν e ντ )/Γ(τ → µν µ ντ ) as mentioned in eq. (5.18). Considering stau mass, close to 280 GeV, translates into an upper limit on tan β ≈ 19. For higher stau mass this upper limit on tan β gets relaxed. The blue dashed line shows the lower bound on tan β, as a function of M2D , arising from the precision measurements of the deviations in the couplings of the Z boson to charged leptons [306] (see eq. (5.17)). 120
50
Htan ΒL m in m ax
tanΒm in from X-ray tanΒm ax from X-ray
40
Up p er bound from R Τ Lower bound from Z l + l -
30
20
10
500
1000
1500
2000
M 2D HGeVL Figure 6.2: Showing the lower and upper limits of tan β from X-ray analysis as a function of M2D for µu = 700 GeV, m3/2 = 10 GeV and ǫ = 10−4 GeV.
We infer from the above discussions, that in a large region of the parameter space, the lower limit on tan β, satisfying the estimated mass and mixing of the sterile neutrino dark matter particle coming from the recent observation of an X-ray line signal at energy 3.5 keV is stronger than the lower limit on tan β coming from the electroweak precision measurements. On the other hand, the upper limit on tan β coming from the X-ray observations becomes stronger than the upper limit arising from the τ Yukawa coupling contribution to Rτ only for higher values of M2D as shown in fig. 6.2 for specific choices of µu and m3/2 . Combining these lower and upper limits on tan β from X-ray observations and the lower bound on tan β obtained from the measurement of Rτ , we can find a range of M2D that is allowed. For smaller values of µu and m3/2 , the lower and upper limits of M2D shift to higher values (see eq. (6.2)). We also observe from fig. 6.1 that the allowed values of f is of the order of 10−4 . Such a small value of f , implies negligible extra contribution to the tree level Higgs boson mass. Therefore, to elevate the Higgs boson mass to 125 GeV, we have to rely on the loop corrections. Sizable radiative corrections are obtained if λS , λT are large (O(1)) and this would imply nearly degenerate Dirac gaugino masses (ǫ ∼ 10−4 GeV) in order to have the active-sterile mixing sin2 2θ14 ∼ 10−11 and a tree level active < neutrino mass ∼ 0.05 eV. The other case, which can relax this strong degeneracy between Dirac gaugino masses, corresponds to the case of small λS , λT ∼ O(10−4 ), which implies multi-TeV top squark to fit the Higgs boson mass. Therefore, this model
121
provides a very interesting possibility where we can connect the Higgs sector with the neutrino sector (both active as well as sterile neutrino).
6.3 Right handed neutrino as a keV warm dark matter To accommodate sterile neutrino as a warm dark matter candidate, it is very important to make sure that the active sterile mixing is very small [285–290] and within the valid range of different X-ray experiments, as discussed in section (5.3.2). Along with the strict bound coming from different X-ray experiments, the keV sterile neutrino must produce the correct relic density ΩN h2 ∼ 0.1, in order to identify itself with the warm dark matter. An approximate formula for the relic density of sterile neutrinos, produced in the early universe with negligible lepton asymmetry via non-resonant oscillations with active neutrinos, known as the Dodelson-Widrow (DW) [284] mechanism is derived in eq. (3.37). Different experimental observations have also put lower limits on the mass of the keV warm dark matter. A very robust bound on the mass of fermionic dark matter comes from the Tremaine-Gunn bound [282, 286] which is MNR > 0.41 keV. Model dependent bounds on the mass of the warm dark matter are much more stringent and obtained from analysing Lyman-α experiment [407, 408]. In fig. 6.3 we present a scatter plot by scanning the parameter space of our model and also show the compatibility of those points with the current experimental findings. The red circles are the points obtained by varying the parameters as 500 GeV < MD 1 < −5 −3 1.2 TeV, 10 < f < 10 , 2.7 < tan β < 17, 400 GeV < met1 ,et2 < 1.2 TeV, keeping ǫ ≡ (M2D − M1D ) ∼ 10−4 GeV. µu and λS are fixed at 750 GeV and 1.1 respectively (λT = λS tan θW ∼ 0.6). All these points respect a Higgs boson mass in between 124.7 GeV and 126.2 GeV avoiding any tachyonic scalar states. Similar plot can also be generated where λT ∼ 10−5. Therefore, to fit the Higgs boson mass in that case, one requires met > 5 TeV. However, the degeneracy between > M1D and M2D is somewhat lifted where ǫ ∼ 1 GeV. The horizontal yellow band in fig. 6.3 is ruled out by the Tremaine Gunn bound, which implies MNR < 0.4 keV [282,286]. The blue region is excluded by taking into consideration the diffuse X-ray background [301]. Cluster X-ray bound rules out a region in the mass-mixing plane by taking into consideration XMM-Newton observations from the Coma and Virgo clusters [302]. Constraints from the cosmic X-ray background (CXB) rules out the region in red stripes [301]. Chandra observation of M31 [303] rules out the region in grey. The light blue line corresponds to the correct relic density provided by the sterile neutrino warm dark matter via DW mechanism. The light blue 122
Diffuse X-ray background
10
Cluster X-ray
1
CXB Milky Way
★
MR N (keV)
M31
DW
100
Tremaine Gunn 10-1 -13 10
10-12
10-11
10-10
2
10-9
10-8
10-7
sin 2θ14 Figure 6.3: The red (grey) points in the mass-mixing plane are obtained by scanning the parameter space as mentioned in the text. The relevant bounds shown in this figure are subsequently discussed in the text.
region above this line marked as DW is ruled out because of the over abundance of sterile neutrino dark matter. The horizontal and vertical lines show the region in the mass and mixing plane consistent with the observed 3.5 keV X-ray line with more than 3σ significance. The black star corresponds to the best fit point. It is clearly evident from this figure that such a small mixing is completely in conflict with the DW production mechanism of sterile neutrinos. However, resonant production of sterile neutrinos in the presence of a lepton asymmetry in primordial plasma, more commonly known as the Shi-Fuller mechanism as discussed earlier in section 3.6.2, can be very important and produce correct relic abundance of the keV sterile neutrinos [291, 292]. Recent studies have shown that a cosmological lepton asymmetry L ∼ O(10−3 ) is capable of producing correct relic density of 0.119 [415]. It was explained in [293–300] that active-sterile neutrino oscillations can themselves create a cosmological lepton number of this magnitude, assuming that the number of sterile neutrinos is negligible to start with. Such a possibility can be easily conceived in our model to generate a large lepton asymmetry. 123
We note in passing that sterile neutrino production in non-standard cosmology with low reheating temperature (∼ a few MeV) has also been discussed in the literature [435– 437]. If the universe has undergone inflation and was never reheated to a temperature above a few MeV then the relic abundance of the sterile neutrinos can be written as 3 sin2 2θ M R T R 14 N , (6.4) ΩN h2 = 10−7dα 10−10 10 keV 5 MeV
where dα = 1.13, assuming that the sterile neutrino couples only with νe as in our case. It is obvious from the above expression that for allowed values of sin2 2θ14 and MNR (from the recent X-ray observation) this production mechanism will give rise to severe under abundance of sterile neutrinos.
1030 1029
τN(sec)
1028 1027 1026 1025 1024 1023
0
5
10
15
20
MRN(keV) Figure 6.4: The red scattered points show the sterile neutrino lifetime as a function of the sterile neutrino mass. The black vertical line represents the 7 keV mass of the sterile neutrino.
6.3.1
Sterile neutrino decay
The most dominant decay mode of the sterile neutrino is N → 3ν. The corresponding decay rate for this process is given by [285] Γ3ν = 8.7 × 10−31 sec−1
sin2 2θ M R 5 14 N . 10−10 1 keV
124
(6.5)
The principal radiative decay mode of the sterile neutrino which is of concern here is N → νγ and the decay width is sin2 2θ M R 5 14 N . (6.6) Γνγ = 1.38 × 10−32 sec−1 10−10 1 keV This decay produces a monochromatic photon line at Eγ = MNR /2. From fig. (6.4) we can see that the lifetime of the sterile neutrino is much larger than the age of the universe, and hence is certainly a good candidate for DM.
6.4 Gravitino cosmology As mentioned earlier, the gravitino mass is the order parameter of R-breaking. If the mass is around a few GeV, it can be a candidate for cold dark matter [438]. However, in our scenario, the gravitino is an unstable particle and decays to an active/sterile neutrino and a monochromatic photon. The tree level decay mode into an active e → γνe is suppressed by the very small mixing Ue (∼ 10−7 ) neutrino final state G bνe between the bino and active neutrino νe [439]. Interestingly, in our model the most e → Nγ) dominant decay mode of gravitino is into a photon and a sterile neutrino (G and the decay width is given as ΓG→N e γ ∼
|UebN |2 m33/2 32πMP2
,
(6.7)
where UebN is the bino sterile neutrino mixing angle. Because of the presence of the b c Sb in the superpotential and the bino Dirac mass term in the Lagrangian, term MR N the tree level bino sterile neutrino mixing is not strongly suppressed (∼ 10−2 ). For the sake of completeness, let us mention that at the one loop level the decay e → γνe occurs [440–443] via trilinear R-parity violating coupling λ′133 which we have G identified with the bottom Yukawa coupling. We have checked that this process is also e → Nγ. The one-loop contribution to suppressed compared to the tree level decay G e → Nγ is negligible because of small active-sterile mixing. the decay G Taking into account the most dominant decay mode of the gravitino in the sterile neutrino plus photon final state, for a 10 GeV gravitino mass, the lifetime is close to 1015 sec. Therefore, to satisfy the experimental constraints coming from the diffuse photon background, one has to consider a scenario where the gravitino density is very much diluted. In order to provide a quantitative analysis we note that for a gravitino of mass 10 GeV the limit on the diffuse photon flux is around 6.89 × 10−7GeVcm−2 sec−1 [444]. This can be translated into a bound on the gravitino relic density and we find −2 2 10 2 −13 , (6.8) Ω3/2 h < 4.34 × 10 UebN 125
for a 10 GeV gravitino. Note that this bound depends strongly on the mass of the gravitino and will get relaxed for a smaller gravitino mass. To satisfy such a strong bound on the gravitino relic density, one must account for a very low reheating temperature. If the reheating temperature is above the SUSY scale, the gravitino relic density would be too large [445]. Therefore, the reheating temperature must lie much below the SUSY threshold. Following [304], we see that if the reheating temperature is below the SUSY threshold, the gravitinos are produced by thermal scattering with neutrinos and bottom quarks. Using the analysis of [304] for production of gravitinos, we obtain an upper bound on the reheating temperature for a 10 GeV gravitino as TR
va 2/7 meb 4/7 < 127 30GeV 500GeV
10−2 UebN
2/7
GeV. (6.9)
Such a low reheating temperature might have important implications in the context of different baryogenesis and leptogenesis scenarios. However, a detailed study in this context is beyond the scope of this chapter.
6.5 Concluding remarks In this chapter we have studied a weak X-ray line of energy around 3.5 keV as observed by XMM-Newton telescope in the light of our model. This signature may come from a decaying DM with mass around 7 keV. As explained earlier, our model provide a keV sterile neutrino naturally which can explain this interesting result. Such a precise measurement also helps to constrain the parameter space in our scenario. In fact, we have been able to show valid regions of tan β as a function of the Dirac wino mass M2D . In order to produce a 7 keV sterile neutrino DM, we also require a gravitino of mass around a few GeV. We have discussed some cosmological implications of a 10 GeV decaying gravitino. When studied in conjunction with the constraints coming from diffuse photon background, this result puts a very stringent bound on the gravitino relic density. This again eventually imposes an upper limit (≤ 130 GeV) on the reheating temperature of the universe. On the other hand, the collider based experiments are making precision measurements in the Higgs sector. However, the recent results seem to be consistent with the predictions of the SM. In this context it is of utmost importance to study the properties of the Higgs boson and see if such a situation can be conceivable in our scenario or not. 126
Chapter 7 h → γγ in U (1)R− lepton number model with a right-handed neutrino 7.1 Introduction Precision measurements in the Higgs sector are of utmost importance as it is the only way to certify if the observed boson is the SM Higgs boson or not. Hence, it is essential to measure the different decay modes of the Higgs boson and validate the result with the SM expectation. Experimentally almost all the decay channels have been probed with reasonable precision. Out of these, results in the h → γγ channel have attracted a lot of attention in recent times. The reason is two-fold: first, this is the discovery mode of the Higgs boson and second, being a loop induced process it may potentially carry indirect hints of new physics. For example, the ATLAS σ(pp→h→γγ) collaboration reported µγγ = 1.17±0.27 [446], where µγγ = σ(pp→h→γγ) SM . On the other hand, CMS collaboration reported a best-fit signal strength in their main analysis [447] where, µγγ = 1.14+0.26 −0.23 . Moreover, a cut-based analysis by CMS produced a slightly different value, which is quoted as µγγ = 1.29+0.29 −0.26 . Although the best fit values appear to deviate from the SM expectations, they are still in agreement with the latter within experimental uncertainties. Therefore, it provides an opportunity to constrain several BSM physics scenarios and to find if such a scenario is still consistent with the data. Detailed studies have already been carried out for this particular channel. For example, h → γγ is studied in a wide variety of supersymmetric (SUSY) models namely, the minimal supersymmetric standard model (MSSM) [448–469], its next-tominimal version (NMSSM) [470–478], the constrained MSSM (CMSSM) [479–484] and also in (B-L)SSM [485–488], µνSSM [489], left-right supersymmetric models [490], and in U(1)′ extension of MSSM [491]. In [492], a triplet-singlet extension of MSSM has
127
been studied and µγγ is computed. In ref. [493], the correlation between h → γγ and h → Zγ in MSSM, NMSSM, CMSSM and nMSSM models is elucidated. Motivated by these results, we would like to investigate the Higgs to diphoton mode in the context of U(1)R -lepton number model, which is augmented by a single righthanded neutrino superfield. As we shall see later in this chapter, the scenario under consideration would have significant bearing on µγγ . This is because one can now afford rather light top squarks which potentially affect the resonant production rate of the lightest Higgs boson. Furthermore, presence of a very light neutralino opens up new decay modes of the Higgs bosons which in turn may suppress its diphoton branching fraction. Also, in general, presence of new particle states and their involved couplings would affect the proceedings. The chapter is organised as follows. A thorough analysis of µγγ requires the knowledge of both production and decays of the Higgs boson. In section 7.2 issues pertaining to the production of Higgs boson in the present scenario is discussed in some detail. Analytical expressions of Higgs boson decaying to two photons in our model are also presented. A dedicated analysis to the computation of the total decay width of the Higgs boson is also performed. In section 7.3, we compute µγγ and show its variation with relevant model parameters, along with the points representing the 7 keV sterile neutrino warm dark matter in this model. We make some concluding remarks in section 7.4. The Higgs boson couplings and partial decay widths to charginos and neutralinos in this model are shown explicitly in the Appendix B and C.
7.2 Contributions to µγγ The resonant production of the Higgs boson at the LHC, with the dominant contribution coming from gluon fusion, is related to its decay to gluons by σ ˆ (gg → h) = 2 3 π Γ(h → gg)/8mh. Thus, µγγ can be expressed entirely in terms of various decay widths of the Higgs boson as follows [466, 467]: σ(pp → h → γγ) , σ(pp → h → γγ)SM Γ(h → gg) ΓSM Γ(h → γγ) = , TOT . . SM Γ(h → gg) ΓTOT Γ(h → γγ)SM −1 = kgg .kTOT .kγγ ,
µγγ =
(7.1)
Γ(h→gg) σ ˆ (gg→h) ΓTOT where we use kgg ≡ σˆ(gg→h) SM = Γ(h→gg)SM and kTOT = ΓSM , ΓTOT being the total decay TOT width of the Higgs boson in the present scenario. The decay of h → γγ is mediated mainly by the top quark and the W ± -loops in the SM and in addition, by top squark,
128
charged Higgs and chargino loops in our scenario. In the subsequent discussion we investigate these widths in some detail. As discussed in chapter 4, note that in this model we have integrated out the b d ) superfield and the sneutrino νea (a = 1(e)) plays the role of the down type Higgs (H down type Higgs boson acquiring a large non-zero vev. The sneutrino (e νa ) couples to charged leptons (second and third generation) and down type quarks via R-parity violating couplings which are identified with the standard Yukawa couplings. Thus, the couplings of the Higgs boson to charged leptons and quarks remain the same as in the MSSM. This is apparent from the first term given in eq. (4.11).
7.2.1
The decay h → gg
The partial width of the Higgs boson decaying to a pair of gluons via loops involving quarks and squarks is given by 2 X GF αs2 m3h X h h h h √ gQ AQ (τq ) + gQe AQe (τQe ) , Γ(h → gg) = 3 36 2π Q e
(7.2)
Q
where τi = m2h /4m2i , GF is the Fermi constant, αs is the strong coupling constant and 3 [τ + (τ − 1)f (τ )] /τ 2 , 2 3 AhQe (τ ) = − [τ − f (τ )] /τ 2 , 4 AhQ (τ ) =
(7.3)
with f (τ ) given by √ 2 τ ≤ 1, arcsin τ √ 2 −1 f (τ ) = 1 1+ 1−τ √ τ > 1. − iπ − log 4 1 − 1 − τ −1
(7.4)
The couplings are given by
cos α , sin β sin α h , gQ (d) = − cos β m2f h m2Z f h gQe = g ∓ 2 (I3 − ef sin2 θW ) sin(α + β), 2 Q mQe mQe
h gQ (u) =
(7.5)
where the angle α is defined in eq. (4.28) and tan β = vu /va . The couplings of the Higgs boson with the left- and the right-handed squarks are exactly the same as in the
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MSSM. However, one can neglect the mixing between the left- and the right-handed squarks due to the absence of the µ-term and the A-terms1 . As far as the production of the Higgs boson is concerned, we shall show later that a rather light top squark with mass around 200 − 300 GeV enhances the value of kgg compared to the SM. The SM and the MSSM results for the decay h → gg can be found in [97, 494, 495].
7.2.2
The decay h → γγ
In the SM, the primary contribution to the decay h → γγ comes from the W boson loop and the top quark loop with the former playing the dominant role. In supersymmetric models, the charged Higgs (H ± ), top squark (e t) and the chargino (e χ± ) provide extra contributions in addition to the W boson and the top quark loop. The authors of ref. [467] have noted that the relative strengths of the loop contributions involving the vector bosons, the fermions and the scalars with mass around 100 GeV follow a rough ratio of 8 : 1.5 : 0.4. Nonetheless, a light charged Higgs boson (H ± ) could contribute substantially if one considers a large hH + H − coupling. However, since the triplet vev is small, the contribution of the triplet to the charged Higgs state is negligible. On the other hand, charginos in loop could enhance the h → γγ decay width, in particular, when they are light and/or diagrams involving them interfere constructively with the W -mediated loop diagram. The Higgs to diphoton decay rate can be written down as [97] X GF α2 m3h X √ gheχ+ χe− Ah1/2 Nc Q2f gfh Ah1/2 + ghW + W − Ah1 + ghH + H − Ah0 + Γ(h → γγ) = i j 3 128 2π f c e 2 X 2 h + (7.6) Nc efeghfefeA0 , fe
where
Ah1 = −[2τ 2 + 3τ + 3(2τ − 1)f (τ )]/τ 2 ,
Ah1/2 = 2[τ + (τ − 1)f (τ )]/τ 2 , Ah0 = −[τ − f (τ )]/τ 2 ,
(7.7)
with the loop functions already defined in eq. (7.4). The relevant couplings are given 1
Actually, tiny ‘A’-terms are generated because of the breaking of R-symmetry but we can neglect them in the present context.
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by, cos α , sin β sin α , ghdd = − cos β ghW W = sin(β − α), m2W cos 2β sin(β + α) ghH + H − = ], [sin(β − α) + 2 mH ± 2 cos2 θW m2f m2Z f ghfefe = g ∓ [I − ef sin2 θw ] sin(α + β), hf f m2fe m2fe 3 mW (ξij sin α − ηij cos α). (7.8) gheχ+i χe−j = 2 meck √ 2λT Here ξij = − √12 Vi1 Uj4 and ηij = − √12 U V + U V i3 j2 i1 j3 . The masses which appear g in the denominator of the couplings given above, represent physical masses propagating in the loop. For example, meck are the physical chargino masses, mfe are the physical masses of the sfermions and so on. We present the complete set of Higgs-charginochargino interaction vertices in Appendix B2 . As noted earlier, the largest contribution in the Higgs decay rate to two photons comes from the W boson loop. Similar to the MSSM, the hW W coupling gets modified by the factor sin(β − α). Hence, in order to have a significant contribution from the W boson loop in our model, the angles α and β need to be aligned in such a way that one obtains a large value of sin(β − α), which can be achieved in the decoupling regime, i.e., the coupling to the lightest Higgs boson becomes SM like. In fig. 7.1, we illustrate the variations of the couplings ghW +W − and gheχ+3 χe−3 , which might play important roles in the decay h → γγ. We choose M1D = 1.5 TeV, µu = 200 GeV, m3/2 = 10 GeV, met = 500 GeV, vS = 10−4 GeV, vT = 10−3 GeV and retain a near degeneracy between the Dirac gaugino masses with ǫ ≡ (M2D − M1D ) = 10−1 GeV, with f = 0.8 and BµL = −(400)2 (GeV)2 . From the left panel of fig. 7.1 we observe that the hW W coupling is almost SM like as we are essentially working in the decoupling limit. This implies that the W -loop contribution in the h → γγ process D remains almost unchanged with varying tan β. On the other hand, as µu 1 while kγγ < 1, all through. We have also checked that the illustrated variations of kgg and kγγ are following their respective gross trends in the MSSM closely in the limit of zero left-right mixing in the scalar sector. Note that for this plot we have not incorporated the constraints from the mass of the Higgs boson and the requirement of having no tachyonic scalar states. In section 7.3, while discussing the quantitative impact of 132
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tanΒ Figure 7.2: Couplings of the Higgs boson to heavier charginos. The thick black line represents the coupling to the heaviest chargino (e χ± 1 ) whereas the blue dashed one represents the same to the chargino immediately lighter to it (e χ± 2) .
the recent LHC results on such a scenario, we present results of detailed scan of the parameter space by including all these constraints. All the previous plots consider a large values of ‘f ’ (f ∼ O(1)) for which one obtains a large tree level correction to the Higgs boson mass as well as an appropriate mass for the active neutrino at the tree level. We adopt such a scenario with relatively large values of ‘f ’ in our study of the Higgs boson decay rates which we present in the next subsection.
7.2.3
Higgs boson decaying to charginos and neutralinos
In the presence of much lighter charginos and neutralinos (as discussed in sections 4.4.2 and 4.4.3), an SM-like Higgs boson with mass around 125 GeV could undergo decays to a pair of these states. It is important to consider such possibilities as these could potentially contribute to the total decay width of the Higgs boson (ΓTOT ) appearing in the expression of µγγ in eq. (7.1). It has been noted in section 4.4.2, that the smallest eigenvalue (mχe08 ) of the neutralino mass matrix corresponds to the neutrino mass. The next-to-lightest neutralino (e χ07 ) turns out to be a bino-like neutralino (the sterile neutrino) for large (small) values of ‘f ’. Moreover, the mass of the next-to-next-to-lightest neutralino state (e χ06 ) is mostly controlled by µu . Since we have chosen µu to be very close to the electroweak 133
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mt HGeVL Figure 7.3: Variations of kgg (in red) and kγγ (in blue) as functions of Met for tan β = 4, 10, 35.
scale, the Higgs boson decay to a pair of χ e06 is not possible. The presence of light neutralino states may enhance the total decay width of the Higgs boson considerably. Amongst them, the most dominant contribution comes from h-e χ07 -e χ08 coupling. However, a detailed study reveals that the contribution of this coupling is not substantial and hence the corresponding decay width is rather small. It is clear from fig. 7.4 that the h-e χ07 -e χ08 coupling grows for small tan β. This is essentially because for smaller values of tan β, the sneutrino component of the lightest Higgs boson mass eigenstate is large, which results in a slightly larger value of this coupling. This fact also shows up for the invisible decay widths of the Higgs boson, which we will discuss later. On the other hand, the lightest chargino eigenstate (e χ± 4 ) corresponds to the electron. ± The mass of the next-to-lightest chargino (e χ3 ) is again controlled by µu if µu < M2D . Thus, decay of the Higgs boson to a pair of χ e± 3 is not possible. The most general expressions for the partial widths of the Higgs boson decaying to a pair of neutralinos (Γ(h → χ e0i χ e0j )) or a pair of charginos (Γ(h → χ e+ e− i χ j )) can be found in the Appendix C and B.
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7.2.4
The total decay width of the Higgs boson
In this section we collect the partial decay widths of the lightest Higgs boson that dominantly contribute to its total decay width. The latter is thus given by3 ΓTOT = Γ(h → bb) + Γ(h → τ τ ) + Γ(h → gg) + Γ(h → W W ∗ ) + Γ(h → ZZ ∗ ) + Γ(h → γγ) + Γ(h → χ e0i χ e0j ) + Γ(h → χ e+ e− i χ j ).
(7.9)
For completeness, we present here the analytical expressions for all the decay rates which go into our analysis but were not presented earlier. These are as follows: 4m2 i3/2 3GF m2b mh sin α 2 h √ 1 − 2b , cos β mh 4π 2 4m2 i3/2 GF m2τ mh sin α 2 h √ 1 − 2τ Γ(h → τ τ ) = , cos β mh 4π 2 m2 3G2F m4W mh W 2 Γ(h → W W ∗ ) = sin (α − β)R , 16π 3 m2h 3G2F m4Z mh h 7 10 2 40 4 i m2Z ∗ Γ(h → ZZ ) = − sin θW + sin θW R . (7.10) 16π 3 12 9 27 m2h Γ(h → bb) =
3
We neglect the rare decay modes like H → Zγ, γ ∗ γ, µ+ µ− , e+ e− etc.
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The function R(x) is defined as [495, 497, 498] R(x) = 3 p
1 − x (1 − 8x + 20x2 ) 3 − (2 − 13x + 47x2 ) − (1 − 6x + 4x2 ) log x. 3x−1 2x 2 (4x − 1) arccos 2x 3/2
(7.11)
The Higgs boson decay rates to charginos and neutralinos are shown in Appendix B and C respectively. In the subsequent sections we present the numerical results of our analysis pertaining to the diphoton signal strength µγγ and subject this to important experimental findings.
7.3 Impact of the LHC results In this section, we discuss the impact of the findings from the LHC pertaining to the Higgs sector on the scenario under discussion. As pointed out earlier, two broad scenarios based on the magnitude of ‘f ’ worth special attention: the scenario with large ‘f ’ (∼ O(1)) and the one for which ‘f ’ is rather small.
7.3.1
The case of large neutrino Yukawa coupling, f ∼ O(1)
A large neutrino Yukawa coupling (f ∼ O(1)) already enhances the tree level Higgs boson mass. Thus, such a scenario banks less on large radiative contributions from the top squark loops to uplift the same. Further, an appropriately small tree level Majorana neutrino mass (the lightest neutralino) can be obtained along with a light bino-like neutralino (e χ07 , the next-to-lightest neutralino) once R-symmetry is broken explicitly, via anomaly mediation. The mass of this neutralino is essentially controlled by the R-symmetry breaking Majorana mass term of the U(1) gaugino (the bino), i.e., M1 , and hence related to the gravitino mass m3/2 . Since we assume m3/2 ∼ 10 GeV, the next-to-lightest neutralino acquires a mass of the order of a few hundred MeV. The presence of such a light bino like neutralino implies an additional contribution to the total decay width of the Higgs boson. We also looked at the diphoton signal strength µγγ and compared it with the latest ATLAS and CMS results. Invisible branching ratio of the Higgs boson To take into account the constraints coming from the invisible decay branching ratio of the lightest Higgs boson we have fixed M1D = M2D = 1.5 TeV, µu = 200 GeV, (i.e., M1 , M2 invisibleL%
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tanΒ Figure 7.5: The lightest Higgs boson invisible branching ratio as a function of tan β for different values of λT . The horizontal line corresponds to the upper limit on the invisible branching ratio from model independent analysis [499].
the Higgs boson decaying to a neutrino and a neutralino (h → χ e07 χ e08 ) could essentially contribute to the invisible final state. This can be understood from the fact that although χ e07 would undergo R-parity violating decays χ e07 → qqν, e+ e− ν, ννν, qq′ e− , where q, q ′ are the SM light quark states from the first two generations, these decay modes involve very small couplings thus and as a result, the decay length happens to be much larger than the collider dimension. Therefore, the LSP neutralino contributes to missing energy (MET) signals [224]. Note that Γ(h → χ e07 χ e07 ) is negligibly small because of suppressed h-e χ07 -e χ07 coupling for a bino-dominated, χ e07 . We observe from fig. 7.5 that this partial decay width is comparatively larger for smaller values of tan β and λT . However, it is clear that the presence of a bino-like neutralino state is not yet constrained from the invisible decay mode of the Higgs boson in our scenario with all the curves staying well below the experimentally derived [499] upper bound of ∼ 20% for the invisible branching fraction of the Higgs boson. The signal strength µγγ It is now important to analyse the signal strength corresponding to the h → γγ channel. In fig. 7.6 we fix λT = 0.45, and f = 0.8, with all other parameters held at the values mentioned in section 7.2.2. The red dashed lines represent the contours of mh = 124 137
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5
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mt Figure 7.6: Contours of various fixed values of mh (124 GeV and 126.2 GeV), µγγ and kT OT in the met –tan β plane. λT and f are fixed at 0.45 and 0.8, respectively. Other parameters are set at the values as mentioned in the text.
GeV and 126.2 GeV respectively and enclose the experimentally allowed range of mh . The black thick lines are the contours of fixed µγγ with values 1.15, 1.1, 1.05 and 1.03 respectively. Figure 7.6 shows that there is an available region of parameter space consistent with the latest experimental findings involving mh and µγγ . Relatively low values of the top squark mass results in an increase of the cross section for the resonant Higgs boson production through gluon fusion and thus enhances µγγ . On the other hand µγγ is almost insensitive to tan β for tan β ≥ 15. This is because hbb coupling (which controls the total decay width of the Higgs boson in a significant way) becomes independent of tan β for larger values of this parameter. Figure 7.7 addresses the same issue but with f = 1 and λT = 0.5. Since a larger value of λT already provides a significant contribution to the Higgs boson mass via radiative correction, only light top squarks are compatible with the measured range of mh . Moreover, a larger value of ‘f ’ implies a larger tan β to have the Higgs boson mass in the correct range. It is pertinent to mention that these plots use spectra of particles which are consistent with the lower bound on the lightest chargino mass (> 104 GeV, from the LEP experiments) and are also free from tachyonic scalar states.
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mt Figure 7.7: Same as in fig. 7.6 but with λT = 0.5 and f = 1.
Relative signal strengths in different final states In this subsection we briefly discuss how other final states arising from the lightest Higgs boson are expected to be affected in our scenario relative to the γγ final state and where they stand vis-a-vis the experimental results. Such a study of relative strengths over the parameter space of our scenario would be indicative of how well the same is compatible with the experimental observations in the Higgs sector, in a global sense. The recent results from the ATLAS and the CMS collaborations on different decay modes of the lightest Higgs boson are presented in table 7.1. In fig. 7.8, we present the µ-values reported by the ATLAS and the CMS collaborations for different final states in the so-called signature (ratio) space, in reference to µγγ . In each plot, the blue circle (green square) represents the experimentally reported central values for a given pair of observables from ATLAS and CMS collaborations, respectively. The solid grey lines show the range of µ values as observed by the CMS experiment while the dashed ones delineate the same as obtained by the ATLAS experiment. In order to generate fig. 7.8 we vary tan β within the range 10 < tan β < 40. We have also varied the mass of the top squark within the range 350 GeV < met < 1.5 TeV with 0.1 < f < 1 and 0.1 < λT < 0.55. All other parameters are kept fixed at the previously mentioned values in section 7.2.2. While scanning, care has been taken to reject spectra with tachyonic scalar states and to conform with the lower bound on the lightest chargino mass of 104 GeV as obtained from the LEP experiment. Also, the 139
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Figure 7.8: Bands representing mutual variation of relative signal strengths in various possible final states arising from the decay of the lightest Higgs boson as obtained by scanning the parameter space of the scenario under consideration. The ranges of different parameters used in the scan are as follows: 10 < tan β < 40, 350 GeV < met < 1.5 TeV, 0.1 < f < 1 and 0.1 < λT < 0.55. The solid grey lines give 1-σ ranges from the MVA based analysis (main analysis) performed by the CMS collaboration (blue circles represent the respective central values) whereas the dashed grey lines represent the corresponding results from the ATLAS collaboration (green squares represent the respective central values).
140
Channel h → γγ ZZ ∗
h −−→ 4l WW∗
h −−−→ 2l2ν h → bb h → ττ
µ (CMS) 1.14+ − 0.93+ −
0.72+ − + 1.0− 0.78+ −
0.26 0.23 [447] 0.39 0.32 [500] 0.20 0.18 [501] 0.5 0.5 [503] 0.27 0.27 [505]
µ (ATLAS) 1.17+ −
0.27 0.27 [446] 0.40 1.44+ −0.33 [446] 1.0+0.30 −0.30 [502] + 0.70 0.2− 0.60 [504] 0.5 1.4+ − 0.4 [506]
Table 7.1: Signal strengths (µ) in different decay final states of the SM-like Higgs boson as reported by the CMS and the ATLAS collaborations (with the corresponding references). scan required mh to be within the range of 124.0 − 126.2 GeV as reported by the LHC experiments. The spread in the upper two plots in fig. 7.8 are due to the variation of f which affects µbb and µγγ whereas µW W and µZZ remain unaffected. The values of µγγ is very much consistent with the recent ATLAS and CMS findings. Finally, in order to have an idea of the mass-spectra of the light neutralino and the chargino states, we provide a few benchmark points in table 8.1, for the large ‘f ’ scenario.
7.3.2
The case of small Yukawa coupling, f ∼ O(10−4)
In the limit when the Yukawa coupling is small (f ∼ 10−4 ), the next-to-lightest neutralino state becomes the sterile neutrino with negligible active-sterile mixing. The lightest neutralino state is again the active neutrino. The tree level Majorana mass of the active neutrino is given by eq. (4.63) whereas the sterile neutrino mass and the mixing angle between the active and the sterile neutrino are given by eqs. (4.66) and (4.67). We have mentioned in the previous chapter that an X-ray line at around 3.5 keV was observed in the X-ray spectra of the Andromeda galaxy and in the same from various other galaxy clusters including the Perseus cluster. The observed flux and the best fit energy peak are shown in eq. (6.1) [411, 412]. The origin of this line is disputed since atomic transitions in the thermal plasma may also be responsible for this energy line. Nevertheless, a possible explanation can be provided by taking into account a 7 keV dark matter, in this case a sterile neutrino [411, 412]. As discussed earlier, the observed flux and the peak of the energy can be translated to an activesterile mixing in the range 2.2 × 10−11 < sin2 2θ14 < 2 × 10−10 . To satisfy such small active sterile mixing, the tree level neutrino mass turns out to be very small (O(10−5 ) eV). Therefore, in order to explain the neutrino mass and mixing, one needs to invoke radiative corrections. For a detailed discussion, we refer the reader to chapter 5. It is
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Parameters
BP-1
BP-2
BP-3
M1D M2D µu m3/2 tan β met f λT vS vT BµL Observables
1500 GeV 1500.1 GeV 200 GeV 20 GeV 25 500 GeV 0.8 0.5 −4 10 GeV 10−3 GeV −(400)2 (GeV)2 BP-1
1000 GeV 1000.1 GeV 200 GeV 20 GeV 35 400 GeV 0.8 0.52 −4 10 GeV 10−3 GeV −(400)2 (GeV)2 BP-2
1200 GeV 1200.1 GeV 200 GeV 10 GeV 40 400 GeV 0.8 0.52 −4 10 GeV 10−3 GeV −(400)2 (GeV)2 BP-3
mh (mν )Tree mχe07 mχe06 mχe05 mχe04 mχe03 mχe02 mχe01 mχe+3
124.98 GeV 0.04 eV 168 MeV 208.73 GeV 208.74 GeV 1504.17 GeV 1504.23 GeV 1.19×105 GeV 1.19×105 GeV 208.13 GeV
125.45 GeV 0.1 eV 169 MeV 210.58 GeV 210.59 GeV 1006.13 GeV 1006.19 GeV 1.11×105 GeV 1.11×105 GeV 211.91 GeV
125.73 GeV 0.08 eV 84 MeV 209.75 GeV 209.76 GeV 1205.29 GeV 1205.31 GeV 1.33 × 105 GeV 1.33 × 105 GeV 210.24 GeV
mχe+2 mχe+1
1500.11 GeV 1508.27 GeV
1000.11 GeV 1012.15 GeV
1200.1 GeV 1210.45 GeV
µγγ
1.07
1.11
1.11
Table 7.2: Benchmark sets of input parameters in the large Yukawa coupling (f ) scenario and the resulting mass-values for some relevant excitations. The Higgs signal strength in the diphoton final state (µγγ ) is also indicated.
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also important to study the signal strength of h → γγ in the light of this 7 keV sterile neutrino with appropriate active-sterile mixing. 40
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f Figure 7.9: Contours of fixed values of mh , µγγ , MNR and sin2 2θ14 in the f − tan β parameter space. The respective values of the contour lines are as shown in the figure. The shaded region in grey corresponds to the experimentally allowed band of the lightest Higgs boson mass. Other parameters are fixed at values mentioned in the text.
In fig. 7.9 we present the contours of mh , µγγ , MNR and sin2 2θ14 in the f –tan β plane. The contour of the sterile neutrino mass of 7 keV is shown with the thick black line. The red dashed lines represent the contours of active-sterile mixing fixed at 2.2 × 10−11 and 2 × 10−10 . We have fixed M1D at 1 TeV, maintaining a degeneracy ǫ = (M2D − M1D ) = 10−4 GeV. µu is fixed at 500 GeV. The other fixed parameters are m3/2 = 10 GeV, met = 400 GeV, λT = 0.57, vS = −0.01 GeV, vT = 0.01 GeV and BµL = −(400)2 (GeV)2 . The not so heavy top squark, as justified in section 7.3.1, enhances µγγ considerably and we show the contours of µγγ at 1.1 and 1.114 respectively with blue dashed lines. Finally, the grey shaded region is the parameter space consistent with the observed Higgs boson mass 124.0 GeV < mh < 126.2 GeV. Figure 7.9 clearly > shows that for this choice of parameters µγγ ∼ 1.1 is completely consistent with a 7 keV sterile neutrino dark matter and the experimentally allowed range of Higgs boson mass. We have seen that charginos do not provide much enhancement to µγγ due to its very suppressed couplings under the present set-up. Furthermore, avoiding possible appearance of tachyonic scalar states restricts the vev of the singlet from becoming 143
large. Therefore, expecting an enhancement in µγγ via suppression of the hbb coupling because of the singlet admixture seems unrealistic. Thus, the only enhancement in µγγ can come from light top squarks. In addition, large radiative corrections from λS and λT reduces the necessity of having heavy top squarks. In the scatter plot of fig. 7.10 we show the possible range of variation of µγγ with varying met . To generate this plot we 1.12 1.10
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mt HGeVL Figure 7.10: Scatter plot showing possible range of variation of µγγ with varying met . The blue points are consistent with 7.01 keV < MR N < 7.11 keV. All points satisfy 124.0 GeV < mh < 126.2 GeV.
have chosen relevant parameters over the following ranges: 1 GeV < m3/2 < 20 GeV, 5 < tan β < 40, 300 GeV < met < 1.5 TeV, 10−5 < f < 3 × 10−4 , 0.1 < λT < 1 and −0.01 GeV < vS < −1 GeV. Other parameters are retained at their previously mentioned values (used to obtain fig. 7.9), maintaining the degeneracy between the Dirac gaugino masses as already mentioned. Again, all these points are consistent with 124.0 GeV < mh < 126.2 GeV and free from any tachyonic scalar states. The effects of the light top squarks results in some enhancement in µγγ . The blue points are consistent with a keV sterile neutrino with mass ranging between 7.01 keV < MR N < 7.11 keV and is known to be a fit warm dark matter candidate having the right relic density. Finally, it is again very relevant to check the relative signal strengths for different decay modes of the lightest Higgs boson in such a scenario with small ‘f ’; similar to what we have done in section 7.3.1 for the large ‘f ’ scenario. Figure 7.11 shows scattered points consistent with the CMS or/and the ATLAS results at 1σ level. However, note 144
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Figure 7.11: Same as in figure 7.8 except for a small input value of f . that the scatter plot in the µγγ –µW W plane is consistent only with the results from the ATLAS experiments at the 1σ level whereas the the scatter plot in the µγγ –µbb plane is consistent only with the results from the CMS experiments at the 1σ level. In the near future, a more precise measurement together with an improved analysis is likely to become more decisive on this issue. Finally, for the sake of completeness, in table 8.6 we provide three more benchmark sets comprising of the input parameters of the small Yukawa coupling scenario (with (f ∼ 10−4 )), the corresponding mass-values of the relevant excitations and the Higgs signal strengths in the diphoton final state (µγγ ).
7.4 Concluding remarks In this chapter we study the h → γγ channel in the U(1)R lepton number model with a right handed neutrino. We show that the recent results from ATLAS and CMS on µγγ is very much consistent with our outcomes for both the cases, i.e., f ∼ O(1) and f ∼ O(10−4 ). We also show for large neutrino Yukawa coupling, f the light bino-like neutralino state is not yet constrained from the invisible branching fraction of the Higgs boson. So far we have seen that the model under consideration have already demonstrated 145
Parameters
BP-4
BP-5
BP-6
M1D µu m3/2 tan β met f λT vS vT Observables
1000 GeV 300 GeV 4 GeV 35 500 GeV 9.9×10−5 0.55 −2 -10 GeV 10−2 GeV BP-4
900 GeV 600 GeV 10 GeV 25 500 GeV 8.9×10−5 0.55 −2 -10 GeV 10−2 GeV BP-5
1200 GeV 600 GeV 15 GeV 15 500 GeV 1.21×10−4 0.55 −2 -10 GeV 10−2 GeV BP-6
mh mR N mχe06 mχe05 mχe04 mχe03 mχe02 mχe01 mχe+3
125 GeV 7.03 keV 292.375 GeV 292.376 GeV 1004.06 GeV 1004.07 GeV 1022.03 GeV 1022.72 GeV 311.56 GeV
124.257 GeV 7.09 keV 571.91 GeV 571.92 GeV 904.16 GeV 904.19 GeV 939.91 GeV 939.83 GeV 609.77 GeV
124.448 GeV 7.03 keV 587.24 GeV 587.25 GeV 1203.24 GeV 1203.28 GeV 1222.84 GeV 1222.72 GeV 608.27 GeV
mχe+2 mχe+1
1000.01 GeV 1011.93 GeV
900.01 GeV 910.62 GeV
1200.02 GeV 1208.7 GeV
sin2 2θ14 µγγ
1.56 × 10−10 1.07
4.7 × 10−11 1.06
2.8 × 10−11 1.06
Table 7.3: Same as in table 8.1 but for small Yukawa coupling with f ∼ O(10−4 ). In all three cases we have chosen ǫ = 10−4 GeV. Neutrino mass at the tree level is very small (O(10−5 ) eV) and not shown in the table (See text for more details).
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its ability to attract constraints from recent experiments in diverse areas ranging from the neutrino to astro-particle physics and finally from the LHC experiments pertaining to the Higgs sector and other BSM searches. It will be really interesting to see if the model can provide any novel signatures as far as the collider experiments are concerned. Therefore, in the next chapter, we take the obviously natural direction of connecting to the top squark sector which is very much in the focus of the current LHC programme and thus could be put to test in a straight-forward way.
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Chapter 8 Light top squarks in U (1)R -lepton number model with a right handed neutrino and the LHC 8.1 Introduction In this chapter we look into the issues regarding top squarks and their probable signatures. In the context of the model studied in this thesis, both the top squarks can be light and devoid on any left-right mixing. As a result, different interesting signatures are possible in this class of models. In our scenario, the squarks carry a non-zero R-charge (R = 1) and hence a non-zero lepton number since lepton number is identified with the R-charges. Top squarks can naturally be light in this model and can have novel signatures at the LHC. In the present work, we take the obviously natural direction of connecting to the top squark sector which is very much in the focus of the current LHC programme and thus could be put to test in a straight forward way. For example, pair production of top squarks and their subsequent decays via R-parity conserving channels, or a combination of both, might lead to signals similar to top quark pair production and provide important information on the model and, in particular, on the scenario with an order one neutrino Yukawa coupling ‘f ’. Note that in the latter case, the presence of the light bino-like neutralino gives rise to signals with missing transverse energy (MET). Furthermore, characteristic signatures for these light top squark states at the LHC even have the potential to discriminate between competing scenarios that may give rise to such a light pair of top squarks. This chapter is organised as follows. The focus area of this work, that is to say, the top squark sector is described in section 8.2. Expressions for the decay rates in various
148
relevant modes are presented. The latest bounds on the the masses of the top squarks as reported by the LHC collaborations are also discussed. The model is incorporated in SARAH (v4.4.1) [507–509]. In section 8.3 we present a few benchmark points that reflect the characteristic decay patterns of the two top squarks and are found to be instrumental in shaping the interesting signatures at the LHC. Section 8.4 is devoted to the actual simulation study of the signals and the most relevant backgrounds using event generators. Estimations of the reaches in the masses of the the top squarks are also presented. In section 8.5 we briefly analyse the issue of the ‘stealth’ top squark which arises naturally in our scenario. We summarise with some concluding remarks in section 8.6.
8.2 The top squark sector In this section we concentrate on the third generation squarks, mainly the top squarks, which play important roles in lifting the Higgs boson mass. Scenarios with light top squarks draw their motivations from the ‘naturalness’ argument. They also provide rich and interesting collider signatures. As discussed previously, the model which we consider here gives us the opportunity to study such light top squarks. Furthermore, R-symmetry prohibits any trilinear scalar couplings (the ‘A’ terms) and Higgsino mass parameter (the µ term). We will later investigate a situation where both the top squarks are light (∼500 GeV) and have negligible chiral mixing, which originates from small R-breaking. The relevant terms in the top squark mass matrix are generated from the F -term, the D-term and the soft terms. The SU(2)L and U(1)Y contributions to the D-fields are given in eq. (4.16). From this point, it is straightforward to calculate the elements of the mass-squared matrix in the top squark sector, which in the basis (e tL , e tR ) ≡ (e t1 , e t2 ) turn out to be √ √ 1 2 2 2 2 2 2 2 (Met )11 = mQe3 + mt + mZ cos 2β − sin θW + vS M1D + 2gM2D vT , 2 3 3 2 2 (Met )12 = (Met )21 = 0, √ 4 2 ′ D 2 2 2 2 2 2 g M1 vS . (8.1) (Met )22 = mue3 + mt + mZ sin θW cos 2β − 3 3 Note that in absence of ‘A’ terms and the µ term, the off-diagonal entries vanish and hence the top squark sector is devoid of any chiral mixing. Thus, the left- and rightchiral states are equivalent to the mass eigenstates. Such a ‘zero’ mixing situation can be contrasted with the MSSM, in which, a substantial mixing is generally required to obtain the observed value of the Higgs mass. 149
2000 1800 1600
met2 (GeV)
1400 1200 1000 0.8 < λS ≤ 0.9 0.9 < λS ≤ 1.0 1.0 < λS ≤ 1.1 1.1 < λS ≤ 1.2 1.2 < λS ≤ 1.3
800 600 400
124.7 GeV < mh < 126.2 GeV
200 200
400
600
800
1000
1200
1400
1600
met1 (GeV)
Figure 8.1: Allowed region in (met1 − met2 ) plane compatible with 124.7 GeV < mh < 126.2 GeV Higgs boson mass after taking into consideration the full one loop corrections. The grey bands indicate the values of met1 and met2 ruled out by the LHC (relevant to our scenario as discussed later). The narrow vertical strip over 197 GeV . met1 . 205 GeV refers to the ‘stealth’ top squark regime. We have discussed in detail, that this model can ameliorate the Higgs naturalness problem because of the additional tree level enhancement in the Higgs boson mass or from one loop quartic corrections. Hence, light top squarks are a natural outcome. To get a more elaborate picture on the one-loop corrections, we first implemented the model in SARAH (v4.4.1) [507–509]. This takes into account the full one-loop corrections computed in the effective potential approach to the neutral scalar potential. Figure 8.1 illustrates the region in the plane of met1 and met2 compatible with 124.7 GeV < mh < 126.2 GeV and various slices of λS over the ranges shown. For this scattered plot we use tan β = 23, M1D = M2D = 1.2 TeV, M3D = 1.5 TeV, µu = 200 GeV, f = 1, vS = vT = 10−4 GeV, BµL = −(200 GeV)2 , tS = (174 GeV)3 and vary λS in the range 0.8 < λS ≤ 1.3. We also vary the soft scalar masses (m2Q )33 , (m2u )33 in the range −8 × 106 (GeV)2 to 8 × 106 (GeV)2 to vary the top squark mass. We observe that higher values of the superpotential coupling λS (hence larger λT ) provide larger corrections to the Higgs boson mass at the loop level and hence the requirement of having multi-TeV top squarks can be avoided. The largest values considered for met1 and met2 in this plot are close to 1660 GeV and 1830 GeV, respectively. As can be seen from fig. 8.1, relatively light top squarks with met1 ≃ met2 may be generic to our scenario (though such a situation could attract more aggressive con150
straints from the LHC). Furthermore, both of them can have sub-TeV masses simultaneously and can still be consistent with the observed value of the SM-like Higgs mass if λS > 1. In the absence of any appreciable chiral mixing (left-right (L-R) mixing), top squarks as heavy as values considered here, could only raise the Higgs boson mass up to around 114 GeV in the MSSM. Hence the model under consideration has a very interesting and a distinct feature where light to moderately heavy (∼ 1 TeV) top squarks with negligible L-R mixing can be compatible with observed mass of the Higgs boson. Similar possibilities are discussed earlier in generic setups [510] and more recently, in a specific SUSY scenario like the Next-to-Minimal SUSY extension of the SM (NMSSM) [511]. At the LHC, top squarks are being searched in their direct production, pp → e te t∗ , followed by their subsequent decays in various possible modes. Out of these, the decays that are relevant to our scenario [512–515] are the R-parity conserving ones e t → be χ+
and
e t → te χ0 ,
(8.2)
and the modes that violate R-parity when a top squark could decay to a bottom quark and a charged lepton, e t → bℓ+ [516, 517]. These channels find major relevance in the context of our model. The reasons are twofold: first, in the large ‘f ’ scenario, we obtain a light bino-like neutralino with mass around a few hundred MeV, in addition to an active neutrino. Therefore, the channels with top squark decaying to a top quark and a bino-like neutralino and/or an active neutrino open up. Secondly, top squark decaying to a bottom quark and a chargino is also important. Additionally, top squark decaying to a bottom quark and an electron becomes an interesting channel to look for. This decay mode is predominantly controlled by the R-parity violating operator λ′133 . In the framework of the MSSM with R-parity violation, strong limit on q this particular b [80]. Hence, coupling exists from the neutrino Majorana mass, |λ′133 | ∼ 3.4 × 10−3 m me b
in such a scenario, the resulting decay rate becomes highly suppressed. However, in the present context, λ′133 is identified with the bottom Yukawa coupling yb . The smallness of the neutrino mass is then explained through small R-breaking effect, parametrised in terms of a small gravitino mass [308], discussed in detail in chapter 5. Thus, a large decay rate for e t → be+ becomes a generic feature in our model. In principle, e t could + + ′ ′ also decay to bµ and bτ via RPV couplings λ233 and λ333 . However, these decays are subdominant compared to e t → be+ because of the stringent constraints on the relevant e is also a possibility but couplings as discussed later. We note in passing that e t → tG highly suppressed [518] for a gravitino of mass ∼ 20 GeV in the present context. The relevant Lagrangians are worked out in the four component notation following
151
[101, 519] and are given by )# ( " " # ′ g′ 1 4g tL χ e0i + t √ PL Ni1 − yt PR Ni6 e Letteχ0 = −t yt PL Ni6 + √ gPR Ni3 + PR Ni1 e tR χ e0i 3 2 3 2 + h.c.,
and
(8.3)
"
#
"
#
tL χ eci + b yt PR Vi3 e tR χ eci + λ′133e tL PL Ui4 χ eci b + h.c., (8.4) Letbeχ+ = b − gPL Ui1 e
where, λ′133 = yb = mb /v cos β, the bottom Yukawa coupling and yt = mt /v sin β is the top Yukawa coupling and mt and mb are the top and the bottom quark masses, respectively. The neutralino and the chargino mixing matrices Nij , Uij and Vij are as defined earlier. Note that, for i = 1 (corresponding to χ eci ≡ e− ) the mixing matrix elements U11 and V13 are suppressed1 . In the following subsections we briefly discuss the salient decay modes of the lighter (e t1 ≈ e tR ) and the heavier (e t2 ≈ e tL ) top squarks.
8.2.1
Decay rates of e t1 (≈ e tR )
The partial decay widths of e t1 in the te χ0i and be χ+ i modes are given by # " 1 2 2 Γ(e t1 → te χ0i ) = me2t1 − m2t − m2χe0 − 4ηRi ζRi mt mχe0i ηRi + ζRi i 16πme3t 1 21 4 2 2 2 2 2 2 4 4 × met1 + mχe0 + mt − 2met1 mt − 2mχe0 mt − 2mχe0 met1 , (8.5) i
i
i
and
# " 1 2 2 Γ(e t1 → be χ+ αRi + βRi me2t1 − m2b − m2χe+ − 4αRi βRi mb mχe+ i ) = i i 16πme3t 1 21 4 4 4 2 2 2 2 2 2 × met1 + mχe+ + mb − 2met1 mb − 2mχe+ mb − 2mχe+ met1 , (8.6) i
i
i
where, 4g ′ √ Ni1 , 3 2 = yt Ni6 ,
ηRi = ζRi
αRi = 0, βRi = yt Vi3 . 1
(8.7)
In contrast to the previous chapters, the lightest neutralino, i.e., the active neutrino is designated by χ e01 , whereas the next-to-lightest state is χ e02 . We follow the same assignments for the charginos also. This is done in order to be consistent with the standard outputs of the spectrum generators used here.
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We note down a few important observations below. • As discussed in section (4.4.2), in the large ‘f ’ case we obtain a light (∼ few hundred MeV, governed by the R-breaking Majorana mass M1 ) bino-like neutralino (e χ02 ). This is because of the presence of MR N c Se term in the Lagrangian, where the coefficient MR becomes very large (∼ 105 GeV) for an order one ‘f ’. This results in forming a heavy pseudo-Dirac pair with mass ∼ MR and makes the lightest eigenvalue very small and predominantly bino-like. • The Dirac wino mass M2D is considered to be heavy to evade bounds from Z boson coupling to electrons [306]. The µu parameter, which controls the mass of the Higgsino (both neutral and charged) can vary between the electroweak scale (∼ 200 GeV) and a much larger value, i.e., a few TeV. • Based on the above discussion and with the help of eqs. (8.3)-(8.7), we find that e t1 +2 0 χ2 . The neutralino can be both bino- or higgsinowould decay into te χ2,3,4 and be like whereas the chargino would only be higgsino-like, assuming the higgsino mass parameter µu (< met1 ) met1 , the top squark cannot decay to an on-shell top quark and a higgsino-like neutralino or a bottom quark and a higgsino-like chargino due to phase space constraints. Therefore, the dominant channel would only be e t1 → te χ02 , where χ e02 is the bino-like MeV neutralino. Moreover, e t1 → tνe would also contribute to MET, although the branching is suppressed due to the small neutralino-neutrino mixing.
8.2.2
Decay rates of e t2 (≈ e tL )
The partial decay widths of e t2 are given by # " 1 2 2 Γ(e t2 → te χ0i ) = me2t2 − m2t − m2χe0 − 4ηLi ζLi mt mχe0i ηLi + ζLi i 16πme3t 2 12 × me4t2 + m4χe0 + m4t − 2me2t2 m2t − 2m2χe0 m2t − 2m2χe0 me2t2 , (8.8) i
i
i
+ We again note that χ e+ and χ e+ 1 is actually e 2 is designated to be the next-to-lightest chargino state. 2
153
and # " 1 2 2 Γ(e t2 → be χ+ αLi + βLi me2t2 − m2b − m2χe+ − 4αLi βLi mb mχe+i i ) = i 16πme3t 2 21 2 2 2 4 2 2 2 4 4 × met2 + mχe+ + mb − 2met2 mb − 2mχe+ mb − 2mχe+ met2 , (8.9) i
i
i
where, ηLi = yt Ni6 , ζLi αLi
! 1 g′ = √ gNi3 + Ni1 , 3 2 = −gUi1 ,
βLi = 0.
(8.10)
In addition, for e t2 we also have the interesting possibility of e t2 → be+ , which is not present in the decays of e t1 [312]. The corresponding partial decay width is given by y 2|Ui4 |2 met2 . Γ(e t2 → be+ ) = b 16π
(8.11)
Some features of e t2 decays are as follows:
• The decay e t2 → be+ is an interesting possibility. This faces no suppression from the phase space and the decay rate is proportional to the bottom Yukawa coupling, yb which grows with tan β. Hence a substantial branching fraction in this mode is expected at large tan β and for a fixed top squark mass. • When µu < met2 , e t2 would decay to higgsino-like chargino and neutralinos. Also, decay to a bino-like neutralino is a possibility. However, a quick look at the couplings in eq. (8.10) would suggest that the decay to higgsino-like neutralinos (e χ03 , χ e04 ) is ηLi ∼ yt -enhanced and hence, is more probable than a decay to a binolike neutralino (suppressed by g ′/3 in the coupling) or to a higgsino-like chargino (suppressed by g times the wino component of the lighter chargino, Ui1 ). • Again, for µu > met2 , decays of e t2 to higgsino-like neutralinos and charginos are kinematically barred. Under such a circumstance, e t2 mainly decays to a bottom quark and an electron (positron). The decay mode e t2 (e tL ) → te χ02 is again suppressed because of a (comparatively) small involved coupling. Finally, b d in the Lagrangian, which has been αRi = βLi = 0 reflects the absence of H integrated out from the theory. 154
8.2.3
Bounds on top squarks
• Recently ATLAS measured the spin correlation in the top-antitop quark events √ and searched for top squark pair production [520] in the pp collisions at s = 8 TeV centre of mass energy and integrated luminosity (L) of 20.3 fb−1 . This particular search has ruled out top squarks with masses between the top quark mass and 191 GeV with 95% confidence level. A very recent study [111] reveals that the window of 197 GeV . met1 . 205 GeV, in the so-called ‘stealth’ regime (i.e., with vanishing LSP mass), cannot yet be ruled out. • Dedicated searches for pair-produced top squarks decaying 100% of the time to bottom quarks and lighter charginos have been performed [517, 521] within the framework of the MSSM. For a chargino with mass close to 200 GeV, the top √ squark below 470 GeV has been ruled out at the s = 8 TeV run of the LHC. In our scenario, e tR decays to this particular channel if µu < metR . Although in our model the corresponding branching fraction is less than 100%, we take a conservative approach and respect this bound. In addition, this search gives the most relaxed bound on the mass of the top squark which is relevant to our analysis. Hence in the present study we choose e tR to be the lighter top squark, i.e., te1 ≈ teR .
• Another decay mode of the top squark relevant to our scenario is e t → te χ01 , where χ e01 implies the lightest supersymmetric particle in the MSSM (in this chapter, however, χ e01 is identified with the active neutrino and χ e02 represents the lightest bino-like neutralino). At the 8 TeV run of the LHC, top squark with mass below 550 GeV is ruled out at 95% confidence level [517, 521] with the assumption BR (e t1 → te χ01 ) = 100%. This bound applies for a massless neutralino (mχe01 = 0). For heavier neutralinos in the final states, this lower bound on the top squark mass can be relaxed further. Note that in the large ‘f ’ scenario we find a super-light bino-like neutralino with mass around a few hundred MeV which thus attracts this bound on the mass of the lighter top squark. • A top squark decaying via R-parity violating mode has also been probed by the LHC experiments. If a top squark undergoes an R-parity violating decay only to a bottom quark and an electron, a stringent lower bound [516,517] exists on the top squark mass with met > 900 GeV.3 Accommodating an even lighter top squark,
Note however, that if e t → bτ + opens up, the corresponding lower bound on the top squark mass can be relaxed. The decay e t → bτ + is mostly controlled by the R-parity violating coupling λ′333 . The existing bound on this particular coupling is much relaxed: λ′333 < 1.4 cos β [306] and can be 3
155
which is central to our present work, thus requires a situation where such a bound is preferentially applicable to the heavier top squark state (te2 ) of the scenario. As described in section 8.2.2, only e tL could decay to a bottom quark and an electron (positron). Hence we choose the heavier top squark e t2 to be the e tL , i.e., te2 ≈ teL . Note again that our consideration is pretty conservative and, as we would find in section 8.3, for generic scenarios where such a decay can have a branching fraction below 50%, the bounds can get considerably weaker thus allowing for an even lighter te2 . Phenomenological discussions on top squarks undergoing such an R-parity violating decay can be found in references [231, 522–527]. The squarks from the third generation have understandably attracted a lot of attention in the recent times. The flavor changing decay of the top squark, e t1 → ce χ01 has been analysed in great detail in [511, 528–531]. Recent searches performed by both ATLAS and CMS collaborations have looked into this channel extensively and ruled out top squark masses below 300 GeV [111, 532, 533]. Top squarks decaying to a top quark along with a neutralino (LSP or NLSP) has also been probed in various SUSY models. A lower limit close to 1 TeV for the top squark mass can be obtained √ at s =14 TeV and with the high luminosity option [534–536]. In addition, thorough phenomenological studies have also been performed in the decay of top squark into a bottom quark and a chargino [537, 538]. We note in passing that interesting final state signatures can be obtained for the decays of bottom squarks as well [539]. For example ebL decays to a bottom quark and the bino-like neutralino with a branching ratio close to 71% and ebR decays to a top quark and an electron with a branching close to 47% in the BP-I scenario. Such a branching would imply 2 b-jets+✚ E✚ T or 2 b-jets+4 leptons+E T in the final states, respectively. The most stringent limit on the mass of the bottom squark comes from the search where it decays to a bottom quark and the lightest neutralino (LSP) with BR (eb → be χ01 )=100%. Bottom squark mass up to 700 GeV has been excluded at 95% confidence level for neutralino mass less than 50 GeV [540].
8.3 The benchmarks and the final states In this section we discuss a few benchmark scenarios that would be broadly representative of the phenomenology that is expected of the framework under consideration. We embed the model in SARAH (v4.4.1) [507–509]. We use the low energy output of saturated for small values of tan β (≤ 5). However, in the present scenario we confine ourselves in the > 20, which renders the decay e t → bτ + insignificant. On the other hand, the decay limit where tan β ∼ e t → bµ+ is also negligible because of the strong constraint |λ′233 | < 6.8 × 10−3 cos β [210].
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SARAH (v4.4.1) and generate the SUSY spectrum using SPheno (v3.3.3) [541, 542]. FlavorKit [543] is used to ensure benchmark points are consistent with all relevant flavor violating constraints. Higgs boson cross-sections and signal strengths are computed using HiggsBounds [544–547] and HiggsSignals [548,549]. As discussed earlier, we will mainly consider two regimes, viz., µu > met1,2 and µu < met1,2 . For each case, we point out the dominant decay modes of both e t1 and e t2 . These dictate the types of interesting signatures at the LHC for each of these cases.
8.3.1
Case 1: µu < met1,2
Two benchmark points for this case are shown in table 8.1. A relatively low value of µu (= 200 GeV) is chosen for both the benchmark points. The masses of the higgsinolike chargino and the neutralinos are mainly controlled by µu . We assume the singlet and the triplet vevs to be small; roughly to be around 10−4 GeV. The Dirac gluino mass (M3D ) is considered to be 1.5 TeV. Since we are considering a small amount of Rbreaking, the Majorana gaugino masses are roughly around a few hundred MeV. Fixing the order parameter of R-breaking, i.e., the gravitino mass O(10 GeV), fixes these soft SUSY breaking parameters. Both λS and λT are considered to be large, which for large tan β (> 20), provide significant radiative corrections to the Higgs boson mass through one loop quartic terms. Such a choice allows us to have a situation where both the top squarks are moderately light. Note that the chosen values of met1 (∼ 470GeV) for BP-I and BP-II are expected to be consistent with the latest LHC bounds discussed in section 8.2.3. The bound assuming BR(e t1 → be χ+ 1 ) = 100% is evidently satisfied e while the one (met1 > 550 GeV) that assumes BR(t1 → te χ01 ) = 100% is not applicable here (see table 8.2). Furthermore, the neutrino Yukawa coupling ‘f ’ is chosen to be 1. Hence, to have the active neutrino mass in the right ballpark, we need to consider the Dirac bino and wino masses to be almost degenerate. As conspicuous from eq. (4.63), this degeneracy provides a suitable suppression to the neutrino Majorana mass when the Dirac gaugino masses themselves are roughly around a TeV or so. Some of the low energy flavor violating branching ratios (which satisfy the respective experimental constraints) are also shown in table 8.1. • Decay branching fractions of e t1
The dominant decay branching fractions of e t1 for BP-I and BP-II are indicated in + table 8.2. χ e2 is the higgsino-like chargino and χ e03,4 are the higgsino-like neutralinos. χ e02 is the bino-like neutralino with mass in the ballpark of a few hundred MeV. As can be seen from table 8.2, the top squarks, once produced in pairs, can undergo both symmetric as well as asymmetric decays. Table 8.3 lists all possible final state 157
Parameters
BP-I
BP-II
M1D M2D M3D µu m3/2 tan β (m2u )33 (m2Q )33 f vS vT λS BµL tS Observables
1200 GeV 1200.1 GeV 1500 GeV 200 GeV 20 GeV 23 1.5×105 GeV2 5.5×105 GeV2 1 −4 10 GeV 10−4 GeV 1.130 −(200 GeV)2 (174 GeV)3 BP-I
800 GeV 800.1 GeV 1500 GeV 200 GeV 20 GeV 35 1.5 × 105 GeV2 6.1 × 105 GeV2 1 −4 10 GeV 10−4 GeV 1.116 −(200 GeV)2 (174 GeV)3 BP-II
mh met1 met2 mχe01 ≡ mνe mχe02 (bino-like) mχe03 mχe04 mχe+1 ≡ me
125.6 GeV 484.7 GeV 918.9 GeV 0.02 eV 167.9 MeV 211.5 GeV 211.5 GeV 0.51 MeV
125.0 GeV 478.0 GeV 905.3 GeV 0.04 eV 168.3 MeV 213.8 GeV 213.8 GeV 0.51 MeV
mχe+2
243.8 GeV
247.1 GeV
Flavor Observables
BP-I
BP-II
BR(B → XS γ) BR(BS0 → µµ) BR(µ → eγ) BR(µ → 3e) µγγ
3.4 × 10−4 2.4 × 10−9 6.1 × 10−24 6.6 × 10−26 1.11
3.3 × 10−4 2.5 × 10−9 3.6 × 10−24 3.0 × 10−26 1.12
Table 8.1: Benchmark sets of input parameters in the large neutrino Yukawa coupling (f ) scenario and the resulting mass-values for some relevant excitations for µu < met1,2 (case 1). M3D denotes the Dirac gluino mass. Also indicated are some of the relevant flavor observables and their values, all of which are currently allowed by experiments. The corresponding values of µγγ (the estimated Higgs di-photon rate compared to its SM expectation) are also mentioned. 158
Decay modes e t1 → b χ e+ 2 e t1 → t χ e03 e t1 → t χ e04 e t1 → t χ e02
BR for BP-I 56.9% 17.3% 17.3% 7.2%
BR for BP-II 57.1% 17.0% 17.0% 8.2%
Table 8.2: Decay branching fractions of e t1 in BP-I and BP-II for µu < met1 .
topologies. However, in the present context, we will mainly consider the dilepton final states accompanied by b-jets and MET. Such a final state might arise when the top squarks, on being pair-produced, undergo the decay e t1 → be χ+ e+ 2. χ 2 in turn, decays + 0 0 to a W and a χ e2 (e χ1 ) with a branching ratio close to 90% (10%) followed by W -s decaying leptonically. Although a semileptonic (ℓνjj) final state from W -decays is a good compromise between the rate and the cleanliness of the signal, we go for a cleaner channel where both the W bosons decay leptonically. As shown in fig. 8.2c, such a topology leads to a final state 2 b-jets + 2 leptons + ✚ E✚ T . A similar final state could also arise from other decays of the lighter top squark, such as those involving e t1 → te χ02 as shown in figs. 8.2a and 8.2b. However, the effective branching ratio is rather suppressed. We note in passing that the various decay combinations shown in table 8.2 could also provide exotic multilepton and multijet final states depending on the leptonic or hadronic decays of both W ± or Z boson. For example, decays such as e t1 → te χ03/4 could give rise to a 2 b-jets + 6 leptons + ✚ E✚ T final state. Some relevant final states arising from the decays of e t1 are tabulated in table 8.3. The branching fractions in BP-I and BP-II are rather similar since we are dealing with similar top squark masses. Also, top Yukawa coupling is practically insensitive to larger values of tan β, as considered in our study. This results in similar branching fractions in the be χ+ mode in BP-I and BP-II. The dynamics of other decays are essentially controlled by the gauge couplings and therefore, they remain similar. • Decay branching fractions of e t2
The dominant decay branching fractions of e t2 for BP-I and BP-II are shown in table 8.4. The pattern can be justified following the discussion in section 8.2.2. The branching fractions of te2 to the three modes indicated are comparable. These lead to distinct final state signatures with appreciable strength. Possible final states arising from the decays of e t2 are listed in table 8.5. The decay channels e t2 → e t1 Z(h) are absent due to negligibly small mixing between the left and the right chiral states of the top squark. A remarkable point to note here is the significant decay branching fraction of e t2 to a 159
µu < met1 : Decays of e t1
d \ \ + e0 bW −χ be χ− e02 → 2b + 2W + ✚ E✚ T 2 → bW χ 2 c 0 0 0 \ \ + − ✚ te χ2 → bW χ e2 bW χ e2 → 2b + 2W + ✚ ET \ d c \ +χ −χ pp → e t1 te∗1 → be χ+ χ02 + h.c. → bW e02 bW e02 + h.c. → 2b + 2W + ✚ E✚ T 2 te \ c [ \ +Z χ −χ pp → e t1 te∗ → te χ0 te χ0 + h.c. → bW e0 bW e0 + h.c. → 2b + 2W + Z + E✚ T d pp → e t1 te∗1 → be χ+ 2 ∗ c e e pp → t1 t1 → te χ02
1
3/4
2
2
✚
2
\ d [ \ +χ −Z χ pp → → be χ+ χ03/4 + h.c. → bW e02 bW e02 + h.c. → 2b + 2W + Z + ✚ E✚ T 2 te \ [ [ \ +Z χ pp → e t1 te∗1 → te χ03/4 te χ03/4 → bW e02 bW − Z χ e02 → 2b + 2W + 2Z + ✚ E✚ T e t1 te∗1
Table 8.3: Possible final states arising out of various decay modes of e t1 when µu < met1 . Decay modes e t2 → b e+ e t2 → t χ e03 e t2 → t χ e04
BR for BP-I 27.8% 35.7% 35.7%
BR for BP-II 47.2% 26.2% 26.2%
Table 8.4: Decay branching fractions of e t2 in BP-I and BP-II for which µu < met2 . µu < met2 : Decays of e t2
d + be − → 2b + e+ e− d pp → e t2 te∗2 → be \ [ + bW −Z χ d pp → e t2 te∗2 → d be+ te χ03/4 + h.c. → be e02 + h.c. → 2b + W + Z + e+ + ✚ E✚ T \ [ [ \ +Z χ −Z χ pp → e t te∗ → te χ0 te χ0 → bW e0 bW e0 → 2b + 2W + 2Z + E✚ 2 2
3/4
3/4
2
2
✚T
Table 8.5: Possible final states arising out of various decay modes of e t2 when µu < m e t2 .
bottom quark and an electron. To reiterate, this decay rate is proportional to λ′133 , which is identified with yb . Therefore, the corresponding decay rate is large. Also, because of the large difference between the mass of the decaying particle (e t2 ) and the total mass of the particles in the final state (mb +me ), the final state electron is expected to be hard. The schematic diagram for such a process is presented in fig. 8.3. Other decay modes, presented in table 8.5, are similar to the previous case where the decay products of e t2 are a top quark and a higgsino-like neutralino. These would further decay to give a final state comprising of 6 leptons+2 b-jets +E T . The variations in the branching fractions as we go from BP-I to BP-II, as can be seen in table 8.4, are due to changing bottom Yukawa coupling as tan β changes.
160
ℓ+
b
te1
p
t
W+
p
νℓ
χe + 2
χe 02
te1
χe 02 t¯
W+
p
χe 02
te∗1
ℓ+
b
p
νℓ
W−
χe 02
te∗1 t¯
¯b
W ¯b
ℓ−
νℓ
νℓ
−
ℓ−
(a) Asymmetric decays of the top squarks with (b) Symmetric decays of top squarks with both one decaying to be χ+ χ02 , thus leading to 2b + 2ℓ + E T 2 while the other decaying decaying to te to te χ02 , thus leading to 2b + 2ℓ + E T final state. final state.
ℓ+
b W+
p te1
p
te∗1
χe + 2
νℓ χe 02 χe 02
χe − 2
νℓ
W− ¯b
ℓ−
(c) Symmetric decays of top squarks with both decaying to be χ± , thus leading to 2b + 2ℓ + E T final state.
Figure 8.2: Final state topologies yielding 2b + 2ℓ + E T in the
8.3.2
Case 2: µu > met1,2
As opposed to the previous case, we consider the situation where µu > met1,2 . To have the Higgs boson mass in the right range we tweak λS . The soft masses (mu )233 and (mQ )233 are modified to get different top squark masses satisfying relevant LHC constraints. vT just takes a different sign when compared to BP-I and BP-II only to exclude tachyonic states, as illustrated in table 8.6. All the other parameters are kept fixed to their values in table 8.1. Due to such a choice of µu , a top squark cannot decay to higgsino-like chargino and neutralinos. Note that, in the present case as well, the
161
b p te2
e+ ¯b
p
te∗2 e−
Figure 8.3: 2b+e+ e− final state arising from both e t2 -s decaying directly to a bottom (anti-bottom) quark and a positron (electron). masses of the top squarks are chosen in such a manner that they satisfy the present experimental bounds. Values of flavor observables are checked to satisfy experimental constraints. However, those are not shown explicitly this time. • Decay branching fractions of e t1
The decay branching fractions for e t1 when µu > met1,2 are shown in table 8.7 for the benchmark points BP-III and BP-IV. An interesting point to note here is that the top squark decays only to a top quark accompanied either by a χ e02 (fig. 8.2b) or a νe (fig. 8.4) both of which are carriers of MET while the former being the dominant one. Again, both symmetric and asymmetric decays of the pair produced e t1 -s are possible. These would lead to 2 b-jets+2ℓ + E T final states. Note that more exotic final states with a larger lepton multiplicity would be absent as heavier higgsino-like neutralino(s) will now be missing in the cascades of e t1 . This is in sharp contrast with what is expected for µu < met1 (case 1) as discussed in section 8.3.1 and thus, may be exploited to distinguish between these two broad scenarios. • Decay branching fractions of e t2
Similarly, the absence of a light higgsino-like chargino and neutralinos implies e t2 would dominantly decay to a bottom quark and an electron (positron). The branching fractions of e t2 under such a circumstance are presented in table 8.8.
8.4 Collider (LHC) analysis
In this section we present the setup and the results of the simulation we carry out at the 13 TeV LHC for pair-produced top squarks that eventually cascade to the final states discussed in section 8.3. 162
Parameters
BP-III
BP-IV
µu vT (m2u )33 (m2Q )33 λS Observables
1500 GeV −10−4 GeV 2×105 (GeV)2 5.2×105 (GeV)2 1.09 BP-III
1100 GeV −10−4 GeV 3×105 (GeV)2 6×105 (GeV)2 1.06 BP-IV
mh met1 met2 mχe01 ≡ mνe mχe02 (bino-like) mχe03 mχe04 mχe+1 ≡ me
125.8 GeV 556.4 GeV 909.0 GeV 0.05 eV 175.8 MeV 1202.1 GeV 1202.2 GeV 0.51 MeV
124.7 GeV 652.8 GeV 908.5 GeV 0.10 eV 175.6 MeV 804.3 GeV 804.3 GeV 0.51 MeV
1304.2 GeV
877.4 GeV
mχe+2
Table 8.6: Same as in 8.1 but for an extra sign on vT and for BP-III and BP-IV for both of which µu > met1,2 (case 2). Values of flavor observables, not shown here explicitly, satisfy all the experimental constraints. Decay modes e t1 → te χ02 e t1 → tνe
BR for BP-III 87.8% 12.2%
BR for BP-IV 94.6% 5.3%
Table 8.7: Decay branching fractions of e t1 in BP-III and BP-IV for which µu > met1 .
8.4.1
The simulation setup and reconstructing the physics objects
We have implemented the model in MadGraph5 aMC@NLO [550]. Events for both signals and backgrounds are generated using the same. We use the parton distribution function √ CTEQ6L1 [551] and a factorisation/renormalisation scale of met1 met2 for generating events at the lowest order. The inclusive rates are then normalised to their respective values obtained after higher order corrections as given by MadGraph5 aMC@NLO in the cases for the SM background and Prospino2 (v2.1) [552, 553] for the case of the SUSY productions. Appropriate branching fractions are obtained from the spectrum generator SPheno [541–543] which, in the first place, is generated by SARAH [507–509]. 163
Decay modes e t2 → b e+ e t2 → t χ e02
BR for BP-III 97.8% 1.8%
BR for BP-IV 98.7% 0.9%
Table 8.8: Decay branching fractions of e t2 for BP-III and BP-IV for which µu > met2 .
We note in passing that the production cross-section for the top squarks (at the tree level) in this model is same as in the MSSM, considering only the dominant strong interaction. Events in the LHE format are fed into Pythia-6.4.28 [554] for showering, hadronisation and jet formation. Clustering of jets is performed with the built-in Pythia module PYCELL which employs a cone algorithm and incorporates appropriate smearing of the momenta. In PYCELL we allowed for an angular coverage of |η| < 5 for the hadron calorimeter with a cell-segmentation of ∆η × ∆φ = 0.1 × 0.1 which resembles a generic LHC detector. A cell is required to have a minimum value of deposited ET = 1 GeV for it to be considered. A jet-cone radius of ∆R(j, j) = 0.4 is employed for finding jets. A minimum summed ET of 20 GeV is required within such a geometry for the configuration to be considered as a jet. Ultimately, formed jets within |η| < 2.5 are considered in our analysis. Care has been taken to isolate final state leptons by imposing the following cuts and isolation criteria: • To select leptonic events we have used pℓT > 10 GeV and |ηℓ | < 2.4. • Lepton-lepton separation has been done by choosing ∆R(ℓ′ , ℓ) > 0.2, where ∆R = p (∆η)2 + (∆φ)2 .
• Subsequently, to separate leptons from jets we have used ∆R(j, ℓ) > 0.5.
• Finally, the sum of the energy deposits of the hadrons which fall within a cone of ∆R ≤ 0.2 around a lepton, must be less than 10 GeV. In this context, by leptons we mean only electrons and muons for which the detection efficiencies are generally very high, unlike the τ lepton. We have used a minimum pT cut of 10 GeV and 17 GeV to isolate muons and electrons, respectively. To estimate the number of b-jets in the final state, a flat (but somewhat conservative) b-tagging efficiency of 60% has been used.
8.4.2
Top squark pair-production cross section
The phenomenology we discuss in this work crucially depends on the rate of pairproduction of the top squarks. It is to be noted that at the lowest order these rates are 164
the same as in the MSSM. Considering the dominant strong contributions in the rates, the variation of the same is only dependent on the mass of the top squark, irrespective of its chiral content. As a quick reckoner for this basic rate, we present the same as a function of met in fig. 8.5 for the 13 TeV run of the LHC. Appropriate K-factors as obtained from the package Prospino2 [552, 553] are already folded in. To this end, we fix all the parameters as given in BP-IV except for the right handed soft squark mass, which we varied from −4 × 104 GeV2 < (m2u )33 < 8 × 105 GeV2 . Such a choice would surely move the Higgs mass away from the allowed range. However, we are here merely concentrating on study of the production cross-section for e te t∗ . The parameters such as λS , λT can be adjusted to fit the Higgs mass, which is unlikely to affect the production rate any significantly (via unknown higher order effects). b t
ℓ+ W+
p
νe
te1
p
νe
te∗1
νℓ
νℓ
W−
t¯ ¯b
ℓ−
Figure 8.4: Symmetric decays of e t1 and e t∗1 (via tνe ) mode leading to 2b+2ℓ + E T .
8.4.3
The Standard Model backgrounds
As mentioned earlier, we would mostly concentrate on the final states with 2 b-jets+2 leptons+E T and 2 b-jets+e+ e− . In the first case, we consider only the most dominant background coming from tt production when both the top quarks decay leptonically. In order to have a realistic normalisation of this background, tt events generated at the lowest order (LO) using MadGraph5 aMC@NLO and the size of the event sample is scaled appropriately to correspond to its next-to-leading order (NLO) + next-to-nextto-leading log (NNLL) cross-section (≈ 816 pb) [555]. The heavier top squark undergoes a significant decay to a bottom quark and an electron. The dominant background comes from the direct production of a pair of bottom quarks with one of them radiating a Z or γ ∗ which subsequently produces a pair of e+ e− . This background can be largely suppressed by putting an on-shell Z veto for the e+ e− pair. To be conservative and for the robustness of the estimate, the NLO computation [556–559] is done with two 165
103
σ (pp → e te t ∗)(pb)
102 101 100
10−1
10−2
10−3 10−4
200
300
400
500
600
700
800
900
1000
met (GeV)
Figure 8.5: Variation of production cross section for a pair of top squarks at the 13 TeV LHC. Other parameters are kept fixed at values given for BP-IV in table 8.6 (see text). Rates include appropriate K-factors obtained from the package Prospino2. additional jets (10 GeV < pjet T < 60 GeV) in the final state. The SM cross-section for + − pp → bbe e +jets we used is 9.43 pb.
8.4.4
Event selection
To optimise the signal to background ratios, we now have to adopt a set of event selection criteria. Towards this, various appropriate kinematic distributions are studied for both signals and the backgrounds. We present our study for two broad scenarios discussed in section 8.3, i.e., for µu < met1,2 and µu > met1,2 . For each of these cases, two different final states are considered, viz., 2 b-jets+2 leptons +E T and bbe+ e− , arising from e t1 and e t2 decays, respectively.
Case 1: µu < met1,2
Here we discuss the decays of both the top squarks pertaining to the case where µu < met1,2 . χ02 → 2 b-jets + 2 leptons + E T (fig. 8.2) • pp → e t1e t∗1 → be χ+ 2 /te
Such a final state could arise from top squarks decaying to be χ± χ02 . 2 and/or te The final state leptons arise from the decays of W bosons. In addition, χ e± 2 is somewhat heavier than the top quark for both BP-I and BP-II. Hence, on an average, one would expect the leptons to be a little harder compared to the 166
(GeV−1)
∗ ¯ + ℓ− E T pp → te1te1 → bbℓ ℓ ≡ e, µ
100
13 TeV Background 13 TeV BP1 13 TeV BP2
µu < met1
1 dN N d pℓ
T
10−1
−1 1 dN N d E T (GeV )
100
10−2
10−3
50
100
150
200
250
300
350
pℓT (GeV)
∗ ¯ + ℓ− E T pp → te1te1 → bbℓ ℓ ≡ e, µ
13 TeV Background 13 TeV BP1 13 TeV BP2
µu < met1
10−1
10−2
10−3
100
200
300
400
500
600
E T (GeV)
Figure 8.6: pT distributions of the harder lepton (left) and the E T distributions (right) for the background and the signals (in benchmark scenarios BP-I and BP-II) in the 2 b-jet +2 lepton+E T final state arising from decays of e t1 for the case µu < met1 .
background leptons originating in the cascades of the top quarks. This can be seen from the left panel of fig. 8.6, where the pT distributions of the harder lepton in the signal in both the benchmarks have extended tails compared to a similar lepton originating from the SM background. The signal distributions for BP-I and BP-II are similar because of similar values of top squark masses in the two benchmarks. The signal E T distributions are different from the corresponding distribution for the SM background. This may be attributed to the much larger mass of the top squark (compared to mt ) and the presence of extra carriers of E T , i.e., the bino-like MeV neutralino (e χ02 ) and the active neutrino (νe ) emerging from top squark decays. It is evident from the right panel of fig. 8.6 that the SM background can be effectively suppressed by applying a hard enough E T cut, viz., E T > 200 GeV.
• pp → e t2e t∗2 → bbe+ e− (fig. 8.3)
As has been pointed out earlier, e t2 could have a significant decay branching fraction to a bottom quark and an electron, which is a characteristic of such a scenario. Along with the enlarged phase space available to this decay mode, a moderately large coupling (∼ λ′133 ≡ yb ) does boost the decay rate. Naturally, we expect electrons (positrons) with high pT . In the absence of a genuine carrier of E T in such a final state, low or at most a moderate E T is expected from mis-measured momenta of the involved physics objects. The leptons are also expected to have uncorrelated momenta. Such events are rare in the SM. The left of panel fig. 8.7 illustrates the hardest electron (positron) pT distribution in the scenario where e t2 decays to a bottom quark and an electron (positron). We 167
10−1
13 TeV Background 13 TeV BP1 13 TeV BP2 ∗ ¯ + e− pp → te2te2 → bbe
∗
−1 1 dN N d E T (GeV )
−1 1 dN N d peT (GeV )
100
µu < met2
10−2
10−3
10−1
400
600
800
1000
1200
peT (GeV)
13 TeV Background 13 TeV BP1 13 TeV BP2
µu < met2
10−2
10−3 200
¯ + e− pp → te2te2 → bbe
0
50
100
150
200
250
300
350
400
E T (GeV)
Figure 8.7: pT distributions of the harder electron (positron) from both background and signal (left) and E T distributions (right) (in benchmark scenarios BP-I and BP-II) in the bbe+ e− final state arising from decays of a pair of e t2 -s for the case µu < met2 .
impose a minimum pT cut of 200 GeV to reduce the SM background substantially. Since met2 is very similar for BP-I and BP-II and so is its kinematics for these two benchmark points, the distributions look very similar. In the right panel of fig. 8.7, we present the MET distribution which arises in this case from mismeasurements of momenta of visible entities in the final state. As expected, the MET distributions peak at small MET (≈ 25 GeV).
Note that eventually, one should be able to reconstruct te2 -s in the invariant mass spectra of appropriately chosen b-jet-electron (positron) systems which would show peaks at met2 . Clearly, the efficiency of reconstructing e t2 would be limited by various detector effects and a close study of the kinematic distributions discussed above would surely be of crucial help. Nonetheless, it appears that the peaks cannot be missed and a reasonable estimation of met2 would thus be possible. Case 2: µu > met1,2 For µu > met1,2 , the top squarks decay mostly in a symmetric manner with e t1 → te χ02 and e t2 → be+ as can be seen from table 8.8. χ02 → 2 b-jets + 2 leptons + E T (fig. 8.2b) • pp → e t1e t∗1 → te χ02 te
For µu > met1 , e t1 decays mostly to a top quark and a bino-like neutralino (see table 8.7). The top quark would subsequently decay to a W boson and a b-jet via cascades. A pair of W ’s can then decay leptonically, semi-leptonically or hadronically. We confine ourselves to leptonic decays of W -bosons for cleaner signals. The final state would then be comprised of 2 b-jets+2 leptons+E T . The pT distributions for the harder of the final state leptons are shown in the left 168
(GeV−1)
∗ ¯ + ℓ− E T pp → te1te1 → bbℓ ℓ ≡ e, µ
100
13 TeV Background 13 TeV BP3 13 TeV BP4
µu > met1
1 dN N d pℓ
T
10−1
−1 1 dN N d E T (GeV )
100
10−2
10−3
50
100
150
200
250
300
350
pℓT (GeV)
∗ ¯ + ℓ− E T pp → te1te1 → bbℓ ℓ ≡ e, µ
13 TeV Background 13 TeV BP3 13 TeV BP4
µu > met1
10−1
10−2
10−3
100
200
300
400
500
600
E T (GeV)
Figure 8.8: pT distributions of the harder lepton (left) and the E T distributions (right) for the background and the signals (in benchmark scenarios BP-III and BP-IV) in the 2 b-jet +2 lepton+E T final state arising from decays of e t1 for the case µu > met1 . panel of fig. 8.8. The presence of an additional source of E T and the heavier mass of e t1 in the signal are behind harder E T distributions (see right panel of fig. (8.8)) when compared to the SM background. To optimise the signal significance, we have again used the same E T cut (> 200 GeV) as for Case 1.
• pp → e t2e t∗2 → bbe+ e− (fig. 8.3)
In this case the overwhelmingly dominant decay mode is e t2 → be+ . As mentioned earlier, the emitted electron (positron) could have a very high pT as is evident from the left panel of fig. 8.9. A strong pT cut (>200 GeV) on the electron can thus be easily afforded to suppress the SM background effectively. Similar to the case of fig. 8.7, the distributions of MET (of spurious origin) for the present case are presented in the right panel of fig. 8.9. Again, the MET distributions peak at small values (≈ 25 GeV), as expected and explained earlier. Again, possible reconstructions of te2 -s in the invariant mass distributions of suitable pairs of bjet-electron (positron) systems are on the cards. We would touch briefly on this issue later in this section.
Before going into the assessment of the signal significance, we mention below some issues of interest/importance pertaining to possible final states in these two cases. • For both µu < met1 (section 8.3.1) and µu > met1 (section 8.3.2), we have only looked into the 2 b-jet+2 lepton+E T final state arising from te1 pair production. However, the first scenario is phenomenologically richer as it can yield multilepton signals with 4-6 leptons in the final states when te1 -s and te2 -s decay via cascades involving the heavier neutralinos and charginos that in turn decay to 169
10−1
13 TeV Background 13 TeV BP3 13 TeV BP4 ∗ ¯ + e− pp → te2te2 → bbe
∗
−1 1 dN N d E T (GeV )
−1 1 dN N d peT (GeV )
100
µu > met2
10−2
10−3
10−1
400
600
800
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peT (GeV)
13 TeV Background 13 TeV BP3 13 TeV BP4
µu > met2
10−2
10−3 200
¯ + e− pp → te2te2 → bbe
0
50
100
150
200
250
300
350
400
E T (GeV)
Figure 8.9: Same as in fig. 8.7 but for benchmark scenarios BP-III and BP-IV and for the case µu > met2 . SM Z bosons (see tables 8.2 and 8.4). Some corroborative analyses can take advantage of such inclusive final states comprising of 4 to 6 leptons along with b-jets and MET. • Furthermore, such a possibility could help differentiate e t2 from the two distinct scenarios considered in this work. For µu < met2 , in addition to the 2 b-jets + e+ e− final state out of which a pair of te2 could be reconstructed, there would also be multi-lepton final states where leptons other than e+ (e− ) would appear. This is in sharp contrast to the regime with µu > met2 . • A final state like 2 b-jets + e+ e− arising from the decays of te2 -s would be ideally free from any MET. However, as pointed out earlier, in reality, mis-measurements of various momenta may give rise to low to moderate amount of MET thus rendering the final state arising from a pair of te2 -s to be similar to that is obtained from te1 pair-production in a part of the phase space. This gives rise to some legitimate concern as to how efficiently the signature of te2 -s could be deciphered, given the rates for such a final state originating in te1 pair production would be, in general, large thanks to smaller mass of te1 .
Such contaminations, however, can be avoided to a reasonable extent by imposing hard cuts on the minimum pT of the leading electrons as guided by the lepton pT distributions in the left panels of figs. 8.6 and 8.7 (for µu < met1 ) and figs. 8.8 and 8.9 (for µu > met1 ). In addition, imposition of a cut on the maximum allowed MET could effectively restrict the contamination thus allowing for a more efficient reconstruction of te2 . By studying the MET distributions presented in the right panels of figs. 8.7 and 8.9, we find an optimal value of this cut to be 170
13 TeV BP1 13 TeV BP2
0.12
0.08
∗ ¯ + e− pp → te2te2 → bbe
0.1
µu < met2
−1 1 dN N dmbe (GeV )
−1 1 dN N dmbe (GeV )
0.1
0.06 0.04
0.08
∗ ¯ + e− pp → te2te2 → bbe
µu > met2
0.06 0.04 0.02
0.02
300
13 TeV BP3 13 TeV BP4
0.12
400
500
600
700
800
900
1000 1100 1200
mbe(GeV)
300
400
500
600
700
800
900
1000 1100 1200
mbe(GeV)
Figure 8.10: Invariant mass distributions for the appropriate pairs of b-jet-eletron (positron) systems (with low MET characteristic of R-parity violating decays of te2 to a bottom quark and an electron (positron)) for µu < mte2 (left) and µu > mte2 (right). The distributions are obtained by imposing pT > 200 GeV for the leading electron and E T < 50 GeV. E T < 50 GeV that helps retain a healthy number of ‘signal’ events with low E T , a characteristic of such a final state originating in the decays of te2 .
In fig. 8.10 we present the invariant mass distributions of appropriately chosen pairs of b-jet and an electron (positron). Guided by figs. 8.7 and 8.9, a high pT threshold of 200 GeV for the leading electron is demanded along with requiring a E T < 50 GeV to ensure that we mostly confine ourselves to the signal region. The left (right) panel of fig. 8.10 represent the case with µu < mte2 (µu > mte2 ). We find that in both cases clear peaks at mte2 show up thus raising the hope that not only te2 -s could be discovered in this mode but also a reliable estimate of its mass would be possible.
Before we close this subsection we like to mention that although we have only discussed two broad scenarios, i.e., µu < mte2 and µu > mte2 , other intermediate situations are all a priori viable. However, the expectations under those scenarios could be substantiated in a straightforward manner from the two cases we present. For example, an increase in value of µu from that in Case 1 would result in suppression of the branching fractions to higgsino-like neutralinos and charginos. With increasing µu , at some point, these decay-modes (see table 3) would be closed for te1 and BR(te1 → tχ˜02 ) = 1. At the same time, branching fractions to the higgsino-like states for te2 would also get suppressed before these decay-modes get completely closed as it happens in Case 2. A detail study of possible correlations among the event rates in various final states could, in principle, shed light on the relative value of µu with respect to mte1 and mte2 . 171
However, this is beyond the scope of the present work.
8.4.5
Signal significance and the reach
The signal significance (σ) is estimated using the expression [560] r h i S −S σ = 2 (S + B)ln 1 + B
(8.12)
which is appropriate for the situation with small number of events (in particular when the number of background events is less than 50). Equation 8.12 is based on likelihoodratios and follows from the Poisson distribution. Here, S and B stand for the numbers of the signal and the background events, respectively after imposition of the set of optimal cuts discussed in section 8.4.4. The K-factors for e te t∗ are computed using Prospino2 (v2.1) [552, 553]. We now estimate the required integrated luminosities for a 5σ reach of te1 and te2 in the four benchmark scenarios we consider. The final states we focus on are 2 b-jets+2 leptons+E T and bbe+ e− , which stem from the decay of e t1 and e t2 , respectively. For pp → e t1e t∗1 → 2 b-jets + 2 leptons + E T σ(pp → e t1e t∗1 ) (fb) Cut acceptance for signal (E T > 200 GeV) Required L (fb−1 ) for 5σ significance
BP-I
BP-II
BP-III
BP-IV
428.9 10−2
463.4 1.7×10−2
193.0 9.7×10−3
73.6 1.1 × 10−2
100.0
82.6
526.0
2830.3
Table 8.9: Required values of integrated luminosities (L) to obtain a 5σ significance in √ the final state at s = 13 TeV. The most important SM background arising from tt pair production is normalised to a cross section of ≈ 816 pb obtained at the NLO+NNLL level (see section 8.4.3). The cut acceptance for the background is 2.3×10−4 . A flat b-tagging efficiency of 60% is used. the first case (see table 8.9) the dominant background comes from tt pair production which subsequently decays to the 2 b-jet+2 lepton+E T final state. An appropriate K-factor of ≈ 1.6 is used to derive the NLO cross sections from the LO ones for e t-pair production. We note that a 5σ signal significance can be achieved for BP-I and BP-II, with an integrated luminosity around 100 fb−1 . To achieve a similar significance for BP-III and BP-IV, one has to wait for a much higher accumulated luminosity, for example, 500 √ fb−1 and 3000 fb−1 , respectively at s = 13 TeV. 172
pp → e t2e t∗2 → 2 b-jets + e+ e−
σ(pp → e t2e t∗2 ) (fb) Cut acceptance for signal (pT > 200 GeV, E T < 50 GeV) Required L (fb−1 ) for 5σ significance
BP-I
BP-II
BP-III
BP-IV
7.83 1.96×10−2
8.69 5.26×10−2
8.45 1.9×10−1
8.48 1.9×10−1
501.23
63.92
6.85
6.61
Table 8.10: Required values of integrated luminosities (L) to obtain a 5σ significance √ in the 2 b-jets + e+ e− final state at s = 13 TeV. The SM background (see section 8.4.3) at NLO is found to be 9.43 pb (see section 8.4.3). The cut acceptance for the background is 1.25×10−4 . A flat b-tagging efficiency of 60% is used. In table 8.10 we present the required luminosities for a 5σ reach of te2 for the four benchmark points. The dominant SM background comes from bbZ/γ ∗ production followed by Z/γ ∗ giving rise to e+ e− pairs. This can be efficiently suppressed by using an on-shell Z-veto for the e+ e− pairs, as discussed in section 8.4.3. Thus, as can be seen from this table, a 5σ significance can be obtained with an integrated luminosity √ as low as < 10 fb−1 for the benchmark scenarios BP-III and BP-IV with s = 13 TeV. In addition, we also study the HT distribution, i.e., the scalar sum of the pT of the e+ e− pair and the reconstructed b-jets and the improvements are marginal. The wildly varying integrated luminosities across the benchmark points are the artifact of varying branching fractions that are instrumental, as has been pointed out in section 8.3. Figure 8.11 summarises the mass-reach for the two top squarks with varying accumulated integrated luminosities (or, in other words, luminosity required to probe a certain top squark mass) at the 13 TeV LHC. The left panel illustrates the case for e t1 in the final state 2b+2 lepton+E T in BP-II while the right one does the same for e t2 via 2b+e+ e− final state in BP-IV. As indicated by tables 8.9 and 8.10, fig. 8.11 also reveals that e t2 has a significantly better reach compared to e t1 with the final states under consideration. This may lead to a tantalising possibility of discovering e t2 of such e a scenario much earlier than t1 and the former could guide us to find the latter. We observe that at the 13 TeV LHC the mass-reaches for e t1 and e t2 are around 575 (750) GeV and 1.2 (1.4) TeV respectively, with an integrated luminosity of 300 (3000) fb−1 .
8.5 The ‘stealth’ top squark scenario The SUSY model under consideration, with super-light carriers of MET like χ e01 ≡ νe and an MeV neutralino LSP (e χ02 ) can easily conceive a rather low mass top squark lying right in the so-called ‘stealth’ window of 197 GeV . met1 . 205 GeV [111]. 173
Figure 8.11: Density plot reflecting the reach for top squark masses via 2 b-jet+2 lepton+E T final state in BP-II in the met1 -L plane (left) and via 2 b-jet+ e+ e− final state in BP-IV in the met2 -L plane (right). The thick black curves are contours of 5σ significance above and on the left of which the masses can be explored with ≥ 5σ significance. The cuts are kept fixed at values mentioned in the text for the respective cases. As discussed in section 8.2.3, the experimental lower bound on metL is more stringent considering its decay modes. Hence we choose e tR to be the lightest top squark (te1 ). A benchmark point can be obtained by choosing (m2u )33 = −2.5 × 104 GeV2 . This results in met1 ∼ 200 GeV. Such a light top squark cannot provide enough correction to the Higgs mass. Hence we choose a relatively large value of λS (=1.28) so that the radiatively generated additional quartic contributions could lift the Higgs boson mass to the observed range. All other parameters are fixed at the values mentioned in BP-I (see table 8.1). Note that the additional tree level contribution proportional to the neutrino Yukawa coupling ‘f ’ remains small (even for its order one value) because of large values of tan β that we require. As a result, te1 mostly decays to te χ02 and e t1 → tνe with ∼ 85% and ∼ 15% branching fractions, respectively. The possible final state topologies are exactly the same as those result from top quark pair production. We again analyse the final state with 2 b-jets+ 2 leptons+E T . We checked that the distributions of various kinematic observables look very similar for the signal and the tt background, which is something literally expected of a ‘stealth’ top squark and what makes it so elusive. In fig. 8.12 we present the pT distribution of the harder lepton (left panel) and the E T distribution (right panel) which clearly demonstrate how similar the behaviors of the SM background and the signal could get. In this context, techniques te t∗ systems [561], use of to exploit differences in spin-correlations inherent to tt and e various transverse mass variables [562] including the one like mT2 in the dileptonic 174
0.2
0.16 0.14
∗
¯ + ℓ− E T pp → te1te1 → bbℓ
0.12
ℓ ≡ eµ
0.1
13 TeV Background met1 = 200 GeV
0.1 −1 1 dN ) N d E T (GeV
0.18
−1 1 dN ) N d pℓ (GeV T
0.12
13 TeV Background met1 = 200 GeV
Stealth top squark scenario
0.08 0.06 0.04
0.08
∗ ¯ + ℓ− E T pp → te1te1 → bbℓ
ℓ ≡ eµ
0.06
Stealth top squark scenario 0.04 0.02
0.02 20
40
60
80
100
120
140
160
180
200
pℓT (GeV)
50
100
150
200
250
E T (GeV)
Figure 8.12: pT distributions of the harder lepton (left) and the E T distributions (right) for the background and the signal in the 2b-jet+2 lepton + E T final state arising from the decays of a pair of e t1 -s in the ‘stealth’ top squark scenario. decay channel [563], incorporating a new variable like ‘topness’ [564] using asymmetric decays of the top squarks have been proposed to study the ‘stealth’ top squark regime in search for an improved sensitivity. Clearly the issue demands dedicated addressal which is beyond the scope of the present discussion.
8.6 Concluding remarks In this chapter we study a scenario where both the top squarks are light, which is an artifact of this model as previously discussed. Again a very light bino-like neutralino comes out naturally in this scenario along with an active neutrino endowed with an appropriately small Majorana mass. Therefore, rich and interesting collider signatures are expected in such a scenario. Further, no L-R mixing in the top squark sector also provide novel signatures at the LHC where both the top squarks decay in a different manner. The signature of a top squark decaying to a top quark and a neutralino and/or a neutrino is similar to the top quark pair production in the SM. Under favorable circumstances, top squark could also decay to a bottom quark and a chargino leading to a similar final state containing 2 b-jets and 2 leptons along with MET. In this mode, top squark mass of around 575 (750) GeV can be probed with 300 (3000) fb−1 of integrated luminosity. Furthermore, in an R-parity violating scenario such as ours, the charginos mix with the electron. In addition, the decay width of the top squark to a bottom quark and an electron (positron) is enhanced because of the enhanced coupling λ′133 as well as an unsuppressed phase space. Hence we study in detail the final state with 2 b-jets 175
accompanied by an e+ e− pair arising from such a dominant decay. We show that even < when the top squark is heavy (met2 ∼ 1.2(1.4) TeV), this particular channel could deliver a large signal significance with 300 (3000) fb−1 of integrated luminosity. Therefore, we could have a tantalising situation where the heavier top squark can be detected earlier than the lighter one. Finally, we also mention the idea of stealth top squark which can be easily conceived in our model.
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Chapter 9 Summary and Conclusion In chapter 1 we described the framework of the Standard model of particle physics and elaborated its immense success. Over the years SM has been tested and tried in several precision measurements and it passed all of them with flying colours. This has been shown explicitly in terms of the pull variables and the oblique parameters. Finally, the discovery of the Higgs boson in 2012 has made the SM complete. All the decay modes of this Higgs boson have been probed with moderate accuracy and the data seem to be consistent with the SM prediction. However, SM suffers from a few drawbacks. From an aesthetic perspective, the naturalness argument remains a major issue for which SM is silent. In addition, recent experimental evidences in favour of neutrino masses and dark matter (DM) cannot be accounted for in the framework of SM. Hence, one needs to go beyond the SM. In this light supersymmetry (SUSY) remains one of most celebrated beyond standard model theories to date. It can provide an answer to the shortcomings of the SM and hence it has been revered by the particle physics fraternity. Thus in chapter 2, we start with the solution of the naturalness problem in a SUSY field theory and then give a brief account of the concepts of SUSY algebra, superfields and the SUSY Lagrangian. We also discuss the issues pertaining to SUSY breaking. This is followed by a detail discussion on the minimal supersymmetric standard model (MSSM) and the latest bounds on the superpartner masses obtained from the 7-8 TeV run of the LHC. However, the two CERN based experiments namely ATLAS and CMS have found no significant excess over the SM background and this has put stringent constraints on the superpartner masses. Chapter 3 is dedicated to neutrino physics and DM, which are natural testing grounds for physics beyond the SM. We look into these issues from the perspective of a SUSY theory. Firstly, we revisited the idea of generating neutrino masses and mixing with or without R-parity violation. We then moved on to the DM sector and discussed 177
in detail the advantages of having a slightly warm dark matter candidate (WDM) as opposed to a cold dark matter (CDM). We took sterile neutrino as a potential candidate for WDM and discussed some of its production mechanisms. In this light, models with U(1)R symmetry and Dirac gauginos are well motivated since they can address the issues pertaining to the Higgs boson mass, DM and neutrino mass generation. Most importantly, the presence of Dirac gluinos can relax the bound on the first two generation squark masses considerably. We introduce the U(1)R symmetric model with a right handed neutrino superfield in chapter 4. In our scenario R-symmetry is identified with lepton number in such a way that the lepton numbers of the standard model fermions are the same as their R-charges but with a negative sign. This model is known as U(1)R -lepton number model. The neutral gauginos are Dirac fermions in this model and one needs to introduce additional chiral superfields in the adjoint representations of the gauge groups. The right-handed neutrino with an appropriate R-charge allows one to write down neutrino Yukawa interactions respecting the U(1)R symmetry. After the electroweak symmetry breaking one of the sneutrinos (we choose it to be the electron sneutrino) develops a non-zero vacuum expectation value, which can be significant because it is not constrained by small neutrino masses. b d can be integrated out from the theory (along with the The down type Higgs field H bu ) and hence the sneutrino plays the role of a down type Higgs inert doublet field R field. We present a detailed description of the scalar sector in this model and obtained a few interesting sum rules. We observe that the tree level Higgs boson mass receives an additional contribution which is proportional to the neutrino Yukawa coupling f . Therefore, the situation with f ∼ O(1) is important as this can ameliorate the Higgs naturalness argument. In the neutral fermion sector we have mixing among the neutralinos, the electron-neutrino and the right handed neutrino consistent with the R-symmetry and that results in a small Dirac neutrino mass at the tree level. A very important property of this R-symmetric model is the existence of a subset of R-parity violating interactions in the superpotential parametrised by λ and λ′ in the literature. However, R-symmetry is not an exact symmetry and should be broken. There are two massless active neutrinos at the tree level, which acquire non-zero masses through one-loop radiative corrections when small R-symmetry breaking effects are turned on through a small gravitino mass. In this chapter we confine ourselves in a situation where the breaking of R-symmetry is communicated to the visible sector through anomaly mediated supersymmetry breaking. This results in small Majorana gaugino masses as well as trilinear scalar couplings, which were zero in the R-conserving limit. We also obtained analytical expressions for the tree level masses of the active and sterile neutrino in the R-violating scenario.
178
In chapter 5 we quantitatively looked into some of the phenomenological aspects of this model. Depending on the size of the R-symmetry breaking order parameter (gravitino mass m3/2 in this case), we show that one can either generate a pair of almost degenerate neutrinos forming a pseudo-Dirac neutrino or two distinct light Majorana neutrinos from the neutralino-neutrino mass matrix at the tree level. Our analysis shows that none of these situations can accommodate the results from LSND experiments with a possible neutrino mass eigenstate at ∼ 1.2 eV. On the other hand, there exists a possibility of having a Majorana sterile neutrino with a mass of the order of a few keV, which can be a good candidate for warm dark matter. A detail scan of our parameter space shows that there are allowed regions where the constraints on this keV dark matter coming from X-ray observations can be satisfied and this keV sterile neutrino can account for the dark matter relic density measured at the PLANCK and WMAP experiments. At the same time there exists an active neutrino acquiring a very small mass at the tree level. All these allowed points in the parameter space are consistent with a ∼ 125 GeV light Higgs boson. We have also identified two distinct cases of heavy and light top squark masses consistent with the Higgs boson mass, dark matter relic density constraint and a small tree level mass of the neutrino. We also investigate the light active neutrino sector and try to fit the three flavor global neutrino data by incorporating one loop radiative corrections to the (3 × 3) light neutrino mass matrix. We choose certain benchmark points for our numerical analysis and show that one can obtain bounds on the trilinear R-parity violating couplings in the superpotential from neutrino data as a function of the R-symmetry breaking order parameter (m3/2 ). We further pay a special attention to the situation with a large Dirac neutrino Yukawa coupling f and demonstrate that a large f can induce additional tree level contribution to the lightest Higgs boson mass to be consistent with the Higgs boson mass measurement at the LHC experiments. Even with such a large value of f , a small Majorana mass for the light active neutrino can be generated at the tree level. A very interesting feature of this scenario is the existence of a few hundred MeV lightest neutralino LSP which is predominantly bino-like. In this Rparity violating scenario, this MeV lightest neutralino LSP can decay into final states involving standard model fermions and can avoid the constraints on such a light MeV neutralino from its overproduction in the early universe. The gravitino is the NLSP in this case with a mass m3/2 ∼ 10 GeV and it is a stable particle in the collider time scale with a lifetime of ∼ 1012 sec. This is cosmologically consistent as long as the < 106 GeV. reheating temperature TR ∼ In chapter 6 we analysed the case with f ∼ 10−4 in the light of the recent observation of a weak X-ray line around Eγ = 3.5 keV by XMM-Newton telescope coming from 179
Andromeda galaxy and various galaxy clusters. We have shown explicitly that a sterile neutrino of mass about 7 keV and with appropriate active-sterile mixing can easily be obtained in our model. We briefly mention different production mechanisms of the sterile neutrino. Allowed ranges of the mass and mixing helped us to put bounds on tan β as a function of the Dirac wino mass M2D . Combining these bounds with the limits coming from the measurements of the τ Yukawa coupling contribution to the ratio Rτ ≡ Γ(τ → e¯ νe ντ )/Γ(τ → µ¯ νµ ντ ), one obtains strong lower bound on D M2 . Finally, gravitino is the LSP in our model with a mass about a few GeV. The gravitino can decay into a photon plus active or sterile neutrino. Therefore, we have also presented a short discussion on the cosmological implications of this light gravitino. We have taken into account the most robust constraint coming from the diffuse photon background, which readily puts a very stringent bound on the gravitino relic density. < This eventually imposes an upper limit ( ∼ 130 GeV) on the reheating temperature of the universe. In chapter 7, a detailed analysis of the h → γγ channel in this U(1)R −lepton number model has been performed. Experimental results reported for other final states arising from the decay of the Higgs boson are also put in context. Compatibility of the scenario with the results reported by the LHC collaborations pertaining to h → γγ channel is demonstrated by studying the parton level production of the Higgs boson and its subsequent decays. It is observed that for large values of ‘f ’, a smaller top squark mass along with a moderately large tan β (∼ 10) provides an enhancement in the production cross section compared to their values predicted by the SM. Contribution to Γ(h → γγ) from charginos in the loop is found to be insignificant due to their very weak coupling with the Higgs boson. In the present scenario, hW W coupling is modified by the factor sin(β − α). It is demonstrated that Γ(h → γγ) may receive a significant contribution from the W boson loop when ‘f ’ and BµL are large. Moreover, a heavy charged Higgs boson does not provide any enhancement to the h → γγ rate either. We have seen that the large ‘f ’ case is accompanied by a very light bino-like neutralino, which contributes to the total decay width of the Higgs boson. However, we show that such a light bino-like neutralino is not yet constrained from the invisible decay branching fraction of the Higgs boson. This leaves a large parameter space open. For f ∼ O(1), the signal strength µγγ matches very well with the main analysis performed by the CMS collaboration as well as the observation made by the ATLAS collaboration. Subsequently, we have also studied the case of small values of ‘f ’. The scenario is characterized by the presence of a sterile neutrino with mass of 7 keV which is a potent warm dark matter candidate. We present the variation of µγγ with the model parameters varied simultaneously over appropriate ranges. We show that the
180
values of µγγ are also compatible with the results of the main analysis performed by the CMS collaboration and also conforms with the observations made by the ATLAS collaboration at the 1σ level. In chapter 8 we took a natural direction and study the LHC phenomenology of this model. In our scenario, light top squarks can be accommodated easily. In addition, a very light bino-like neutralino comes out naturally in this scenario along with an active neutrino endowed with an appropriately small Majorana mass. Therefore, rich and interesting collider signatures are expected in such a scenario. The signature of a top squark decaying to a top quark and a neutralino and/or a neutrino is similar to the top quark pair production in the SM. Under favorable circumstances, top squark could also decay to a bottom quark and a chargino leading to a similar final state containing 2 b-jets and 2 leptons along with MET. In this mode, top squark mass of around 575 (750) GeV can be probed with 300 (3000) fb−1 of integrated luminosity. Furthermore, in an R-parity violating scenario such as ours, the charginos mix with the electron. In addition, the decay width of the top squark to a bottom quark and an electron (positron) is enhanced because of the enhanced coupling λ′133 as well as an unsuppressed phase space. Hence we study in detail the final state with 2 b-jets accompanied by an e+ e− pair arising from such a dominant decay. We show that < even when the top squark is heavy (met2 ∼ 1.2(1.4) TeV), this particular channel could deliver a large signal significance with 300 (3000) fb−1 of integrated luminosity. In the model discussed in this chapter, only e t2 (≈ e tL ) decays to be+ . This is an artifact of no L-R mixing in the top squark sector. Also, note that the scenario prohibits e t2 + + decaying to bµ or bτ . The final state arising from the pair produced top squarks (e t2 ) decaying to be+ (¯be− ) mode could carry MET which can only be of spurious origin (mis-measurements of various visible momenta, defects in the detector, etc.) and hence is characteristically small. This feature can be used to establish such a model and differentiate it from other competing ones. Due to a relatively clean final state and hence, a possibility to reconstruct the heavier top squark mass reasonably efficiently, such a state could be within an easier reach of the current LHC run when compared to its lighter peer. Such a scenario thus, gives rise to an interesting possibility that e t2 can be found much earlier than e t1 at the LHC and could carry a reliable hint as to where exactly to look for the latter. The signal region for te1 is attributed with a much larger MET as is usual in searches for new heavy states in scenarios with a stable charge- and color-neutral particle(s). This is in sharp contrast with the case of e t2 in such a scenario. Although the analyses in this chapter are presented in terms of two broad scenarios, viz., µu < mte2 and µu > mte2 , it is pointed out that the signatures discussed are robust under intermediate situations except for some obvious quantitative issues getting in.
181
Simultaneous searches in various channels described in this chapter are expected to shed light on the detailed aspect of the spectrum and the involved new couplings of such a scenario. Finally, we have demonstrated how the ‘stealth’ top squark can appear in our model naturally.
182
Appendix A
A.1 Quartic terms in the potential From the minimisation equations of the scalar potential described in eqs. (4.23)-(4.24), we can replace the vev of the singlet and the triplet fields. They are h i g′ f MR 2 D √ µu λS sin β + tS − 2 M1 cos 2β − 2 sin 2β h i vS = −v 2 . (A.1) m2S + bS + 4(M1D )2
Similarly
vT = −v 2
h
i cos 2β i . h m2T + bT + 4(M2D )2
µu λT sin2 β 2
+
√g M D 2 2
Quartic terms in the superpotential are generated from i2 g 2 h i2 √ h i g ′2 h 2 Vquartic = hu − νea2 + h2u − νea2 + 2g ′M1D hSR i h2u − νea2 8 h 8 i √ D + 2gM2 hTR i νea2 − h2u .
(A.2)
(A.3)
Plugging eqs. (A.1) and (A.2) in eq. (A.3) we obtain a simpler from of the quartic terms in the potential given as i2 g 2 (M2D )2 [h2u − νea2 ]2 g ′2 (M1D )2 [h2u − νea2 ]2 (g 2 + g ′2) h 2 hu − νea2 − − . Vquartic = 8 2[m2S + bS + 4(M1D )2 ] 2[m2T + bT + 4(M2D )2 ] (A.4) One can now consider two limiting conditions. 1/2
• In the limit when M1D , M2D ≫ mS , bS
[317, 323] (supersoft-SUSY breaking)
The quartic term given in eq. (A.4) reduces to i2 (g 2 + g ′2 ) h i2 (g 2 + g ′2 ) h 2 2 2 2 hu − νea − hu − νea = 0. Vquartic = 8 8 183
(A.5)
• In the limit when mS , mT ≫ M1D , M2D [306] (our case) Vquartic =
i2 (g 2 + g ′2 ) h 2 hu − νea2 . 8
(A.6)
Hence, we observe that eq. (A.5) reflects the ‘supersoft’ limit in which the SUSY breaking is of D-type only. In such a scenario the Dirac gaugino masses are considered to be much larger than the soft scalar masses. As a result, the quartic term vanishes. However, in our scenario, the presence of soft (but not supersoft) parameters (m2S , m2T , bS , bT ) are generated from the F -terms of the spurion superfield. In the limit when these terms are dominating compared to the Dirac gaugino masses, one gets back the full quartic term as shown in eq. (A.6).
184
Appendix B
B.1 The Higgs-chargino-chargino coupling In this appendix we work out the Higgs-chargino-chargino coupling in the scenario under discussion and present the analytical expression for the width of the lightest Higgs boson decaying into a pair of charginos. The relevant Lagrangian in the twocomponent notation containing the Higgs-chargino-chargino interaction is given by √ Si2 S i1 + − + e− e eL + 2λT Teu vu + √ hi R Lheχ+ χe− = g va + √ hi w d 2 2 Si1 Si3 + − e e +R e− + g vu + √ hi Hu w e − λS vS + √ hi H u d 2 2 Si4 Si4 e +R e− ˜+ ˜ − + λT vT + √ hi H u d + g vT + √ hi Tu w 2 2 Si4 (B.1) − g vT + √ hi w˜ + T˜d− + h.c., 2 where the matrix S connects the mass and gauge eigenstates of the CP even scalar mass squared matrix, written in the basis (hR , ν˜R , SR , TR ). To be more precise the physical CP-even scalar states are related to the gauge eigenstates in the following manner: h1 S11 S12 S13 S14 hR h S 2 21 S22 S23 S24 ν˜R (B.2) = . h3 S31 S32 S33 S34 SR h4 S41 S42 S43 S44 TR
In our notation the lightest physical state (h4 ) of the CP even scalar mass matrix corresponds to the physical Higgs boson, h. Moreover, the charginos χ ˜± i are four component Dirac fermions which arise due to the mixing between the charged gauginos and higgsinos as well as the charged lepton of first generation. In order to evaluate find out the Higgs-chargino-chargino coupling and to evaluate the Higgs boson partial decay 185
width to a pair of charginos, it is pertinent to write down the interaction Lagrangian in the four-component notation. We now define the 4-component spinors as ! ! ! ! e+ + c H Teu+ w e e u R (4) e f= e W (B.3) ¯e− , T = ¯e− , Le = e¯− . ¯e− , H = R w T L d d Using the transformation relations,
¯ (4) f w e+ e− L = Le PL W e L Te e− = HP Teu+ R d
− e +w f PL H e H = W u e e L H, e e +R e− = HP H u d
(B.4)
the Lagrangian in eq. (B.1) can be expressed in the four component notation as S41 f e S43 e e S42 (4) f √ S41 e e (4) + 2λT √ hHP Lheχ+ χe− = g √ hLe PL W L T + g √ hW PL H − λS √ hHPL H 2 2 2 2 S44 e e S44 f e S44 e f + λT √ hHPL H + g √ hW PL T − g √ T PL W + h.c. (B.5) 2 2 2
The chargino masses can have any sign. By demanding that the four component Lagrangian contains only positive masses for the charginos, we define the chargino states in the following manner [101] ! + χ i χ e+ = (ǫi PL + PR ) , i = 1, ..., 4 (B.6) i χ ¯− i
where ǫi carries the sign of the chargino masses, which can be ±1. When ǫ = −1, PR − PL = γ5 , which essentially implies a γ5 rotation to the four component spinors to absorb the sign. Hence, the transformation relations involving only PL changes, which modifies the Feynman rules. The two-component mass eigenstates (χ± i ) of the charginos are related to the gauge eigenstates in a manner shown in eq. (4.70). Using the following set of relations f = PL Vi1∗ ǫi χ PL W ei PL Te = PL Vi2∗ ǫi χ ei e = PL V ∗ ǫi χ ei PL H i3
f = PR Ui1 χ PR W ei e = PR Ui3 χ PR H ei PR Te = PR Ui2 χ ei
PR L(4) = PR Ui4 χ ei , e 186
(B.7)
we rewrite eq. (B.5) in the mass eigenstate basis as (4)m ˜j , Lhχ˜+χ˜− = ghχ˜i ζij∗ PL + ζji PR χ i
where
√ λT S41 S41 λS S43 S42 √ Ui4 Vj1 + 2 √ Ui3 Vj2 + √ Ui1 Vj3 − √ Ui3 Vj3 g 2 g 2 2 2 λT S44 S44 S44 √ Ui3 Vj3 + √ Ui1 Vj2 − √ Ui2 Vj1 ǫi . + g 2 2 2
ζij =
(B.8)
j
(B.9)
The coupling is obtained from Eq. (B.8) as χ ˜+ i h
χ ˜− j
Figure B.1: The Higgs-chargino-chargino vertex.
g ∗ ζij (1 − γ5 ) + ζji (1 + γ5 ) . 2
(B.10)
It is now straightforward to compute the lightest Higgs boson decay width to a pair of charginos, which we find as Γh→eχ+i χe−j =
i1/2 2 g 2 h 2 2 2 2 2 m ) − 4m + m m − (m − + − + h χ ej χ ei χ ej χ ei 16πm3h i h (ζij2 + ζji2 )(m2h − m2χe+ − m2χe− ) − 4ζij ζji mχe+i mχe−j . i
j
(B.11)
Finally, if we assume the singlet and the triplet vev’s to be very small, this would imply that the singlet and triplet mixing in the light CP-even Higgs boson states become negligible. Under such an assumption, the CP even states can be written as 1 ν˜R ≃ va + √ (H cos α − h sin α) 2 1 hR ≃ vu + √ (H sin α + h cos α) , 2 187
(B.12)
where we have chosen S41 = cos α, S42 = − sin α, and S43 ∼ S44 ∼ 0. With this simplification we can write √ cos α sin α 2λT Ui3 Vj2 + Ui1 Vj3 ǫi ζij = − √ Ui4 Vj1 + √ g 2 2 = ξij sin α − ηij cos α, (B.13) where Ui4 Vj1 ξij = − √ ǫi 2 √ 2λT 1 Ui3 Vj2 + Ui1 Vj3 ǫi . ηij = √ g 2
188
(B.14)
Appendix C
C.1 The Higgs-neutralino-neutralino coupling In a similar manner the interaction of the Higgs boson with neutralinos can be constructed from the following (two-component) Lagrangian ′ g′ Si1 S g S i2 i1 0 e0 e − √ va + √ hi ebνe + λS vu + √ hi SeR Lheχ0 χe0 = √ vu + √ hi ebH d u 2 2 2 2 2 g Si1 Si2 Si1 e u0 + √g va + √ ed0 − √ vu + √ hi w eH hi wν e e + λT vu + √ hi TeR 2 2 2 2 2 Si3 Si4 Si2 e0 H e0 e 0N c + λS vs + √ hi + λT vT + √ hi R va + √ hi H d u −f u 2 2 2 Si1 − f vu + √ hi N c νe + h.c. 2 (C.1) We stick to the notation for the lightest CP even physical scalar state being denoted by h4 and identified with the lightest Higgs boson h. We again define the 4-component spinors as [59] ! ! ! ! ˜b ˜0 ˜0 S˜ R H u d ˜ = ˜ ˜ ˜ B ¯T , S = ¯eT , Rd = ¯e0T , Hu = ¯e 0T , e S Rd Hu b ! ! ! ! ˜ c W T˜ N ν e c ˜ T˜ = . ¯ T , νe = ν¯T , N = N ¯eT , W = f ¯ cT T W e
(C.2)
189
In terms of these spinors the 4-component Lagrangian takes the following form g ′ S42 ¯e S41 ¯e e g S41 f g ′ S41 ¯e e ¯ (4) ˜ √ √ hBPL νe + λS √ hSP H − Lheχ0χe0 = √ √ hBP L u L Rd − √ √ hW PL Hu 2 2 2 2 2 2 2 g S42 f S41 ¯ e S43 ¯e S44 ¯e ¯ e e + √ √ hW PL νe + λT √ hTePL R d + λS √ hRd PL Hu + λT √ hRd PL Hu 2 2 2 2 2 S42 e¯ S 41 ¯ c PL νe + h.c. − f √ hHu PL N c − f √ hN 2 2 (C.3) Eq. (C.3) represents the interactions in the gauge eigenstate basis. Neutralinos are physical Majorana spinors, arising due to the mixing of the neutral gauginos, higgsinos as well as the active (first generation) and sterile neutrino states. The four component neutralino state is defined as ! 0 χ i , i = 1, ..., 8 (C.4) χ e0i = (ǫi PL + PR ) χ ¯0i
where χ0i are two component neutralino mass eigenstates and they are related to the gauge eigenstates as χ0i = Nij ψj0 ,
i, j = 1, ..., 8
(C.5)
e W f , Te, R ed , H e u , N c , νe T . As presented in Appendix B, in a similar where ψ 0 = eb, S, fashion we use the following transformation relations to write down the interaction Lagrangian given in Eq. (C.3) in the mass eigenstate basis e = N ∗ PL ǫi χ e = Ni1 PR χ PL B e0i , PR B e0i i1 PL Se = Ni2∗ PL ǫi χ e0i , PR Se = Ni2 PR χ e0i f = Ni3∗ PL ǫi χ f = Ni3 PR χ PL W e0i , PR W e0i
PL Te = Ni4∗ PL ǫi χ e0i , PR Te = Ni4 PR χ e0i ed = N ∗ PL ǫi χ ed = Ni5 PR χ PL R e0i , PR R e0i i5 e u = Ni6∗ PL ǫi χ e u = Ni6 PR χ PL H e0i , PR H e0i PL N c = Ni7∗ PL ǫi χ e0i , PR N c = Ni7 PR χ e0i PL νe = Ni8∗ PL ǫi χ e0i , PR νe = Ni8 PR χ e0i .
(C.6)
It is now straightforward to write down the Higgs-neutralino-neutralino interaction in the 4-component notation as 0 (4)m ¯˜0i h ζij′∗ PL + ζji′ PR χ Lhχ˜0 χ˜0 = g χ ˜j , 190
(C.7)
where h g′ N N λS Ni2 Nj5 Ni3 Nj6 λT Ni4 Nj5 f Ni7 Nj8 i i1 j6 √ √ √ + + ǫi − − g 2 g 2 g g 2 2 2 hN N hλ N N i g ′ Ni1 Nj8 f Ni6 Nj7 i i5 j6 i3 j8 S √ √ − − + S42 ǫi + S43 ǫi 2 g 2 g g 2 2 hλ N N i i5 j6 T √ ǫi + (i ↔ j). (C.8) + S44 g 2
ζij′ = S41
χ ˜0i h
χ ˜0j
Figure C.1: The Higgs-neutralino-neutralino vertex. Finally, the partial decay width Γ(h → χ e0i χ e0j ) is given as Γh→eχ0i χe0j =
i1/2 h g2 2 2 2 2 2 2 {m − (m × + m )} − 4m m h χ e0i χ e0j χ e0i χ e0j 16πm3h (1 + δij ) i h 2 ′ ′ 2 2 ′2 ′2 ζij + ζji mh − mχe0 − mχe0 − 4ζij ζji mχe0i mχe0j . i
j
(C.9)
Again in the limit where the singlet and triplet vev’s are very small, we can safely ignore the contributions from S43 and S44 . Furthermore, replacing S41 by cos α and S42 by -sin α, we can write ζij′ = ηij′ cos α + ξij′ sin α,
(C.10)
where, ηij′ ξij′
g ′ Ni1 Nj6 λS Ni2 Nj5 Ni3 Nj6 λT Ni4 Nj5 f Ni7 Nj8 √ √ √ = + + ǫi + (i ↔ j), − − g 2 g 2 g g 2 2 2 ′ g Ni1 Nj8 f Ni6 Nj7 Ni3 Nj8 √ ǫi + (i ↔ j). (C.11) + = − g 2 g 2 2
191
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