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Gravitino and Scalar Τ-Lepton Decays in Supersymmetric Models with Broken R-Parity

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Gravitino and Scalar Τ-Lepton Decays in Supersymmetric Models with Broken R-Parity Universit¨at Hamburg Deutsches Elektronen-Synchroton Department Physik Theory Group Gravitino and Scalar τ -Lepton Decays in Supersymmetric Models with Broken R-Parity Diploma Thesis by Jan Hajer Hamburg January 2010 Betreuer: Prof. Dr. Wilfried Buchmuller¨ (DESY) Zweitgutachter: Prof. Dr. Jan Louis (Universit¨at Hamburg) Abstract Mildly broken R-parity is known to provide a solution to the cosmological gravitino problem in supergravity extensions of the Standard Model. In this work we consider new effects occurring in the R-parity breaking Minimal Supersymmetric Standard Model including right-handed neutrino superfields. We calculate the most general vacuum expectation values of neutral scalar fields including left- and right-handed scalar neutrinos. Ad- ditionally, we derive the corresponding mass mixing matrices of the scalar sector. We recalculate the neutrino mass generation mechanisms due to right- handed neutrinos as well as by cause of R-parity breaking. Furthermore, we obtain a, so far, unknown formula for the neutrino masses for the case where both mechanisms are effective. We then constrain the couplings to bilinear R-parity violating couplings in order to accommodate R-parity breaking to experimental results. In order to constrain the family structure with a U(1)Q flavor symmetry we furthermore embed the particle content into an SU(5)b Grand Unified Theory. In this model we calculate the signal of decaying gravitino dark matter as well as the dominant decay channel of a likely NLSP, the scalar τ-lepton. Comparing the gravitino signal with results of the Fermi Large Area Telescope enables us to find a lower bound on the decay length of scalar τ-leptons in collider experiments. Zusammenfassung Es ist bekannt, dass schwach gebrochene R-Parit¨at eine L¨osungsm¨oglichkeit fur¨ das kosmologische Gravitino Problem in Supergravitations-Erweiterungen des Standardmodells darstellt. Daher berechnen wir in dieser Arbeit die Effekte, die im R-Parit¨ats- brechenden minimalen supersymmetrischen Standardmodell unter Beruck-¨ sichtigung von rechtsh¨andigen Neutrinos auftreten. Wir berechnen die allge- meinsten Vakuumerwartungswerte der neutralen Felder inklusive der links- und rechtsh¨andigen skalaren Neutrinos. Zus¨atzlich berechnen wir die Massen- mischungsmatrizen der skalaren Felder. Im fermionischen Sektor berechnen wir sowohl die Neutrinomasse, die durch die rechtsh¨andigen Neutrinos er- zeugt wird, als auch diejenige, die durch die R-Parit¨ats-Brechung erzeugt wird. Dadurch werden wir in die Lage versetzt eine bisher unver¨offentlichte Formel fur¨ die Neutrinomasse herzuleiten, die gilt, wenn beide Mechanismen wirken. Um den experimentellen Ergebnissen Rechnung zu tragen fuhren¨ wir da- nach bilineare R-Parit¨ats-Brechung ein. Um die Familienstruktur durch eine U(1)Q-Familiensymmetrie einzuschr¨anken betten wir den Teilcheninhalt in eine SU(5)b große vereinheitlichte Theorie ein. In diesem Modell berechnen wir ein m¨ogliches Signal von zerfallender dunkler Materie, sowie den do- minanten Zerfallskanal eines m¨oglichen zweitleichtesten supersymmetrischen Teilchens, des skalaren τ-Leptons. Durch Vergleich unserer Rechnung fur¨ das Gravitino mit den Ergebnissen des Fermi Large Area Telescope k¨onnen wir eine untere Schranke fur¨ die Zerfallsl¨ange des skalaren τ-Leptons in Beschleu- nigerexperimenten vorhersagen. Anna, Anton, Beate, Milena und Miro gewidmet vii Contents Introduction 1 1 Theoretical Foundation 5 1.1 The Standard Model and Beyond . 5 1.2 Supersymmetry .......................... 7 1.2.1 GlobalSupersymmetry . 7 1.2.2 Local Supersymmetry – Supergravity . 9 1.2.3 Super Higgs Mechanism . 10 1.2.4 R-Parity.......................... 11 1.2.5 Supersymmetry Breaking . 12 1.3 U(1)FlavorSymmetry . 12 1.4 RenormalizationGroup. 13 2 Supersymmetric Standard Models 15 2.1 Superpotential and Chiral Superfields . 15 2.2 Choice of the Weak Interaction Basis . 17 2.3 VectorSuperfields. 18 2.4 SoftSUSYBreakingLagrangian . 19 3 Symmetry Breaking: Higgs and Sneutrinos 21 3.1 ScalarBosons ........................... 22 3.1.1 The Minimal Supersymmetric Standard Model . 22 3.1.2 Adding Majorana Sneutrinos . 23 3.1.3 MSSM Including R-Parity Violating Couplings . 24 3.1.4 Majorana Sneutrinos and R-Parity Violation . 26 3.2 GaugeBosons........................... 27 3.3 NeutralinosandNeutrinos . 28 3.3.1 Neutralinos in the MSSM . 28 3.3.2 Neutrino Mass Generation via Majorana Masses . 30 3.3.3 Neutrino Mass Generation via R-Parity Violation . 30 3.3.4 Combining Majorana Neutrinos and R-Parity Violation 32 viii Contents 4 Constraining SUSY with Bilinear R-Parity Breaking 35 4.1 ProtonDecay ........................... 35 4.2 GravitinoDarkMatter . 36 4.3 Bilinear R-ParityBreaking. 37 4.4 Gravitino Decay into Photon and Neutrino . 39 4.5 Scalar τ-Lepton Decay into τ-Lepton and Neutrino . 42 5 SU(5) GUT with U(1)Qˆ Flavor Symmetry 45 5.1 U(1)Q Flavor Charge Assignment . 45 5.2 SuperpotentialParametersb . 46 5.2.1 R-Parity Conserving Parameters . 46 5.2.2 R-Parity Breaking Parameters . 47 5.3 Soft Supersymmetry Breaking Masses . 48 5.3.1 R-Parity Conserving Masses . 48 5.3.2 R-Parity Breaking Masses . 48 5.4 GUT with Flavor Symmetry and Bilinear Breaking . 49 5.5 Flavor Structure of τ-SleptonDecays . 50 Conclusion and Outlook e 53 Appendices 55 A Units and Physical Constants in the Standard Model 57 A.1 Cabibbo-Kobayashi-Maskawa Matrix . 57 B Notation 59 B.1 Two-Component Spinor Notation . 59 B.2 Four-Component Spinor Notation . 61 C Addendum to Neutral Field Mixing 63 C.1 Neutral Scalar Mass Mixing . 63 C.2 Neutralino Mass Mixing Matrix . 65 C.2.1 Solving Quartic Equations . 66 D Field Mixing in the Charged Sector 69 D.1 ScalarFields............................ 69 D.1.1 R-Parity Conserving Case . 69 D.1.2 R-ParityViolatingCase . 70 D.2 FermionicFields. 71 D.2.1 R-Parity Conserving Case . 71 D.2.2 R-ParityViolatingCase . 72 Contents ix E Renormalization Group Equations 73 E.1 RGEs in the Minimal Supersymmetric Standard Model . 73 E.2 RGEs in the R-Parity Breaking MSSM . 74 E.3 Bilinear R-ParityBreakingRGEs . 75 E.4 RGEs in the SU(5) Flavor Model . 76 F Cosmology 77 G Matrix Diagonalization 79 G.1 HierarchicalMatrices . 79 Acknowledgments 81 Bibliography 83 x List of Figures 1.1 GaugeCouplingRunning. 6 4.1 FeynmanDiagramofProtonDecay . 35 4.2 Feynman Diagram of Gravitino Decay . 40 4.3 Feynman Diagram of Scalar τ-LeptonDecay . 42 5.1 Scalar τ-LeptonDecayLength . 51 5.2 Scalar τ-LeptonBranchingRatio . 51 List of Tables 1.1 Gravity Supermultiplet . 9 2.1 ChiralSupermultiplets . 16 2.2 GaugeSupermultiplets . 19 5.1 U(1)Q Flavor Symmetry Charge Distribution . 45 5.2 Orderb of the Scalar τ-Lepton Decay in Flavor Parameter . 50 A.1 PhysicalConstants . 58 B.1 FermionNames .......................... 61 1 Introduction The Standard Model (SM) of particle physics describes the behavior of high energy physics up to the electroweak scale very well [1, 2]. Up to now all particles predicted by the SM, except for the Higgs boson, are found and most parameters it depends on are measured to a high degree of accuracy. Hence it is expected that the Higgs boson will be found in the near future at the Large Hadron Collider (LHC). In the SM fermions couple to the Vacuum Expectation Value (VEV) of the Higgs boson in order to acquire mass. However, despite all its successes, it is a well-established fact that the SM must be extended in order to have a full description of nature. One reason is that neutrinos are known to have a tiny mass, as is required to account for neutrino oscillations [3]. This small left-handed neutrino masses can be explained by introducing super-heavy right-handed Majorana neutrino singlets that generate the light neutrino masses via the seesaw mechanism [4, 5]. Furthermore the SM suffers from the hierarchy problem, namely the observation that the Higgs boson is 16 orders of magnitude lighter than the fundamental Planck scale, despite dominant ultraviolet contributions at loop level [6]. The hierarchy problem can be circumvented by imposing Supersymmetry (SUSY), which doubles the particle content by introducing new supersym- metric particles whose spins differ by one half from the known particles and which cancel the large loop contributions [6] to the mass of the Higgs bo- son. In this thesis we consider local SUSY, i.e. Supergravity (SUGRA). It inevitably predicts the existence of a spin-3/2 particle, the gravitino, as the supersymmetric partner of the gauge boson of gravity, the graviton [7, 8]. With imposed SUSY new problems emerge, for instance matter instability and the µ-problem [6]. The latter is a naturalness problem or, in other words, a moderate hierarchy problem, but it is not the topic of this thesis. Matter stability, on the other hand, can be assured by introducing R-parity [9]. However, this symmetry is not governed by a fundamental mechanism and consequently we may take a different approach. The supersymmetric low-energy mass spectrum is unknown and deter- 2 Introduction mined by the chosen SUSY breaking model. Hence, there are numerous different possibilities for the Lightest Supersymmetric Particle (LSP), for in- stance the scalar τ-lepton, the neutralino or the scalar neutrino. Provided the LSP is uncharged, it could amount, on cosmic scales, to the observed Dark Matter (DM) density which is crucial to the SM of cosmology [10]. Consistent cosmology needs in addition an inflationary phase, caused by an scalar inflaton field [11]. The decay of the inflaton field can lead to re- heating [12]. If one explains the observed baryon asymmetry via leptogenesis one has to deal with a high reheating temperature [13]. This constrains the low-energy mass spectrum. Since the gravitino decays are suppressed by the Planck mass, the late decays of the gravitino into lighter supersymmetric particles spoil Big Bang Nuclesynthesis (BBN) predictions [14]. Hence the gravitino must be the LSP, but then the Next-to-Lightest Supersymmetric Particle (NLSP) can only decay into the gravitino.
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